Performance of Modern Eddy-Viscosity Turbulence Models Von der Fakult¨ at f¨ ur Luft- und Raumfahrttechnik und Geod¨asie der Universit¨ at Stuttgart zur Erlangung der W¨ urde eines Doktor-Ingenieurs (Dr.-Ing.) genehmigte Abhandlung
Vorgelegt von Alan Celi´ c geboren in T¨ ubingen
Hauptberichter: Prof. Dr.-Ing. habil. Ernst H. Hirschel 1. Mitberichter: Prof. Dr.-Ing. Siegfried Wagner 2. Mitberichter: Prof. Peter Bradshaw Tag der m¨ undlichen Pr¨ ufung: 23.07.2004
Institut f¨ ur Aerodynamik und Gasdynamik Universit¨at Stuttgart 2004
Berichte aus der Luft- und Raumfahrttechnik
Alan Celi ´c
Performance of Modern Eddy-Viscosity Turbulence Models
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D 93 (Diss. Universität Stuttgart)
Shaker Verlag Aachen 2004
Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the internet at http://dnb.ddb.de. Zugl.: Stuttgart, Univ., Diss., 2004
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Copyright Shaker Verlag 2004 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the publishers. Printed in Germany. ISBN 3-8322-3517-5 ISSN 0945-2214 Shaker Verlag GmbH • P.O. BOX 101818 • D-52018 Aachen Phone: 0049/2407/9596-0 • Telefax: 0049/2407/9596-9 Internet: www.shaker.de • eMail:
[email protected] Acknowledgments This work was conducted during my time as a research engineer at the Institut f¨ ur Aerodynamik und Gasdynamik (IAG) of the University of Stuttgart, Germany, and was funded by the Deutsche Forschungsgemeinschaft (Grants Hi 342/4-1 to 342/4-4). I am deeply grateful to my thesis adviser Professor Dr.-Ing. habil. Ernst H. Hirschel for his great personal support and help, and technical advice. Professor Hirschel always believed in my work, which gave me confidence especially during difficult periods when things did not go as smoothly as hoped. I highly appreciate that I could be one of his doctoral students. I am also deeply grateful to Professor Peter Bradshaw from Stanford University who was a distant adviser and a “Mitberichter” (co-referee) for this Ph.D. thesis. Professor Bradshaw’s invaluable professional and linguistic advice as well as his personal support made this work a great experience and joy for me. I have learned so much from him. I also wish to thank Professor Dr.-Ing. Siegfried Wagner for his commitment as a Mitberichter and for offering me the opportunity to perform this study at his institute. The IAG is a great place to work at and I also thank all my former colleagues for creating such a great atmosphere. Very special thanks go to Dr.-Ing. Werner Haase from EADS Munich. Dr. Haase put me on track at the beginning of this work by supplying his CFD code and his great experience in turbulence modeling. He always took the time to teach me about CFD and turbulence modeling which I sincerely appreciate. I am likewise grateful to Dr.-Ing. Markus Kloker from the IAG for providing his invaluable advice when I had questions concerning numerics. I am indebted to Professor Stefan Staudacher who, on short notice, attended my Ph.D. exam in place of Professor Bradshaw who unfortunately could not take the burden of the long travel to Stuttgart because at that time he had not completely recovered from an accident. Last but not least, I wish to thank my family and my close friends who have supported me in many ways and without whom I would have not been able to master this work.
Alan Celi´c Toulouse, October 24th, 2004
Contents Notation
9
Abstract
15
Zusammenfassung
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1 Introduction 1.1 The Present Study . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Contents and Organization of the Thesis . . . . . . . . . . . .
25 33 34
I
37
Topological Approach to Turbulence Modeling
2 Basic Considerations 3 Governing Equations and Numerical Method 3.1 Governing Equations of the Mean Flow . . . . 3.2 The Baldwin-Lomax Model . . . . . . . . . . . 3.3 The Johnson-King Model . . . . . . . . . . . . 3.4 Numerical Method (I) . . . . . . . . . . . . . .
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42 42 44 46 48
4 Demonstration 4.1 Description of Flow Case . . . . . . . . . . . . . . . . . . . . 4.2 Computational Grid . . . . . . . . . . . . . . . . . . . . . . . 4.3 Computational Results and Discussion . . . . . . . . . . . . . 4.3.1 Topology of the Velocity Field . . . . . . . . . . . . . 4.3.2 Pressure and Skin-Friction Distributions . . . . . . . . 4.3.3 Boundary-Layer Profiles . . . . . . . . . . . . . . . . . 4.3.4 Numerical Experiment in the Recirculation Zone . . . 4.3.5 Comments Regarding Hidden Three-Dimensional Effects in Nominally Two-Dimensional Flows . . . . . .
50 50 51 53 53 55 60 67
II
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Analysis of Modern Turbulence Models
5 Numerical Method (II)
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6 Models Investigated 6.1 The k, ω Models of Wilcox . . . . . . . . . . . . . . . . . . . 6.2 The k, ω Shear-Stress Transport (SST) Model of Menter . . . 6.3 The Turbulent/Non-turbulent (TNT) k, ω Model of Kok . . . 6.4 The Local Linear Realizable (LLR) k, ω Model of Rung . . . 6.5 The Explicit Algebraic Reynolds-Stress Model (EARSM) of Wallin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Boundary Conditions for the k, ω Models . . . . . . . . . . . 6.6.1 Free-Stream Boundary Conditions . . . . . . . . . . . 6.6.2 Wall Boundary Conditions . . . . . . . . . . . . . . . 6.7 The One-Equation Model of Spalart & Allmaras . . . . . . . 6.8 The One-Equation Model of Edwards & Chandra . . . . . . . 6.9 The Strain-Adaptive Linear Spalart-Allmaras (SALSA) Model 6.10 Boundary Conditions for the One-Equation Models . . . . . .
75 76 79 81 82
7 Test Cases Selected 7.1 Flat-Plate Boundary Layer (Case FPBL) . . . . . . . . . . . 7.1.1 Computational Setup . . . . . . . . . . . . . . . . . . 7.1.2 Computational Results and Discussion . . . . . . . . . 7.1.3 Some Modifications of the k, ω SST Model . . . . . . 7.1.4 Effects of Low-Reynolds-Number Modifications . . . . 7.2 Boundary Layer with Adverse Pressure Gradient (Case BS0) 7.2.1 Computational Setup . . . . . . . . . . . . . . . . . . 7.2.2 Computational Results and Discussion . . . . . . . . . 7.3 Boundary Layer with Pressure-Induced Separation (Case CS0) 7.3.1 Computational Results and Discussion . . . . . . . . . 7.4 Separated Airfoil Flow (Case AAA) . . . . . . . . . . . . . . 7.4.1 Computational Results and Discussion . . . . . . . . .
94 95 95 96 101 105 107 108 111 118 119 127 127
8 Numerical Issues 8.1 Grid Convergence . . . . . . . . . . . . . . . . . 8.2 Local Preconditioning for Low Mach Numbers . 8.3 Transition . . . . . . . . . . . . . . . . . . . . . 8.4 Artificial Damping in Boundary Layers . . . . . 8.5 Boundary-Value Dependences . . . . . . . . . . 8.5.1 Dependences on Wall Value of ω . . . . 8.5.2 Dependence on Free-Stream Value of ω
137 137 141 146 149 153 153 154
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83 87 87 88 89 91 91 93
9 Summary and General Conclusions
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10 Outlook
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III
161
Appendices
A RANSLESS – A New Approach to RANS/LES Coupling A.1 Brief Review of Turbulence Physics at Turbulent Separation A.2 RANS/LES Coupling for Separated Flows . . . . . . . . . . A.2.1 Inflow Conditions for LES . . . . . . . . . . . . . . . A.2.2 Outflow Conditions for LES . . . . . . . . . . . . . . A.2.3 Inflow Conditions for RANS . . . . . . . . . . . . . . A.2.4 Outflow Conditions for RANS . . . . . . . . . . . . . A.3 Closing Note . . . . . . . . . . . . . . . . . . . . . . . . . .
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162 162 162 164 167 167 167 167
B Details of the Johnson-King Model
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C Graphs of Computational Results C.1 Boundary Layer with Adverse Pressure Gradient (Case BS0) C.2 Boundary Layer with Pressure-Induced Separation (Case CS0) C.3 Separated Airfoil Flow (Case AAA) . . . . . . . . . . . . . .
172 172 185 207
D Overview of Algorithmic Accomplishments
232
E Typical FLOWer Input Decks 234 E.1 Typical FLOWer Input Deck for Case FPBL . . . . . . . . . 234 E.2 Typical FLOWer Input Deck for Case BS0 and CS0 . . . . . 238 E.3 Typical FLOWer Input Deck for Case AAA . . . . . . . . . . 242 Bibliography
254
Notation General Conventions • Scalar variables are italicized; vector, matrix and tensor variables are bold and italicized. • To avoid departing too much from conventions usually used in literature on turbulence modeling and general fluid mechanics, some symbols denote more than one quantity. • Following the usual Einstein summation convention, repetition of an index in a single term denotes summation with respect to that index: ui ui ≡
3
ui ui = u1 u1 + u2 u2 + u3 u3
i=1
Latin Symbols a a1 aex , aex ij A A+ b c cf cv cp cD cL cM D e E FKleb (y; ymax /CKleb )
√ speed of sound, γRT Townsend’s constant; Bradshaw’s constant extra-anisotropy tensor wing area Van Driest damping coefficient wingspan chord length skin-friction coefficient, τw /(0.5ρ∞ |v∞ |2 ) specific heat at constant volume specific heat at constant pressure; pressure coefficient, p − p∞ /(0.5ρ∞ |v∞ |2 ) global drag coefficient global lift coefficient global moment coefficient cylinder diameter; Van Driest damping function volume-specific inner energy volume-specific total energy Klebanoff intermittency function
H I i, j, k k ks lmix M m ˙ p P Pr P rt q R Re Ret S, Sij t T Tη u, v, w ui ui uj
u+ uτ v vmix w x, y, z y+
volume-specific total enthalpy identity matrix grid-point index in the ξ, η, ζ direction kinetic energy of turbulent fluctuations per unit mass surface roughness height mixing length Mach number mass per unit time mean static pressure preconditioning matrix Prandtl number turbulent Prandtl number heat-flux vector perfect-gas constant Reynolds number turbulent Reynolds number mean strain-rate tensor time static temperature Kolmogorov time scale of dissipation Cartesian components of v Cartesian components of v in index notation tensor of one-point central moments of fluctuating velocity components, specific Reynolds-stress tensor dimensionless, sublayer-scaled velocity, u/uτ friction velocity, τw /ρ mean velocity in vector notation mixing velocity vector of dependent flow variables Cartesian coordinates dimensionless, sublayer-scaled wall distance, yuτ /ν
Greek Symbols α γ δ δv∗ ∆ λ Λ ν µ µt µ ti µ to ψ ξ, η, ζ ζ θ κ ρ σ(x) σ, σij τ , τij τw ω |ω| Ω Ω, Ωij
angle of attack specific-heat ratio, cp /cv boundary-layer thickness δ kinematic displacement thickness, 0 (1 − u/ue ) dy relative difference between computed and measured values dissipation rate per unit mass heat-conductivity coefficient wing aspect ratio, b2 /A kinematic viscosity, µ/ρ dynamic viscosity eddy viscosity inner-layer eddy viscosity outer-layer eddy viscosity streamfunction curvilinear coordinates second viscosity coefficient δ momentum thickness, 0 ρeρuue (1 − u/ue ) dy K´ arm´ an constant fluid density non-equilibrium parameter viscous stress tensor Reynolds-stress tensor shear stress at the wall specific dissipation rate per unit mass magnitude of the vorticity vector absolute value of the vorticity mean rotation tensor
Subscripts e exp
value at boundary-layer edge measured value, value from experiment
local m max min neq r, ref s w ∞
local value value at position of maximum Reynolds shear stress maximum value minimum value non-equilibrium reference quantity streamline wall (surface) value free-stream value
Superscripts + −
fluctuating part of a flow variable sublayer-scaled value Reynolds-averaged value, time-averaged value
Acronyms AAA CFD DNS DES EADS EARSM FPBL GCI IAG LES LLR MBC ONERA RANS RANSLESS
Aerospatiale-A airfoil computational fluid dynamics direct numerical simulation detached-eddy simulation European Aeronautic Defence and Space Company explicit algebraic Reynolds-stress model flat-plate boundary layer grid-convergence index Institut f¨ ur Aerodynamik und Gasdynamik large-eddy simulation local linear realizable Menter’s boundary condition (for ωw ) Office National d’Etudes et de Recherches Aerospatiales Reynolds-averaged Navier-Stokes (computation) RANS surrounded LES scenario
RBC RSTM SALSA SST TNT WBC
Rudnik’s boundary condition (for ωw ) Reynolds-stress transport model strain-adaptive linear Spalart Allmaras shear-stress transport turbulent/non-turbulent Wilcox’s boundary condition (for ωw )
Abstract Turbulent flows of engineering interest are frequently computed by solving the Reynolds-averaged Navier-Stokes equations in combination with an eddyviscosity turbulence model. In situations where the turbulent boundary layer separates from the surface of the body under consideration poor predictive accuracy of the computational results is encountered more often than not. In this work, the topology of velocity fields serves as a basis to generally classify separated flows. It is speculated whether the computation of a single flow class can be improved if turbulence modeling is adjusted to topological structures. A separated airfoil flow is investigated in detail using the flow topology as a guideline for identifying flow regions of possible importance for turbulence modeling. It is found that modifications of the turbulence model downstream of separation do not improve or influence computational results. It appears that boundary-layer development upstream of separation is the key issue for accurately predicting the primary separation. Therefore, different boundary-layer flows with increasing physical complexity are selected to study the effect of adverse pressure gradient on model predictions. A broad set of modern turbulence models including some recently developed models is employed for the computation of the test cases selected. Computational results are compared to experimental data. None of the turbulence models investigated is able to predict important flow quantities for all test cases in good agreement with experimental data. The “best” turbulence model changes from flow case to flow case. The predictions of models based on one transport equation for the eddy viscosity are much closer to each other than predictions of the k, ω models employed in this work. Compared to experiment, transport-equation models are found to respond qualitatively better to pressure gradient than the algebraic model investigated. No general conclusion can be drawn on what model to use for computing pressure-induced separated flows. Besides turbulence model performance, several additional important issues for the computation of separated flows are investigated. These range from purely numerical issues, like local preconditioning for low Mach numbers and artificial damping of the numerical scheme, to issues regarding boundary conditions for the turbulence variables and specification of transition locations. It is found that local preconditioning is essential for obtaining accurate flow solutions for small Mach numbers with a compressible flow solver. Moreover, it proves to be necessary to reduce standard artificial damping terms in the
direction normal to the wall in boundary layers in order to prevent spurious momentum loss. The separated airfoil flow is seen to be very sensitive to the prescribed location of transition and to the rate at which the turbulence model reaches the fully turbulent state. A partly novel method for coupling RANS and LES for the computation of turbulent flows is proposed. It is intended to form the basis for future work.
Zusammenfassung Einleitung Abgel¨ oste turbulente Str¨ omungen spielen eine zentrale Rolle in vielen ingenieurtechnischen Anwendungen. Die zuverl¨ assige und exakte Berechnung solcher Str¨ omungen stellt trotz jahrzehntelanger intensiver Forschungsbem¨ uhungen auf diesem Gebiet eine große Herausforderung dar. Viele verschiedene statistische Turbulenzmodelle wurden zu diesem Zweck entwickelt und vorgeschlagen. In dieser Arbeit wurde zum einen der topologieorientierte Ansatz nach Hirschel (1999) f¨ ur die Berechnung von abgel¨ osten turbulenten Str¨ omungen diskutiert und anhand eines Beispieles demonstriert. Zum anderen wurden elf moderne Wirbelviskosit¨ atsmodelle, wie sie in der europ¨ aischen Luft- und Raumfahrtindustrie heutzutage zum Einsatz kommen, an verschiedenen Str¨ omungsf¨ allen unterschiedlicher physikalischer Komplexit¨ at untersucht. Außerdem wurden wichtige numerische Aspekte untersucht und dargestellt.
Ein topologieorientierter Ansatz zur Turbulenzmodellierung Grundlegende Ideen Nach Hirschel (1999) wurde die Topologie von Geschwindigkeitsfeldern als Ordnungsprinzip benutzt, um eine Einteilung abgel¨ oster aerodynamischer Str¨ omungen in verschiedene Klassen vorzunehmen. Es wurden zwei Hauptklassen definiert, die sich beide auf starre, unbewegte K¨ orper bei station¨ arer Anstr¨ omung beziehen. Zur Klasse 1 geh¨ oren statistisch station¨ are, abgel¨ oste turbulente Str¨ omungen, w¨ ahrend in Klasse 2 statistisch instation¨ are Str¨ omungen zusammengefasst sind. Diese Hauptklassen unterteilen sich in weitere Unterklassen, welche sich aus den topologischen Strukturen des gemittelten Geschwindigkeitsfeldes ergeben. Beispielsweise besteht Unterklasse 1.1 aus statistisch zweidimensionalen abgel¨ osten Str¨ omungen mit einem oder zwei Rezirkulationsgebieten. Wie von Hirschel (1999) vorgeschlagen, wurde untersucht, ob die Berechenbarkeit von Str¨ omungen einer Klasse verbessert werden kann, wenn das verwendete Turbulenzmodell innerhalb der ausgezeichneten topologischen Strukturen modifiziert wird. Dies wird als topologieorientierte Turbulenzmodellierung bezeichnet. Des Weiteren regte Hirschel (1999) an, die Str¨ omungstopologie als einen Leitfaden zur Analyse von abgel¨ osten Str¨ omungen heranzuziehen. Beide Aspekte wurden im ersten Teil dieser Arbeit behandelt.
18
Zusammenfassung
Erhaltungsgleichungen und numerisches Verfahren F¨ ur die Berechnung der Str¨ omung wurden die kompressiblen Reynolds-gemittelten NavierStokes Gleichungen in konservativer und integraler Schreibweise numerisch gel¨ ost. Das numerische Verfahren ist ein so genanntes Jameson-Verfahren, das eine zentrale Raumdiskretisierung zweiter Ordnung verwendet. Die Zeitintegration erfolgt durch ein f¨ unfstufiges Runge-Kutta-Zeitschrittverfahren, wobei Konvergenzbeschleunigung mit Hilfe lokaler Zeitschritte, Mehrgittertechnik und implizitem Residuum-Gl¨ atten erzielt wird. K¨ unstliche viskose Terme zweiter und vierter Ordnung erh¨ ohen die Stabilit¨ at des Verfahrens, indem sie die Wiggle-Moden d¨ ampfen. Um eine Verbesserung der Genauigkeit und der Konvergenz bei niedrigen Machzahlen zu erzielen, wird ein so genanntes Local Preconditioning“ verwendet. Der Reynods-Spannungstensor, der ” den Einfluss der Turbulenz auf die gemittelte Str¨ omung beschreibt, wurde im ersten Teil der Arbeit mit Hilfe des Johnson-King Turbulenzmodells berechnet. Johnson & King (1985) entwickelten dieses Modell gezielt f¨ ur die Berechnung von abgel¨ osten Profilumstr¨ omungen. Es basiert auf dem algebraischen Modell von Baldwin & Lomax (1978), wobei zus¨ atzlich eine gew¨ ohnliche Differentialgleichung f¨ ur den Transport der Reynods-Schubspannung gel¨ ost wird, um Konvektionseffekte in der Turbulenz zu ber¨ ucksichtigen. Im zweiten Teil der Arbeit wurden mehrere, verschiedene Turbulenzmodelle untersucht, die weiter unten aufgef¨ uhrt sind. Demonstration an einer abgel¨ osten Profilumstr¨ omung Betrachtet wurde die zweidimensionale, abgel¨ oste Umstr¨ omung des Aerospatiale-AProfils bei M = 0.15, Re = 2.0 · 106 und α = 13.3◦ . Dabei bildete sich ein einzelnes Rezirkulationsgebiet in der N¨ ahe der Hinterkante aus. Somit geh¨ ort diese Str¨ omung zu der oben diskutierten Unterklasse 1.1. Ausgehend von der Profilnase wurden auf der Oberseite folgende topologische Strukturen identifiziert: Staupunkt (der auch ein Anlegepunkt ist), laminare Grenzschicht, transitionale Abl¨ oseblase, turbulente Grenzschicht, Abl¨ osepunkt, freie Scherschicht, Wiederanlegepunkt, Wirbelfokus und ,,r¨ ucklaufende” Grenzschicht. Die berechneten Ergebnisse wurden mit experimentellen Daten qualitativ und quantitativ verglichen. W¨ ahrend die Simulation die Str¨ omungstopologie qualitativ richtig berechnete, stellten sich deutliche quantitative Unterschiede zwischen experimentellen und berechneten Ergebnissen heraus. Insbesondere stimmte die vorhergesagte Lage des Abl¨ osepunktes nicht mit der gemessenen Lage u ¨berein. Bei den berechneten Geschwindigkeitsprofilen in der turbulenten Grenzschicht und im Rezirkulationsgebiet gab es zunehmen-
Zusammenfassung
19
de Abweichungen der Berechnungsergebnisse von den gemessenen Daten bei Ann¨ aherung an die Profilhinterkante. Des Weiteren berechnete das Turbulenzmodell von Null verschiedene Werte f¨ ur die Reynods-Spannungen im Rezirkulationsgebiet, obwohl das Experiment dort nur sehr geringe Werte zeigte. Ausgehend von diesem Ergebnis wurde das Johnson-King-Modell derart modifiziert, dass in Rezirkulationsgebieten der Unterklasse 1.1 keine ReynodsSpannungen mehr vorhergesagt werden. Es zeigte sich, dass der Einfluss dieses numerischen Experimentes auf die Umstr¨ omung des Profils nahezu nicht existent war. Folglich haben Modifikationen an der Turbulenzmodellierung stromab der prim¨ aren Abl¨ osung im Rezirkulationsgebiet keinen Einfluss auf den Abl¨ osepunkt. Vielmehr ist die Entwicklung und der Zustand der Grenzschicht, und deshalb auch deren korrekte Vorhersage, stromauf der Abl¨ osestelle von zentraler Bedeutung. Dieses Ergebnis war nicht unmittelbar zu er¨ warten, denn bei einer Profilumstr¨ omung k¨ onnen kleine Anderungen im Hinterkantenbereich aufgrund der Zirkulation um das Profil einen Einfluss auf Bereiche, die weit stromauf liegen, aus¨ uben. Die gegenseitige Abh¨ angigkeit zwischen Grenzschicht, Zirkulation um das Profil und Druckgradient entlang der Grenzschicht erschwert bei Profilumstr¨ omungen allerdings die systematische Untersuchung von Ursachen f¨ ur das Versagen eines Turbulenzmodells. Deshalb wurde die anschließende vergleichende Untersuchung moderner Turbulenzmodelle f¨ ur abgel¨ oste Str¨ omungen anhand von Str¨ omungsf¨ allen durchgef¨ uhrt, bei denen keine enge Kopplung des gesamten Str¨ omungsfeldes vorliegt und bei welchen der Druckgradient direkt oder indirekt vorgegeben werden kann. Die einzige Ausnahme bildete dabei die Umstr¨ omung des Aerospatiale-A-Profils, da diese bereits mit dem Johnson-King-Modell untersucht wurde. Auch war diese Str¨ omung bereits Gegenstand in vielen anderen vergleichenden Untersuchungen. Die genaue Vorhersage abgel¨ oster Profilumstr¨ omungen ist außerdem ein zentraler Punkt bei der Entwicklung von Hochauftriebsystemen im Flugzeugbau.
Vergleichende Untersuchung moderner Turbulenzmodelle Eingesetzte Turbulenzmodelle Bei den untersuchten Turbulenzmodellen handelte es sich um ein algebraisches Turbulenzmodell sowie um mehrere Ein- und Zweigleichungsmodelle. Ein nichtlineares, explizites, algebraisches Reynods-Spannungsmodell wurde ebenfalls untersucht. Im Einzelnen kamen folgende Turbulenzmodelle f¨ ur die vergleichende Untersuchung zum Einsatz: • das algebraische Modell von Baldwin & Lomax (1978)
20
Zusammenfassung
• die beiden k-ω-Transportgleichungsmodelle nach Wilcox (1988, 1998) • das k-ω-SST-Transportgleichungsmodell von Menter (1993) • das k-ω-TNT-Transportgleichungsmodell von Kok (2000) • das k-ω-LLR-Transportgleichungsmodell von Rung & Thiele (1996) • das explizite, algebraische Reynods-Spannungsmodell (EARSM) von Wallin & Johansson (2000) • das Eingleichungsmodell von Spalart & Allmaras (1992) • das Eingleichungsmodell von Edwards & Chandra (1996) • das Eingleichungsmodell SALSA von Rung et al. (2003) Zus¨ atzlich zu den genannten Modellen wurde das k-ω-SST-Modell von Menter im Rahmen dieser Arbeit modifiziert. Ergebnisse mit diesem Modell wurden ebenfalls f¨ ur die vergleichende Untersuchung verwendet. Untersuchte Str¨ omungsf¨ alle Die Str¨ omungsf¨ alle f¨ ur die vergleichende Untersuchung der Turbulenzmodelle wurden so gew¨ ahlt, dass mit jedem Str¨ omungsfall eine Zunahme an physikalischer Komplexit¨ at erzielt wurde. Dadurch sollte gepr¨ uft werden, bei welchem ,,Komplexit¨ atsgrad” ein Turbulenzmodell im Vergleich zum Experiment eine unzureichende VorhersageGenauigkeit liefert. Außerdem wurden nur solche Str¨ omungsf¨ alle ausgew¨ ahlt, f¨ ur welche umfangreiche und zuverl¨ assige experimentelle Daten zur Verf¨ ugung standen. Als erstes wurde eine turbulente Grenzschicht an der ebenen Platte, das heißt mit verschwindend geringem Druckgradienten, berechnet. Hier zeigte sich, dass insbesondere bei der Vorhersage des Reibungsbeiwertes die einzelnen Modelle sehr unterschiedliche Ergebnisse lieferten. Das Gleiche galt auch f¨ ur die Vorhersage des Geschwindigkeitsprofils in der Grenzschicht. Dabei hatten die Ergebnisse des k-ω-TNT-Modells von Kok und des modifizierten ¨ k-ω-SST-Modells die besten Ubereinstimmungen mit dem Experiment von DeGraaff & Eaton (2000) und dem logarithmischen Wandgesetz von Coles. Als n¨ achster Str¨ omungsfall wurde eine stark verz¨ ogerte, anliegende Grenzschicht untersucht, bei welcher der Druckgradient ein starkes Nichtgleichgewicht zwischen mittlerer Str¨ omung und Turbulenz verursachte. Die Vorhersage solcher Str¨ omungen stellt eine schwierige Aufgabe f¨ ur die Turbulenzmodelle dar. (Das zugeh¨ orige Experiment wurde von Driver & Johnston
Zusammenfassung
21
(1990); Driver (1991) durchgef¨ uhrt und ausf¨ uhrlich dokumentiert.) Auch f¨ ur diesen Str¨ omungsfall lieferten die Turbulenzmodelle deutlich unterschiedliche Vorhersagen. Im Vergleich zum Experiment zeigte das k-ω-Modell nach Wilcox von 1998 die besten Ergebnisse, insbesondere f¨ ur den Verlauf der Wandschubspannung. Das Baldwin-Lomax-Modell lieferte f¨ ur alle untersuchten Str¨ omungsgr¨ oßen die gr¨ oßten Abweichungen von den gemessenen Werten. Der n¨ achste Str¨ omungsfall war im Prinzip identisch mit dem zuletzt diskutierten mit der Ausnahme, dass der Druckgradient gr¨ oßer war und somit Abl¨ osung eintrat. (Das Experiment ist ebenfalls von Driver & Johnston, 1990). Wieder zeigte sich ein große Streuung der Berechnungsergebnisse der verschiedenen Turbulenzmodelle. Dabei war die Streuung zwischen den Ergebnissen der k-ω-Modelle gr¨ oßer als zwischen den Resultaten der Eingleichungsmodelle. Insbesondere die Position des Abl¨ osepunktes und des Wiederanlegepunktes wurde von allen Modellen unterschiedlich wiedergegeben. Keines der Modelle war in der Lage, die Position des Abl¨ osepunktes in gu¨ ter Ubereinstimmung mit dem Experiment vorherzusagen. Die Stromabentwicklung der maximalen Reynods-Schubspannung in der Grenzschicht wurde ¨ ebenfalls von keinem der Turbulenzmodelle in Ubereinstimmung mit dem Experiment wiedergegeben: Alle Modelle lieferten das Maximum zu weit stromauf und berechneten stromab davon einen Abfall der maximalen ReynodsSpannung. Im Gegensatz dazu zeigte das Experiment einen durchgehenden Anstieg der maximalen Reynods-Spannung. Die Ergebnisse der Transportgleichungsmodelle f¨ ur den Verlauf der maximalen Reynods-Schubspannung lagen allerdings n¨ aher an den experimentellen Daten als die des BaldwinLomax-Modells. Dies wird auf die ansatzweise Ber¨ ucksichtigung von Konvektionseffekten bei den Transportgleichungsmodellen zur¨ uckgef¨ uhrt. Insgesamt lagen die Ergebnisse des modifizierten k-ω-SST-Modells am n¨ achsten bei den experimentellen Daten f¨ ur diesen Str¨ omungsfall. Als letzter Fall wurde wieder die abgel¨ oste Str¨ omung um das AerospatialeA-Profil betrachtet, die jetzt mit allen untersuchten Modellen berechnet wurde. Bei der Auswertung der Str¨ omungstopologie des Abl¨ osegebietes zeigte sich, dass alle Modelle ein sehr viel kleineres Rezirkulationsgebiet voraussagten, als gemessen worden ist. Im Vergleich zum Experiment wurde der Abl¨ osepunkt von allen Modellen zu weit stromab berechnet. Das BaldwinLomax-Modell und das k-ω-Modell von Wilcox von 1988 berechneten fast keine Abl¨ osung, w¨ ahrend das Johnson-King-Modell, das k-ω-SST-Modell und das SALSA-Modell die gr¨ oßten Abl¨ osegebiete voraussagten. Aufgrund der zu klein berechneten Abl¨ osegebiete lieferten die Turbulenzmodelle im Vergleich
22
Zusammenfassung
zu den gemessenen Daten zu große Auftriebs- und zu geringe Widerstandsbeiwerte. Auch wurde wieder eine große Streuung zwischen den Ergebnissen der einzelnen Modelle beobachtet. Das Johnson-King-Modell lieferte Ergebnisse mit den geringsten Abweichungen zum Experiment. Bei allen untersuchten Str¨ omungsf¨ allen war festzustellen, dass die Vorhersage-Genauigkeit eines Turbulenzmodells f¨ ur verschiedene Str¨ omungsgr¨ oßen unterschiedlich gut war. Außerdem lieferte bei jedem Str¨ omungsfall ¨ ein anderes Turbulenzmodell die beste Ubereinstimmung mit den jeweiligen experimentellen Daten. Somit konnte keines der untersuchten Turbulenzmodelle als zuverl¨ assig bez¨ uglich der erzielten Vorhersage-Genauigkeit bewertet werden. Numerische Aspekte Die Empfindlichkeit der berechneten L¨ osungen gegen¨ uber Netzverfeinerung beziehungsweise -vergr¨ oberung wurde anhand ausgiebiger und systematischer Gitterkonvergenzstudien u uft. F¨ ur alle ¨berpr¨ Str¨ omungsf¨ alle konnte eine ausreichende Netzfeinheit mit Hilfe des so genannten Grid-Convergence Index“, der auf der Richardson-Extrapolations” Methode basiert, nachgewiesen werden. Der Einfluss des eingesetzten Local Preconditioning f¨ ur kleine Machzahlen wurde anhand einer Profilumstr¨ omung untersucht. Es zeigte sich, dass der Einsatz von Local Preconditioning eine deutliche Erh¨ ohung der Berechungsgenauigkeit erzielte. Beispielsweise wurden unerw¨ unschte Druckoszillationen in der N¨ ahe der Hinterkante von Profilen eliminiert. (Diese Druckoszillationen bei geringen Machzahlen sind typisch f¨ ur numerische Verfahren zur Berechnung kompressibler Str¨ omungen.) Auch wurde korrektes asymptotisches Verhalten des Reibungsbeiwertes f¨ ur verschwindend geringe Machzahlen festgestellt. Folglich ist bei Verwendung eines Verfahrens zur Berechnung von kompressiblen Str¨ omungen, in denen Abl¨ osegebiete mit geringen lokalen Machzahlen existieren, der Einsatz von Local Preconditioning notwendig. Der Einfluss der expliziten k¨ unstlichen D¨ ampfung wurde ebenfalls untersucht. Es stellte sich heraus, dass die k¨ unstliche D¨ ampfung in Grenzschichten normal zur Wand reduziert werden musste. Dort waren die physikalischen viskosen Fl¨ usse ausreichend groß, um das numerische Verfahren zu stabilisieren. Ein zus¨ atzliches D¨ ampfen erzeugte insbesondere im laminaren Teil der Grenzschicht einen zu hohen Impulsverlust. Beim Aerospatiale-A-Profil wurde der Einfluss der Transitionsvorgabe in der Simulation untersucht. Es zeigte sich eine hohe Empfindlichkeit der Str¨ omungsl¨ osung gegen¨ uber den vorgegebenen Transitionslagen am Profil.
Zusammenfassung
23
Wurde zum Beispiel das Einschalten des Turbulenzmodells um ein Prozent der Profiltiefe stromab verschoben, stellte sich ein stark abgel¨ ostes Str¨ omungsgebiet mit zwei gegensinnig rotierenden Rezirkulationsgebieten ein. Schlussfolgerungen Modifikationen des Turbulenzmodells, die stromab und innerhalb eines Rezirkulationsgebietes greifen, haben keinen Einfluss auf die Lage der prim¨ aren Abl¨ osung und die Gr¨ oße des berechneten Rezirkulationsgebietes. Die untersuchten Turbulenzmodelle lieferten unterschiedliche Ergebnisse f¨ ur gleiche Str¨ omungsf¨ alle. Dies traf selbst f¨ ur Modelle zu, die der gleichen Modellklasse angeh¨ oren, wie zum Beispiel die k-ω-Modelle. Bei jedem Str¨ omungs¨ fall lieferte ein anderes Modell die beste Ubereinstimmung mit dem Experiment. Transportgleichungsmodelle hatten die grunds¨ atzliche Tendenz, bei abgel¨ osten Str¨ omungen eine h¨ ohere Vorhersage-Genauigkeit f¨ ur die untersuchten Str¨ omungsgr¨ oßen zu erzielen als das algebraische Baldwin-Lomax-Modell. Es konnte keine allgemeine Aussage dar¨ uber gemacht werden, welches der untersuchten Turbulenzmodelle sich f¨ ur die Berechnung von abgel¨ osten turbulenten Str¨ omungen am besten eignet. Jedoch wurde nach Hirschel (1999) der Versuch unternommen, eine solche Aussage f¨ ur einzelne Klassen abgel¨ oster turbulenter Str¨ omungen zu erhalten. Die Lage der f¨ ur die Berechnung u ¨blicherweise vorzugebenden Transitionsorte kann große Auswirkungen auf die erzielten Rechenergebnisse haben. Eine sorgf¨ altige Modellierung der Transition im Rahmen eines Turbulenzmodells, dass heißt die Art, wie ein Turbulenzmodell den vollturbulenten Zustand erreicht, erscheint insbesondere f¨ ur abgel¨ oste Profilumstr¨ omungen von großer Bedeutung zu sein. Nur mit Hilfe eines geeigneten Local Preconditioning kann mit einem kompressiblen Verfahren eine befriedigende Genauigkeit der Str¨ omungsl¨ osung bei kleinen Machzahlen (z.B. M ≤ 0.15) erzielt werden. Des Weiteren m¨ ussen die u unstlichen D¨ ampfungsglieder in Grenzschichten reduziert werden, ¨blichen k¨ da sonst unphysikalisch hohe Impulsverluste in der Grenzschicht auftreten k¨ onnen. In Anbetracht der teilweise entt¨ auschenden Vorhersage-Genauigkeit, die mit den betrachteten Turbulenzmodellen erzielt wurde (auch in anderen Arbeiten), erscheint eine zuverl¨ assige und genaue Vorhersage von abgel¨ osten turbulenten Str¨ omungen im Allgemeinen nicht m¨ oglich. Um die heutige Modellierung der Turbulenz und damit die Berechenbarkeit abgel¨ oster turbulenter Str¨ omungen zu verbessern, erscheint ein streng kombinierter und systema-
24
Zusammenfassung
tisch koordinierter Einsatz von Experiment, direkter numerischer Simulation, Large-Eddy-Simulation und Reynolds-gemittelter Simulation einzig erfolgversprechend. Die einzelnen Disziplinen sollten dabei in direkter gegenseitiger Wechselwirkung stehen. Ausblick Eine m¨ ogliche Verbesserung der Vorhersage-Genauigkeit f¨ ur turbulente Str¨ omungen bei gleichzeitig vertretbarem Rechenaufwand kann eine Kopplung von Reynolds-gemittelter Simulation (RANS) und Large-EddySimulation (LES) ergeben. Die LES kommt dabei nur in Str¨ omungsbereichen zum Einsatz, in denen eine physikalisch genauere Erfassung der Turbulenz erzielt werden soll, als dies mit RANS m¨ oglich ist. Ein kritischer Punkt bei dieser Methode ist die Erzeugung von physikalisch sinnvollen turbulenten ¨ Schwankungen beim Ubergang vom RANS-Gebiet zum LES-Bereich. Zu diesem Zweck wurde ein teilweise neuer Ansatz vorgeschlagen, der als Basis f¨ ur k¨ unftige Arbeiten dienen soll.
1
Introduction
Turbulence plays a vital role in many flows of engineering interest. Fast and accurate computations of turbulent flow fields are therefore demanded by engineers in a wide variety of technical fields. It is commonly accepted that the Navier-Stokes equations permit accurate description of laminar, turbulent and transitional flows of simple fluids (Bradshaw, 1999; Krause, 1999). Resolving the entire spectrum of the fluid’s turbulent motion in space and time by solving the Navier-Stokes equations, i.e. performing a direct numerical simulation (DNS), can yield a highly accurate solution of the flow field. Yet, the smallest turbulent length and time scales decrease with increasing Reynolds number (Re) and, as a consequence, computational power needed for the resolution of all turbulent scales is approximately proportional to Re3 (Rodi, 2000; Bradshaw, 1999). Despite ever-increasing computational power of modern supercomputers DNS of flows at high Reynolds number is still not feasible. In fact, no matter how powerful a computer may be there will always be a limit to the achievable Reynolds number. In most practical cases computations of flows at high Reynolds numbers are performed using some form of statistical turbulence model to account for the effect of turbulence on the flow. In the statistical approach to turbulence modeling, mean values are studied, which vary relatively smoothly with time and space. This idea dates back to 1895 when Reynolds introduced his concepts of averaging. In the averaging process instantaneous flow variables are expressed as the sum of a mean and a fluctuating part. This sum is inserted into the conservation equations of mass, momentum and energy of the flow. Finally, Reynolds averaging of the equations is performed and the Reynoldsaveraged Navier-Stokes equations (RANS), which describe the mean motion of the fluid, are obtained. Due to the non-linearity of the convection terms, so-called Reynolds stresses appear as additional terms in the mean momentum equations. Since smoothly-varying mean values are computed, resolution demands are relaxed by several orders of magnitude compared to DNS, and a solution of the mean motion of the flow is obtained at much lower computational cost. Before Reynolds introduced his theory of averaging, it was recognized by some researchers that turbulence acts very much like additional stresses. In order to develop a mathematical description of turbulent stresses, Boussinesq suggested in 1877 the concept of eddy viscosity. In this framework, turbulent
26
1 Introduction
stresses are treated analogously to molecular stresses, with eddy viscosity being the turbulent counterpart of molecular viscosity. Many different turbulence models have been proposed for the computation of Reynolds stresses since the days of Reynolds and Boussinesq and a complete list of all published models cannot be given. Instead, only the most important or best known “landmarks” in statistical turbulence modeling shall be mentioned here. In 1925, Prandtl introduced his mixing length theory for the computation of eddy viscosity. He did not assume the eddy viscosity to be constant, as was usually done at that time. Instead, Prandtl made it a function of local flow quantities, namely the gradient of mean flow velocity, and a characteristic turbulent length scale, the mixing length. The rationale behind Prandtl’s ideas is that the eddy viscosity can, on dimensional grounds, be regarded as a product of a suitable length and velocity scale. Today, virtually all of the so-called algebraic turbulence models in use originate from Prandtl’s mixing length theory. The Baldwin-Lomax (Baldwin & Lomax, 1978) model is one of the most common models for computational fluid dynamics belonging to this category. Its strengths are the capability to deliver reasonable results for many boundary-layer flows while being numerically robust and computationally inexpensive and fast. More advanced eddy-viscosity models solve partial differential transport equations for either one or two turbulent quantities in order to incorporate non-local and flow-history effects into the eddy viscosity. Prandtl was the first to suggest a one-equation model where the desired velocity scale is computed from the turbulent-kinetic-energy transport equation. In this model, an algebraic prescription for computing the length scale is still needed. In the one-equation model of Spalart & Allmaras (1992), a partial differential equation for the eddy viscosity itself is solved and the need to compute the length scale separately is circumvented. In two-equation models, the eddy viscosity is related solely to turbulent quantities, in contrast to algebraic models where it is related purely to mean flow quantities. In almost all two-equation models the turbulent kinetic energy k serves as the velocity scale. In order to compute an appropriate length scale any expression of the form km n may be employed, with being the turbulent dissipation rate. (m and n don’t have to be integers.) Pioneering work on two-equation models was done as early as 1942 by Kolmogorov (see Wilcox, 1998) but due to the computational demands of such models computers had to come into general engineering use in order to be able to solve the models’ equations for flows of interest. By
27
far the most popular two-equation model is the k, (m = 0, n = 1) model that was suggested by Jones & Launder (1972). It has often been modified and re-tuned and became a quasi-standard model in industrial use although its defects, like even poorer performance than algebraic models for boundary layers in adverse pressure gradients, cannot be dismissed. In addition, models based on k and cannot be integrated through the viscous sublayer without modifications. To permit integration through the sublayer, so-called low-Reynolds-number modifications, which usually consist of viscous damping functions, have to be incorporated into the model. Alternatively, wall functions are applied in order to entirely “bridge” the near-wall region in the solution of the model equations. Besides , the specific dissipation rate ω has gained increasing popularity as the second variable in two-equation models. This corresponds to setting m = −1, n = 1. As noted by Wilcox (1998), k, ω models offer very appealing advantages over the k, models. First, k, ω models perform better in boundary-layer flows, especially with adverse pressure gradients, and, secondly, they can be integrated through the viscous sublayer without any near-wall modifications. It was recognized very early that the eddy-viscosity assumption has major shortcomings and there is still very active research underway to entirely avoid the use of eddy viscosity. This can be done, in principle, by solving the exact Reynolds-stress transport equations, which can be deduced from the Navier-Stokes equations. However, additional unknown terms like pressurestrain correlation and turbulent diffusion, which require modeling, appear in the Reynolds-stress equations and, hence, introduce new uncertainties. Rotta (1951) was the first to introduce a complete Reynolds-stress transport model (RSTM), but as with two-equation models, computers at that time did not offer sufficient computational power to solve the model equations. One of the first RSTMs that was computationally “affordable” at the time of its development is the model by Bradshaw, Ferriss & Atwell (1967). In this model, a direct proportionality between Reynolds shear stress and turbulent kinetic energy k in a two-dimensional boundary layer is assumed. k is computed from a differential transport equation and, hence, the concept of eddy viscosity is avoided in the computation of the Reynolds stress. Another model that is also mandatory to mention in connexion with RSTMs is the model devised by Launder, Reece & Rodi (1975). It is the best known and most extensively tested model that computes the complete Reynolds-stress tensor from modeled Reynolds-stress equations.
28
1 Introduction
A different approach, which is expected to give a more accurate description of the Reynolds-stress tensor than linear eddy-viscosity models, is to assume a non-linear constitutive relation between the Reynolds-stress tensor and the strain-rate and rotation tensors. Several different ways have been proposed to derive such relations (e.g. Lumley, 1972; Saffman, 1976; Speziale, 1987; Rodi, 1976; Gatski & Speziale, 1993) and, especially for flows where system rotation and anisotropy of the Reynolds-stress tensor play an important role in computing the effect of turbulence on the mean flow, these kind of models improve flow predictions, at least qualitatively. A recent survey and analysis of models belonging to this category can be found in Rung (2000). All models mentioned so far focus on the computation of Reynolds stresses. The latter are a result of averaging the entire turbulent wave-number spectrum. In large-eddy simulations (LES), averaging is performed only for high wave numbers belonging to small eddies while large and energy-bearing eddies are resolved. This is done by applying low-pass filtering to the Navier-Stokes equations. The filtering process introduces subgrid stresses, which account for the interaction between resolved turbulent structures and subgrid scales. Because most of turbulent energy is carried by larger eddies, modeling the high wave-number part of the spectrum seems to be an attractive alternative to full Reynolds-stress modeling on the one hand and DNS on the other. However, LES is only about one order of magnitude “cheaper” in computing costs than DNS for wall bounded flows. Approaching the wall, large turbulent scales decrease in size and hence a “quasi” DNS must be utilized to resolve the energy bearing eddies. Therefore, qualitatively similar restrictions on achievable Reynolds numbers hold in both LES and DNS, although being less restrictive for LES. There are attempts to use “off-the-wall” boundary conditions so that the viscous wall region can be excluded from the main computation. These boundary conditions often rely in some form on the law of the wall and are not trustworthy for flows where the law of the wall does not hold, as for separated boundary layers. Besides development of suitable numerical methods for LES, derivation and application of subgrid-scale models and appropriate near-wall treatment are subject to current research in LES. Recently, a lot of research on hybrid methods combining RANS simulation in one part of a flow and LES in another has been performed. Detachededdy simulation (DES) proposed by Spalart, Jou, Strelets & Allmaras (1997), for example, uses a conventional eddy-viscosity model, namely the SpalartAllmaras model, for the near wall region. Away from the wall, the computa-
29
tion converts to LES and the Spalart-Allmaras model acts as a subgrid-scale model. This is ensured by the modified formulation of the underlying SpalartAllmaras model, which “detects” whether the grid is fine enough or not to use LES, and not by explicit switching from RANS to LES. Other hybrid LES/RANS approaches are of explicit zonal character. There, LES is performed in regions of the flow where a richer flow description is necessary and classical RANS in other regions. This method has been applied in the computation of aero-engine gas-turbines where LES has been used in the combustion chamber while RANS models have been applied in the compressor and turbine sections (see Schl¨ uter & Pitsch, 2001). Despite several decades of research on statistical turbulence models and the wide variety of models proposed the computed solution of a flow field frequently does not meet desired engineering accuracy. In most such situations, the failure can be attributed to poor performance of the employed turbulence model(s). This is primarily the case for separated flows and for types of flows where the applied model was not tested or calibrated before. An important fact in regard to occasionally disappointing performance of turbulence models is the general lack of useful possibilities to make estimates of accuracy of results a priori, i.e. without knowing the “real” solution. In light of the remarks in the above paragraph and considering documented experience of the research community, it may be deduced that, whatever model is considered, one can always draw the same conclusion about a model’s performance: There exists a variety of flow situations where the model performs reasonably well. However, important flow scenarios will be encountered where the model will unexpectedly produce results of insufficient and unpredictable accuracy (compared to experiment or DNS). This indicates that, although a single universal turbulence model remains the ultimate goal in turbulence modeling, development of such an universal model is highly uncertain. From this premise, several researchers have abandoned development of universal models in favor of construction of models that are superior in a rather limited number of flow classes. Johnson & King (1985), for example, developed a very successful model for the treatment of two-dimensional, subsonic, pressure-driven separated flows and shock-induced separated flows. Straightforward extension of this concept is the combination of “optimal” models in order to broaden the field of application: In different regions of a flow different models are applied using for each region an as optimal model as possible. The model-selection process is guided by the flow type encountered in the considered region. This concept of zonal modeling was frequently dis-
30
1 Introduction
cussed, see for example Kline et al. (1981), but Ferziger et al. (1988) were one of the first to rigorously apply the zonal approach. They regard a complex flow field as a compound of several different structural flow zones where each flow zone comprises a single part of the flow with similar turbulent structures. Ferziger et al. (1988) argue that a turbulence model should be linked to local properties of the flow; the turbulence model itself should vary as a function of relevant flow parameters. This is based on the fact that modeling flows with a single kind of turbulent structure can be successfully accomplished with existing turbulence models. To demonstrate their approach, they use the same baseline model throughout the entire domain and adapt the model’s closure constants and functions from zone to zone using so-called bridges. Because a structural flow zone is supposed to contain similar turbulent structures, zoning of the flow is guided by physical insight of and knowledge about turbulence structure. The topological approach to turbulence modeling for separated flows according to Hirschel (1999) suggests a classification of separated aerodynamic flow fields guided by the observed or expected flow topology. The term “flow topology” denotes in this case the topology of streamlines of the Reynoldsaveraged velocity field. (More frequently, in turbulence research, the term “topology” is used in the context of coherent turbulent structures.) An excerpt of a possible classification of separated flows according to Hirschel (1999) is shown in Table 1.1. Following this line of reasoning, separated turbulent flows around rigid fixed bodies in steady flow at infinity are sub-divided into flow classes with and without vortex shedding denoted as class 1 and 2, respectively. Sub-classes with two- and three-dimensional flows are defined in each major class using topological and geometrical arguments. For example, class 1.1 in Table 1.1 comprises two-dimensional separated flows with one or two recirculation zones featuring so-called critical or nodal points. These are points within the flow domain where all velocity components are zero and the streamlines’ slopes are indeterminate (see Chong et al., 1990). The flow topologies of class 1.1 are frequently encountered in RANS solutions of flows past airfoils at high angle of attack near or at stall conditions. Indeed, many test cases for turbulence models are of this kind, (see Haase et al., 1993, 1997; Dervieux et al., 1998). However, it is not clear whether, or in what parameter range, flows of class 1.1 exist in a statistically steady sense. Trailing-edge separation of an airfoil flow may exist statistically only in an unsteady, vortex-shedding (sub-class 2.1) or vortex-flapping motion. In this case, sub-class 1.1 comprises time-averaged topologies which are not coin-
31
cident with the time-varying statistical ensemble. One the one hand, it is generally questionable whether a RANS model is able to yield the statistical ensemble of a statistically unsteady flow. On the other hand, for engineering purposes, it may be sufficient to define a time-average solution and postulate that solving the RANS equations together with a turbulence model should deliver this time average. In other situations, it may be necessary to predict the low-frequency part of a solution. Here, comparison of the computed solution with experimental data is especially difficult since consistent distinctions between low- and high-frequency parts are needed for both the turbulence model and the measurement. The main distinctive factors between class 1.2 and 1.3 stem from different geometrical properties of the bodies under consideration. Slender bodies lead to different flow topologies than bodies with high aspect ratios, and it has been observed that flows of class 1.3 are free from large-scale unsteadiness for a wide range of parameters even if the vortices become turbulent. Based on this classification of flow fields, it is hypothesized that computation of a single class can be improved if turbulence modeling is adjusted to topological structures like recirculation zones, separating boundary layers, free shear layers etc. In this aspect, the topological approach is quite similar to the zonal idea discussed above, the main difference being the general classification and partition of separated flows by means of the topology of the velocity field. In addition, the flow topology serves as a guideline for discussion of results and model performance.
32
1 Introduction
Table 1.1: Possible classes of separated aerodynamic flows (schematic, averaged structures; excerpt from Hirschel, 1999) Rigid fixed bodies in steady flow at infinity 1. Flows without vortex shedding
2. Flows with vortex shedding
1.1 Steady separated flow past airfoils with recirculation area(s) – 2D
2.1 Unsteady separated flow past airfoils – 2D
a)
α
V∞ b)
α
α
V∞
V∞ 1.2 Steady separated flow past wings with large aspect ratio – 3D
2.2 Unsteady separated flow past wings with large aspect ratio – 3D
? 1.3 Steady separated flow past wings and slender bodies with longitudinal vortices – 3D
2.3 Unsteady separated flow with longitudinal vorices past slender bodies – 3D
1.1
The Present Study
1.1
33
The Present Study
The motivation for this work was twofold: The one aspect was testing and demonstration of the topological approach to turbulence modeling for separated flows suggested by Hirschel (1999). The other aspect was to conduct a comparative study of various modern eddy-viscosity turbulence models in current production use in European aerospace industry with the objective to assess the models’ performance for separated turbulent flows. Embarking on the topological concepts of Hirschel (1999) the starting point for this work was the group of flows belonging to class 1.1 a) in Table 1.1. The computed flow around an airfoil at high angle of attack with a single recirculation zone near the trailing edge was investigated in detail and compared with data from experiment. Subsequently, the applied turbulence model was modified in the recirculation zone according to arguments of Hirschel (1999) in order to study the feasibility of this approach. The major outcome of this feasibility test was that a modification of a turbulence model does not improve computational results if it is restricted to the recirculation zone. In particular, the point of separation is not affected by modifications inside of the recirculation zone and the size of the computed recirculation zone does not change. Consequently, following conclusions were drawn: 1. The boundary-layer development influenced by pressure gradient upstream of separation has to be considered in order to correctly capture the position of the primary separation point (or separation line in threedimensional flows). 2. One major challenge in computing separated flow fields is that the physical nature of turbulence structure changes from that of a boundary layer to that of a free shear layer. Turbulence models have to properly mimic the effect of this change on the mean flow. In order to investigate a model’s ability to perform this task, several flow cases have to be studied which feature a successively increased strength of change in the flow structure from case to case. Based on these conclusions the following strategy was chosen: First, the performance of several modern turbulence models was investigated in a classical flat-plate boundary layer with zero pressure gradient. This was deemed to be necessary to basically assess the ability of turbulence models to correctly compute boundary layers in the frame of full Navier-Stokes solutions
34
1 Introduction
as opposed to boundary-layer methods where these models typically have been tested. Secondly, an attached boundary layer under substantial adverse pressure gradient was computed using the same models; results obtained with the different models were compared with each other and with experimental data. This flow case is very well suited to test the models in a non-equilibrium boundary layer without separation. By increasing adverse pressure gradient a separating boundary layer was obtained. As before, computations with all models were performed; experimental data served for comparison of results. Finally, the separated airfoil flow was again analyzed utilizing the available set of turbulence models. Questions regarding surface boundary conditions and dependence on freestream values for the turbulence equations were addressed as well as purely numerical issues like grid convergence and low-Mach-number preconditioning. The sensitivity of the computational solution to the choice of transition location was also investigated. The major contribution of this work is seen in thorough testing and performance assessment of various modern turbulence models for computing flows with pressure-induced boundary-layer separation. The applied models range from the algebraic model of Baldwin & Lomax (1978) to the explicit algebraic Reynolds-stress model of Wallin & Johansson (2000), covering also a range of one- and two-equation eddy-viscosity models. Although the selected flow cases may seem somewhat simple, they offer a natural increase in complexity. It is evident from results obtained that the computation of the considered cases poses a challenging task. In particular, variances among results computed with different models show that the task of successful turbulence modeling for such flows is not completed, and further improvements are necessary. For this purpose, a partially new method of coupling RANS and LES was proposed. The method is presented in the appendix of this thesis.
1.2
Contents and Organization of the Thesis
The present thesis is organized in three major parts. Part I is concerned with ideas proposed by Hirschel (1999). Accordingly, in Section 2, the flow topology of velocity fields is used as a guideline for classifying separated turbulent flows and for identifying flow regions of possible importance to turbulence modeling. In Section 3, the governing equations of the flow, the numerical method, and the turbulence model employed in Part I of this work are briefly discussed. Subsequently, in Section 4, the flow field of a separated airfoil flow
1.2
Contents and Organization of the Thesis
35
is investigated using the flow topology as a guideline for the analysis of results. In addition, a numerical experiment is presented where the turbulence model was modified in the separated region of the airfoil flow. Comments on hidden three-dimensional effects in nominally two-dimensional flows are placed in Section 4 at the end of Part I. Part II deals with a comparative study of various modern turbulence models for computing separated boundary-layer flows. First, in Section 5, the numerical method employed in Part II is briefly discussed. Secondly, in Section 6, the turbulence models investigated are presented. Thirdly, in Section 7, the test cases selected are introduced and computational results obtained with the models are compared with experimental data. Numerical issues, like grid convergence, local preconditioning and artificial damping are subject of Section 8. Sections 9 and 10 contain the conclusions and the future work, respectively. Part III consists of the appendices. In the first section of the appendices, Section A, a partly novel approach to coupling RANS and LES for separated flows is proposed. In Section B, details of the Johnson-King model are presented and Section C contains comparative graphs of computational and experimental results. A brief overview of the work performed concerning algorithmic issues is given in Section D. In the last section, Section E, typical input decks used for the computations with the FLOWer code are presented.
Part I
Topological Approach to Turbulence Modeling
38
2
2 Basic Considerations
Basic Considerations
It was already mentioned in the Introduction, that, more often than not, the computed solution of a turbulent flow field does not offer the required accuracy. The reasons for unsatisfactory results are manifold; numerous possibilities have to be kept in mind when considering a particular flow case. The lack of reliable models for determining locations of transition from laminar to turbulent in general flow situations is one such possibility. Due to this lack, mostly, the locations have to be explicitly specified by the user before starting the computation. In some cases, transition locations on the surface of a wind-tunnel model are known from experiments, but in general they are unknown. However, even if this kind of transition is given from experimental data this may not comprise enough information about all the transition mechanisms potentially encountered in the flow case considered. For instance, in complex, three-dimensional, separated flow fields, like the flow around a delta wing at high angle of attack, transition can occur in shear layers away from the surface before the turbulent fluid “hits” the surface of the body. In this way, transition can significantly influence the momentum and heat transfer at the surface of the body as well as the overall flow field. This type of transition and its location can hardly, if possible at all, be identified in experiments. One more example where the modeling of transition can have major impact on the flow solution is dynamic stall of airfoil flows. Ekaterinaris & Menter (1994) point out that correct modeling of leading edge transition is a key issue in the computation of such flows. Clearly, in all turbulent flow fields where transition plays a crucial role incorrect specification of transition location(s) may become a major source of error. Another possible source of error in the computation of separated flow fields is “hidden” three-dimensionality in nominally two-dimensional flows. Frequently, in the computation and analysis of a particular flow case it is implicitly assumed that the flow field is two-dimensional in the usual sense of RANS. This assumption may not be correct, especially when considering a massively-separated flow. Similarly, it is often assumed that the flow field is statistically steady and that time-averaged mean values are a valid and accurate representation of probability mean values. Neither may be the case, and this can lead to discrepancies between measured and computed results. (For a more detailed discussion of statistical concepts for describing turbulent flows and the problems arising in this context the reader is referred to Monin & Yaglom (1977); Rotta (1951); Celi´c & Hirschel (1999).)
39
Numerical issues, like accuracy of the numerical method, iterative convergence and grid convergence, have to be treated with appropriate thoroughness in order to eliminate numerical effects on the results to greatest possible extent. These questions will be addressed in more detail in Section 8. The following list summarizes some of the most important sources of errors when computing turbulent aerodynamic flows: • specification of transition locations and transition modeling • hidden three-dimensional effects in nominally two-dimensional flows • the flow case considered is statistically unsteady; probability mean values and time-averaged mean values differ (it is questionable whether the concept of Reynolds averaging is applicable to unsteady flows) • strong non-equilibrium between mean flow and turbulence – the Boussinesq assumption becomes really wrong (this applies only to eddyviscosity turbulence models) In order to identify possible turbulence-model failures, only those flow cases should be considered where all other possible sources of errors can be regarded as relatively small or non-existent. This is postulated of the group of flow fields belonging to class 1.1 in Table 1.1, which were considered first in this work. At this point, the validity of this assumption is not questioned but it will be discussed again later. The schematic flow topology is repeated in Figure 2.1 and is discussed in the following; topological structures shown in the figure are listed in Table 2.1. Following the stagnation streamline, the flow reaches the surface of the airfoil at the stagnation point A1, where it is split into two attached laminar boundary layers a1 and a4 on the upper and lower sides, respectively (see Figure 2.1). If the Reynolds number of the flow is high enough the laminar boundary layers will become turbulent at some position downstream of the stagnation point. In Figure 2.1, the region of laminar-turbulent transition T on the upper side of the airfoil is denoted by a2; on the lower side it is denoted by a5. Generally, transition can be forced by some tripping device or it can occur “naturally”, i.e. free transition takes place. In the latter case, a small transitional bubble with laminar separation and turbulent reattachment may be found on the upper side of the airfoil, depending on the type of pressure distribution. Downstream of the transition region a2 the turbulent boundary layer a3 separates from the surface of the body at the separation
40
2 Basic Considerations
transitional separation bubble (time averaged)
a2 T
a1
a3 A1 a4
b1
S1 T a5
a7 a6
F1 A2 a8 S2
b2
Figure 2.1: Schematic flow topology of airfoil flows with one recirculation zone at high angle of attack (class 1.1 a) of Table 2.1.
Table 2.1: Two-dimensional topological structures shown in Figure 2.1 with a classification according to Peake & Tobak (1980) Symbol
Phenomena
A1, A2
attachment point (stagnation point)
Classification half saddle
S1
(primary) squeeze-off separation point
half saddle
S2
flow-off separation point
half saddle
F1
recirculation zone around focus
focus
a1, a4
attached laminar boundary layer
a2, a5
laminar-turbulent transition region (T)
–
attached turbulent boundary layer
–
attached boundary layer in recirculation zone
–
shear-layer skeleton streamline
–
a3, a6, a8 a7 b1, b2
–
point S1. This type of separation is herein called “squeeze-off separation” following Hirschel (1986) since the two boundary layers a3 and a7 converge and “squeeze” each other off the surface at S1. Enclosed by the dividing streamline b1 a recirculation zone with focus F1 containing slowly moving fluid is present at the trailing edge. The situation in the vicinity of the reattachment point A2 (and the boundary layer a8) is not as clear as for the other nodal points A1, S1, F1 and S2. Topologically speaking, A2 is simply a half-saddle. However, the fluid is moving very slowly and the streamlines
41
found in the computations are highly curved in the vicinity of A2 in order to satisfy the Reynolds-averaged continuity equation after the turbulent shear layer b1 has deflected away from the wall (Collins & Simpson, 1978). Following Hirschel (1986), the turbulent boundary layer a6 on the lower side of the airfoil remains attached until it reaches the “flow-off separation point” S2 at the trailing-edge . The formation of a transitional bubble at the upper surface depends on the shape of the airfoil and angle of attack, i.e. pressure distribution, and the Reynolds number. Detailed numerical and experimental investigations of a transitional bubble were performed for example by Lang et al. (2000). The time-averaged streamline topology of the transitional bubble, as shown in Figure 2.1, is very similar to the recirculation zone located further downstream at the trailing edge of the airfoil. However, underlying flow physics differs substantially. The separation point of a laminar boundary layer is welldefined while the separation point S1 of the turbulent boundary layer exists only in an averaged sense. (The latter is “artificially” obtained through an averaging procedure.): In a time-accurate view, the bubble sheds spanwise vortices at its rear end and these are the beginning of the turbulent boundary layer. The time-accurate behavior of the recirculation zone at the trailing edge is not very well understood. It is assumed that some kind of non-periodic vortex-flapping or vortex-shedding mechanisms combined with small scale turbulent motion take place. Experiments show strong interaction between wakes of the pressure and suction sides (Collins & Simpson, 1978). Since turbulence models for the Reynolds-averaged Navier-Stokes equations are the subject of this work, the subtle time-accurate features of the flow are not considered and only time-averaged flow topologies are discussed. Nevertheless, one should keep the complex time-accurate flow structure in mind when considering computational solutions of such flow fields.
42
3 3.1
3 Governing Equations and Numerical Method
Governing Equations and Numerical Method Governing Equations of the Mean Flow
For the mathematical description of turbulent flows the Reynolds-averaged Navier-Stokes equations in integral and conservation form are considered. Strictly speaking, Favre- or mass-averaged equations are employed in order to eliminate density fluctuations from the equations. However, for simplicity, and because all flows computed in this work are at low Mach numbers where compressibility effects are negligible, the term “Reynolds-averaged” is used. Consequently, fluctuating quantities are denoted by a single prime as is common practice in Reynolds averaging as opposed to double primes in Favre averaging. In contrast to the notation frequently encountered in text books about turbulence, the overbar indicating an averaged quantity is omitted for all variables but the Reynolds stresses. This is in compliance with the usual notation of the Reynolds- or Favre-averaged equations used in most texts about computational fluid dynamics. Results of the present work are not affected by these conventions. Equations for mass, momentum and energy transport derived with the help of a finite control volume fixed in space read, in symbolic notation, ∂ w dV + F · n dS = 0. ∂t V S In the above equation, n denotes the outward normal vector of the surface element dS of the control volume V . The vector w contains the conserved variables, w = (ρ, ρu, ρv, ρw, ρE)T , (3.1) and the generalized flux vector is written as ⎡ ⎤ ρv ⎢ ⎥ F=⎣ ρv ⊗ v + pI − σ − τ ⎦ ρvH − σ · v − τ · v − λ∇T
(3.2)
with v = (u, v, w)T being the flow-velocity vector and I the identity matrix. (⊗ denotes the dyadic product of two vectors.) The total energy E is the sum of internal and kinetic energy, E =e+
1 2 |v| , 2
3.1
Governing Equations of the Mean Flow
43
where for a calorically perfect gas the internal energy becomes e = cv T . T denotes static temperature, p static pressure and cv the specific heat coefficient at constant volume. The relation between total energy E and total enthalpy H is given by p H=E+ . ρ For a Newtonian fluid and with the usual bulk-viscosity assumption ζ = − 23 µ the viscous stress tensor is defined as 1 (3.3) σ = 2µ S − (∇ · v) · I , 3 with the strain-rate tensor
1 ∇ ⊗ v + (∇ ⊗ v)T . 2 Kinetic gas theory relates the coefficient of dynamic viscosity µ to static temperature. This relation can be expressed through Sutherland’s formula in conjunction with a reference viscosity µref at a reference temperature Tref : 3 2 T T ref + 110K µ = µref . Tref T + 110K S=
While the specific formulas by which the Reynolds-stress tensor τ in Equation 3.2 is computed depend on the turbulence model employed, it is generally defined by one-point central moments of fluctuating velocity components and reads in index notation τ ≡ τij = −ρui uj . (3.4) This is a result of averaging the Navier-Stokes equations. The computation of τ will be addressed when discussing the various turbulence models. All models investigated compute an eddy viscosity µt which, in turn, is used to derive an expression for the effective heat-conductivity coefficient λ: µ γR µt λ= + . γ − 1 Pr P rt R denotes the perfect-gas constant. While the molecular Prandtl number P r is a fluid property and can be regarded as approximately constant for most gases, with P r = 0.72 for air at standard conditions, this is not the case for the turbulent Prandtl number P rt . However, in the framework of eddy-viscosity models it is assumed that P rt = 0.9 throughout the flow field. γ denotes the ratio of specific heat coefficients with γ = 1.4 for air.
44
3.2
3 Governing Equations and Numerical Method
The Baldwin-Lomax Model
Although all computations for this chapter were performed using the model of Johnson and King, the Baldwin-Lomax Model is presented first since it can be regarded as a “pre-requisite” for the former. The Reynolds-stress tensor in Equation 3.4 is modeled with the help of Boussinesq’s assumption, which states that the turbulent stresses are proportional to the strain-rate tensor, 1 τ = 2µt S − (∇ · v) · I . (3.5) 3 This is in analogy to the molecular stresses in Equation 3.3 but with the fundamental difference that the eddy viscosity µt is not a fluid property. It is dependent on the particular flow case and has to be computed by the turbulence model. Equation 3.5 is also referred to as a “constitutive relation”. Bearing in mind thin-shear-layer flows, like boundary layers and wakes, Baldwin & Lomax (1978) suggested a two-layer eddy-viscosity model with separate algebraic expressions in each layer. Hence, the eddy viscosity is defined by µti , y ≤ ymin µt = (3.6) µto , y > ymin with y being the coordinate in the wall-normal direction in case of boundarylayer flows. In wake flows, the wake center line is used as the origin for y. ymin denotes the smallest value of y for which µti = µto . The inner and outer viscosities are computed as follows: Inner Layer: 2 µti = ρ lmix |ω|,
lmix = κyD, , κ = 0.40,
−y + /A+ 0
D = 1−e
(3.7) A+ 0
= 26.
Expression 3.7 is based on Prandtl’s mixing length lmix = 0.40y multiplied by Van Driest’s damping function D. This damping function is only active in the close vicinity of solid walls in order to account for the damping effect of the wall on the eddy viscosity. The dimensionless, sublayer-scaled wall-distance is defined as τw yuτ y+ = with uτ = ν ρ
3.2
45
The Baldwin-Lomax Model
where the friction velocity uτ serves as a velocity scale; it is related to the shear stress at the wall. ν is the kinematic viscosity of the fluid and |ω| denotes the magnitude of the vorticity vector. Outer Layer: µto = αρCcp Fwake FKleb , 2 Cwake ymax Udif Fwake = min ymax Fmax ; , (3.8) Fmax Fmax =
1 max (lmix |ω|) , κ y
FKleb (y; ymax /CKleb ) = 1 + 5.5 α = 0.0168,
Ccp = 1.6,
6 −1
y ymax CKleb
CKleb = 0.3,
,
(3.9)
Cwake = 1.
In the above relations, ymax is the value of y at which lmix |ω| achieves its maximum. Udif is the maximum velocity for boundary-layer flows, and for free shear layers it is the difference between the maximum value of |v| in the layer and the value of |v| at ymax . Corrsin & Kistler (1954) as well as Klebanoff (1954) experimentally showed that the flow at the boundary-layer edge is intermittent. This means that at a fixed location in space close to the boundary-layer edge, the flow is sometimes non-turbulent and sometimes turbulent. The reason for this is that the shape of the sharp interface between laminar and turbulent regions is highly distorted and moving. In order to account for intermittency and its effect on the outer eddy viscosity, µto is multiplied by the empirical function FKleb computed from Equation 3.9. Figure 3.1 shows a typical eddy-viscosity profile computed with the Baldwin-Lomax model for a flat-plate boundary layer. As a final comment, the Baldwin-Lomax model can be viewed as a reformulated and extended model of Cebeci & Smith (1974). The main difference is that the latter requires the computation of boundary-layer properties which are difficult to determine in general Navier-Stokes computations. Specifically, for the Cebeci-Smith model, the product Ccp Fmax in Equation 3.8 is replaced by ue δv∗ , where ue stands for the velocity at the boundary-layer edge and δv∗ denotes the kinematic displacement thickness. In boundarylayer methods, where these properties are readily available, the Cebeci-Smith model can be easily applied.
46
3 Governing Equations and Numerical Method
10-2
-3
y/L
10
-4
10
0
20
40
µturb/µ∞
60
Figure 3.1: Typical Baldwin-Lomax eddy-viscosity profile for a flat-plate boundary layer.
3.3
The Johnson-King Model
For the investigation of the topological concepts of Hirschel (1999) the turbulence model devised by Johnson & King (1985) was employed. This model has been extensively tested for separated airfoil flows and has shown either equivalent or even superior performance for such flows compared to many other models (see Haase, 1997; Haase & Fritz, 1993). Its development was guided by physical insight while avoiding significant increase of mathematical complexity in comparison with the Cebeci-Smith or Baldwin-Lomax models. The main physical aspects that inspired the development of the JohnsonKing model are the following: 1. For rapidly-changing turbulent flows, like separating boundary layers, convection of turbulence is an essential effect and must be properly accounted for in the model (Bradshaw et al., 1967; Johnson & King, 1985). 2. Perry & Schofield (1973) carried out experiments of adverse pressure gradient boundary layers near separation. They found that descriptions for determining the mean velocity profile based on velocity scales related to the maximum shearing stress and its distance from the wall gave
3.3
47
The Johnson-King Model
the best results. Hence, the maximum of (−u v )1/2 is used as the controlling velocity scale in the Johnson-King model. In the style of the Cebeci-Smith and Baldwin-Lomax models, the JohnsonKing model uses a two-layer approach for the eddy viscosity and Boussinesq’s Equation 3.5 for the Reynolds-stress tensor. However, instead of the switch in Equation 3.6, a smooth exponential blending between the inner and outer eddy viscosities is employed:
µt = µto 1 − e(−µti /µto ) . (3.10) Inner Layer: µti = ρD2 κy(−u v m )1/2 , −y(−u v m )1/2 D = 1 − exp , νA+ κ = 0.40,
(3.11) (3.12)
A+ = 15.
It can be seen from the above equations that the inner eddy viscosity µti is strongly dependent on the velocity scale (−u v m )1/2 . Additionally, the resulting eddy viscosity µt is functionally dependent on µto for the greatest part of the boundary layer, see Equation 3.10. The re-formulation of Van Driest’s damping function D in terms of the velocity scale (−u v m )1/2 offers computational advantages compared to the original expression used in Equation 3.7 in cases where uτ = 0 or uτ < 0, as occurs for boundary-layer separation. Outer Layer: (3.13) µto = σ(x)0.0168ρue δv∗ FKleb Here, the intermittency function FKleb is basically the one given in Equation 3.9 with ymax /CKleb being replaced by the boundary-layer thickness δ. σ(x) is a modeling parameter that varies with the streamwise position x. It allows for adaption of the resulting eddy-viscosity distribution to non-equilibrium conditions. The heart of the model is an ordinary differential transport equation for (−u v m )1/2 which accounts for convection, diffusion, production and dissipation effects. This equation is a simplified form of the shear-stress transportequation developed by Bradshaw et al. (1967). It assumes that the path of maximum turbulent kinetic energy km coincides with the downstream direction x and, additionally, the ratio of maximum shear stress to maximum
48
3 Governing Equations and Numerical Method
turbulent kinetic energy is constant (−u v m /km = constant) in boundary layers, which is experimentally supported. The equation is given by Lm um d(−u v m ) Lm Dm , − (−u v m )1/2 = (−u v m, eq )1/2 − dx a1 (−u v m ) (−u v m ) dissipation production convection
diffusion
(3.14)
ym /δ ≤ 0.225 , Lm = ym /δ > 0.225 Cdif (−u v m )1/2 Dm = 1 − σ(x) , a1 δ(0.7 − (y/δ)m ) a1 = 0.25, Cdif = 0.5. 0.4ym , 0.09δ,
(3.15)
The subscript m denotes that the quantity is evaluated where −u v assumes its maximum in the wall-normal direction. Lm is the dissipation length scale. It is constructed such that it resembles Prandtl’s expression for the mixing length for ym /δ ≤ 0.225, and Escudier’s expression for ym /δ > 0.225. Expression 3.15 for Dm is a diffusion model based on an “effective” velocity at which turbulent energy is transported by turbulent diffusion effects. This concept differs from the gradient-diffusion model frequently employed. It is based on observations discussed by Townsend (1976) and supported by experiments of Bradshaw which suggest that most of the diffusion transport depends on the large eddies. Following this line of reasoning, turbulent diffusion can be viewed as either convection of smaller eddies by the motion of larger eddies or, simply, as a transfer of energy from one part of a large eddy to another. In this case, modeling diffusion by an “effective” or “convective” velocity seems to be more appropriate than the usual gradient-diffusion model employing the eddy viscosity as a diffusion coefficient. In the Johnson-King model, (−u v m )1/2 serves as the “effective” velocity scale. Johnson & King (1985) point out that turbulent diffusion plays an important role especially in flow recovery regions after reattachment of a separated boundary layer while its contributions to the rate equations of turbulence upstream of separation are secondary. Further details of the Johnson-King model will be given in Appendix B.
3.4
Numerical Method (I)
The RANS equations were solved with a customized version of the Jamesontype flow solver MUFLO, which was originally developed by Haase at the
3.4
Numerical Method (I)
49
European Aeronautic Defence and Space Company (EADS). MUFLO uses a two-dimensional cell centered finite-volume method for structured grids. Both convective and diffusive/dissipative fluxes in the Navier-Stokes equations are approximated by a second-order central space discretization. To stabilize the central scheme and prevent spurious oscillations, a blend of second- and fourth-order artificial damping terms is utilized. An optimized five-stage Runge-Kutta method combined with a multigrid scheme is employed for the explicit time integration. Additional convergence acceleration to steady state is achieved through local time stepping and implicit residual smoothing. A detailed description of the numerical method is given in Haase & Fritz (1993); Jameson et al. (1981); Jameson & Baker (1983, 1984). In order to increase the physical accuracy of the computed solution for low-Mach-number flows, local preconditioning of the Navier-Stokes equations was implemented in this work into MUFLO. Especially for two-dimensional separated flows, where relatively large and slowly moving recirculation zones exist, local preconditioning offers an improved accuracy of the flow solution in these regions. The preconditioning method and the influence on the computational results will be presented in Section 8. Various algebraic and half-equation turbulence models, including the models of Baldwin-Lomax and Johnson-King, are implemented in MUFLO. However, the Johnson-King model produced computational results which were closest to the experimental data for the investigated flow case compared to the results obtained with the other models. Therefore, it is the only model which was applied for the demonstration of the topological approach.
50
4 4.1
4 Demonstration
Demonstration Description of Flow Case
The flow around the Aerospatiale-A airfoil at a Reynolds number of 2 · 106 , a Mach-number of 0.15 and an angle of attack of 13.3◦ was investigated. This flow case exhibits the time-averaged flow topology of class 1.1 a) shown in Figure 2.1. Experiments in two different wind tunnels (F1 and F2) were conducted at ONERA (Office National d’Etudes et de Recherches Aerospatiales) and a complete database of the experimental results is available in Chaput (1997). In the experiments, transition at the lower surface was fixed with a trip at 30 percent of chord length. On the upper side, free transition was allowed and a laminar separation bubble with turbulent re-attachment at 0.12 chord length was observed during the tests. The experimental values for the lift (cL ), drag (cD ), pressure (cp ) and skin friction (cf ) coefficients obtained from measurements in ONERA’s wind tunnel F1 are judged to be more accurate than those obtained in wind tunnel F2 (Chaput, 1997). This is mainly attributed to the larger cross-sectional area of wind tunnel F1. However, measurements of boundary-layer profiles of the flow variables including turbulence quantities require measurement techniques like Laser Doppler Velocimetry (LDV) which were not available in wind tunnel F1. These quantities were measured using LDV in experiments conducted in wind tunnel F2. The differences between the data measured in the two wind tunnels can lead to various discrepancies. For example, the cf distribution obtained from measurements in wind tunnel F1 is not “compatible” with the cf values that can be determined from the velocity profiles since the latter were measured in wind tunnel F2. In particular, the locations of the separation point (cf = 0) determined in the two wind tunnels differ by almost 10 percent of the chord length. Consequently, all computational results, like profiles of flow variables and turbulence statistics as well as coefficients for pressure and skin friction, are compared to the values measured in wind tunnel F2. Only the lift and drag coefficients are additionally compared to those obtained from experiments in wind tunnel F1 following the recommendations in Chaput (1997). Although complicating the validation process, this situation is unavoidable. However, while many other experimental data of separating airfoil flows exist, frequently, these data do not comprise detailed measurements of the turbulent quantities. These
4.2
Computational Grid
51
quantities are available for the selected flow case which makes it especially attractive for the validation of turbulence models for separating airfoil flows.
4.2
Computational Grid
The computation of the flow over airfoil A was performed on a structured, body-aligned grid with a so-called C topology. Special care was taken in the generation of the computational grid. In particular, it was focused on resolving all topological structures discussed in Section 2. In order to reduce the required number of cells, grid points were clustered around the stagnation point A1 as well as in the region of the transitional bubble on the upper side and at the transition location at the lower surface. At the trailing edge, where the recirculation zone was encountered and where the converging boundary layers from the upper and lower sides experience a discontinuity in the boundary condition from no-slip to wake treatment, grid-point clustering was also applied. In regions and in the directions where the spatial variations of the flow field were not severe, that is, where second derivatives of the flow variables with respect to the spatial coordinates were relatively small, coarser grid spacing was allowed. In addition to ensuring fine grid spacing where needed, a smooth distribution of the metric terms was achieved by a combination of algebraic grid generation, using geometrical stretching, and grid smoothing. In boundary layers, the grid lines were forced to be normal and parallel to the surface, resulting in almost rectangular cells, which minimizes discretization errors. In the wall-normal direction, geometrical stretching of the grid point spacing was applied in order to further reduce the number of cells without sacrificing the fidelity of the solution. During the current study, computations were performed with many different grids in order to find the most suitable configuration. In order to investigate grid convergence of the computational solution, an extremely fine grid was created, which was coarsened by successively skipping every other grid node. The “standard” grid level used in this section has the following dimensions: • a total of 512 cells in the wraparound direction ξ with 384 cells placed on the airfoil surface, and in each case 64 cells located in the upper and lower wake (ξ = (i − 1)/(imax − 1) with 1 ≤ i ≤ 513 being the grid line index and imax = 513 the total number of grid lines in that direction)
52
4 Demonstration
• a total of 128 cells were used in the wall-normal direction η with 64 cells in the boundary layer (η = (j − 1)/(jmax − 1) with 1 ≤ j ≤ 129 being the grid line index and jmax = 129 the total number of grid lines in the considered direction; since a C-mesh is used there are, in fact, 256 cells across the wake region) • farfield boundaries were located at least 18 chords away from the nearest surface in order to minimize errors resulting from the numerical farfield boundary treatment. In addition, vortex correction was applied at farfield boundaries to account for the effect of circulation associated with the lift produced by the airfoil. The distribution of the metric term ∂ξ/∂x along the upper surface of the airfoil is shown in Figure 4.1. It can be seen that particularly in the convection direction and inside the boundary layer the distribution of the metric term is smooth and continuous. The same is true for the other metric terms ∂ξ/∂y, ∂η/∂x and ∂η/∂y (not shown). The grid point clustering in the above-mentioned areas is clearly recognizable in Figure 4.2 where the grid in the vicinity of the airfoil is shown.
flow direction in the boundary layer
400
∂ξ/∂x
300 200 100 0.85
0.8
0.75
0.7
ξ
0 0 0.65
0.6
0.55
0.5 0.5 1
η
Figure 4.1: Distribution of ∂ξ/∂x metric along upper surface of airfoil A.
4.3
53
Computational Results and Discussion
0.4
x/c
0.2 0
-0.2 0
0.5
y/c
1
Figure 4.2: Grid with 512 × 128 cells for computation of flow over airfoil A (only a cutout of the grid in the vicinity of the airfoil is shown).
4.3
Computational Results and Discussion
To mimic the transition in the computation, the turbulence model was activated only downstream of x/c = 0.12 on the upper side and downstream of x/c = 0.3 on the lower side as well as in the wake. (This appeared to be common practice in all publications found where this flow case was computed.) 4.3.1
Topology of the Velocity Field
The topology of the computed flow field is shown in Figure 4.3. A very thin laminar separation bubble and a small, single recirculation zone close to the trailing edge are predicted by the code. It is noted that the streamlines shown in Figure 4.3 are iso-contour lines of the streamfunction ψ, which was evaluated from the total differential dψ = with
∂ψ ∂ψ dx + dy ∂x ∂y
∂ψ = −ρv, ∂x
(4.1)
∂ψ = ρu. ∂y
Setting ψ = 0 at the airfoil surface as the initial condition for the integration of Equation 4.1 leads to the straightforward definition of the dividing streamline enclosing the recirculation zone (denoted b1 in Figure 2.1): Since
54
y/c
4 Demonstration
0.087
y/c
0.02
0.086
0
-0.02
0.085 0.115
0.118
0.121
x/c
0.9
0.95
1
x/c
y/c
0.1 0
-0.1
0
0.2
0.4
x/c
0.6
0.8
1
Figure 4.3: Computed topology of flow around airfoil A (JohnsonKing model).
a dividing streamline is identical to the separating streamline, it “carries” the same value of ψ as the surface, namely ψ = 0. An additional advantage of representing streamlines as iso-contour lines of ψ is the inherent avoidance of spiraling streamlines inside two-dimensional recirculation zones. This is called a repelling or attracting focus and is not possible in two dimensions without sources or sinks of fluid. In contrast, the common method to compute streamlines, which is also used in three-dimensional flows, is to integrate the path of imaginary particles “released” in the velocity vector field. However, this method can lead to a summation of the numerical errors along the path of integration and, frequently, results in weakly-spiraling streamlines even in purely two-dimensional flows. In Figure 4.4, dividing streamlines of the rear recirculation zone obtained from experimental data (experiment F2) and from the computation with the Johnson-King model are compared. It can be seen that the recirculation zone obtained from the Johnson-King computation is much smaller than the one evaluated from measured data. Although the “grid” of the experimental data is very coarse compared to the computational grid and, hence, the integration of Equation 4.1 based on the measured data is much more inaccurate, the
4.3
55
Computational Results and Discussion
0.04
S1exp
experiment
S1comp
Johnson-King
y/c
0.02 0 trailing edge -0.02 -0.04
A2exp 0.85
0.9
x/c
0.95
A2comp 1
Figure 4.4: Comparison of recirculation zones for airfoil A (dashed lines: experiment F2; solid lines: computed with JohnsonKing model).
large differences in the size of the recirculation zone cannot be attributed to inaccuracies in the determination of the experimental streamlines alone. Besides possible errors produced solely by the turbulence model, weak threedimensional effects are believed to be present in this region which add to discrepancies between the computed and measured solution. In order to get some basic information about possible three-dimensional effects, the experimental streamlines computed from imaginary particle traces and the iso-contour line ψ = 0 are compared in Figure 4.5. The separating streamlines coincide well up to x/c ≈ 0.98 for the two methods. Downstream of x/c ≈ 0.98, however, large differences in the patterns of the streamlines are encountered. On the one hand, this is attributed to the discussed inaccuracies in the evaluation of streamlines. But on the other hand, the pronounced spiraling motion of the particle traces is taken as a strong indication of threedimensional effects in the experiment which are not accounted for in the computation. 4.3.2
Pressure and Skin-Friction Distributions
In Figure 4.6, the computed pressure distribution is compared with the measured values. The overall agreement is fair. Significant differences are encountered only downstream of x/c ≈ 0.8 where the larger separation region found in the experiment leads to a more pronounced plateau in the experimental pressure distribution. The small transitional separation bubble, clearly recog-
56
4 Demonstration
0.125 0.1
S1
0.075
y/c
0.05 0.025 0
trailing edge
-0.025 -0.05
ψ=0
A2
-0.075 0.8
0.85
0.9
x/c
0.95
1
1.05
Figure 4.5: Streamlines constructed from experimental data for flow around airfoil A; solid, thin lines: particle traces; dashed, thick lines: ψ = 0. nizable in the computed pressure distribution at x/c ≈ 0.12, is not visible in the experimental data due to the inherently coarse positioning of the pressure orifices at the airfoil surface. The corresponding distributions of the skin-friction coefficients on the upper side of the airfoil are shown in Figure 4.7. Note that the oscillations in the computed skin friction at the trailing edge are mainly due to inaccuracies of the numerical scheme. This will be discussed in more detail in Section 8. A relatively large negative peak at x/c ≈ 0.12 and a subsequent steep rise of the skin friction indicate the transitional separation bubble and the onset of the turbulent boundary layer, respectively. Computed cf values compare well with measured data in the aft part of the airfoil (x/c > 0.8). This is somewhat surprising since the computed pressure distribution (Figure 4.6) and the size of the recirculation zone (Figure 4.4) found in the computation do not agree well with the corresponding experimental data in this region. In particular, the separation point S1 estimated from the iso-contour line ψ = 0 is located at x/c ≈ 0.825 in the experiment while in the computed solution it is found to be at x/c ≈ 0.90 (Figure 4.4). Note that there are also inconsistencies within the measured data itself: The position of the separation S1 obtained from the skin-friction distribution (x/c ≈ 0.87) and the one obtained from ψ = 0 (x/c ≈ 0.825) differ by approximately 0.045
4.3
57
Computational Results and Discussion
-4 -3
Experiment Johnson-King
cp
-2 -1 S1 0 1
0
0.25
0.5
x/c
0.75
1
Figure 4.6: Pressure distribution on surface of airfoil A; S1 relates to cf = 0 inferred from experiment F2, see Figure 4.7.
chord lengths, although both are evaluated using data measured in the same wind tunnel (F2). These deviations are believed to result from measurement uncertainties. In particular, even small uncertainties in the cf distribution favor large differences in the location of the separation point cf = 0 due to the small slope of the cf distribution in this region. In fact, measurements of cf are subject to higher measurement uncertainties than measurements of the velocity. Therefore, the separation position x/c ≈ 0.825 is taken to be the more accurate one. In addition, looking at the velocity profiles in Figure 4.11, it is evident that the separation point must be located upstream of x/c ≈ 0.87 since x/c ≈ 0.87 is the first downstream position where backflow in the boundary layer, i.e. negative tangential velocity u, is encountered. The global lift and drag coefficients measured in wind tunnel F1 and F2, as well as the ones computed with the Johnson-King model, are listed in Table 4.1. Since no experimental data regarding the moment coefficient cM are available, only the computed cM is reported in the table. The good agreement of the computed lift coefficient with that measured in wind tunnel F2 is consistent with the close agreement of the corresponding pressure distributions (Figure 4.6).
58
4 Demonstration
0.015 Experiment Johnson-King
cf
0.01
0.005 S1ψ
S1cf
0 0
0.25
0.5
x/c
0.75
1
Figure 4.7: Skin-friction distribution on upper side of airfoil A; S1ψ denotes the position of separation evaluated from the streamfunction ψ, and S1cf denotes the position of separation evaluated from the skin-friction distribution (cf = 0).
Table 4.1: Force and moment coefficients for Airfoil A experiment F1
experiment F2
component
total
total
Johnson-King computation total
pressure
friction
cL
1.56
1.52
1.53
1.531
−0.001
cD
0.021
0.031
0.026
0.020
0.006
cM
−
−
0.0055
0.0050
0.0005
The value of cD measured in wind tunnel F2 is larger than the one obtained in wind tunnel F1 indicating that the recirculation zone at the trailing edge is also larger in the wind tunnel F2. This is also supported by comparing the cf distributions from the two experiments: The separation point S1 (cf = 0) in experiment F1 is located further downstream than in case F2 (Figure 4.8). Since the drag coefficient for this separated flow case is mainly due to pressure drag (see Table 4.1) the size of the recirculation zone at the
4.3
59
Computational Results and Discussion
trailing edge greatly influences cD . Hence, the smaller recirculation zone in the experiment F1 leads to a significantly lower drag coefficient compared to the value from experiment F2. Significant differences exist in the computed and measured drag coefficients with the computed value being approximately 16 percent lower than the one obtained from the experiment F2 and approximately 24 percent larger than cD from F1. This can again be explained with the different sizes of the trailing edge recirculation zone: The computed length of the separated region, and hence the size of the recirculation zone, is larger than the one found in experiment F1 but shorter than in experiment F2. The separation location in experiment F2 was discussed above by means of the streamline topology; the separation point was found to be at 0.825 ≤ x/c ≤ 0.87 (Figure 4.4). The separation location S1 (cf = 0) for case F1, on the other hand, can be inferred from Figure 4.8. Using a best-fit polynomial representation of the measured values for this purpose yields cf = 0 at x/c ≈ 0.94. Hence the position of the separation point obtained from computational results is in between the positions of the two separation points evaluated from experimental data.
0.002 Experiment F1 Experiment F2
cf
0.001 S1F2
S1F1
0
-0.001 0.6
0.7
0.8
x/c
0.9
1
Figure 4.8: Skin-friction distributions from experiment F1 and F2 for airfoil A.
60
4 Demonstration
The reattachment point A2 could not be evaluated from the experimental cf distribution because no change of sign from negative to positive values of cf is seen in the experimental data. The reason for this is that the spacing of the skin-friction measurements was too coarse to sufficiently resolve the cf distribution in the immediate vicinity of the trailing edge in order to capture the change of sign. 4.3.3
Boundary-Layer Profiles
Boundary-layer profiles at different downstream positions are investigated in order to compare the computed and measured flow data in more detail. The goal is to gain a deeper insight into the deviations already encountered between the computed and measured results. Since no measurements were taken on the lower side of the airfoil, only profiles on the upper side can be compared. The positions of the profiles on the airfoil are shown in Figure 4.9. The local coordinate systems are oriented such that the coordinate axes are parallel and normal, respectively, to the surface of the airfoil at the indicated downstream position x/c. For this purpose, a post-processing tool called newmono was developed that performs the appropriate coordinate transformations of the velocity vectors and the stress tensors of the computed flow solution. See Section D in the appendices for a short description of newmono.
0.87 0.9 0.93 0.96 0.99
0.775 0.825
0.7
0.6
y/c
0.1
0. 5
0. 4
0.3
0.2
0 -0.1 0.2
0.4
0.6
x/c
0.8
1
Figure 4.9: Locations of measurement stations at the upper surface of airfoil A. The profiles of the normalized tangential velocity component u/U∞ are plotted in Figures 4.10, 4.11 and 4.12. In the figures, z/c denotes the wallnormal coordinate with z/c = 0 at the wall. Note, while in Figure 4.10 all
4.3
61
Computational Results and Discussion
profiles have the same origin, in Figure 4.11 the profiles for x/c = 0.825 and x/c = 0.87 are shifted by constant values ∆u = 0.6 and ∆u = 0.3, respectively. Similarly, in Figure 4.12, the profiles for x/c = 0.93 and x/c = 0.96 are shiftedb y ∆u = 0.6 and ∆u = 0.3, respectively. This is done for better recognizability while still being able to show several profiles in the same diagram.
1.6
0.3
1.4
0.4 0.5
1.2
0.6
0.7
u/U∞
1
0.775
0.8 0.6 0.4
Experiment Johnson-King
0.2 0
0
0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04
z/c
Figure 4.10: Velocity profiles for airfoil A at x/c = 0.3, x/c = 0.4, x/c = 0.5, x/c = 0.6, x/c = 0.7, and x/c = 0.775.
Up to the position x/c = 0.5 the computed profiles match the measured data very well. Downstream of x/c = 0.5 the computed profiles do not appropriately adjust to the pressure gradient; they are “fuller” in shape than the measured profiles. Due to the fuller velocity profiles of the computed flow solution, more momentum is transported normal to the wall leading to a separation location that is located more downstream than in the experiment. As a consequence, the recirculation zone predicted is shorter and the backflow region (u < 0) does not extend as far away from the wall as in the experimental data. The discrepancies in the velocity profiles increase considerably after x/c = 0.6. This increase is far too large to be attributed only to inaccuracies of the
62
4 Demonstration
1.6
0.825 (shifted by ∆u=0.6)
1.4 1.2
0.87 (shifted by ∆u=0.3)
u/U∞
1 0.9
0.8 0.6 0.4
Experiment Johnson-King
0.2 0 0
0.01
0.02
0.03
0.04
z/c
0.05
0.06
0.07
Figure 4.11: Velocity profiles for airfoil A at x/c = 0.825, x/c = 0.87, and x/c = 0.9 (the graphs for x/c = 0.825 and x/c = 0.87 are shifted by ∆u = +0.6 and ∆u = +0.3, respectively).
turbulence model. Hence, it corresponds to loss of momentum and mass flux in the experiment evidently due to spanwise outflow. Regarding the profiles of Reynolds shear stresses (Figures 4.13 to 4.15), it is noted that Reynolds shear stresses are subject to much larger measurement uncertainties than mean velocities. With this in mind, one can say that the profiles of the computed Reynolds shear stresses match the measured ones rather well up to a downstream position of x/c = 0.6. At x/c = 0.7 large differences between the computed and measured values arise in the outer part of the boundary layer (Figure 4.13 d)). Especially the kink found in the measured profile at a wall distance 0.025 ≤ z/c ≤ 0.035 is not reproduced at all by the computation. This kink is also visible in the experimental profiles further downstream, its position moving further away from the wall with increasing boundary-layer thickness. So, it is not believed to be an artifact due to measurement errors. Looking at the profiles further downstream reveals that the agreement between the measured and computed Reynolds shear stresses becomes successively poorer when approaching the trailing edge (Figures 4.14 and 4.15).
4.3
63
Computational Results and Discussion
1.6
0.93 (shifted by ∆u=0.6)
1.4 1.2
0.96 (shifted by ∆u=0.3)
1 0.8
u/U∞
0.99
0.6 0.4 Experiment Johnson-King
0.2 0 0
0.02
0.04
0.06
z/c
0.08
0.1
Figure 4.12: Velocity profiles for airfoil A at x/c = 0.93, x/c = 0.96, and x/c = 0.99 (the graphs for x/c = 0.93 and x/c = 0.96 are shifted by ∆u = +0.6 and ∆u = +0.3, respectively).
The maximum level of the Reynolds stress is underpredicted by the model for x/c ≥ 0.825 with an increasing deviation from the experimental values when approaching the trailing edge. In contrast, the Reynolds-stress level inside the recirculation zones is clearly overpredicted by the computation. In the experiment, the Reynolds shear stress is negligible up to a wall distance of z/c = 0.015, i.e. up to the center of the recirculation zone (Figure 4.15 d)). The turbulence model, however, predicts significant Reynolds shear stress in this region. This finding led to a numerical experiment, i.e. a “topological modification” of the turbulence model which is discussed in the next section.
0.004
2
2
0.003
0.004
0
0.001
0.002
0.003
-/U∞
z/c
0.015
0.02
0.025
Experiment Johnson-King
0.015
0.002
0.003
0.004
0
0
0
0.01
0.02
0.005
z/c
0.03
0.01
0.04
0.06
Experiment Johnson-King
0.015
Experiment Johnson-King
0.05
d) x/c=0.7
z/c
b) x/c=0.5
Figure 4.13: Reynolds-shear-stress profiles for airfoil A at x/c = 0.4 (a), x/c = 0.5 (b), x/c = 0.6 (c), and x/c = 0.7 (d).
0 0.01
c) x/c=0.6
z/c
0.01
0.001
0.002
0.003
0.004
0.005
0
0.005
0.005
Experiment Johnson-King
0.001
0
0
a) x/c=0.4
0.001
0.002
-/U∞
2
-/U∞
2
-/U∞
0.005
64 4 Demonstration
0.004
2
2
0.004
0.005
0
0.001
0.002
0.003
-/U∞
0.04
z/c
0.04
0.05
0.06
0.08
Experiment Johnson-King
0.06
c) x/c=0.87
z/c
0.03
0.002
0.003
0.004
0.005
0.006
0
0
0
0.01
0.02
0.02
z/c
0.04
0.04
0.06
0.06
0.08
Experiment Johnson-King
0.05
d) x/c=0.9
z/c
0.03
Experiment Johnson-King
b) x/c=0.825
Computational Results and Discussion
Figure 4.14: Reynolds-shear-stress profiles for airfoil A at x/c = 0.775 (a), x/c = 0.825 (b), x/c = 0.87 (c), and x/c = 0.9 (d).
0 0.02
0.02
0
0.01
0.001
0.002
0.003
0.004
0.001
0
0
Experiment Johnson-King
a) x/c=0.775
0.001
0.002
0.003
-/U∞
2
-/U∞
2
-/U∞
4.3 65
0.005
2
2
0.005
0.006
0
0.001
0.002
0.003
0.004
-/U∞
0
0.001
0.002
0.003
0.004
-/U∞
0
0
0.04
0.04
0.06
z/c
0.06
0.08
0.1
Experiment Johnson-King
0.08
c) x/c=0.99
z/c
Experiment Johnson-King
0
0.001
0.002
0.003
0.004
0.005
0
0.001
0.002
0.003
0.004
0.005
0.006
0
0
0.01
Experiment Johnson-King
z/c
0.04
z/c
0.08
Experiment
0.06
Experiment Johnson-King
position of separating streamline b1 0.02 0.03
Johnson-King
d) x/c=0.99
0.02
b) x/c=0.96
Figure 4.15: Reynolds-shear-stress profiles for airfoil A at x/c = 0.93 (a), x/c = 0.96 (b), x/c = 0.99. (d) shows cut-out of (c) in recirculation zone.
0.02
area shown in figure d)
0.02
a) x/c=0.93
2
-/U∞
2
-/U∞
0.006
66 4 Demonstration
4.3
67
Computational Results and Discussion
4.3.4
Numerical Experiment in the Recirculation Zone
The original Equation 3.10 for the eddy viscosity was modified to yield a damping of the effective eddy viscosity in the recirculation zone. This was accomplished by scaling the eddy viscosity with an appropriate non-dimensional function which depends on the streamfunction.
min(ψ, 0) ζ (−µti /µto ) , µ t = µ to 1 − e (4.2) 1− ψmin ζ = 0.1 − 0.3. Since the streamfunction is negative in the recirculation zone with its minimum value at the focus F1, Equation 4.2 leads to a modification of the eddy viscosity only inside the recirculation zone. The modeling parameter ζ controls the functional behavior close to the edges of the recirculation zone. A low value for ζ, for example ζ = 0.1, gives an abrupt onset of the damping when ψ changes from positive to negative values, i.e. when “entering” the recirculation zone. Increasing ζ results in a less abrupt onset of the damping. Results Iso-contours of the Reynolds shear stress obtained with the original model cross the separating streamline without being affected by the recirculation zone (Figure 4.16). In contrast, employing Equation 4.2 for the
y/c
0.02 0
-0.02 0.9
x/c
0.95
1
Figure 4.16: Iso-contour lines of Reynolds shear stress for the standard Johnson-King model near the trailing edge of airfoil A on its upper side. computation of the eddy viscosity yields iso-contour lines that end abruptly at the separating streamline (Figure 4.17). The Reynolds shear stress obtained with the modified model is nearly zero inside the recirculation zone and it
68
4 Demonstration
changes quickly to a finite value when crossing the separating streamline, see 2 Figure 4.18. Outside of the recirculation zone the value of (−u v )/U∞ computed with the modified model is very close to the value obtained with the standard model formulation (Figure 4.18).
y/c
0.02 0
-0.02 0.9
x/c
0.95
1
Figure 4.17: Iso-contour lines of Reynolds shear stress for the topologically-modified Johnson-King model near the trailing edge of airfoil A on its upper side.
0.005 Experiment Johnson-King modified Johnson-King original
2
-/U∞
0.004 0.003
Experiment
Johnson-King modified
0.002 position of separating streamline b1
0.001 0 0
0.01
0.02
z/c
0.03
Figure 4.18: Reynolds-shear-stress profiles for airfoil A at x/c = 0.99 with and without modification of the Johnson-King model.
4.3
69
Computational Results and Discussion
However, the modification of the Johnson-King model has a very small effect on the velocity distribution (Figure 4.19). Similarly, the influence on the global force coefficients and the position of separation S1 is virtually non-existent (Table 4.2). 1 0.8
u/U∞
0.6 0.4 Experiment Johnson-King modified Johnson-King original
0.2 0 0
0.02
0.04
0.06
z/c
0.08
0.1
0.12
Figure 4.19: Velocity profiles for airfoil A at x/c = 0.99 with and without modification of the Johnson-King model.
Table 4.2: Effects of the topological modification of the model on the global force coefficients for airfoil A Johnson-King model
standard
cL
1.5324
modified 1.5324
cD
0.025995
0.025996
cM
0.0055042
0.0055045
separation location (x/c)
0.898898
0.898898
The results obtained with the topologically modified Johnson-King model show that the deviations between the computational and experimental results encountered in the preceding sections are not mainly due to inaccurate
70
4 Demonstration
turbulence modeling inside the recirculation zone. In fact, the correct response of the turbulence model to the adverse pressure gradient upstream of the separation point S1 is the key to accurately predicting the boundary-layer development and the separation point. On the one hand, this seems to be obvious since a boundary layer is governed by parabolic differential equations saying that the state of a boundary layer is influenced only by the upstream region and the local pressure gradient. On the other hand, the considered flow case is an airfoil flow where small changes at the trailing edge can have strong influence on the overall circulation. Hence, changes at the trailing edge can easily affect the boundary-layer development further upstream by modifying the pressure distribution via the circulation. Bearing these considerations in mind, it was decided to perform the envisaged comparative study of turbulence models for separating flows by means of flow cases where the pressure distribution is held “fixed”. This ensures that the pressure gradient is basically the same for each model and is not sensitive to the computed flow solution which may change with the employed model. Thus, an objective study of the effect of pressure gradient on the turbulence models is possible.
4.3.5
Comments Regarding Hidden Three-Dimensional Effects in Nominally Two-Dimensional Flows
W¨ urz & Althaus (1995) investigated various cases of separated flows around airfoil FX 63-137 close to, and at, stall conditions. In particular, they varied the aspect ratio of the wing section and the Reynolds number to study the effect on the trailing-edge separation. Oil-flow patterns on the model’s surface were evaluated to visualize wall streamlines and discuss three-dimensional effects. A standard scenario in their investigations comprised a wing aspect ratio of Λ = 1.46 and a Reynolds number of Re = 1·106 . For these parameters, the separation line was a straight line in the spanwise direction indicating a basically two-dimensional separation zone. Increasing the aspect ratio of the wing section resulted in a wavy shape of the separation line and triggered the formation of spanwise cellular vortex patterns leading to a three-dimensional flow topology. For example, W¨ urz and Althaus found two spanwise vortex cells in a flow past a wing section with Λ = 2.92 and Re = 1 · 106 (Figure 4.20). The formation of such cellular vortex patterns was also discussed by Weihs & Katz (1983).
4.3
Computational Results and Discussion
71
Figure 4.20: Two vortex cells at airfoil FX 63-137 for Λ = 2.92 and Re = 1 · 106 (W¨ urz & Althaus, 1995, the picture is courtesy of Werner W¨ urz, IAG, University of Stuttgart).
For the Aerospatiale-A airfoil, the aspect ratio of the wing in wind tunnel F1 was Λ = 2.5; in wind tunnel F2 it was Λ = 2.33. Both values are just slightly lower than Λ = 2.92 in W¨ urz & Althaus (1995). According to the results of W¨ urz and Althaus this favors the formation of three-dimensional vortex cells or, at least, a wavy shape of the separation line in the spanwise direction. In light of this discussion, the spiraling streamlines found at the trailing edge of airfoil A (Figure 4.5) strongly suggest that three-dimensional effects are, in fact, present in this flow case. However, due to the much smaller separation region at airfoil A, compared to the separation regions found in the flows past airfoil FX 63-137 investigated by W¨ urz and Althaus, the three-dimensionality was probably less pronounced for our case. Note, that these conjectures are in contradiction to the statement “... the flow was two-dimensional up to an incidence of 13◦ ...” which is made in the discussion of the experimental results for the flow over airfoil A in Chaput (1997). Possibly, the measurements of the flow over airfoil A were taken close to a center plane of a vortex cell which would reduce three-dimensional effects in the measured data. Nevertheless, it remains an open question what solution can be expected of a purely two-dimensional computation and whether it is legit-
72
4 Demonstration
imate to expect that such a computation can reproduce the flow field found in the symmetry plane of a three-dimensional vortex system. This question can only be answered by comparing two- and three-dimensional computations with each other and with experimental data. For this purpose, it is mandatory that the experimental data comprise accurate information about the three-dimensionality of the flow field.
Part II
Analysis of Modern Turbulence Models
74
5
5 Numerical Method (II)
Numerical Method (II)
The flow computations for the comparative study of the turbulence models were performed using the FLOWer code of the MEGAFLOW software system. This software system has been, and is still, developed at the Deutsches Zentrum f¨ ur Luft- und Raumfahrt (DLR) in Braunschweig, Germany, and is becoming a standard CFD tool in the European aircraft industry. FLOWer solves the three-dimensional unsteady and compressible Reynolds-averaged Navier-Stokes equations in integral form. For this purpose, a cell vertex or, optionally, a cell-centered finite-volume formulation on block-structured grids is utilized. On user input, either a second-order central scheme or one of the various available flux-difference or flux-vector upwind schemes is applied for the space discretization of the convective fluxes. In either case, the diffusive fluxes are centrally discretized. The central discretization of the convective fluxes is augmented by a blend of second- and fourth-order artificial damping terms in order to prevent odd-even decoupling, damp spurious oscillations and allow for sharp shock resolution. For the time integration, an explicit five-stage Runge-Kutta scheme with optimized damping properties for multigrid is employed. Convergence to steady state is accelerated by means of local time stepping, implicit residual smoothing, simple or full multigrid and local preconditioning for low Mach numbers. For the computation of unsteady flows a dual-time-stepping procedure is available. The effect of turbulence on the mean flow is modeled by algebraic or transport-equation turbulence models. In the transport-equation models the convective terms are discretized using either a first-order upwind scheme or a second-order central scheme with artificial damping. The time integration of the turbulence equations is performed explicitly, according to the solution method of the mean flow equations, or implicitly. In the latter case, two different point-implicit schemes, where only the source terms are treated implicitly, or a line-implicit and a fully implicit treatment are available. All computations performed for this work were done using the central space discretization for the convective fluxes of the mean flow and the first-order Roe type upwind scheme for the convective terms in the turbulence-transport equations. Additionally, a point-implicit treatment of the turbulence-transport equations was selected. A detailed presentation of the numerical algorithms utilized in FLOWer can be found in Kroll et al. (1995) and Aumann et al. (2000); a more general discussion of the MEGAFLOW project is available in Kroll et al. (2000).
75
6
Models Investigated
Transport-equation models, in general, have been developed to account for non-local and non-equilibrium effects, also called flow-history effects. There are two main types of transport-equation models, one in which differential equations are solved for the transport of the Reynolds stresses and one in which the transport equations yield the eddy viscosity. Stress-transport models are not in very common use. In the present thesis we therefore focus on eddy-viscosity-transport models. (For the sake of brevity, we will also denote “eddy-viscosity-transport models” simply by “transport-equation models”.) These models enjoy large popularity in today’s computational fluid dynamics (CFD) applications. In this work, two different categories of eddy-viscosity-transport models were investigated. The first category comprises two-equation models which utilize the square root of the turbulent kinetic energy k as a velocity scale of turbulence while the specific dissipation rate ω is used as a reciprocal time scale. A partial differential transport equation must be solved for both k and ω. The eddy viscosity is a linear function of these two parameters. The other category consists of one-equation models that solve a partial differential transport equation for the eddy viscosity itself. For the comparative study, a total of eleven different eddy-viscosity turbulence models from three different model classes were considered. They range from the algebraic Baldwin-Lomax model over various one- and two-equation models to a non-linear, explicit algebraic Reynolds-stress model. The model of Baldwin & Lomax was already presented in detail in Section 3.2 and no description will be repeated in this section. The Johnson-King model in its original formulation is confined to two-dimensional applications. Although extensions to three-dimensional flows exist (see Abid et al., 1989) the resulting performance does not reach the level of the two-dimensional version (Haase, 1997). Consequently, an implementation of the Johnson-King model into the three-dimensional FLOWer code was not performed in the MEGAFLOW project nor in the present work. Since most of the turbulence models employed in this work are discussed in great detail in the referenced literature only short descriptions, basically consisting of the important equations, are presented in this section. The models investigated are listed in Table 6.1.
76
6 Models Investigated
Table 6.1: Turbulence models investigated in this work
6.1
Model
Developer(s) / Reference
Baldwin-Lomax
Baldwin & Lomax (1978)
see Subsection
Johnson-King
Johnson & King (1985)
k, ω 1988
Wilcox (1988)
6.1
k, ω 1998
Wilcox (1998)
6.1
k, ω SST
Menter (1993)
6.2
k, ω TNT
Kok (2000)
6.3
k, ω LLR
Rung & Thiele (1996)
6.4
EARSM of Wallin
Wallin & Johansson (2000)
6.5
Spalart-Allmaras
Spalart & Allmaras (1992)
6.7
3.2 3.3, B
Edwards-Chandra
Edwards & Chandra (1996)
6.8
SALSA
Rung et al. (2003)
6.9
k, ω SST modified
Celi´c
7.1.3
The k, ω Models of Wilcox
Wilcox developed two different versions of a k, ω type of model. The first model (Wilcox, 1988) proved to be superior to the k, models of Jones & Launder (1972) or Launder & Sharma (1974) for boundary-layer flows under adverse pressure gradients. In these flows, k, models typically produce too high levels of turbulence. Since the 1988 model of Wilcox does not yield the correct spreading rate of free shear layers in all cases Wilcox refined his model in 1998 to further improve the predictive accuracy in such situations (Wilcox, 1998). Unlike turbulence models based on the transport equations for k and , the k, ω models of Wilcox do not require viscous damping, that is, low-Reynoldsnumber modifications, to predict a realistic value of the additive constant in the law of the wall. Nevertheless, Wilcox suggested low-Reynolds-number modifications for his models to enhance the models’ solutions for k in the viscous sublayer. An additional and intended outcome of the low-Reynoldsnumber modifications devised by Wilcox is the possibility to predict laminarturbulent transition. This desirable feature, however, is restricted to flat-plate boundary layers since the modifications were validated based on the minimal critical Reynolds number obtained from linear stability theory in the Blasius boundary layer. The models are unable to predict transition in general flows
6.1
77
The k, ω Models of Wilcox
which is no surprise if one bears in mind that transition is a very complicated and still not fully-understood unsteady physical process. Since only the 1988 k, ω model of Wilcox was available in FLOWer, the 1998 model and the low-Reynolds-number extensions for both models were implemented into the FLOWer code during this work. Before presenting the models’ equations, it is noted that from the definition of the turbulent kinetic energy, k=
1 1 u u + v v + w w = ui ui , 2 2
it follows that the trace of the Reynolds-stress tensor is −2ρk. In order to insure this condition in the framework of the applied two-equation models, the constitutive relation, Equation 3.5, must be appropriately extended: 2 1 −ρui uj ≡ τ = 2µt S − (∇ · v) · I − ρk · I. 3 3 Although the model equations were discretized and solved in integral form they are presented for simplicity and space reasons in the usual differential form and in index notation. In some situations, the equivalent tensor notation is also shown in order to provide the link to equations which were above introduced in tensor notation. Both model versions of Wilcox are based on the same transport equations for k and ω. The models’ equations read as follows: Eddy Viscosity: k µt = ρ . (6.1) ω Turbulent Kinetic Energy: ∂ρk ∂ui ∂ ∂k ∂ρkuj = −ρui uj − β ∗ ρkω + + (µ + σ ∗ µt ) . (6.2) ∂t ∂xj ∂xj ∂xj ∂xj Specific Dissipation Rate: ω ∂ui ∂ ∂ω ∂ρωuj ∂ρω = α (−ρui uj ) − βρω 2 + + (µ + σµt ) . ∂t ∂xj k ∂xj ∂xj ∂xj However, the two models apply different closure coefficients and auxiliary functions. 1988 Closure Coefficients: α=
5 , 9
β=
3 , 40
β∗ =
9 , 100
σ=
1 , 2
σ∗ =
1 . 2
78
6 Models Investigated
1998 Closure Coefficients: α=
13 , 25
9 , 100
σ=
1 , 2
σ∗ =
1 . 2
Ωij Ωjk Ski 1 + 70χω , , χω = 1 + 80χω (β0∗ ω)3 1 ∇ ⊗ v − (∇ ⊗ v)T , Ωij ≡ Ω = 2 1, χk ≤ 0 1 ∂ω ∂k . , χk = 3 fβ∗ = 1+680χ2 k ω ∂xj ∂xj , χ > 0 k 1+400χ2
β0 =
β0∗ =
β ∗ = β0∗ fβ ∗ ,
β = β0 fβ , 9 , 125
fβ =
(6.3)
(6.4)
k
One can see from the above equations that the two models of Wilcox differ in the destruction terms. The destruction term in the equation for ω is altered in the 1998 version only for three-dimensional flows since the product Ωij Ωjk Ski in Equation 6.3 is zero in two dimensions. In addition, the crossdiffusion parameter χk is only active in the outer part of turbulent shear layers. Hence, for two-dimensional boundary-layer flows the two models are expected to yield very similar results. For the low-Reynolds-number versions of the two models Wilcox suggested the following modifications of the eddy viscosity and the closure coefficients: k µt = α ∗ . ω 1988 Low-Reynolds-Number Coefficients: α∗ = α= β∗ =
α0∗ + Ret /Rek , 1 + Ret /Rek
5 α0 + Ret /Reω 1 · , · 9 1 + Ret /Reω α∗ 5/18 + (Ret /Reβ )4 9 , · 100 1 + (Ret /Reβ )4
1 β 1 3 , σ ∗ = σ = , α0∗ = , α0 = , 40 2 3 10 k , Reβ = 8, Rek = 6, Reω = 2.7. Ret = ων 1998 Low-Reynolds-Number Coefficients: β=
α∗ =
α0∗ + Ret /Rek , 1 + Ret /Rek
6.2
79
The k, ω Shear-Stress Transport (SST) Model of Menter
α= β∗ = β=
9 fβ , 125
Ret =
k , ων
13 α0 + Ret /Reω 1 · , · 25 1 + Ret /Reω α∗
4/15 + (Ret /Reβ )4 9 · fβ ∗ , · 100 1 + (Ret /Reβ )4 σ∗ = σ = Reβ = 8,
1 , 2
α0∗ =
Rek = 6,
β0 , 3
α0 =
1 , 9
Reω = 2.95.
Ret is frequently referred to as the turbulent Reynolds number.
6.2
The k, ω Shear-Stress Transport (SST) Model of Menter
Menter (1993) modified the 1988 model of Wilcox in two ways. First, in order to remove the model’s sensitivity to the free-stream value of ω, Menter incorporated a blending function that converts the k, ω model to the k, model of Jones & Launder in the outer part of a boundary layer and in the free stream. This is plausible since the k, model shows almost no sensitivity to the free-stream value of . Since the k, ω model is employed for the largest part of the boundary layer the resulting model retains the favorable features of the k, ω model for boundary-layer flows. The change between the two models is accomplished with the help of a blending function which turns on a so-called cross-diffusion term in the ω equation of the k, ω model in the outer part of the boundary layer. This cross-diffusion term renders the k, ω model a k, model. It is obtained when re-writing the equation in the k, model in terms of ω. Additionally, the closure coefficients of the k, ω model are converted to the appropriate values of the k, model also by employing the blending function. The second modification applied by Menter is based on Townsends’ and Bradshaw’s assumption that the Reynolds shear stress in a boundary layer is proportional to the turbulent kinetic energy, τxy = ρa1 k. This concept has already been introduced in the discussion of the Johnson-King model in Section 3.3. Menter uses this assumption to limit the eddy viscosity in regions where production of k exceeds dissipation of k which would normally lead to too high levels of µt . The model equations read in detail: Eddy Viscosity: a1 ρk µt = , max (a1 ω; ΩF2 )
80
6 Models Investigated
F2 = tanh (arg2 ) ,
arg2 = max
√ 2 k 500ν ; . 0.09ωd d2 ω
d is the wall-normal distance and Ω denotes the absolute value of the vorticity. Turbulent Kinetic Energy: ∂ρk ∂ui ∂ ∂k ∂ρkuj = −ρui uj − β ∗ ρkω + + (µ + σ ∗ µt ) . ∂t ∂xj ∂xj ∂xj ∂xj Specific Dissipation Rate: ρ ∂ui ∂ ∂ω ∂ρω ∂ρωuj = α (−ρui uj ) −βρω 2 + + (µ + σµt ) +(1−F1 )CD . ∂t ∂xj µt ∂xj ∂xj ∂xj CD is the cross-diffusion term. It is obtained from the k, model written in k, ω formulation and reads CD = 2ρσ2
1 ∂k ∂ω . ω ∂xk ∂xk
Global Coefficients: a1 = 0.31,
κ = 0.41,
β∗ =
9 . 100
σ1∗ = 0.85,
α1 =
κ2 β1 √ − σ . 1 β∗ β∗
k, ω Coefficients (set 1): β1 =
3 , 40
σ1 =
1 , 2
k, Coefficients (set 2): β2 = 0.0828,
σ2 = 0.856,
σ2∗ = 1.0,
α1 =
κ2 β2 − σ2 √ ∗ . ∗ β β
The blending between set 1 and set 2 of the closure coefficients is performed with the following function: φ = F1 φ1 + (1 − F1 ) φ2 . φ1 stands for any of the closure coefficients from set 1 and φ2 , respectively, for closure coefficients from set 2. The blending function F1 is defined as follows: F1 = tanh arg14 ,
6.3
81
The Turbulent/Non-turbulent (TNT) k, ω Model of Kok
arg1 = min max
√
k 500ν ; 0.09ωd d2
4ρσ2 k ; max(CD ; 10−20 ) d2
.
Both blending functions F1 and F2 are unity inside the boundary layer and fall to zero approaching the boundary-layer edge. In order to insure the desired behavior of the model it is important that F1 goes to zero well inside the boundary layer, i.e. at about 50 percent of the boundary-layer thickness, while F2 is unity for most of the shear layer. Both functions are zero in free shear layers away from walls. The reader is referred to the original paper of Menter (1993) for a detailed discussion of the model and its blending functions.
6.3
The Turbulent/Non-turbulent (TNT) k, ω Model of Kok
The k, ω TNT model was developed by Kok (2000) with a goal similar to that Menter had in mind for the development of the SST model, namely to reduce the dependence of Wilcox’s models on the free-stream value of ω. For this purpose, Kok added the cross-diffusion term obtained from the equation re-written in terms of ω to the ω equation in Wilcox’s model. But instead of using a blending function, which requires computation of the wall distance, the cross diffusion is taken into account only if it is positive. This makes the use of a blending function superfluous. In addition, Kok re-tuned the diffusion coefficients of the model in order to correct the model behavior at turbulent/non-turbulent interfaces. The model uses Equation 6.1 for the eddy viscosity and Equation 6.2 for the turbulent kinetic energy. The modified transport equation for ω is given by: Specific Dissipation Rate: ∂ρω ω ∂ui ∂ ∂ω ∂ρωuj = α (−ρui uj ) − βρω 2 + + (µ + σµt ) + CD . ∂t ∂xj k ∂xj ∂xj ∂xj The cross-diffusion term CD is defined as: ∂k ∂ω ρ CD = σD max ;0 . ω ∂xk ∂xk The following closure coefficients were suggested by Kok: TNT Closure Coefficients: α=
5 , 9
β=
3 , 40
β∗ =
9 , 100
σ=
1 , 2
σ∗ =
2 , 3
σD =
1 . 2
82
6 Models Investigated
6.4
The Local Linear Realizable (LLR) k, ω Model of Rung
Another k, ω type of turbulence model which was tested in this work is the LLR model of Rung & Thiele (1996). It is closely related to the realizable k, model developed by Shih et al. (1995). The rationale behind these models is to explicitly secure realizability, i.e. 2 2 2 u2 i ≥ 0 and ui uj ≥ (ui uj ) . Conventional two-equation transport models, for example, can yield u2 i < 0 for large strain rates, which is not physical. In addition to incorporating the realizability constraints in their model, Shih et al. developed a novel transport equation for . Instead of directly deriving a transport equation for they chose to first model the exact equation for the mean-square vorticity fluctuation ωi ωi . By multiplying the model equation for ωi ωi by ν and considering that for large Reynolds numbers = ν ωi ωi Shih et al. then obtain a new transport equation for . Rung, in turn, reformulated the model of Shih et al. in terms of ω and applied further modifications. These mainly comprise low-Reynolds-number damping functions in order to achieve the correct asymptotic near-wall behavior of the turbulent kinetic energy, k ∝ y 2 . The equations of the resulting k, ω LLR model are given in the following: Eddy Viscosity: k µt = cµ ρ . 0.09 ω Turbulent Kinetic Energy: µt ∂k ∂ρk ∂ui ∂ ∂ρkuj µ+ = −ρui uj − βk ρkω + + . ∂t ∂xj ∂xj ∂xj 2 ∂xj Specific Dissipation Rate: µt ∂ω ω ∂ui ∂ ∂ρω ∂ρωuj µ+ = αω (−ρui uj ) − βω ρω 2 + + . ∂t ∂xj k ∂xj ∂xj 2 ∂xj Auxiliary Relations for the Eddy Viscosity: cµ = fµ c∗µ , c∗µ Rµ =
Rt 70
α ,
fµ =
1/80 + Rµ , 1 + Rµ
1 , 0.12 , = max 0.04, min ˜ (b0 + As )U 2 3 Rt Rt k 1 Rt = +2 , , α = + 1.6 3 0.09 ων 2 150 150
6.5
The Explicit Algebraic Reynolds-Stress Model (EARSM) of Wallin
83
√ III 1 181 a0 , b0 = max 48 , As = 3 cos arccos , ˜ 140 + S 2 2 S U 0.09 ω 1 (Ω2 + S 2 ) 2 S ˜ = U , a0 = 8.0 − 4.1 tanh , 0.09 ω 1.8 · 0.09 ω S = 2Sij Sij , Ω = 2Ωij Ωij , III = Sij Sjk Ski . Auxiliary Relations for the k-Destruction Term: 0.83/3 + Rk , βk = 0.09 1 + Rk
∗
Rk = A
A∗ = tanh 4
Rt 100
2.5
∗
+ (1 − A )
Rt , 100
Rt 100
.
Auxiliary Relations for the ω-Production Term: αω =
fβ =
5 5 0.09 , (1 − fβ ) + 9 9 c∗µ fβ
1/80 + Rtf , 1 + Rtf
Rtf =
Rt 10000
2 .
Auxiliary Relation for the ω-Destruction Term: 1.83 βω = 0.09 . 1 + µcµ /(µ + µt ) It is noted that the above LLR model equations are not identical to the ones published in Rung & Thiele (1996). Rather, the present equations describe the model version that was implemented by Rung into the FLOWer code. This version of the model yields improved numerical robustness without affecting the underlying physical assumptions that were employed for the model development outlined in Rung & Thiele (1996); Shih et al. (1995).
6.5
The Explicit Algebraic Reynolds-Stress Model (EARSM) of Wallin
All models, except the EARSM model of Wallin & Johansson (2000), evaluated in this work belong to the category of linear eddy-viscosity models.
84
6 Models Investigated
They rely on a constitutive relation, the well-known Boussinesq approximation, that is linear with respect to the mean strain-rate tensor S. There are many types of applications where this assumption is not a powerful enough model to yield even qualitatively correct results. In flows with strong influence of streamline curvature, system rotation, pressure gradient or secondary flows of Prandtl’s second kind the Boussinesq approximation can badly fail. A general Reynolds-averaged approach to computing these effects would be based on modeling the full Reynolds-stress transport equations. These Reynoldsstress transport models promise a great potential for high predictive accuracy since they naturally incorporate such complex phenomena as inter-component transfer of Reynolds stresses, non-equilibrium between mean flow and turbulence, anisotropy of the Reynolds-stress tensor and system rotation, to name only a few. Yet, besides questions pertaining to appropriate modeling of unknown terms in the Reynolds-stress transport equations, flow computations with such models bear non-trivial numerical difficulties in complex three-dimensional flows. The latter issue, especially, motivated development of turbulence models that do not introduce additional differential equations compared to two-equation models but still offer the physical advantages of Reynolds-stress transport models yet at a lower computational effort. One of the first models for this purpose was suggested by Rodi (1976) who converted the differential transport equations of the Reynolds-stresses into implicit, algebraic expressions. Rodi assumed that the sum of all terms containing derivatives of the Reynolds stresses, that is, the convection and diffusion terms, can be written as the sum of the convection and diffusion of the turbulent kinetic energy k multiplied with the individual Reynolds stress and normalized with k. This algebraic Reynolds-stress model “inherits” most of the desirable features of the “parent” differential Reynolds-stress transport model. The main physical simplification is that the advection and diffusion of the Reynolds-stress anisotropy tensor a ≡ aij = ui uj /k − 2δij /3 are neglected. This is frequently referred to as the algebraic Reynolds-stress model assumption. Due to the implicit character of the resulting equations the model has a very difficult mathematical behavior like multiple solutions or singularities (see Wilcox, 1998). Therefore, explicit formulations are highly desirable and have been suggested for example by Gatski & Speziale (1993). In order to derive an explicit formulation of the algebraic Reynolds-stress model rewritten in terms of the anisotropy a, Gatski & Speziale embarked on ideas which were pioneered by Spencer (1971) and Pope (1975) and are based on integrity basis methods and the Cayley-Hamilton theorem. The
6.5
The Explicit Algebraic Reynolds-Stress Model (EARSM) of Wallin
85
resulting model yields non-linear, anisotropic constitutive relations that are closely related to the underlying Reynolds-stress transport model. This route is also taken in the explicit algebraic model of Wallin & Johansson (2000). Additionally, new developments like a near-wall treatment that ensures the realizability constraints and a formulation for compressible flows are incorporated into this model. Other ingredients are the model of Rotta (1951) for the slow pressure strain, the rapid pressure-strain model of Launder et al. (1975), and the assumption of an isotropic dissipation rate tensor. The most important advantage of the explicit algebraic Reynolds-stress model over linear eddy-viscosity models is the ability to predict the anisotropy of normal Reynolds-stresses. The resulting model equations read as follows: Constitutive Relation: 2 ∗ −ρui uj = −ρk δij + 2µt Sij − ρkaex ij . 3 Eddy Viscosity: 1 µt = − f1 (β1 + IIΩ β6 )ρkτ, 2 1 µ τ = max , 20 . 0.09ω ρωk Extra-Anisotropy Tensor: aex ij
=
3B2 − 4 1 (1 − f12 ) S − δ S II S ij ik kj max (IIS , IISeq ) 3 1 2 +f1 β3 Ωik Ωkj − IIΩ δij 3 B2 + f12 β4 − (1 − f12 ) (Sik Ωkj − Ωik Skj ) eq 2 max (IIS , IIS ) 2 +f1 β6 Sik Ωkl Ωlj + Ωik Ωkl Slj − IIΩ Sij − IV δij 3 +f12 β9 (Ωik Skl Ωlm Ωmj − Ωik Ωkl Slm Ωmj ) .
Invariants: IIS = tr{Sik Skj },
IIΩ = tr{Ωik Ωkj },
IISeq =
405c21 , 216c1 − 160
IV = tr{Sik Ωkl Ωlj },
c1 = 1.8.
86
6 Models Investigated
Normalized Mean Strain Rate and Rotation Tensors: τ ∂ui ∂uj 2 ∂uk τ ∂ui ∂uj + − δij , Ωij = − Sij = . 2 ∂xj ∂xi 3 ∂xk 2 ∂xj ∂xi β-Coefficients: β1 = − β4 = −
N (2N 2 − 7IIΩ ) , Q
2(N 2 − 2IIΩ ) , Q
β3 = −
β6 = −
12N −1 IV , Q
6N , Q
β9 =
6 . Q
Auxiliary Relations:
f1 = 1 − exp −CyA Rey − CyB Re2y , √ ρ kd Rey = , µ
CyA = 0.092,
CyB = 0.00012,
5 2 N − 2IIΩ 2N 2 − IIΩ , 6 √ 1/3 √ 1/3 + P1 + P2 + P1 − P2 1/6 + 2 P12 − P2 cos 13 arccos √ P21 Q=
⎧ ⎨ N= P1 =
⎩
C1 3 C1 3
P1 −P2
C12 2 9 + IIS − IIΩ C1 , 27 20 3 C1 =
P2 = P12 −
,P2 ≥ 0 , ,P2 < 0
C12 2 9 + IIS + IIΩ 9 10 3
3
9 (c1 − 1). 4
Turbulent Kinetic Energy: ∂ui ∂ρk ∂ ∂k ∂ρkuj = −ρui uj − β ∗ ρkω + + (µ + σ ∗ µt ) . ∂t ∂xj ∂xj ∂xj ∂xj Specific Dissipation Rate: ω ∂ui ∂ ∂ω ∂ρωuj ∂ρω = α (−ρui uj ) − βρω 2 + + (µ + σµt ) + CD , ∂t ∂xj k ∂xj ∂xj ∂xj CD = σD
ρ max ω
∂k ∂ω ;0 . ∂xk ∂xk
,
6.6
87
Boundary Conditions for the k, ω Models
Low-Reynolds-Number Coefficients: α∗ =
5 α0 + Ret /Reω 1 · , · 9 1 + Ret /Reω α∗
α= β∗ = β=
3 , 40
σ∗ =
α0∗ + Ret /Rek , 1 + Ret /Rek
5/18 + (Ret /Reβ )4 9 , · 100 1 + (Ret /Reβ )4
2 , 3
σ = σD =
1 , 2
α0∗ =
β , 3
α0 =
1 , 10
k , Reβ = 8, Rek = 6, Reω = 2.7. ων One can see from the above equations that the EARSM model investigated in this work relies on the transport equations for k and ω pertaining to the TNT model of Kok in combination with the low-Reynolds-number modifications suggested by Wilcox for the 1988 model. The application of the low-Reynolds-number coefficients is essential in order to recover the correct asymptotic near-wall behavior of the Reynolds stresses and the turbulent kinetic energy. The low-Reynolds number extensions for the Wallin model were implemented into FLOWer during this work and all presented results were computed using the low-Reynolds number version. Ret =
6.6 6.6.1
Boundary Conditions for the k, ω Models Free-Stream Boundary Conditions
The free-stream boundary conditions applied to the turbulence variables were the same for all k, ω models: k∞ =
3 (0.005 |v∞ |)2 , 2
µt∞ = 10−3 µ,
ω∞ =
ρ∞ k∞ . µt∞
The above conditions were specified at the farfield inflow boundaries of the computational domain. Generally speaking, k∞ and ω∞ are transported streamwise through the irrotational outer flow field by the corresponding transport equations towards the body under consideration. Actual values of the turbulence variables at the boundary-layer edge can be quite different from specified free-stream values since k and ω decay during the streamwise transport process. Hence, the values of k and ω at the boundary-layer edge
88
6 Models Investigated
depend on the flow and on the distance between the farfield boundary and the body. In situations where the farfield inflow boundary is located far away from the body a variation of the free-stream values has a weak effect on the computed values at the boundary-layer edge. However, it is ω∞ on which results obtained with the k, ω models of Wilcox can depend on. It was encountered that the dependence was especially an issue if the farfield inflow boundary could not be positioned far enough, that is 1.5 to 2.0 characteristic lengths, from the body. This will be discussed in detail in Section 8.5.2. On outflow boundaries standard convective outflow boundary conditions were applied. These are boundary conditions where zero streamwise gradients of the flow variables are specified. (However, special outflow treatment was applied for the test cases BS0 and CS0, see Subsection 7.2.1.) 6.6.2
Wall Boundary Conditions
The boundary condition for the turbulent kinetic energy at a solid surface is straightforward; k is set to zero at all no-slip walls. The specification of ω, however, is not unique and several different methods for setting the wall value of ω exist. Three of them were tested in this work. 1. In the method according to Wilcox a so-called slightly-rough-surface boundary condition is assumed. This bears the advantage that surface roughness is simply modeled by adjusting the surface value of ω accordingly. The resulting boundary condition for ω at the wall reads: 2 2500 uτ uτ ks ωw = , ks+ = . (6.5) + νw νw ks ks is the surface roughness height and for hydraulically smooth surfaces ks+ ≤ 5. In the present work ks+ = 5 was used. A disadvantage of this method is that the wall value of ω depends on uτ and hence on the skin friction. This means that in points where the skin friction is zero, as for separation and attachment points in a two-dimensional flow, the wall value of ω is zero, too. However, there is no physical reason why the wall value of ω should vanish in such points. 2. Menter (1993) suggested a method in which the wall value of ω depends on the distance of the first grid point from the wall, ∆y: ωw =
60νw . 0.075∆y 2
(6.6)
6.7
The One-Equation Model of Spalart & Allmaras
89
This expression mimics the fact that ω ∼ y −2 approaching the wall as pointed out by Wilcox (1998). 3. A very similar method to the one proposed by Menter is the procedure used by Rudnik (1997). However, in contrast to Menter’s approach Rudnik uses a fixed reference length scale yr and the surface boundary condition for ω is then given by ωw =
60νw . 0.075yr2
(6.7)
Rudnik recommends yref = 10−5 as an appropriate value for the reference length scale for a large variety of airfoil flow cases (see Rudnik, 1997). This is also the value adopted in the present work if not otherwise noted. In FLOWer, one of the above boundary conditions is specified by user input.
6.7
The One-Equation Model of Spalart & Allmaras
Early one-equation models were based on the transport equation of the turbulent kinetic energy which serves as the velocity scale for the computation of the eddy viscosity. These models are “incomplete” in a sense that they require the specification of a turbulent length scale which varies from flow case to flow case and must be specified a priori, i.e. the length scale is not part of the obtained flow solution. By contrast, one-equation models solving a transport equation for the eddy viscosity itself inherently provide the necessary velocity and time scale and are thus complete. One of the main motivations for the development of turbulence models employing one transport equation for the eddy viscosity is to reduce the computational effort which is required for the solution of two-equation transport models. This comes in addition to the intention to utilize the principal advantages of transport-equation models over algebraic ones. As noted earlier, the latter do not account for flow-history effects and rely on physical assumptions that are in the spirit of equilibrium boundary layers and become incorrect when separated and multiple shear layers are present. Besides these physical arguments various issues at the implementation level favor the use of transport equation models. For example, algebraic models typically evaluate the velocity and/or vorticity profile normal to the wall; additionally,
90
6 Models Investigated
some of the models, like the Cebeci-Smith model, require the computation of the boundary-layer thickness. Reliable generalizations of these procedures for three-dimensional flows around complex geometries and, possibly, on unstructured grids are extremely difficult to develop and computationally expensive. Spalart & Allmaras (1992) proposed a very successful one-equation eddyviscosity-transport model. Unlike other attempts, for example the model of Baldwin & Barth (1991) or, more recently, Menter’s one-equation model (Menter, 1997), they did not derive their one-equation model by simplifying an existing k, two-equation model. Rather, they constructed a transport equation term by term using “empiricism, arguments of dimensional analysis, Galilean invariance and selective dependence on the molecular viscosity”. The model constants and closure functions were calibrated using building block flow cases like different kinds of free shear flows and boundary layers. The interested reader is referred to the papers of Spalart & Allmaras (1992, 1994) for the details of the model derivation. The model uses the same constitutive relation as the Baldwin-Lomax model, Equation 3.5. The additional model equation read as follows: Effective Eddy Viscosity: µt = ρ ν˜fv1 . Transport of Eddy Viscosity: ∂ρ˜ ν ∂ ∂ρ˜ ν uj = Pν˜ − Dν˜ + + ∂t ∂xj ∂xj
2 µ + ρ˜ ν ∂ ν˜ cb2 ∂ ν˜ +ρ σ ∂xj σ ∂xj
(6.8)
Diffusion
with the production and destruction terms Pν˜ = cb1 ρS˜ν˜,
Dν˜ = cw1 fw ρ
2 ν˜ . d
Closure Coefficients: cb1 = 0.1355,
cb2 = 0.622,
cb1 1 + cb2 + , κ2 σ Auxiliary Relations: cw1 =
fv1 =
χ3 , χ3 + c3v1
cv1 = 7.1,
cw2 = 0.3,
fv2 = 1 −
χ , 1 + χfv1
σ=
cw3 = 2,
κ = 0.41.
fw = g
2 , 3
1 + c6w3 g 6 + c6w3
1/6 ,
6.8
χ=
ν˜ , ν
g = r + cw2 r6 − r ,
S˜ = S +
6.8
91
The One-Equation Model of Edwards & Chandra
ν˜ fv2 , κ2 d2
S=
r=
ν˜ , ˜ 2 d2 Sκ
2Ωij Ωij .
(6.9)
The One-Equation Model of Edwards & Chandra
Edwards & Chandra (1996) performed a comparative study of several oneequation transport models including the Spalart-Allmaras model. They found that the original formulation of the strain-rate norm S˜ in Equation 6.9 can lead to a singular behavior of S˜ in the near-wall region. The achievable level of residual convergence of the numerical scheme is limited when this happens. In order to increase the numerical robustness of the Spalart-Allmaras model for such cases they suggested the following more stable way of computing S˜ as well as the near-parameter r which is used for the computation of the wall-blockage function fw : √ 1 ν˜ S˜ = S + fv1 , r = tanh / tanh(1.0), ˜ 2 d2 χ Sκ 2 ∂ui ∂uj ∂ui 2 ∂uk + − . S= ∂xj ∂xi ∂xj 3 ∂xk All other model equations including the transport of the eddy viscosity, Equation 6.8, are identical to the original Spalart-Allmaras model. Since the modifications of Edwards and Chandra are purely numerically motivated the new model is expected to yield results very similar to the original SpalartAllmaras model.
6.9
The Strain-Adaptive Linear Spalart-Allmaras (SALSA) Model
It is often reported in the literature that the Johnson-King model and the k, ω SST model of Menter show significant improvements of predictive accuracy for separating boundary-layer flows compared to their “parent” models, namely the Baldwin-Lomax model and the 1988 k, ω model of Wilcox, respectively. These improvements are mainly attributed to a forced limitation of the eddy viscosity in separated flow regions. The limitation mechanisms used in the two models are based on Bradshaw’s assumption of constant ratio between the turbulent kinetic energy and the Reynolds shear stress in a boundary-layer flow. Motivated by the success of limiting the eddy viscosity
92
6 Models Investigated
in regions of excessive production, Rung et al. (2003) transfered the limitation concept to the framework of the one-equation model of Spalart & Allmaras. They developed an eddy-viscosity-transport model that is based on the transport equation of the original Spalart-Allmaras model, Equation 6.8, with a modified production term. In detail, following relations are employed in the SALSA model: Constitutive Relation: 2 ∗ −ρui uj = µt Sij − ρkδij 3 where S ∗ µt ρk = √ , cµ = 0.09 cµ and 1 ∂ui ∂uj 1 ∂uk ∗ ∗ ∗ = + δij , S ∗ = 2Sij Sij . Sij − 2 ∂xj ∂xi 3 ∂xk The same closure coefficients and closure function are used as for the original Spalart-Allmaras model. However, the near-wall parameter r and the strain-rate norm S˜ are redefined following a route similar to that of Edwards & Chandra: ν˜ ρ0 r = 1.6 tanh 0.7 , ˜ 2 d2 ρ Sκ 1 S˜ = 1.04 S ∗ + fv1 . χ The key feature of the SALSA model is the reduction of the production term Pν˜ for excessive strains. For this purpose, the ratio between ν˜ obtained from the transport equation, Equation 6.8, and an eddy viscosity based on Prandtl’s mixing-length theory, νP randtl = lmix · vmix = κd · κd S ∗ , is used to yield the non-equilibrium factor ν˜ . κ2 d2 S ∗ This ratio is typically less than unity for separated boundary layers and is used to damp the production term. The employed expressions are: σneq =
cb1
√ = 0.1355 Γ,
Pν˜ = cb1 ρS˜ν˜,
Γ = min(1.25, max(γ, 0.75)), γ = max(α1 , α2 ).
χ . α1 = (1.01 σneq )0.65 , α2 = max 0, 1 − tanh 68
6.10
6.10
Boundary Conditions for the One-Equation Models
93
Boundary Conditions for the One-Equation Models
The specification of boundary conditions for the models based on the transport equation for the eddy viscosity, Equation 6.8, is straightforward compared to procedures applied for the two-equation models. Two aspects must be considered: First, the free-stream value of µt must be sufficiently small in order not to influence the irrotational part of the flow solution. This is ensured by setting the free-stream value of µt to a small fraction of the fluid’s viscosity. In this work the following relation was employed: ν˜∞ = 10−5 ν∞ . Secondly, since all fluctuations are zero at the wall and, hence, all Reynolds stresses vanish, the eddy viscosity at the wall must be zero as well: ν˜w = 0.
94
7
7 Test Cases Selected
Test Cases Selected
The flow cases investigated have, on the one hand, a simple geometry in order to simplify grid generation, to exclude numerical errors associated with extremely skewed grids, and to be easily able to study grid-convergence effects by simply doubling the number of grid points. On the other hand, they are well suited for validation purposes since for each flow case accurate experimental data are available and each flow case offers the necessary physical complexity. Table 7.1 gives an overview of the considered flow cases including the basic flow parameters which were specified for the computations. A closer description of the investigated flow will be given in the corresponding section.
Table 7.1: Test cases selected including basic flow parameters Test case
Case ID
Flow class
Mref
Reref
Tref [K]
Boundary layer with dp/dx = 0
FPBL
BL 1.1 (Table 7.2)
0.02848
4.47 · 106
296.4
Non-equilibrium boundary layer with dp/dx > 0
BS0
BL 2.1 (Table 7.2)
0.08772
280000
291.1
Non-equilibrium boundary layer with dp/dx > 0 and separation
CS0
BL 3.1 (Table 7.2)
0.08772
280000
291.1
Separated flow around airfoil A
AAA
1.1 (Table 1.1)
0.15
2.00 · 106
294.4
In the introduction, aerodynamic flows are classified using the flow topology as a guideline. Similarly, following Hirschel (2003), the boundary-layer flows investigated in this section can also be classified using the topology of the velocity fields (see Table 7.2). Of course, the topologies shown in Table 7.2 are very simple and the classification is straightforward. However, the table is intended for again demonstrating the concept of classifying flows on the basis of flow topology. The purpose of the classification is to investigate the performance of the present turbulence models for each class of flows. It is hoped that future investigations of this kind will show whether general conclusions can be drawn on what model to use for a given flow class.
7.1
95
Flat-Plate Boundary Layer (Case FPBL)
Table 7.2: Possible classes of two-dimensional turbulent boundary layers on flat surfaces following Hirschel (2003) (flows with favorable pressure gradient are not considered in the present work) Class BL 1.1 ∂p ∂x
∂p ∂x
=0
Case FPBL
7.1
Class BL 2.1 >0
Case BS0
Class BL 3.1 ∂p ∂x
> 0 and separation
Case CS0
Flat-Plate Boundary Layer (Case FPBL)
A flat-plate boundary-layer flow with zero pressure gradient was studied in order to investigate the models’ ability to predict the law of the wall. New and very accurate experiments were performed by DeGraaff & Eaton (2000) and are available via the Internet. The advantage of DeGraaff’s data over earlier measurements is that the first measurement point was located very close to the wall, well within the viscous sublayer, so that model predictions of the sublayer can be checked in detail. In the experiments, the boundary-layer profiles were evaluated at a downstream position corresponding to Reθ = 2900, where Reθ denotes the Reynolds number based on the momentum thickness θ. The free-stream velocity in the experiments was u∞ = 9.83 m/s. In order to resemble this value for u∞ in the computation the following input parameters were specified: Mref = 0.02848 and Reref = 4471085 in combination with L = 7 m. The transition location was specified at x/L = 0.04. Reθ = 2900 was then obtained at approximately 40 percent of the plate. 7.1.1
Computational Setup
The grid for the flat-plate boundary-layer computations is shown in Figure 7.1. The plate is located at the lower boundary between x/L = 0 and x/L = 1 where no-slip and adiabatic-wall conditions were applied. In the front part of the lower boundary (−0.5 ≤ x/L < 0) symmetry was enforced by setting all derivatives in the y-direction to zero. The entire upper boundary was modeled by characteristic boundary conditions while free-stream conditions
96
7 Test Cases Selected
were prescribed at the left inflow boundary. At the right boundary of the computational domain the pressure was fixed while zero streamwise gradients of the velocity and the density were specified.
y/L
0.3 0.2 0.1 0 -0.5
0
x/L
0.5
1
Figure 7.1: Grid for flat-plate boundary-layer computations. The grid contained a total of 144 cells in the x-direction, where 96 cells were located along the no-slip boundary. 64 cells were used in the wall-normal direction. In order to resolve the boundary layer algebraic grid clustering normal to the wall was employed ensuring that the first grid point above the surface was located below y + = 1. However, the grid spacing was uniform at the entry of the computational domain leading to the downwards slope of the “horizontal” grid lines in the region −0.5 ≤ x/L ≤ 0. Additionally, grid clustering in the x-direction was performed at the leading and trailing edges of the plate. These regions pose discontinuities in the boundary conditions resulting in large streamwise gradients. A high grid resolution is necessary in these regions in order to limit the influence of these discontinuities and to accurately resolve the large gradients. It is noted that the grid employed has highly smooth distributions of all metric terms which is deemed to be an essential prerequisite for obtaining accurate numerical solutions, especially in conjunction with numerical methods using a central space discretization. 7.1.2
Computational Results and Discussion
Pressure Distribution One can see from the wall-pressure distribution shown in Figure 7.2 that, except at the leading edge and around the transition location (x/L = 0.04), a virtually zero pressure gradient with cp = 0 along the wall is predicted. The presented cp distribution is taken from the computation with the Baldwin-Lomax model. However, all other turbulence
7.1
97
Flat-Plate Boundary Layer (Case FPBL)
models applied in this work show very similar results confirming that no significant pressure gradient acts on the boundary layer. 0.01
cp
0.005 0
-0.005 -0.01
0
0.25
0.5
x/L
0.75
1
Figure 7.2: Pressure coefficient along flat plate.
Velocity Profiles and Skin Friction To compare the computed and measured velocity profiles a typical semi-logarithmic presentation was chosen in which the velocity is plotted over the wall distance in dimensionless wall coordinates. 1 Coles’ logarithmic law of the wall (u+ = 0.41 ln y + + 5.0) reproduces the experimental velocity profile in the logarithmic region (30 ≤ y + ≤ 250) with high accuracy (Figure 7.3). As it is expected from theory, in the lower part of the viscous sublayer, that is for y + ≤ 5, the linear relation u+ = y + yields a good approximation to the experimental data. The velocity profiles obtained with the different turbulence models are compared to the experimental data in Figures 7.4a) to 7.4d). WBC, MBC and RBC denote the specification of ω at the wall according to Wilcox, Menter or Rudnik, respectively (see Subsection 6.6.2). In the case of the Wilcox and SST models the wall value of ω was determined following the procedure suggested by the respective model developer. For the TNT model, LLR model and the model of Wallin, Rudnik’s method for determining ωw was adopted. This is because the method of Rudnik is recommended by the FLOWer implementation team and the model developers do not recommend any specific surface boundary treatment for ω for the latter models. One can see from Figures 7.4b) and 7.4d) that the EARSM model of Wallin and the SST model of Menter overpredict the velocity in the outer region of the boundary layer. Both models of Wilcox, however, underpredict u+ compared to the experimental data for the largest part of the boundary
98
7 Test Cases Selected
25
Experiment (DeGraaff) u+=1/0.41 ln(y+) + 5.0 u+=y+
15
u
+
20
10 5 0 0 10
y+
101
102
103
Figure 7.3: The log-law compared with the experiment of DeGraaff at cf = 3.362 × 10−3 , that is at Reθ = 2900.
layer (Figure 7.4a)); they slightly overpredict the slope of the velocity profile in the logarithmic region. Note that the Wilcox 1998 model yields a velocity profile that is closer to the measured data in the wake region of the boundary layer than the profile obtained with the 1988 model. This difference is due to the function χk in Equation 6.4 of the 1998 model: In the wake region, ω is small compared to values encountered close to the wall, and gradients of both turbulence variables are non-zero. This combination leads to large values of χk which, in turn, increases the dissipation coefficient β ∗ in the transport equation for k (Equation 6.2) and, hence, reduces k and the eddy viscosity. With the exception of the models of Menter, Wallin, and Wilcox, all of the models employed yield velocity profiles which are in very close agreement with the experiment. Both over- and underprediction of the sublayer-scaled velocity u+ are frequently rooted in the prediction of the shear stress at the wall. This is best seen when writing u+ and y + in terms of τw : u+ =
u =u uτ
ρw , τw
y+ =
yuτ y = ν ν
τw . ρw
7.1
99
Flat-Plate Boundary Layer (Case FPBL)
It follows from these relations that an overprediction of the wall shear stress τw leads to an underprediction of u+ in Figures 7.4a) to 7.4d) (and vice versa) even if the absolute velocity u was correctly predicted by the computation. Hence, the presentation of the velocity profiles in wall coordinates must always be discussed in conjunction with the corresponding skin-friction coefficients. Table 7.3: Local cf at Reθ = 2900 (case FPBL) Source
cf × 103
∆
Experiment (DeGraaff & Eaton, 2000)
3.362
–
k, ω TNT (RBC)
3.2636
−2.9%
Spalart-Allmaras
3.2502
−3.3%
SALSA
3.2093
−4.6%
k, ω LLR (RBC)
3.1788
−5.4%
Baldwin-Lomax
3.1474
−6.4%
Edwards-Chandra
3.1315
−6.8%
k, ω 98 (WBC)
3.6128
+7.5%
Wallin (RBC)
3.0739
−8.6%
k, ω 88 (WBC)
3.7262
+10.8%
k, ω SST (MBC)
2.9009
−13.7%
Results obtained for cf (Table 7.3) are consistent with the discussion about over- and underprediction of u+ profiles in Figure 7.4: The model of Wallin and the k, ω SST model of Menter yield skin-friction coefficients that are significantly lower than the experimental value and they overpredict u+ . Both models of Wilcox compute cf that is significantly larger than measured and they underpredict u+ . The k, ω TNT model gives the best prediction of the skin-friction coefficient; it agrees within three percent with the experiment. Note that the Spalart-Allmaras model yields a very similar value for the skin-friction coefficient. However, the velocity profile in the transition region between the sublayer and the logarithmic part, that is for 15 ≤ y + ≤ 45, agrees even better with the experimental data than the velocity profile obtained with the TNT model. All other model predictions of the skin friction range in between these extrema.
+
u
+
u
103
d)
b)
101
101
102
y+
102
Experiment (DeGraaff) kω SST MBC
y+
Experiment (DeGraaff) kω TNT RBC Wallin RBC kω LLR RBC
Figure 7.4: Velocity profiles for flat plate at Reθ = 2900.
0 0 10
102
0 0 10
15
20
25
0 0 10
5
y+
103
5 101
102
Experiment (DeGraaff) Spalart-Allmaras Edwards-Chandra SALSA
y+
5
10
15
20
25
10
c)
101
Experiment (DeGraaff) Baldwin-Lomax kω 88 WBC kω 98 WBC
10
15
20
25
0 0 10
5
10
15
20
a)
+
u +
u
25
103
103
100 7 Test Cases Selected
7.1
101
Flat-Plate Boundary Layer (Case FPBL)
7.1.3
Some Modifications of the k, ω SST Model
In order to improve the model predictions for the flat-plate boundary layer two simple ad hoc modifications of the k, ω SST model were tested. These are discussed in the following. Eddy Viscosity in the Viscous Sublayer All linear k, ω models investigated in this work set the eddy viscosity proportional to k/ω. It can be shown that this gives the following asymptotic sublayer behavior: µt ∝ y n
for
y→0
(7.1)
with n ≈ 5.28 for the Wilcox 1998 model and n ≈ 5.23 for all the other linear k, ω models. Expanding the fluctuating velocities in Taylor series near the wall and employing the no-slip condition and the continuity equation yields u v ∝ y 3
for
y → 0.
Since the velocity gradient in the sublayer is constant (du+ /dy + = 1) and µt = −ρu v /(du/dy), the exact exponent for the eddy viscosity following from this asymptotic analysis is n = 3. Hence, a different functional dependence of µt on k and ω that yields an exponent closer to the theoretical value of 3 in the viscous sublayer would be more appropriate. This is discussed below. In the viscous sublayer the turbulent motion is strongly affected by the presence of the solid surface; the turbulent eddies decrease in size approaching the wall. Since small scale turbulent motions are diffused and dissipated by viscosity it is therefore appropriate to base the characteristic turbulence time scale in the viscous sublayer on the viscosity and the turbulence dissipation: ν ν Tη = ∝ . kω Tη is typically referred to as the Kolmogorov time scale. Since the eddy viscosity can be written as a product of k and a time scale, it seems to be reasonable to use Tη to compute the eddy viscosity in the viscous sublayer. For this purpose the following expressions were implemented into the SST model: 1−F3 a1 ρk µt = (SL ρν)F3 , max (a1 ω; ΩF2 )
102
7 Test Cases Selected
k/ω arg3 = max 1 − ,0 , SL νk/ω
1 F3 = tanh arg34 , 2
SL = 2. SL is a sublayer model constant and F3 is an additional blending function that ensures a smooth transition from the sublayer expression µt = SL kν/ω to the standard SST formulation for the eddy viscosity. This reformulation of the eddy viscosity in terms of Tη yields for the asymptotic behavior approaching the wall n ≈ 2.62 in Equation 7.1. This is much closer to the theoretical value n = 3 than in the original model. Computations show that the differences between the absolute values of µt obtained with the original expression and the sublayer formulation are small with respect to µt . Due to the large velocity gradient in the sublayer, however, the effect on the Reynolds shear stress and, hence, on the momentum equation is significant. Reducing ω in the Logarithmic Region Analysis of the eddy-viscosity profile at Reθ = 2900 reveals that µt computed with the original k, ω SST model is lower than µt inferred from the experimental data in the region 40 ≤ y + ≤ 200 (Figure 7.5).
0.025
Experiment (DeGraaff) kω SST MBC
νturb/(θUe)
0.02 0.015 0.01 0.005 0 0 10
101
y+
102
103
Figure 7.5: Eddy viscosity at Reθ = 2900 (case FPBL).
7.1
103
Flat-Plate Boundary Layer (Case FPBL)
To investigate to what extent this difference does have an influence on the computed velocity profile the eddy viscosity was increased in the region of interest. For this purpose a local damping of ω around y + = 75 was introduced which was achieved by increasing the destruction term in the ω equation. In particular, the destruction coefficient β1 of the SST model was modified as follows: 3 1 β1 = , (1 − 1.3F4 ) 40 2 1 75 − y + − 0.7, 0 . (7.2) F4 = max exp − 2 85 Figure 7.6 shows the resulting distribution of β1 across the boundary layer. At y + = 75 a peak value of β1 ≈ 0.123 is obtained leading to a strong amplification of the destruction term in the ω transport equation in this region. The original value β1 = 3/40 is maintained below y + = 3.2 and above y + = 147.
0.14
β1
0.12
0.1
0.08
0.06 0 10
101
y+
102
103
Figure 7.6: Profile of modified destruction coefficient β1 at Reθ = 2900 (case FPBL). Like all functions depending on y + , and hence on the skin friction, Equation 7.2 bears the disadvantage that it is not defined for vanishing wall shear
104
7 Test Cases Selected
stress. Theoretically, in such situations, y + = 75 is located infinitely far away from the wall. Due to the construction of Equation 7.2, however, this does not pose any numerical difficulties since then the original value β1 = 3/40 is recovered. Results The modifications of the k, ω SST model lead to excellent agreement of computed and measured velocity profiles (Figure 7.7). Similarly, the skin-friction coefficient obtained with the modified model agrees within roughly one percent with the experimental value (Table 7.4). This comes as no surprise since the modifications were tailored in order to yield best possible agreement of computational results with measurements for this flow case. It will be shown, however, that these modifications improve model predictions for separated flows as well.
25 20
Experiment (DeGraaff) kω SST original kω SST modified
u+
15 10 5 0 0 10
101
y+
102
103
Figure 7.7: Comparison of velocity profiles computed with the original and modified k, ω SST model (case FPBL). It is noted that the focus of this work was not the development of new turbulence models. Therefore, a further improvement and a generalization of the simple modifications presented was not pursued.
7.1
105
Flat-Plate Boundary Layer (Case FPBL)
Table 7.4: Local cf at Reθ = 2900 obtained with the modified k, ω SST model (case FPBL) Source
7.1.4
cf × 103
∆
Experiment (DeGraaff)
3.362
–
k, ω SST original
2.9009
−13.7%
k, ω SST modified
3.3253
−1.1%
Effects of Low-Reynolds-Number Modifications
Wilcox (1998) developed the low-Reynolds-number modifications for his k, ω models with two aspects in mind. One was modeling of laminar-turbulent transition which will be discussed in Section 8.3. The other aspect was improvement of near-wall behavior of k and ω. Yet, model predictions for skin friction and velocity profiles obtained with and without application of lowReynolds-number terms were virtually identical (Table 7.5, Figure 7.8); differences were encountered only in the k and ω profiles. Figure 7.9 shows that in the computation including low-Reynolds-number modifications a near-wall peak in the k profile is obtained. This is consistent with results from DNS (Wilcox, 1998).
Table 7.5: Local cf at Reθ = 2900 obtained with Wilcox’s 1988 k, ω model in combination with Rudnik’s surface boundary condition for ω (k, ω 88 RBC) with and without low-Reynolds-number modifications (case FPBL) Source
cf × 103
∆
Experiment (DeGraaff)
3.362
–
k, ω 88 (RBC) standard
3.4383
+2.3%
k, ω 88 (RBC) including viscous corrections
3.4327
+2.1%
Except for the Aerospatiale-A airfoil, the test cases considered in this work were computed without low-Reynolds-number modifications. This was done for the following reasons: • Viscous correction suggested by Wilcox do not alter model predictions for skin friction and velocity profiles in fully-turbulent flows. Yet, for
106
7 Test Cases Selected
30 25
kω 88 RBC kω 88 RBC low Reynolds number
u
+
20 15 10 5 0 0 10
10
1
y+
2
10
10
3
Figure 7.8: Effect of low-Reynolds-number modifications on u+ profile of a flat-plate boundary layer; computation performed with the 1988 k, ω model of Wilcox in combination with Rudnik’s surface condition for ω.
the test cases considered, no experimental data are available for k (and ω) in the near-wall region. Hence, direct validation of the low-Reynoldsnumber modifications was not possible. • Typically, in production codes, the simpler high-Reynolds-number versions of the k, ω models are implemented. • Viscous correction do not contribute to the solution of the core problems of turbulence modeling since these are Reynolds averaging and the assumption of an eddy viscosity. However, in cases where viscous effects like transition are significant the viscous corrections developed by Wilcox can have a large influence on computed results even in the turbulent part of the flow. As noted above, this will be discussed in Section 8.3.
7.2
107
Boundary Layer with Adverse Pressure Gradient (Case BS0)
1.2E-05 1E-05
kω 88 RBC standard kω 88 RBC low Reynolds number
k/(p∞/ρ∞)
8E-06 6E-06 4E-06 2E-06 0
10-4
y/L
10-3
10-2
Figure 7.9: Effect of low-Reynolds-number modifications on the k profile of a flat-plate boundary layer; computation were performed with the 1988 k, ω model of Wilcox in combination with Rudnik’s surface condition for ω.
7.2
Boundary Layer with Adverse Pressure Gradient (Case BS0)
The next flow case considered is a turbulent boundary layer with a strong adverse pressure gradient that was experimentally investigated by Driver (see Driver & Johnston, 1990; Driver, 1991). In this flow, an axisymmetric boundary layer developed in the axial direction on a circular cylinder; the cylinder was lengthwise mounted along the center of the wind tunnel (Figure 7.10). An adverse pressure gradient was imposed at the cylinder by diverging all four wind tunnel walls. To reduce separation of the tunnel wall boundary layers, boundary-layer suction was applied at the tunnel side walls. The level of suction was used to control boundary-layer thickness at the tunnel walls and, thus, to adjust the pressure gradient at the boundary layer of the cylinder surface. No separation of the boundary layer at the cylinder occurred in case BS0, which is the case discussed in this section. The pressure gradient, how-
108
7 Test Cases Selected
Figure 7.10: Driver’s cylinder flow; case CS0 with separation (schematic figure is courtesy of David Driver, NASA-Ames Research Center, used with permission).
ever, was strong enough to cause significant non-equilibrium between mean flow and turbulence. Case CS0 with separation will be presented in Section 7.3. Driver’s extensive experimental data for these flow cases are highly accurate and self-consistent and form an excellent basis to assess performance of turbulence models for adverse pressure-gradient flows without and with separation.
7.2.1
Computational Setup
For the computation of the cylinder flow three grid planes were located closely spaced in the circumferential direction ψ, as indicated in Figure 7.11. The spacing between the planes was ∆ψ = 0.5◦ . Each plane contained 256 cells in the axial and 64 cells in the radial direction. Additionally, algebraic grid clustering was employed normal to the wall ensuring that the position of the first grid point above the surface was below y + = 1. To achieve axial symmetry a three-dimensional computation was performed and appropriate boundary conditions were set in the circumferential direction. For this purpose, the two outer grid planes served as ghost layers in which symmetry boundary
7.2
109
Boundary Layer with Adverse Pressure Gradient (Case BS0)
conditions were applied. Hence, only the flow solution obtained on the center grid plane is compared to experimental data.
1.4
axissymmetric boundary conditions in ϕ-direction
y/D
1.2 inviscid streamline (Euler wall)
1 0.8 0.6 0.4 -6
outflow
prescribed inflow no slip wall -4
-2
0
x/D
2
4
6
Figure 7.11: Computational grid and boundary conditions for Driver’s cylinder flow (not to scale). The computational grid and the boundary conditions applied are shown in Figure 7.11. Note that, for better recognizability, the x to y ratio is not unity in the figure. In addition, only every other grid line in the x-direction (axial) and every fourth grid line in the y-direction (radial) is shown. The usual no-slip and adiabatic wall conditions were applied at the cylinder surface. At the inflow boundary all flow variables were prescribed and held fixed during the computations. In particular, the vector of the conserved variables was determined from the experimental velocity profile assuming constant density across the boundary layer. Regarding the turbulence variables, k was inferred from the experimental Reynolds-stress profile and ω was computed with the usual definition of eddy viscosity, e.g. Equation 6.1. (The eddy viscosity had been determined from the shear-stress and velocity profiles.) The Mach number specified for the computation was Mref = 0.08872, and the Reynolds number based on the diameter of the cylinder was Reref = 280000. To impose the experimental pressure distribution in the computation, a procedure was applied that is based on modeling the upper boundary opposite to the cylinder surface as an inviscid streamline: An inviscid streamline was used to determine the shape and position of the upper-boundary grid line. The streamline was inferred from experimental data by applying the
110
7 Test Cases Selected
requirement of mass preservation: y1 +ys (x) u(x, y)y dy = constant. m ˙ = 2πρ
(7.3)
y1
m ˙ is mass per unit time, y1 denotes the cylinder radius, and ys (x) is the unknown coordinate of the inviscid streamline. At the inflow boundary ys (x) was set equal to the wall-normal position of the last measurement point on the experimental velocity profile. The experimental velocity profile was then integrated using Equation 7.3 to yield m. ˙ Subsequently, at all other downstream stations, Equation 7.3 was employed to compute ys (y) which determined the shape and position of the upper-boundary grid line. Along the upper boundary, a perfect slip-wall condition was applied which finally rendered the grid line an inviscid streamline. (Note that the cylinder surface and the perfect slip-wall were much closer to each other than the cylinder surface and the wind tunnel walls.) The inviscid-streamline method was applied also by Menter (1993) who showed that this procedure yields very similar results to the direct specification of pressure. Directly specifying surface pressure, however, is numerically less consistent in the framework of full Navier-Stokes computations. As a result of the inviscid-streamline procedure, an internal flow is obtained. For such flows special attention must be paid to specification of pressure at the inflow and outflow boundaries. Since the pressure at the inflow boundary was set fixed to the measured value, the outflow pressure could not be independently set. In such situations, the outflow pressure depends on the total-pressure loss and, hence, on the particular flow solution. Consequently, one can not simply specify the measured value since the totalpressure losses can differ between computation and experiment. Therefore, the outflow pressure must be adapted to the flow solution such that a smooth pressure distribution is obtained throughout the flow field. In particular, fixed inflow pressure in combination with inappropriate outflow pressure can yield a non-physical pressure “jump” between the inflow boundary and the first interior grid line in the inflow region. In addition, in the compressible RANS equations pressure carries the thermodynamic information: Merely specifying a pressure difference between inflow and outflow is not applicable in this context, as can be done for incompressible Navier-Stokes computations. For these reasons, the pressure at the outflow was adapted during the computation such that the surface pressure obtained from the flow computation resembled the experimental value at the first grid point downstream of the
7.2
Boundary Layer with Adverse Pressure Gradient (Case BS0)
111
inflow boundary. For the other flow variables, standard convective outflow conditions were specified. 7.2.2
Computational Results and Discussion
Pressure Distribution Figure 7.12 shows computed and measured pressure distributions along the cylinder surface. The graphs of the pressure distributions obtained with different models lie very close to each other up to x ≈ 0. Downstream of this point differences among the pressure distributions are evident; all investigated models tend to overpredict surface pressure, though to different extent. The Baldwin-Lomax model yields the highest surface-pressure level while the 1998 k, ω model of Wilcox, the original SST model, the SALSA model, and the model of Wallin predict pressure values that are also somewhat too high but in fairly close agreement with the measurements. Note that due to the small distance between the lower and upper “walls” in the computation, even small changes in the momentum thickness predicted by the models significantly affect the surface pressure. Skin-Friction Distribution Corresponding skin-friction distributions are shown in Figure 7.13. Relatively larger variances are encountered among skin-friction distributions obtained with different models compared to the variances in results for the pressure distributions. Differences in cf obtained with different one-equation models are smaller than between computational results from k, ω models. Regarding the prediction of measured skin friction for this flow case, the 1998 k, ω model of Wilcox and the modified k, ω SST model give the best results. Comparing the graphs in Figure 7.12 and Figure 7.13 one can see that if a turbulence model yields accurate predictions of surface pressure (with regard to experimental data) it does not necessarily yield skin-friction predictions of comparable accuracy. This is best recognized from results obtained with the Wallin model: The Wallin model yields cp that is very close to measurements while it predicts cf in modest agreement with the experiment.
0.4
0.4
0.6
0
0.2
cp
0
0.2
cp
c)
a)
-1
-1
x/D
x/D
1
1
2
Experiment Spalart-Allmaras Edwards-Chandra SALSA
2
3
3
0.4
0.6
0
0.2
0
0.2
d)
b)
-1
-1
0
0
x/D
x/D
1
1
2
Experiment kω SST MBC kω SST modified
2
Experiment kω TNT RBC Wallin RBC kω LLR RBC
Figure 7.12: Pressure distributions at cylinder surface for case BS0 (arrows show positions of measured and computed boundary-layer profiles).
0
0
Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC
0.4
0.6
cp cp
0.6
3
3
112 7 Test Cases Selected
cf
cf
2
x/D
1
2
3
3
0.002
0.003
0
-1
-1
d)
b)
0
0
1
2
x/D
1
2
Experiment kω SST MBC kω SST modified
x/D
Experiment kω TNT RBC Wallin RBC kω LLR RBC
Figure 7.13: Skin-friction distributions at cylinder surface for case BS0 (arrows show positions of measured and computed boundary-layer profiles).
0
0
1
Experiment Spalart-Allmaras Edwards-Chandra SALSA
x/D
0.001
0.002
0.003
0
c)
0
Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC
0.001
-1
-1
a)
0.001
0.002
0.003
0
0.001
0.002
0.003
cf cf
3
3
7.2 Boundary Layer with Adverse Pressure Gradient (Case BS0) 113
114
7 Test Cases Selected
Velocity Profiles Velocity profiles are compared at different downstream positions. These positions are given by the experiment and are marked with arrows in Figures 7.12a) and 7.13a). So as not to overload this section with an excessively large number of graphs the corresponding figures are placed in Appendix C.1. All models but the Baldwin-Lomax model yield velocity profiles that are in fair agreement with experimental results (Figures C.1 to C.5). With increasing downstream position x/D, however, the agreement decreases and the differences among the results increase (Figure C.6). (Note that the measurement points are sparsely distributed in the direction normal to the wall at all downstream positions except x/D = 1.08857. This makes a thorough comparison of computed and measured results somewhat difficult.) Results obtained with the Baldwin-Lomax model are in poor agreement with experimental data at x/D = 1.08857 and x/D = 1.63286. Inspection of the velocity profiles and the pressure distributions shows that the velocity at the boundary-layer edge and the surface pressure are obviously linked by Bernoulli’s equation (p + 0.5(u2 + v 2 ) = constant). Reynolds-Shear-Stress Profiles Computed and measured Reynolds shear stresses are compared in Figures C.7 to C.12; the Reynolds stresses were evaluated at the same downstream positions as the velocity profiles discussed above. It is noted that measurement uncertainties are much larger in case of Reynolds stresses than for mean velocities. This should be kept in mind when comparing computational and experimental results. The Baldwin-Lomax model and Wilcox’s 1988 k, ω model generally overpredict the level of Reynolds shear stress while other models yield Reynolds stresses that are either close to or below measured values. Best overall agreement of computed u v with measurements is obtained with the oneequation models. Note that the largest differences between computed and measured Reynolds-stress profiles are found at the last downstream station (x/D = 1.63286). The reason for this is that in the experiment the Reynolds-stress level significantly increases between x/D = 1.08857 and x/D = 1.63286. In the computations, this increase is much less pronounced, if at all existent. This can be also seen in Figure 7.14 where the streamwise development of maximum Reynolds shear stress is shown. In the figure, the experimental data are only crude estimates inferred from the graphs of u v ; an accurate determination of measured u v m is not possible for most downstream positions
7.2
Boundary Layer with Adverse Pressure Gradient (Case BS0)
115
because of sparse data. However, the qualitative development of u v m is assumed to be correctly captured by the graph and it may serve for comparing the computational results with the experiment. The Baldwin-Lomax model computes streamwise development of u v m that is in very poor qualitative agreement with the experimental estimate (Figure 7.14). In particular, the total maximum of u v is obtained too far upstream at x/D = 0.0907143. Downstream of this point, u v decreases which indicates that the Baldwin-Lomax model instantaneously responds to the change of pressure gradient. This behavior is not seen in the results obtained with the other models; they yield total maxima of u v further downstream at x/D = 1.63286, as it is encountered in the measurements. These models are based on transport equations which do, to some extend, account for flow-history effects. Flow-history effects are believed to be the reason why in this flow case Reynolds stresses do not immediately respond to changes in the pressure gradient. Yet, even transport-equation models do not sufficiently account for flow-history effects. This leads to the general underprediction of Reynolds shear stress at the last downstream position considered (Figure C.12).
2
m /0.001Ur
2
m /0.001Ur
-1
-1.2
-1.4
-1.6
-1.8
-2
-2.2
-2.4
-1
-1.2
-1.4
-1.6
-1.8
-2
-2.2
-1
-1
x/D
x/D
1
Experiment Spalart-Allmaras Edwards-Chandra SALSA
1
-1
-1.2
-1.4
-1.6
-1.8
-2
-2.2
-2.4
-1
-1.2
-1.4
-1.6
-1.8
-2
-2.2
-2.4
-1
-1
x/D
0
x/D
Experiment kω SST MBC kω SST modified
0
Experiment kω TNT RBC Wallin RBC kω LLR RBC
Figure 7.14: Streamwise development of u v m for case BS0.
0
0
Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC
2
m /0.001Ur 2
m /0.001Ur
-2.4
1
1
116 7 Test Cases Selected
7.2
117
Boundary Layer with Adverse Pressure Gradient (Case BS0)
Summary The performance of each turbulence model for computing case BS0 is summarized in Table 7.7. In the Table, models’ performances are rated according to the predictive accuracy of computational results in comparison with corresponding experimental data. Each flow variable discussed above is separately considered.
Table 7.6: Symbols used to rate turbulence-model performance Symbol
Meaning
+++
very good agreement with measurements
++ +
good agreement with measurements fair agreement with measurements
modest agreement with measurements
−
weak agreement with measurements
−− −−−
poor agreement with measurements very poor agreement with measurements
Table 7.7: Turbulence-model performance for case BS0 Model k, ω 98 WBC k, ω SST modified Edwards-Chandra k, ω SST MBC Spalart-Allmaras
−u v m
"
cp
cf
u
2(+ + +)
2(+ + +)
+++
+
16+
2(++)
2(++)
++
+
11+
2(++)
2(+)
+++
++
11+
2(+ + +)
2(+)
+
+
10+
2(++)
2(+)
++
++
10+
SALSA
2(+ + +)
2(+)
++
10+
Wallin RBC
2(+ + +)
2(−)
++
+
7+
k, ω TNT RBC
2(+)
−
+
2+
k, ω LLR RBC
2(++)
2(− − −)
+++
+
2+
2(+)
2(−−)
−−
−−
6−
2(−−)
−−−
−−−
10−
k, ω 88 WBC Baldwin-Lomax
From an engineering point of view, cp and cf are frequently more important than u or u v . Therefore, ratings for cp and cf are weighted by a factor of two, as indicated in the Table 7.7. The last column in this table contains
118
7 Test Cases Selected
the sum of ratings given for each model while symbols used for the rating procedure are defined in Table 7.6. Of course, this rating system is somewhat subjective rather than purely quantitive. However, it is meant to give qualitative conclusions about the models’ performances and to enable the reader to quickly obtain an overview about the various models. This approach indicates that Wilcox’s k, ω 1998 model gives the best overall results for the BS0 flow case. Moreover, the model yields good or very good agreement with measurements for all flow variables considered (Table 7.7).
7.3
Boundary Layer with Pressure-Induced Separation (Case CS0)
To investigate the performance of turbulence models for predicting flow separation, a separated boundary-layer flow was computed. For this purpose, the cylinder flow of Driver (1991) was again considered. In the experiment, Driver used the same experimental setup as for case BS0 without separation (Section 7.2). However, the suction level at the wind-tunnel walls was increased to increase the adverse pressure gradient at the cylinder surface and to force boundary-layer separation. Consequently, for computing this flow case, the same computational setup was utilized as for case BS0; the upper boundary was again modeled by an inviscid streamline inferred from measured velocity profiles. Details about the computational setup and grid employed can be found in Section 7.2 and will not be repeated here. The only difference compared to the procedure presented in Section 7.2 is that the inviscid streamline was obtained from experimental data for case CS0. The resulting (simple) topology of the separated flow field is schematically shown in Figure 7.15. It belongs to sub-class 1.1 a) in Table 1.1, i.e. statistically steady, two-dimensional separated flows with a single recirculation zone. b1 S1
F1
A1
cylinder surface
Figure 7.15: Flow topology of separated flow field along Driver’s cylinder (case CS0; symbols as in Table 1.1).
7.3
Boundary Layer with Pressure-Induced Separation (Case CS0)
7.3.1
119
Computational Results and Discussion
Flow Topology and Velocity Profiles The computed and measured topologies of the dividing streamlines are shown in Figure 7.16. S1exp and A1exp denote the separation and reattachment points, respectively, inferred from the measured streamfunction. Note that in the figures different scales are used for the x and y axes for better recognizability; the recirculation zone is much thinner in reality than suggested by the figures. Large differences among the computational results for the flow topology of the recirculation zone are encountered. The Baldwin-Lomax, the k, ω 1988 and the k, ω TNT model predict recirculation zones that are much thinner than the one inferred from experimental data. In fact, the k, ω 1988 model yields such a thin separation zone that the corresponding dividing streamline is not visible in Figure 7.16a). All models but the k, ω 1988 model have in common that they yield recirculation zones that extend further in the streamwise direction than in the experiment, i.e the distance between S1 and A1 is larger in the computational results than in the measurements. This is also seen from the skin-friction distributions discussed below (see also Figure 7.18). For the k, ω 1998 model, predictions are in fair overall agreement with the experimental flow topology. The SALSA model predicts a much larger recirculation zone compared to the experiment and to results obtained with other one-equation models. Modifications applied to the k, ω SST model improve streamline predictions in the upstream half of the recirculation zone while at the downstream end the modifications yield only slight improvements. The velocity profiles of the separated boundary-layer flow (Figures C.13 to C.22) are placed in the Appendix. Analogously to the procedure applied in Subsection 7.2 for the attached boundary layer, the profiles were evaluated at different downstream stations. x/D positions of the stations are indicated by arrows in Figures 7.17a) and 7.18a). At the first station, that is at x/D = −1.08857, all models yield velocity profiles that are in close agreement with measurements. Only the BaldwinLomax model shows a tendency to compute a velocity profile that is fuller in shape than the measured one. This tendency is amplified with increasing downstream distance. To compensate high mass flux due to fuller profiles, the Baldwin-Lomax model predicts a lower velocity level in the outer region of the boundary layer than in the experiment. All other models are able to predict velocity profiles in reasonable agreement with measurements. Differences between computed and measured data are seen mainly in the inner region of
120
7 Test Cases Selected
0.7
Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC
y/D
0.65 0.6 0.55 0.5 0.45
S1exp 0
0.7
y/D
1
2
Experiment kω TNT RBC Wallin RBC kω LLR RBC
0.6
b)
0.55 0.5 0.45
S1exp 0
0.7
A1exp
x/D
1
2
Experiment Spalart-Allmaras Edwards-Chandra SALSA
0.65
y/D
A1exp
x/D
0.65
0.6
c)
0.55 0.5 0.45
S1exp 0
A1exp
x/D
0.7
1
2
Experiment kω SST MBC kω SST modified
0.65
y/D
a)
0.6
d)
0.55 0.5 0.45
S1exp 0
A1exp
x/D
1
2
Figure 7.16: Topology of recirculation zones for case CS0; dividing streamlines are shown.
the boundary layer. Variances among computed results are, however, larger than for the attached case (Subsection 7.2) and they are more pronounced at stations located further downstream. The modified SST model yields best overall agreement with experimental velocity profiles.
7.3
Boundary Layer with Pressure-Induced Separation (Case CS0)
121
Pressure Distribution Up to x/D ≈ −0.4, all turbulence models yield surface-pressure distributions which are in close agreement with measurements (Figure 7.17). Downstream of x ≈ 0, however, large variances are found between pressure distributions obtained with different models: While most models overpredict surface pressure in the region −0.4 ≤ x/D ≤ 3.0, the model of Wallin virtually duplicates experimental results. The BaldwinLomax model yields worst agreement of predicted and measured surface pressure; all other model predictions are in between these extrema. Skin-Friction Distribution Corresponding skin-friction distributions are shown in Figure 7.18. Boundary-layer separation S1 and reattachment A1 are defined at points where cf = 0. (In the experiment, due to the high accuracy of the experimental data of Driver, the positions of S1 and A1 inferred from the streamfunction and the ones obtained from the cf distribution were found to be almost identical.) From inspection of Figure 7.18, large variances among computational results are found. Note, however, that the predictions of the one-equation models are closer to each other than those of the k, ω type of models. Wilcox’s 1988 k, ω model is the only turbulence model that has cf = 0 in close agreement with the experimental separation point. Furthermore, the model yields a skin-friction distribution that is close to measurements up to separation. However, it computes a tiny separation zone which is barely existent. Both Wilcox models predict skin-friction distributions that show a small “kink” in the vicinity of cf = 0. This behavior is a result of the applied wall treatment for ω: Wilcox’s procedure (Equation 6.5) yields ωw = 0 for vanishing uτ and this leads to the small kink around cf = 0. While the SALSA model yields cp that is clearly below the predictions obtained with the other one-equation models, it predicts cf that is very close to the predictions obtained with the latter models (Figures 7.17 and 7.18). The modified k, ω SST model yields the best overall agreement of computational results with measured skin-friction data. From looking at the pressure and skin-friction results, the same conclusion is drawn as in Subsection 7.2: An accurate prediction of surface pressure does not necessarily mean that skin friction is predicted with comparable accuracy.
0.4
0.4
0.6
0
0.2
cp
0
0.2
cp
c)
a)
-1
-1
x/D
S1exp
x/D
1
1
2
Experiment Spalart-Allmaras Edwards-Chandra SALSA
A1exp
2
Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC
3
3
0.4
0.4
0.6
0
0.2
0
0.2
d)
b)
-1
-1
0
0
x/D
S1exp
x/D
S1exp
1
1
Figure 7.17: Pressure distributions at cylinder surface for case CS0 (arrows show positions of measured and computed boundary-layer profiles).
0
0
S1exp
A1exp
0.6
cp cp
0.6
2
Experiment kω SST MBC kω SST modified
A1exp
2
Experiment kω TNT RBC Wallin RBC kω LLR RBC
A1exp
3
3
122 7 Test Cases Selected
cf
cf
0
0.001
0.002
0.003
0
0.001
0.002
0.003
c)
a)
0
0
x/D
S1exp
2
1
A1exp
2
Experiment Spalart-Allmaras Edwards-Chandra SALSA
x/D
1
A1exp
3
3
0
0.001
0.002
0.003
0
0.001
0.002
0.003
-1
d)
-1
b)
0
0
x/D
S1exp
1
A1exp
2
1
A1exp
2
Experiment kω SST MBC kω SST modified
x/D
S1exp
Experiment kω TNT RBC Wallin RBC kω LLR RBC
Figure 7.18: Skin-friction distributions at cylinder surface for case CS0 (arrows show positions of measured and computed boundary-layer profiles).
-1
-1
S1exp
Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC
cf cf
3
3
7.3 Boundary Layer with Pressure-Induced Separation (Case CS0) 123
124
7 Test Cases Selected
Reynolds-Shear-Stress Profiles As was done for case BS0, profiles of computed and measured Reynolds shear stresses are compared at different downstream stations (Figures C.23 to C.32); the Reynolds stresses were evaluated at the same downstream positions as the velocity profiles discussed above. Upstream of reattachment A1 in the experiment (x/D < 1.63286) and as for case BS0, the Baldwin-Lomax model and the k, ω 1988 model of Wilcox yield Reynolds shear stress that is too high compared to measurements. However, in contrast to results obtained for case BS0, also Wilcox’s 1998 model, the k, ω TNT model, and all one-equation models compute too high levels of Reynolds stress. This overprediction is most pronounced just upstream of separation S1 in the experiment, that is around x/D = 0.181429. Differences in the results obtained with one-equation models are again generally less pronounced than differences among results obtained with k, ω models. The modified SST model and, to less extend, the k, ω LLR model yield especially good agreement between measured and computed Reynolds shear stress. For x/D ≥ 1.63286, i.e. downstream of reattachment A1 in the experiment, the models tend to underpredict the Reynolds stress level compared to measured values. Regarding streamwise development of maximum Reynolds shear stress, the same conclusions are drawn as for the attached boundary-layer flow discussed in Subsection 7.2: The Baldwin-Lomax model predicts the maximum of u v m too far upstream, thus responding too quickly to the decreasing pressure gradient compared to experimental data; the other models yield results in better qualitative agreement with the experiment and they predict the maximum of u v m further downstream than the Baldwin-Lomax model (Figure 7.19). This is attributed to the fact that all models but the BaldwinLomax model are based on transport equations and are therefore able to qualitatively account for flow history effects. However, even the transportequation models predict the maximum of u v m too far upstream compared with experimental results. Downstream of its maximum, all models yield a decrease of u v m in downstream direction. This is in contrast to the experiment where an increase in u v m up to the last measurement station is found. Hence, downstream of reattachment A1 in the experiment (x/D ≥ 1.63286) all model predictions are in poor agreement with the measured u v m . Up to x/D = 1.08857, the modified k, ω SST model yields the best agreement of streamwise development of u v m with the experimental results.
7.3
125
Boundary Layer with Pressure-Induced Separation (Case CS0)
The non-linear model of Wallin is able to correctly predict anisotropy of the Reynolds stress tensor in the separation region (Figures C.33). However, model predictions for u u and v v are below experimental results. Underprediction of u u is especially pronounced near the wall while underprediction of v v is larger in the outer region than near the wall; predictions for v v are in better overall agreement with the experiment than for u u . Note that in the flow case considered, u u and v v contribute only marginally to the momentum equations and a correct prediction is not important to get this flow “right”. This statement is true for all flow cases considered in this work and, therefore, u u and v v are not included in the discussions of results. Summary To gain a qualitative overview of turbulence-model performance for case CS0, the computational results are rated against the experimental results. Table 7.8 reports ratings in the same manner as for case BS0 shown in Subsection 7.2. In addition, the last column contains the sum of the ratings for both cases BS0 and CS0. Note that for the ratings regarding predictions for u not only the velocity profiles but also the predictions of the flow topology are considered. For case CS0, the best overall results yield the modified k, ω SST model and the model of Wallin. Table 7.8: Turbulence-model performance for case CS0 Model k, ω SST modified Wallin RBC k, ω SST MBC k, ω 98 WBC
cp
cf
u
−u v m
"
BS0 + CS0
+++
+
4+
15+
2(+ + +)
2(−−)
+
3+
10+
2(++)
2(−−)
+
+
2+
12+
2(+)
2(−−)
++
−
1−
15+
k, ω LLR RBC
2(++)
2(− − −)
2−
SALSA
2(++)
2(− − −)
−
3−
7+
Spalart-Allmaras
2(−−)
+
−
4−
6+
Edwards-Chandra
2(−−)
+
−
4−
7+
2(−−)
−−
−−
8−
14−
k, ω 88 WBC k, ω TNT RBC
2(−−)
2(−)
−−
−
9−
7−
Baldwin-Lomax
2(− − −)
2(− − −)
−−−
−−−
18−
28−
2
m /0.001Ur
2
m /0.001Ur
-1.5
-2
-2.5
-3
-3.5
-4
-4.5
-5
-1.5
-2
-2.5
-3
-3.5
-4
-4.5
-1
c)
-1
a)
-0.5
-0.5
0.5
x/D
x/D
0.5
1
1
1.5
A1exp
1.5
A1exp
2
2
-1.5
-2
-2.5
-3
-3.5
-4
-4.5
-5
-1.5
-2
-2.5
-3
-1
d)
-1
-0.5
-0.5
0.5
x/D
0
0.5
x/D
S1exp
Experiment kω SST MBC kω SST modified
0
S1exp
b)
-3.5
-4
Experiment kω TNT RBC Wallin RBC kω LLR RBC
-4.5
-5
Figure 7.19: Streamwise development of u v m for case CS0.
0
S1exp
Experiment Spalart-Allmaras Edwards-Chandra SALSA
0
S1exp
Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC 2
m /0.001Ur 2
m /0.001Ur
-5
1
1
1.5
A1exp
1.5
A1exp
2
2
126 7 Test Cases Selected
7.4
Separated Airfoil Flow (Case AAA)
7.4
127
Separated Airfoil Flow (Case AAA)
This subsection focuses again on the separated flow around the AerospatialeA airfoil that was discussed in detail in Section 4. Results computed with different turbulence models are now compared; the same computational grid and setup as the one presented in Subsection 4.2 were employed to perform the computations. Although results obtained with the Johnson-King model are presented in Section 4 they are included in this subsection for reference. A similar procedure for presenting computational results is pursued as in Subsection 4.3: Profiles of streamwise velocity and Reynolds shear stress were evaluated at twelve different downstream stations from x/c = 0.3 to x/c = 0.99 (see Figure 4.9) to investigate streamwise development. The graphs are placed, however, in Appendix C.3 so as not to overload this section with a large number of figures. 7.4.1
Computational Results and Discussion
Flow Topology and Velocity Profiles Evaluating streamlines at the trailing edge reveals that none of the turbulence models is able to predict a recirculation zone that is of comparable size with the one found in the experiment: While the Johnson-King model, the k, ω SST model and the SALSA model yield the largest separation zones, recirculation zones predicted by the models are generally smaller than the measured one (Figure 7.20). (Note that S1exp and A2exp shown in Figure 7.20 were evaluated from the measured streamfunction while in Figures 7.21, 7.22 and 7.23, S1exp relates to the position of zero skin friction. A2exp could not be evaluated with the latter method; see Subsection 4.3.2.) Regarding prediction of streamwise velocity, a consistent trend is encountered for all turbulence models: All models predict too high values of u in the boundary layer resulting in fuller velocity profiles compared with measurements (Figures C.34 – C.45). Overall predictive accuracy is poor and differences between measured and computed velocity profiles increase when approaching the trailing edge of the airfoil. The models do not sufficiently respond to decreasing pressure gradient. This finding is in contrast to the results for the separated boundary layer (case CS0) discussed in Subsection 7.3. There, it was found that models responded too quickly to changes in pressure gradient compared to measurements. Regarding the airfoil flow, the BaldwinLomax model yields velocity profiles in poorest agreement with experimental results while the Johnson-King results are closest to measurements.
128
7 Test Cases Selected
y/c
0.02 0
S1exp Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC
-0.02 -0.04 0.85
0.9
x/c
A2exp 0.95
a) 1
y/c
0.02 0
S1exp Experiment kω TNT RBC Wallin RBC kω LLR RBC
-0.02 -0.04 0.85
0.9
x/c
A2exp 0.95
b) 1
y/c
0.02 0
S1exp Experiment Spalart-Allmaras Edwards-Chandra SALSA
-0.02 -0.04 0.85
0.9
x/c
A2exp
0.95
c) 1
y/c
0.02 0
S1exp Experiment kω SST MBC kω SST modified Johnson-King
-0.02 -0.04 0.85
0.9
x/c
0.95
A2exp
d) 1
Figure 7.20: Topology of separation zones for airfoil A; separating streamlines are shown.
7.4
Separated Airfoil Flow (Case AAA)
129
Pressure and Skin-Friction Distribution For all models, computed cp distributions are above measured values at the rear part of the airfoil, that is for x/c > 0.8 (Figure 7.21). (Note that in the figure the direction of the cp axis is reversed and graphs of larger cp values appear below graphs of lower cp values. This way of presenting cp distributions is common practice for airfoil flows and is therefore applied in this work.) The plateau in the experimental pressure distribution encountered at the trailing edge on the upper surface of the airfoil is characteristic for separated flow regions; none of the turbulence models is able to reproduce this pressure plateau. Differences between computed surface-pressure distributions are visible mainly in the region of the suction peak on the upper surface: While results obtained with the Johnson-King model are in close agreement with measurements, all other models overpredict the suction peak compared to experimental values. The Baldwin-Lomax model yields the strongest suction peak and is therefore in poorest agreement with experimental results. This overprediction of suction peak correlates with predicted values for lift coefficient: The model of Baldwin & Lomax yields the highest value for cL while the Johnson-King model predicts the lowest lift (Table 7.9); the latter model yields also the lowest suction peak. All models have in common that they overpredict cL compared to F2 measurements. Figure 7.22 compares skin-friction distributions on the upper surface of the airfoil for the region where experimental values are available (0.3 ≤ x/c ≤ 0.99). One can see that all models tend to predict cf = 0 further downstream than found in the experiment. However, for the SALSA model, the oneequation model of Edwards & Chandra, and Menter’s SST model, the x/c position where cf = 0 is in very close agreement with measurements; the models of Wilcox and Wallin do not predict separation at all. While both Menter’s SST model and the Johnson-King model yield similar separation zones, they yield clearly different skin-friction results.
cp
cp
0.5
x/c
0.75
S1exp
Experiment Spalart-Allmaras Edwards-Chandra SALSA
0.75
1
1
-1
-2
-3
-4
-5
1
0
0
d)
b)
0.25
0.25
Figure 7.21: Pressure distributions for airfoil A.
1
0.25
x/c
0.5
1
c)
0.25
0
-1
-2
-3
-4
0
0
0
S1exp
Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC
-5
0
-1
-2
-3
-4
-5
1
0
-1
-2
-3
-4
a)
cp cp
-5
0.5
x/c
0.5
x/c
S1exp
0.75
S1exp
Experiment kω SST MBC kω SST modified Johnson-King
0.75
Experiment kω TNT RBC Wallin RBC kω LLR RBC
1
1
130 7 Test Cases Selected
0
0.002
0.004
0.006
0.5
x/c
0.6
x/c
0.6
0.7
0.7
0.9
0.8
0.002
0.004
0.006
-0.002
0.3
0.3
d)
b)
0.4
0.4
0.5
0.5
Figure 7.22: Skin-friction distributions for airfoil A.
0.9
S1exp
Experiment Spalart-Allmaras Edwards-Chandra SALSA
0.8
-0.002 0.4
0.5
0
0.002
-0.002
c)
0.4
S1exp
0.004
0.006
0
0.3
0.3
a)
Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC
0
0.002
0.004
0.006
-0.002
cf
cf
cf cf
x/c
0.6
x/c
0.6
0.7
0.7
0.9
0.8
0.9
S1exp
Experiment kω SST MBC kω SST modified Johnson-King
0.8
S1exp
Experiment kω TNT RBC Wallin RBC kω LLR RBC
7.4 Separated Airfoil Flow (Case AAA) 131
132
7 Test Cases Selected
Global Aerodynamic Coefficients Computed and measured aerodynamic coefficients are compared in Table 7.9. Because of separation, total drag consists mainly of pressure drag. Hence, the size of the recirculation zone is qualitatively reflected in the drag coefficient: Turbulence models that yield larger recirculation zones yield also larger values for cD (Figure 7.20 and Table 7.9). For example, the Johnson-King model computes the largest separation zone and the largest drag coefficient among all model predictions. On the contrary, the Baldwin-Lomax yields a tiny recirculation zone that is not recognizable in Figure 7.20 and, at the same time, it predicts the lowest drag of all turbulence models (Table 7.9). All models predict significantly lower drag coefficients than found in the experiment F2. This is consistent with the fact that all models yield recirculation zones that are smaller in size than the measured one. Besides influencing drag, trailing-edge separation also influences the lift coefficient: Large separation regions reduce the circulation around the airfoil, and therefore lift, compared to attached flow. Consistently, turbulence models, which yield large separation zones, predict low values of lift, the Johnson-King model being the most prominent example (Figure 7.20 and Table 7.9). In addition, circulation around the airfoil influences the pressure distribution which, in turn, has a direct impact on the separation itself. This complex viscous/inviscid interaction between boundary-layer separation and inviscid flow makes separated airfoil flow a demanding test case for turbulence models. Yet, thorough identification of reasons for model failure are difficult to perform due to the strong coupling between flow phenomena. The coupling between boundary layer separation and pressure distribution also effects the computational results for the moment coefficient cM : A positive value for cM is predicted by the Johnson-King model and the two k, ω SST models (Table 7.9). These models yield the best agreement of cp with the measurements at the suction peak and the trailing edge, see Figure 7.21. Additionally, they yield also the largest separation regions (Figure 7.20). In contrast, negative values for cM are predicted by all other models. They compute cp in relatively poorer agreement with the experiment and yield a smaller recirculation zone. Due to the lack of experimental data for cM , no general statement about the computational results can be made. Summarizing, one can say that the turbulence models investigated yield too high a value of cL , show no characteristic trailing-edge plateau in the cp
7.4
133
Separated Airfoil Flow (Case AAA)
Table 7.9: Force and moment coefficients for airfoil A cL
cD
cM
total
total
pressure
friction
total
Experiment (F1)
1.56
0.021
–
–
–
Experiment (F2)
1.52
0.031
–
–
–
Baldwin-Lomax
1.725
0.0158
0.0097
0.0061
−0.0165
Johnson-King
1.531
0.0260
0.0197
0.0065
+0.0055
k, ω 88 WBC
1.647
0.0175
0.0108
0.0067
−0.0044
k, ω 98 WBC
1.630
0.0176
0.0112
0.0065
−0.0022
k, ω SST MBC
1.568
0.0181
0.0127
0.0054
+0.0052
k, ω TNT RBC
1.627
0.0166
0.0108
0.0058
−0.0014
k, ω LLR RBC
1.650
0.0168
0.0102
0.0066
−0.0045
Wallin RBC
1.657
0.0153
0.0099
0.0054
−0.0044
Spalart-Allmaras
1.671
0.0166
0.0108
0.0058
−0.0088
Edwards-Chandra
1.643
0.0166
0.0114
0.0052
−0.0050
SALSA
1.624
0.0173
0.0119
0.0054
−0.0029
k, ω SST modified
1.604
0.0167
0.0103
0.0064
+0.0021
distribution, and predict cf = 0 too far downstream (compared to measurements). Reynolds-Shear-Stress Profiles For all turbulence models, agreement between predicted and measured Reynolds shear stresses is fair up to x/c = 0.6 (see Figures C.46 to C.49). However, clear differences among computed Reynolds-shear-stress profiles are seen in the outer part of the boundary layer. Compare, for example, results obtained with the Wallin and k, ω LLR model in Figure C.46. One can see, that the Wallin model yields significantly lower Reynolds stresses than the LLR model in the outer part of the boundary layer. Downstream of x/c = 0.6 basically three things are encountered (see Figures C.50 to C.57): 1. All turbulence models predict less Reynolds stress than measured. This can also be seen in Figure 7.23 where streamwise development of maximum Reynolds shear stress is shown. The model of Wallin yields the lowest values of −u v m while the Baldwin-Lomax model and Menter’s SST Model predict the highest values.
134
7 Test Cases Selected
2. In the experimental results downstream of x/c = 0.7, a second local maximum in the −u v profiles is found in the outer part of the boundary layer. This kink is not reproduced by any turbulence model, not even qualitatively, and it may be an additional hint for three-dimensional effects in the flow. 3. In the experimental Reynolds-stress profiles, the global maximum of −u v is moving further away from the wall with increasing downstream position. The SST models and the Johnson-King model are the only turbulence models that are able to predict this effect in fair agreement with measurements. All other models underpredict outward movement of −u v m with streamwise position. As a final note, turbulence models that make use of Bradshaw’s assumption (−u v /k = constant) yield Reynolds-shear-stress profiles in better agreement with measured profiles than models which do not build on this assumption. However, this holds only for −u v ; for example, the one-equation models yield better agreement of cf with the measurements than the SST models or the Johnson-King model.
2
2
0.002 0.3
0.003
0.004
0.005
0.006
0.007
0.002 0.3
0.003
0.004
0.005
0.006
0.007
0.4
c)
0.4
a)
x/c
0.7
0.8
0.9
0.6
x/c
0.7
0.8
0.9
Experiment S1exp Spalart-Allmaras Edwards-Chandra SALSA
0.6
S1exp
1
1
0.002 0.3
0.003
0.004
0.005
0.006
0.007
0.002 0.3
0.003
0.004
0.005
0.006
0.007
0.4
d)
0.4
b)
0.5
0.5
x/c
0.7
0.8
0.6
x/c
0.7
0.8
Experiment kω SST MBC kω SST modified Johnson-King
0.6
Experiment kω TNT RBC Wallin RBC kω LLR RBC
Figure 7.23: Streamwise development of −u v m for airfoil A.
0.5
0.5
Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC 2 2
-m /U∞
-m /U∞
-m /U∞
-m /U∞
0.9
S1exp
0.9
S1exp
1
1
7.4 Separated Airfoil Flow (Case AAA) 135
136
7 Test Cases Selected
Summary of Model Performance A summary of predictive accuracy of turbulence models is given in the same manner as introduced in Subsections 7.2 and 7.3. Corresponding ratings for the flow over airfoil A are listed in Table 7.10. According to the table, the Johnson-King model offers the highest predictive accuracy for this flow case followed by Menter’s original SST model. Generally, the performance of the models for the present flow case is poor compared with the performance achieved for the flow cases discussed above (see Tables 7.7 and 7.8).
Table 7.10: Turbulence-model performance for airfoil A Model Johnson-King k, ω SST MBC
cp
cf
u
−u v m
"
2(++)
−
+
4+
2(+)
−
++
3+
2(−)
2(+ + +)
−−
−−
Edwards-Chandra
2(−−)
2(++)
−−−
−−
5−
Spalart-Allmaras
2(−−)
2(+)
−−−
−−
7−
2(−)
2(−)
−−
−
7−
SALSA
k, ω SST modified k, ω TNT RBC
2(−−)
−−
−−
8−
Baldwin-Lomax
2(− − −)
2(−−)
−−−
+
12−
k, ω 98 WBC
2(−−)
2(−−)
−−−
−−−
14−
Wallin RBC
2(−−)
2(−−)
−−−
−−−
14−
k, ω 88 WBC
2(−−)
2(− − −)
−−−
−−−
16−
k, ω LLR RBC
2(−−)
2(− − −)
−−−
−−−
16−
8
Numerical Issues
8.1
Grid Convergence
Most of this work was concerned with the predictive accuracy of the turbulence models employed. This section, however, deals with an equally important issue, namely computational accuracy. Assessment of model accuracy is subject to validation while determination of computational accuracy is known as verification. (For an in-depth discussion about verification and validation the reader is referred to Roache (1998).) Note that rigorous grid-dependence analysis is mandatory for any computational study; computational results obtained without a thorough investigation of grid dependence cannot be trusted. Grid-convergence studies presented in the following rely on generalized Richardson extrapolation. The latter was used to obtain a grid-convergence index (GCI) which relates results from any grid-convergence test to the expected results from grid doubling using a second-order accurate method. This yields an “objective asymptotic approach to quantification of uncertainty of grid convergence” (Roache, 1998). In addition, by performing computations on three different grids the observed order of accuracy was determined. The following relations were applied: Observed Order of Accuracy: f3 − f2 p = ln / ln(2), (8.1) f2 − f1 Grid-Convergence Index: GCI [grid level 1] = 1.25
GCI [grid level 2] = 1.25
2 | | f1f−f 1
2p − 1
3 | | f2f−f 2
2p − 1
,
(8.2)
.
(8.3)
p denotes the observed order of accuracy; f1 , f2 and f3 are the solutions on the fine, intermediate and coarse grid, respectively. Any flow variable can be utilized for this purpose but in the present work skin-friction coefficients or related variables are utilized since these are a good indicator for accuracy. The grid refinement ratio between grids employed for the convergence tests was always two. This means that grid level one had twice the resolution of grid level two. Hence, grid level two was obtained by omitting every other grid
138
8 Numerical Issues
point from level one. Correspondingly, level three was obtained by omitting every other grid point from level two. Using GCI values obtained with Equations 8.2 and 8.3 asymptotic range of (grid) convergence is achieved if the following relation holds: GCI [grid level 2] ≈ 2p GCI [grid level 1].
(8.4)
In addition, if Equation 8.4 holds, that means if all solutions obtained are in the asymptotic range, then the GCI of grid level 3 is approximated by GCI [grid level 3] ≈ 2p GCI [grid level 2].
(8.5)
Grid-convergence studies were performed for every flow case discussed in this work to assess computational accuracy of the flow solutions obtained. However, due to the high computational effort necessary for computing flow solutions on fine grids grid convergence was verified in each case with only one turbulence model; transport-equation models were exclusively utilized for this purpose. In some cases, grid-convergence tests with different models were pursued, and qualitatively similar results were obtained when using different models. This indicates that a grid-convergence index obtained from computations with a given turbulence model is also representative for computations on the same grids with other models. Flat Plate To conduct the analysis of predictive accuracy presented in Section 7.1 the computational grid contained 144 × 64 cells. In order to investigate grid convergence, additional solutions were computed on two finer grids consisting of 288 × 128 and 576 × 256 cells. Skin-friction coefficients evaluated at x/L = 0.361511 served to compute the grid-convergence indices. (x/L = 0.361511 is the grid point closest to the position where Reθ = 2900 was found on the standard grid, that is grid level 3.) The results are reported in Table 8.1. Two conclusions follow from the results shown in Table 8.1: a) Asymptotic convergence of results is achieved and b) the GCI for the standard grid is approximately 3.78 percent. This means that the estimated error compared with the exact solution is 3.78 percent including 1.25 as a factor of safety. (This is a rather sloppy definition of GCI and it is used here for simplicity – see Roache (1998) for a more precise description of GCI). While the numerical method applied is formally second-order accurate, the observed order is as low as 0.82 for this flow case. This shows that in
8.1
139
Grid Convergence
Table 8.1: Grid-convergence results for flat plate determined from solutions obtained with Wilcox’s 1988 k, ω model in combination with Rudnik’s surface condition for ω (k, ω 88 RBC) Variable
cf at x/L = 0.361511
p
0.82
GCI [576 × 256]
1.21%
GCI [288 × 128]
2.14%
2p GCI [576 × 256]
2.13%
2p GCI [288 × 128]
3.78%
actual applications of numerical schemes the formal order is not necessarily achieved. This is especially true on extremely stretched or skewed grids. Even larger values of p than the formal order can be observed if explicit artificial damping terms of high order are used in the numerical scheme. This will be shown below.
Driver’s Cylinder Flow Separated flows are much more sensitive to grid refinement or coarsening than attached flows. Therefore, and for reasons of brevity, only grid-convergence results for case CS0 (with separation) are reported for the cylinder flow of Driver. (For case BS0, smaller values of GCIs were obtained and, therefore, better grid convergence was achieved compared to case CS0.) For the cylinder flow, local skin-friction coefficients were the basis for the GCI evaluation. The coefficients were evaluated at x/D = −0.511. On the standard grid, this is the coordinate of the surface grid point located closest to the boundary-layer profiles at −0.544286 (see for example Figure 7.18). The standard grid contained 256 × 64 cells and it was used as level two for the grid-convergence studies. This is also the grid which was used for validation of turbulence models discussed in Subsection 7.3. A coarser grid with 128×32 cells and a finer grid featuring 512 × 128 cells were utilized as level three and one, respectively. From looking at Table 8.2, one can see that the asymptotic range is achieved and that the GCI on the standard grid is 5.32 percent demonstrating that good computational accuracy is obtained on the standard grid. Like
140
8 Numerical Issues
Table 8.2: Grid-convergence results for the separated cylinder flow (case CS0) determined from solutions obtained with Wilcox’s 1998 k, ω model in combination with Wilcox’s surface condition for ω (k, ω 98 WBC) Variable
cf at x/D = −0.54428
p
1.44
GCI [512 × 128]
2.01%
GCI [256 × 64]
5.32%
2p GCI [512 × 128]
5.45%
in the case of the flat plate computations, the observed order of accuracy is lower than the formal order of the numerical scheme. To confirm the consistency of the GCI results shown in Table 8.2, gridconvergence indices for the minimum value of cf and for cD due to skin-friction (both not shown) were evaluated. Very similar results to the ones in Table 8.2 were obtained. Airfoil A For airfoil A, grid convergence for two different variables was investigated, viz. global drag coefficient due to friction, and global lift coefficient. The finest grid level employed consisted of 512 × 128 cells, being also the grid which was used for studying the performance of turbulence models for this flow case. Two coarser grids were obtained by keeping every other and every fourth grid point resulting in 256 × 64 and 144 × 32 cells for level two and three, respectively. Observed orders of convergence and GCIs differ significantly for cD and cL (Table 8.3). At the first glance these larger disparities seem surprising but the following discussion is believed to offer a plausible explanation: Computed skin friction is much more sensitive to artificial damping of the numerical scheme than computed pressure. (The latter contributes by far to the largest part to the lift coefficient.) In the current flow case exclusively fourth-order damping terms are active since no shocks or discontinuities are present. The artificial damping terms are designed to damp so-called Wiggle-modes which are the smallest resolvable wave lengths; influence of damping is therefore bound to the grid spacing. While the effect of damping does not change relative to the grid spacing when refining the grid it is successively reduced
8.2
141
Local Preconditioning for Low Mach Numbers
compared to gradients of flow variables. Therefore, increased grid resolution and decay of damping terms both promote a “sharper” resolution of flow gradients. Since gradients of velocity normal to the wall provide skin friction, decay of artificial damping terms consequently contributes to the accuracy of computed skin friction. This means that the effect of refining the grid from level two to level one is more pronounced for skin friction than for pressure; (f3 −f2 )/(f2 −f1 ) is much smaller for cD than for cL (Equation 8.1). However, good computational accuracy as well as asymptotic range of the solution is achieved on the grid with 512 × 128 cells for airfoil A for both quantities (Table 8.3). Table 8.3: Grid-convergence results for airfoil A determined from solutions obtained with the SALSA model
8.2
Variable
cD due to friction
cL
p
0.367
4.29
GCI [512 × 128]
5.98%
0.016%
GCI [256 × 64]
7.82%
0.309%
2p GCI [512 × 128]
7.71%
0.308%
Local Preconditioning for Low Mach Numbers
All computations for this work were performed with numerical methods designed for the solution of the compressible Navier-Stokes equations. In the low-subsonic Mach-number regime, however, such methods generally do not perform well. When the magnitude of flow velocity becomes small compared to speed of sound, convective terms render the flow equations stiff. This leads to numerical difficulties. The reason is that the numerical scheme has to account for the large disparity between the fluid velocity u and the acoustic wave speed u + a at which pressure signals are transported downstream. In addition, in explicit methods the time step is restricted and set proportional to 1/(u + a). Hence, in low-Mach-number regions of a flow field, the local time step is mainly controlled through speed of sound. This means that the numerical solution of the time marching scheme is advanced using time steps well-adapted to the propagation speed of acoustic waves. However, waves associated with u are excessively “over-resolved”. For the latter, a much larger time step would be possible.
142
8 Numerical Issues
A solution to this problem is the application of preconditioning for the Navier-Stokes equations, which was implemented into the MUFLO code during this work. In the following, the Euler equations are discussed for simplicity but the same arguments apply to the full Navier-Stokes equations. According to the preconditioning procedure presented in Turkel, Radespiel & Kroll (1997) the Euler equations are considered in two dimensions and in differential form: ∂w ∂w ∂w P −1 +A +B =0. (8.6) ∂t ∂x ∂y Matrices A and B depend on the set of variables w. In the numerical methods considered, w usually contains the conserved variables, i.e. w = (ρ, ρu, ρv, ρE)T . In the framework of preconditioning, sets of variables that yield sparse and (nearly) diagonal forms of matrices A and B are preferred. Analysis of eigenvalues, that is propagation speeds, and construction of an appropriate preconditioning matrix P are significantly eased in this case. The task is to find a matrix P −1 that substantially reduces or removes the disparity in the eigenvalues of the system. However, the choice of P −1 is not unique. Note that P −1 does not alter the solution in the limit of steady state since the time derivative vanishes. It does alter the way in which this limit is reached, though, and this is exactly what is wanted. Clearly, preconditioning can be used only for the solution of steady flow problems or within dual-time-stepping schemes for unsteady flows. The preconditioning method adapted in this work is closely patterned after the work of Turkel et al. (1997). In particular, the inverse of the preconditioning matrix P −1 is chosen to be ⎞ ⎛ βMr2 βM 2 0 0 − a2r δ 2 a ⎟ ⎜ αu ⎜ − αu2 1 0 δ ⎟ ρa ρa2 ⎟ , (8.7) P=⎜ ⎜ − αv 0 1 αv δ ⎟ ⎠ ⎝ ρa2 ρa2 0 0 0 1 and w = (p, u, v, T )T is the set of variables. The parameters α and δ are used to generalize the preconditioning matrix. For δ = 0 the preconditioner suggested by Turkel (1987) is obtained, while for δ = 1 and α = 0 the preconditioner introduced by Choi & Merkle (1993) is recovered. In the present study, best results were obtained when setting δ = 1 and α = 0. Special attention must be paid to element (1,1) of P when the local Mach 2 number approaches zero. Normally, βMr2 = a2local Mlocal but when the local
8.2
Local Preconditioning for Low Mach Numbers
143
Mach number becomes very small, for example around stagnation points or critical points in general, it is necessary to bound βMr2 away from zero, since βMr2 = 0 leads to a singular preconditioning matrix. This is done by setting 2 βMr2 = max(a2local Mlocal , εa2local M 2 ). The value of ε has to be specified by the user and it was found to be dependent on Reynolds number and grid spacing. For example, for an airfoil flow, a Reynolds number based on chord length of a few millions and a very fine grid may require ε values of six and higher in order to weaken the effect of preconditioning and to stabilize the computation. This reflects the fact that although local preconditioning provides advantages like accuracy improvements and very often substantial convergence speed-up, it leads to reduced robustness. The reader may be referred to Lee (1996) for a detailed theoretical discussion of preconditioning and its implications. To study the effect of preconditioning on the accuracy of the numerical solution, laminar flow around a NACA 0012 airfoil at zero incidence and Re = 5000 was used as a test case. The flow over the airfoil was computed for two different Mach numbers, M = 0.15 and M = 0.05. In both cases the numerical solutions were obtained with and without local preconditioning. Without preconditioning the surface-pressure distribution along the airfoil degenerates with decreasing Mach number. Following the Prandtl-Glauert √ rule (cp = cp,0 / 1 − M 2 ), the curve of the pressure distribution for M = 0.15 should be slightly above the one for M = 0.05; this is obviously not the case (Figure 8.1). Lowering the Mach number also increases pressure oscillations near the trailing edge, for which compressible flow solvers are known in general. These spurious oscillations are due to artificial damping terms which are scaled with the “acoustic” eigenvalues u + a and v + a. 2 With preconditioning (and setting βMr2 = max(a2local Mlocal , 0.05a2local M 2 )) the curves for cp differ only by the Prandtl-Glauert factor (Figure 8.2). In addition, pressure oscillations near the trailing edge are absent due to correct scaling of artificial damping terms. The skin-friction distributions along the airfoil computed without preconditioning are presented in Figure 8.3. Results discussed by Schlichting (1982) indicate that, in general, cf should decrease for increasing Mach numbers if an adiabatic wall is assumed. This trend is reproduced qualitatively correct by the numerical results (Figure 8.3). However, for the very low Mach numbers considered in this study, cf depends only marginally on M and the curves for cf should be indistinguishable, which is obviously not correctly reproduced. In contrast, and according to theory, the curves obtained with preconditioning lie on top of each other and the small unphysical increase in
144
8 Numerical Issues
-0.5 M = 0.05 M = 0.15
-0.4 -0.3
cp
-0.2 -0.1 0 0.1 0.2 0
0.25
0.5
0.75
x/c Figure 8.1: Pressure distributions for NACA 0012, Re = 5000, α = 0◦ , laminar flow, without preconditioning.
cf at the trailing edge (Figure 8.3) is not present in the solution obtained with preconditioning (Figure 8.4).
-0.5 M = 0.05 M = 0.15
-0.4 -0.3
cp
-0.2 -0.1 0 0.1 0.2 0.3
0
0.25
0.5
0.75
1
x/c Figure 8.2: Pressure distributions for NACA 0012, Re = 5000, α = 0◦ , laminar flow, with preconditioning.
8.2
145
Local Preconditioning for Low Mach Numbers
Note that results presented in this section are not confined to the present numerical methods with central differencing in combination with artificial damping terms. Numerical methods based on upwind schemes show similar difficulties in obtaining accurate solutions when the Mach number becomes small. 0.15 0.125
M = 0.05 M = 0.15
0.1
cf
0.075 0.05 0.025 0 -0.025 -0.05
0
0.25
0.5
0.75
1
x/c Figure 8.3: Skin-friction distributions for NACA 0012, Re = 5000, α = 0◦ , laminar flow, without preconditioning. For the flow over airfoil A computed with the Johnson-King model, viz. with the MUFLO code, ε had to be set to four in order to stabilize the computation. This explains the oscillations in cf obtained with the JohnsonKing model at the trailing edge (see Figure 7.22 d)): In this region, due to the relatively high value of ε, the effect of preconditioning was not strong enough to completely remove the numerical artifacts. For this reason, preconditioning in FLOWer was modified and locally different values of ε were employed. This procedure yielded nearly optimal preconditioning at the trailing edge of the airfoil. In order to completely resolve numerical difficulties associated with low Mach numbers, application of an alternative preconditioner might be considered, but this was out of scope of the present work. The rise of the friction coefficient at the trailing edge, like that shown in Figure 8.3, is amplified if metric terms of the computational grid are not smooth in the streamwise direction across the trailing edge. Therefore, it is mandatory to use computational grids with a smooth distribution of metric
146
8 Numerical Issues
0.15 0.125
M = 0.05 M = 0.15
0.1
cf
0.075 0.05 0.025 0 -0.025 -0.05
0
0.25
0.5
0.75
1
x/c Figure 8.4: Skin-friction distributions for NACA 0012, Re = 5000, α = 0◦ , laminar flow, with preconditioning.
terms everywhere in the domain. Many grid generators, however, produce grids with smooth metrics along the airfoil and along the wake but not also across the trailing edge.
8.3
Transition
Transition in the RANS computations was achieved by “activating” the turbulence models downstream of a prescribed plane in the flow field. In the case of algebraic models, this was performed by simply setting µt = 0 upstream of the transition location. In the framework of transport-equation models, the production term was explicitly set to a very small value (virtually to zero) in the laminar region. This limiter was released downstream of transition letting production grow to “standard” levels. Figure 8.5 shows iso-contours of k in the transition region computed by Wilcox’s 1988 k, ω model in combination with Wilcox’s wall treatment for ω. One can see that significant k values are obtained only downstream of the first turbulent “i-plane” in the region where the limiter on the production term in the k equation is disabled. Computations of the flow around the Aerospatiale-A airfoil on very fine grids showed that it is difficult to obtain steady-state solutions on grids with more than 512 × 128 cells. Analysis of computations revealed that unsteadiness is triggered from the transitional separation bubble encountered on the
8.3
147
Transition
0.087
0.0005
y/c
1E-05
1E-05
0.086 first turbulent i-plane 0.085
0.115
0.117
x/c
0.119
0.121
Figure 8.5: Modeling of transition for flow over airfoil A; isocontours of k are shown in the transition region on the upper surface of the airfoil; Wilcox k, ω 1988 model, WBC.
upper surface of the airfoil. On fine grids, the negative surge in skin friction (see for example Figure 4.7) randomly and rapidly moved up- and downstream and also changed in magnitude. On the one hand, this behavior is rooted in the inherent unsteady physics of the bubble, which is resolved to some degree on the fine grid. On the other hand, however, Reynolds averaging should be active from the beginning of the turbulent boundary layer and therefore suppress unsteadiness arising from transition processes. Spalart & Strelets (1997) performed DNS and RANS computations of a transitional bubble. They showed that in the RANS computation, transition occurred at a lower rate than in the DNS. The turbulence model yielded a skin-friction distribution whose recovery to positive values was too slow compared with DNS. Regarding the flow over airfoil A, numerical experiments revealed that setting the prescribed transition location a little bit more upstream of the experimentally-observed transition location improved steadiness of the flow solution. In particular, the location of transition was prescribed such that cf = 0 in the recovery region was obtained at the experimentallyobserved transition location (x/c = 0.12). Good results were obtained using x/c = 0.1167 for the transition location. Thus, delayed transition of the kind reported by Spalart & Strelets is believed to be the reason for the unsteadiness encountered for airfoil A. As mentioned in Subsection 6.1, Wilcox developed low-Reynolds number modifications for his models. Computations of the flow around airfoil A using
148
8 Numerical Issues
Wilcox’s 1988 k, ω model were performed with and without application of lowReynolds-number terms. The main effect of the modifications is that the rate at which the model approaches the fully-turbulent state is slightly increased compared to computations performed without low-Reynolds-number terms. By including viscous corrections in the computation, skin-friction recovery from negative values in the transitional separation bubble to the positive peak just downstream of transition is more rapid (Figure 8.6). This means that delay of transition is somewhat reduced by applying low-Reynolds-number terms. Thus, if transition is important it is recommended to include the low-Reynolds-number modifications in computations performed with Wilcox’s k, ω models. However, Figure 8.6 shows that despite the use of viscous corrections the turbulence model is not able to correctly capture the trailingedge separation.
0.015
Experiment kω 88 RBC kω 88 RBC low Reynolds number
cf
0.01
0.005
0 0.2
0.4
x/c
0.6
0.8
1
Figure 8.6: Influence of low-Reynolds-number modifications on skin-friction; flow over airfoil A computed with Wilcox’s 1988 k, ω model and Rudnik’s surface boundary condition for ω (k, ω 88 RBC). Great sensitivity of the flow solution to transition location in general was found. For example, prescribing transition at x/c = 0.13 instead of
8.4
Artificial Damping in Boundary Layers
149
x/c = 0.1167 yielded in one case of the computations for airfoil A a massively separated flow field with two counter-rotating recirculation zones (Figure 8.8). Therefore, in flow situations where location of transition plays an important role, like separated airfoil flow or dynamic stall to name only two, careful modeling of transition is mandatory. In particular, correctly predicting the increase of skin-friction as well as the change of momentum thickness between the laminar and the turbulent part of the boundary layer is essential. To perform this task a special “transition strip” procedure is necessary where the turbulence model is modified in order to reproduce the high levels of turbulence, and the associated rapid increase in cf and θ just downstream of transition, found in the experiment. For k, ω models, in addition to viscous corrections, this could be accomplished through special treatment of ω at the wall in the transition region. For example, substantially reducing ωw increases k throughout the boundary layer which, in turn, increases the rate at which transition occurs. Transition modeling is a broad field and is out of the scope of this work. However, it is noted and emphasized that transition and how it is modeled can significantly influence computational results even for RANS computations. This cannot be ignored as is often done in commercial CFD codes. (However, many of the latter are intended for internal flows where transition is less important.)
8.4
Artificial Damping in Boundary Layers
It was noted in Section 5 that the numerical method employed for this work uses second- and fourth-order artificial damping terms to prevent odd-even decoupling and to damp spurious oscillations. For subsonic flows, only the damping terms based on fourth-order differences are active. Although these terms are necessary to stabilize the central scheme they can pollute the solution through introduction of too much diffusion. In boundary layers, for example, large gradients of tangential momentum flux normal to the wall lead to high values of damping terms. However, viscous momentum flux normal to the wall is sufficiently high to provide the necessary damping of the numerical scheme in this direction. Therefore, artificial damping in boundary layers in the wall-normal direction can be reduced or even entirely omitted. This was accomplished by pre-multiplying the corresponding artificial damping term by (M/M∞ )2 in the FLOWer code.
150
8 Numerical Issues
To study effects of reducing artificial viscosity in boundary layers, solutions of the flow over airfoil A computed with full and reduced damping were analyzed. In Figure 8.7, corresponding skin-friction distributions on the upper side of the airfoil are shown.
0.015 full artificial damping reduced artificial damping
cf
0.01
0.005 kω 88 RBC model transition at x/c=0.1113 0 0
0.25
x/c
0.5
0.75
Figure 8.7: Influence of explicit artificial damping on skinfriction; flow over airfoil A computed with Wilcox’s 1988 k, ω model and Rudnik’s surface boundary condition for ω (k, ω 88 RBC). In the computation with reduced artificial damping the laminar part of the boundary layer remains attached while in the case with full damping the laminar boundary separates and a transitional bubble is encountered. It is noted that all computations of the flow over airfoil A presented in the sections above show laminar separation despite the fact that reduction of artificial damping was applied in each case. The reason is, that in the current case transition was prescribed further upstream, namely at x/c = 0.1113, than in the computations presented above. This was necessary to yield a steadystate solution for the computation with full damping. (Specifying transition at x/c = 0.1167 led to an unsteady transitional bubble as discussed in Section 8.3.)
8.4
Artificial Damping in Boundary Layers
151
Reducing artificial damping obviously leads to an increased lateral momentum flux towards the wall in the laminar boundary layer. This yields a higher level of cf compared to the computation with full artificial damping (Figure 8.7). Regarding the turbulent boundary layer, effects of reducing artificial damping enter mainly through transition. In the case with full damping a transitional bubble is computed: The bubble gives rise to increased production of turbulence in the model compared with the case where no transitional bubble is encountered. More production of turbulence, in turn, yields a larger cf immediately downstream of transition than without a transitional bubble (Figure 8.7). Hence, reducing the amount of artificial damping increases cf in the laminar part of the boundary layer but reduces cf in the turbulent boundary layer just downstream of transition. Further downstream in the turbulent boundary layer, the differences in cf computed with and without reduced artificial viscosity are small. This means that reducing artificial damping in boundary layers is especially important when laminar and transitional flow is present. Combined effects of artificial viscosity and location of transition were also investigated. In the computation of the flow over airfoil A with full artificial damping, shifting transition from x/c = 0.1113 to x/c = 0.13 yields a massively separated flow with two counter-rotating recirculation zones (Figure 8.8). With reduced artificial damping, however, shift of transition does not lead to such a fundamental change in flow topology: Both transition locations x/c = 0.1167 and x/c = 0.13 yield a single recirculation zone at the trailing edge. Moreover, the rear part of the turbulent boundary layer is barely affected by changing the transition location (Figure 8.9). The sensitivity to transition and artificial damping of flow computations for airfoil A demonstrates that transition modeling and numerical dissipation must be carefully treated since both can have a major impact on computational results. Note, however, that in the fully turbulent flows investigated (cases BS0 and CS0), reducing artificial viscosity in the boundary layer had very little effect on the flow solution.
152
8 Numerical Issues
0.015 transition at x/c=0.1113 transition at x/c=0.13 full artificial damping kω 88 RBC
cf
0.01
0.005
0 0.2
0.4
x/c
0.6
0.8
1
Figure 8.8: Skin-friction distributions for airfoil A: full artificial damping in combination with shift of transition location from x/c = 0.1113 to x/c = 0.13. 0.015 transition at x/c=0.1167 transition at x/c=0.13 0.01
cf
reduced artificial damping kω 88 RBC
0.005
0 0.2
0.4
x/c
0.6
0.8
1
Figure 8.9: Skin-friction distributions for airfoil A: reduced artificial damping in combination with shift of transition location from x/c = 0.1167 to x/c = 0.13.
8.5
153
Boundary-Value Dependences
8.5 8.5.1
Boundary-Value Dependences Dependences on Wall Value of ω
Regarding the k, ω models, it was noted in Subsection 7.1.2 that the various model developers suggest different procedures for specifying ω at the wall. In order to investigate the effect of ωw on the flow solution, computations were performed of the flat-plate boundary-layer flow with a given turbulence model but with different specification procedures for ωw . For this purpose, the wall treatment of ω according to Wilcox (Equation 6.5) and to Rudnik (Equation 6.5) were applied in combination with the 1998 k, ω model of Wilcox. From looking at Table 8.4 one can see that Rudnik’s method yields a much larger wall value for ωw and a lower skin-friction coefficient compared to Wilcox’s method. As implied by the discussion in Section 7.1.2, a decrease in wall shear gives rise to an increased u+ distribution (Figure 8.10).
Table 8.4: Influence of ωw on cf at Reθ = 2900 (case FPBL) ωw /(L p∞ /ρ∞ )
cf × 103
-
3.362
–
k, ω 98 (WBC)
28280
3.6128
+7.5%
k, ω 98 (RBC)
628233
3.3345
−0.8%
Source Experiment (DeGraaff)
∆
It was found by explicitly increasing the wall value of ω that the influence on the skin friction decreases with increasing ωw . However, the order of magnitude of ωw that was necessary to achieve a virtually ωw -independent solution is very sensitive to the grid resolution. On fine grids, ωw had to be much larger to yield a cf that was independent of ωw than on coarse grids. This favors the application of Menter’s procedure for determining ω at the wall since it is grid dependent (Equation 6.6). In addition, variations of ωw had negligible influence on the flow solutions for boundary layers under strong pressure gradients. It was further found that extremely large values of ωw cause numerical difficulties and can prevent iterative convergence. It is therefore recommended to compute a flow solution with, say, Rudnik’s method and then successively increase ωw until the variation in the skin-friction coefficient with ωw is below some predefined limit.
154
8 Numerical Issues
25
Experiment (DeGraaff) kω 98 RBC kω 98 WBC
15
u
+
20
10 5 0 0 10
101
y+
102
103
Figure 8.10: Influence of ωw on the computed velocity profile of the flat-plate boundary layer (WBC stands for Wilcox’s surface boundary condition, RBC denotes Rudnik’s surface boundary condition).
8.5.2
Dependence on Free-Stream Value of ω
It is mentioned in the literature (see, for example, Kok, 2000; Menter, 1992) that results obtained with Wilcox’s k, ω models are sensitive to the value of ω specified in the farfield. In the current study, this sensitivity was noticed only if the farfield boundary was located very close to the body. For example, for the flat-plate computations presented in Subsection 7.1.2 the closest farfield boundary was located one half plate length away from the body. To evaluate the influence of ω∞ two computations were performed with ω∞ = 0.565 and ω∞ = 5650. Note that k∞ was varied correspondingly so that µT in the free stream was not affected. (In both computations, k∞ was very small compared to u2∞ .) Despite the larger difference of ω∞ specified in the two computations, the skin-friction coefficients obtained differ less than 1.5 percent (Table 8.5). For airfoil A, the farfield boundary was located 18 chords away from the body; in one computation ω∞ = 1.331 and in the other case ω∞ = 13310 was
8.5
155
Boundary-Value Dependences
specified. Virtually no effects of changing ω∞ is encountered (Table 8.6). (In the computations of the cylinder flows there was no boundary where farfield values for ω had to be specified.)
Table 8.5: Influence of ω∞ on cf at Reθ = 2900 (case FPBL) ω∞ /(L p∞ /ρ∞ )
cf × 103
Experiment (DeGraaff)
-
3.362
–
k, ω 88 (RBC) low ω∞
0.565
3.4890
+3.8%
k, ω 88 (RBC) high ω∞
5650
3.4383
+2.3%
Source
∆
From looking at the results presented above, the following conclusions regarding the free-stream value of ω are drawn: Specification of ω∞ is not crucial for boundary layer flows in the framework of full Navier-Stokes computations if the farfield boundary is sufficiently far away. Sufficiently in this context means at least one characteristic length scale of the body. However, for boundary-layer methods where ω∞ is specified at the boundary-layer edge the value of ω∞ can have a significant influence on computational results. Table 8.6: Influence of ω∞ on drag of airfoil A due to friction ω∞ /(L p∞ /ρ∞ )
cD × 102 due to friction
k, ω 98 (WBC) low ω∞
1.331
0.6469
k, ω 98 (WBC) high ω∞
13310
0.6468
Source
9
Summary and General Conclusions
Following Hirschel (1999), flow topology was used as a guideline to turbulence modeling for separated flows. This approach was investigated on the basis of a separated airfoil flow. It was found that consideration of flow topology of the mean velocity field of separated flows permits identifying the topological flow structures of possible importance for turbulence modeling. However, modifications of the turbulence model confined to the recirculation zone of separated flows do not improve predictive accuracy of the computation. The key issue for the accurate prediction of turbulent separation is correctly capturing boundary-layer development upstream of the primary separation and the boundary-layer state at separation. It was also found that the nominally two-dimensional flow around the airfoil showed strong evidence of three-dimensionality. In addition, it was argued that separated airfoil flows in general are problematic for the assessment of turbulence models due to the strong interaction between primary separation and pressure gradient via the circulation around the airfoil. In the following, assessment of turbulence-model performance was pursued for the prediction of boundary-layer development with strong adverse pressure gradient. For this purpose, a comparative study of eleven modern eddy-viscosity turbulence models was performed, with special regard to turbulent boundary-layer separation caused by an adverse pressure gradient. Four flow cases with increasing physical complexity were selected and a classification of the cases based on the topology of the mean velocity fields was performed. The cases consisted of a flat-plate boundary layer (case FPBL), a non-equilibrium boundary layer on an axial cylinder with strong pressure gradient without separation (case BS0), a non-equilibrium boundary layer on an axial cylinder with pressure-induced separation (case CS0), and again the separated airfoil flow (case AAA). For each case extensive experimental data including Reynolds stresses are available. Computational results were compared with experimental data. A large variation in predictive accuracy of the models was encountered even for the simplest flow case. In addition, predictive accuracy obtained with a given model for a given flow case varied significantly between the variables evaluated. For the separated boundary-layer flow CS0 for example, it happened that with the 1998 k, ω model of Wilcox good results were obtained for streamwise velocity while poor results were obtained for skin friction. This kind of behavior was not consistent throughout the flow cases
157
considered. Moreover, the “best” turbulence model changed from flow case to flow case. The performance of the turbulence models investigated in this work is summarized in Tables 9.1 and 9.2. Containing the flow classes defined, the tables are also meant to serve as an attempt for recommending a turbulence model for a given flow class. For example, the 1998 k, ω model of Wilcox may be recommended for boundary-layer flows with strong adverse pressure gradient without separation. However, many more computations of flows belonging to a given class must be performed before this kind of recommendation can be made on a thorough basis. For the separated airfoil flow, it was found that turbulence models yield higher predictive accuracy if they limit the eddy viscosity in boundary layers using the assumption of constant ratio between turbulent kinetic energy and Reynolds shear stress. However, this finding does not hold for the two nonequilibrium boundary-layer cases BS0 and CS0. For example, the 1998 k, ω model of Wilcox does not build on this assumption but it yielded the best agreement of computational results with measurements for case BS0. For the flow cases with adverse pressure gradient, models using transport equations for the eddy viscosity typically showed better response to pressure gradient than the algebraic model of Baldwin & Lomax (1978). In addition, predictions of the one-equation models were generally much closer to each other than those of the k, ω models. Regarding transitional flows, it was found that user-specified transition location and the rate at which turbulence models approach fully-turbulent state can have major impact on computational results. In particular, transportequation models generally show too slow transition when compared with experiment or DNS. For the airfoil flow, which featured a transitional bubble, delay of transition and/or specifying transition too far downstream prevented convergence to steady state and yielded unrealistic flow solutions. The reason was that the turbulence models did not introduce sufficient levels of Reynolds averaging in the transition region in order to suppress unsteadiness resulting from transition mechanisms. Several numerical issues were also addressed in this work. For example, grid convergence of computational results was verified based on systematic evaluation of grid-convergence indices. Regarding the application of the numerical scheme utilized for this work, the use of local preconditioning for low Mach numbers proved to be mandatory. Numerical tests showed that standard artificial damping must be reduced in boundary layers to minimize the
158
9 Summary and General Conclusions
effect of numerical diffusion on the laminar part of the boundary layers and to prevent spurious momentum loss. For models based on k and ω, dependence of computational results on the specification of ω at the wall was investigated: Extremely large values of ω at the surface have to be prescribed in order to minimize solution dependence on ωw . Regarding the free-stream value of ω, only a very small influence of ω∞ on the computation was seen for the cases considered. As a closing note, the current work shows also that a unified approach consisting of combined and coordinated efforts of experiment, DNS, LES and RANS is necessary to attack the problem of turbulence modeling. In addition, it is highly desirable that experiments supply detailed information on possible three-dimensional effects in nominally two-dimensional flows, on unsteadiness and the procedures applied to average the measured data, and on transition locations. Regarding the transition process, detailed information not only about the skin-friction coefficient but also about the change of momentum thickness in the transition region are important for the validation of turbulence models. Following Hirschel (2003), experiments should also give conclusive results for determining the flow class which the flow under consideration belongs to.
Table 9.1: Summary of performance of turbulence models investigated in the present work for the separated flow over the Aerospatiale-A airfoil (case AAA) Flow class (see Table 1.1) Test case See Section Good performance Medium performance Poor performance
1.1 a) AAA separated airfoil flow 7.4 – Johnson-King; k, ω SST; SALSA; Edwards-Chandra k, ω SST modified; Spalart-Allmaras; k, ω TNT; Baldwin-Lomax; k, ω 1998; Wallin; k, ω 1988; k, ω LLR
159
Table 9.2: Summary of performance of turbulence models investigated in the present work for the turbulent boundary-layer cases (cases FPBL, BS0 and CS0) Flow class (see Table 7.2) Test case
BL 1.1 FPBL flat plate
BL 2.1 BS0 ≥0
∂p ∂x
BL 3.1 CS0 ≥ 0 with separation
∂p ∂x
See Subsection
7.1
7.2
Good performance
k, ω SST mod. k, ω TNT Spalart-Allmaras SALSA
k, ω 1998 k, ω SST mod. Edwards-Chandra k, ω SST Spalart-Allmaras SALSA Wallin
7.3
Medium performance
k, ω LLR Baldwin-Lomax Edwards-Chandra k, ω 1998 Wallin
k, ω LLR k, ω TNT
k, ω SST mod. Wallin k, ω SST k, ω 1998 k, ω LLR SALSA Spalart-Allmaras Edwards-Chandra
Poor performance
k, ω 1988 k, ω SST
k, ω 1988 Baldwin-Lomax
k, ω 1988 k, ω TNT Baldwin-Lomax
–
160
10
10 Outlook
Outlook
Computational results for separated flows presented in this work demonstrate that the turbulence models investigated are not a reliable tool for computing such flows. In addition, it is greatly to be feared that this statement will, more or less, hold for every turbulence model based on classical Reynolds averaging. This gives rise to the fundamental question whether separated turbulent flows are generally amenable to the concepts of Reynolds averaging. In order to correctly predict mean variables of separated turbulent flow fields, the state of turbulent boundary layer at primary separation must be accurately reproduced by the computational method. A richer description of the boundary-layer flow in the vicinity and upstream of separation than offered by RANS might be necessary to perform this task. In principle, two approaches can be utilized in this regard: Large-eddy simulation or direct numerical simulation. Yet, as with DNS, LES of aerodynamic flows at high Reynolds numbers including large regions of thin boundary layers is not feasible today or in the foreseeable future (Spalart, 2000). However, great potential is seen for hybrid methods combining RANS and LES to substantially reduce computational costs compared to performing LES for the entire flow domain. Compared to pure RANS, hybrid methods can yield a better description of turbulence in regions where necessary. A partially new idea of RANS/LES coupling will be proposed in Section A of the appendices. It is intended to form the basis for future work.
Part III
Appendices
162
A A.1
A RANSLESS – A New Approach to RANS/LES Coupling
RANSLESS – A Partially New Approach to Coupling RANS and LES for Turbulent Flows Brief Review of Turbulence Physics at Turbulent Separation
Before proceeding with the discussion of a computational approach offering an alternative to classical RANS for separated flows, a brief summary of turbulence physics of boundary-layer separation is given in the following. Simpson and his co-workers have performed extensive experimental and theoretical investigations of separating turbulent boundary layers (see Wetzel & Simpson, 1998; Chesnakas & Simpson, 1997; Simpson, 1996; Chesnakas & Simpson, 1996; Simpson, 1991; Agarwal & Simpson, 1990; Simpson et al., 1977). Recapitulating their main findings, one can say that large-scale turbulent structures rapidly grow in all directions when approaching separation of a turbulent boundary layer. Due to the large eddies intermittent flow is observed far away from the wall at the position of maximum Reynolds shear stress in the separated shear layer immediately downstream of separation. The eddies supply high levels of turbulence to the backflow region underneath the separated shear layer. This means that in the backflow region velocity fluctuations such as u and v are large, but effective Reynolds shear stress like −u v is low. Moreover, fluid elements in the backflow do not come from far downstream as is suggested by mean streamlines. Instead, they are intermittently transported to and away from the wall through turbulent diffusion mechanisms induced by the large eddies in the separated shear layer. Hence, mean streamlines do not represent mean path lines of fluid elements in the backflow region. In summary, large-scale turbulent structures, high levels of turbulence and slow mean motion characterize turbulent separation. Therefore, modeling Reynolds stresses by means of mean velocity gradients is not appropriate in this flow region. Given this premise, it is believed that well-resolved LES is better suited to compute turbulent separation and will yield higher predictive accuracy compared to RANS.
A.2
RANS/LES Coupling for Separated Flows
Several researchers have introduced methods for RANS/LES coupling or related techniques. For an overview, the reader is referred to Sagaut (2002);
A.2
RANS/LES Coupling for Separated Flows
163
Friedrich & Rodi (2000); Chung & Hyung (1997). In the present work, a hybrid RANS/LES approach is considered where turbulent separation and a small part of the boundary layer upstream of separation will be computed by LES. The relatively small LES domain, in turn, will be surrounded by conventional RANS. The latter will be utilized to compute the remaining flow field and thereby provide mean flow variables used as a basis for the necessary boundary conditions for the LES. This concept will be called RANSLESS which stands for RANS-surrounded LES Scenario. Figure A.1 shows a schematic overview of the current approach for a mildly separated airfoil flow. Two possible LES domains are shown in the figure because it must be determined what shape and size of LES domains are necessary to achieve sufficiently accurate results. Location, shape and size of the LES domain will be specified by the user based on flow solutions obtained from pure RANS. This means that in a first step conventional RANS will be used to yield an approximate solution of the flow field. In the long run, this process will happen in a self organizing manner. possible LES domains surrounded by RANS
Figure A.1: Overview of RANS/LES coupling for mildly separated airfoil flow. Standard RANS and LES methods will be employed in the corresponding domains and coupling between the RANS and LES codes will take place by exchange of flow variables evaluated at the interfaces of the domains. To obtain a physically consistent and tightly coupled flow-field solution proper boundary conditions for both numerical methods must be specified at the interfaces. Hence, the key issue in the proposed work will be the development of methods for generating physically and numerically accurate boundary conditions at the interfaces. We propose an approach, novel to the knowledge
164
A RANSLESS – A New Approach to RANS/LES Coupling
of the author, for specifying inflow boundary conditions for LES from mean velocities supplied by a surrounding RANS. This is discussed in the following. A.2.1
Inflow Conditions for LES
One of the most delicate issues in RANS/LES coupling is the application of proper inflow conditions for LES on boundaries where the flow enters the LES domain coming from the RANS region. The reason is that LES requires unsteady inflow conditions containing turbulent fluctuations and, by definition, the latter are not available from RANS. Flow variables computed by RANS are mean values, and turbulent fluctuations must be “artificially” created. These must be superposed on the RANS solution to yield unsteady inflow boundary conditions for the LES, with mean values matched to corresponding variables of the upstream RANS. Often, random fluctuations were employed for this purpose (Lund et al., 1998). However, it is important to create physically meaningful turbulent fluctuations because influence of poor inflow conditions can persist over a long downstream distance (Chung & Hyung, 1997). A new method is proposed for creating turbulent inflow conditions from RANS data. It is derived from a procedure for generating turbulent inflow data for simulations of spatiallydeveloping boundary layers proposed in Lund et al. (1998). It is believed that the method outlined in this work is able to generate fluctuations with amplitude and phase information closely linked to realistic turbulent structures. Consequently, the method reduces influence of the inflow interface to a minimum and minimizes the length of the section necessary for the development of organized turbulent motion. Figure A.2 shows a schematic overview of coupling LES with RANS at the LES inflow boundary; the following discussion relates to the figure. The idea is to link RANS and LES by an overlap region. The upstream end of the overlap is referred to as “LES inflow boundary” and the downstream end is termed “RANS outflow boundary”. Inside the overlap, a recycle station and a forcing region will be present. The latter is discussed below because it is used for prescribing outflow boundary conditions for the RANS computation. (This subsection is concerned with inflow conditions for LES.) As mentioned above, first, a RANS solution for the entire flow field will be performed. Then, a mean velocity profile will be extracted from the (steady) RANS solution at the LES inflow boundary and random fluctuations will be superposed on this mean profile. The amplitudes and covariances of the random fluctua-
A.2
165
RANS/LES Coupling for Separated Flows
tions will be constructed to satisfy the Reynolds-stress tensor of the RANS computation at the LES inflow boundary. The unsteady velocities obtained in this manner will be prescribed as inflow boundary conditions for the LES computation. It is noted that random fluctuations serve only to “induce” fluctuations within the LES domain and to trigger transition. Next, velocity fluctuations obtained with LES at the recycle station will be re-scaled and subsequently superposed on the mean velocity profile of the RANS solution at the LES inflow boundary. Thus, re-scaled fluctuations from inside the LES domain replace random fluctuations used for startup. This re-scaling and superposition process will be constantly repeated and after an initial transient, realistic turbulent structure will be obtained at the recycle station as well as at the LES inflow boundary.
shear layer edge
RANS
Overlap
LES turbulent boundary layer
u’in= γ u’rec
forcing region
ψ=0 Detachment
1111111111111111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000000000000000 1111111111111111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000000000000000
LES inflow boundary
recycle station
RANS outflow boundary
Figure A.2: Generation of turbulent fluctuations at LES inflow boundary (schematic view).
Scaling Laws To apply the LES inflow procedure outlined above appropriate scaling laws for re-scaling turbulent fluctuations are required. A simple method patterned after the work of Lund et al. (1998) will be utilized for this purpose. Fluctuations at the recycle stations will be simply multiplied by a scaling factor: ui |LES inflow = χui |recycle station .
(A.1)
166
A RANSLESS – A New Approach to RANS/LES Coupling
χ can be determined in different ways. Lund et al., for example, suggested using the ratio of friction velocities at the recycle station and the LES inflow boundary: uτ |recycle station χ= . (A.2) uτ |LES inflow However, in derivation of Equation A.2 Lund et al. assumed an equilibrium boundary layer. Since we are envisaging separation where the flow is not in equilibrium new prescriptions for computation of χ might prove to be necessary. For this purpose, ratios of maximum shear stresses and maximum kinetic energies will be investigated: χ=
u v m |recycle station u v m |LES inflow
,
χ=
kmax |recycle station . kmax |LES inflow
(A.3)
It is believed that these or similar relations will lead to promising results in non-equilibrium situations. Superposition In addition to the development of new scaling laws special care must be taken for the superposition of mean and fluctuating velocities. As a first test, mean velocities computed by RANS and velocity fluctuations obtained with Equation A.1 will be summed. The inflow boundary condition for the LES domain then reads: ui,LES |LES inflow = Ui,RANS |LES inflow + ui |LES inflow .
(A.4)
ui and Ui denote unsteady velocity components and Reynolds-averaged velocity components, respectively. This procedure is called inflow-velocity modulation. However, Kaufmann et al. (2002) performed an analysis of inflow velocity modulation in terms of acoustic waves. They showed that specifying inflow conditions for LES of gas burners based on velocity modulation can lead to uncontrolled pressure waves and resonances. In their work, Kaufmann et al. presented a new method called inflow-wave modulation. Amplitudes of ingoing waves were imposed without interacting with outgoing waves, and undesired reflections and interactions between entering and leaving waves were suppressed. Therefore, the inflow-wave modulation procedure of Kaufmann et al., eventually modified, is proposed for imposing turbulent fluctuations on RANS velocities.
A.3
167
Closing Note
A.2.2
Outflow Conditions for LES
Specification of LES outflow condition will be closely patterned after the work of Schl¨ uter & Pitsch (2001) and is therefore not further discussed. A.2.3
Inflow Conditions for RANS
On boundaries where the mean flow moves from the LES to the RANS domain, time averaged LES data evaluated at the interface will be specified as inflow boundary conditions for the RANS. As pointed out in Q´em´er´e et al. (2000) and Schl¨ uter et al. (2003) some special care must be taken for prescribing turbulence variables for the turbulence model employed. Because applying inflow conditions for RANS computations was successfully performed for computing the cylinder flows discussed in Subsections 7.2 and 7.3, application to RANS/LES coupling is deemed to be straightforward. A.2.4
Outflow Conditions for RANS
Outflow conditions for RANS have to account for upstream influence of the mean LES solution. For this purpose, a forcing region inside the overlap of the LES and RANS domains will be used (Figure A.2). In the forcing region, body forces will be applied to the RANS computation in order to drive RANS velocity profiles to match mean LES profiles in the forcing region. The body forces will be formulated such that they are functionally dependent on differences inside the forcing region between mean LES velocity components and corresponding velocities components in the RANS computation: fi ∝
1 (Ui,RANS − Ui,LES ) . tc
(A.5)
tc is some characteristic time scale. It can, to first approximation, be determined from the following relation: tc =
A.3
length of the forcing region . U∞
Closing Note
It is yet to see whether RANSLESS will improve predictive accuracy of computations of pressure-induced turbulent separation. However, by concept, it offers the following advantages over existing methods:
168
A RANSLESS – A New Approach to RANS/LES Coupling
• Physically meaningful turbulent fluctuations are prescribed at the LES inflow. • Unlike in detached-eddy simulation (DES), no transition region between RANS and LES with an undefined turbulent state of the flow exists. • No extra LES or DNS need to be performed to create a database of flow variables used for specification of inflow conditions for the LES domain. • Application of RANSLESS is possible to boundary-layer flows as well as to free shear layers or swirl flows. RANSLESS will be applied first to a simple flat-plate boundary layer to test the interface treatments proposed. After successful validation of RANSLESS in this flow the same separated flow cases which were used in this work for studying turbulence-model performance will be computed. In order to investigate the gain of predictive accuracy accomplished by RANSLESS, results obtained will be compared to results from RANS computations and to experimental results. Maturing of RANSLESS for application to general complex flows is the long-term goal.
169
B
Details of the Johnson-King Model
In order to solve the Johnson-King model equations, an initial distribution of (−u v )1/2 is required. For this purpose, two preliminary steps are performed: 1. A converged flow field solution is computed with the help of a simple algebraic turbulence model, for example the Cebeci-Smith or BaldwinLomax model. 2. Next, the mean strain rate and eddy viscosity obtained from step one are used to evaluate (−u v m )1/2 employing the following relation: ) µt ∂u ∂v 1/2 = (B.1) + (−u v m ) . ρ ∂y ∂x m
(−u v m )1/2 is then inserted into Equations 3.10 – 3.13 to yield a first approximation of the so-called equilibrium eddy viscosity µt, eq . For this purpose, the non-equilibrium parameter σ(x) is set constant and equal to unity, σ(x) = 1. Through Equation B.1, µt, eq gives rise to a new value of (−u v m )1/2 . The final value of µt, eq is iteratively found such that it satisfies the equilibrium form of the model summarized as follows:
µt, eq = µto , eq 1 − e(−µti , eq /µto , eq ) , (B.2) µti , eq = ρD2 κy(−u v m, eq )1/2 , −y(−u v m, eq )1/2 D = 1 − exp , νA+ µto , eq = 0.0168ρue δv∗ FKleb ) µt , eq ∂u ∂v 1/2 = (−u v m, eq ) + ρ ∂y ∂x
(B.3) (B.4) (B.5) .
(B.6)
m
The iteration of µt, eq can, for example, be performed by a simple Aitken method. From comparing definitions of µto and µto , eq (Equations 3.13 and B.5) it follows that the parameter σ(x) is defined as the ratio of non-equilibrium and equilibrium outer eddy viscosity: σ(x) = µto /µto , eq .
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B Details of the Johnson-King Model
It is further noted that the flow solution is held fixed during the iteration for µt, eq . Once the desired value for µt, eq is found, a new flow field is computed. The procedure pertaining to step 2 is repeated until a converged solution of the flow field is obtained. To solve the non-equilibrium Johnson-King model, results from step two, above, with σ(x) = 1, are used as starting conditions. If, however, the solution process has already proceeded to the stage where a non-equilibrium solution is available, then the latter is used as the starting condition instead. Solving the non-equilibrium model means iteratively establishing a distribution for σ(x) so that values of (−u v m )1/2 independently obtained from Equation 3.14 and Equation B.1 coincide. This can be accomplished by the following steps: 1. From the starting conditions, compute an equilibrium eddy viscosity µt, eq from Equations B.2 – B.6, as discussed in step two above but without updating the flow solution. Of course, at the very first step in the solution of the Johnson-King model, this eddy viscosity will be identical to the one obtained in step 2 above. It differs, however, from the latter when a new (−u v )1/2 distribution is available. 2. Similarly, compute a non-equilibrium eddy viscosity µt using Equations 3.10 – 3.13 and B.1. Again, at the beginning, this viscosity will have the same value as µt, eq since σ(x) = 1. However, in the course of the computation σ(x) = 1 for non-equilibrium flows and µt will, in general, differ from µt, eq . 3. The rate equation for the streamwise development of the maximum Reynolds shear stress is solved next. More precisely, the square root of the maximum Reynolds shear stress divided by the density is obtained but for sake of brevity (−u v m )1/2 is referred to as the maximum Reynolds shear stress. Inserting (−u v m )1/2 into Equation B.1 yields an additional eddy viscosity which is denoted by µ ˜t, m : µ ˜t, m =
ρ(−u v m ) ∂u ∂v + ∂x ∂y
. m
4. In order to make the maximum Reynolds shear stress obtained from the rate equation and the one obtained from Equation B.1 coincide, which is equivalent to µ ˜t, m = µt, m , an appropriate value for σ(x) must
171
be found. Since σ(x) acts on the outer eddy viscosity, this can be accomplished by iterating µto , m such that µt, m = µ ˜t, m . σ(x) can then be determined by the ratio of the value of µto , m when µt, m = µ ˜t, m to the initial value of µto , m . This procedure can be expressed by a simplified Newton method: !
f (µto , m ) = µt, m (µto , m ) − µ ˜t, m = 0 , (n)
(n+1)
(n)
µ to , m = µ to , m −
f (µto , m ) (n)
f (µto , m )
,
with f (µto , m ) = (n)
(n)
∂f (µto , m ) (n) ∂µto , m
(n)
≈ 1 − e(−µti /µto
)
.
(1)
With µ ˜t, m from step 3, µt, m and µti , m from step 2 at hand, the simplified Newton algorithm for the iteration of µto , m reads: (n+1)
(n)
µ to , m = µ to , m (n+1)
µt, m
(n+1)
= µ to , m
µ ˜t, m (n)
µt, m
,
(n+1)
1 − e−µti , m /µto , m
,
with n = 1, 2 . Finally, the single-step update for σ(x) is given by: (3)
σ(x)new = σ(x)old
µ to , m (1)
µ to , m
.
5. The last step is to solve the rate equation for (−u v m )1/2 with the newly obtained σ(x) distribution and to compute the final non-equilibrium eddy viscosity from Equations 3.10 – 3.13 using the new value for (−u v m )1/2 .
172
C C.1
C Graphs of Computational Results
Graphs of Computational Results Boundary Layer with Adverse Pressure Gradient (Case BS0)
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C.1 Boundary Layer with Adverse Pressure Gradient (Case BS0) 179
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Figure C.8: Reynolds-shear-stress profiles for case BS0 at x/D = −0.544286.
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Figure C.9: Reynolds-shear-stress profiles for case BS0 at x/D = −0.0907143.
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Boundary Layer with Pressure-Induced Separation (Case CS0)
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Boundary Layer with Pressure-Induced Separation (Case CS0)
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Figure C.18: Velocity profiles for case CS0 at x/D = 0.362857.
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0
0
c)
a)
10
10
30
40
40
y [mm]
30
y [mm]
60
50
60
Experiment Spalart-Allmaras Edwards-Chandra SALSA
50
0
0.2
0.4
0.6
0.8
0
0.2
0.4
0.6
0.8
0
0
d)
b)
10
10
20
20
30
40
40
y [mm]
30
y [mm]
Figure C.19: Velocity profiles for case CS0 at x/D = 0.725714.
20
20
Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC
U/Ur U/Ur
U/Ur
U/Ur
60
50
60
Experiment kω SST MBC kω SST modified
50
Experiment kω TNT RBC Wallin RBC kω LLR RBC
192 C Graphs of Computational Results
U/Ur
U/Ur
0
0.2
0.4
0.6
0
0.2
0.4
0.6
0
0
10
c)
10
a)
20
20
60
70
50
60
70
Experiment Spalart-Allmaras Edwards-Chandra SALSA
50
0
0.2
0.4
0.6
0
0.2
0.4
0.6
0
0
10
d)
10
b)
20
20
40
y [mm]
30
40
y [mm]
30
Figure C.20: Velocity profiles for case CS0 at x/D = 1.08857.
40
y [mm]
30
40
y [mm]
30
Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC
U/Ur U/Ur
60
70
50
60
70
Experiment kω SST MBC kω SST modified
50
Experiment kω TNT RBC Wallin RBC kω LLR RBC
C.2 Boundary Layer with Pressure-Induced Separation (Case CS0) 193
0
0.2
0.4
0.6
0
0.2
0.4
0.6
0
0
10
c)
10
a)
20
20
60
70
50
60
70
Experiment Spalart-Allmaras Edwards-Chandra SALSA
50
0
0.2
0.4
0.6
0
0.2
0.4
0.6
0
0
10
d)
10
b)
20
20
40
y [mm]
30
40
y [mm]
30
Figure C.21: Velocity profiles for case CS0 at x/D = 1.63286.
40
y [mm]
30
40
y [mm]
30
Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC
U/Ur U/Ur
U/Ur
U/Ur
60
70
50
60
70
Experiment kω SST MBC kω SST modified
50
Experiment kω TNT RBC Wallin RBC kω LLR RBC
194 C Graphs of Computational Results
U/Ur
U/Ur
0
0.2
0.4
0.6
0
0.2
0.4
0.6
0
0
10
c)
10
a)
20
20
60
70
50
60
70
Experiment Spalart-Allmaras Edwards-Chandra SALSA
50
0
0.2
0.4
0.6
0
0.2
0.4
0.6
0
0
10
d)
10
b)
20
20
40
y [mm]
30
40
y [mm]
30
Figure C.22: Velocity profiles for case CS0 at x/D = 2.17714.
40
y [mm]
30
40
y [mm]
30
Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC
U/Ur U/Ur
60
70
50
60
70
Experiment kω SST MBC kω SST modified
50
Experiment kω TNT RBC Wallin RBC kω LLR RBC
C.2 Boundary Layer with Pressure-Induced Separation (Case CS0) 195
2
/0.001Ur
2
/0.001Ur
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
0
0
10
10
y [mm]
y [mm]
20
Experiment Spalart-Allmaras Edwards-Chandra SALSA
20
Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
0
0
d)
b)
10
10
y [mm]
y [mm]
20
Experiment kω SST MBC kω SST modified
20
Experiment kω TNT RBC Wallin RBC kω LLR RBC
Figure C.23: Reynolds-shear-stress profiles for case CS0 at x/D = −1.08857.
c)
a) 2
/0.001Ur 2
/0.001Ur
-2
196 C Graphs of Computational Results
2
/0.001Ur
2
/0.001Ur
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
0
0
c)
a)
y [mm]
20
y [mm]
20
30
Experiment Spalart-Allmaras Edwards-Chandra SALSA
30
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
0
0
d)
b)
10
10
20
y [mm]
20
y [mm]
30
Experiment kω SST MBC kω SST modified
30
Experiment kω TNT RBC Wallin RBC kω LLR RBC
Figure C.24: Reynolds-shear-stress profiles for case CS0 at x/D = −0.544286.
10
10
Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC 2
/0.001Ur 2
/0.001Ur
C.2 Boundary Layer with Pressure-Induced Separation (Case CS0) 197
2
/0.001Ur
2
/0.001Ur
0
-0.5
-1
-1.5
-2
-2.5
-3
0
-0.5
-1
-1.5
-2
-2.5
0
0
c)
a)
20
20
30
y [mm]
y [mm]
30
50
40
50
Experiment Spalart-Allmaras Edwards-Chandra SALSA
40
0
-0.5
-1
-1.5
-2
-2.5
-3
0
-0.5
-1
-1.5
-2
-2.5
-3
0
0
d)
b)
10
10
20
20
30
y [mm]
30
y [mm]
50
40
50
Experiment kω SST MBC kω SST modified
40
Experiment kω TNT RBC Wallin RBC kω LLR RBC
Figure C.25: Reynolds-shear-stress profiles for case CS0 at x/D = −0.0907143.
10
10
Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC 2
/0.001Ur 2
/0.001Ur
-3
198 C Graphs of Computational Results
2
/0.001Ur
2
/0.001Ur
0
-0.5
-1
-1.5
-2
-2.5
-3
0
-0.5
-1
-1.5
-2
-2.5
-3
0
0
c)
a)
20
20
30
y [mm]
30
y [mm]
50
40
50
Experiment Spalart-Allmaras Edwards-Chandra SALSA
40
0
-0.5
-1
-1.5
-2
-2.5
-3
0
-0.5
-1
-1.5
-2
-2.5
-3
0
0
d)
b)
10
10
20
20
30
y [mm]
30
y [mm]
50
40
50
Experiment kω SST MBC kω SST modified
40
Experiment kω TNT RBC Wallin RBC kω LLR RBC
Figure C.26: Reynolds-shear-stress profiles for case CS0 at x/D = 0.0907143.
10
10
Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC 2
/0.001Ur 2
/0.001Ur
C.2 Boundary Layer with Pressure-Induced Separation (Case CS0) 199
2
/0.001Ur
2
/0.001Ur
0
-0.5
-1
-1.5
-2
-2.5
-3
0
-0.5
-1
-1.5
-2
-2.5
0
0
c)
a)
20
20
30
y [mm]
y [mm]
30
50
40
50
Experiment Spalart-Allmaras Edwards-Chandra SALSA
40
0
-0.5
-1
-1.5
-2
-2.5
-3
0
-0.5
-1
-1.5
-2
-2.5
-3
0
0
d)
b)
10
10
20
20
30
y [mm]
30
y [mm]
50
40
50
Experiment kω SST MBC kω SST modified
40
Experiment kω TNT RBC Wallin RBC kω LLR RBC
Figure C.27: Reynolds-shear-stress profiles for case CS0 at x/D = 0.181429.
10
10
Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC 2
/0.001Ur 2
/0.001Ur
-3
200 C Graphs of Computational Results
2
/0.001Ur
2
/0.001Ur
-4
20
30
y [mm]
y [mm]
30
50
40
50
Experiment Spalart-Allmaras Edwards-Chandra SALSA
40
-3
-1
-1.5
-2
-2.5
-3
-3.5
-4
0
-0.5
-1
-1.5
-2
-2.5
0
0
d)
b)
10
10
20
20
30
y [mm]
30
y [mm]
50
40
50
Experiment kω SST MBC kω SST modified
40
Experiment kω TNT RBC Wallin RBC kω LLR RBC
Figure C.28: Reynolds-shear-stress profiles for case CS0 at x/D = 0.362857.
0
10
20
-4 -3.5
0
c)
10
Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC
-0.5
0
0
a)
-0.5
-1
-1.5
-2
-2.5
-3
-3.5
-4
0
-0.5
-1
-1.5
-2
-2.5
-3
-3.5 2
/0.001Ur 2
/0.001Ur
C.2 Boundary Layer with Pressure-Induced Separation (Case CS0) 201
2
/0.001Ur
2
/0.001Ur
-4
-3
20
30
40
y [mm]
40
y [mm]
30
60
50
60
Experiment Spalart-Allmaras Edwards-Chandra SALSA
50
-2
-1
-1.5
-2
-2.5
-3
-3.5
-4
0
-0.5
-1
-1.5
0
0
d)
b)
10
10
20
20
30
40
40
y [mm]
30
y [mm]
60
50
60
Experiment kω SST MBC kω SST modified
50
Experiment kω TNT RBC Wallin RBC kω LLR RBC
Figure C.29: Reynolds-shear-stress profiles for case CS0 at x/D = 0.725714.
0
10
20
-3 -2.5
0
c)
10
-4 -3.5
-0.5
0
0
Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC
-0.5
-1
-1.5
-2
-2.5
-3
-3.5
-4
0
-0.5
-1
-1.5
-2
-2.5
a) 2
/0.001Ur 2
/0.001Ur
-3.5
202 C Graphs of Computational Results
2
/0.001Ur
2
/0.001Ur
-4
20
40
y [mm]
30
40
y [mm]
30
60
70
50
60
70
Experiment Spalart-Allmaras Edwards-Chandra SALSA
50
-3
-1
-1.5
-2
-2.5
-3
-3.5
-4
0
-0.5
-1
-1.5
-2
-2.5
0
0
d)
b)
10
10
20
20
40
y [mm]
30
40
y [mm]
30
50
50
Figure C.30: Reynolds-shear-stress profiles for case CS0 at x/D = 1.08857.
0
10
20
-4 -3.5
0
c)
10
Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC
-0.5
0
0
a)
-0.5
-1
-1.5
-2
-2.5
-3
-3.5
-4
0
-0.5
-1
-1.5
-2
-2.5
-3
-3.5 2
/0.001Ur 2
/0.001Ur
70
60
70
Experiment kω SST MBC kω SST modified
60
Experiment kω TNT RBC Wallin RBC kω LLR RBC
C.2 Boundary Layer with Pressure-Induced Separation (Case CS0) 203
2
/0.001Ur
2
/0.001Ur
0
-1
-2
-3
-4
-5
0
-1
-2
-3
-4
0
0
c)
a)
10
10
30
30
40
y [mm]
y [mm]
40
50
50
70
60
70
Experiment Spalart-Allmaras Edwards-Chandra SALSA
60
0
-1
-2
-3
-4
-5
0
-1
-2
-3
-4
-5
0
0
d)
b)
10
10
20
20
30
30
40
y [mm]
40
y [mm]
50
50
Figure C.31: Reynolds-shear-stress profiles for case CS0 at x/D = 1.63286.
20
20
Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC 2
/0.001Ur 2
/0.001Ur
-5
70
60
70
Experiment kω SST MBC kω SST modified
60
Experiment kω TNT RBC Wallin RBC kω LLR RBC
204 C Graphs of Computational Results
2
/0.001Ur
2
/0.001Ur
0
-1
-2
-3
-4
-5
0
-1
-2
-3
-4
-5
0
0
c)
a)
10
10
30
30
40
y [mm]
y [mm]
40
50
50
70
80
60
70
80
Experiment Spalart-Allmaras Edwards-Chandra SALSA
60
0
-1
-2
-3
-4
-5
0
-1
-2
-3
-4
-5
0
0
d)
b)
10
10
20
20
30
30
40
y [mm]
40
y [mm]
50
50
Figure C.32: Reynolds-shear-stress profiles for case CS0 at x/D = 2.17714.
20
20
Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC 2
/0.001Ur 2
/0.001Ur
70
80
60
70
80
Experiment kω SST MBC kω SST modified
60
Experiment kω TNT RBC Wallin RBC kω LLR RBC
C.2 Boundary Layer with Pressure-Induced Separation (Case CS0) 205
206
C Graphs of Computational Results
a)
/0.001Ur
2
14
10 8 6 4 2 0
0
2
10
20
30
40
y [mm]
50
60
b)
14
/0.001Ur
Experiment Wallin RBC
12
12 Experiment Wallin RBC
10 8 6 4 2 0
0
10
20
30
40
y [mm]
50
60
Figure C.33: Profiles of normal Reynolds stresses for case CS0 at x/D = 0.725714.
C.3
C.3
Separated Airfoil Flow (Case AAA)
Separated Airfoil Flow (Case AAA)
The rest of this page has been deliberately left blank
207
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0
0
c)
a)
0.01
Experiment Spalart-Allmaras Edwards-Chandra SALSA
0.01
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0
0
d)
b)
z/c
0.005
z/c
0.005
Figure C.34: Velocity profiles for airfoil A at x/c = 0.3.
z/c
0.005
z/c
0.005
Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC
u/U∞ u/U∞
u/U∞
u/U∞
0.01
0.01
Experiment kω SST MBC kω SST modified Johnson-King
Experiment kω TNT RBC Wallin RBC kω LLR RBC
208 C Graphs of Computational Results
u/U∞
u/U∞
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0
0
c)
a)
0.005
0.005
0.015
0.01
0.015
Experiment Spalart-Allmaras Edwards-Chandra SALSA
0.01
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0
0
d)
b)
0.005
0.005
Figure C.35: Velocity profiles for airfoil A at x/c = 0.4.
z/c
z/c
Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC
u/U∞ u/U∞
z/c
z/c
0.015
0.01
0.015
Experiment kω SST MBC kω SST modified Johnson-King
0.01
Experiment kω TNT RBC Wallin RBC kω LLR RBC
C.3 Separated Airfoil Flow (Case AAA) 209
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0
0
c)
a)
0.005
0.005
0.02
0.015
0.02
Experiment Spalart-Allmaras Edwards-Chandra SALSA
0.015
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0
0
d)
b)
0.005
0.005
z/c
0.01
z/c
0.01
Figure C.36: Velocity profiles for airfoil A at x/c = 0.5.
z/c
0.01
z/c
0.01
Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC
u/U∞ u/U∞
u/U∞
u/U∞
0.02
0.015
0.02
Experiment kω SST MBC kω SST modified Johnson-King
0.015
Experiment kω TNT RBC Wallin RBC kω LLR RBC
210 C Graphs of Computational Results
u/U∞
u/U∞
0
0.2
0.4
0.6
0.8
1
1.2
0
0.2
0.4
0.6
0.8
1
1.2
0
0
c)
a)
0.005
0.005
0.015
z/c
0.015
z/c
0.025
0.02
0.025
Experiment Spalart-Allmaras Edwards-Chandra SALSA
0.02
0
0.2
0.4
0.6
0.8
1
1.2
0
0.2
0.4
0.6
0.8
1
1.2
0
0
d)
b)
0.005
0.005
0.01
0.01
Figure C.37: Velocity profiles for airfoil A at x/c = 0.6.
0.01
0.01
Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC
u/U∞ u/U∞
0.015
z/c
0.015
z/c
0.025
0.02
0.025
Experiment kω SST MBC kω SST modified Johnson-King
0.02
Experiment kω TNT RBC Wallin RBC kω LLR RBC
C.3 Separated Airfoil Flow (Case AAA) 211
0
0.2
0.4
0.6
0.8
1
1.2
0
0.2
0.4
0.6
0.8
1
1.2
0
0
0.005
c)
0.005
a)
0.01
0.01
0.03
0.035
0.025
0.03
0.035
Experiment Spalart-Allmaras Edwards-Chandra SALSA
0.025
0
0.2
0.4
0.6
0.8
1
1.2
0
0.2
0.4
0.6
0.8
1
1.2
0
0
0.005
d)
0.005
b)
0.01
0.01
0.02
z/c
0.015
0.02
z/c
0.015
Figure C.38: Velocity profiles for airfoil A at x/c = 0.7.
0.02
z/c
0.015
0.02
z/c
0.015
Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC
u/U∞ u/U∞
u/U∞
u/U∞
0.03
0.035
0.025
0.03
0.035
Experiment kω SST MBC kω SST modified Johnson-King
0.025
Experiment kω TNT RBC Wallin RBC kω LLR RBC
212 C Graphs of Computational Results
u/U∞
u/U∞
0
0.2
0.4
0.6
0.8
1
1.2
0
0.2
0.4
0.6
0.8
1
1.2
0
0
c)
a)
0.01
0.01
z/c
z/c
0.03
0.03
0.05
0.04
0.05
Experiment Spalart-Allmaras Edwards-Chandra SALSA
0.04
0
0.2
0.4
0.6
0.8
1
1.2
0
0.2
0.4
0.6
0.8
1
1.2
0
0
d)
b)
0.01
0.01
0.02
0.02
z/c
z/c
Figure C.39: Velocity profiles for airfoil A at x/c = 0.775.
0.02
0.02
Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC
u/U∞ u/U∞
0.03
0.03
0.04
Experiment kω SST MBC kω SST modified Johnson-King
0.04
Experiment kω TNT RBC Wallin RBC kω LLR RBC
0.05
0.05
C.3 Separated Airfoil Flow (Case AAA) 213
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0
c)
a)
0.01
0.01
0.02
0.02
0.05
0.06
0.04
0.05
0.06
Experiment Spalart-Allmaras Edwards-Chandra SALSA
0.04
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0
d)
b)
0.01
0.01
0.02
0.02
z/c
0.03
z/c
0.03
Figure C.40: Velocity profiles for airfoil A at x/c = 0.825.
z/c
0.03
z/c
0.03
Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC
u/U∞ u/U∞
u/U∞
u/U∞
0.05
0.06
0.04
0.05
0.06
Experiment kω SST MBC kω SST modified Johnson-King
0.04
Experiment kω TNT RBC Wallin RBC kω LLR RBC
214 C Graphs of Computational Results
u/U∞
u/U∞
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0
c)
a)
0.01
0.01
0.02
0.02
z/c
0.04
z/c
0.04
0.05
0.05
0.07
0.06
0.07
Experiment Spalart-Allmaras Edwards-Chandra SALSA
0.06
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0
0.01
d)
0.01
b)
0.02
0.02
0.03
0.03
0.04
z/c
0.04
z/c
Figure C.41: Velocity profiles for airfoil A at x/c = 0.87.
0.03
0.03
Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC
u/U∞ u/U∞
0.05
0.05
0.07
0.06
0.07
Experiment kω SST MBC kω SST modified Johnson-King
0.06
Experiment kω TNT RBC Wallin RBC kω LLR RBC
C.3 Separated Airfoil Flow (Case AAA) 215
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0
z/c
Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC
z/c
Experiment Spalart-Allmaras Edwards-Chandra SALSA 0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0
z/c
Experiment kω TNT RBC Wallin RBC kω LLR RBC
z/c
Experiment kω SST MBC kω SST modified Johnson-King 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
d)
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
b)
Figure C.42: Velocity profiles for airfoil A at x/c = 0.9.
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
c)
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
a)
u/U∞ u/U∞
u/U∞
u/U∞
216 C Graphs of Computational Results
u/U∞
u/U∞
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0
z/c
Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC
z/c
Experiment Spalart-Allmaras Edwards-Chandra SALSA 0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0
z/c
Experiment kω TNT RBC Wallin RBC kω LLR RBC
z/c
Experiment kω SST MBC kω SST modified Johnson-King 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
d)
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
b)
Figure C.43: Velocity profiles for airfoil A at x/c = 0.93.
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
c)
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
a)
u/U∞ u/U∞
C.3 Separated Airfoil Flow (Case AAA) 217
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0
c)
a)
0.02
0.02
z/c
z/c
0.06
0.06
0.08
Experiment Spalart-Allmaras Edwards-Chandra SALSA
0.08
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0
d)
b)
0.02
0.02
0.04
0.04
z/c
z/c
Figure C.44: Velocity profiles for airfoil A at x/c = 0.96.
0.04
0.04
Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC
u/U∞ u/U∞
u/U∞
u/U∞
0.06
0.06
0.08
Experiment kω SST MBC kω SST modified Johnson-King
0.08
Experiment kω TNT RBC Wallin RBC kω LLR RBC
218 C Graphs of Computational Results
u/U∞
u/U∞
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0
c)
a)
0.025
0.025
0.1
0.075
0.1
Experiment Spalart-Allmaras Edwards-Chandra SALSA
0.075
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0
d)
b)
0.025
0.025
z/c
0.05
z/c
0.05
Figure C.45: Velocity profiles for airfoil A at x/c = 0.99.
z/c
0.05
z/c
0.05
Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC
u/U∞ u/U∞
0.1
0.075
0.1
Experiment kω SST MBC kω SST modified Johnson-King
0.075
Experiment kω TNT RBC Wallin RBC kω LLR RBC
C.3 Separated Airfoil Flow (Case AAA) 219
0.006
2
2
0.006
0.007
0
0.001
0.002
0.003
0.004
0.005
-/U∞
0
0
d)
b)
z/c
0.005
z/c
0.005
Figure C.46: Reynolds-shear-stress profiles for airfoil A at x/c = 0.3.
0
0
z/c
0.001 0.01
0.002
0.001 0.005
0.003
0.005
0.006
0.007
0
0.002
Experiment Spalart-Allmaras Edwards-Chandra SALSA
0.01
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.003
c)
z/c
0.005
Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC
0.004
0
0
a)
0.004
0.005
-/U∞
2
-/U∞
2
-/U∞
0.007
0.01
Experiment kω SST MBC kω SST modified Johnson-King
0.01
Experiment kω TNT RBC Wallin RBC kω LLR RBC
220 C Graphs of Computational Results
0.004
0.005
2
2
0.004
0.005
0
0.001
0.002
0.003
-/U∞
z/c
0
0
d)
b)
0.005
0.005
z/c
z/c
0.01
0.01
Figure C.47: Reynolds-shear-stress profiles for airfoil A at x/c = 0.4.
0 0.015
0 0.01
0.001
0.004
0.005
0
0.001
Experiment Spalart-Allmaras Edwards-Chandra SALSA
0.015
0.002
0.005
z/c
0.01
0.001
0.002
0.003
0.004
0.005
0.002
c)
0.005
Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC
0.003
0
0
a)
0.003
-/U∞
2
-/U∞
2
-/U∞
0.015
Experiment kω SST MBC kω SST modified Johnson-King
0.015
Experiment kω TNT RBC Wallin RBC kω LLR RBC
C.3 Separated Airfoil Flow (Case AAA) 221
0.004
2
2
0.004
0.005
0
0.001
0.002
0.003
-/U∞
z/c
0
0
d)
b)
0.005
0.005
z/c
0.01
z/c
0.01
Figure C.48: Reynolds-shear-stress profiles for airfoil A at x/c = 0.5.
0 0.02
0 0.015
0.001
0.004
0.005
0
0.001
0.01
0.02
Experiment Spalart-Allmaras Edwards-Chandra SALSA
0.015
0.002
0.005
z/c
0.01
0.001
0.002
0.003
0.004
0.005
0.002
c)
0.005
Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC
0.003
0
0
a)
0.003
-/U∞
2
-/U∞
2
-/U∞
0.005
0.02
0.015
0.02
Experiment kω SST MBC kω SST modified Johnson-King
0.015
Experiment kω TNT RBC Wallin RBC kω LLR RBC
222 C Graphs of Computational Results
0.004
2
2
0.004
0
0.001
0.002
0.003
-/U∞
z/c
0.025
0
0
d)
b)
0.005
0.005
0.01
0.01
0.015
z/c
0.015
z/c
Figure C.49: Reynolds-shear-stress profiles for airfoil A at x/c = 0.6.
0.02
0.003
0.004
0
0 0.015
0.025
Experiment Spalart-Allmaras Edwards-Chandra SALSA
0.02
0 0.01
z/c
0.015
0.001
0.005
0.01
0.001
0.002
0.003
0.004
0.001
c)
0.005
Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC
0.002
0
0
a)
0.002
0.003
-/U∞
2
-/U∞
2
-/U∞
0.025
0.02
0.025
Experiment kω SST MBC kω SST modified Johnson-King
0.02
Experiment kω TNT RBC Wallin RBC kω LLR RBC
C.3 Separated Airfoil Flow (Case AAA) 223
2
2
0.004
0
0.001
0.002
0.003
-/U∞
z/c
0
0
d)
b)
0.01
0.01
z/c
0.02
z/c
0.02
Figure C.50: Reynolds-shear-stress profiles for airfoil A at x/c = 0.7.
0.04
0 0.03
0.003
0.004
0
0 0.02
0.04
Experiment Spalart-Allmaras Edwards-Chandra SALSA
0.03
0.001
0.01
z/c
0.02
0.001
0.002
0.003
0.004
0.001
c)
0.01
Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC
0.002
0
0
a)
0.002
0.003
-/U∞
2
-/U∞
2
-/U∞
0.004
0.04
0.03
0.04
Experiment kω SST MBC kω SST modified Johnson-King
0.03
Experiment kω TNT RBC Wallin RBC kω LLR RBC
224 C Graphs of Computational Results
0.004
0.005
2
2
0.004
0.005
0
0.001
0.002
0.003
-/U∞
z/c
0
0
d)
b)
0.01
0.01
0.02
0.02
z/c
z/c
0.03
0.03
Figure C.51: Reynolds-shear-stress profiles for airfoil A at x/c = 0.775.
0.05
0 0.04
0
0.004
0.005
0
0.001
0.03
0.05
Experiment Spalart-Allmaras Edwards-Chandra SALSA
0.04
0.001
0.02
z/c
0.03
0.002
0.01
0.02
0.001
0.002
0.003
0.004
0.005
0.002
c)
0.01
Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC
0.003
0
0
a)
0.003
-/U∞
2
-/U∞
2
-/U∞
0.05
0.04
0.05
Experiment kω SST MBC kω SST modified Johnson-King
0.04
Experiment kω TNT RBC Wallin RBC kω LLR RBC
C.3 Separated Airfoil Flow (Case AAA) 225
0.004
2
2
0.004
0.005
0
0.001
0.002
0.003
-/U∞
z/c
0
0
d)
b)
0.01
0.01
0.02
0.02
z/c
0.03
z/c
0.03
0.04
0.04
Figure C.52: Reynolds-shear-stress profiles for airfoil A at x/c = 0.825.
0.06
0 0.05
0.004
0.005
0
0 0.04
0.06
Experiment Spalart-Allmaras Edwards-Chandra SALSA
0.05
0.001
0.03
0.04
0.001
0.02
z/c
0.03
0.002
0.01
0.02
0.001
0.002
0.003
0.004
0.005
0.002
c)
0.01
Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC
0.003
0
0
a)
0.003
-/U∞
2
-/U∞
2
-/U∞
0.005
0.06
0.05
0.06
Experiment kω SST MBC kω SST modified Johnson-King
0.05
Experiment kω TNT RBC Wallin RBC kω LLR RBC
226 C Graphs of Computational Results
0.005
0.006
2
2
0.005
0.006
0
0.001
0.002
0.003
0.004
-/U∞
0.06
0.07
0
0
d)
b)
0.01
0.01
0.02
0.02
0.03
0.03
0.04
z/c
0.04
z/c
Figure C.53: Reynolds-shear-stress profiles for airfoil A at x/c = 0.87.
0.05
0.004
0.005
0.006
0
0
z/c
0.07
0 0.04
0.06
Experiment Spalart-Allmaras Edwards-Chandra SALSA
0.05
0.001 0.03
z/c
0.04
0.001 0.02
0.03
0.002
0.01
0.02
0.001
0.002
0.003
0.004
0.005
0.006
0.002
c)
0.01
Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC
0.003
0
0
a)
0.003
0.004
-/U∞
2
-/U∞
2
-/U∞
0.06
0.07
0.05
0.06
0.07
Experiment kω SST MBC kω SST modified Johnson-King
0.05
Experiment kω TNT RBC Wallin RBC kω LLR RBC
C.3 Separated Airfoil Flow (Case AAA) 227
0.005
0.006
2
2
0.005
0.006
0.007
0
0.001
0.002
0.003
0.004
-/U∞
z/c
0
0 0
0
z/c
Experiment kω TNT RBC Wallin RBC kω LLR RBC
z/c
Experiment kω SST MBC kω SST modified Johnson-King
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
d)
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
b)
Figure C.54: Reynolds-shear-stress profiles for airfoil A at x/c = 0.9.
0.001
0.001
z/c
0.002
0.002
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
0.003
0.005
0.006
0.007
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.004
c)
Experiment Spalart-Allmaras Edwards-Chandra SALSA
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC
0.003
0
0
a)
0.004
-/U∞
2
-/U∞
2
-/U∞
0.007
228 C Graphs of Computational Results
0.005
0.006
0.007
2
2
0.005
0.006
0.007
0
0.001
0.002
0.003
0.004
-/U∞
z/c
0
0 0
0
z/c
Experiment kω TNT RBC Wallin RBC kω LLR RBC
z/c
Experiment kω SST MBC kω SST modified Johnson-King
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
d)
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
b)
Figure C.55: Reynolds-shear-stress profiles for airfoil A at x/c = 0.93.
0.001
0.001
z/c
0.002
0.002
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
0.003
0.005
0.006
0.007
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.004
c)
Experiment Spalart-Allmaras Edwards-Chandra SALSA
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC
0.003
0
0
a)
0.004
-/U∞
2
-/U∞
2
-/U∞
C.3 Separated Airfoil Flow (Case AAA) 229
0.006
2
2
0.006
0.007
0
0.001
0.002
0.003
0.004
0.005
-/U∞
0
0
d)
b)
0.02
0.02
0.04
0.04
0.06
z/c
0.06
z/c
Figure C.56: Reynolds-shear-stress profiles for airfoil A at x/c = 0.96.
0
0.1
0 0.08
0.001 0.06
0.002
0.005
0.006
0.007
0
0.001
z/c
0.1
Experiment Spalart-Allmaras Edwards-Chandra SALSA
0.08
0.002
0.04
z/c
0.06
0.003
0.02
0.04
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.004
c)
0.02
Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC
0.003
0
0
a)
0.004
0.005
-/U∞
2
-/U∞
2
-/U∞
0.007
0.1
0.08
0.1
Experiment kω SST MBC kω SST modified Johnson-King
0.08
Experiment kω TNT RBC Wallin RBC kω LLR RBC
230 C Graphs of Computational Results
0.006
0.007
2
2
0.006
0.007
0
0.001
0.002
0.003
0.004
0.005
-/U∞
0
0
d)
b)
0.025
0.025
z/c
0.05
z/c
0.05
0.075
0.075
Figure C.57: Reynolds-shear-stress profiles for airfoil A at x/c = 0.99.
0
0.1
0 0.075
0.001
z/c
0.002
0.005
0.006
0.007
0
0.001
Experiment Spalart-Allmaras Edwards-Chandra SALSA
0.1
0.002
0.05
0.075
0.003
0.025
z/c
0.05
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.004
c)
0.025
Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC
0.003
0
0
a)
0.004
0.005
-/U∞
2
-/U∞
2
-/U∞
0.1
Experiment kω SST MBC kω SST modified Johnson-King
0.1
Experiment kω TNT RBC Wallin RBC kω LLR RBC
C.3 Separated Airfoil Flow (Case AAA) 231
232
D
D Overview of Algorithmic Accomplishments
Overview of Algorithmic Accomplishments
In the course of the work, the following major algorithmic issues were accomplished: • A flow analysis tool, called newmono, was designed and implemented. With this tool, flow quantities can be extracted from the flow solution and plotted in coordinate systems that are locally normal and parallel, that is “monocline”, to selected topological skeleton lines. In particular, vectors and tensors are transfered to the locally monocline system by according transformation rules. For reasons of brevity, the details of newmono and its implementation are not presented in this thesis. • Local preconditioning for low Mach numbers was implemented into the MUFLO flow solver (Subsection 8.2). (Several different preconditioning methods were investigated; not reported here.) • The Johnson-King model was modified according to topological arguments (Subsection 4.3.4). • The 1998 k, ω model of Wilcox was implemented into the FLOWer code (Subsection 6.1). • Low-Reynolds-number modifications for the 1988 and 1998 k, ω models of Wilcox and the k, ω TNT model of Kok were implemented into the FLOWer code. In this vein, the implementation of the explicit algebraic Reynolds-stress model of Wallin in the FLOWer code was extended to yield the non-linear form of the model (Section 6). • The artificial damping terms in the FLOWer code were modified such that the damping is reduced in the direction normal to the wall in boundary layers (Subsection 8.4). • Local preconditioning in the FLOWer code was modified for airfoil flows: In the vicinity of stagnation points, local preconditioning is reduced compared to other regions of the flow. Maximum local preconditioning is achieved around trailing edges. This procedure proved to yield best possible results in two aspects: First, stability problems of the preconditioning method which are related to large flow-angle changes in the vicinity of the leading-edge stagnation point are resolved by reducing preconditioning in this region. Secondly, pressure oscillations at the
233
trailing edge due to incorrect scaling of artificial damping terms at low Mach number are substantially reduced by maximal local preconditioning (Subsection 8.2). • A modified version of the k, ω SST model of Menter was implemented into the FLOWer code (Subsection 7.1.3). • An adaption technique for the exit pressure of internal flows was designed and implemented into the FLOWer code (Subsection 7.2). • A new boundary treatment was implemented into the FLOWer code to allow for prescribing measured inflow conditions.
234
E E.1
E Typical FLOWer Input Decks
Typical FLOWer Input Decks Typical FLOWer Input Deck for Case FPBL
$$ $$ Input 116.10 $$ $$---------------------------------------------------------------------$$ Testcase $$ -------$$ STRING 2D flat plate DeGraaff & Eaton $$ $$---------------------------------------------------------------------$$ General Control Data $$ -------------------$$ I2D3D 2 ILAG 0 INCORE 1 RESTOL 1.0e-06 FOTOL 1.0e-13 PEXIT 1.0 ISTEPOUT 10 NSAVE 200 ITLNS 1 $$ $$---------------------------------------------------------------------$$ Flow Data $$ --------$$ MACH 0.02848 ALPHA 0.00 $$ Bezugslaenge fuer Re-Zahl L=7.0m -> Re RENO 4471085.12002 RELEN 1.00 TINF 296.4 $$ $$---------------------------------------------------------------------$$ Geometrical Data $$ ---------------$$ AREF 1.0000
E.1
Typical FLOWer Input Deck for Case FPBL
235
XYZREF 0.2500 0.0000 0.0000 CREF 1.0000 SREF 1.0000 $$ $$---------------------------------------------------------------------$$ Space and Time Discretization Data $$ ---------------------------------$$ TUSPACE 11 $$ $$---------------------------------------------------------------------$$ Boundary Treatment Control Data $$ ------------------------------$$ BCV 0 $$ $$---------------------------------------------------------------------$$ Turbulence Model Data $$ --------------------$$ $$----<Wallin EARSM Model>--ITURB 27 $$ $$----<nonlinear, 2D>--ITU27LIN 0 ITU27DIM 2 $$ $$----<Wilcox values for 1988 Model>--ITUKWSET 1 $$ $$----<Wilcox’ wall boundary condition for omega>--BCTURBKW 0 $$ $$----<use lo Reynolds-number modifications>---ILORENO 1 $$ $$------$$---- KINFLU 2 $$ KPRDLIM 1000000. $$
236
E Typical FLOWer Input Decks
$$---------------------------------------------------------------------$$ Transition Data $$---------------$$ NTRAN 1 XTRANU 0.04 XTRANL 0.04 ZTRAN 1.00 $$ $$---------------------------------------------------------------------$$ Multigrid Control Data $$ ---------------------$$ LEVEL 6 NGIT 3 3 ISTART 3 ITYPC 1 MAXLEV 3 NEND 12000 6000 15000 0 0 0 NDUM 1 1 1 1 1 1 EPSC 0.2 DTVI 0.00 $$ $$---------------------------------------------------------------------$$ RUNGE-KUTTA CONTROL PARAMETERS FOR MESH LEVEL $$ --------------------------------------------$$ LEVPAR 2 $$...................................................................... GRIDF CFL 6.50 CFLS 3.75 RVIS4 64. RVIS2 2. 2. 2. RVIS0 16. ZETA 0.8 FILTYPE 1 ISMOO 2 EPSXYZ 0.2 1.0 0.0 SMS 1 1 1 1 1 $$...................................................................... GRIDC
E.1
Typical FLOWer Input Deck for Case FPBL
237
CFL 6.50 CFLS 3.75 RVIS4 64. RVIS2 2. 2. 2. RVIS0 16. ZETA 0.8 FILTYPE 1 ISMOO 2 EPSXYZ 0.2 1.0 0.0 SMS 1 1 1 1 1 $$...................................................................... GRIDEND $$ $$---------------------------------------------------------------------$$ Block Dependent Data $$ -------------------$$ MBLM 1 $$ BLOCK001 IVIS 40 IJKDIR 3 1 0 $$ BLOCKEND $$ $$---------------------------------------------------------------------$$ Time Accurate Data $$ -----------------STEPTYPE 0 TIMESTEP 0.2 NMAX2T 100 MAXIT2T 250 OUTWHAT 0 NOUTSURF -1 SURFVAL 1 1 20 $$ $$ $$---------------------------------------------------------------------$$ Preconditioning $$ --------------$$ IPREC 1
238
E Typical FLOWer Input Decks
UPC EPSLOCM
E.2
1.0 4.0
Typical FLOWer Input Deck for Case BS0 and CS0
$$ $$ Input 116.10 $$ $$---------------------------------------------------------------------$$ Testcase $$ -------$$ STRING Driver Cylinder case BS0 $$ $$ $$---------------------------------------------------------------------$$ General Control Data $$ -------------------$$ I2D3D 2 IROSY 1 ILAG 0 PHI 0.5 INCORE 1 RESTOL 3.0e-06 FOTOL 1.0e-13 ISTEPOUT 10 ISTEPADP 300 $$PEXIT 1.0024 PEXIT 1.0024874853 PADAPT 1 NSAVE 300 ITLNS 1 $$ $$---------------------------------------------------------------------$$ Flow Data $$ --------$$ MACH 0.08772 ALPHA 0.00 RENO 280000.
E.2
Typical FLOWer Input Deck for Case BS0 and CS0
239
RELEN 1. TINF 291.00 $$ $$---------------------------------------------------------------------$$ Geometrical Data $$ ---------------$$ AREF 1.0000 XYZREF 0.2500 0.0000 0.0000 CREF 1.0000 SREF 1.0000 $$ $$---------------------------------------------------------------------$$ Space and Time Discretization Data $$ ---------------------------------$$ TUSPACE 11 $$ $$---------------------------------------------------------------------$$ Boundary Treatment Control Data $$ ------------------------------$$ BCV 0 $$ $$---------------------------------------------------------------------$$ Turbulence Model Data $$ --------------------$$ $$----<Wallin EARSM Model>--ITURB 27 $$ $$----<nonlinear, 2D>--ITU27LIN 0 ITU27DIM 2 $$ $$----<Wilcox values for 1988 Model>--ITUKWSET 1 $$ $$------BCTURBKW 1 $$ $$----<use lo Reynolds-number modifications>----
240
E Typical FLOWer Input Decks
ILORENO 1 $$ $$------$$---- KINFLU 2 $$ KPRDLIM 1000000. $$ $$---------------------------------------------------------------------$$ Transition Data $$ --------------$$ NTRAN 0 $$ $$---------------------------------------------------------------------$$ Multigrid Control Data $$ ---------------------$$ LEVEL 6 NGIT 3 3 ISTART 3 ITYPC 2 MAXLEV 5 NEND 12000 6000 10000 0 0 0 NDUM 1 1 1 1 1 1 EPSC 0.2 DTVI 8.0 $$---------------------------------------------------------------------$$ RUNGE-KUTTA CONTROL PARAMETERS FOR MESH LEVEL $$ --------------------------------------------$$ LEVPAR 2 $$...................................................................... GRIDF CFL 6.5 CFLS 3.75 CFLTU 6.5 RVIS4 64. RVIS2 2. 2. 2. RVIS0 16. ZETA 0.5 ISMOO 2
E.2
Typical FLOWer Input Deck for Case BS0 and CS0
241
EPSXYZ 0.2 1.0 0.0 FILTYPE 1 SMS 1 1 1 1 1 $$...................................................................... GRIDC CFL 6.5 CFLS 3.75 CFLTU 6.5 RVIS4 64. RVIS2 2. 2. 2. RVIS0 16. ZETA 0.5 ISMOO 2 EPSXYZ 0.2 1.0 0.0 FILTYPE 1 SMS 1 1 1 1 1 $$...................................................................... GRIDEND $$ $$---------------------------------------------------------------------$$ Block Dependent Data $$ -------------------$$ MBLM 1 $$ BLOCK001 IVIS 40 IJKDIR 0 0 0 $$ BLOCKEND $$ $$---------------------------------------------------------------------$$ Preconditioning $$ --------------$$ IPREC 1 UPC 1.0 EPSLOCM 2.0
242
E.3
E Typical FLOWer Input Decks
Typical FLOWer Input Deck for Case AAA
$$ $$ Input 116.10 $$ $$---------------------------------------------------------------------$$ Testcase $$ -------$$ STRING ONERA A-AIRFOIL 13.3 DEG AOA, F2 CASE $$ $$---------------------------------------------------------------------$$ General Control Data $$ -------------------$$ I2D3D 2 ILAG 0 INCORE 1 RESTOL 1.0e-06 FOTOL 1.0e-16 ISTEPOUT 10 NSAVE 500 ITLNS 1 $$ $$---------------------------------------------------------------------$$ Flow Data $$ --------$$ MACH 0.15 ALPHA 13.3 RENO 2.00E+6 RELEN 1.00 TINF 294.4 $$ $$---------------------------------------------------------------------$$ Geometrical Data $$ ---------------$$ AREF 1.0000 XYZREF 0.2500 0.0000 0.0000 CREF 1.0000 SREF 1.0000 $$
E.3
Typical FLOWer Input Deck for Case AAA
243
$$---------------------------------------------------------------------$$ Space and Time Discretization Data $$ ---------------------------------$$ TUSPACE 11 $$ $$---------------------------------------------------------------------$$ Boundary Treatment Control Data $$ ------------------------------$$ BCV 1 $$ $$---------------------------------------------------------------------$$ Turbulence Model Data $$ --------------------$$ $$------ITURB 28 $$ $$----<Wilcox wall boundary condition for omega>--BCTURBKW 0 $$ $$-------ILORENO 1 $$ $$------$$---- KINFLU 2 $$ KPRDLIM 1000000. $$ $$---------------------------------------------------------------------$$ Transition Data $$---------------$$ NTRAN 1 XTRANU 0.1167 XTRANL 0.30 ZTRAN 1.00 $$ $$---------------------------------------------------------------------$$ Multigrid Control Data
244
E Typical FLOWer Input Decks
$$ ---------------------$$ LEVEL 6 NGIT 4 2 ISTART 0 ITYPC 1 MAXLEV 4 NEND 0 10000 1000 500 1 0 NDUM 1 1 1 1 1 1 EPSC 0.2 DTVI 4.0 $$ $$---------------------------------------------------------------------$$ RUNGE-KUTTA CONTROL PARAMETERS FOR MESH LEVEL $$ --------------------------------------------$$ LEVPAR 2 $$...................................................................... GRIDF CFL 6.50 CFLS 3.75 RVIS4 64. RVIS2 2. 2. 2. RVIS0 16. ZETA 0.70 FILTYPE 1 ISMOO 2 EPSXYZ 0.2 1.0 0.0 SMS 1 1 1 1 1 $$...................................................................... GRIDC CFL 6.50 CFLS 3.75 RVIS4 64. RVIS2 2. 2. 2. RVIS0 16. ZETA 0.70 FILTYPE 1 ISMOO 2 EPSXYZ 0.2 1.0 0.0 SMS 1 1 1 1 1 $$......................................................................
E.3
Typical FLOWer Input Deck for Case AAA
245
GRIDEND $$ $$---------------------------------------------------------------------$$ Block Dependent Data $$ -------------------$$ MBLM 1 $$ BLOCK001 IVIS 40 IJKDIR 3 0 0 $$ BLOCKEND $$ $$---------------------------------------------------------------------$$ Time Accurate Data $$ -----------------STEPTYPE 0 TIMESTEP 0.2 NMAX2T 100 MAXIT2T 250 OUTWHAT 0 NOUTSURF -1 SURFVAL 1 1 20 $$ $$ $$---------------------------------------------------------------------$$ Preconditioning $$ --------------$$ IPREC 1 UPC 1.0 EPSLOCM 0.25
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Lebenslauf Pers¨ onliche Daten Name: Geburtsdatum: Geburtsort:
Alan Celi´c 21.05.1971 T¨ ubingen
Beruf seit 09/03
Post-doctoral Research Engineer am Centre Europ´een de Recherche et de Formation Avanc´ee en Calcul Scientifique (CERFACS), Toulouse, Frankreich
04/1998 - 08/03
Wissenschaftlicher Mitarbeiter am Institut f¨ ur Aerodynamik und Gasdynamik (IAG) der Universit¨ at Stuttgart
Studium 10/1990 - 03/1998
Studium der Luft- und Raumfahrttechnik an der Universit¨ at Stuttgart mit den Vertiefungsf¨ achern Flugzeugbau/Leichtbau und Str¨ omungslehre Diplomarbeit am NASA Ames Research Center (Moffett Field, Kalifornien, USA) mit dem Titel Computational Study of Surface Tension and ” Wall Adhesion Effects on an Oil Film Flow Underneath an Air Boundary Layer“
09/1996 - 04/1997
NASA Ames Research Associate for CFD
10/1995 - 04/1996
Werksstudent bei der Firma Simons & Susslin Manufacturing Inc., San Jose, Kalifornien, USA
Schule 08/1981 - 05/1990
Gymnasium Spaichingen
08/1977 - 07/1981
Schillerschule Spaichingen (Grundschule)
