Advanced Computational Methods in Heat Transfer IX

Advanced Computational Methods in Heat Transfer IX WIT Press publishes leading books in Science and Technology. Visit...

Author: B. Sunden | C. A. Brebbia | B. Sunden | C. A. Brebbia

365 downloads 1581 Views 10MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

Advanced Computational Methods in

Heat Transfer IX

WIT Press publishes leading books in Science and Technology. Visit our website for the current list of titles. www.witpress.com

WITeLibrary Home of the Transactions of the Wessex Institute. Papers presented at Heat Transfer 2006 are archived in the WIT elibrary in volume 53 of WIT Transactions on Engineering Sciences (ISSN 1743-3533). The WIT electronic-library provides the international scientific community with immediate and permanent access to individual papers presented at WIT conferences. http://library.witpress.com

This page intentionally left blank

NINTH INTERNATIONAL CONFERENCE ON ADVANCED COMPUTATIONAL METHODS IN HEAT TRANSFER

HEAT TRANSFER IX CONFERENCE CHAIRMEN B. Sundén Lund Institute of Technology, Sweden C. A. Brebbia Wessex Institute of Technology, UK

INTERNATIONAL SCIENTIFIC ADVISORY COMMITTEE R. Amano A. Buikis L. De Biase G. De Mey S. del Giudice K. Domke J. Gylys

P. Heggs C. Herman D. B. Ingham Y. Jaluria P. S. Larsen X. Luo A. Mendes K. Onishi H. Oosthuizen

B. Pavkovic W. Roetzel B. Sarler S. Sinkunas A. C. M. Sousa J. Szmyd S. Yanniotis

Organised by Wessex Institute of Technology, UK Lund University of Technology, Sweden Sponsored by The Development in Heat Transfer Book Series


Transactions Editor Carlos Brebbia Wessex Institute of Technology Ashurst Lodge, Ashurst Southampton SO40 7AA, UK Email: [email protected]

WIT Transactions on Engineering Sciences Editorial Board B. Abersek University of Maribor Slovenia

B Alzahabi Kettering University USA

K S Al Jabri Sultan Qaboos University Oman

A G Atkins University of Reading UK

J A C Ambrosio IDMEC Portugal

A F M Azevedo University of Porto Portugal

H Azegami Toyohashi University of Technology Japan

R Belmans Katholieke Universiteit Leuven Belgium

G Belingardi Politecnico di Torino Italy

E Blums Latvian Academy of Sciences Latvia

S K Bhattacharyya Indian Institute of Technology India

F-G Buchholz Universitat Gesanthochschule Paderborn Germany

A R Bretones University of Granada Spain

W Cantwell Liverpool University UK

J Byrne University of Portsmouth UK

S K Chakrabarti Offshore Structure Analysis USA

D J Cartwright Bucknell University USA

H Choi Kangnung National University Korea

A Chakrabarti Indian Institute of Science India

L De Biase University of Milan Italy

J J Connor Massachusetts Institute of Technology USA

R de Borst Delft University of Technology Netherlands

L Debnath University of Texas-Pan American USA

G De Mey Ghent State University Belgium

S del Giudice University of Udine Italy

M Domaszewski Universite de Technologie de Belfort-Montbeliard France

I Doltsinis University of Stuttgart Germany

W Dover University College London UK

J Dominguez University of Seville Spain

K M Elawadly Alexandria University Egypt

J P du Plessis University of Stellenbosch South Africa

F Erdogan Lehigh University USA

M E M El-Sayed Kettering University USA

H J S Fernando Arizona State University USA

M Faghri University of Rhode Island USA

E E Gdoutos Democritus University of Thrace Greece

C J Gantes National Technical University of Athens Greece

D Goulias University of Maryland USA

R Gomez Martin University of Granada Spain

D Gross Technische Hochschule Darmstadt Germany

R H J Grimshaw Loughborough University UK

R C Gupta National University of Singapore

R Grundmann Technische Universitat Dresden Germany

K Hameyer Katholieke Universiteit Leuven Belgium

J M Hale University of Newcastle UK

P J Heggs UMIST UK

L Haydock Newage International Limited UK

D A Hills University of Oxford UK

C Herman John Hopkins University USA

T H Hyde University of Nottingham UK

M Y Hussaini Florida State University USA

N Ishikawa National Defence Academy Japan

D B Ingham The University of Leeds UK

N Jones The University of Liverpool UK

Y Jaluria Rutgers University USA

T Katayama Doshisha University Japan

D R H Jones University of Cambridge UK

E Kita Nagoya University Japan

S Kim University of WisconsinMadison USA

A Konrad University of Toronto Canada

A S Kobayashi University of Washington USA

T Krauthammer Penn State University USA

S Kotake University of Tokyo Japan

F Lattarulo Politecnico di Bari Italy

M Langseth Norwegian University of Science and Technology Norway

Y-W Mai University of Sydney Australia

S Lomov Katholieke Universiteit Leuven Belgium

B N Mandal Indian Statistical Institute India

G Manara University of Pisa Italy

T Matsui Nagoya University Japan

H A Mang Technische Universitat Wien Austria

R A W Mines The University of Liverpool UK

A C Mendes Univ. de Beira Interior Portugal

T Miyoshi Kobe University Japan

A Miyamoto Yamaguchi University Japan

T B Moodie University of Alberta Canada

G Molinari University of Genoa Italy

D Necsulescu University of Ottawa Canada

D B Murray Trinity College Dublin Ireland

H Nisitani Kyushu Sangyo University Japan

S-I Nishida Saga University Japan

P O'Donoghue University College Dublin Ireland

B Notaros University of Massachusetts USA

K Onishi Ibaraki University Japan

M Ohkusu Kyushu University Japan

E Outa Waseda University Japan

P H Oosthuizen Queens University Canada

W Perrie Bedford Institute of Oceanography Canada

G Pelosi University of Florence Italy

D Poljak University of Split Croatia

H Pina Instituto Superior Tecnico Portugal

H Power University of Nottingham UK

L P Pook University College London UK

I S Putra Institute of Technology Bandung Indonesia

D Prandle Proudman Oceanographic Laboratory UK

M Rahman Dalhousie University Canada

F Rachidi EMC Group Switzerland

T Rang Tallinn Technical University Estonia

K R Rajagopal Texas A & M University USA

B Ribas Spanish National Centre for Environmental Health Spain

D N Riahi University of IlliniosUrbana USA

W Roetzel Universitaet der Bundeswehr Hamburg Germany

K Richter Graz University of Technology Austria

S Russenchuck Magnet Group Switzerland

V Roje University of Split Croatia

B Sarler Nova Gorica Polytechnic Slovenia

H Ryssel Fraunhofer Institut Integrierte Schaltungen Germany

R Schmidt RWTH Aachen Germany

A Savini Universita de Pavia Italy

A P S Selvadurai McGill University Canada

B Scholtes Universitaet of Kassel Germany

L C Simoes University of Coimbra Portugal

G C Sih Lehigh University USA

J Sladek Slovak Academy of Sciences Slovakia

P Skerget University of Maribor Slovenia

D B Spalding CHAM UK

A C M Sousa University of New Brunswick Canada

G E Swaters University of Alberta Canada

T S Srivatsan University of Akron USA

C-L Tan Carleton University Canada

J Szmyd University of Mining and Metallurgy Poland

A Terranova Politecnico di Milano Italy

S Tanimura Aichi University of Technology Japan

S Tkachenko Otto-von-Guericke-University Germany

A G Tijhuis Technische Universiteit Eindhoven Netherlands

E Van den Bulck Katholieke Universiteit Leuven Belgium

I Tsukrov University of New Hampshire USA

R Verhoeven Ghent University Belgium

P Vas University of Aberdeen UK

B Weiss University of Vienna Austria

S Walker Imperial College UK

T X Yu Hong Kong University of Science & Technology Hong Kong

S Yanniotis Agricultural University of Athens Greece

M Zamir The University of Western Ontario Canada

K Zakrzewski Politechnika Lodzka Poland


Advanced Computational Methods in

Heat Transfer IX Editors B. Sundén Lund University of Technology, Sweden C. A. Brebbia Wessex Institute of Technology, UK

B. Sundén Lund Institute of Technology, Sweden C. A. Brebbia Wessex Institute of Technology, UK

Published by WIT Press Ashurst Lodge, Ashurst, Southampton, SO40 7AA, UK Tel: 44 (0) 238 029 3223; Fax: 44 (0) 238 029 2853 E-Mail: [email protected] http://www.witpress.com For USA, Canada and Mexico Computational Mechanics Inc 25 Bridge Street, Billerica, MA 01821, USA Tel: 978 667 5841; Fax: 978 667 7582 E-Mail: [email protected] http://www.witpress.com British Library Cataloguing-in-Publication Data A Catalogue record for this book is available from the British Library ISBN: 1-84564-176-0 ISSN: 1746-4471 (print)

ISSN: 1743-3533 (online) The texts of the papers in this volume were set individually by the authors or under their supervision. Only minor corrections to the text may have been carried out by the publisher. No responsibility is assumed by the Publisher, the Editors and Authors for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. © WIT Press 2006 Printed and bound in Great Britain by Athenaeum Press Ltd., Gateshead. 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 written permission of the Publisher.

Preface Research and developments of computational methods for solving and understanding heat transfer problems continue to be important because heat transfer topics are commonly of a complex nature and different mechanisms like heat conduction, convection, turbulence, thermal radiation and phase change may occur simultaneously. Although heat transfer might be regarded as an established and mature scientific discipline, its role and relevance in sustainable development and reduction of greenhouse gases as well as for micro- and nanoscale structures and bio-engineering have been identified recently. Non-linear phenomena may, besides the momentum transfer, appear due to temperature-dependent thermophysical properties. In engineering design and development works, reliable and accurate computational methods are requested to replace or complement expensive and time comsuming experimental trial and error work. Tremendous advancements have been achieved during recent years due to improved numerical solution approaches of non-linear partial differential equations and computer developments to achieve efficient and rapid calculations by parallelised computations on, e.g., PC-clusters. Nevertheless, to further progress computational methods in heat transfer, developments in theoretical and predictive procedures, both basic and innovative, and applied research are needed. To validate the numerical calculations accurate experimental investigations are needed. This book contains the edited versions of the papers presented at the Ninth International Conference on Advanced Computational Methods and Experimental Measurements in Heat Transfer and Mass Transfer held in the New Forest, Ashurst Lodge, Ashurst, UK in July 2006. The objective of this conference series is to provide a forum for presentation and discussion of advanced topics, new approaches and application of advanced computational methods and experimental measurements to heat and mass transfer problems. All papers have been reproduced directly from material submitted by the authors but an attempt has been made to use a unified outline and methods of presentation for each paper. The contributed papers are grouped in appropriate sections to provide better access for readers. The selected sections show the wide range of applied and fundamental problems in the heat and mass transfer field. The editors would like to thank all the distinguished and wellknown scientists who supported our efforts by serving in the International Scientific Advisory

Committee, reviewing the submitted abstracts and papers. The excellent administrative work of the conference secretariat at WIT is greatly appreciated and the efficient cooperation and encouragement by the staff at WIT Press contributed significantly in producing the conference proceedings. The Editors, May 2006

Contents Section 1: Natural and forced convection A discussion on finite-difference schemes for low Prandtl number Rayleigh-Bénard convection X. Luo & W.-K. Chen ............................................................................................3 A numerical study of the convective heat transfer between a room and a window covered by a partially open plane blind with a gap at the top P. H. Oosthuizen .................................................................................................13 An analytical solution to the Graetz problem with viscous dissipation for non-Newtonian fluids R. Chiba, M. Izumi & Y. Sugano .........................................................................23 Heat transfer by unsteady laminar mixed convection in 2-D ventilated enclosures using the vorticity-stream function formulation S. Boudebous & Z. Nemouchi .............................................................................33 Effect of roughness shape on heat transfer and flow friction characteristics of solar air heater with roughened absorber plate A. Chaube, P. K. Sahoo & S. C. Solanki .............................................................43 Study of conjugate heat transfer accompanying mixed convection in a vertical tube submitted to a step of entry temperature O. Kholai, M. Kadja & T. H. Mai .......................................................................53 Section 2: Advances in computational methods An interaction of a sonic injection jet with a supersonic turbulent flow approaching a re-entry vehicle to atmosphere D. Sun & R. S. Amano .........................................................................................65

Simulation of coupled nonlinear electromagnetic heating with the Green element method A. E. Taigbenu.....................................................................................................77 Lattice Boltzmann simulation of vortices merging in a two-phase mixing layer Y. Y. Yan & Y. Q. Zu............................................................................................87 A numerical solution of the NS equations based on the mean value theorem with applications to aerothermodynamics F. Ferguson & G. Elamin ...................................................................................97 Solution of the radiative transfer problems in two-dimensional participating cylindrical medium with isotropic scattering using the SKN approximation N. Döner & Z. Altaç ..........................................................................................109 Presentation of the hemisphere method P. Vueghs & P. Beckers ....................................................................................121 Temperature identification based on pointwise transient measurements A. Nassiopoulos & F. Bourquin ........................................................................131 Section 3: Heat and mass transfer Heat transfer in 3D water and ice basins S. Ceci, L. De Biase & G. Fossati.....................................................................143 Coupled heat and moisture transport in a building envelope on cast gypsum basis J. Maděra, P. Tesárek & R. Černý....................................................................153 Spray water cooling heat transfer under oxide scale formation conditions R. Viscorova, R. Scholz, K.-H. Spitzer & J. Wendelstorf ..................................163 How to design compact mass transfer packing for maximum efficiency H. Goshayshi .....................................................................................................173 Autowave regimes of heat and mass transfer in the non-isothermal through-reactors A. M. Brener & L. M. Musabekova ...................................................................181

Experimental studies of heat transfer between crystal, crucible elements, and surrounding media when growing large-size alkali halide ingots with melt feeding V. I. Goriletsky, B. V. Grinyov, O. Ts. Sidletskiy, V. V. Vasilyev, M. M. Tymoshenko & V. I. Sumin .....................................................................191 Modelling of heat and mass transfer in water pool type storages for spent nuclear fuel E. Fedorovich, A. Pletnev & V. Talalov............................................................199 Section 4: Modelling and experiments Experimental study of in-line tube bundle heat transfer in vertical foam flow J. Gylys, S. Sinkunas, V. Giedraitis & T. Zdankus............................................213 Use of graphics software in radiative heat transfer simulation K. Domke...........................................................................................................221 On heat transfer variation in film flow related with surface cross curvature S. Sinkunas, J. Gylys & A. Kiela .......................................................................231 Experimental investigation of enhanced heat transfer of self-exciting mode oscillating flow heat pipe with non-uniform profile under laser heating F. Shang, H. Xian, D. Liu, X. Du & Y. Yang.....................................................241 Investigation of heat transfer in the cup-cast method by experiment, and analytical method F. Pahlevani, J. Yaokawa & K. Anzai...............................................................249 Atwood number effects in buoyancy driven flows M. J. Andrews & F. F. Jebrail ..........................................................................259 Testing of the vapour chamber used in electronics cooling A. Haddad, R. Boukhanouf & C. Buffone .........................................................269 Modeling a real backdraft incident fire A. Tinaburri & M. Mazzaro ..............................................................................279 Numerical and experimental studies in the development of new clothing materials E. L. Correia, S. F. C. F. Teixeira & M. M. Neves ...........................................289 Sensitivity analysis of a computer code for modelling confined fires P. Ciambelli, M. G. Meo, P. Russo & S. Vaccaro.............................................299

Conservative averaging as an approximate method for solution of some direct and inverse heat transfer problems A. Buikis ............................................................................................................311 Section 5: Heat exchangers and equipment Estimating number of shells and determining the log mean temperature difference correction factor of shell and tube heat exchangers S. K. Bhatti, Ch. M. Krishna, Ch. Vundru, M. L. Neelapu & I. N. Niranjan Kumar ....................................................................................323 The re-commissioned thermosyphon reboiler research facility in the Morton Laboratory A. Alane & P. J. Heggs .....................................................................................337 Analysis of water condensation and two-phase flow in a channel relevant for plate heat exchangers J. Yuan, C. Wilhelmsson & B. Sundén ..............................................................351 Numerical heat transfer modelling of staggered array impinging jets A. Ramezanpour, I. Mirzaee, R. Rahmani & H. Shirvani .................................361 The re-commissioning of a vent and reflux condensation research facility for vacuum and atmospheric operation J. C. Sacramento Rivero & P. J. Heggs ............................................................371 Heat transfer modelling in double pipes for domestic hot water systems I. Gabrielaitiene, B. Sunden & J. Wollerstrand................................................381 Convective heat transfer investigations at parts of a generator circuit breaker T. Magier, H. Löbl, S. Großmann, M. Lakner & T. Schoenemann ...................391 Simplified 3-D FE model of thermal conditions inside a shoe H. Raval, Z. W. Guan, M. Bailey & D. G. Covill..............................................401 Section 6: Energy systems Radiative heat transfer in a model gas turbine combustor M. C. Paul & W. P. Jones .................................................................................413 Thermo-economics of an irreversible solar driven heat engine K. M. Pandey & R. Deb ....................................................................................423

Analysis of a new solar chimney plant design for mountainous regions M. A. Serag-Eldin..............................................................................................437 Section 7: Micro and nano scale heat and mass transfer Sinusoidal regime analysis of heat transfer in microelectronic systems B. Vermeersch & G. De Mey.............................................................................449 Viscous dissipation and temperature dependent viscosity effects in simultaneously developing flows in flat microchannels with convective boundary conditions S. Del Giudice, S. Savino & C. Nonino.............................................................457 Experimental study of water evaporation from nanoporous cylinder surface in natural convective airflow S. Hara ..............................................................................................................467 Author index ....................................................................................................477


Section 1 Natural and forced convection


Advanced Computational Methods in Heat Transfer IX

3

A discussion on finite-difference schemes for low Prandtl number Rayleigh-Bénard convection X. Luo & W.-K. Chen Institute of Thermal Engineering, University of Shanghai for Science and Technology, People’s Republic of China Dedicated to Prof. Dr.-Ing. Wilfried Roetzel on the occasion of his 70th birthday

Abstract The natural convection in a horizontal fluid layer heated from below has complex dynamic behaviour. For the Rayleigh-Bénard convection of low Prandtl number fluids, the calculated flow and temperature fields are very sensitive to the truncation error of numerical algorithms. Different kinds of finite-difference schemes might yield different numerical results. In the present work the error analysis of the upwind scheme and QUICK scheme for the Rayleigh-Bénard convection of low Prandtl number fluid was conducted. It shows that the upwind scheme will introduce numerical dispersion. This effect enlarges the viscosity term of the momentum equations and therefore no oscillation could be predicted. The QUICK scheme has higher calculation accuracy. However, it introduces an additional third-order differential term which might overestimate the oscillation effect. Keywords: Rayleigh-Bénard convection, low Prandtl number fluid, two-dimensional roll, finite-difference scheme, QUICK scheme.

1

Introduction

The natural convection in a horizontal layer confined by two rigid boundaries and heated from below is well known as Rayleigh-Bénard convection. This phenomenon reveals series non-linear characteristics and complex dynamic behaviour and has been well investigated [1−2]. The studies of low Prandtl WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/HT060011

4 Advanced Computational Methods in Heat Transfer IX number Rayleigh-Bénard convection have been motivated not only by its astrophysical applications but also by its special flow and heat transfer characteristics because in this case the non-linear inertial terms become significant. Clever and Busse [3] used the Galerkin technique to obtain the Nusselt number for low Prandtl number convection at various Rayleigh numbers. According to the linear stability analysis, the critical Rayleigh number Rac and critical wave number kc for two rigid boundaries are given as Rac = 1707.762 and kc = 3.117, respectively [1]. Bertin and Ozoe [4] calculated the problem with a finite-element method and found that Rac increases with the decrease of Pr. The computed critical Rayleigh number for Pr = 0.01 and k = 3.14 is given as Rac = 2095.0. Later Ozoe et al. [5] used a more accurate second-order central difference scheme to solve the problem and the calculated Nusselt number agreed closely with those of Clever and Busse [3]. The computed value of Rac for all values of Pr also agreed well with the theoretical value. Their results show that the Rayleigh-Bénard convection of low Prandtl number fluid is very sensitive to the numerical algorithm. As the Rayleigh number is larger than its critical value, the fluid begins to move and forms a steady flow pattern of two-dimensional rolls parallel to each other. For higher Rayleigh number the rolls become unsteady and the bending of the rolls propagates along the roll axis in time. By means of stability analysis for steady convection rolls Clever and Busse [6] discussed the oscillation instability and the critical Rayleigh number Rat for the onset of oscillation. Later they found that the transition from thermal convection in the form of rolls in a fluid layer heated from below to travelling-wave convection occurs at Rat = 1854 in the limit of low Prandtl numbers and in the presence of rigid boundaries [7]. Ozoe and Hara [8] carried out numerical computations with a second-order central difference approximation to predict Rat. The computing region is 4 times as large as the height, i.e., the aspect ratio A = 4. For the grid size of 0.02, the oscillation occurs at Rat ≈ 4500 for 4-roll pattern and Pr = 0.01. By extrapolation to zero grid size they inferred that the critical oscillatory Rayleigh number Rat was less than 2000. Yang et al. [9] and Wang et al. [10] used the SIMPLE algorithm with QUICK (quadratic upwind interpolation of convective kinematics) scheme to solve the same problem as that of Ozoe and Hara [8] with A = 4 and Pr = 0.01 and found that the fluid flow and heat transfer is steady and stable for Ra ≤ 2200. Their numerical calculation showed that the oscillation occurs at Ra = 2500. As has been mentioned by Yang et al. [11], for low Prandtl number fluid the calculated Nusselt numbers with different numerical schemes are quite different. They suggested a possible explanation that the problem might have bifurcations and the results from different numerical schemes might lie at different branches of the solution. In the present work, this problem was solved numerically with the upwind scheme, power law scheme and QUICK scheme. The Taylor series expansion was used to analyse the truncation errors of the schemes. It was found that the upwind scheme would introduce fictitious viscosity and underestimate the value of Nusselt number. The QUICK scheme was more suitable for the low Prandtl

WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


5

number natural convection. However, it introduces a third-order differential term which might affect the oscillation characteristics of the physical model.

Figure 1:

2

Schematic diagram of the horizontal fluid layer heated from below.

Mathematical model

Let us consider a horizontal fluid layer with two rigid boundaries at its upper and lower surfaces, as shown in fig. 1. The fluid layer is heated from below and cooled from above. The temperatures at the upper and lower rigid boundaries are Th and Tc, respectively, and Th > Tc. The fluid expands when it is heated, therefore it suffers an upward buoyancy force. The problem is simplified with the following assumptions: (1) The flow pattern is assumed to be two-dimensional; (2) All properties of the fluid are constant except the density in the buoyancy term of the momentum equation which is a linear function of the fluid temperature; (3) The left and right boundaries are two symmetric boundaries, that means that the wave number in this model is fixed. The following dimensionless variables and parameters are used for the modeling: X = x H , Y = y H , τ = UR t H , U = u UR , V = v UR , A = L H , UR =

a H

RaPr , DV =

Pr , DT = Ra

1

RaPr

, Pr =

ν a

, Ra =

gβ (Th − Tc )H 3 . νa

We assume that at first there is no motion in the fluid, and the temperature distribution is in a steady state. The steady-state temperature and reduced pressure distributions can be expressed as Tτ =0 = Tc + (Th − Tc )(1 − Y )

(1)

pτ =0 = ρU R2Y (1 − Y 2) .

(2)

and


6 Advanced Computational Methods in Heat Transfer IX It is convenient to introduce the dimensionless temperature and pressure perturbations,

θ=

T − Tτ =0 T − Tc = − (1 − Y ) , Th − Tc Th − Tc

(3)

P=

p − pτ =0 p  Y = − Y 1 −  . 2 ρU R2 ρU R2 

(4)

The dimensionless governing equation system can then be written as:  ∂ 2U ∂ 2U ∂U ∂U ∂U ∂P +U +V =− + DV  + 2 ∂τ ∂X ∂Y ∂X ∂Y 2  ∂X  ∂ 2V ∂ 2V ∂V ∂V ∂V ∂P +U +V =− + DV  + 2 ∂τ ∂X ∂Y ∂Y ∂Y 2  ∂X  ∂ 2θ ∂ 2θ ∂θ ∂θ ∂θ +U +V = DT  + 2 ∂τ ∂X ∂Y ∂Y 2  ∂X

  , 

(5)

  + θ , 

(6)

  + V , 

∂U ∂V + = 0, ∂X ∂Y

(7)

(8)

with the initial and boundary conditions

τ = 0:

θ =P=0;

X = 0 and X = A :

U=

Y = 0 and Y = 1 :

U =V =θ = 0.

∂V ∂θ = =0; ∂X ∂X

(9) (10) (11)

The average Nusselt number is calculated at the lower plate, A

Nu = 1 −

1 ∂θ A ∫0 ∂Y

dX

(12)

Y =0

The zero solution of the governing equation system (5)−(11) corresponds to the motionless state which is stable for Ra < Rac. Even the fluid is stirred before, the disturbance will gradually vanish and finally the fluid approaches the steady state again. However, If Ra > Rac, this zero solution becomes unstable. Any WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


7

small disturbance will be enlarged gradually and finally the Rayleigh-Bénard convection will be set up. For small value of Ra − Rac, the motion is twodimensional. With the increase of the Rayleigh number, the flow becomes steady three-dimensional flow, unsteady flow, and finally, turbulent flow [12]. In this paper eqns. (5)−(11) for Ra > Rac were solved with finite-difference methods. The research was emphasised on the oscillatory characteristics of the two-dimensional rolls, which reflects some special features of the upwind scheme and QUICK scheme.

3

Comparison of finite-different schemes

Ozoe et al. [5] used the vorticity and stream function to eliminate the pressure gradient terms in eqns. (5)−(11) and solved the problem by central differences for the derivatives in space and alternating-direction implicit differences for the derivatives in time. They calculated the Nusselt number for Ra ≤ 3000 with the aspect ratio A = 1. They found that the grid size had significant effect on the calculated Nusselt number and had to extrapolate their results to the case of (∆X, ∆Y) → 0. As shown in table 1, their results agree well with those of Clever and Busse [3] obtained by Galerkin method. No oscillation is found in their calculation. Table 1:

Ra

2000 2300 2500 3000 4000 5000 6000 10000

The calculated Nusselt numbers from different sources.

Clever and Busse [3] k = 3.11

Nu (Pr = 0.01) Ozoe and Ozoe and Ukeba [5] Hara [8] k = 3.14 k = 3.14 (1 roll) (4 rolls)

1.01955 1.17335 1.33978 1.59614 1.89397 2.22264

1.105 ** 1.18 ** 1.35 ** 1.259 1.289 * 1.502 *

Yang et al. [9] and Wang et al. [10] k = 3.14 k = 3.93 (4 rolls) (5 rolls) 1.018  1.094 1.104 1.167 * 1.176 * 1.269 * 1.355 * * 1.511 1.609 * 1.793 * 1.651 * * 1.764 1.933 * 2.052 * 2.316 *

(*: The solution is oscillating; **: Data are extrapolated to zero grid size.) Ozoe and Hara [8] further calculated the Nusselt number for higher Rayleigh numbers, 4000 ≤ Ra ≤ 2.8×105. The data listed in table 1 are taken from fig. 2 of [8]. These data are calculated under the grid size of 0.02. Their


8 Advanced Computational Methods in Heat Transfer IX calculation shows that the oscillation appears when Ra ≥ 6000. However, their predicted values of Nu are much lower than those of Clever and Busse [3]. We calculated the problem by the SIMPLE algorithm [13] and upwind scheme with Ra = 5000, Pr = 0.01, A = 4 and found that the grid size of 0.02 would yield Nu = 1.29 which is a little higher than that of Ozoe and Hara [8]. This value is not correct because the grid size of 0.01 would yield another value of Nu = 1.46. No oscillation can be found in the numerical solutions. The power law scheme yields the same results as the upwind scheme. Applying the Taylor series expansion to the upwind scheme, we can find that the upwind scheme introduces an additional dimensionless viscosity U∆X/2. In the above example, because of the low Prandtl number, we have DV = 1.414×10−3. The maximum dimensionless velocity Umax = 0.7776. Thus, for ∆X = 0.02, this fictitious viscosity could be 5.5 times as large as the real viscosity and therefore leads to a lower value of Nu. Yang et al. [11] compared the QUICK scheme with the power law scheme and found that for natural convection of low Prandtl number fluids the QUICK scheme provides higher accuracy. Therefore, the QUICK scheme was used in the calculations of Yang et al. [9] and Wang et al. [10]. Some results for Pr = 0.01, A = 4 and ∆X = ∆Y = 0.02 are shown in table 1. The calculated values of Nu for 4 rolls are close to those of Clever and Busse [3]. Applying the Taylor series expansion to the QUICK scheme, 3 7 φ ( X ,τ ) − φ ( X ,τ − ∆τ ) U  3 + φ ( X ,τ ) + φ ( X + ∆X ,τ ) − φ ( X − ∆X ,τ ) ∆τ ∆X  8 8 8

1 φ ( X + ∆X ,τ ) + φ ( X − ∆X ,τ ) − 2φ ( X ,τ )  + φ ( X − 2∆X ,τ ) = D , 8 ∆X 2 

(13)

we have, ∂φ ∂φ ∂ 2φ + O(∆τ ) + U =D + R ( ∆X 2 ) , ∂τ ∂X ∂X 2

(14)

in which R (∆X 2 ) =

∆X 2  ∂ 4φ ∂ 3φ   D  + O(∆X 3 ) . − U 24  ∂X 4 ∂X 3 

(15)

It shows that the QUICK scheme does not introduce the fictitious viscosity term and has the truncation error of O(∆X 2), therefore the predicted Nu values are more accurate. Wang et al. [10] also used the QUICK scheme to predict the roll oscillation. They found that for Pr = 0.01, A = 4 and ∆X = ∆Y = 0.02 the oscillation occurs at Ra = 2500, which is much higher than the theoretical value of the critical WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


9

oscillatory Rayleigh number, Rat =1854. However, they neither tested the effect of the grid size on the oscillation characteristics nor analysed the stability characteristics of the QUICK scheme. For low Prandtl number fluid, we have D/U ~ 10−2. Thus, eqn. (15) is simplified as, R (∆X 2 ) = −

U∆X 2 ∂ 3φ + O(∆X 3 ) . 24 ∂X 3

(16)

In such a case, the QUICK scheme will introduce an additional term into the governing equations which affects the oscillation characteristics of the RayleighBénard convection. In fact, the stability of the QUICK scheme is conditional. The stability condition is given as [14], P∆ =

U∆X 8 ≤ . D 3

(17)

For Ra = 2500, Pr = 0.01 and ∆X = 0.02, the maximum dimensionless velocity Umax = 0.867, which yields P∆ = 8.67 and the condition (17) can no longer be valid. Although an oscillation solution can be obtained with the QUICK scheme, but it is not sure whether this oscillation is caused by the instability of the numerical scheme or by the characteristics of the physical problem or by both of them. To verify the numerical schemes, two critical values should be considered. One is the critical oscillatory Rayleigh number Rat, which is difficult to be determined with direct numerical simulation. By extrapolation to zero grid size, Ozoe and Hara [8] estimated that Rat < 2000, which is near to the theoretical value, Rat = 1854. We tested a calculation with QUICK scheme for Ra = 2000, Pr = 0.01 and ∆X = 0.005. And contrary to our expectation, after several thousand hours of computing time of a PC with a 2.8 GHz CPU, we obtained a steady convection (The relative velocity disturbance ∆U/Umax was less than 10−9). When we added a small disturbance in Rayleigh number, an oscillatory disturbance was set up. The disturbance of the maximum vertical velocity component is shown in fig. 2. At τ = 30, there is some change in the frequency. After that point the oscillation becomes violent. Then, at about τ = 55, we increased the iteration accuracy and set the minimum iteration times to 3. And the oscillation gradually vanished. The solution reached the steady convection again. The other criterion is the oscillation frequency. According to fig. 12 of Clever and Busse [6], the theoretical frequency of the oscillatory disturbance for Pr = 0.01 and Ra = 2500 is about 0.01 (after the conversion between different definitions of dimensionless time). But the QUICK scheme of Yang et al. [11] with the grid size of 0.02 resulted a frequency of 0.3125 (see fig. 2 of [11]). Our calculation with the QUICK scheme gave a frequency of 0.3344. Both of them are much larger than the theoretical value given by Clever and Busse [6]. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

10 Advanced Computational Methods in Heat Transfer IX

Figure 2:

Velocity disturbance due to a disturbance in Rayleigh number (Ra = 2000, Pr = 0.01, A = 4 and ∆X = 0.005).

These examples show that the Rayleigh-Bénard convection of low Prandtl number fluid is very sensitive to the oscillation characteristics of the finitedifference schemes. Therefore, more accurate schemes should be developed for the direct simulation of such a problem.

4

Conclusions

The Rayleigh-Bénard convection of low Prandtl number fluid is difficult to be solved because of its nonlinear properties and special dynamic characteristics. The two-dimensional transient numerical calculations with SIMPLE algorithm were carried out to simulate such a problem. Different numerical schemes such as upwind scheme, power law scheme and QUICK scheme were used in the calculation. We find that the upwind scheme and power law scheme are not suitable for the direct simulation of low Prandtl number Rayleigh-Bénard convection because they will introduce a fictitious dispersion term. The QUICK scheme provides an accuracy of the second order and will not introduce the numerical dispersion into the problem to be solved. The predicted values of Nusselt number are reasonably accurate. However, it fails to predict the critical oscillatory Rayleigh number and the oscillation frequency because of the stability and oscillation characteristics of the QUICK scheme itself. New numerical schemes should be developed to simulate the low Prandtl number Rayleigh-Bénard convection. In the present work we have restricted our calculation with a fixed aspect ratio A = 4, i.e., the wave number k = 3.14. In fact, the critical Rayleigh numbers also depend on the wave number k and the number of rolls to be considered. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


11

Therefore, there remains a lot of unsolved problems in the numerical simulation of the Rayleigh-Bénard convection.

Acknowledgements The present research was sponsored by the National Natural Science Foundation of China (No. 50478113), Shanghai Leading Academic Discipline Project (No. T0503) and Shanghai Pujiang Program (No. 05PJ14078).

Nomenclature a A DT

thermal diffusivity, m2/s Aspect ratio, A = L/H dimensionless thermal diffusivity, DT = 1 / RaPr

DV g H k

dimensionless viscosity, DV = Pr / Ra acceleration constant due to gravity, m/s2 height of the fluid layer, m wave number; heat conductivity, W/m2K width of the computing region, m Nusselt number, Nu = qH/[(Th − Tc)k] reduced pressure, N/m2 dimensionless pressure perturbation, eqn. (4) Prandtl number, Pr = ν /a heat flux, W/m2 Rayleigh number, Ra = gβ (Th − Tc)H 3/(νa) the critical oscillatory Rayleigh number time, s temperature of the upper plate, K temperature of the lower plate, K velocity component in the x direction, m/s dimensionless velocity component in the x direction, U = u/UR

L Nu p P Pr q Ra Rat t Tc Th u U

reference velocity, U R = a RaPr / H , m/s UR v velocity component in the y direction, m/s V dimensionless velocity component in the y direction, V = v/UR x horizontal spatial coordinates, m X dimensionless horizontal spatial coordinates, X = x/H y vertical spatial coordinates, m Y dimensionless vertical spatial coordinates, Y = y/H Greek symbols β volumetric coefficient of expansion, 1/K ν kinematic viscosity, m2/s θ dimensionless temperature perturbation, eqn. (3) τ dimensionless time, τ = URt/H Subscript c critical value WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


References [1] Getling, A.V., Rayleigh-Benard Convection—Structures and Dynamics, World Scientific: Singapore, 1998. [2] Koschmieder, E.L., Benard Cells and Taylor Vortices, Cambridge University Press, 1993. [3] Clever, R.M. & Busse, F.H., Low-Prandtl-number convection in a layer heated from below, Journal of Fluid Mechanics, 102, 61−74, 1981. [4] Bertin, H. & Ozoe, H., Numerical study of two-dimensional natural convection in a horizontal fluid layer heated from below, by finite-element method: influence of Prandtl number, International Journal of Heat and Mass Transfer, 29(3), 439−449, 1986. [5] Ozoe, H., Ukeba, H. & Churchill, S.W., Numerical analysis of natural convection of low Prandtl number fluids heated from below, Numerical Heat Transfer, Part A, 26, 363−374, 1994. [6] Clever, R.M. & Busse, F.H., Transition to time-dependent convection, Journal of Fluid Mechanics, 65, part 4, 625−645, 1974. [7] Clever, R.M. & Busse, F.H., Convection at very low Prandtl numbers, Physics of Fluids, Series A, 2(3), 334−339, 1990. [8] Ozoe, H. & Hara, T., Numerical analysis for oscillatory natural convection of low Prandtl number fluid heated from below, Numerical Heat Transfer, Part A, 27, 307−317, 1995. [9] Yang, M., Chui, X.-Y., Tao, W.-Q. & Ozoe, H., Bifurcation and oscillation of natural convection in a horizontal layer of low Prandtl number fluid, Journal of Engineering Thermophysics, 21(4), 461−465, 2000 (in Chinese). [10] Wang, J.-G., Yang, M., Zhao, M., Cui, X.-Y. & Zhang, L.-X., Bifurcation of natural convection for low Prandtl number fluid heated from below, Journal of Engineering Thermophysics, 24(1), 76−78, 2003 (in Chinese). [11] Yang, M., Li, X.-H., Tao, W.-Q. & Ozoe, H., Computation and Comparison for heat and flow using a QUICK and other difference schemes, Journal of Engineering Thermophysics, 20(5), 593−597, 1999 (in Chinese). [12] Krishnamurti, R., Some further studies on the transition to turbulent convection, Journal of Fluid Mechanics, 60(3), 285-303, 1973. [13] Patankar, S.V., Numerical Heat Transfer and Fluid Flow, Hemisphere: New York, 1980. [14] Tao, W.-Q., Numerical Heat Transfer, Xi’an Jiaotong University Press: Xi’an, pp. 220−231, 1988 (in Chinese).



13

A numerical study of the convective heat transfer between a room and a window covered by a partially open plane blind with a gap at the top P. H. Oosthuizen Department of Mechanical and Materials Engineering, Queen’s University, Canada

Abstract An approximate model of a window covered by a partially open plane blind has been considered. The window is represented by a vertical isothermal wall section which is exposed to a large surrounding room in which the mean temperature is lower than the window temperature. The blind is represented by a thin straight vertical wall which offers no resistance to heat transfer across it. The top of this thin section is aligned with the top of the heated wall section. There is a thin horizontal wall section at the top of the “blind”. This horizontal section does not fully reach to the vertical wall with the result that there is a small gap between the blind system and the vertical wall. The main purpose of this study was to determine the effect of the size of this gap on the heat transfer rate from the “window” to the room. The length of the thin vertical wall section is, in general, less than the height of the window and thus represents a partially open blind. Attention has only been given to the convective heat transfer from the window. The governing equations, written in dimensionless form, have been solved using a commercial finite-element based code. The solution has the following parameters: Rayleigh number, Prandtl number, dimensionless horizontal distance between the window and the blind, dimensionless distance of the bottom of the blind above the bottom of the window and dimensionless size of the gap at the top of the window. Results have only been obtained for a Prandtl number of 0.7. Keywords: heat transfer, convection, windows, shading, blinds, numerical.

WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/HT060021


1

Introduction

Blinds and other forms of covering can be used to reduce building energy use and improved models for the effects of blinds on window heat transfer are needed to assist in the development of systems that make the maximum use of blinds for this purpose. The situation considered in the present study is an approximate model of a window covered by a partially open plane blind. The present work was undertaken as part of a wider study of the effect of window coverings on the heat transfer rate from windows, particularly for the case where the window is hotter than the room air, i.e., for the case where air-conditioning is being used. The situation considered is shown in fig. 1. In this situation, the window is represented by a vertical isothermal wall section with parallel adiabatic wall sections above and below the heated section. This heated wall section (the “window”) is exposed to a large surrounding room in which the mean temperature is assumed to be known and lower than the window temperature. The plane blind is represented by a thin straight vertical wall which offers no resistance to heat transfer across it and in which conductive heat transfer is negligible. The top of this thin wall section (i.e., of the “blind”) is aligned with the top of the heated wall section (i.e., with the top of the “window. There is a thin horizontal wall section at the top of the “blind” which is thus normal to the “blind”. This horizontal section does not fully reach to the vertical wall with the result that there is a small gap between the blind system and the vertical wall.

Figure 1:

Situation considered. The two limiting cases of a fully open (H=1) and a fully closed (H=0) “blind” are shown on the right.

The main purpose of this study was to determine the effect of the dimensionless size of this gap on the heat transfer rate from the “window” to the room. The length of the thin vertical wall section (i.e., of the “blind”) is, in WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


15

general, less than the height of the window and thus represents a partially open plane or roller blind. Attention has only been given to the convective heat transfer from the window, i.e., radiative heat transfer and the effects of solar radiation have not been considered. Although the model used here is only an approximation of the real situation, the results obtained with this model will give an indication of the effect of the governing parameters on the convective heat transfer rate from an actual window. The present work was undertaken as part of a wider study of the effect of window coverings on the heat transfer rate from windows particularly for the case where the window is hotter than the room air, i.e. where air-conditioning is being used, see for example Collins et al. [1–3], Machin et al. [4], Shahid et al. [5]. These studies and those described by Duarte et al. [6] and Phillips et al. [7] have concentrated on Venetian blinds. Some studies involving vertical blinds have also been undertaken, e.g., see Oosthuizen et al. [8–10]. Some studies of situations involving plane blinds have been undertaken, e.g., see Oosthuizen [11–13]. However these studies have not considered the effect of a gap between the wall and the top of the window-blind system. The present study, as is the case in many of the previous studies mentioned above, considers only the convective heat transfer. In window heat transfer situations the radiant heat transfer can however be very important and can interact with the convective flow, e.g. see Collins et al. [1] and Phillips et al. [14].

2

Solution procedure

The flow has been assumed to be laminar and two-dimensional. Fluid properties have been assumed constant except for the density change with temperature that gives rise to the buoyancy forces, this being treated by means of the Boussinesq type approximation. The covering over the heated wall section (the “blind”) has been assumed to offer no resistance to heat transfer and to have negligible thickness so that conduction along it is negligible. The effects of radiative heat transfer have been neglected. The governing equations have been written in terms of dimensionless variables using the height, L’, of the heated wall section (the “window”) as the length scale and the overall temperature difference (Tw – Ta) as the temperature scale, Ta being the air temperature in the “room” to which the window is exposed. It has been assumed that the “window” temperature, Tw, is higher than the temperature of the air in the “room”. The resultant dimensionless equations have been solved using a commercial finite-element based code, FIDAP. Only the mean heat transfer rate from the isothermal surface (the “blind”) will be considered here. This has been expressed in terms of a mean Nusselt number, Nu, based on the window height, L’, and on the overall temperature difference (Tw – Ta).



3

Results

The dimensionless governing equations have the following parameters: 1. 2. 3. 4. 5.

the Rayleigh number based on the window height, L’, and on (Tw – Ta), Ra the Prandtl number, Pr the dimensionless horizontal distance between the window and the blind, W = W’ / L’ the dimensionless distance of the bottom of the blind above the bottom of the window, H = H’ / L’. the dimensionless gap between the horizontal top section of the blind system and the top of the window, G =G’ / L’.

Here, as shown in fig. 1, G’ is the size of gap between the top of “blind system” and “wall”, H’ is the height of bottom of thin wall section (i.e., of the “blind”) above bottom of heated wall section (i.e., of the “window”) and W’ is the distance of thin vertical wall section (i.e., the “blind”) from the heated wall section (i.e., the “window”). Because of the application being considered, results have only been obtained for a Prandtl number of 0.7. Rayleigh numbers of between 10 3 and 10 8, H values of between 0 and 1, W values of between 0.02 and 0.12, and G values of between 0 and W have been considered. An H value of 0 corresponds to a “fullyclosed blind” while an H value of 1 corresponds to a “fully-open blind”, these two cases being shown in fig. 1. Results were also obtained for the no-blind case and the mean Nusselt numbers given for this case were found to be in excellent agreement with values given by empirical equations for the mean natural convective heat transfer rate from an isothermal vertical flat plate. The effect of the dimensionless gap G on the mean Nusselt number for various values of H and for Rayleigh numbers of 105, 106 and 107 is shown in figs. 1, 2 and 3 respectively. These results are all for W = 0.1. It will be seen from these figures that the gap size has the biggest effect on the heat transfer rate when the Rayleigh number is low and the gap size is small. For example, it will be seen from fig. 2 that for a Rayleigh number of 105 the value of G has quite a significant effect on the Nusselt number at all values of G whereas it will be seen from fig. 4 that for a Rayleigh number of 107 the value of G only has a significant effect on the Nusselt number when G is less than greater than about 0.03. This is because at low Rayleigh numbers the thickness of the boundary layer on the “window” is relatively large and significantly greater than the gap size whereas at high Rayleigh numbers the thickness of the boundary layer on the “window” is relatively small and can be significantly less than G with the result that the gap size has only a small effect on the flow and therefore on the heat transfer rate. This is illustrated by the typical streamline patterns shown in fig. 5.



17

Figure 2:

Variation of mean Nusselt number with dimensionless gap size G for various values of H for W = 0.1 and Ra =105.

Figure 3:




Figure 4:


Figure 5:

Streamline patterns for W = 0.1, H = 0.8 and Ra =107 for G values of from left to right of 0.02, 0.04, 0.06, and 0.08.



19

The effect of the dimensionless “blind – window” spacing W on the mean Nusselt number is illustrated by the results given in figs. 6 and 7. It will be seen from these results that as W increases the mean Nusselt decreases but it then passes through a minimum and then increases with increasing W. This form of behaviour arises because when W is small then is very little convective motion in the air between the blind and the window and the heat transfer is mainly by conduction and heat transfer rate decreases as the thickness of the air layer increases. However as W increases further, convective motion begins between the blind and the window and this tends to increase the heat transfer rate and thus leading the minimum in the Nu – W variation. It will also be seen from the results given in figs. 6 and 7 that the value of W at which the minimum heat transfer rate occurs increases with H at the smaller value of H considered. The form of the variation of the Nusselt number with dimensionless blind opening, H, is illustrated by the results given in fig. 8. It will be seen that as the result of the presence of the top gap, the mean Nusselt number does vary linearly with H as is sometimes assumed.

Figure 6:

Variation of mean Nusselt number with dimensionless window-blind spacing, W, for a Rayleigh number of 107 and H = 0.4 for two values of G.



Figure 7:

Variation of mean Nusselt number with dimensionless window-blind spacing, W, for a Rayleigh number of 107 and H = 0.8 for two values of G.

Figure 8:

Variation of mean Nusselt number with dimensionless blind opening H for a Rayleigh number of 107 and W = 0.1 for two values of G.



4

21

Conclusions

The dimensionless blind opening, H, has been shown to have a very significant affect on the mean Nusselt number, the results given here allowing an estimate to be made of this effect for various values of Ra and W.

Acknowledgment This work was supported by the Natural Sciences and Engineering Research Council of Canada.

References [1]

[2]

[3] [4] [5]

[6] [7] [8]

[9]

Collins, M., Harrison, S.J., Oosthuizen, P.H. & Naylor, D., Sensitivity analysis of heat transfer from an irradiated window and horizontal louvered blind assembly. American Society of Heating, Refrigerating and Air-Conditioning Engineers (ASHRAE) Trans., 108, pp. 1-8, 2002. Collins, M.R., Harrison, S.J., Naylor, D., & Oosthuizen, P.H., Heat transfer from an isothermal vertical plate with adjacent heated horizontal louvers: Numerical analysis. J. Heat Transfer, 124(6), pp. 1072-1077, 2002. Collins, M.R., Harrison, S.J., Naylor, D., & Oosthuizen, P.H., Heat transfer from an isothermal vertical plate with adjacent heated horizontal louvers: Validation. J. Heat Transfer, 124(6), pp. 1078-1087, 2002. Machin, A.D., Naylor, D., Oosthuizen, P.H., & Harrison, S.J., Experimental study of free convection at an indoor glazing surface with a Venetian blind. J. HVAC&R Research, 4(2), pp. 153-166, 1998. Shahid, H., Naylor, D., Oosthuizen, P.H., & Harrison, S.J., A numerical study of the effect of horizontal louvered blinds on window thermal performance. Paper SH2. Proc. of the 2nd Int. Conf. on Heat Transfer, Fluid Mechanics and Thermodynamics (HEFAT), ed. J.P. Meyer, pp. 1-6, 2003 Duarte, N., Naylor, D., Oosthuizen, P.H. & Harrison, S.J., An interferometric study of free convection at a window glazing with a heated Venetian blind. Int. J. HVAC&R Research, 7(2), pp. 169-184, 2001. Phillips, J., Naylor, D., Harrison, S.J., & Oosthuizen, P.H., Free convection from a window glazing with a Venetian blind: Numerical model development. Trans. CSME, 23(1B), pp. 159-172, 1999. Oosthuizen, P.H., Sun, L., & Naylor, D., A numerical study of the effect of normal adiabatic surfaces on natural convective heat transfer from a vertical isothermal plate. Progress in Transport Phenomena, Proc. of the 3rd Int. Symposium on Transport Phenomena, eds. S. Dost, H. Struchtrup, & I Dincer, Elsevier: Paris, pp. 327-331, 2002. Oosthuizen, P.H., Sun, L., & Naylor, D., The effect of inclined vertical slats on natural convective heat transfer from an isothermal heated vertical WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


[10]

[11]

[12]

[13]

[14]

plate. Proc. Of the 10th Annual Conf. of the CFD Society of Canada, pp. 515-519, 2002. Oosthuizen, P.H., Sun, L., & Naylor, D., The effect of heat generation in inclined slats in natural convection from an isothermal heated vertical plate. Proc. Of the 11th Annual Conf. of the CFD Society of Canada, pp. 1-8, 2003. Oosthuizen, P.H., Natural convection in a square enclosure with a partially heated wall section covered by a blind-like attachment with heat generation in the attachment. Proc. Of the 1st Int. Conf, on Heat Transfer, Fluid Mechanics and Thermodynamics (HEFAT), Skukuza, Kruger National Park, South Africa, Vol. 1, Part 2, pp. 702-707, 2002. Oosthuizen, P.H., Natural convection in a square enclosure with a partially heated wall section covered by a blind-like attachment with nonuniform heat generation in the attachment. Progress in Transport Phenomena, Proc. of the 3rd Int. Symposium on Transport Phenomena, eds. S. Dost, H. Struchtrup & I. Dincer, Elsevier: Paris, pp. 29-34, 2002. Oosthuizen, P.H., Natural convection in a square enclosure with a partially heated wall section covered by a blind-like attachment in which there is a linearly varying heat transfer rate, Paper IMECE2002-32968, Proc. On CD of IMECE2002, Vol. 1, 2002. Phillips, J., Naylor, D., Harrison, S.J., & Oosthuizen, P.H., Numerical study of convective and radiative heat transfer from a window glazing with a Venetian blind. Int. J. of HVAC&R Research, 7(4), pp. 383-402, 2001.



23

An analytical solution to the Graetz problem with viscous dissipation for non-Newtonian fluids R. Chiba1, M. Izumi2 & Y. Sugano3 1

Miyagi National College of Technology, Japan Department of Mechanical Engineering, Ishinomaki Sensyu University, Japan 3 Department of Mechanical Engineering, Iwate University, Japan 2

Abstract Forced convection heat transfer in a non-Newtonian fluid flow between parallel plates subjected to convective cooling on the external surfaces is investigated analytically. Fully developed laminar velocity distributions obtained by a power law fluid rheology model are used, and viscous dissipation is taken into account. The effect of heat conduction in the direction of fluid flow is considered negligible. The physical properties are assumed to be constant. We approximate the smooth change in the velocity distribution between the plates as a piecewise constant velocity. The theoretical analysis of the heat transfer is performed using an integral transform technique—Vodicka’s method. An important feature of the approach is that an arbitrary distribution of the temperatures of the surrounding media in the direction of fluid flow and an arbitrary velocity distribution of the fluid can be permitted. A comparison with the existing results provides a verification of this technique. The effects of the Brinkman number, Biot number and rheological properties on the distributions of the fluid temperature and the local Nusselt number are illustrated. Moreover, the effects of these parameters on the length of the freeze-free zone are discussed in the case where the temperatures of the surrounding media are below the solidification temperature of the fluid. Keywords: heat transfer, forced convection, non-Newtonian fluid, analytical solution, viscous dissipation, Graetz problem, channel flow.



1

Introduction

An understanding of the convection heat transfer in non-Newtonian fluids inside conduits is of great importance in the design of several types of thermal equipment. From this viewpoint, heat transfer problems of this type have been investigated by a large number of researchers since the 1880s. The problem pertaining to the derivation of the local Nusselt number in the thermal entry region when an incompressible fluid flows through a conduit with a fully developed velocity distribution is of particular interest; this problem is referred to as the Graetz problem. It has attracted the interest of not only engineers but also applied mathematicians because of the difficulties involved in deriving the solution. The Graetz problem is normally solved by using a quasi-analytical method. That is, the separation of variables approach is applied to the governing equation, and only the resultant eigenvalue problem is solved numerically. It is very difficult to obtain an exact solution to the eigenvalue problem except in some special cases. Therefore, a completely explicit analytical solution to the heat transfer problem for a fluid with an arbitrary velocity distribution has not been reported thus far. The objective of this study is to solve mathematically the forced convection heat transfer problem in a conduit between parallel plates subjected to heat loads from the surrounding by Vodicka’s method [1], which is a type of integral transform method, and to derive completely explicit analytical solutions of the fluid temperature and local Nusselt number. Heat conduction in the direction of fluid flow is considered negligible since the present study focuses on heat transfer with a sufficiently large Peclet number. However, viscous dissipation is taken into account. Numerical calculations are used to illustrate the effects of the Brinkman number, Biot number and rheological properties on the distributions of the fluid temperature and local Nusselt number. Moreover, the effects of these parameters on the length of the freeze-free zone are discussed for the case where the temperatures of the surrounding media are below the solidification temperature of the fluid.

2

Analysis

2.1 Analytical model and formulation Figure 1 shows the physical model and coordinate system. A non-Newtonian fluid with a fully developed velocity distribution u(y) flows into a conduit between parallel plates. The fluid temperature at the entrance is T0(y). The conduit of width 2L contacts the surrounding media with temperature T1∞(x) and T2∞(x) at y = L and y = –L, respectively. The heat transfer coefficients on the external surfaces of the plates are h1 and h2. In this study, the flowing fluid has no analytical restriction in its velocity distribution form. With regard to the type of fluid flowing inside the conduit, a power law fluid, which can approximate the non-Newtonian viscosity of many WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


25

types of fluids with a good accuracy over a wide range of shear rates, is considered here. The shear stress acting on the viscous fluid τyx is given as follows: τ yx = κ

du dy

ν −1

du , dy

(1)

where κ and ν are the power law model parameter and power law model index, respectively. ν < 1 indicates a pseudoplastic fluid, ν > 1 indicates a dilatant fluid and ν = 1 is equivalent to a Newtonian fluid. The fully developed velocity distribution is expressed as follows [2]: 1+ν   2ν + 1  y ν  u ( y ) = um 1− , ν + 1  L   

(2)

where um is the mean velocity. y

T1∞(x)

h1

u = u(y) T = T0(y) O

2L

x Thin parallel plates

h2

T2∞(x)

Figure 1: Physical model and coordinate system. The following assumptions are introduced: (i) material properties are independent of temperature and are therefore constant, (ii) heat resistance of the parallel plates is negligible, (iii) heat conduction in the direction of fluid flow is negligible, (iv) mode of flow is always laminar. In this case, the steady-state heat balance taking viscous dissipation into account is expressed as follows: ρ cu

∂T ∂ 2T du = λ 2 + τ yx , ∂x dy ∂y

(3)

where ρ, c and λ are the density, specific heat and thermal conductivity, respectively. The boundary conditions are given as follows: T (0, y ) = T0 ( y ) ,

λ

∂T ( x, L) + h1 T ( x, L) − T1∞ ( x) = 0 , ∂y


(4) (5)

26 Advanced Computational Methods in Heat Transfer IX λ

∂T ( x, − L) − h2 T ( x, − L) − T2∞ ( x) = 0 . ∂y

(6)

Taking the generality of the analysis into account, the expressions of eqns (3)–(6) in dimensionless form yield the following equations: ν +1

U (η )

∂θ ∂ 2θ  1 +ν  = + Br   ∂ξ ∂η 2  ν  θ (0,η ) = θ 0 (η ) ,

η

ν +1 ν

,

(7) (8)

∂θ (ξ ,1) + H1 θ (ξ ,1) − θ1∞ (ξ ) = 0 , ∂η ∂θ (ξ , −1) − H 2 θ (ξ , −1) − θ 2∞ (ξ ) = 0 , ∂η

(9) (10)

+ν 1−ν where U(η) = (2ν + 1)/(ν + 1)[1–|η|(1+ν)/ν], Br = κ u1max L /[λ (T0b − Ts )] , umax = (1 + 2ν)um/(1 + ν), θ = (T–Ts)/(T0b–Ts), η = y/L, ξ = λx/(umL2ρc) and H = hL/λ. Eqn (7) is a partial differential equation with variable coefficients; therefore, it is very difficult to obtain the exact solution. In order to solve eqn (7), we divide the conduit into n regions in the η direction and approximate U(η) as a constant Ui in each region, as shown in fig. 2. In this case, the dimensionless energy equation in the ith region (i = 1, 2, …, n) is obtained as follows:

Ui

∂θi ∂ 2θi = + Q(η ) , ∂ξ ∂η 2

(11)

where Q(η) = Br[(1+ν)/ν]ν+1|η|(1+ν )/ν. The continuous conditions at the imaginary interfaces and boundary conditions are expressed by the following equations: θi (ξ ,ηi ) = θi +1 (ξ ,ηi ) ,

(12)

∂θi (ξ ,ηi ) ∂θi +1 (ξ ,ηi ) = , ∂η ∂η θi (0,η ) = θ0 (η ) ,

(13) (14)

U(η) Region number 1

2

3

i

i+1

n-1 n

U3 U2

U1 η0=-1 η1 η2

ηi

ηn-1ηn=1

η

Figure 2: Virtual division inside the conduit. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

Advanced Computational Methods in Heat Transfer IX ∂θ n (ξ ,1) + H1 θ n (ξ ,1) − θ1∞ (ξ ) = 0 , ∂η ∂θ1 (ξ , −1) − H 2 θ1 (ξ , −1) − θ 2∞ (ξ ) = 0 . ∂η

27

(15) (16)

2.2 Vodicka’s method The solution to eqns (11)–(16) is obtained as ∞

2

m =1

j =1

θi (ξ ,η ) = ∑ φm (ξ ) X im (η ) + ∑ L j (η ) Pj (ξ ) ,

(17)

P1 (ξ ) = −θ 2∞ (ξ ) , P2 (ξ ) = θ1∞ (ξ ) .

(18)

where

Lj(η) (j = 1, 2) is the solution to the boundary value problem expressed as follows: d2L j dη

2

=0,

dL j (−1) dη

− H 2 L j (−1) = H 2 ⋅ (2 − j ) ,

dL j (1) dη

+ H1L j (1) = H1 ⋅ ( j − 1) . (19)

Xim(η) is the solution to the eigenvalue problem expressed as follows: d 2 X im dX im (ηi ) dX (i +1) m (ηi ) + γ m2U i X im = 0 , X im (ηi ) = X (i +1) m (ηi ) , = , dη dη dη 2 dX 1m (−1) dX nm (1) − H 2 X 1m ( −1) = 0 , + H1 X nm (1) = 0 . dη dη

(20)

Lj(η) and Xim(η) are given as L1 (η ) =

H1H 2η − (1 + H1 ) H 2 H H η + (1 + H 2 ) H1 , L2 (η ) = 1 2 , H1 + H 2 + 2 H1H 2 H1 + H 2 + 2 H1 H 2

(21)

X im (η ) = Aim cos( U i γ mη ) + Bim sin( U i γ mη ) .

(22)

The conditions necessary to determine the unknown coefficients Aim and Bim can be obtained by substituting eqn (22) into the continuous and boundary conditions in eqn (20). Eigenvalues γm (m = 1, 2, ...) are obtained from the condition under which Aim and Bim are both non-zero and are therefore positive roots of the following transcendental equation: G e1 ⋅ Ee1 ⋅ Ee2 " Ee( n−1) ⋅ aen = 0 ,

where G e1 = d1m sin( d1m ) − H 2 cos( d1m ) d1m cos( d1m ) + H 2 sin(d1m ) , dim = U i γ m , cos[d (i +1) mηi ] sin[d (i +1) mηi ]   d cos(dimηi ) − sin(dimηi )   Eei =  im ,   −d sin[ d ] d η d sin( d ) cos( d ) η η ( i +1) m i ( i +1) m cos[ d ( i +1) mηi ]  (i +1) m  im i im i    im WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

(23)

28 Advanced Computational Methods in Heat Transfer IX  d cos(d nm ) + H1 sin(d nm )  aen =  nm .  d nm sin( d nm ) − H1 cos(d nm ) 

(24)

By substituting eqn (17) into eqn (14), the following equation is obtained: ∞

2

m =1

j =1

G (η ) = ∑ φm (0) X im (η ) = θ 0 (η ) − ∑ L j (η ) Pj (0) .

(25)

The eigenvalue function Xim(η) obtained from eqns (20) and (22) has an orthogonal relationship with discontinuous weight functions; this is expressed as follows: n

const. (m = k ) . (m ≠ k )  0

η ∑ U i ∫η X im (η ) X ik (η )dη =  i

i −1

i =1

(26)

Lj(η), Q(η) and G(η) can be expanded into an infinite series by Xim(η) as follows: ∞

L j (η ) = ∑ lmj X im (η ) ,

(27)

m =1

∞

Q(η ) = U i ∑ qm X im (η ) ,

(28)

m =1

∞

G (η ) = ∑ g m X im (η ) ,

(29)

m =1

where n

n

η

η

lmj = M m−1 ∑ U i ∫η i L j (η ) X im (η )dη , qm = M m−1 ∑ ∫η i Q(η ) X im (η )dη , i −1

i =1

n

i −1

i =1

n

η

η

g m = M m−1 ∑ U i ∫η i G (η ) X im (η )dη , M m = ∑ U i ∫η i [ X im (η )]2 dη . i −1

i =1

i −1

i =1

(30)

Taking eqns (19) and (20) into account, we substitute eqns (17), (27) and (28) into eqn (11). This yields a first-order linear ordinary differential equation for φm(ξ) as follows: 2 dPj dφm + γ m2 φm = qm − ∑ lmj . dξ dξ j =1

(31)

Solving eqn (31) with the condition φm(0) = gm, which is obtained from the comparison between eqns (25) and (29), we obtain φm(ξ) as φm (ξ ) = g m e−γ mξ + 2

qm

γ m2

ξ

2

dPj (t )

j =1

dt

(1 − e−γ mξ ) − e−γ mξ ∫0 eγ m t ∑ lmj 2

2

2

dt .

(32)

Finally, the temperature solution for the ith region inside the conduit θi(ξ, η) is derived as follows: ∞

θi (ξ ,η ) = ∑ φm (ξ )[Aim cos( U i γ mη ) + Bim sin( U i γ mη )] m =1

[(1 + H1 ) H 2 − H1H 2η ]θ 2∞ (ξ ) + [(1 + H 2 ) H1 + H1H 2η ]θ1∞ (ξ ) + H1 + H 2 + 2 H1 H 2 WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

.

(33)


29

The dimensionless bulk temperature of the fluid is given as θ B (ξ ) =

1 1 ∫ U (η )θ (ξ ,η )dη . 2 −1

(34)

Consequently, the local Nusselt numbers at η = 1 and –1 are expressed by the following equations: ∂θ n (ξ ,1) −1 ∂η Nu1 (ξ ) = = = λ θ n (ξ ,1) − θ B (ξ ) θ n (ξ ,1) − θ B (ξ ) h 1L

(35)

H H [θ (ξ ) − θ 2∞ (ξ )]   ∞ ×  ∑ φm (ξ ) U n γ m [Anm sin( U n γ m ) − Bnm cos( U n γ m )] − 1 2 1∞ , H1 + H 2 + 2 H1H 2   m =1 ∂θ1 (ξ , −1) −1 ∂η Nu2 (ξ ) = = = λ θ1 (ξ , −1) − θ B (ξ ) θ1 (ξ , −1) − θ B (ξ ) h 2 L

−

(36)

 ∞ H H [θ (ξ ) − θ 2∞ (ξ )]  ×  ∑ φm (ξ ) U1 γ m [A1m sin( U1 γ m ) + B1m cos( U1 γ m )] + 1 2 1∞ . H1 + H 2 + 2 H1H 2   m =1

3

Numerical calculation

As a numerical example, we consider the case of θ0(η) = 1, θ1∞(ξ) = θ2∞(ξ) = 0 and H1 = H2 = H. In this case, the local Nusselt numbers of eqns (35) and (36) equate with each other since the temperature field in the conduit is symmetric with respect to the x-axis (ξ-axis). The number of terms in the infinite series in eqn (33) is 500, unless otherwise specified. Note that this value is used under the verification of a sufficient convergence of the numerical results.

4

Results and discussion

4.1 Examination of the number of partitions In order to estimate the accuracy and usefulness of the present analytical solution, which is obtained from the approximation of continuous change as a piecewise constant in the fluid velocity distribution, we first consider the most basic Graetz problem of Br = 0, H = ∞ and ν = 1, or the case in which a Newtonian fluid flows without viscous dissipation between parallel plates that are maintained at a constant temperature. This case has already been analyzed quasi-analytically by Rosales et al. [3]. Figure 3 shows a comparison of the fluid temperature distribution calculated by the present solution with that in reference [3]. A smaller value of ξ has a larger effect of the number of partitions n on the temperature distribution. For n = 5, the temperature distribution at ξ = 10–4 fluctuates widely. The temperature distribution obtained from the present analytical solution with n = 20 WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

30 Advanced Computational Methods in Heat Transfer IX is in good agreement with that of reference [3] for all ξ. Table 1 shows a comparison between the local Nusselt numbers of the present solution and reference [3]. For n = 20, the difference between them is less than 0.1% in the range of 10–4 < ξ < 100. Consequently, as long as the local Nusselt number is discussed in this range, n = 20 provides sufficiently accurate results. Note that all of the following results are obtained from the analytical solution with n = 20. Table 1:

Comparison of local Nusselt numbers.

ξ

Number of eigenvalues used

10–4 10–3 10–2 10–1 100

500 200 200 200 200

Nu [Present (n = 20)] 16.661 5 7.749 2 3.692 6 2.047 66 1.885 14

[Rosales et al.] 16.668 7 7.751 3 3.693 4 2.047 82 1.885 17

1 10-4

0.8 n=10,20

0.6 θ

n=5,10,20 n=5 n=10 n=20 Rosales et al. [3]

0.4 0.2 0

n=5,10,20

0

0.2

10-3

10-2 0.1

ξ=1

0.4

0.6

0.8

1

η

Figure 3: Relationship between the number of partitions n and convergence of dimensionless temperature θ. 4.2 Effects of parameters on local Nusselt number distribution Figure 4 illustrates the effects of the viscous dissipation, heat transfer coefficients of the external surfaces and rheological character of the fluid on the local Nusselt number around the entrance of the conduit. The Nusselt numbers at ξ = 10–4 are the largest in figs. 4(a) and (b). For the Brinkman number Br = 0, the Nusselt numbers decrease simply with an increase in ξ and finally converge on the Nusselt numbers in the developed temperature field. On the other hand, for Br ≠ 0, the Nusselt numbers do not necessarily decrease monotonously, and they show a change from a decreasing to an increasing trend, especially for small values of Br. However, they converge to a certain value, regardless of the value of Br. While the Nusselt numbers at ξ = 10–4 increase with a decrease in the Biot number H, the convergent value of the Nusselt number for Br ≠ 0 is WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


31

independent of the Biot number, only depending on ν. Smaller values of ν tend to result in higher local Nusselt numbers throughout the conduit. 40

25 H=1

35

15

20

Nu

Nu

25 1

15 10 5 0 -4 10 40

10-3

10-2

10-1

100

0 -4 10 25

101

H=10

0.5

10-3

10-2

10-1

Br=0

100

101

H=10

20

25

15

20

Nu

Nu

1

Br=0

30

15 1

10-3

10-2

10 5

5

10 5

10-1

5

0.5

Br=0

100

0 -4 10 25

101

H=∞

0.5

1

10-3

10-2

Br=0

10-1

100

101

H=∞

20 15

Nu

Nu

10 5

0.5

35

0 -4 10 40 35 30 25 20 15 10 5 0 -4 10

H=1

20

30

5

10

-3

10

-2

ξ

1

10

-1

Br=0

10

10

0

10

1

0 -4 10

(a) ν = 0.3 Figure 4:

5 1

5

0.5

10

-3

10

-2

ξ

10

-1

0.5 Br=0

10

0

10

(b) ν = 3

Nusselt number around the channel entrance.

4.3 Critical fluid velocity diagram The critical fluid velocity for avoiding solidification within the conduit is obtained in the case where the temperatures of the surrounding media are below the solidification temperature of the flowing fluid; this highlights the applicability of the present analytical solution in the field of industry. Figure 5 shows the relationship between the dimensionless solidification temperature and the dimensionless length of the freeze-free zone with different Biot numbers and Brinkman numbers. This figure makes use of the temperature distribution in the direction of the ξ coordinate at η = 1. For Br = 0, it can be seen that θf is asymptotic to 1 as ξf approaches 0 and θf is asymptotic to 0 as ξf approaches infinity. The extent of the approach depends on the value of H. This result is the same as that reported by Sadeghipour et al. [4], who investigated the critical fluid velocity of a pipe flow. For Br ≠ 0, the effect of Br becomes significant as H decreases and ξf increases. The effect of ν on the critical fluid velocity is considerably smaller than that of H or Br.


1

32 Advanced Computational Methods in Heat Transfer IX 1.5

1.5 Br=0 Br=0.5 Br=1

Br=0 Br=0.5 Br=1

0.1

0.1

1

1

0.5

θf

θf

0.5 1

1

5

0.5

0.5

5

10

10

H=50

0 -4 10

H=50

-3

10

10

-2

ξf

10

-1

0

10

1

10

0 -4 10

10

(a) ν = 0.3 Figure 5:

5

-3

-2

10

-1

ξf

10

0

10

1

10

(b) ν = 3

Variation of freeze-free zone with dimensionless solidification temperature for different Biot numbers and Brinkman numbers.

Conclusions

The forced convection heat transfer problem with viscous dissipation in a conduit between parallel plates subjected to heat loads from the surrounding has been solved mathematically by Vodicka’s method, which is a type of integral transform method, and completely explicit analytical solutions of the fluid temperature and local Nusselt number have been derived. The conclusions obtained through numerical calculations are summarised as follows: (1) With regard to the Graetz problem in the case of a conduit between parallel plates, the number of partitions in the conduit should be over 20 in order to obtain a sufficiently accurate local Nusselt number. (2) The local Nusselt number in the thermal entry region tends to increase with a decrease in the Biot number and power law model index. (3) The effect of the power law model index i.e., velocity distribution of the fluid, on the critical fluid velocity for avoiding solidification is considerably smaller than that of the Biot number or Brinkman number.

References [1] Vodicka, V., Linear heat conduction in laminated bodies. Mathematische Nachrichten, 14(1), pp. 47-55, 1955. (in German) [2] Mikhailov, M.D. & Ozisik, M.N., Unified analysis and solutions of heat and mass diffusion, Dover Publications: New York, pp. 344, 1994. [3] Rosales, M.A. & Frederick, R.L., Semi analytic solution to the Cartesian Graetz problem: results for the entrance region. International Communications in Heat and Mass Transfer, 31(5), pp. 733-740, 2004. [4] Sadeghipour, M.S., Ozisik, M.N. & Mulligan, J.C., Transient freezing of a liquid in a convectively cooled tube. Trans. ASME Journal of Heat Transfer, 104(2), pp. 316-322, 1982.



33

Heat transfer by unsteady laminar mixed convection in 2-D ventilated enclosures using the vorticity-stream function formulation S. Boudebous & Z. Nemouchi Département de Génie Mécanique, Université Mentouri de Constantine Algéria, Algeria

Abstract In this work, a numerical study is presented of mixed laminar convection in ventilated enclosures. The left vertical wall of enclosure is maintained at a constant temperature greater than that of the fluid at entry, while the other walls are adiabatic. Two cases of ventilation are considered. In the first case, the fluid enters from the bottom left corner and leaves the domain through the right upper corner. In the second case, the fluid enters from the bottom left corner and leaves the domain through the middle of the upper wall. The equations governing the phenomenon are discretised using the finitedifference method. A computer programme is developed to simulate the flow behaviour and the heat transfer in the enclosure. Velocity and temperature fields are obtained. These numerical simulations are performed for a Grashof number of 106, a Prandlt number of 0.7 and three values of the Richardson number (0.5, 5, and 25) Keywords: mixed convection, vorticity-stream function formulation, square enclosure, numerical method.

1

Introduction

An interesting review on natural convection in enclosures can be found in Fusegi and Hyun [1]. Concerning the subject of the present work, numerous investigations have been reported. We can cite as examples: Unsteady 2-D hot water flow for energy extraction from a storage system, Cha and Jaluria [2] with cold water inlet at the bottom of the left vertical wall and hot water exit at the top of the same or opposite wall. Unsteady 2-D flow of air, Raji WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/HT060041

34 Advanced Computational Methods in Heat Transfer IX and Hasnaoui [3] with inlet at the top of the left vertical wall and exit at the top or bottom of the opposite wall. Transient mixed convection, Omri and Ben Nasrallah [4] with air inlet at the top (at the bottom in a second case) of the hot left wall and exit at the bottom (at the top in the second case) of the right cold wall. Air flow with openings at the top of the vertical walls with one or two heat sources of zero thickness imbedded on a vertical board of finite thickness placed on the bottom wall, Hsu and Wang [5]. In the present study, laminar mixed convection flow of air in a ventilated cavity has been investigated numerically. The physical model under consideration and coordinates chosen are depicted in figure 1. Outlet

Outlet Adiabatic wall

Y,V

g

Adiabatc wall

Case A

Hot wall

Adiabatc wall

Hot wall

Case B

X,U

Inlet

Inlet

Adiabatic wall

Xo

Xo

Figure 1:

2

Adiabatic wall

Geometrical configuration.

Governing equations

The flow and heat transfer phenomena to be investigated here are described by the complete Navier-Stokes and energy equations for two-dimensional laminar incompressible flows. The viscous dissipation term in the energy equation is neglected and the Boussinesq approximation is invoked for the buoyancy induced body force term in the Navier-Stokes equations. From the governing equations of mass, momentum conservations, the vorticity-stream function formulation may be obtained by defining the stream function and vorticity, as, respectively, ∂ψ ∂ψ ∂V ∂U − ω= (1) U= V =− ∂Y ∂X ∂X ∂Y Hence, the equations in dimensionless form can be written as follows. Stream function equation ∂ 2ψ ∂X

2

+

∂ 2ψ ∂Y 2

= −ω


(2)


35

Vorticity-transport equation ∂θ ∂ω 1  ∂ 2 ω ∂ 2 ω  ∂ω ∂ω = + + Ri +V +U 2 2   ∂X ∂Y Re  ∂X ∂X ∂τ ∂Y 

(3)

Energy equation 1  ∂ 2θ ∂ 2θ ∂θ ∂θ ∂θ = + +V +U ∂Y Re Pr  ∂X 2 ∂Y 2 ∂X ∂τ

   

(4)

where Re, Ri and Pr denote, respectively, Reynolds number, Richardson number and Prandlt number. They are defined as Re =

Vo L

Pr =

ν

ν α

Ri =

Gr

Re 2

Gr =

gβ (Tw − To )L3

(5)

ν2

Here Gr is the Grashof number. The other dimensionless parameters are defined as follows:

X =

x L

Y=

y L

U=

u Vo

V=

v Vo

θ=

T − To T w − To

τ=

Vo t L

(6)

where L is the characteristic length of the cavity, Vo is the inlet velocity, ν is the kinematic viscosity, α is the thermal diffusivity, g is the acceleration of gravity, β is the thermal expansion coefficient, Tw is the wall temperature, To is the temperature of the fluid at the entry and t is the time. For the problem geometry, the following boundary conditions are specified:

U = 0. V = 0. θ = 1. Ψ = 0. at X = 0.

U = 0. V = 0.

∂θ = 0. Ψ = Ψ xo ∂X

at

X = 1.

0 ≺ Y ≺ 1.

0 ≺ Y ≺ 1.

U = 0. V = 1. θ = 0. Ψ = − X ω = 0. at Y = 0. 0 ≺ X ≺ XO


(7)

(8)

(9)

36 Advanced Computational Methods in Heat Transfer IX ∂θ = 0. Ψ = Ψ xo ∂Y

U = 0. V = 0.

U = 0. V = 0.

at Y = 0.

∂θ = 0. Ψ = 0. at ∂Y

∂ 2Φ ∂Y 2

= 0. at

Y = 1.

Y = 1.

X O ≺ X ≺ 1.

(10)

0 ≺ X ≺ 1. − X O

(11)

1. − X O ≺ X ≺ 1.

(12)

where Φ stands for θ , ω , ψ , U and V . The width of the inlet and the outlet Xo is equal to 0.1. In general, the value of vorticity on a solid boundary is deduced from Taylor series expansion of the stream function around the solid point and can be expressed mathematically as

ω wall = −

∂ 2ψ

(13)

∂n 2

where n is the outward drawn normal of the surface. The convective heat transfer from the heated wall can be characterized by an average Nusselt number, Num, defined as Num =

3

1

∂θ 

∫ − ∂X  0

dY X =0

Numerical procedure

The governing equations (2-4) along with the boundary conditions (7-13) are solved numerically, employing finite-difference techniques. The Alternating Direction Implicit (A.D.I.) method of Peaceman and Rachford in [6] is used for time marching. The buoyancy and diffusive terms are discretized by using central differencing while the use of a third-order upwind scheme [7] is preferred for convective terms. Convergence of iteration for the stream function solution is obtained at each time step. The resulting set of finite difference equations is then solved by using the Non Linear Over Relaxation (N.L.O.R.) method [8]. All computations are performed using non-uniform grids with denser clustering near the walls where boundary layers develop and high gradients are expected [9]. The solutions were initially tested with mesh sizes of 81 x 81, 101 x 101, 121 x 121 and 151 x 151. It was found that variations in the solution fields were not significant (of the order of 1% in the mean Nusselt number obtained) between



37

mesh sizes of 121 x 121 and 151 x 151. As a compromise between accuracy and CPU time, the mesh size of 121 x 121 is used for all calculations. The validity of the computer code developed has been checked for the sudden expansion of an oblique velocity field in a cavity [10] and the buoyancy-driven cavity flow [11]. The various sets of results compare very well and are nearly identical, confirming the credibility of the code.

4

Numerical results and discussion

The main characteristics of the flow and energy transport for each Richardson number (Ri) and for each case of ventilation will be shown in the following. The mean Nusselt numbers (Num) are shown as plots versus the time (figure 2). Flow and temperature fields are shown in terms of stream traces, isotherms, velocity and temperature profiles (figures 3-6). 40

40 Ri=0.5

Case A

Ri=0.5

Case B

Ri=5.0

Ri=5.0

Ri=25.0

Ri=25.0

Num

30

Num

30

20

20

10 0E+0

2E+5

4E+5

6E+5

Figure 2:

8E+5

10 1E+6 0E+0

2E+5

4E+5

6E+5

8E+5

1E+6

Mean Nusselt numbers versus time.

In both cases A and B, figure 2 shows that as Ri increases, Num decreases implying that the more the forced convection is dominant, the more important is the heat flux from the hot wall. Figure 3 shows the stream traces and the profiles of the velocity components U and V in the X and Y directions respectively. As Ri increases, a boundary layer forms along the hot vertical wall. The relatively fast circulation in the center of the enclosure for Ri=0.5 becomes slower for Ri=5 and 25. Consistently with this result, the isotherms and the temperature profiles (figure 4) show that an essentially conductive heat transfer for Ri=25 takes over a forced convection for Ri=0.5. It is clear that in case B the effect of Ri on the flow and the heat transfer is very similar to that in case A (figures 5 and 6)



0.9

0.9

0.8

0.8

0.7

0.7

X=0.25

0.6

0.6

X=0.50

0.5

0.5

X=0.75

Y

1.0

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0.0

0.0

V

0.4

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 X

-0.1 -0.2

Y=0.25

-0.3

Y=0.50

-0.4

Y=0.75 -0.5 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 U

Ri=0.5 1.0

1.0

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

Y

V

1.1

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0.0

0.0

-0.1 -0.1

Ri=5

0.0

0.1

0.2

0.3

0.4

0.5 X

0.6

0.7

0.8

0.9

1.0

0.0

0.1

0.2

0.3

0.4

0.5 U

0.6

0.7

0.8

0.9

1.0

0.8

0.9

1.0

1.1

1.5 1.4 1.3 1.2 1.0

1.0

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

Y

V

1.1

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0.0

0.0

-0.1

0.0

0.1

0.2

0.3

0.4

0.5 X

0.6

0.7

-0.2 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 U

Ri=25 Figure 3:

Stream traces (left) and Velocity profiles (right) case A.



39

1.0 X=0.25 0.9

X=0.5 X=0.75

0.8 0.7

Y

0.6 0.5 0.4 0.3 Y=0.25 0.2

Y=0.5 Y=0.75

0.1 0.0

Ri=0.5

0.0

0.1

0.2

0.3

0.4

0.5 X

0.6

0.7

0.8

0.9

1.0

θx

1.0 0.9 0.8 0.7

θy

Y

0.6 0.5 0.4 0.3 0.2 0.1 0.0

Ri=5

0.0

0.1

0.2

0.3

0.4

0.5 X

0.6

0.7

0.8

0.9

1.0

0.0

0.1

0.2

0.3

0.4

0.5 X

0.6

0.7

0.8

0.9

1.0

1.0 0.9 0.8 0.7

Y

0.6 0.5 0.4 0.3 0.2 0.1 0.0

Ri=25 Figure 4:

Isotherms (left) and Temperature profiles (right) case A.


V


0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

Y

1.0

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0.0

0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 X

-0.1 -0.2 -0.3

-0.4 -0.6-0.5-0.4-0.3-0.2-0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 U

Ri=0.5

1.1 1.0 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0.0

0.0

V

Y

1.0 0.9

-0.1

0.0

0.1

0.2

0.3

0.4

0.5 X

0.6

0.7

0.8

0.9

1.0

-0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 U

Ri=5 1.6 1.5 1.4 1.3 1.2 1.0

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

Y

V

1.1 1.0

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0.0

0.0

-0.1

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 X

-0.2 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 U

Ri=25 Figure 5:

Stream traces (left) and Velocity profiles (right) case B.



41

1.0 0.9 0.8 0.7

Y

0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.0

Ri=0.5

0.1

0.2

0.3

0.4

0.5 X

0.6

0.7

0.8

0.9

1.0

θx 1.0 0.9 0.8 0.7

Y

θy0.6 0.5 0.4 0.3 0.2 0.1 0.0

Ri=5

0.0

0.1

0.2

0.3

0.4

0.5 X

0.6

0.7

0.8

0.9

1.0

0.0

0.1

0.2

0.3

0.4

0.5 X

0.6

0.7

0.8

0.9

1.0

1.0 0.9 0.8 0.7

Y

0.6 0.5 0.4 0.3 0.2 0.1 0.0

Ri=25 Figure 6:

Isotherms (left) and Temperature profiles (right) case B.



References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]

Fusegi, T. & Hyun, J.M., Laminar and transitional natural convection in an enclosure with complex and realistic conditions, Int. J. Heat Fluid Flow. 3, pp. 258-268, 1994 Cha, C. K. & Jaluria, Y., Recirculating mixed convection flow for energy extraction. Int. J. Heat Mass Transfer. 27, pp. 1801-1812, 1984 Raji, A. & Hasnaoui, M., Correlations en convection mixte dans des cavités ventilées. Rev. Gen. Therm. 37, pp. 874-884, 1998 Omri, A. & Ben Nasrallah, S., Control volume finite element numerical simulation of mixed convection in an air-cooled cavity. Numerical Heat Transfer, Part A, vol.36, pp. 615-637, 1999. Hsu, T. H. & Wang, S. G., Mixed convection in a rectangular enclosure with discrete heat sources. Numerical Heat Transfer, Part A, vol.38, pp. 627-652, 2000. Peaceman, D.W. & Rachford, H.H., Numerical solution of parabolic and elliptic differential equations, J. Soc. Indust. Appl. Math. 3, pp. 28-41, 1955. Kawamura, T. & Kawamura, K., Computations of high Reynolds number flow around a circular cylinder with surface roughness, AIAA Paper No 84-0340, 1984. Sewell, G., The numerical solution of ordinary and partial differential equations, Academic Press, INC, New-York, 1988. Manole, D.M. & Lage, L.L., Nonuniform grid accuracy test applied to the natural convection flow within a porous medium cavity, Numerical Heat Transfer, Part B,vol.23,pp. 351-368, 1993. Song, B., Liu, G.R. & Lam, K.Y., Four-point interpolation schemes for convective fluxes, Numerical Heat Transfer, Part B,vol.35,pp. 23-39, 1999. Rahman, M.M., Miettinen, A. & Siikonen, T., Modified SIMPLE formulation on a collocated grid with an assessment of the simplified Quick scheme, Numerical Heat Transfer, Part B,vol.30,pp. 291-314, 1996.



43

Effect of roughness shape on heat transfer and flow friction characteristics of solar air heater with roughened absorber plate A. Chaube1, P. K. Sahoo2 & S. C. Solanki2 1 2

GEC, Jabalpur, India IIT Roorkee, India

Abstract A 2-D computational analysis is carried out to assess the comparative performance of the absorber plate of a solar air heater with different roughness elements using commercial software package Fluent 6.1. The assessment is based on heat transfer enhancement with minimum pressure penalty. Ten different rib shapes (viz. Rectangular, Square, Chamfered, Triangular, Semicircle etc) are investigated at the Reynolds number range from about 3000–20000, in which solar air heaters normally operate. The SST kω turbulence model is selected by comparing the predictions of different turbulence models with experimental results available in the literature. Using the selected turbulence model a computational analysis is carried out to predict the heat transfer performance and flow friction characteristics of absorber plates with 10 different rib roughness elements. The analysis is carried out for both the flow regimes i.e. transitional flow regime (5 ≤ e+ ≤ 70) and fully rough regime (e+ ≥ 70). Keywords: turbulence, roughness Reynolds number, aspect ratio, friction factor, heat transfer coefficient, Nusselt number, chamfer angle, pitch etc.

1

Introduction

Artificial roughness up to laminar sub-layer to enhance heat transfer coefficient is used in various applications like gas turbine blade cooling channels, heat exchangers, nuclear reactors and solar air heaters. A number of experimental studies [1, 2] in this area have been carried out but very few attempts of numerical investigation have been made so far due to complexity of flow pattern and computational limitations. In the present work, an attempt is made to predict WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/HT060051

44 Advanced Computational Methods in Heat Transfer IX numerically, the details of both the velocity and temperature fields responsible for heat transfer enhancement. The presence of rib may enhance heat transfer because of interruption of the viscous sub layer, which yields flow turbulence, separation and reattachment leading to a higher heat transfer coefficient. The enhancement of heat transfer by flow separation and reattachment caused by ribs is significantly higher compared to that by the increased heat transfer area due to ribs (fin-effect) [3]. The heat transfer measurements results for two different steps, p/e = 14 and p/e = 8, indicate the importance of roughness geometry [4]. Liou et al. [6] have performed both the numerical analysis and experimental study to investigate the heat transfer and fluid flow behavior in a rectangular channel flow with stream wise periodic ribs mounted on one of the principal walls. They have concluded that the flow acceleration and the turbulence intensity are two major factors that influence the heat transfer coefficient. The combined effect is found to be optimum for the pitch to rib height ratio equal to 10, which results in the maximum value of average heat transfer coefficient. Rau et al. [5] experimentally found optimum pitch to rib height ratio to be equal to 9. Hence these investigations reveal that not only the rib geometry but also its geometrical arrangement play vital role in enhancing the heat transfer coefficient. Karwa [7] has reported an experimental investigation for the same configuration for the Reynolds number range of 4000-16000. Tanda [9] has reported experimental investigation of heat transfer in a rectangular channel with transverse and V-shaped broken ribs using liquid crystal thermography. He concluded that features of the inter-rib distributions of the heat transfer coefficient are strongly related to rib shape and geometry; a relative maximum is typically attained down stream of each rib for continuous transverse ribs (due to flow reattachment). The main aim of the present analysis is to investigate the flow and heat transfer characteristics of a 2 dimensional rib roughened rectangular duct with only one principal (broad) wall subjected to uniform heat flux by making use of computer simulation. The ribs are provided only on the heated wall. The other three walls are smooth (without ribs) and insulated. Such a case is encountered in solar air heaters with artificially roughened absorber plate.

2

Solution domain

The solution domain shown in Fig. 1 has been selected as per the experimental details given by Karwa [7]. A rectangular duct with the duct height(H) of 40 mm, rib height (e) of 3.4 mm, rib width of 5.8 mm and pitch (p) of 34 mm has been taken for analysis. The uniform heat flux of 4 kW/ m2 is given on ribbed surface. A 2-D CFD analysis of heat transfer and fluid flow through a high aspect ratio (7.5) rectangular duct with transverse ribs provided on a broad, heated wall and other three walls smooth and insulated, is carried out using Fluent 6.1 software. A non-uniform rectangular mesh with grid adoption for y+=1 at adjacent wall region is applied as shown in Fig. 2. Similar analysis is carried out for a smooth duct of same dimensions for similar range of Reynolds number 3000 to 15000 to



45

find out the ratio of Stanton number for ribbed duct and smooth duct as experimentally determined by Karwa [7].

Figure 1:

Duct geometry and solution domain as per experimental details of Karwa [7].

Figure 2:

Rectangular mesh with grid adoption for y+=1 at top and bottom wall.

2.1 Selection of turbulence model To select the turbulence model, the previous experimental study is simulated using different low Reynolds number models such as Standard kω model, Renormalization-group kε model, Realizable kε model and Shear stress transport kω model. The results of different models are compared with experimental results. The shear stress transport kω model is selected on the basis of its closer results to the experimental results as shown in Fig. 3. 2.2 Thermo hydraulic assessment of different roughness elements After finding out satisfactory simulation capability of the SST kω turbulence model, the performance of different rib shapes as shown in Fig.4 have been WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

46 Advanced Computational Methods in Heat Transfer IX assessed. The similar flow domain used for the predictions is as shown in Fig.1. The other parameters used are as follows (for transitionally rough flow regime): Reynolds Number =2900 to e+ Range = 7 to 84 Range 19500 Duct height (H) =20 mm Pitch (p) = 10 mm Inlet length =225 mm Rib Height (e) = 1 mm Length of test section =121 mm Aspect Ratio of Duct = 5 (AR) Outlet length =115 mm Uniform Heat Flux = 1000W/m2

2.3 For fully rough flow regime e+ Range

= 35 to 400

Rib Height

= 3 mm

Turbulence Model Selection 0.006 0.0055 Stanton No.

0.005 0.0045 0.004 0.0035 0.003 0.0025 0.002 0

5000

10000 15000 20000 25000 Re

Experimental Results[7] Standard kω Realizable kε Figure 3:

Shear Stress Transpot kω Renormalization-group kε

Comparison between experimental and computational predictions from different low Re turbulence models.

2.4 Boundary conditions The following boundary conditions are given through the boundary conditions panel: (i) Velocity at inlet (ii) Turbulence intensity at inlet (20% taken from literature) WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


(iii) (iv) (v) (vi)

Figure 4:

3

47

Hydraulic diameter of duct Out let pressure (Atmospheric pressure) Constant heat flux on broad bottom surface of test section No heat transfer from other walls of the duct.

Different roughness elements under investigation.


The comparative performance of different roughness elements are obtained on the basis of heat transfer enhancement at constant pumping power requirement or performance index as described below:

η=

St Sto

( ) f fo

1

3


48 Advanced Computational Methods in Heat Transfer IX Performance index Vs Re in Transitionally Rough Flow Regime 1.5 1.4

Performance Index

1.3 1.2 1.1 1 0.9 0.8 0

5000

10000

15000

20000

25000

Re Square 1x1

Rectangular1.5x1

Rectangular2x1

Rectangular2.5x1

Chamfered 9

Chamfered 11

Chamfered 13

Chamfered 15

Semicircular r=1

Triangular h=1, b=1

Figure 5:

Variation of heat transfer enhancement for constant pumping power requirement with Reynolds number.

Performance index Vs Re in Fully Rough Regime 1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 0

5000

10000

15000

20000

25000

Re

Rectangular 2x3

Square 3x3

Rectangular 4x3

Rectangular 5x3

Chamfered 11

Chamfered 13

Chamfered 15

Chamfered 17

Semicircular r=3

Triangular h=3, b=3

Figure 6:

Variation of heat transfer enhancement at constant pumping power with Reynolds number.



49

The comparison of the performance by different rib geometries for equal pumping power would be more useful for engineering applications. The performances of different roughness elements on the basis of equal pumping power for transitionally rough flow regime and fully rough flow regime are shown in Figs. 5 and 6 respectively. In transitionally rough flow a substantial enhancement in heat transfer is found along with increase in friction factor, but in fully rough flow a marginal enhancement in heat transfer over that in the transitionally rough flow is found with sharp increase (more than double in comparison to transitionally rough flow regime) in friction factor. Therefore a low value of performance index is found in fully rough flow regime in comparison to transitionally rough flow regime. In the transitional flow regime rectangular rib of 1 mm height and 2 mm width shows best performance among all rib shapes under investigation. Among the rest of ribs, the triangular rib of 1mm height and 1 mm base gave the best performance. In the fully rough flow regime rectangular rib of 3 mm height and 4 mm width is found as best performer under constant pumping power conditions. Although the square rib (3x3) gives the best heat transfer characteristics for constant mass flow rate condition. The effects of various flow and roughness parameters on heat transfer and friction characteristics for flow of air in a rectangular duct of aspect ratio 5 under the present investigation are being discussed below: 3.1 Variation of Nusselt number with Reynolds number In all cases Nusselt number increases with the increase of Reynolds number. The rate of increase of Nusselt number with Reynolds number is substantially higher in roughed duct in comparison to that of in smooth duct of similar dimension. The effect of roughness on variation of Nusselt number with the Reynolds number may be explained as under: At low Reynolds number in transitionally rough flow the roughened surface Nusselt numbers are nearly those of smooth surfaces. It is because the roughness elements lie within the laminar sub layer, which is the major component of the heat transfer resistance. As the Reynolds number increases, the roughness elements begin to project beyond the laminar sub layer because the boundary layer thickness decreases with an increase in Reynolds number. This reduction in boundary layer thickness increases the heat transfer rate. In addition to this, there is local contribution to the heat removal by the vortices originated from the roughness element. Thus the Nusselt number curve deviate from the smooth duct turbulent Nusselt number curve. When the Reynolds number further increases, the roughness elements project deeper in to the turbulent region. Finally, with the increase of Reynolds number, the thickness of laminar sub layer becomes very small and energy loss due to the vortices now attains a constant value and is independent of viscous effect.


50 Advanced Computational Methods in Heat Transfer IX 3.2 Variation of friction factor with Reynolds number The shedding of vortices also causes additional loss of energy resulting in increased friction factor. Thus the friction factor curves deviate from the smooth duct friction factor curve. Except semicircular rib, in all the cases friction factor increases with increase of Reynolds number in both the flow regimes. In almost all the cases friction factor becomes independent of Reynolds number at high flow rates. The Reynolds number corresponding to the start of the region of independence of friction factor is termed as critical Reynolds number. 3.3 Effect of rib width In the transitionally rough flow regime increasing the width of square rib enhancement of heat transfer and reduction in friction factor is observed. Where as in fully rough flow regime also the best performance index is found for 4x3 rib. Hence for every flow condition there is an optimum w/e ratio. In case of transitionally rough flow 2x1 rib gives the best performance index. The further increase in width gives negligible or adverse effect. 3.4 Effect of chamfer angle It is observed that both the flow regimes, transitionally rough and fully rough, the effect of chamfer angle in enhancement of heat transfer is very small in comparison to the enhancement of friction factor. That is why chamfered rib roughness did not exhibit the comparable performance index with other types of roughness.

4

Conclusions

The following conclusions are drawn from the present analysis: The Shear Stress Transport kω turbulence model predicted very close results to the experimental results, which yields confidence in the predictions done by CFD analysis in the present study. In transitionally rough flow a substantial enhancement in heat transfer is found along with increase in friction factor, but in fully rough flow a marginal enhancement in heat transfer over that in the transitionally rough flow is found with sharp increase (more than double in comparison to transitionally rough flow regime) in friction factor. The reattachment point and point of maximum heat transfer coincide, which shows the great influence of reattachment of flow on convective heat transfer coefficient. The results show that in rectangular ribs, there is an optimum width of rib, at which it gives maximum heat transfer with minimum pressure drop penalty.



51

In continuous transverse ribs chamfering is not much effective because although it gives small enhancement in heat transfer but at the cost of steep rise in friction factor. The thermohyraulic analysis for constant pumping power shows that rectangular rib (1mm x 2mm) gives the best performance in comparison to other shapes under consideration.

References [1]

[2] [3] [4] [5] [6]

[7]

[8] [9] [10] [11]

Muluwork, K.B., Solanki, S.C., and Saini, J.S., 2000, “ Study of heat transfer and friction in solar air heaters roughened with staggered discrete ribs,” Proc. 4th ISHMT- ASME Heat and Mass Transfer Conf., Pune, India, pp.391-398. Karwa, R., Solanki, S.C., and Saini, J.S., 2001, “ Thermo-hydraulic performance of solar air-heaters having integral chamfered rib roughness on absorber plates,” Energy 26, 161-176 (2001). Lee, C.K., and Abdel-Moneim, S.A., 2001, “ Computational Analysis of Heat Transfer in Turbulent Flow Past a Horizontal Surface with 2-D Ribs,” Int. Comm. Heat Mass Transfer, Vol. 28 No.2, pp. 161-170. Slanciauskas, A., 2001, “ Two friendly rules for the turbulent heat transfer enhancement” Int. Jr. of Heat and Mass transfer 44, 2155-2161. Rau, G., Cakan, M., Moeller, D., and Arts, T., 1998, “The Effect of Periodic Ribs on the Local Aerodynamic and Heat Transfer Performance of a Straight Cooling Channel,” ASME Vol. 120, pp. 368-375. Tong-Miin Liou, Jenn-Jiang Hwang and Shih-Hui Chen, “Simulation and measurement of enhanced turbulent heat transfer in a channel with periodic ribs on one principal wall” Int. Jr. Heat Mass Transfer, Vol. 36, pp. 507-517 (1993). Karwa, R., 2003, “ Experimental Studies of Augmented Heat Transfer and Friction in Asymmetrically heated Rectangular Ducts with Ribs on the Heated Wall in Transverse, Inclined, V-Continuous and V-Discrete Pattern,” Int. Comm. Heat Mass Transfer, Vol. 30, No. 2, pp. 241-250. Fluent 6.0 User’s Guide Vol. 2. Tanda, G., “ Heat transfer in rectangular channels with transverse and Vshaped broken ribs” Int. Jr. Heat Mass Transfer, Vol. 47 pp. 229-243, (2004). Davidson, Lars, 1997 “An Introduction to Turbulence models,” Chalmers publication. 11.Versteeg, H.K., and Malalasekera, W., 1995, “An Introduction to Computational Fluid Dynamics”, Longman publication.




53

Study of conjugate heat transfer accompanying mixed convection in a vertical tube submitted to a step of entry temperature O. Kholai1, 2, M. Kadja2 & T. H. Mai1 1

Laboratoire de Thermomécanique, Faculté des sciences, Reims, France Département de Génie Mécanique, Université de Constantine, Constantine, Algeria 2

Abstract The paper presents the results of a numerical study on transient mixed laminar convection for the case of an ascending flow inside a vertical tube in which the wall conduction is significant. The outer surface of the tube is submitted to a convective heat exchange. The transient regime is provoked by a step type perturbation in temperature at entry of the tube. The governing equations are solved using a finite volume method. Results are presented for water (Pr=5) and for two values of Grashof numbers of 104 and 105. This method shows that no matter what the value of the imposed step is, it provokes the birth of a big recirculation cell next to the wall, and gives rise to an instability situated between the cool and the hot regions of the flow. Keywords: transient convection, temperature step, convective heat exchange, finite volume method, instabilities.

1

Introduction

Conjugate heat transfer (mixed convection + conduction) in transient regime in internal flows is present in many industrial installations: such as compact heat exchangers, solar collectors, cooling of electronic components etc. As a matter of fact, the importance and complexity of this phenomenon and the consequences have stimulated the interest of many researchers: so that a very important number of studies have been done in this field. Nguyen et al. [1] performed a numerical study on transient mixed convection in vertical tubes having a wall, which is submitted to variable heat flux. Their WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/HT060061

54 Advanced Computational Methods in Heat Transfer IX results indicate that the increase in heat flux gives rise to a recirculation zone, which increases in intensity and size as a function of time. Experimentally Morton et al. [2] have confirmed the existence of a recirculation cell near the tube axis. For conjugate heat transfer problems (conduction in the wall and convection in the fluid, inside a tube), Bilir and Ates [3] performed a parametric study in a vertical tube submitted to convective exchange at the outer surface. Bernier and Baliga [4, 5] studied the problem of mixed convection by taking into account wall conduction for an ascending flow where the tube is submitted to a uniform heat flux. They have shown that axial diffusion of heat through the tube wall becomes very big for high conductivity and thickness of the wall and for a low Peclet number. Recirculation zones and instability have been shown to occur by Zgal et al. [6] and Su and Chung [7]. More recently Mai et al. [8], Popa and Mai [9] and Mai et al. [10] have focussed attention on a more complex problem of heat transfer instability. They have shown that a perturbation of mixed convection in a vertical tube via a step rise in temperature (of mass flow rate) at entry, gives rise to instabilities of thermo-hydraulic flow structure and to the creation of recirculation cells inside the fluid. The present study consists in analysing the phenomenon of transient instability of a flow in mixed laminar convection in a circular vertical duct when the entry and the outer surface of the duct are submitted to a step change in temperature (positive or negative) and to a convective exchange with ambient air, respectively. The influence of these two conditions and also the ratio of thermal diffusivities (wall/fluid) on the transient behaviour of the flow are studied in detail.

2

Problem formulation

The considered problem consists in studying conjugate heat transfer for a laminar flow inside a vertical tube submitted to convective exchange with air through a section of the outer wall. The wall thickness is taken into consideration and is equal to δ=0.1D. Adiabatic sections have been added upwind and downwind of the cooled section so as to permit the study of longitudinal thermal diffusion in the fluid and the wall. The configuration and the coordinate system of the tube are given in Fig. 1. The governing equations of this problem are: the continuity equation, the Navier-Stokes equations and the equation of energy. The study is based on the following simplifying assumptions: -The fluid is Newtonian and incompressible -The thermo-physical properties of the fluid and the solid are constant -The fluid obeys the Boussinesq approximation: density is constant except in the natural convection generating terms (density varies linearly with temperature) -The flow is laminar and in transient regime -Viscous dissipation and radiation heat transfer are negligible. By taking these assumptions into account and adopting the following dimensionless variables:



R=

r , D

Z=

z , D

U=

55

T−T u v 0 , V= , θ= U U T −T av av in 0

p + ρ gz t α 0 , τ= , and α * = 2 D U α ρ U av f 0 av where u and v are the velocity components in the z and r directions respectively, T is the temperature, p is the pressure, D is the diameter of the tube, t is the time, Uav is the mean axial velocity at inlet. g is the acceleration due to gravity. T0 is the ambient temperature. ρ0 is the fluid density at T0. The index ‘in’ stands for inlet and ‘f’ for fluid. The governing equations can be written in the following form in cylindrical coordinates: ∂U 1  ∂RV  +  (1) =0 ∂Z R  ∂R  P=

(

)

∂U ∂U ∂U ∂P 1  ∂ 2 U 1 ∂  ∂U  Gr +U +V =− + + θ R  +  ∂Z ∂R ∂Z Re  ∂Z 2 R ∂R  ∂R  Re 2 ∂τ

(2)

∂V ∂V ∂V ∂P 1  ∂ 2 V 1 ∂  ∂V  V  + +U +V =− + R −   ∂τ ∂Z ∂R ∂R Re  ∂Z 2 R ∂R  ∂R  R 2 

(3)

 ∂ 2 θ 1 ∂  ∂θ  (4) R   2 + R ∂R  ∂R   ∂Z where U, V, P, θ are the adimensional variables (adimensional velocity components, adimensional pressure and adimensional temperature), R and Z are the adimensional coordinates and Re, Pr, Gr and α* are the dimensionless parameters controlling the problem: U D Reynolds number Re = av ν ν Pr = Prandtl number α gβ ∆T D 3 Gr = Grashof number based on the heat flux ν2 α α* = Ratio of thermal diffusivities solid/fluid αf where β is the thermal expansion coefficient and ν is the kinematic viscosity of the fluid. Initially (i.e. at τ=0), the common boundary conditions are based on a parabolic profile of velocity and a uniform temperature profile at entry, a symmetry of the hydrodynamic and temperature fields with respect to the tube axis and a completely developed profile at exit. ∂θ ∂θ ∂θ α* +U +V = ∂Z ∂R RePr ∂τ


56 Advanced Computational Methods in Heat Transfer IX On the other hand the thermal condition on the wall (r=D/2+δ) is: ∂θ =0 If Z(L1+L2 ) then

wI wZ Adiabatic . L3

Axis

Figure 1:

3

Geometric configuration of the studied case.

Numerical procedure

The equations presented above have been integrated and discretized on a staggered grid following the finite volume approach explained in [11]. We have chosen the SIMPLE algorithm in order to solve for pressure. The power law-differencing scheme has been employed to calculate the convective fluxes in the transport equations for momentum and energy . Temporal discretization has been achieved using an implicit unconditionally stable scheme. The physical domain (solid + fluid) has been treated as a heterogeneous medium, the kinematic viscosity being supposed equal to infinity in the solid region. At the interface, the thermal diffusivity was evaluated using the harmonic mean between that of the fluid and that of the solid [11]. The resulting algebraic equations have been solved using the iterative line-by-line method associated with the tri-diagonal matrix algorithm (TDMA). Convergence is considered to be reached when the maximum relative change in velocities and temperatures WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


57

during two consecutive time increments is less than 10-4. On the basis of many tests, the mesh adopted is non-uniform and consists of 102 nodes in the axial direction and 38 nodes in the radial direction, from which 31 nodes are in the fluid and 7 nodes in the solid wall.

4 Results and discussions In this study, the results obtained correspond to the mixed convection of water (Pr=5) in a vertical tube submitted on one hand to a convective exchange and on the other hand perturbed at entry via a step change in temperature (positive or negative). Many numerical simulations have been done. They concern the influence of the step change in temperature at entry on the stability of the thermal and hydrodynamic fields for two different Grashof numbers: 104 and 105, and also the influence of the thermal diffusivity ratio (wall/fluid) on the temperature of the interface. 4.1 Validation The results of the model used in the present study have been compared with the corresponding analytic results obtained by Kakac and Yener [12]. Fig. 2 illustrates the axial evolution of the Nusselt number Nuz for the case of forced convection (Gr=0) in a tube submitted to a uniform heat flux. One can notice a very good accord between the model and the analytic solution. 24 Analytical Solution Used Model

20

Nu

16 12 8 4 0

0

20

40

60

80

100

120

140

160

Z

Figure 2:

Validation of the model for an average Nusselt number (forced convection).

4.2 Effect of the thermal diffusivities ratio The effect of the thermal diffusivity ratio α* (wall/fluid) on the axial evolution of the interfacial temperature is presented in Fig. 3. One can notice that the increase of this ratio improves the heat transfer by conduction through the wall, which causes a remarkable decrease of the temperature at the interface. The value of the latter tends to that of the ambient temperature when the ratio becomes equal to infinity. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

58 Advanced Computational Methods in Heat Transfer IX 1,0 0,8 *

α =0,1

0,6

*

α =1

Tint

*

α =10

0,4

*

α =100

0,2 0,0 -0,2

0

1

2

3

4

5

6

7

8

9

10

Z

Figure 3:

Effect of the ratio of thermal diffusivities on the temperature at the interface.

4.3 Structure of the thermal and flow fields Fig. 4(a) shows the variation of the dynamic and thermal fields as a function of the increase in the temperature step for a given time (τ=200) and a Grashof number of 104. One can notice the presence of a recirculation zone stuck on the wall. This zone is due to the increase in ascending forces caused by natural convection after having perturbed the flow with a positive temperature step at entry. The appearance of this zone is due to the fact that the flow is accelerated in the central region of the tube and decelerated next to the wall. In effect, due to reheating, a density differential establishes itself inside the fluid, the density of the fluid situated in the central zone diminishes with respect to that near to the wall (Boussinesq hypothesis). As the temperature step increases, the recirculation cell intensifies and its size becomes bigger and ends by occupying the totality of the tube. In Fig. 4(b), corresponding to the thermal field, one can notice a stratification of the isotherms. They are thinner near to the entry of the tube and become larger and larger as one approaches the fluid exit. On the wall one can notice that the radial temperature gradient is negligible due to the importance of the thermal diffusivity of the material with respect to that of the fluid. On the other hand one can clearly see a significant axial temperature gradient, as shown by the isotherms, which are parallel. The increase of the temperature step causes a more condensed stratification and a high heat transfer between the middle of the tube and the wall. The transient behaviour of the flow is shown in Figs. 5 and 6 for the case of a positive step of 0.6 (reheating) or a negative step of -0.6 (cooling). If a negative step is imposed at entry, one notices that the intensity of the recirculation zone becomes weaker as time increases. This causes a decrease of the important fluid current at the centre of the tube. The dynamic instability equally entrains a WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


59

thermal disequilibrium inside the flow. This is due to the interaction between the fluid cooled next to the wall and the fluid heated next to the tube axis. This instability is clearly remarked for the times τ= 12, 19, 29 and 39.

(a) Figure 4:

Effect of the temperature step on the streamfunction and the isotherms, Gr=104, ∆θ=0.2, 0.4, 0.6, 0.8, 1.

(a) Figure 5:

(b)

Evolution of the dynamic field (a) and thermal field (b) as a function of time τ=0, 12, 19, 29, 39 and 130 for Gr=105 and ∆θ=-0.6.

(a) Figure 6:

(b)

(b)

Evolution of the dynamic field (a) and thermal field (b) as a function of time τ=0, 8, 16, 24, and 39 for Gr=105 and ∆θ=0.6.


60 Advanced Computational Methods in Heat Transfer IX After a sudden perturbation of the flow via a positive step (reheating) of fig. 6, the fluid undergoes a local acceleration at entry and especially near the axis of symmetry, consisting of the reverse current next to the tube wall so as to obey the law of conservation of mass. This can be explained by the fact that convective exchange applied to the outer surface cools the entire region neighbouring the wall. On the other hand, the central region of the tube is heated due to the temperature step. The increase of time τ=0, 8, 16, 24 and 39 shows that the region touched by the instability is situated between the cooled region and the heated one inside the fluid.

5

Conclusion

In this paper a numerical study has been performed of the transient behaviour of a laminar flow inside a vertical tube submitted to a sudden change of temperature at entry. An implicit method has been used to solve the governing equations of mixed convection. The results allowed the assessment of the influence of a step change in the imposed temperature at entry of the tube and also of the ratio of thermal diffusivities on the velocity and temperature fields. The existence of a transient instability region was put in evidence, situated in the middle of the tube, between the cold and the hot fluid. The results equally show the axial diffusion in the wall.

Acknowledgments The authors are deeply grateful to both the Algerian and the French higher Education Ministries for the BAF grant within the framework of which the present research was done.

References [1] [2] [3] [4] [5]

Nguyen, C. T., Maïga, S. B., Landry, M., Galanis, N., & Roy, G., Numerical investigation of flow reversal and instability in mixed laminar vertical tube flow. Int. J. Therm. Sc., 43(8), pp.797-808, 2004. Morton, B. R., Ingham, D. B., Keen, D. J., & Heggs, P. J., Recirculating combined convection in laminar pipe flow. ASME. J.Heat Transfer, 111, pp.106-113, 1989. Bilir, S. & Ates, A., Transient conjugated heat transfer in thick walled pipes with convective boundary conditions. Int. J. Heat Mass Transfer, 46(14), pp.2701-2709, 2003. Bernier, M. A. & Baliga, B. R., Conjugate conduction and laminar mixed convection in vertical pipes for upward flow and uniform wall heat flow. Num. Heat Transfer part A, 21, pp.313-332, 1992. Bernier, M. A.& Baliga, B. R., Visualization of mixed convections flows in vertical pipes using a thin semi-transparent gold-film heater and dye injection.. Int. J. Heat Fluid Flow, 13(3), pp.241-249, 1992. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


[6] [7] [8] [9]

[10] [11] [12]

61

Zgal, M., Galanis, N., & Nguyen, C. T., Developing mixed convection with aiding buoyancy in vertical tubes: a numerical investigation of different flow regimes. Int. J. Therm. Sc., 40, pp.816-824, 2001. Su, Y. C. & Chung, J. N., Linear stability analysis of mixed convection flow in vertical pipe. J. Fluid Mech., 422, pp.141-166, 2000. Mai, T. H., Zebiri, R. & Lorenzo, T., Convection mixte en régime transitoire de couche limite laminaire sur une plaque vertical. C. R. Acad. Sci. Paris, t.329, Série IIb, pp.627-631, 2001. Mai, T. H. & Popa, C. V., Numerical study of transient mixed convection in vertical pipe flows. Proc. Of the Conf. On Advances in Fluid Mechanics IV, eds. M. Rahman, R. Verhoven & C. A. Brebbia, WIT Press, pp.75-84, 2002. Mai, T. H., Popa, C. V. & Polidori, G., Transient mixed convection flow instabilities in vertical pipe. Heat Mass Transfer, 41, pp.216-225, 2001. Patankar, S. V., Numerical Heat Transfer and Fluid Flow. Hemisphere, Washington, D. C., 1980 Kakac, S. & Yener, Y., Convectif Heat Transfer. CRC Press, Boca Raton, 1995.



Section 2 Advances in computational methods



65

An interaction of a sonic injection jet with a supersonic turbulent flow approaching a re-entry vehicle to atmosphere D. Sun & R. S. Amano Department of Mechanical Engineering, University of Wisconsin-Milwaukee, USA

Abstract A shock-interaction flow field generated by transverse sonic injection into a supersonic flow was simulated by solving the Favre-averaged Navier-Stokes equations using the weighted essentially non-oscillatory (WENO) schemes. Three-dimensional results indicate that there appear four pairs of vortices around the secondary injection. In the upstream of the square injection there exist two main recirculation regions and the primary vortex induces the horseshoe vortex region. After the secondary injection flow ejects from the square hole, it is forced by the supersonic main flow and then it becomes a pair of counter rotation vortices towards downstream. In the downstream there is a low-pressure region that conduces a pair of helical vortices. Keywords: supersonic flow, bow shock wave, horse-shoe vortex, CFD.

1

Introduction

Shock-interaction flowfield generated from a sonic gaseous flow injected transversely into a supersonic freestream is encountered in practical applications such as space shuttle reentry atmosphere, rocket motor thrust vector control systems, supersonic combustion, high-speed flight vehicle reaction control jets, and gas-turbine cooling systems [1-3]. One example is the case when re-entry vehicles or reusable rockets enter atmosphere, its attitude has to be controlled to endure large aerodynamic heating. Because of a high angle of attack on re-entering, the flow separates from the control surface. If the jet is injected into a hypersonic flow, complicated interaction between the jet and the flow occurs. It results in boundary layer separation, shock waves, and vortices, which are WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/HT060071

66 Advanced Computational Methods in Heat Transfer IX schematically shown in Fig. 1. The mixing flow field is very complex, which includes a bow shock wave in front of the injection, boundary layer separation and vortices. Therefore a higher-order scheme is needed when computing such a flow field. In 1980s, a new high-order scheme, essentially non-oscillatory scheme (ENO), was introduced by Harten et al. [4] Later, Wilcoxson and Manousiouthakis [5] and Jiang and Shu [6] developed the weighted ENO schemes based on ENO schemes, which are simpler and more efficient. In 3-D flowfield, a three-dimensional bow shock forms ahead of the injection and interacts with the approaching boundary layer, resulting in a separation bubble. A barrel shock also occurs as the under expanded jet accelerates into the free stream. Acceleration of the jet core flow continues until a normal shock, or Mach disk, forms. Directly downstream of the jet plume another separated zone develops in the region between the jet exit and the boundary layer reattachment point. A pair of counter-rotating vortices generated within the jet fluid and a horseshoe vortex region also forms near the jet exit and wraps around the injector as illustrated in the schematic. In this paper, the fifth-order WENO scheme of Jiang and Shu [6] and the k-ε turbulent model are used to calculate the supersonic flowfield with secondary injection. The freestream Mach number is 3.7 and 3.0 in two and threedimensional flowfield, respectively, and the injections in both flowfield are sonic. The slot width of the injection in two-dimensional flowfield is 1mm and the orific of the injection in three-dimensional flow is a cubic whose width is 1mm. The conditions of the two-dimensional flowfield is the same with reference 4.

Figure 1:

Bow shock in a mixing jet-stream.

The schematic of the transverse jet injected into a supersonic cross flow is shown in Fig. 1. A three-dimensional bow shock is formed ahead of the injected stream and it interacts with the approaching boundary layer, resulting in a separation bubble. A barrel shock also occurs as the under-expanded jet accelerates into the cross flow. Acceleration of the jet core flow continues until a normal shock, or Mach disk, forms. Directly downstream of the jet plume another separated zone develops in the region between the jet exit and the boundary layer reattachment point. A pair of counter-rotating vortices generated within the jet fluid and a horseshoe vortex region also forms near the jet exit and wraps around the injector as illustrated in the schematic. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


2

67

Governing equations

The three-dimensional Favre-averaged Navier-Stokes equations and the low-Reynolds number k-ε turbulent model are given as follows: ∂Q ∂ ∂ ∂ + (E − Ev ) + (F − Fv ) + (G − Gv ) = H ∂z ∂y ∂t ∂x

 ρu    2  ρu + p   ρuv    E =  ρuw     u (e + p )   ρuk     ρuε  0     τ xx     τ xy   τ xz     Ev =  uτ xx + vτ xy + wτ xz − qx     µt  ∂k    µ +  σ k  ∂x       µt  ∂ε    µ +    σ ε  ∂x   

 ρ     ρu   ρv    Q =  ρw     e   ρk     ρε 

(1)

 ρv     ρuv   ρv 2 + p    F =  ρvw     v(e + p )  ρvk     ρvε 

0     0  τ xz    0   τ yz     0 τ zz     uτ + vτ + wτ − q  0  yz zz z Gv =  xz  H = 0     µt  ∂k    µ +  µt P − ρε  σ k  ∂z     2   c f µ ε P − c f ρ ε   µt ∂ε  1 1 t 2 2   k k   µ +    σ ε  ∂z   

 ρw     ρuw   ρvw    G =  ρw 2 + p     w(e + p )  ρwk     ρwε  0     τ xy     τ yy   τ yz     Fv =  uτ xy + vτ yy + wτ yz − q y     µt  ∂k    µ +  σ k  ∂y       µt  ∂ε    µ +    σ ε  ∂y   

            

where P represents the production of kinetic energy and the following form is used:   ∂u  ∂u  2 ∂u j 2 ∂u k − P =  µt  i + δ ij  − ρkδ ij  i  3   ∂x j ∂xi 3 ∂xk  ∂x j    


(2)

68 Advanced Computational Methods in Heat Transfer IX µ t = Cµ f µ ρ

[

{

f µ = 1 − exp − 0.0165 R y

k2

ε

, C µ = 0.09 ,

]} (1 + 20.5 R ) 2

(3)

t

c1 = 1.44 , c2 = 1.92 , σ k = 1.0 , σ ε = 1.3

Rt = ρk

2

µε , R y = ρ k y µ , f1 = 1 + (0.05 f µ ) ,

(

f 2 = 1 − exp − Rt2

Figure 2:

3

(4)

3

)

(5)

A flow field around an injected jet.

Numerical method

3.1 Spatial discretization The semi-discrete form of Eq. (1) can be written as:

[

∂Q = − (E − Ev )i +1 2 , j ,k − (E − Ev )i −1 2 , j ,k ∂t − (F − Fv )i , j + 1 2 ,k − (F − Fv )i , j −1 2 ,k

[ − [(G − G )

v i , j ,k + 1 2

]

] ]

(6)

− (G − Gv )i , j ,k −1 2

The spatial differencing of numerical fluxes adopts the fifth-order accurate WENO scheme of Jiang and Shu [6] for the inviscid convective fluxes and the fourth-order central differencing for the viscous fluxes. The key idea of the WENO is to use a combination of all the candidate stencils to approximate the fluxes at the boundaries to a higher-order accuracy and at the same time to avoid spurious numerical oscillations near shocks instead of using only one of the candidate stencils. By adopting the WENO scheme, we split the physical fluxes (say, Fˆ ) locally into positive and negative parts as ˆ = Fˆ + Q ˆ + Fˆ − Q ˆ (7) Fˆ Q

()

()

()

where ∂Fˆ + ∂Qˆ ≥ 0 and ∂Fˆ − ∂Qˆ ≤ 0 . In this paper, the local Lax-Friedrichs flux splitting method is used.

() (()

ˆ = 1 Fˆ Q ˆ ± ΛQ ˆ Fˆ ± Q 2

)


(8)


69

where Λ = diag ( λ1 , λ2 , λ3 , λ4 , λ5 ) , and λ1 , λ 2 , λ3 , λ4 , λ5 are the local eigenvalues. We first consider the one-dimensional scalar conservation laws. For example, u t + f (u )x = 0 (9) Let us discretize the space into the uniform interval of size ∆x and denote x j = j∆x . The spatial operator of the WENO schemes, which approximates

− f (u )x at x j , will take the conservative form L=− ~

~

(

~ 1 ~ f j +1 2 − f j −1 2 ∆x

)

(10) ~

~

where f j +1 2 and f j −1 2 are numerical fluxes. Designate f j++1 2 and f j−+1 2 , ~

respectively, the positive and negative parts of numerical flux, f j +1 2 , we have

~ ~ ~ f j +1 2 = f j++1 2 + f j−+1 2 (11) ~+ Here we only describe how to compute f j +1 2 on the basis of the WENO.

~+ f j +1 2 can be shown symmetrically as follows: ~+ 7 11 +  2 f j +1 2 = ω0+  f j+− 2 − f j+−1 + fj  6 6 6   5 + 2 +  + 1 + + ω1  − f j −1 + f j + f j + 1  6 6  6  5 1 2  + ω 2+  f j+ + f j++1 − f j++ 2  6 6 6 

(12)

where ω k+ = α 0+ =

and

(

1 ε + IS0+ 10

)

−2

α 0+

α k+ , k = 0 ,1,2 + α 1+ + α 2+

, α 1+ =

(

6 ε + IS1+ 10

(

)

(

(

)

(

(

)

(

)

−2

α 2+ =

2 1 13 + f j − 2 − 2 f j+−1 + f j+ + f j+− 2 − 4 f j+−1 + 3 f j+ 4 12 2 2 1 13 + IS1+ = f j −1 − 2 f j+ + f j++1 + f j+−1 − f j++ 1 12 4 2 1 13 + f j − 2 f j++1 + f j++ 2 + 3 f j+ − 4 f j++1 + f j++ 2 IS2+ = 4 12

IS0+ =

(

3 ε + IS2+ 10

)

−2

)

2

)

(13)

)

2

Equation (13) represents the smoothness measurement of stencils. Through the smoothness measurement, the interpolation polynomial on each stencil is assigned a weight from which we can construct a polynomial to approximate the numerical fluxes by combining all the polynomials.


70 Advanced Computational Methods in Heat Transfer IX 3.2 Time discretization The time discretization of the WENO scheme can be implemented by the third-order Runge-Kutta method. To solve the ordinary differential equation du = L(u ) dt

(14)

where L is a discretization of the spatial operator, the third-order Runge-Kutta is simply u (1) = u n + ∆tL u n 3 1 1 u (2 ) = u n + u (1) + ∆tL u (1) (15) 4 4 4 2 2 1 u n +1 = u n + u (2 ) + ∆tL u (2 ) 3 3 3

( )

( )

( )

4 Presentation of results and discussion 4.1 Cross-flow over two-dimensional flat surface The computed flow field resulting from the transverse injection of two-dimensional sonic jets into a supersonic turbulent flow at a Mach number of 3.71 and a unit Reynolds number of 2.01×107 are simulated by employing the WENO scheme. The result is compared with the experiment and published calculations at the same condition [7]. The computational domain is 88mm×50mm with the slot width of 1mm. The free-stream Mach number is 3.71 and total pressure is 1atm. The jet is sonic and the total pressure is 0.31atm. The law of the wall coordinates y+ for the first mesh point has been maintained as y+σ σ is the effective diameter of molecular;.


90 Advanced Computational Methods in Heat Transfer IX The kinematical viscosity ν = (τ − 1 / 2) RTδt and density ρ are calculated according to the index function:

φ − φl ( ρ − ρl ) φh − φl h φ − φl ν (φ ) = ν l + (ν − ν ) φh − φl h l ρ (φ ) = ρ l +

(14) (15)

where, ρ h and ρ l are the density of heavy and light fluids respectively. ν h and ν l denote the kinematical viscosity of heavy and light fluids respectively. φ h and φ l are the maximum and minimum values of the index function, respectively.

Figure 1: Discrete velocity set of 2-D nine-velocity (D2Q9) model

3

Figure 2: Initial velocity distribution in side the computational domain

Numerical simulation

To simulate vortices merging in a two-phase spatially growing mixing layer, a rectangular domain of the flow-field is considered as: D = [0, Lx] × [-Ly/2, Ly/2], which is surrounded by an inflow boundary at the left, a free outflow boundary at the right, and a slip boundary at two other upper and lower sides. Unless otherwise mentioned, the channel length Lx and width Ly are set at 250 and 50 respectively. The initial interface between two phases is evenly distributed around the midpoint in vertical direction, the corresponding values of index function are given as φ h = 0.259 and φ l = 0.04 , and the initial velocity field in side the computational domain consists of a hyperbolic tangent profile (see Fig. 2) defined as u ( x, y ) = 1 + Ra ⋅ tanh( y / 2) ; v( x, y ) = 0 The velocity profile at the left boundary (x = 0) is given as u ( y ) = 1 + Ra ⋅ tanh( y / 2) + u * ( y ) ; v( y ) = v * ( y )


(16) (17)


91

where, u and v are the velocity components in the x and y directions, u* and v* are the corresponding velocity perturbations; while, Ra is the velocity ratio. Assuming that U1 and U2 are the free-stream velocity in the upper and lower layers, respectively, the average velocity, U = (U1+U2)/2, and the momentum thickness at x = 0, named as θ 0 here, are used as the reference scales for the velocity and length, respectively. Henceforth, all quantities will be normalised with respect to appropriate combinations of U and θ 0 . Reynolds number for the mixing layer can be defined as Re = Uθ 0 / ν , in which ν is kinematical viscosity. It is shown that the influence of Reynolds number upon the kinetic characteristic of the large-scale coherent structures is small enough if Reynolds number is larger than 200. Thus, Reynolds number is set at 200 in the following simulations. The velocity perturbation u* and v* can be expressed as u* = ∑ An Fn' ( y ) cos(α n x − ω n t + ϕ n ) / α n

(18)

n

v* = ∑ An Fn ( y ) sin(α n x − ω n t + ϕ n )

(19)

n

where A, α 1 , ω1 , ϕ1 and Ai (i ≥ 2) , α i (i ≥ 2) , ω i (i ≥ 2) , ϕ i (i ≥ 2) are amplitudes, wave number, frequency and phase of the basic wave and subharmonic waves, respectively. Fn(y) are normalized characteristic mode determined from a linear stability theory. The frequency of basic wave that can lead to the most unstable mode of mixing lay is 0.225 with the velocity ratio Ra = 0.5 (Monkewitz and Patrick, 1982). So, ω1 is given as 0.225, meanwhile Ra = (U1-U2)/(2U)=0.5, Fn(y) = 1/(1+y2), A1 = 0.02 , Ai = 0.01 (i ≥ 2) , ω 2 = ω1 / 2 , ω 3 = ω1 / 3 , ω 4 = ω1 / 4 , α n = ω n , ϕ1 = 0 . In what follows, for simplicity, the subharmonic waves with the frequencies of ω 2 , ω 3 and ω 4 are donated as the second, the third and the fourth subharmonic waves respectively. By superimposing multi-harmonics and changing their phase shifts, one can study the effects of the velocity perturbation to mixing layer.

4


Because the basic wave and several sub-harmonic waves can be forced synchronously, there are many choices of the parameters in equations (18) and (19). Therefore, some typical combinations of perturbation waves are chosen for the simulation. Under some specific initial and boundary conditions, the instabilities will grow and vortices will appear; two or more vortices start to spiral around one another and then merge into a new vortex. The spiralling behaviour and the merging behaviour may repeat itself with the newly formed vortices. In the following, the results of simulation will be visualized by means of plots for phase distribution, vortices contour and corresponding frequency spectrum.


92 Advanced Computational Methods in Heat Transfer IX 4.1 Non-vortex-merging In this section, the forced perturbation is only a basic wave with the parameters of ϕ1 and ω1 . Fig. 3 shows the distributions of two immiscible fluids flow at t = 350 for two different values of surface tension. It shows that the vortices appear clearly but no merging takes place under a single-frequency perturbation. This is also be proved by the plots of corresponding frequency spectrum shown in Fig. 4, in which the frequency spectrum for three different cases at x=120 is presented. Similar plots are obtained at the other location of the flow-fields, which indicates that the vortices do not experience the merging anywhere. The effects of surface tension can be judged in Fig. 3. With a zero surface tension, the interface is elongated at upstream. The elongated interface rolls up and forms vortices during the flow. The newly formed vortex spins and migrates continuously towards downstream and sweeps more and more layers of interfaces into it. For k = 0.01, the flow is qualitatively similar to the zero surface tension case except for the shape of interface in the corresponding vortices; the rollup of the interface in the vortices is slower because the extension of the interface is limited by the surface tension. When k increases to 0.1, the rolling up of interface at up- and mid-stream is similar to the previous two cases, but the interface evolution is delayed more by the much stronger surface tension. Differences can be identified at the downstream of the mixing layer, where the interface is pinched and broken.

(a) k = 0

Figure 3:

(b) k = 0.01

Phase distributions with different values of surface tension at t = 350.

(a) k=0

Figure 4:

(c) k = 0.1

(b) k=0.01

(c) k=0.1

Frequency spectrum plots with different values of surface tension at x = 120.

4.2 Two-vortex-merging The interaction of the basic wave and its second sub-harmonic wave of ϕ 2 = 0 in the mixing layer is first considered. The surface tension parameter, k, is chosen as 0 and 0.1 respectively. The phase distribution and corresponding frequency spectrum for these two cases with different surface tension are shown in Figs. 5 and 6 respectively. It can be seen that the vortices formed at the beginning of the mixing layer are nearly of the same frequency as that of the basic wave. Under WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


93

the action of the second sub-harmonic wave, every two of these vortices will wind with each other and merge into a bigger vortex. As shown in Fig. 6, the frequency of the big vortices becomes half of the small ones. The merged vortices start to shear when they move downstream and no further merging takes place. However, with the effect of surface tension, although the rollup of the interface in vortices is slower, the interfacial ends are blunter and the interface pinching and breaking behaviours take place; nevertheless, the basic mode of vortices merging is the same as that of zero surface tension case.

(a) k = 0

Figure 5:

(b) k = 0.1

Phase distributions with different values of surface tension at t = 350.

Fig. 6 shows the frequency spectrum of different surface tension at different position. At x = 40, the main frequency approximates to ω1 , which indicates that no merging takes place. When vortices move to x = 120, the frequency of ω1 / 2 substitutes ω1 to dominate over the flow-field, while the basic frequency, ω1 , is still obvious. Such a phenomenon reveals that two vortices are merging at this location and the structures of small vortices have not disappeared completely. Some frequencies which have never been added to the perturbation, such as 3ω1 / 2 and 2ω1 , can be identified in Fig. 6. They are the sums or differences of the frequency between the basic wave and sub-harmonic waves. Moreover, the sameness of the frequency spectrum evolution for different cases can further prove that the strength of surface tension does not affect vortices forming, the mode of merging and the migration velocity of the vortices. (a) k=0

(b) k =0.1

Figure 6:

Frequency spectrum plots with different values of surface tension.

4.3 Three-vortex-merging Fig. 7 shows the results in which the perturbation consists of the basic wave and the third sub-harmonic wave of ϕ 3 = π / 3 . With the interactions of the basic wave and sub-harmonic wave, three vortices merging takes place in the mixing layer. Two downstream vortices merge in the first place, and then the newly WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

94 Advanced Computational Methods in Heat Transfer IX formed vortex merges with the third one upstream. In fact, as it is shown, no matter what parameter of surface tension is chosen, the basic mode of vortices merging is almost the same. Furthermore, as shown in Fig. 8, the evolutions of frequency spectrum in three cases of different surface tension are quite synchronous.

(a) k = 0 Figure 7:

(b) k = 0.1

Phase distributions with different values of surface tension at t = 350. (a) k=0

(b) k = 0.1

Figure 8:

Figure 9:

Frequency spectrum plots with different values of surface tension

Interface distributions and vortices contours with different surface tension.

Fig. 9 shows a comparison of corresponding vortices contours with the same velocity perturbation and three different values of surface tension within the region of x ∈ [200, 250] at t = 350; in each case, three main vortices merge into a larger one. It is shown that, with surface tension effect, more small vortices appear in the region of main vortices; and with the increase of surface tension, many small vortices are formed, and the cores of the three main vortices can even disappear. In addition, with zero surface tension at the two fluids interface (indicated by the dished line), the high vortices concentrate in the cores of the vortices; the structures of vortices match the corresponding phase distributions very well. However, with surface tension effect, the vorticity field is disturbed; the vorticity concentrations appear on the interfaces. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


95

4.4 More vortex-merging Further to the previous discussion, Fig. 10 shows the vortex-merging in a triple-frequency forcing mixing layer. The channel length Lx and width Ly are given as 375 and 125 respectively. The perturbation consists of a basic wave, a second sub-harmonic wave and a fourth sub-harmonic wave; and all phase shifts equal to zero. Based on this type of perturbation, a phenomenon of four vortices merging takes place in the mixing layer. In the first place, smaller basic vortices are formed upstream in the mixing layer. Thus, every two of these vortices will finish the primary merging and form to a bigger vortex. Then in the flow field downstream, the secondary merging takes place, i.e. every two newly merged vortices repeat the merging process. In this way, the whole process of four-vortexmerging is completed and no further merging behaviour takes place. It is noted from the frequency spectrum plots shown in Fig. 11 that, during the process of vortices merging, the main frequency transfers from ω1 to ω1 / 4 via ω1 / 2 .

Figure 10:

Figure 11:

5

Phase distributions with zero surface tension at t = 400.

Frequency spectrum plots with zero surface tension.

Conclusion

The behaviour of vortices merging in two-dimensional two-phase spatially growing mixing layer is numerically studied by the LBM. By changing the strength of surface tension and the combinations of perturbation waves, the effects of the surface tension and the velocity perturbation on the vortices merging are investigated. With a single-frequency forcing, vortices appear clearly but no merging takes place in the mixing layer. When the mixing layer is forced by the two- or threefrequency perturbation, the vortices start to merge. The results show that the lower frequency of sub-harmonic wave applied, the more vortices are merged. The scale of the large vortex is directly proportional to the number of basic WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

96 Advanced Computational Methods in Heat Transfer IX vortices being involved in the process. Through vortices merging, the interfacial surface area is enlarged quickly downstream in the flow-field. In addition, the effect of surface tension strength on vortices formation, the pattern of merging, and the migration velocity of the vortices is limited. Although the rollup of the interface in vortices is slow, the surface tension does have effect on interfacial ends, pinching and breaking. Based on this, it can be assumed that, no matter how strong of the surface tension is, the vortices evolution in two-phase mixing layer should be controlled positively by forcing the suitable perturbation upstream in the flow-field and obtaining expectant flow patterns consequently.

Acknowledgement The project is partly supported by Royal Society international joint project Ref. 15127 and British EPSRC under grant EP/D500125/1.

References [1] [2]

[3] [4] [5] [6] [7] [8] [9] [10] [11]

Andrew, K. Gunstensen, Daniel H. Rothman, Stéphane Zaleski and Gianluigi Zanetti. 1991. Lattice Boltzmann model of immiscible fluids. Physical Review A, 43(8): 4320-4327. He, X., Chen, S. and Zhang, R. 1999. A lattice Boltzmann scheme for incompressible multiphase flow and its application in simulation of Rayleigh-Taylor instability. Journal of Computational Physics, 152(2): 642-663. Ho C-M., Huang L-S. 1982. Subharmonics and vortex merging in mixing layers. Journal of Fluid Mechanics, 119: 443-473. Inoue, O., 1992. Double-frequency forcing spatially growing mixing layers. Journal of Fluid Mechanics, 234: 553-581. Inoue, O., 1995. Note on multiple-frequency forcing on mixing layers, Fluid Dynamics Research, 16(2/3): 161-172. Lifshitz, Y. and Degani, D., 2004. Mathematical model for turbulent mixing layer with harmonic perturbations. European Congress on Computational Methods in Applied Sciences and Engineering, 1-19. Monkewitz, P.A. and Patrick Huerre. 1982. Influence of the velocity ratio on the spatial instability of mixing layers. Physics of Fluids, 25(7): 11371143. Shan, X. and Chen, H. 1993. Lattice Boltzmann model for simulating flows with multiple phases and components. Physical Review E, 47(3): 1815-1819. Silvestrini, J.H. 2000. Dynamics of coherent vortices in mixing layers using direct numerical and large-eddy simulations. Journal of the Brazilian Society of Mechanical Sciences, 22 (1): 53-67. Swift, M.R., Osborn, W.R. and J. M. Yeomans. 1995. Lattice Boltzmann simulation of nonideal fluids. Physical Review Letters, 75(5): 830-833. Swift, M.R., Orlandini, E., W. R. Osborn, and J. M. Yeomans. 1996. Lattice Boltzmann simulations of liquid-gas and binary fluid systems. Physical Review E, 54(5): 5041-5052. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


97

A numerical solution of the NS equations based on the mean value theorem with applications to aerothermodynamics F. Ferguson & G. Elamin Department of Mechanical Engineering, North Carolina A & T State University, USA

Abstract An innovative and robust algorithm capable of solving a variety of complex fluid dynamics problems is developed. This so-called, Integro-Differential Scheme (IDS) is designed to overcome known limitations of well-established schemes. The IDS implements a smart approach in transforming 3-D computational flowfields of fluid dynamic problems into their 2-D counterparts, while preserving their physical attributes. The strength of IDS rests on the implementation of the mean value theorem to the integral form of the conservation laws. This process transforms the integral equations into a finite difference scheme that lends itself to efficient numerical implementation. Preliminary solutions generated by IDS demonstrated its accuracy in terms of its ability to capture complex flowfield behaviours. In this paper, the results obtained from the application of the IDS to two problems; namely, the hypersonic flat plate problem, and the shock/boundary layer interaction problem, are documented and discussed. In both cases, the results showed very good agreement with the physical expectation of these problems. In an effort to this new algorithm, IDS solution to the shock/boundary layer interaction problem was compared to the experimental findings described in NASA Mem., No., 2-1859W, March, 1959. The results obtained by IDS show excellent agreement with the experimental data. Keywords: Integro-Differential Scheme, mean value theorem, hypersonic boundary layer, finite volume, control volume, numerical scheme.



1

Introduction

The Navier-Stokes equations governing fluid flows can either be highly elliptic or highly hyperbolic, or both, depending on the applicable boundary conditions. As a result, the Navier-Stokes equations are very complicated and, in general, do not lend themselves to analytic solutions. In addition, aerospace designers are currently demanding solutions to fluid flow problems under conditions that cannot be duplicated with existing experimental facilities. Hence, the only way to obtain reasonable, complete information on fluid flows and their characteristics lies in computational fluid dynamic (CFD) methods. A literature survey indicated that there are many well-established numerical schemes available to aerospace designers. Anderson [1] and Chung [2] presented a wide variety of these schemes in their books. Akwaboa [3] used MacCormack technique to solve the supersonic flow over a rearward-facing step problem. Chang et al. [4], Zhang et al. [5] and Changh [6] introduced different versions of the space-time conservation element and solution element method for solving fluid flow problems. Even though, these schemes have led to significant improvements in the state of the art in CFD, they have many drawbacks, and therefore still not adequate to handle certain CFD demands. 1.1 Research objective This research focuses on the development of a robust, efficient, and accurate numerical framework that is capable of solving complex fluid flow problems, and one that is capable of overcoming most of the limitations generated by existing schemes. The proposed scheme is based on a clever approach to the merging of the traditional finite volume and the finite difference schemes. In the process of creating a new numerical scheme, the mean value theorem is used to evaluate the rates of change of fluxes at the center of the control volume.

2

The governing equations

When defining any numerical solution to a fluid dynamic problem, the conservation laws must be satisfied for an appropriate set of boundary conditions. As known in fluid dynamics, the conservation laws can be applied in two basic forms; the differential form and the integral form. However, experience has shown that when the integral form of the conservation laws is applied to fluid dynamics problems, high fidelity numerical solutions can be obtained. It is therefore no surprise that the Integro-Differential Scheme (IDS) is based on the integral formulation of the conservation laws described in subsections 2.2, 2.2 and 2.3. 2.1 Conservation of mass equation Consider the conservation of mass equation in the following form,



∫∫∫ v

∂ρ dv + ∫∫ ρ V d s = 0 . ∂t s

99 (1)

In eqn (1) ρ , v, t , represent density, volume, and time, respectively. The symbols, ds and V , represent the surface of the control volume and the fluid velocity, respectively. These quantities are defined through the use of the following vectors:

ds = dydzi + dxdz j + dxdyk

(2)

and V =ui +

v j+

wk .

(3)

2.2 Conservation of momentum equation Consider the conservation of momentum equation in the following form, ∂ ρ V dv + ∫∫ ρ V .d s V = − ∫∫ Pd s + ∫∫ τˆd s ∂ t ∫∫∫ v s s S

(

)

(4)

where the symbol, P, represents pressure and the symbol, τˆ, is the tensor that defines the various components of the local viscous stresses. This tensor can be described by the following equation:

τ xx  τˆ = τ yx τ  zx and the symbols,

τ xy

τ xz 

τ yy

τ yz 

(5)

τ xx ,τ xy ,τ yy ,τ yx ,τ zx ,τ zy and τ zz ,

are the local shear stress



τ zz 

τ zy

components. 2.3 Conservation of energy equation Consider the conservation of energy equation in the following form, ∂ ∂t

∫∫∫ ρEdv + ∫∫ ρEV .ds = − ∫∫ PV .ds + ∫∫τˆ.V ds + ∫∫ q ds v

s

s

s

(6)

s

where the symbol, E, represents the total energy per unit mass of fluid. The vector, q , represents the rate of heat conducted per unit area through the surface of the control volume. In general, the vector, q , can be written in Cartesian coordinate format, such that q vis = q x i + q y j + q z k (7) where q x , q y , and q z represent the rate of heat conducted per unit area in x ,y, and z coordinate directions, respectively. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


3

The Integro-Differential Scheme

The Integro-Differential Scheme (IDS) combines two schemes; namely, the finite volume scheme and finite difference scheme. IDS relies on the coupled behavior of discretized cells and their corresponding nodes. The numerical process is conducted in two alternating fashions, and for the sake of simplicity, only the two-dimension form of the IDS is explained in this paper. A typical control volume, illustrated in Figure 1, describes the numerical details associated with the finite volume formulation. Similarly, numerical details associated with the finite difference formulation are described through the use of Figure 2. 4’

3’

out

i,j+1

i-1,j+1 4

dy

3

1’

in

d 2’

1

2

dx Figure 1:

i-1,j

a

out dz

a

i-1,j-1

i+1,j+1

c i,j

b

i,j-1

i+1,j

i+1,j-1

in Finite volume model.

Figure 2:

Finite difference model.

Moreover, as illustrated in Figure 2, the center of any four neighboring control volumes, namely, control volumes; a, b, c, and d, is defined by the indices i and j. Any control volume will be defined locally by the nodes (1, 2, 3, and 4) as shown in Figure 1, and globally by its relative location to the point i,j as in Figure 2. 3.1 Application of the conservation laws to the control volume To demonstrate the utility of this numerical approach to fluid dynamic problems, consider a typical flow through the surfaces of an infinitesimal control volume, as illustrated in Figure 1. Even though the IDS has the potential to solve any 2D or 3D fluid-flow problem, for the purpose of simplicity, the discussions conducted in this paper are limited to 2D fluid flow problems. However, when describing the 2D approach, a major challenge involves the conversion of the naturally 3D conservation laws into their 2D counterparts that maintain the integrity of the 3D flowfield and its associated effects. To achieve this goal, the control volumes are chosen as infinitesimal rectangular prisms, with unit normal, ñ, in the x, y, and z directions. Also it was assumed that, the dimension, dz, of a typical control volume is always a single unit. These assumptions led to the fact that the fluid properties in the z-direction across any control volume are constants and the net flow of mass, momentum, and energy in the z-direction is WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


101

always zero. Armed with these assumptions, the algebraic forms of the rate of change of mass, momentum, and energy at the center of each control volume are formulated as follows:  ∂ρ     ∂t 

average

 =  

((ρ u )1

average

)3 )

+ (ρ u

 + 

2∆x

((ρ v )1

    ∂E     ∂t 

)4 ) − ((ρ u )2

+ (ρ u

  (ρ Eu )1 + (ρ Eu =  2∆x 

+ (ρ v )2 ) − ((ρ v )3 + (ρ v )4 )   2∆y    (ρ Eu )2 + (ρ Eu  −  2∆x  

)4

  (ρ Ev )1 + (ρ Ev   2∆y   2 ∞

2 ∞

1   (τ Re L  1   τ xy Re L 

( )lower

1  τ Re L 

( yy )lower

1  τ Re L 

)2

  (ρ Tv )3 + (ρ Tv  −  2∆y  

)4

xx

 v3 + v4   2∆y

  − τ 

( yy )upper

Re

 ∂ (ρ u  ∂t 

)  average  

 =  

L



(

2 ∞

Pr M

((ρ u ) 2

1

)left

− (q xx

(ρ u ) ) − ((ρ u ) 2

+

2

4

 u1 + u 4     +  2∆x  lower

yy

∆x

+

2

2∆x

(ρ u ) ) 2

 ∂ (ρ v  ∂t 

)  average  

 =  

((ρ v ) 2

1

((τ

   

L

+

xx

)left

3



− (τ

xx

(ρ v ) ) − ((ρ v ) 2

2

2

3

(

+ ρv

2

2∆y   

((ρ uv )1

+

(ρ uv )4 ) − ((ρ uv )2

1 2 ∞

1 Re

  

L

((ρ T )1

((

 τ   

+

(

− τ ∆y

yy

)3

− (τ

xx

)upper

)  

∆y

4

(11)

(ρ uv )3 ) 

(ρ T )2 ) − ((ρ T )3

)lower

(10)

 +  ) −  



+

2∆y yy

 

) ) +  + 

2∆x

γM

(9)

)

)4 ) 

lower

xy

)upper

+

) + ((τ )

)right

∆x

(

− q yy ∆y

 ((ρ vu )1 + (ρ vu )2 ) − ((ρ vu )3 + (ρ vu  2∆y  1  ((ρ T )1 + (ρ T )4 ) − ((ρ T )2 + (ρ T 2∆x γ M ∞2  1 Re

   −  

   −  

 v1 + v 2   2∆y

) + ((q )

)right

   + 

 u1 + u 4   −   2∆x   v1 + v 2     −   2 ∆ y  

)right

( xy )upper

  − τ 

 (q xx   

   +  

)3

 v3 + v4   2∆y



)4

  (ρ Tu )2 + (ρ Tu  −  2∆x  

( xy )left  u 22 ∆+ xu 3  − (τ xy )right

1

   + 

)4

 u2 + u3    − (τ  2∆x 

)left

xx

)3

  (ρ Ev )3 + (ρ Ev  −  2∆y  

  (ρ Tv )1 + (ρ Tv   2∆y  

1

γM

)2

  (ρ Tu )1 + (ρ Tu  2∆x 

1

γM

(8)

)upper

+

( ρ T )4 ) 

) + ((τ


xy

)left

 − 

(

− τ ∆x

xy

)right

)  

102 Advanced Computational Methods in Heat Transfer IX The non-dimensional total energy, E, is E =

T γ (γ − 1 )M

2 ∞

+

u

2

+ v2 . 2

(12)

3.2 Flowfield Construction A careful examination of the governing eqns (8) – (11), indicates that the system is closed relative to four unknown variables, namely ρ , u , v, and T . These unknowns are included in a solution vector, U m , such that U

U U =  U  U

1 2 3 4

 ρ    = ρu  ρv   E 

     

(13)

Using Taylor’s expansion, the solution can be constructed based of the following time marching scheme:  dU m  (14) (U m ) it ,+j∆ t = (U m ) ti , j +   ∆t .  dt  i , j 3.3 IDS Marching Steps Eqn (14) represents a typical explicit time marching scheme. Like most established numerical schemes the IDS uses eqn (14). However, the major differences in the IDS as compared to the so-called established explicit schemes, is the way it handles the right side of eqn (14), namely, the old values of the m solution flux vector, (U m ) ti , j , the time derivative vector, (dU dt ) , and the i, j

time step, ∆t . 3.3.1 Evaluation of the time derivative The evaluation of the time derivatives, (dU

dt

)m , , is accomplished through the i, j

use of the mass, momentum, and energy equations. Eqns (8) – (11) are m implemented globally to obtain the time derivative (dU dt ) , at the center of i, j

each cell, a, b, c, and d. In another consistence averaging process, the time derivative at node, (i,j), is obtained as an arithmetic average of the time derivatives at the cell centers. 1  dU m   dU m   dU m   dU m   dU m     =   +  +  +   dt 4 dt dt   i, j a   b  dt  c  dt  d  i , j 

(15)

3.3.2 Evaluation of the solution vector As indicated in Figure 3, information at the point of interest, (i, j), is updated solely based on the values of the point in question along with all its eight immediate neighbors. All required fluxes and derivatives are evaluated based on arithmetic averages of the primitive variables. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


i-1,j+1

i,j+1 d

i+1,j+1 c

i,j

i-1,j a

i+1,j b i+1,j-1

i,j-1

i-1,j-1 Figure 3:

103

Illustration of the IDS stencil.

3.3.3 Evaluation of the time increment Since the IDS is an explicit scheme, the time increment, ∆t , is subject to a stability criterion. To determine the size of the time step, the Courant-FriedrichsLewy (CFL) criterion, documented in Anderson [1], is used.

4 Results and discussions In this paper, the IDS is employed to solve two problems; namely, the hypersonic flat plate problem, and the shock/boundary layer interaction problem. Grid(101x101)

Grid(151x151)

Grid(201x201)

0.8 0.7 0.6

y/L

0.5 0.4 0.3 0.2 0.1 0.0 0.0

0.5

1.0

1.5

u/u_inf

Figure 4:

Grid independence studies.

4.1 The supersonic flow over a flat plate problem The hypersonic flow over a flat plate is a classical fluid dynamics problem, and in the past it has received considerable attention [1, 3, 7]. However, it has no exact analytical solution. The IDS solver was used to solve the flat plate problem under a variety of conditions, ranging from incompressible to compressible to hypersonic. The results provided in this study are for a Reynolds Number, Re = 1000, and a Mach number of 4.0. The results of validation studies conducted, using grid densities and residual errors are indicated in Figures 4 and 5. Grid studies were conducted over the following grid sizes; namely, 101x101, WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

104 Advanced Computational Methods in Heat Transfer IX 151x151, and 201x201. Convergence studies were conducted on a Dell, Intel based PC until residual errors were in the range of 10-14 – 10-15. The plot in Figures 5 indicates the horizontal velocity profile obtained from the grid density studies. In Figure 6, the maximum residual obtained from the mass, momentum, and the energy fluxes is plotted as a function of the time step. To further strengthen the validity of the algorithm, the reference temperature method was used to evaluate the skin friction coefficient, Cf , and the wall heat transfer coefficient, and Stanton number, Ch (Rasmussen [8]). Data obtained from these studies were also positive.

Figure 5:

Figure 6:

Residual error studies.

Illustration of the shock boundary-layer interaction problem.



Figure 7:

105

Illustration of IDS obtained y-velocity component distribution. Present

Data in NASA Memor. 2-18-59W

P/P_inf

1.4

1.2

1.0 0.40

0.50

0.60

0.70

0.80

0.90

X/L

Figure 8:

Pressure distribution along the wall.

4.2 The shock/boundary layer interaction problem In 1959, Hakkinen et al. [9] studied the shock wave/boundary layer interaction problem experimentally. This problem is illustrated in Figure 6. More recently, due to the massive increase in computer capabilities, studies, [10, 11, 12], investigated this problem numerically. Using the IDS Solver, the inlet, outlet, and far field boundary conditions were set to be same as those of the flat plate problem. However, the flow on the top boundary is specified to form an oblique shock impinging on the wall. The bottom boundary consists of freestream and solid wall boundaries, whose lengths are 0.2 and 0.8 respectively. The flow Reynolds was set to 296000 and the Mach number set to 2.0. Figure 7 illustrates WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

106 Advanced Computational Methods in Heat Transfer IX the carpet plot of the y-velocity component obtained from this study. Figures 8 and 9 compare the pressure and the friction coefficient along the solid wall with the experimental data obtained by Hakkinen et al. [9]. The present results are in good agreement with the experimental data. Present

Data in NASA Memor. 2-18-59W

5

Cf X 1000

4 3 2 1 0 -1

0.2

Figure 9:

5

0.4

0.6

0.8

1

X/L

Skin friction distribution along the wall.

Conclusion

A new numerical scheme for solving equations that govern fluid dynamics problems was developed. This innovative scheme is called the ‘integrodifferential scheme’ and abbreviated as IDS. The scheme name depicts exactly what it says, by combining the integral form of the conservation laws to formulate the governing equations and transforming them in a suitable differential form for appropriate finite difference representation. The concept of the control volume was considered when calculating the integrations and the finite difference held for the numerical implementation of the scheme. In this paper the new scheme is employed to solve the viscous flow over a flat plate problem and the shock/boundary layer interaction problem. In both cases, the results showed very good agreement with the physical expectation of the flow, the empirical formulas, and the experimental data. This agreement solidified the belief that the scheme is robust, efficient, and capable of solving a variety of complex fluid dynamics problems.

References [1]

Anderson, Jr., J. D., “Computational Fluid Dynamics-The basis with applications”, McGraw-Hill, Inc., 1995.



[2] [3] [4]

[5]

[6]

[7] [8] [9] [10] [11] [12]

107

Chung, T. J., “Computational Fluid Dynamics”, Cambridge University Press, 2002. Stephen Akwaboa, “Navier-Stokes Solver for a Supersonic Flow over a Rearward-Facing Step”, M. S. Thesis, (Department of Mechanical Engineering, North Carolina A & T State University, Greensboro, 2004). Chang, S. C., Wang, X. Y., & Chow, C. Y., “The space-time Conservation Element and Solution Element method: A new high-resolution and genuinely multidimensional paradigm for solving conservation law”, Journal of Computational Physics, Vol. 156, 1999, PP 89-136. Zeng-Chan Zhang, John Yu, S. T. & Sin-Chung. Chang, “The space-time Conservation Element and Solution Element Method for Solving the Two and Three Dimensional Unsteady Euler Equations Using Quadrilateral and Hexahedral Meshes”, Journal of Computational Physics, Vol. 175, 2002, PP 168-199. Sin-Chung. Chang, “The Method of space-time Conservation Element and Solution Element- A New Approach for Solving the Navier-Stokes and Euler Equations”, Journal of Computational Physics, Vol. 119, 1995, PP 295-324. MacCormack, R. W., “Current Status of Numerical Solutions of the Navier-Stokes Equations”, AIAA paper no. 85-0032, 1985. Maurice Rasmussen, & David Ross Boyd, “Hypersonic flow”, John Wiley & Sons Inc., 1994. Hakkinen, R. J., Greber, I., Trilling, L. & Abarbanel, S. S., “The interaction of an oblique shock wave with a laminar boundary layer.”, NASA Memor. 2-18-59W, March (1959). Reyhner, T. A. , & Flugge-Lotz, I., “The interaction of a shock wave with a laminar boundary layer”, Int. J. Non-linear mechanics, Vol. 3, PP. 173199. Grasso, F., & Marini, M., “Analysis of Hypersonic Shock-Wave Laminar Boundary-Layer Interaction Phenomena”, Computers and Fluids Journal, Vol. 25, No. 6, PP 561-581, 1996. Moujin Zhang, John Yu, S. T. , & Sin-Chang, “Solving the Navier-Stokes Equations by the CESE Method”. AIAA paper no. 2004-0075, 2004.




109

Solution of the radiative transfer problems in two-dimensional participating cylindrical medium with isotropic scattering using the SKN approximation N. Döner1 & Z. Altaç2 1

Mechanical Engineering Department, Dumlupınar University, Kütahya, Turkey 2 Eskişehir Osmangazi University, Eskişehir, Turkey

Abstract The SKN (Synthetic Kernel) approximation is applied to a two-dimensional homogeneous cylindrical participating medium with isotropic scattering. The SKN equations are tested against benchmark problems consisting of cold homogeneous participating medium. The solutions are compared with those obtained by various methods available in the literature. The SK3 approximation results for geometries of very large and very small aspect ratios are in excellent agreement with those of benchmark solutions. The SK3 solutions with moderate aspect ratios are accurate within several percent. Keywords: synthetic kernel method, participating medium, isotropic scattering, two-dimensional cylindrical medium.

1

Introduction

The SKN approximation which was developed to solve neutron transport equation can also be applied to radiative integral transfer equation (RITE) [1]. The significance of solving the integral equations is that one has to deal with the spatial variables rather than spatial and angular variables that exist in radiative transfer equation (RTE)-Boltzmann’s equation. Thus the solutions obtained from the integral equations do not exhibit ray effect which is a phenomena that plagues discrete ordinates and similar methods. Ray effect is the result of discretization of angular variables. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/HT060111

110 Advanced Computational Methods in Heat Transfer IX The SKN method is applied to the RITE by substituting approximations to its radiative transfer kernels through a sum of exponentials, as in the exponential kernel approximation. Then RITE can be reduced into a set of coupled second-order partial differential equations, which will be referred to as the SKN equations. Physical boundary conditions are embedded in the force functions (surface integrals) of the RITE which are exactly preserved in the SKN equations; however, boundary conditions (BCs) of mathematical in nature are required to solve the SKN equations. Recently, the applicability and validity of the SKN approximation to thermal radiative transfer problems of one dimensional plane parallel and spherical geometries, and two-dimensional participating homogeneous and inhomogeneous medium have been investigated [2-4]. Solutions obtained with the SKN method are very accurate and certainly superior to P1 and differential or modified differential approximations. It has been also demonstrated that in twodimensional geometries the SKN method is a high order approximation which contains no ray effect [4]. Solving second order elliptic differential equations with simple boundary conditions lead to less computational efforts and cpu time. In this study, the SKN method with Gauss quadrature set has been applied to radiative transfer in two-dimensional cylindrical participating homogeneous medium. The accuracy and convergence of the method are investigated.

2

Derivations of SKN equations

The SKN equations for two-dimensional cylindrical medium can written as [9]  2  ∂ 2 ∂ 2   1 ∂ + 2  + 1 Gn (τ r ,τ z ) = S (τ r ,τ z ), − µ n  2 +   ∂τ r τ r ∂τ r ∂τ z   

n = 1, 2,..., N

(1)

where (τ r ,τ z ) are the optical coordinates, Gn (τ r ,τ z ) is a function representing the nth component of the synthetic kernel function, S (τ r ,τ z ) is the isotropic source function for a cold medium and is defined as S (τ r ,τ z ) = ω G (τ r ,τ z )

(2)

and ω is the scattering albedo. The incident radiation and heat flux can be defined as N

G (τ r ,τ z ) = F1 (τ r ,τ z ) + ∑ wn Gn (τ r ,τ z )

(3)

n =1

N

q(τ r ,τ z ) = F2 (τ r ,τ z ) + ∑ wn q n (τ r ,τ z )

(4)

n =1

where ( µ n , wn ) are the Gauss quadratures for µ ∈ (0,1) , q n (τ r ,τ z ) is a vector function defined as q n (τ r ,τ z ) = − µ n2 ∇Gn (τ r ,τ z ) [4], F1 (τ r ,τ z ) and F2 (τ r ,τ z ) WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


111

are the force functions containing physical boundary conditions (BCs). Then the net heat flux components can be also be expressed as N ∂Gn (τ r ,τ z ) (5) qr (τ r ,τ z ) = F2 r (τ r ,τ z ) − ∑ wn µ n2 ∂τ r n =1 N

qz (τ r ,τ z ) = F2 z (τ r ,τ z ) − ∑ wn µ n2 n =1

∂Gn (τ r ,τ z ) ∂τ z

(6)

The SKN equations are subject to the following mathematical BCs regardless of the physical BCs: For τ r = 0 ,

∂Gn (0,τ z ) =0 ∂τ r

(7)

For τ z = 0 ,

∂Gn (τ r , 0) 1 = G (τ , 0) ∂τ z µn n r

(8)

For τ z = L ,

∂Gn (τ r , L) 1 =− G (τ , L) ∂τ z µn n r

(9)

For τ r = R ,

∂Gn ( R,τ z ) K ( R / µn ) =− 1 G ( R, τ z ) ∂τ r µn K0 ( R / µn ) n

(10)

where K 0 ( x) and K1 ( x) are zeroth and first order modified Bessel functions. These BCs are adapted from the exact boundary conditions of the onedimensional SKN derivations. The source term of the SKN equations is given by Eq. (2). If the medium is pure absorber, Eq. (1) yields zero solution for Gn . Then the RITE is no longer an integral equation and the solutions for the incident energy and the net radiative heat flux; respectively, are simply F1 (τ r ,τ z ) and F2 (τ r ,τ z ) which are the exact solutions. As the scattering albedo increases, source coupling becomes stronger. If an iterative scheme is used in the numerical solution of the SKN equations, computation time for pure scattering cases is the highest. Previous studies revealed that the errors in the SKN method are also the highest for the pure scattering medium [2–4].

3 Results and discussions In order to test the applicability and the accuracy of the SKN approximation, we have adapted the following benchmark problems [9]: Benchmark Problem 1 (BP-1): A Short cylindrical medium with large optical radius, aspect ratios of 10 to 20, can be considered as a plane parallel geometry. The medium is homogeneous, cold and pure scattering ( ω = 1 ). The two-dimensional, solid cylinder solutions of SKN method for a combination of WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

112 Advanced Computational Methods in Heat Transfer IX optical radius of 0.2 to 25 mean free path (mfp) and height of 0.002 to 5 mfp are compared with the plane parallel geometry solutions of Crosbie and Viskanta [5] for the incident energy at the top and the incident energy and the net radiative heat flux at bottom. Benchmark Problem 2 (BP-2): Long cylindrical geometry with small aspect rations for a cold medium with scattering albedos of ω = 0.3, 0.5 and 0.9 are considered. The two dimensional cylindrical SKN solutions for the incident energy at the center and the incident energy and the net radiative heat flux at the surface are compared with the one-dimensional cylindrical RITE solutions [6]. Benchmark Problem 3 (BP-3): Radiative transfer in a two-dimensional cylindrical medium (Figure 1) which is subject to collimated unit irradiation on top surface was considered [7]. The other surfaces are cold and transparent. The medium is also cold, homogeneous with pure isotropic scattering ( ω = 1 ). The incident energy and net radiative heat flux solutions using SKN approximation are compared with those of Wu and Wu [7] and Hsu et al. [8]. For BP-3, the surface integrals of the incident energy and the net radiative heat flux in the RITE yield [7]: F1 (τ r ,τ z ) = exp(−τ z )

(11)

F2 r (τ r ,τ z ) = 0

(12)

F2 z (τ r ,τ z ) = exp(−τ z )

τz

(13)

τz = 0

τr

φ τr θ I τz τz = L τr = 0 τr = R

Figure 1:

The geometry and the coordinate system.

This study was carried out on a Pentium III 800 MHz processor with 512 Mb RAM. Computation time with SK1, SK2 and SK3 approximations naturally increase with the order of approximation since the number of differential equations to be solved are also increased. In Table 1, the effects of grid refinement and CPU time on numerical calculations are depicted. It is clear that as the scattering albedo is increased for a WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


113

fixed grid size and the same order of approximation, cpu time increases due to strong coupling in the SKN equations which requires more iterations for convergence. However, SKN equations can be solved directly using a block tri-diagonal solver in which case the CPU time remains nearly the same. Several cases of grid combinations were considered to ensure grid independence of the presented solutions within four significant digits. The grid structure was increased with the optical dimensions such as using M r × M z =50×100, M r × M z =100×200 and M r × M z =200×200 for cylinders with R=0.125 and L=0.25 mfp, R=0.5 and L=1.0 mfp, and R=2.0 and L=4.0 mfp, respectively.

Table 1:

The effect of cpu time and grid refinement for cylinder of R=0.125 mfp and L=0.25 mfp.

ω = 0.5 Mr × Mz

10×10 10×20 25×50 50×50 50×100

SK1 0.016 0.184 1.414 4.75 33.7

SK2 CPU (sec) 0.034 0.635 2.765 9.69 72.5

Table 2:

R

Z =1.0 L J

0.2 0.002 2 0.2 5 0.5 5 1 10 2 12 2.5 25 5.0

20 20 20 10 10 10 10

ω = 1.0 SK3

SK1

0.062 0.671 4.215 31.85 149.3

0.189 0.283 2.084 9.26 68.6

SK2 CPU (sec) 0.262 0.431 4.814 20.98 173.9

SK3 0.412 0.876 8.0 39.12 432.0

Comparisons of the solutions for BP-1.

G ( L)

1.0452 1.2645 1.4968 1.7574 2.0706 2.1729 2.4552

1.0253 1.0742 1.0538 0.9661 0.7807 0.7028 0.4504

SK3

SK2

Exact (Ref. [5])

G (0)

q ( L) G (0,0) G (0, L) q (0, L) G (0,0) G (0, L) q (0, L) 0.9901 0.9087 0.7975 0.6587 0.4825 0.4240 0.2613

1.0284 1.2499 1.5021 1.7672 2.0776 2.1791 2.4645

1.0085 1.0637 1.0659 0.9757 0.7790 0.7002 0.4505

0.9907 0.9089 0.7974 0.6586 0.4829 0.4249 0.2654

1.0343 1.2669 1.5008 1.7569 2.0721 2.1756 2.4673

1.0145 1.0783 1.0572 0.9634 0.7806 0.7033 0.4516

0.9907 0.9089 0.7976 0.6594 0.4834 0.4252 0.2652

In Table 2, the relative errors of G(0,0), G(0,L) and q(0,L) for BP-1 are compared with the exact plane parallel solutions [5]. The agreement of the exact and the SKN solutions, with increasing orders, is excellent. For example, the SK3 solutions for pure scattering medium (γ=20 and L=0.5 mfp) yield relative errors of –0.26%, –0.32% and –0.01% for G(0,0), G(0,L) and q(0,L), respectively. The errors for the same cylinder are 0.07%, 0.10% and 0.03% in—though not given in Table 2—Wu and Wu [7]’s study. For γ=10 and L=1 mfp, the errors for G(0,0), G(0,L) and q(0,L)are 0.03%, 0.27% and –0.12%, respectively, when using SK3 approximation. In Wu and Wu [7]’s study, the errors are found to be 0.28%, 0.52% and 0.23%, respectively. In other cylinder configurations, the errors when using the SK3 approximation have the same order of magnitude. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

114 Advanced Computational Methods in Heat Transfer IX Comparisons of the exact solutions [6], of BP-2 with the SKN solutions are given in Table 3 for various scattering albedos. For a medium with ω=0.9 and R=0.5 mfp, the relative errors of G(0), G(R) and q(R) with SK3 approximation are found to be –0.99%, 0% and –0.63%, respectively. For the same scattering albedo and R=1.0 mfp, the errors yield 0.50%, 0.02% and –0.67% and for R=2.5 mfp, –0.15%, –0.19% and –1.77% for G(0), G(R) and q(R), respectively. The SK2 solutions are in excellent agreement yielding 2 to 3 significant digit accuracies, however, as the medium becomes strongly scattering, higher order, such as SK3 approximation, becomes necessary. Table 3: R=0.5

ω

Comparisons of the solutions for Benchmark Problem 2. L=5

G (0) / 4π G ( R ) / 4π − q ( R ) G (0) / 4π G ( R ) / 4π − q ( R ) G (0) / 4π G ( R ) / 4π − q ( R )

0.3 0.6011 0.5 0.6828 0.9 0.9183 R=1.0 0.3 0.3644 0.5 0.4571 0.9 0.8247

0.7891 0.8348 0.9588 L=5 0.6944 0.7475 0.9249

1.4890 0.6059 1.1707 0.6917 0.2939 0.9388

0.7891 1.4886 0.6026 0.8352 1.1704 0.6858 0.9609 0.2947 0.9274

0.7889 1.4900 0.8346 1.1723 0.9589 0.2957

2.1647 0.3658 1.8014 0.4608 0.5453 0.8440

0.6937 2.1666 0.3636 0.7465 1.8054 0.4554 0.9246 0.5523 0.8205

0.6943 2.1660 0.7474 1.8036 0.9247 0.5490

0.3 0.0811 0.5 0.1266 0.9 0.5298

0.6030 0.6510 0.8548

2.7308 0.0807 2.4317 0.1254 1.0522 0.5238

0.6028 2.7370 0.0812 0.6507 2.4421 0.1268 0.8547 1.0713 0.5306

0.6035 2.7384 0.6518 2.4439 0.8565 1.0709

In Figure 2, the radial distribution of the incident radiation at the top and bottom surfaces of BP-3 are given comparatively with those of Refs [7] and [8] for cylinder with optical dimensions of R=0.125 and L=0.25 mfp. The maximum relative errors with SK3 approximation on the top surface are -3.06% and –1.87% in comparison to the solutions of [7] and [8], respectively. On the other hand, these errors with SK3 approximation on the bottom surface are –2.4% and – 2.1% in comparison to the solutions of [7] and [8], respectively. In Figure 3, the axial distribution of the incident radiation at the center and outer surface are depicted comparatively with those of [7] and [8] for cylinder with optical dimensions of R=0.125 and L=0.25 mfp. The maximum relative errors with SK3 approximation at the centerline are –2.4% and –2.06% in comparison to the solutions of [7] and [8], respectively; whereas, these errors for the outer surface are –2.32% and –1.92%. In Figure 4, the radial distributions of the incident radiation at the top and bottom surfaces are comparatively presented with those of [7] and [8] for cylinder with optical dimensions of R=0.5 and L=1.0 mfp. The maximum relative errors with SK3 approximation on the top surface are –5.36% and – 3.96% in comparison to [7] and [8] solutions, respectively; while the maximum errors with SK3 approximation on the bottom surface are –8.5% and –3.9% when compared to those of [7] and [8] solutions, respectively. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


115

1.2

TOP

G

1.1

1.0

Wu&Wu Hsu&Tan SK3

0.9 BOTTOM

0.8 0.0

Figure 2:

0.2

0.4

r/R

0.6

0.8

1.0

Radial distribution of the incident radiation at the top and bottom surfaces for cylinder with optical dimensions of R=0.125 mfp and L=0.25 mfp. 1.2

1.1

G

CENTERLINE 1.0 RADIAL SURFACE

Wu&Wu

0.9

Hsu&Tan SK3

0.8 0.0

0.2

0.4

0.6

0.8

1.0

z/L

Figure 3:

Axial distribution of the incident radiation at the center and outer surface for cylinder with optical dimensions of R=0.125 mfp and L=0.25 mfp.



TOP

1.2

1.0

Wu&Wu

G

Hsu&Tan SK3

0.8

BOTTOM

0.6

0.4 0.0

Figure 4:

0.2

0.4

r/R

0.6

0.8

1.0

Radial distribution of the incident radiation at the top and bottom Surfaces cylinder with optical dimensions of R=0.5 mfp and L=1 mfp. 1.6

1.2

G

CENTERLINE

RADIAL SURFACE

0.8

Wu&Wu

0.4

Hsu&Tan SK3

0.0 0.0

Figure 5:

0.2

0.4

0.6 z/L

0.8

1.0

Axial distribution of the incident radiation at the center and outer surface for cylinder with optical dimensions of R=0.5 mfp and L=1 mfp.



117

In Figure 5, the axial distribution of the incident radiation at the center and outer surface using SK3 approximation are shown comparatively with those of [7] and [8] for cylinder with R=0.5 and L=1.0 mfp. The maximum relative errors with SK3 are –5.4% and –3.97% at the center and –4.3% and –3.0% at the outer surface with respect to the solutions of [7] and [8], respectively. In Figure 6, the radial distributions of the incident radiation using SK3 approximation at the top and bottom surfaces are comparatively depicted with those of [7] and [8] for R=2 and L=4 mfp cylinder. The maximum relative errors of SK3 approximation on the top surface are –3.03% and 3.14% in comparison to the solutions of [7] and [8], respectively. On the other hand, the maximum errors with SK3 approximation on the bottom surface are –8.5% and 8.5% with respect to those of [7] and [8], respectively.

2.0 TOP

1.6

1.2

G

Wu&Wu Hsu&Tan SK3

0.8

0.4 BOTTOM

0.0 0.0

Figure 6:

0.2

0.4

r/R

0.6

0.8

1.0

Radial distribution of the incident radiation at the top and bottom surfaces for cylinder with optical dimensions of R=2 mfp and L=4 mfp.

In Figure 7, the axial distribution of the incident radiation using SK3 approximation at the center and outer surface are depicted comparatively with those of [7] and [8] for cylinder with R=2 and L=4 mfp. The maximum relative errors with SK3 approximation at the centerline are –3.03% and 3.14% in comparison to the solutions of [7] and [8] respectively; whereas, these errors for the outer surface are –4.31% and 0.83%, respectively. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

118 Advanced Computational Methods in Heat Transfer IX 2.5 Wu&Wu Hsu&Tan SK3

2.0

1.5 G 1.0

CENTERLINE

0.5

RADIAL SURFACE

0.0 0.0

Figure 7:

0.2

0.4

z/L

0.6

0.8

1.0

Axial distribution of the incident radiation at the center and outer surface for cylinder with optical dimensions of R=2 mfp and L=4 mfp. 0.5

0.4

TOP

0.3 q

Wu&Wu Hsu&Tan SK3

0.2

BOTTOM

0.1

0.0 0.0

0.2

0.4

0.6

0.8

1.0

r/R

Figure 8:

Radial distribution of the radiative heat flux at the top and bottom surfaces for cylinder with optical dimensions of R=2 mfp and L=4 mfp.



119

The incoming and/or outgoing radiative heat fluxes using SK3 also compare well within several percent errors, generally yielding underestimated values, with the referred studies. To illustrate the behavior of the radiative heat fluxes, we considered the distribution of cylinder with optical dimensions of R=2 and L=4 mfp. In Figure 8, the radial distribution of the outgoing radiative heat flux using SK3 approximation at the top and bottom surfaces are comparatively depicted with those of [7] and [8]. Solutions at the top and bottom surfaces from Wu and Wu [7] and Hsu et al. [8] almost coincide, and thus two the lines are not distinguishable. The maximum relative errors with SK3 approximation on the bottom surface are –1.92% and –1.87% in comparison to those of [7] and [8]], respectively, while these relative errors are –3.21 and –3.98%, respectively.

4 Conclusions The SKN approximation solutions for various two-dimensional homogeneous cylindrical mediums with isotropic scattering are compared with the solutions available in the literature. This study concludes the following: (i) the method is very accurate yielding 2-3 significant accurate solutions for mostly absorbing medium while the highest errors occur in pure scattering medium, (ii) the SKN equations can be numerically solved very easily with no numerical complexities, (iii) the approximation requires much less then computational effort when compared to the cpu time requirements of the exact RITE of the same grid configuration, (iv) the approximation can be improved especially at low orders by the selection of the synthetic kernel quadratures, (v) the method for moderate aspect ratios yield relative errors of several percent for pure scattering medium; however, medium with some absorption yield less errors, (vi) the method is free of “ray effect”.

References [1] Altaç Z. and Spinrad B.I., The SKN method I: A high order transport approximation to neutron transport problems, Nuclear Science and Engineering, 106, pp. 471-479,1990. [2] Altaç Z., The SKN approximation for solving radiative transfer problems in absorbing, emitting, and isotropically scattering plane-parallel medium: part 1, ASME Journal of Heat Transfer, 124, pp. 674-684, 2002 a. [3] Altaç Z., The SKN approximation for solving radiative transfer problems in absorbing, emitting, and linear anisotropically scattering plane-parallel medium: part 2, ASME Journal of Heat Transfer, 124, pp. 685-695, 2002 b. [4] Altaç Z. and Tekkalmaz M., The SKN approximation for solving radiation transport problems in absorbing, emitting and scattering rectangular geometries, JQSRT, Volume 73, No. 4, pp. 219-230, 2002. [5] Crosbie A.L. and Viskanta R., Nongray Radiative Transfer in a Planar Medium Exposed to a Collimated Flux, JQSRT, Volume 10, pp. 465-485, 1970.


120 Advanced Computational Methods in Heat Transfer IX [6] Altaç Z., Radiative Transfer in Absorbing, Emitting and Linearly Anisotropic-Scattering Inhomogeneous Cylindrical Medium, JQSRT, Volume 77, pp. 177-192, 2003. [7] Wu S.C. and Wu C.Y., Radiative Heat Transfer in a Two-Dimensional Cylindrical Medium Exposed to Collimated Radiation, International Comm. Heat & Mass Transfer, Volume24, No 4, pp. 475-484, 1997. [8] Hsu P.F., Tan Z.M., Wu S.H. and Wu C.Y., Radiative Heat Transfer in Finite Cylindrical Homogeneous and Nonhomogeneous Scattering Media Exposed to Collimated Radiation, Numerical Heat Transfer, Part A, Volume35, pp. 655-679, 1999. [9] Döner N., Calculation of Radiative Heat Transfer in Cylindrical Participating Medium Using the SKN Method, Ph. D. Thesis, Osmangazi University (in Turkish), 2003.



121

Presentation of the hemisphere method P. Vueghs & P. Beckers Aerospace and Mechanical Engineering Department, University of Liege, Belgium

Abstract In the field of image synthesis or thermal radiation, the challenge is to calculate the radiative exchanges between the surfaces of the model. To quantify the interactions between the surfaces, we define a number called view factor, noted Fij. The view factor represents the fraction of the diffusely emitted power from a surface Ai which reaches a surface Aj. The calculation of the view factors is a very difficult problem. A method has been designed in image synthesis and also applied to thermal problems. This method, called hemi cube, uses a cube as a screen onto which all the scene is projected. Here, we present another method that we have named hemisphere, where the surface of projection is the unit sphere surrounded by the hemi cube. This projection, based on the Nusselt’s Analogy, is simpler and more natural than the hemi cube’s projection. We show in this paper that the hemisphere method is faster and more efficient than the hemi cube. In this paper, we present a tessellation of the hemisphere into cells characterized by an equal elementary view factor. This allows us to avoid over sampling and to save computation time. We show that this method is faster and more efficient than the classic hemi cube. We present also some results that illustrate the efficiency of the hemisphere method. Keywords: view factor, hemisphere, projection, spherical coordinates, thermal radiation, Nusselt’s Analogy, ray casting.

1

Definition of the view factors

In the field of image synthesis, it is necessary to calculate the inter reflection of light between all the surfaces that compose the scene [9]. In the same way, in thermic, we have to evaluate all the heat transfers between all the surfaces of the model.


122 Advanced Computational Methods in Heat Transfer IX The exchanges of energy between two surfaces depend on the geometrical configuration of these surfaces. Mathematically, this geometrical dependency is expressed by a function called view factor [2]. The view factor expresses that the exchange of energy between two surfaces depends on their sizes, their relative orientations and the distance between them. The view factor does not depend on the radiative properties of the surfaces, such as absorbance and emittance. By definition, the view factor between two surfaces Ai and Aj is the fraction of the energy emitted diffusely by the surface Ai which reaches the surface Aj. We can note here the ambiguity of this definition based on the notions of emitted and absorbed energy although only the geometrical parameters appear in the definition, the surface properties do not appear. According to [14], if we assume that two surfaces Ai and Aj are fully visible to each other, the diffuse view factor between these surfaces is given by

FAi − Aj =

1 Ai

∫ ∫ Ai

cos θi cos θ j

Aj

π rij2

dAi dAj

(1)

JG

where θ is the angle between the normal vector at the surface and the vector rij

which joins points on the two surfaces. This expression has no analytical solution, except in the case of particular geometrical configurations [3, 6, 7]. It implies a numerical resolution of the integration involved in the computation of the view factors. Several techniques have been designed. If we consider the view factor between a point on the surface i, noted dAi, and the surface Aj, eqn (1) can be simplified. It is the limit of the usual view factor when the area Ai goes to zero. This new view factor is called pointwise view factor and is the inner integral of eqn (1)

FdAi − Aj = ∫

cos θi cos θ j

Aj

π rij2

dAj

(2)

We can evaluate this expression by different analytical methods. The most usual in image synthesis is called hemi cube [13]. The method presented in this article is mainly based on the hemi cube method. The principle is the same. It is also linked to the idea presented by F. Sillion to decompose the projection surface into cells of equal view factors [4]. F. Sillion proposed to project all the environment onto a single plane which takes the place of the top face of the hemi cube. He introduced a quadrangular mesh of this plane. The cells of this mesh are characterized by approximately equal elementary view factors. This idea has also been developed by Vivo et al. [11] and Lluch et al. [16]. They used the polar coordinates to obtain cells of equal view factors in the single plane. With this method, they conserve the axial symmetry of the formula of the view factor. The main disadvantage of the single plane methods is that the portion of the environment situated near the horizon cannot be taken into account. However, if the size of the plane is sufficient, we can reduce the neglected portion to a given threshold.



123

Here, we choose to project the scene onto the hemisphere because it is easier to mesh this surface than a plane of a cube, if we want to obtain cells of equal view factors. We obtain a very regular mesh which respects the axial symmetry of the view factor. We can also guarantee that the entire scene is taken into account in this method, unlike in the single plane methods.

2

The hemi cube

The hemi cube was designed and presented in 1985 by Donald P. Greenberg and Michael F. Cohen [13]. It was first used in image synthesis and after to solve thermal analysis [10]. This method consists of drawing a cube centered around the origin. The

JG

z axis coincides with the normal vector ni . The construction is represented in figure 1 according to references [8, 13].

Figure 1: Principle of the hemi cube. The cube is discretized into a number of cells. Each cell is associated with an element of view factor, called delta view factor. These elements are contributions to the pointwise view factor. Each value depends on the position of the cell on the hemi cube. The cells near the z axis are associated with the highest values and the cells located near the edges are characterized by lower values. This remark explains why we have over sampling. This method presents drawbacks: • an irregular sampling of the space, which implies that some areas are over sampled to guarantee a minimal accuracy. • the contributions to the pointwise view factor are not equal. We need to compute the value and to store it for each cell of the hemi cube. Nevertheless, the main advantage of this technique is that we can easily take occlusions into account by performing a test based on the distance of the surfaces projected onto the hemi cube. If many surfaces are projected onto the same cell of the hemi cube, we only keep the nearest one, the others will not be visible from the origin. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


3

Distribution of elements of equal view factors

In this section, we will present the solution we chose to obtain a regular mesh of cells with equal view factors. First, we will call back the definition of the view factor and the Nusselt’s Analogy. After that, we will mesh the hemisphere in spherical coordinates, in colatitude θ and in longitude ϕ. 3.1 Definition of the view factor and Nusselt’s Analogy According to Puech and Sillion [5], the view factor between a point xi on a surface Ai and a surface Aj (fully visible from xi) is noted FdAi − Aj and is given by eqn (2). Note that to establish this equation, we assumed that patch j is fully visible from xi. In other formulations, the visibility function appears explicitly. This function is defined between two points, it is equal to 1 if the points are mutually visible and 0 otherwise. This function is discontinuous and increases the difficulty of the computation. If we use the Nusselt’s Analogy (cfr [15] cited by [5]), the pointwise view factor FdAi − Aj can be considered as the result of two successive projections: •

a first projection onto the unit sphere centered on xi. This step corresponds to the factor

cos θ j rij2

in the relation (2). The solid angle

subtended by the surface dAj is given by d ω j =

cos θ j dAj rij2

.

•

a second orthogonal projection down onto the plane of the surface Ai, which corresponds to the factor cos θi. Now we can transform the relation (2). We obtain the following expression

FdAi − Aj = ∫

cos θi

Ωj

where Ωj is the solid angle subtended by Aj.

π

dω j

(3)

3.2 Mesh of the hemisphere Now, with the help of the Nusselt’s Analogy, we will decompose the hemisphere into solid angles. Each solid angle will be characterized by the same view factor. The decomposition will be performed following the meridians and parallels, i.e. in terms of latitude and longitude. The total number of cells N will be obtained by multiplying the numbers of subdivisions into the longitude and latitude directions, nlon and nlat.



125

3.2.1 In terms of colatitude In figure 2, if we consider an elementary surface dAi and a spherical cap (of unit radius) defined between θ=0 and θ=θS, the view factor is calculated by

FdAi − Aj =

Figure 2:

1

2π

θS

0

0

π∫ ∫

cos θ sin θ dθ dϕ = sin 2 θ S

(4)

Spherical cap of angular aperture θS.

By extension, we can determine a sequence of values θ which correspond to spherical caps larger and larger. These caps delimit rings centered on the normal vector at the origin. The rings are characterized by a same value of the view factor with respect to the center of the emitting surface. If we fix the resolution in colatitude nlat, we can subdivide the hemisphere in spherical rings with equal view factors. 3.2.2 In terms of longitude After that, we can chose the discretization of the hemisphere in longitude simply by dividing the rings in a determined number nlon of sectors. All the sectors must have the same angular aperture. 3.2.3 Resulting mesh So we have subdivided the hemisphere along the two spherical coordinates θ (colatitude) and ϕ (longitude). The view factor associated to each cell is equal to

1 . Figure 3 shows the hemisphere subdivided in longitude and latitude. nlat nlon

The cells are quadrangular, except the cells which have a vertex vertically above the center of the hemisphere and which are triangular.



Figure 3:

4

Wireframe view of the hemisphere.

Calculation of the view factor

The origin dAi is placed in a more or less complicated scene. The complexity of the scene is linked to the number of surfaces which compose the scene. The number of view factors to compute is equal to the number of surfaces in the scene. After meshing the hemisphere, we cast a ray along each direction {θ, ϕ}. For each ray, we first compute all the possible intersections with all the surfaces of the scene. If no intersection is detected, this means that the model presents an aperture. For example, this is the case of a satellite which orbits around the Earth and evacuates the power dissipated by its equipments to the deep space at 3K. The contribution to the view factor is added to a node which does not belong to the model. This supplementary node represents the deep space and gathers all the energy fluxes emitted by the spacecraft. When only one intersection is detected, the contribution to the view factor is assigned to the intersected surface. If several intersections are detected along the same direction, we store the distance of each intersection. At the end of the process, we only keep the nearest intersected surface. This technique allows us to solve the visibility problem very easily and quickly.

5

Error analysis

The hemisphere method has been programmed and tested. The results are shown below. Among the several tests we have performed, we chose to present the following one. For 2 perpendicular rectangles sharing a common edge, the aspect ratio of the rectangles is D and E, with respect to the common edge. In this case, it is easy to obtain an analytical solution of the view factor between the two surfaces. In accordance to [6], the resulting formula is given below



FAi − Aj =

1  1 1  E arctan   + D arctan   πE  E D

 1 − D 2 + E 2 arctan  2 2  D +E

(

)

 E 2 1 + D2 + E 2  2 2 2  1 + E E + D

(

)(

)

  

E2

(

)(

2 2  1  1 + E 1 + D + ln   2 2  4  1 + D + E

(

)

 D2 1 + D2 + E 2  2 2 2  1 + D D + E

(

)(

)

  

D2

    

127

) (5)

Here, the two geometrical parameters are equal to 1. The exact value is 0.200044. We used analytical solutions to verify the hemisphere method (both point-toarea and area-to-area formulations). We utilized the Gauss quadrature formula to integrate the elementary view factors from the hemisphere onto the area of the square. Here we used 5 Gauss points in each direction (25 points in total). We made use of the norm L1 to compute the relative error, i.e. the difference between the exact solution (VFanalytical=0.200044) and the integrated hemisphere’s view factor VFhemisphere.

ε=

VFanalytical − VFhemisphere

(6)

VFanalytical

We performed this calculation for nlat and nlon varying from 20 to 100 (it is not useful to plot the results for lower resolutions, the calculated error will be too high). The results are plotted at figure 4(a).

(a) Figure 4:

(b)

(a) Error for different resolutions. (b) Theoretical error for different resolutions.


128 Advanced Computational Methods in Heat Transfer IX The abscissa x and y are the values of the parameters nlon and nlat respectively. The z axis represents the logarithm of the relative error. In this figure, the error varies between -2.5 and -4, i.e. 10-2.5 and 10-4. We can observe that the error decreases when the resolution increases. The error can be approximated by ε ≈ 1 . This error is plotted at figure 4(b). For nlon nlat 10 000 casted rays (nlon =100 and nlat =100), the error will be around 10-4 10−4 . Remark: during this test, we found some difficulties with this particular geometry. The vertical edges of the square are projected onto the hemisphere along meridians. When the mesh of the hemisphere and the projected edges are superimposed, nearly all the cells in the meridian are wrong. The error is independent of the colatitude resolution, around

1 . This is called nlon

discretization error. It gives oscillations in the error graphics. To prevent this case, we implemented a variant of the hemisphere, where the rays are randomly casted through each cell. We compute a mean value for each cell and we find an error close to

1 . The mathematical theory which supports this method is nlon nlat

not presented in this article but is developed in the internal report [17]. This random method has an evident drawback. The results are affected by the noise that characterizes all random processes. Nevertheless, this method is more reliable since it is not affected by the discretization error. If we consider the corresponding error plots, the oscillations disappear. Such error plots are shown in reference [17].

6

Results

In order to illustrate the hemisphere method, we implemented the well known scene called the Cornell box [1]. Picture 5 was obtained with a mesh of around 2 000 triangular cells. We used the random version of the hemisphere, with 4 rays per cell. To display this view, we used the tone mapping method called mean value mapping [12] and linear interpolation. The inter reflection of light is correctly modeled. The penumbra caused by the obstacles is also automatically computed. Note that a colour copy of all these pictures is accessible from [18].

7

Conclusions

In this article, we have presented a simple method which allows us to compute the view factors from a point to surfaces of finite dimensions. This method allows us to keep a good physical comprehension of the problem.



129

Figure 5: Cornell box using the hemisphere’s method. Compared to the hemi cube method, this method is faster and requires less memory. This method needs only one projection, the algorithm is simpler. All the elementary view factors are equal, which prevents useless over sampling and the storage of all the contributions to the view factors. Compared to the single plane method, the hemisphere method is as fast. Nevertheless, we can study the whole hemisphere, without neglecting the surfaces close to the horizon.

References [1] [2] [3]

The Cornell Box, http://www.graphics.cornell.edu/online/box/. Donald P. Greenberg, Benneth Bataille, Cindy M. Goral, Kenneth E. Torrance, Modeling the interactions of light between diffuse surfaces. Computer Graphics, 6(2), pp. 213-222, 1984. W. R. Morgan, D. C. Hamilton, Radiant-interchange configuration factors, Technical report 2836, National Advisory Committee for Aeronautics, 1952. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

130 Advanced Computational Methods in Heat Transfer IX [4] [5] [6] [7] [8] [9] [10]

[11] [12] [13] [14] [15] [16] [17] [18]

Claude Puech, François Sillion, A general two-pass method integrating specular and diffuse reflection, Computer Graphics, 23(3), pp. 335-344, 1989. Claude Puech, François Sillion. Radiosity and Global Illumination, Morgan Kaufmann Publishers, Inc., 1994. John R. Howell. A Catalog of Radiation Heat Transfer Configuration Factors, McGraw Hill, 1982. http://www.me.utexas.edu/~howell/. H. Y. Wong. Handbook of Essential Formulae and Data on Heat Transfer for Engineers. Longman, 1976. Stéphane Jardon, Première approche sur l’étude des facteurs de vue, 2003, Mechanical Department, University of Liège, Belgium. James T. Kajiya, The rendering equation, SIGGRAPH, 20(4), 1986. T. G. Gonda, K. R. Johnson, A. R. Curran. Development of a signature supercode, Technical Proceedings 1938: Advances in Sensors, Radiometric Calibration, and Processing of Remotely Sensed Data (SPIE International Symposium on Optical Engineering and Photonics in Aerospace and Remote Sensing): Orlando, FL, 1993. Roberto Vivo, Maria José Vicent, Javier Lluch. Un método para el calculo del factor de forma alternativo al hemi-cubo. Universidad Politécnica de Valencia. Kresimir Matkovic. Tone mapping techniques and color image difference in global illumination. Master’s thesis, Technischen Universität Wien, February 1998. Donald P. Greenberg, Michael F. Cohen, The hemi cube – a radiosity solution for complex environments, Computer Graphics, 19(3), pp. 31-40, 1985. John R. Wallace, Michael F. Cohen. Radiosity and Realistic Image Synthesis, Morgan Kaufmann Publishers, Inc, 1993. W. Nusselt, Graphische Bestimmung des Winkelverhaltnisses bei der Wärmestrahlung, Zeitschrift des Vereines Deutscher Ingenieure, 72(20), pp. 673, 1928. J. Lluch, R. Molla, P. Jorquera, R. Vivo, M. J. Vicent, Study of the form factor calculation by single polar plane. Department of Computer Systems and Computation, Universidad Politécnica de Valencia. P. E. Vueghs, Méthode de l’hémisphère – validation de la méthode, technical report, University of Liège, Belgium, 2005. LTAS – Infographie, Université de Liège, http://www.ulg.ac.be/ltas-cao/.



131

Temperature identification based on pointwise transient measurements A. Nassiopoulos & F. Bourquin French Public Works Research Laboratory (LCPC), Paris, France

Abstract The inverse problem of temperature identification based on discrete transient measurements is considered. An iterative procedure combining the conjugate gradient algorithm with Tikhonov regularization is used to determine unknown boundary heat fluxes. We show numerically and theoretically that the choice of an H 1 -type space for the minimization gives much better results than a L2 -type one. A dual formulation of the problem coupled with a model reduction method is used to construct a fast and accurate algorithm suited for real time computations in the three-dimensional case. Keywords: temperature reconstruction, inverse heat conduction, optimal control theory, adjoint method.

1 Introduction Various methods for damage detection and structural health monitoring exist. Among them, the output-only techniques consist in identifying the low frequency spectrum of a given structure in order to detect modifications with respect to the non-damaged reference state. However, in civil engineering, thermal loading due to environmental factors induces mechanical stress on structures and can cause eigenfrequency shifts of much larger order of magnitude than those caused by structural damage [1]. Thus, the knowledge of the exact thermal state of a given structure and its effects on vibrational properties appears as a bottleneck in this field [2]. The need for temperature identification arises from the partial information on the thermal state of a structure one can get. The most commonly used temperature sensors can only provide local or pointwise measurements. In addition, some parts of a structure can simply be inaccessible to direct measurements. This is the case WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/HT060131

132 Advanced Computational Methods in Heat Transfer IX in civil engineering structures, although the number of sensors that equip modern ones is rapidly increasing. The thermal state at any point of a given structure can only be obtained through a suitable reconstruction procedure. Among possible methods, one will eliminate Kalman filtering techniques in continuous time which would lead to the solution of Riccati equations of great dimension, and that seems inappropriate without preliminary reduction. The temperature identification problem can be seen as an inverse problem consisting in inverting the heat equation. This class of problems are in general mathematically ill-posed and regularization techniques are needed to solve them. They have been studied extensively in literature during the last few decades together with many techniques for their numerical computation [3]. Inverse problems concerning heat conduction have been the topic of numerous works due to the wide range of applications that are concerned [4–7]. A review of literature can be found in [8–10]. This paper deals with the inverse problem of temperature identification based on discrete measurements. In view of easy implementation in general purpose finite element software, the approach is that of optimal control theory [11]. The problem is written in a least squares setting. The adjoint technique [12, 13] is used to determine the gradient of an error functional together with Tikhonov regularization [3]. When standard gradient-based algorithms are put to work, each computational step entails solving the same forward heat equation with changing data. A dual formulation of the problem and a model reduction technique are put to work to obtain a high speed accurate algorithm suited for real-time applications.

2 Problem statement Consider a solid in a multidimensional domain Ω with boundary ∂Ω. Assuming absence of internal heat sources and zero initial temperature, the temperature field inside the solid is given by the heat equation:  ∂θ    ρc ∂t − div(K grad θ) = 0 in Ω × [0, T ] (K grad θ) · n + αθ = Φ on ∂Ω × [0, T ]    θ(x, 0) = 0 in Ω

(1)

Here, Φ(x, t) = g(x, t) + αθext where g denotes an inward heat flux with FourierRobin conditions and θext is the external temperature, n is the outwards normal vector on the boundary, x ∈ Ω is the space variable, t ∈ [0, T ] the time variable, ρ the mass density, c the heat capacity and K the scalar conductivity coefficient of the material taken to be homogeneous and isotropic for the sake of simplicity. Assume that m sensors are available inside the structure at locations xk , k = 1..m. They deliver the data {θkd (t)}m k=1 , t ∈ [0, T ]. Based on these measurements, the aim is to reconstruct the temperature field over the time interval [0, T ], focusing on the accuracy of the temperature field reconstruction at the final instant T . WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


133

One way to achieve this is to seek the boundary flux Φ responsible for the measurements. The problem of recovering the boundary condition Φ can be viewed as an operator inversion problem. Consider operator T defined by T :

U →M T Φ = {θ(xk , t)}m k=1

(2)

where θ(x, t) is the temperature field verifying the heat equation (1). In the above, U is the control space (such as Φ ∈ U ) and M = L2 ([0, T ])m is the measurements space (such that {θkd (t)}m k=1 ∈ M ). Note that we have not explicitly defined so far the nature of space U as this will be the object of section 3. Under these notations, the temperature reconstruction problem consists in finding Φ such that T Φ = {θkd }m k=1

(3)

For the sake of simplicity, only the case of the boundary condition Φ reconstruction is considered in this paper. It is possible to extend all that follows to the case where one wants to reconstruct a couple of functions {θ0 , Φ} with θ0 (x) = θ(x, 0) an unknown initial thermal state in (1). Due to the smoothing properties of operator T , the latter is not invertible and the inverse problem (3) is mathematically ill-posed in the sense of Hadamard. The solution of (3) can only be derived in a least squares sense through a functional minimization procedure. The functional can be for instance a quadratic form of the residual term E = θ(xk , t) − θkd (t), (k = 1..m), which measures the distance between the data θkd (t) and the values at sensor locations of the temperature field determined by Φ. J(Φ)

= =

m 2 1 T θ(xk , t) − θkd (t) dt + Φ2U 2 2 k=1 0 1 d m 2 2 T Φ − {θk }k=1 M + ΦU 2 2

(4)

The last term stands for the Tikhonov regularization, being a small coefficient that guarantees numerical stability even with noisy input data. · 2U and · 2M are suitable norms in U and M respectively.

3 Minimization procedures The results of the minimization process strongly depend on the a priori assumptions on Φ that determine the choice of space U . In this section two different possibilities are considered, showing that better results are obtained within the framework of an H 1 -type space. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

134 Advanced Computational Methods in Heat Transfer IX 3.1 Minimization in L2 The most natural choice consists in looking for a solution Φ belonging to the space of square integrable functions, namely L2 (∂Ω×[0, T ]). This space can be equipped with the scalar product: T

u, v U = uv dγ dt, ∀u, v ∈ U (5) 0

∂Ω

The adjoint technique is employed to write J as a quadratic form. To this end, we introduce the adjoint operator T ∗ whenever it exists: T∗ :

M →U T ∗ {ϑk (t)}m k=1 = ϕ(x, t)

(6)

where φ is the restriction over ∂Ω of the so-called adjoint state p:  M    −ρcpt − div(Kgrad p) = ϑk (t)δxk in Ω × [0, T ]    (Kgrad p) · n + αp = 0     p(x, T ) = 0

k=1

on ∂Ω × [0, T ] in Ω

(7)

ϕ(x, t) = p(x|∂Ω , t) Operators T and T ∗ verify the duality relation which writes m M T Φ, {ϑk }k=1 M

=

U Φ, T

∗

{ϑk }m k=1 U

(8)

In the above, U and M are the dual spaces of U and M respectively and angle brackets stand for the duality pairing. To formally prove this relation, one has to rewrite equations (1) and (7) in variational form. After an integration by parts of the time derivative term and combination of the last two expressions, one obtains m k=1

0

T

θ(xk , t)ϑk (t) dt =

0

T

Φϕ dγ dt

(9)

∂Ω

which is exactly relation (8). Using (8), the functional J can be written as a quadratic form of Φ: J

= = =

1 2 2 T Φ − {θkd }m k=1 M + ΦU 2 2

1 d m 2 2 T Φ2M − 2 T Φ, {θkd }m k=1 M + {θk }k=1 M + ΦU 2 2

1 AΦ, Φ U − Φ, b U + c 2

(10)

d m 2 with A = J T ∗ T + I, b = J T ∗ {θkd }m k=1 and c = {θk }k=1 M . J is the Riesz isomorphism between U and U while I stands for the identity operator. It follows



135

directly that the gradient of J is given by: ∇J(Φ) = J T ∗ (T Φ − θkd (t)) + Φ

(11)

This expression means that the gradient of the functional is the restriction on the boundary ∂Ω of the adjoint field obtained by (7) with a source term {ϑk (t)}m k=1 equal to the residual E = {θ(xk , t) − θkd (t)}m k=1 . This method of obtaining the gradient allows for a gradient-type method to be used. Each iteration will consist of a computation of the direct equation (1), followed by a computation of the adjoint equation (7) with {ϑk (t)}m k=1 = E. The gradient will then be given by p|∂Ω + Φ. Here, the classical conjugate gradient algorithm was used giving very satisfactory results [7]. Note that equations (1) and (7) have the same structure, so that the same numerical procedure can be used to solve them. The whole algorithm can thus be easily implemented in any classical general purpose scientific software, this being one of the main advantages of the method. The results below show the disadvantages of the approach described so far. They concern a one-dimensional beam of length L to which a given flux is prescribed at each end. All material constants are set to 1. Some direct simulations with an arbitrary flux input give measurements on sensors located at L5 , L2 and 4L 5 respectively. These measurements are then used to simulate the reconstruction algorithm. The model is discretized with P1 finite elements and an implicit Euler scheme is used for time integration. In figure 1, the reconstructed flux (dotted line) is compared to the prescribed one (solid line). The flux is well reconstructed on almost the entire time interval, but the accuracy of the results drops near the final instant: the curve exhibits some oscillations and moves away from the target curve to reach zero at t = T . As a consequence, the reconstruction of temperature field is of acceptable accuracy far from t = T , while the reconstruction of θ(x, T ), which is the most interesting output in view of applications, is very unsatisfactory. This phenomenon is due to the property assigned to the adjoint field p to be null at t = T (equation (7)) and is well known in the literature [8, 13]. This final condition on p has to be prescribed in order to verify relation (8). An alternative definition of the adjoint field can be considered, but this implies that the space U has also to be modified. This is the topic of the next section. 3.2 Minimization in H 1 A change of framework for the minimization procedure overcomes this final instant problem. It consists in choosing an H 1 -type space instead of the L2 -type one. Take U = H 1 ([0, T ], L2 (∂Ω)) with the scalar product

u, v

U

=

0

T

uv dγ dt + ∂Ω

0

T

∂t u∂t v dγ dt, ∀u, v ∈ U ∂Ω



Figure 1: Reconstruction in L2 : flux (left) and final temperature reconstruction (right). In order to verify relation (8), the operator T ∗ has to be defined in a different manner. One can easily verify that if ϕ in (6) is defined by 0

T

φp dt =

0

T

φϕ dt +

0

T

∂t φ∂t ϕ dt, ∀φ ∈ H 1 ([0, T ] × ∂Ω)

(12)

where p verifies (7), then operator T ∗ verifies again the duality relation (8) with the new choice of U . As can be seen, the computation of T ∗ needs one more step involving the solution of (12) with p known. This last equation is of the form L(φ) = a(ϕ, φ), ∀φ, a(·, ·) being a quadratic and L(·) a linear form. It can be easily implemented with the finite element method. The great advantage of this is that the end condition on ϕ with respect to time is not a Dirichlet one anymore. As a consequence, the value of Φ at t = T is not fixed and the accuracy of reconstruction near the final instant is much better as can be seen in figure 2 corresponding to the same case as previously. The curves of reconstructed and target final temperatures are identical.

Figure 2: Reconstruction in H 1 : flux (left) and final temperature reconstruction (right). WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


137

4 Dual formulation Consider the self-adjoint operator T T ∗ . Consider next the problem of finding ∗ m d m {Xk (t)}m k=1 ∈ M such that T T {Xk (t)}k=1 = {θk (t)}k=1 . If we set Φ = ∗ m T {Xk (t)}k=1 , then solving (3) is equivalent to finding {Xk (t)}m k=1 ∈ M such that d m (13) T T ∗ {Xk (t)}m k=1 = {θk (t)}k=1 This new problem is called the dual of (3). It exhibits an ill-posed nature and some form of regularization is needed. Applying Tikhonov regularization, one will consider the problem m d m T T ∗ {Xk (t)}m k=1 + {Xk (t)}k=1 = {θk (t)}k=1

(14)

Due to the self-adjoint nature of T T ∗ , the latter is equivalent to minimizing the dual functional J : 1 m A{Xk (t)}m J = − {Xk (t)}m (15) k=1 , {Xk (t)}k=1 k=1 , b 2 M M with A = T T ∗ + I and b = {θkd (t)}m k=1 . The dual formulation exhibits many features of interest. Firstly, the problem is set over space M which is of much smaller size than the corresponding space U . As a consequence, fewer search directions are involved in the minimization procedure and the conjugate gradient method is expected to converge much faster. Secondly, all scalar products that need to be computed are scalar products in M rather than in U so that the complexity of the algorithm depends more on the number of sensors than on the size of domain Ω. This new formulation results in dramatic reduction of computational costs and in greater adaptability to different geometries. Furthermore, it allows some parallelization of the algorithm, this issue being however beyond the scope of this paper.

5 Model reduction ∞ 1 We introduce two distinct function bases {ξi }∞ i=1 , {χi }i=1 that span H (Ω), and a ∞ 2 third function basis {ηi }i=1 that spans the space L (∂Ω) such that we can approximate the fields θ, p and Φ by a reduced order linear combination:

θ p Φ

∼ θr (x, t) = ∼ pr (x, t) = ∼ Φr (x, t) =

αi (t)ξi (x)

i=1 ˇ

βi (t)χi (x)

i=1 ˜

(16)

γi (t)ηi (x)

i=1

These approximations result in a model reduction that enables a considerable speed-up of the computational procedure, since the size of the systems resulting of WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

138 Advanced Computational Methods in Heat Transfer IX the finite element discretization of (1) and (7) only depends on the three parameters ˜ Under these notations, the objective of reconstruction is to recover the , ˇ and . modal coefficients γi (t), i = 1..˜ of the approximate flux Φr . The quality of the results depends on the accurate description of the actual variables θ, p and Φ by the approximate ones. Another crucial point is the choice of the basis functions in order to ensure observability and controllability of the physical system.

6 Numerical test case Hereafter is shown a numerical example on a three-dimensional case based on this model reduction. A solid of dimensions Lx = 1, Ly = 0.5 et Lz = 0.4 and material properties all set to 1 is subject to an unknown heat flux on the face of coordinate x = 0 and zero heat fluxes on all other faces. All boundary conditions are of the Fourier-Robin type. Four sensors are located at points (0.1, 0.4, 0.1), (0.1, 0.1, 0.2), (0.2, 0.1, 0.3) and (0.2, 0.4, 0.4). Like in the previous 1D example, a direct simulation with a prescribed flux on face x = 0 gives measurements that are taken afterwards as input data for a simulation of the reconstruction algorithm. The prescribed flux has the form shown on figure 3 and can be decomposed into two functions η1 (x) and η2 (x) with arbitrary time variation so that Φ = γ1 (t)η1 (x) + γ2 (t)η2 (x). Here, η1 (x) and η2 (x) are the first two eigenvalues of the Laplace operator on a rectangle with Neumann boundary conditions.

Figure 3: Reconstruction of flux coefficients γ1 (t) and γ2 (t) (on the left); prescribed (solid line) and reconstructed (dotted line) flux on face x = 0 (on the right). Figure 3 shows the very accurate reconstruction of the two corresponding coefficients γ1 (t) and γ2 (t). A very accurate reconstruction of the different components αi of the temperature field obtained under this flux input was also observed. Figure 4 shows the isovalues of the final temperature on two sections on the xy-direction and the isovalues of the difference between reconstructed and target fields: there is WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


139

a ratio of 104 between the corresponding orders of magnitude. The reconstruction took 4.4s CPU for 20 iterations.

Figure 4: Isovalues of the temperature field on planes z = 0 and z = 0.2, at t = T : target field (left); difference between target and reconstructed fields (right).

7 Conclusions The adjoint technique for the minimization of a least squares error functional has been applied with success to the problem of temperature reconstruction based on pointwise measurements. The use of the H 1 space has proved to be the best-suited framework for the minimization procedure, and has permitted us to overcome the final instant problem of the classical technique. The dual formulation of the problem and the model reduction using well chosen basis functions enables us to carry very fast computations with great accuracy. The algorithm can thus be adapted for the real time monitoring of structures. In such a case, the reconstruction procedure can be carried out at given intervals with overlapping time domains: the initial temperature condition for each computation will be known from the previous reconstruction computation and added to the reconstructed field. The entire procedure could be compared with an observer of great efficiency and speed.

References [1] Farrar, C., Hemez, F., Shunk, D., Stinemates, D. & Nadler, B., A review of shm literature: 1996-2001. Los Alamos National Laboratory Internal Reports, 2003. [2] Kullaa, J., Elimination of environmental influences from damagesensitive features in a structural health monitoring system. Proceedings of the First WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


[3] [4]

[5]

[6]

[7]

[8] [9] [10] [11] [12]

[13]

European Workshop on Structural Health Monitoring, Paris, July 10-12, 2002, Onera. DEStech Publications, pp. 742749, 2002. Engl, H.W., Hanke, M. & Neubauer, A., Regularization of Inverse Problems. Kluwer Academic Publishers, 1994. Beck, J.V., Blackwell, B. & Haji-sheikh, A., Comparison of some inverse heat conduction methods using experimental data. Int J Heat Mass Transfer, 39(17), pp. 36493657, 1996. ¨ Neto, A.J.S. & Ozisik, M.N., Two-dimensional heat conduction problem of estimating the time-varying strength of a line heat source. J Appl Phys, 71(11), pp. 53575362, 1992. Videcoq, E. & Petit, D., Model reduction for the resolution of multidimensional inverse heat conduction problems. Int J Heat Mass Transfer, 44, pp. 18991911, 2001. Prudhomme, M. & Nguyen, T.H., Fourier analysis of conjugate gradient method applied to inverse heat conduction problems. Int J Heat Mass Transfer, 42, pp. 44474460, 1999. Alifanov, O.M., Inverse Heat Transfer Problems. Springer-Verlag, New York, 1994. Beck, J.V., Blackwell, B. & Clair, C.S., Inverse heat conduction, illposed problems. Wiley Interscience, New York, 1985. ¨ Ozisik, M.N. & Orlande, H.R.B., Inverse Heat Transfer. Taylor and Francis, 2000. Lions, J.L., Optimal control of systems gouverned by PDEs. Dunod, 1968. ¨ Jarny, Y., Ozisik, M.N. & Bardon, J.P., A general optimization method using adjoint equation for solving multidimensional inverse heat conduction. Int J Heat Mass Transfer, 34(11), pp. 29112919, 1991. Huang, C.H. & Wang, S.P., A three-dimensional inverse heat conduction problem in estimating surface heat flux by conjugate gradient method. Int J Heat Mass Transfer, 42, pp. 33873403, 1999.


Section 3 Heat and mass transfer



143

Heat transfer in 3D water and ice basins S. Ceci, L. De Biase & G. Fossati University of Milano Bicocca, Department Environmental Sciences, Italy

Abstract Studying heat transfer in 3D water or ice basins involves the solution of Navier Stokes or Stokes systems of PDE, coupled with a scalar equation for thermal energy transport. The method used in this paper is based on a Finite Volume technique where physical quantities and boundary conditions are approximated by means of high order formulae and the time advance is dealt with by a fractional step technique. This paper is mainly concerned with applications, but, with respect to a preceding version of the method, an evaluation of performance of techniques dealing with the turbulent viscosity in water basins is presented and a study of the temperature field in a glacier is included. The method is applied to a thermal discharge in a water basin of lower temperature and to a portion of the Priestley Glacier (Antarctica). The results are very accurate and coherent with the physical theory and with measured data. Keywords: fluid dynamics, glacier modelling, glacier temperature, thermal discharges.

1

Brief description of the method

In this paper some interesting applications of a method presented in Deponti et al. [1] aimed at modelling meso-scale mass and energy flows in glaciers are presented, together with applications of a similar method (Hagos et al. [2]), modelling thermal discharges in water basins. The method for the study of flow and thermal energy transport in water basins is based on Navier-Stokes equation, coupled with the temperature transport equation.


144 Advanced Computational Methods in Heat Transfer IX In glaciers, convective terms are negligible (LeB et al. [3], Hutter [4], Paterson [5], Colinge and Blatter [6], Blatter et al. [7]). Therefore the problem is modelled by Stokes system, in vector, unsteady conservation form: ∇ ⋅ u = 0  (1)   ∂u − ∇ ⋅ ν (∇u) = −∇p + g + (∇ν ) ⋅ (∇u)T       ∂t T

where (∇ν ) ⋅ (∇u) is a viscous forcing term. Although an acceleration term appears in momentum equation, the final solution verifies a steady Stokes system since, at steady state, the temporal derivative vanishes.

2

Non-dimensionalization

Equations and boundary conditions are non-dimensionalized by means of factors defined on the basis of the velocity norm, the glacier or basin depth and the amplitude of the temperature range.

3

Temporal evolution

A Fractional Step technique deals with the temporal evolution. At time instant n+1, at first, provisional velocities are computed on the basis of a hydrostatic pressure and a surface elevation η relative to the preceding time instant. Subsequently the full pressure is calculated and velocities are corrected. Finally the surface elevation is computed by Saint-Venant equation and the temperature field is found.

4

Space discretization

The domain is subdivided into a set of cell-centred rectangular control volumes with faces orthogonal to the coordinate axes. A Finite Volume method is used to grant both local and global conservation of each physical quantity; the profiles approximating diffusive and convective terms are defined in such a way that a uniform distribution of volume dimensions is not needed. Boundary conditions are therefore approximated by Generalized Finite Difference formulae (De Biase et al. [8]). The vertical dimension of the surface volumes evolves in time.

5

Viscosity

The turbulent viscosity in water flows is modelled by our method in several possible ways (as an evolution of Hagos et al. [2]): constant viscosity, Prandtl method, standard k − ε and low-Reynolds k − ε methods (Mohammadi and Pironneau [9]) In glaciers, molecular viscosity is considered (Gudmundsson [10]). WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


6

145

Domain definition

The reference system is defined in such a way that the free surface and the bedrock or bottom are, in the average, approximated by planes parallel to the x-y coordinate plane, while inflow and outflow boundaries are parallel to the y-z coordinate plane and lateral boundaries are parallel to the x-z coordinate plane.

7

The thermal energy equation in a glacier

This equation is written as in Anderson [11], on the basis of the temperature field: DT ρc − ∇ ⋅ ( K ∇T ) = S (2) Dt where ρ is the fluid density, K is the thermal conductivity and S, the heat production rate for unit volume, generated by ice deformation, is written on the basis of stress components, as: 2 2 + εyy + εzz2 + 2 ( εxy + εxz + εyz ) + ε02  = 2µ 2εII + ε02  S = 2µ εxx (3) where µ is the dynamic viscosity and εII is the second invariant of the stress tensor ε.

8

Case test 1: thermal discharge

The domain for case test 1 simulates the outfall from an industrial plant: in the channel, water of a given temperature flows toward a water basin of lower temperature. The domain is a 3D rectangle of total length 20 m, width 1 m and total height 2 m, with a backward-facing step of length 10 m and height 1 m; it is discretized into 10750 volumes of dimensions ∆x = ∆z = 0.2 m, ∆y=0.1 m . Initial conditions are of null motion and pressure fields on the whole domain and unperturbed free surface. At the inflow boundary, homogeneous Neumann conditions for hydrodynamic pressure and for components v,w of velocity are assigned, while for the u component a value of 5 m/s is given and

T = 26D C ; k = 0.003uˆ 2 ; ε = cµ3 4

k3 2 L3

, with initial viscosity of 3*10-4 m2/s.

At the outflow boundary homogeneous Neumann conditions for velocity, temperature, k and ε and homogeneous Dirichlet for the hydrodynamic pressure are imposed. On lateral walls friction conditions are assigned, with Chezy coefficient of 50 m1/2/s, homogeneous Neumann conditions for temperature, k and ε. On the free surface null stress is assumed with homogeneous Neumann conditions for temperature, k and ε (Barkley et al. [12], Beaudoin et al. [13]).



Figure 1:

Motion fields with constant viscosity, Prandtl technique and a k − ε method.

In figure 1 the motion fields obtained by imposing constant viscosity of 10-6 m2/s, by Prandtl approximation and by a standard k − ε method are compared. Chezy coefficient is normally in the range between 30 and 60 m1/2/s, depending on the substrate of the channel. Since a constant density hypothesis has been formulated in this work, the coefficient must be divided by 1000, the water density.



147

°c

°c

Figure 2:

Temperature distribution after 132 and 600 time steps.

In the application presented above it is evident that the constant viscosity model is not an option for the problem at study and that the best results are obtained by means of the k − ε technique. In figure 2 the temperature distributions obtained by this technique after 132 and 600 time steps are shown.

9

Case test 2: the Priestley Glacier

For the glacier bedrock, the interface ice/rock is implicitly defined as B ≡ −h − z = 0

The kinematic boundary condition imposes the vertical velocity to equal the z component of the melting/freezing rate. On the lateral boundaries, orthogonal to the y axis, the kinematic condition is the same, but the y component must be considered for lateral velocity; dynamic boundary conditions for velocity are:


148 Advanced Computational Methods in Heat Transfer IX  ∂u ∂v  2 = Cµ 2  + u = Cσ xy   ∂y ∂x  v = Cσ

2 yy

2

 ∂v  = C  −P + 2µ  ∂y  

2

 ∂v ∂w  2 = Cµ 2  + w = Cσ yz   ∂z ∂y 

where ^ is the slip coefficient. (4) 2

On the free surface, the kinematic boundary condition is determined by the accumulation/ablation rate vector, while dynamic B.C. are: ∂u ∂w ∂v ∂w ∂w + = 0, + = 0, = 0. (5) ∂z ∂x ∂z ∂y ∂z On the bedrock, a geothermal flux is assigned: ∂T q geo = (6) ∂n K where q geo , the geothermal flux, varies between 46 mWm-2 and 77 mWm-2 depending on the rock age. On lateral solid walls, at ice/ice interfaces, a null flux condition is imposed, while at rock/ice interfaces, the condition ∂T q geo =± (7) ∂y K is imposed. On the free surface Dirichlet conditions can be assigned (Paterson [5], Luthi and Funk [14, 15]), or a Neumann condition of the form (Hutter, [4]): ∂T q atm = (8) ∂n K where q atm is the atmospheric heat flux. n +1  n +1 n +1  ∂T   S n +1 + ∇ ⋅ u i T − κ  , i=1,2,3 (9)  = ∆t c   ∂xi   where S is the heat production term, κ is the thermal diffusivity and c is the thermal capacity. The area of study, built by means of the RAMP database and of aerial images, has dimensions L = 14000 m, D = 7000 m and H = 1000m, discretized into 5466 volumes. Dimensions of internal control volumes are: ∆x=∆y=600 m and ∆z=100 m. Components of the gravity acceleration vector are: gx=0.169 m/s2, gy=0.007 m/s2 and gz=9.909 m/s2. Homogeneous Neumann conditions are imposed for velocity and pressure at the inflow boundary, free exit for velocity and homogeneous Neumann for pressure are assigned at the outflow boundary. The free surface can evolve in space and time. On the bedrock and on the lateral

T n +1 − T n



149

walls a slip condition is imposed, with slip coefficients C =5x10-18 ms-1Pa-2 and C =10-18 ms-1Pa-2, respectively. The variable viscosity is computed by Glen’s constitutive law, with parameter n=3 (Pettit and Waddington [16]). Initial conditions are null motion and pressure fields on the entire domain.

Figure 3:

Velocity field in a longitudinal section and a horizontal section; surface elevation in the glacier.



Pas

kPa

Figure 4:

Viscosity field, temperature distribution and hydrodynamic pressure in a longitudinal section of the Priestley Glacier.



151

10 Conclusion Results presented in this paper are in full agreement with the theory and with physical observations and measurements (Barkley et al. [12], Beaudoin et al. [13] for water applications, Kamb [17], Pattyn [18] for the Priestley application). In future some more work should be done in order to allow a more flexible assignment of boundary conditions on the volumes belonging to two different boundary surfaces. Indeed the rectangular shape of the volumes creates a steplike border of the computational domain and this makes it difficult to follow irregularities of the physical domain and to assign boundary conditions on some volume faces.

References [1]

[2] [3] [4] [5] [6] [7] [8]

[9] [10] [11]

Deponti A., Pennati V., De Biase L., A fully 3D finite volume method for incompressible Navier-Stokes equations, Int. J. for Numerical Methods in Fluids, (in press; available online in Wiley Interscience, DOI: 10.1002/fld.1190), 2006 Hagos S., Deponti A., Pennati V., De Biase L., Applicazione di un nuovo metodo ai volumi finiti a problemi di inquinamento termico, La Termotecnica, Aprile 2005 LeB. Hooke R., Principles of Glacier Mechanics, Prentice Hall, Upper Saddle River, New Jersey 07458, 1998. Hutter K., Theoretical Glaciology, Material Science of Ice and the Mechanics of Glaciers and Ice Sheets, D. Reidel Publishing Company, Dordrecht, Holland, 1983. Paterson W.S.B., The Physics of Glaciers, Third Edition, Pergamon, 1994. Colinge J., Blatter H., Stress and Velocity fields in glaciers: Part I. Finitedifference schemes for higher-order glacier models, Journal of Glaciology, Vol. 44, N. 148, pages 448-456 (a), 1998. Blatter H., Clarke G. K. C., Colinge J., Stress and Velocity fields in glaciers: Part II. Sliding and basal stress distribution, Journal of Glaciology, Vol. 44, N. 148, pages 457-466 (b), 1998. De Biase L., Feraudi F., Pennati V., A Finite Volume Method for the solution of Convection-Diffusion 2D problems by a Quadratic Profile with Smoothing, International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 6, N. 4, pages 3-24, 1996. Mohammadi B., Pironneau O., Analysis of the K-Epsilon Turbulence Model, John Whiley and Sons, 1994. Gudmundsson G. H., Basal-flow characteristics of a non-linear flow sliding frictionless over strongly undulations bedrock, Journal of Glaciology, Vol. 43, N. 143, pages. 80-89 (b) 1997. Anderson J.D., Introduction, Von Karman Institute for Fluids Dynamics Lecture Series: Introduction to Computational Fluid Dynamics, pages 18-22, 1983 WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

152 Advanced Computational Methods in Heat Transfer IX [12] [13] [14] [15] [16] [17] [18]

Barkley D., Gomes M., Henderson R.D., Three-dimensional instability in Flow over a Backward Facing Step. International Journal for Fluid Mechanic, Vol. 473, pages 167-190, 2002. Beaudoin J.F., Cadot O., Aider J.L., Centrifugal Instability in the Backward Facing Step, International Journal for Fluid Mechanic, 2003. Lüthi M., Funk M., Dating ice cores from a high Alpine glacier with a flow model for cold firn, Annals of Glaciology N. 31, pages 69-79, 2000. Lüthi M., Funk M., Modelling heat flow in a cold, high-altitude glacier: interpretation of measurements from Colle Gnifetti, Swiss Alps, Journal of Glaciology, N. 47(157), pages 314-324, 2001. Pettit E.C., Waddington E.D., Ice flow at low deviatoric stress, Journal of Glaciology, vol. 49, N. 166, pages. 359-369, 2003 Kamb B., Sliding motion of glaciers: Theory and observation, Reviews of Geophysics and Space Physics, N. 8, 4, pages 673-728, 1970. Pattyn F., Ice-sheet modelling at different spatial resolutions: focus on the grounding zone, Annals of Glaciology, N. 31, pages 211-216, 2000.



153

Coupled heat and moisture transport in a building envelope on cast gypsum basis J. Maděra, P. Tesárek & R. Černý Department of Mechanics, Faculty of Civil Engineering, Czech Technical University in Prague, Czech Republic

Abstract Flue gas desulphurization (FGD) gypsum can be potentially used as a material for load bearing structures. In this paper, a computational assessment of hygrothermal performance of a building envelope based on several modifications of FGD gypsum is presented. In the computer simulations of temperature and relative humidity fields, three variations of FGD gypsum wall, based on the raw material and on two types of hydrophobized gypsum, with the thickness of 300 mm, are solved. The thermal insulation function of the wall is achieved by exterior thermal insulation boards with the thickness of 100 mm, which are considered in four variants. Insulation I is hydrophilic material with a low value of hygroscopic moisture content on mineral wool basis, Insulation II capillary active material with higher value of hygroscopic moisture content on calciumsilicate basis, Insulation III hydrophobic material with a low value of water vapor resistance factor on mineral wool basis and Insulation IV hydrophobic material with higher value of water vapor resistance factor on polystyrene basis. Lime plaster with the thickness of 10 mm is used on the exterior wall surface. The computational analysis reveals that use of hydrophobization admixtures in the gypsum element does not lead to any improvement of hygrothermal behavior of the envelope provided by an exterior thermal insulation. Therefore, the application of a gypsum element without any hydrophobization seems to be a more favorable solution. The common hydrophobized thermal insulation materials on the basis of polystyrene or mineral wool are found to be satisfactory from the point of view of hygrothermal performance of the analyzed castgypsum based envelope. Keywords: building envelope, flue gas desulphurization (FGD) gypsum, thermal insulation boards, heat and moisture transport. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/HT060151


1

Introduction

Calcined gypsum is a historical binder, which was, used already several thousands years ago. Gypsum was called gatch in Persia, gypsos in Greek, and gypsum in Latin. Iranians, Egyptians, Babylonians, Greeks, and Romans were familiar with the art of working with gypsum plasters; decorated interior walls were found for instance in Pompeii. Gypsum was found in the binder of buildings in the territory of today’s Syria dated 7000 B.C.; it was also used in Cheops pyramid 2650 B.C, in the palace of Knossos etc. Nowadays, calcined gypsum is used in many technological modifications, which should improve its properties, in particular as binder of rendering mortars, for the production of stuccowork and also for plasters [1]. In the second half of the 20th century, new technologies for desulfurization of flue gases in power stations and heating plants appeared which were based on the reaction of sulfur (II) oxide formed during combustion of brown coal with high content of sulfur with limestone. Although these technologies were definitely very suitable from the point of view of the protection of environment, one problem appeared from the very beginnings, namely the very high amount of flue gas desulfurization (FGD) gypsum as waste product. The utilization of FGD gypsum as secondary raw material remained insufficient considering the amount of its production until these days. For instance, in Czech Republic calcined gypsum is produced from FGD gypsum only in one power station (Počerady), the remaining production ends with gypsum that is used only partially as additive retarding the setting of cement. Calcined gypsum is mostly used for the production of gypsum plasterboards [2]. That part of produced gypsum, which is not utilized, is deposited as waste. However, FGD gypsum can be potentially used as a material for load bearing structures as well. Modifications of this material can enhance its original properties and increase its service life. In this paper, a computational assessment of hygrothermal performance of a building envelope based on several modifications of FGD gypsum is done.

2

Materials and building envelopes

In the computer simulations of temperature and relative humidity fields we have solved three variations of FGD gypsum wall, based on the raw material and on two types of hydrophobized gypsum, with the thickness of 300 mm (Fig. 1). The thermal insulation function of the wall was achieved by exterior thermal insulation boards with the thickness of 100 mm, which were considered in four variants. Insulation I was hydrophilic material with low value of hygroscopic moisture content on mineral wool basis, Insulation II capillary active material with higher value of hygroscopic moisture content on calcium-silicate basis, Insulation III hydrophobic material with low value of water vapor resistance factor on mineral wool basis and Insulation IV hydrophobic material with higher value of water vapor resistance factor on polystyrene basis. On the external side of the wall, lime plaster with the thickness of 10 mm was used. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


2

1

155

3

1 – external plaster 2 – thermal insulation 3 – load-bearing structure

10

300

100

Figure 1:

[mm]

Composition of building envelope used to computer simulation.

The basic FGD gypsum material (we will denote it S0 in what follows) was β-form of calcined gypsum with purity higher than 98% of FGD gypsum, produced at the electric power station Počerady, CZ. The water/gypsum ratio was 0.627. After classification according to the Czech standard ČSN 72 2301, the gypsum was categorized as G-13 B III [3]. The first modification of FGD gypsum (S3) contained the admixture IMESTA IBS 47 produced by Imesta Inc., Dubá u České Lípy, CZ. The other (S4) contained the admixture ZONYL 9027 produced by Du Pont, USA. The water/gypsum ratio was the same as for S0. The composition of gypsum materials is shown in Table 1. Table 1: Material S0 S3 S4

Water/gypsum ratio 0.627 0.627 0.627

Composition of gypsum materials. Admixture none IMESTA IBS 47 ZONYL 9027

Concentration of the admixture none 0.5% by mass 5.0% solution

The material properties of non-modified and modified gypsum were measured in the Laboratory of Transport Processes (LTP), Faculty of Civil Engineering, Czech Technical University in Prague [4]. They are given in Table 2, where ρ is the bulk density, c the specific heat capacity, κ the moisture diffusivity, µ the water vapor diffusion resistance factor, θsat the saturated moisture content, θhyg the maximum hygroscopic moisture content, λ the thermal conductivity. The properties of insulation materials and lime plaster were partially obtained from the material database of Delphin computer code [5] and


156 Advanced Computational Methods in Heat Transfer IX partially measured in LTP. The material properties of insulation boards are given in Table 3. Table 2:

S0 S3 S4

c ρ 3 [kg/m ] [J/kgK] 1019 840 942 840 941 840 Table 3:

Basic materials properties of gypsum. κ [m2/s] 2.63e-7 1.47e-7 7.32e-9

µ [-] 5.4 5.4 5.4

θsat θhyg λ 3 3 3 3 [m /m ] [m /m ] [W/mK] 0.6 0.23 0.47 0.61 0.181 0.41 0.62 0.166 0.38

Material parameters of insulation materials.

c ρ 3 [kg/m ] [J/kgK] I 150 840 II 230 1000 III 280 840 IV 30 1300

κ [m2/s] 1.10-7.e0.0485. θ 2.10-8.e0.0523. θ 5.10-13.e0.1486. θ 2.10-11.e0.0475. θ

µ [-] 2 2.5 3 50

θsat [m3/m3] 0.95 0.88 0.31 0.97

θhyg [m3/m3] 0.006 0.22 0.0073 0.001

λ [W/mK] 1.1 0.4 1.2 0.56

3 Numerical solution by TRANSMAT For the calculations we employed the computer simulation tool TRANSMAT 4.3 [6] which was developed in the Department of Mechanics, Faculty of Civil Engineering, Czech Technical University in Prague. The construction of the code is based on the application of the general finite element computer simulation tool SIFEL (SImple Finite ELements) developed in the Department of Mechanics, FCE CTU. The moisture (1) and heat balance (2) equations were formulated according to the Künzel’s model [7], dρ v ∂ϕ = div Dϕ gradϕ + δ p grad (ϕp s ) dϕ ∂t dH ∂T = div(λgradT ) + Lv div δ p grad (ϕp s ) dT ∂t

[

]

[

]

(1) (2)

where ρv is partial moisture density, ϕ the relative humidity, δp the water vapor permeability, ps the partial pressure of saturated water vapor in the air, H the enthalpy density, Lv the latent heat of evaporation of water, λ the thermal conductivity and T is the temperature. The liquid water transport coefficient is defined as Dϕ = κ

dρ v . dϕ


(3)


157

The proper initial and boundary conditions of the model are crucial factor affecting the reliability of the calculations. In our computer simulations, the analyzed building envelopes were exposed from inside to constant conditions (temperature equal to 21°C and relative humidity equal to 55%) and from outside to the climatic conditions corresponding to the reference year for Prague. The 1st of July was chosen as the starting point in the calculations. We have chosen two characteristic profiles in the assessment of the hygrothermal performance of the envelope, A-A´, B-B´ (Fig. 2), where the profile A-A´ was between the insulation board and the load-bearing structure (the distance of 110 mm from the exterior), the profile B-B´ was the cross section of the wall from the exterior to the interior. In these profiles we have compared relative humidity and temperature calculated for the analyzed envelopes.

Exterior TRY for Prague

Interior

A

Constant temperature T = 294.15 K

relative humidity temperature wind rain short wave

Constant relative humidity ϕ = 55% A´ Figure 2:

4

Scheme of typical envelope.

Computational results and discussion

4.1 Non modified gypsum S0 Fig. 3 shows an example of the relative humidity profile in the wall based on non-modified gypsum (S0) for December 15, which can be considered as characteristic for the winter period.


158 Advanced Computational Methods in Heat Transfer IX 1

Relative humidity [-]

0.9 0.8 0.7 0.6 0.5 0.4 0

50

100

150

200

250

300

350

400

Distance [mm] Insulation I

Figure 3:

Insulation II

Insulation III

Insulation IV

Relative humidity, non modified gypsum (S0), B-B´ profile.

Fig. 4 presents the history of relative humidity in the A-A´ profile from January 1 to December 31 for four years simulation.


0.62 0.6 0.58 0.56 0.54 0.52 0.5

1 500

1 550

1 600

1 650

1 700

1 750

1 800

Time [days] Insulation I

Figure 4:

Insulation II

Insulation III

Insulation IV

Relative humidity, non modified gypsum (S0), A-A´ profile.

Fig. 5 shows an example of the temperature profile in the wall based on the non-modified gypsum (S0) for December 15, which can be considered as characteristic for the winter period.



159

Temperature [K]

290 285 280 275 270 0

50

100

150

200

250

300

350

400


Figure 5:

Insulation II

Insulation III

Insulation IV

Temperature, non modified gypsum (S0), B-B´ profile.

4.2 Modified gypsum S3 Fig. 6 shows an example of the relative humidity profile in the wall based on the modified gypsum (S3) for December 15 analogous to Fig. 3. The results obtained for modified gypsum (S3) and non-modified gypsum (S0) were very similar, so that the effect of gypsum hydrofobization was very small. The same similar results were also achieved in the analogs to Figs. 4 and 5. 1


0.9 0.8 0.7 0.6 0.5 0.4 0

50

100

150

200

250

300

350

400


Figure 6:

Insulation II

Insulation III

Insulation IV

Relative humidity, modified gypsum (S3), B-B´ profile.


160 Advanced Computational Methods in Heat Transfer IX 4.3 Modified gypsum S4 Fig. 7 shows an example of the relative humidity profile in the wall based on the modified gypsum (S4) for December 15. Here, some differences in relative humidity (mainly in the insulation layer and partially also in the gypsum element, the highest for Insulation II) compared to the results for S0 in Fig. 3 were observed but they were not very significant because they did not change the overall character of the hygrothermal performance of the gypsum wall. 1


0.9 0.8 0.7 0.6 0.5 0.4 0

50

100

150

200

250

300

350

400


Figure 7:

Insulation II

Insulation III

Insulation IV

Relative humidity, modified gypsum (S4), B-B´ profile.

0.62


0.6 0.58 0.56 0.54 0.52 0.5 0.48 0.46 0.44 0.42 1 500

1 550

1 600

1 650

1 700

1 750

1 800

Time [days] Insulation I

Figure 8:

Insulation II

Insulation III

Insulation IV

Relative humidity, modified gypsum (S4), A-A´ profile.

Fig. 8 presents the relative humidity history in the A-A´ profile from January 1 to December 31 for four years simulation. The differences observed in comparison with Fig. 4 for non-modified gypsum wall were most pronounced for WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


161

Insulation II similarly as in Fig. 7 but in the assessment of the overall hygrothermal performance of the wall, also here they could not be considered as very significant. Fig. 9 shows an example of the temperature profile in the wall based on the modified gypsum (S4) for December 15. A comparison with the corresponding Fig. 5 reveals that the differences from the temperature profiles in the wall based on non-modified gypsum were almost negligible.

Temperature [K]

290 285 280 275 270 0

50

100

150

200

250

300

350

400


Figure 9:

5

Insulation II

Insulation III

Insulation IV

Temperature, modified gypsum (S4), B-B´ profile.

Conclusions

The computational analysis in this paper revealed that the use of hydrophobization admixtures in the cast-gypsum element of building envelopes provided by any of the four very different exterior thermal insulations did not lead to significant improvements of the hygrothermal behavior of the envelope. The hygrothermal performance of the studied envelopes was satisfactory in all analyzed cases. Therefore, an application of a gypsum element without any hydrophobization seems to be the preferential solution, particularly taking into account the substantially lower price. The common hydrophobized thermal insulation materials on the basis of polystyrene or mineral wool were found to be satisfactory from the point of view of hygrothermal performance of the analyzed cast-gypsum based envelope. Therefore, they are supposed to be the preferred materials in this respect.

Acknowledgements This research has been supported partially by the Czech Science Foundation, under grant No. 103/06/P021 and partially by the Ministry of Education of the Czech Republic, under contract No. 6840770003. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


References [1] Wirsching F., Calcium Sulfate. In: Ullmann’s Encyklopedia of Industrial Chemistry, Weinheim, 1985, pp. 555-583. [2] Hanusch, H., Übersicht über Eigenschaften und Anwendung von Gipskartonplatten. Zement-Kalk-Gips, No. 5, 1974, 245-251. [3] ČSN 72 2301 Gypsum binding materials, Czech standard (in Czech). Vydavatelství Úřadu pro normalizaci s měření, Praha 1979. [4] Tesárek P., Kolísko J., Rovnaníková P., Černý R., Properties of Hydrophobized FGD Gypsum. Cement Wapno Beton 10/72 (2005), 255264. [5] Grunewald J., DELPHIN 4.1 - Documentation, Theoretical Fundamentals, TU Dresden, Dresden, 2000. [6] Maděra J., Černý R. TRANSMAT – a Computer Simulation Tool for Modeling Coupled Heat and Moisture Transport in Building Materials. Proceedings of Workshop 2005 - Part A,B, Prague: CTU, 2005, pp. 470-471. [7] Künzel H.M., Simultaneous Heat and Moisture Transport in Building Components, Ph.D. Thesis. IRB Verlag, Stuttgart 1995.



163

Spray water cooling heat transfer under oxide scale formation conditions R. Viscorova1 , R. Scholz2 , K.-H. Spitzer1 and J. Wendelstorf1 1 Institute

of Metallurgy, Clausthal University of Technology, Germany for Energy Process Engineering and Fuel Technology, Clausthal University of Technology, Germany

2 Institute

Abstract Spray water cooling is an important technology used for the cooling of materials from temperatures up to 1800K. The heat transfer coefficient (HTC) in the so-called steady film boiling regime is known to be a function of the water mass flow density. Below a specific surface temperature TL , film boiling becomes unstable and the HTC shows a strong dependence on temperature (Leidenfrost effect). The HTC was measured by an automated cooling test stand (instationary method). Compared to the previous state-of-the-art, an additional temperature dependency in the high temperature regime was found. A new analytic fit formula for the dependence of the heat transfer coefficient on temperature and water impact density is proposed and discussed. Spray water cooling of steel materials at temperature levels above 1000K introduces additional effects due to the formation of oxide layers (scale). These effects and experiments under scale formation conditions will be presented and discussed. Keywords: scale formation, spray water cooling, continuous casting, hot rolling, heat transfer coefficient.

1 Introduction Spray water is used for cooling in steel materials production processes, e.g. as part of the casting and rolling procedure. Due to the high temperatures the steel surface is oxidized. A lack of knowledge about the influence of this oxide layer (scale) on heat transfer conditions may prevent a quantitative prediction of the cooling procedure. For predictable homogenous cooling, the dependence of the heat transfer coefficient (HTC) on its principal parameters, the surface temperature WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/HT060161

164 Advanced Computational Methods in Heat Transfer IX and the water mass flow density, is required even in the case of oxidized steel surfaces. The interrelationship between oxide scale formation and heat transfer also determines the temperature of the metal surface. The layer of the oxide scale forms a resistance for the heat transfer. Additionally, due to the cooling process, mechanical tensions arise between the oxide layer and the bulk material which lead to spallation of the scale and thus again to changes of the heat transfer conditions. For quantitative description, the concept of the Heat Transfer Coefficient (HTC, α) is used, which is defined through its relation to the heat flow density to the surface q [W/m2 ] (see also Figure 1): q = α · (TS − TW )

(1)

For spray water cooling, TW is the water temperature, while TS is the (local) surface temperature. This approach is most suitable for situations were α is constant, i.e. not depending on ∆T ≡ TS − TW . It can also be used for any general heat flow q by using a function for α depending on the same parameters as q. In the so-called steady film boiling regime, α is known to be a function of the water mass flow density [1, 2, 3, 4]. Below a specific surface temperature TS,L , the heat transfer coefficient shows a strong dependence on temperature (Leidenfrost effect [5]). In this paper, the heat transfer coefficient was measured by an automated cooling test stand (instationary method, see Figure 2 and the discussion in [1]) under oxidizing and non-oxidizing sample surface conditions. Additionally, the measurement precision was taken into account (see e.g. [1]). This paper continues the work started in [4].

spray water

Tw surface

S scale heat flow

B q

metal U thermocouple

Figure 1: Heat transfer to an oxidized sample.

2 Experimental procedure The thin sheet specimens of cold rolled metals with a thickness from 1.0-1.5 mm were milled to discs with 70 mm in diameter. Up to 5 thermocouple pairs, one in WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


165

Magnetic valve

Cooling water

Volume flow measuring instrument

Furnace (7,5kW / 1500°C) Nozzle field Steel sample

Electricity supply and furnace temperature control

Sample supporting stand Sample-cross bar D Measuring and control system

Gas supply unit

- temperature measurement - cross bar control

Air Argon

Figure 2: Sketch of the experimental set-up.

the center and the others perpendicular at a radius of 10 mm, were spot welded to the lower side of the sample discs. The temperature measurement was carried out with Ni-CrNi thermocouples with a wire diameter of 0.5 mm. The thermocouple wires were isolated by thin ceramic tubes. The experimental set-up sketched in Figure 2 was used for heating, oxidizing the samples and measurement of the temperature during spray cooling. After installation in the furnace sample holder, the disks were heated up to 1200 ◦ C under protective atmosphere. For the measurements with scale, the samples were oxidized in the furnace by supplying air instead of protective gas for a specific time and temperature. Reaching the test conditions (temperature, oxidation time), the sample was moved automatically from the furnace under the full cone nozzle which takes approximately 4 seconds and subsequently cooled with spray water. The spray water mass flow density VS was determined experimentally and varied in the range of about 4 ± 1 to 30 ± 1 kg m2 s−1 . The water temperature TW was approximately 18 ◦ C. The materials investigated were commercially pure Nickel (99.2% Ni) and Iron (low-carbon steel, AISI 1008, 99.7% Fe).

3 Measurement of the HTC at elevated temperatures For the investigation of the HTC from a non oxidized surface, Nickel material was used, because surface oxidation is minimized, and the physical properties are known within the temperature range being studied. The samples were annealed WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

166 Advanced Computational Methods in Heat Transfer IX at 1200 ◦ C and consequently cooled. The cooling curves were measured at three points for each sample as described above. Figure 3 shows typical cooling curves of the Nickel specimens for different water mass flow densities. The sample reaches the spray water cooling position after about 4.2 s. The cooling starts in the range of stable film boiling. The primary effect of the water mass flow density VS can be seen by comparing the different cooling curves. When increasing VS from 4 up to 30 kg m2 s−1 the sample cooling rate increases. At approximately 300-450 ◦ C, dependent on VS , the cooling curves bend. At this point the Leidenfrost temperature is reached and more rapid cooling begins (unstable film boiling).

1200 Sample: d=1mm, Nickel 99.3

1100 1000 temperature T[°C]

900 800 700 600 500 400 300 200

30

25

18

12

water mass flux density Vs [kg /(m2s)]

8

4

100 4

5

6

7

8

9

10

11

12

13

14

time t[s]

Figure 3: Cooling curve for different water mass flow densities VS (spray water cooling of pure Nickel).

For each cooling curve (VS -value), α(TS ) is obtained from the following equation:

α(TS ) ≈ −

ρ · cp (TU ) · d ∂TU · TU − TW ∂t z=0

with: WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

(2)

Advanced Computational Methods in Heat Transfer IX α

heat transfer coefficient

[W m−2 K−1 ]

cp

specific heat capacity of the sample (= f (T )!)

[J kg K−1 ]

d

thickness of the sample

[m]

z

vertical coordinate (bottom: z = 0,U; top: z = d,S)

[m]

t

time

[s]

ρ

mass density of the sample

[kg m−3 ]

TU

temperature measured at the lower side (U, Fig.1)

[K]

TW

temperature of the spray water

[K]

167

Equation (2) is valid for thin samples such as in this case (d = 1 mm) because of the small temperature difference between the cooled top and the measurement point at the bottom (≈10 ◦ C in the stable film boiling regime). The specific heat capacity cp of the sample materials depending on temperature was calculated with the Thermocalc Software based on the chemical analysis. A fit function describing the α(∆T, VS ) dependence was calculated from the measurement data VS VS · ∆T α(∆T, VS ) = erf × 245 · VS 1 − + (3) 5 58223 ∆T 4.3 · ∆T 2 1 − tanh 115 and is shown in Figure 4. In the open literature, the HTC in the range of stable film boiling (i.e. above ∆T = 600K) is assumed to be independent of the surface temperature. As shown in Figure 4, for the higher spray water densities, VS > 15 kg m2 s−1 there is a decrease in the measured HTC even in the stable film boiling regime. This effect may be explained with an increasing vapor layer thickness and yields to α-contours no longer parallel to the ∆T -axis.

18000

30 a [W/(m K)] 2

20000 15000 10000 5000 0

20

30

250

10

500 DT [K]

Da=2000

10

750 1000

1000

2

Vs [kg/(m s)]

0

DT [K]

0

200

400

600

800 1000

Figure 4: Heat transfer coefficient after eq. (3) in dependence on temperature difference and water mass flow density. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)


4 The HTC for an oxidized surface Following the initial considerations and the basic knowledge on scale properties (e.g. [4]), the determination of the influence of scale on the heat transfer conditions introduces some difficulties due to the following reasons: • Thin scale layers (< 10 µm) may show enough adhesion for measurement but imply a very small heat resistance. • Small gas layers between the scale and the substrate (blistering) imply a big effect on heat resistance (random effect). • The adhesion of medium thickness scale layers (10 . . . 200 µm) may be not sufficient to withstand spray water – spallation and descaling occurs (instationary effect). • Thicker scale layers (> 200 µm) are removed rapidly by the spray water – the thermal effect becomes inhomogeneous. Regardless these difficulties, the practical importance remains and thus the influence of an oxide layer on heat transfer was investigated in a second set of experiments in which the low alloy material AISI 1008 (99.7% Fe) was used. In order to get well defined scale formation, the oxidation kinetics was determined in the first step [6]. These investigations allow to calculate the scale layer thickness from the oxidation temperature and time. 4.1 Modelling of the HTC for an oxidized surface For understanding the influence of an oxide layer on the heat transfer during spray water cooling, the mechanism of heat transfer at an oxidized surface is theoretically described first. Since calculations often can not include thin layer effects, an effective heat transfer coefficient containing all scale layer effects is introduced. The heat flow is thus described by an effective heat transfer coefficient which is using the temperature difference between the steel surface and the temperature of the water (see Figure 1): q = αeff,∆T =TB −TW · (TB − TW ) q

heat flow density

[W m−2 ]

αeff

effective heat transfer coefficient

[W m−2 K−1 ]

TB

temperature at the steel-scale interface

[K]

(4)

The heat transfer from the surface to the spray water is given by q = α∆T =TS −TW · (TS − TW )

(5)

and the heat transfer through the oxide layer can be approximated by q=

λsc · (TB − TS ) δsc


(6)

Advanced Computational Methods in Heat Transfer IX λsc

effective heat conductivity of the scale

[W m−1 K−1 ]

δsc

thickness of the scale layer

[m]

TS

temperature at the surface of the oxide scale

[K]

169

Under quasi-stationary conditions and for uniform cooling, the heat flow (4) from the surface to the cooling water (5) is equal to the heat flow through the scale layer (6) and we get the effective HTC defined by (4): αeff,∆T =TB −TW =

1 α∆T =TS −TW

+

δsc λsc

−1 (7)

The effective heat transfer coefficient αeff contains the influence of the oxide layer (thermal insulation, temperature drop). It can be calculated by (7) using the HTC α(∆T = TS − TW , VS ), which does not depend on bulk material properties. So accurate measurements of α(∆T = TS − TW , VS ) without scale can be used for surfaces with well defined oxide layers. Additionally, a measurement using an oxidized surface and (2) for analysis will yield to a measured αeff . The function αeff (∆T = TB − TW , VS ) calculated from (7) using the fit-formula α(∆T = TS − TW , VS ) from (3) is shown in Figure 5 and discussed below.

15000 2

a [W/(m K)]

-3.5

320µm

10000

-4

100µm

5000

-4.5

0

32µm 200

-5

lg(dz[m])

10µm 400 -5.5

3µm

600

DT [K]

800 -6

1µm

Figure 5: Calculated effective heat transfer coefficient depending on temperature difference ∆T = TB −TW and oxide layer thickness (logarithmic scale).


9000 8000

-2

-1

HTC [W m K ]

7000 6000 5000 4000

Dfrom eq.(3) Deff from eq.(7) & (3) with Gsc=150µm

experimental scatter

170 Advanced Computational Methods in Heat Transfer IX d=1.5mm, AISI-1008 VS=4 kg m-2 s-1

experimental Deff after eq. (2):

oxidation time in air at 1000°C (mean D):

Deff from eq.(7) & (3) with Gsc=300µm or gas gap

300s Ö Gsc~140µm 105s Ö Gsc~83µm 0s Ö Gsc 0 = ∂x 2 α ∂t θ ( x, t ) = θ a at x = 0, t > 0 ∂θ = 0 at x = L, t > 0 ∂x θ ( x, t ) = θ L for t = 0 in 0 ≤ x ≤ L

(9-a) (9-b) (9-c) (9-d)

And solve this problem by using the integral method. For the case δ (t ) < L (the thermal layer thickness is less than the slab thickness) the eqn (9-a) was integrated over the thermal layer thickness and obtain

−α

∂θ 0 ∂x

= x =0

d (γ − θ aδ ) dt

(10-a)

δ

where

(10-b)

γ ≡ ∫ θ 0 ( x, t )dx x =0

A cubic profile for the temperature as given by eqn (11) was chosen and by applying the conditions given by eqn (12) the coefficients were determined and utilize eqn (10-b) to determine the thermal layer thickness.

{θ (x, t ) = a + bx + cx

2

0

θ

x =0

= θa , θ

x =δ

= θL ,

∂θ ∂x

+ dx 3

=0, x =δ

}

∂T ∂x 2

(11)

2

=0

(12)

x =0

The resulting temperature profile becomes

θ 0 ( x, t ) − θ a x x = 1− 3   + 1   2δ  2δ  θL −θa where

δ = 8αt

This solution is valid for 0 ≤ x ≤ L as long as (8) shows the temperature profile.

δ ≤L


3

(13-a) (13-b) and after that eqn


4

255

Comparing the experiment and analytical temperature profile

Fig 4 and 5 shows the difference between the temperature profile from the analytical equation and the experiment. From Fig. 4 it could be summarized that there is good agreement between the experiment and analytical equation for temperature distribution at the center of cup and inside of cup’s wall. Center of cup 800

Temperature (C)

700

Equation 5

600

ha=100W/m2k , hi =1000W/m2k

500

Analytical Equ.

400

Experiment

300 200 100 0 0

20

40

60

80

100

120

140

160

Time (sec)

(a) Inside of cup's wall 700

Equation 6

Temperature (C)

600 500

ha=100W/m2k , hi =1000W/m2k

400

Analytical Equ. Experiment

300 200 100 0 0

20

40

60

80

100

120

140

160

Time (sec)

(b) Figure 4:

Comparing of analytical temperature profile and experiment.


256 Advanced Computational Methods in Heat Transfer IX Outside of cup's wall 600

Temperature (C)

500

Equation 8

400

Equation 7

Analyical Equ.

300

Experiment

ha=100W/m2k , hi =1000W/m2k

200

100

0 0

20

40

60

80

100

120

140

160

180

Time (sec)

(a) Outside of Cup's wall 600

Equation 13 Temperature (C)

500

Equation 8

400

Experiment

ha=100W/m2k , hi =1000W/m2k

300

Analytical Equ.

200

100

0 0

50

100

150

200

250

300

350

Time (sec)

(b) Figure 5:

Comparing of analytical temperature profile and experiment.

Fig. 5(a) shows a combination of equation (8) and (9) could not be a good solution for this problem, but equation (13) for early stage and equation (9) after saturation time (uniform temperature distribution) are in good agreement with experimental result (Fig 5(b)).



5

257

Conclusion

Cup-cast method is the most convenient method for semi-solid casting that has just been developed by the author. This method is based on the nucleation and growth of solid particles, and sufficient rest time (holding the isothermally in semi-solid state), heat transfer phenomena will considerably influence on the semi-solid microstructure of this method. Based on the facts of this method those driven out from the temperature measurement a model for heat transfer in cup cast method was proposed. This analytical equation is in good agreement with experiments at the center and inside of the cup’s wall and is in acceptable agreement with experiment at the outside of the cup’s wall.

6 Index A Table 2:

List of symbols and their definition in analytical model.

θ

Temperature of the melt

D

Diameter of cup

X

Thickness of the cup

h

Heat transfer coefficient Thermal conductivity of cup Density of cup Specific heat coefficient of cup Thermal diffusivity of cup Slab thickness

λ ρ c a L

θa θ0 θi δ

Room temperature Temperature of the cup outside Temperature of cup inside Thermal layer thickness

subscript i a

Between melt and cup Between cup and air

References [1] J. L. Jorstad, Q. Y. Pan, D. A. Pelian, “Solidification microstructure affecting ductility in semi-solid-cast products”, Materials Science and Engineering A, 413-414 (2005) pp. 189-191. [2] D. Brabazon, D. J. Browne, A. J. Carr, “Experimental investigation of the transient and steady state rheological behavior of Al-Si alloys in the mushy state”, Materials Science and Engineering A 356 (2003) pp. 69-80.


258 Advanced Computational Methods in Heat Transfer IX [3] B. P. Gautham, P. C. Kapur, “Rheological model for short duration response of semi-solid metals”, Materials Science and Engineering A 393 (2005) pp. 223-228. [4] J. L. Wang, Y. H. Su, C-Y. A. Tsao, “Structural evolution of conventional cast dendritic and spray-cast non-dendritic structure during isothermal holding in the semi-solid state”, Scripta Materialia (USA), No. 12, 37 (1997), pp. 2003-2007.



259

Atwood number effects in buoyancy driven flows M. J. Andrews & F. F. Jebrail Los Alamos National Laboratory, USA

Abstract Consideration is given to Atwood number (non-dimensional density difference) effects in buoyancy driven flows. Buoyancy driven (natural convection) flows may be treated as Boussinesq for small Atwood number, but as Atwood number increases (>0.1, i.e. large temperature differences) the Boussinesq approximation is no longer valid and the distinct “bubble” and “spike” geometry of Rayleigh– Taylor buoyant plumes is formed. Aside from asymmetry in the flow the Atwood number also affects key turbulent mix parameters such as the molecular mix, and heat transfer coefficients. This paper will present recent experimental work being performed in the buoyancy driven mix laboratory at Texas A&M University with air/helium as mixing components. Corresponding numerical simulations performed at Los Alamos are presented for the experiments, and future directions for the research discussed. Keywords: buoyancy, Boussinesq, Atwood number, natural convection.

1

Introduction

This paper describes experiments and corresponding simulations to investigate non-Boussinesq effects at high Atwood (At) number (At ≡ (ρ1 − ρ 2 ) (ρ1 + ρ 2 ) a non-dimensional density ratio) in buoyancy driven turbulence (Rayleigh-Taylor mixing). The Boussinesq assumption for buoyancy driven flows states that density difference effects need only be accounted for in the gravitational terms, and density may be taken as a constant elsewhere in the equations. At small Atwood number (At0.1. Consequences for natural convection and high temperature gas heating/cooling problems are discussed.

2

Experiments

2.1 Experimental facility A schematic of the experiment is shown in Figure 1. The experimental facility was a wind tunnel, with a splitter plate that separates an upper air stream ( ρ1 ) from a lower air/helium stream ( ρ 2 ). The stream velocities were kept the same to avoid shear, and great care was taken to obtain a constant and controllable helium flow rate. Thus, by pre-mixing air and helium for the lower stream the density difference between the upper and lower streams could be varied from At=0 (air top and bottom), up to At=0.75 (air on top, and helium on the bottom). More details of the experimental set-up can be found in Banerjee and Andrews [2]. For future reference, the vertical depth of the channel is 1.2m, and out-of-plane width is 0.6m. By introducing smoke into the upper channel and using a calibration wedge the density profile can be measured from digital photographs, see Banerjee and Andrews [2]. Splitter Plate Test Section Vents

Exit Plenum

Meshes

Air In

1.2m 2.0 m

1.0 m He- Air Mix In

Wooden ribs

y Air blowers

x

He from Flow metering unit

Adjustable dampers Side View

Figure 1:

Schematic of the experimental facility.



261

2.2 Experimental details Two experiments are reported in this paper corresponding to At=0.035 and At=0.26. Table 1 reports the experimental conditions (note the stream velocity, U, is higher for At=0.26 to keep the spread angle small, and hence a parabolic flow). Use of Taylors’ hypothesis relates distance downstream (x) and time (t) as t=x/U, and this time is also reported in Table 1. Table 1: At 0.035 0.035 0.035 0.26 0.26

3

Experimental parameters.

U (m/s) 0.6 0.6 0.6 1.2 1.2

x (m) 0.75 1.4 1.75 0.5 1.5

t (s) 1.25 2.33 2.92 0.42 1.25

Governing equations and numerical details

3.1 Governing equations The incompressible Euler equations are used in conjunction with the MILES (see below for more details) modeling technique: Volume conservation: Scalar transport: Momentum:

∇•u = 0

Df =0 Dt

D (ρ u ) = −∇p + ρ g Dt

(1) (2) (3)

with the fluid velocity u = (u, v, w ) , density, ρ , pressure, p , and gravity, g = (0,0, g z ) , and scalar f . There are six independent variables and five equations, the seventh equation is a linear equation of state for density such as ρ = L( f ) . In the present work we take f to be the non-dimensional density, or mixture fraction, defined as f = (ρ − ρ 2 ) (ρ1 − ρ 2 ) . 3.2 Numerical solution procedure 3.2.1 Overview For the present experiment we have used MILES, namely, Monotone Integrated LES. MILES modeling involves solving Euler governing equations and using numerical diffusion to model turbulent diffusion. Success with this modeling technique for buoyancy driven flows has been reported by Youngs [1]. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

262 Advanced Computational Methods in Heat Transfer IX The governing equations presented above are a coupled set of partial differential equations for which there exist several solution procedures. The present work solves the governing equations using the RTI3D code described by Andrews [3]. In particular, a fractional time step technique is used in which for each time step an advection calculation is followed by a Lagrangian source term update. The Lagrangian update is presented next, and this is followed by a brief description of the advection step for the scalar f (details may be found in Andrews [3]). 3.2.2 Lagrangian momentum source term updates The Lagrangian w momentum equation is: ∆t w*n = wnn +1 2 + pPn − pTn + g z (4) ρt ∆y The n+1/2 superscript refers to a value from the advection calculation, and * to an intermediate value that does not necessarily satisfy continuity. The subscripts refer to spatial position (north face), typical of the SIMPLE method (Patankar [4]), and a staggered arrangement of momentum and mass cells is used. Following the SIMPLE practice, velocity corrections are defined so that uin,e+1 = ui*,e + ∆ui ,e (and similarly for the other velocities) and a new pressure

(

)

p Pn +1 = p Pn + ∆p p where ∆p is a pressure correction. By substituting these

expressions for n+1 into the volume conservation equation and then subtracting equation (4) evaluated with the * we arrive at the usual Poisson equation for pressure corrections: a P ∆pP + a E ∆pE + aW ∆pW + a N ∆p N + aS ∆pS = − Div

(5)

with Div the divergence of the * velocity values. The Poisson equation (5) is solved using a Full Multi-Grid method, and the pressure corrections are used in a SIMPLE style to provide updated n+1 velocities and pressures that simultaneously satisfy the momentum equations (3) and volume conservation. 3.2.3 Transport procedures The 3D transport procedures are split into x/y/z-steps, this fractional splitting simplifies the calculation to one-dimensional updates that lends itself to high order calculation of cell fluxes with the Van Leer [5] method. There follows a brief description of the scalar x-step advection, the y and z steps being similar, and similar advection steps are performed for the momentum. The x-step advection for the scalar is given by: f P* = f Pn + ∆y∆z∆t (ue f e − u w f w )

(6)

where P refers to the center of a control volume, e the east face, and w the west face. The face values for the u velocities are available, and the face values for the scalar are computed using a second order approximation with Van-Leer limiting to prevent non-physical oscillation as:



f e = f upwind + sign(ε e )

263

(1 − ε e ) ∆xD

(7) e 2 where ε e = ∆t ue ∆x , and upwind values are taken according to the sign of ε e . The derivative is evaluated following Van Leer as:  2 ∆w 2 ∆e  (8) , De = S min  D ,  ∆x ∆x    1 if ∆ e and ∆ w > 0  where ∆ w = f Pn − fWn , ∆ e = f En − f Pn and S = − 1 if ∆ e and ∆ w < 0 . 0 otherwise  Van Leer limiters have been used in equation (8) to limit the gradient of the volume fraction profile, thereby preventing spurious oscillations. The representation for the gradient of the cell profile D determines the accuracy of the representation. In the present work D = (∆ e + ∆ w ) (2∆x ) , so the gradient is computed with a central difference so this scheme is referred to as “2nd order”. 3.3 Computational details for simulation of the experiments The experiments are performed in a statistically steady gas channel, but are modelled on the computer using transient simulations that are related to the experiment through the Taylor hypothesis described above. The computational domain is taken to be 1.2m high (z), and 0.6m square (x & y), with a computational grid of 64x64x128 (x-y-z). The computational time step is selected by the computer program to keep the Courant number below 0.25. Initial conditions for the simulations are prescribed to fit the density interface off the splitter plate. Here we use the following initial density interface perturbations: h( x, y ) = hw + ∑ a k cos(k x x )cos k y y + bk cos(k x x )sin k y y + k x ,k y (9) ck sin(k x x )cos k y y + d k sin(k x x )sin k y y

( )

( )

( )

( )

where the spectral amplitudes are chosen randomly but give an rms amplitude of 0.6m/100, and the wave numbers range from modes 8 to 16 (Dimonte et al. [6]). The wake off the splitter plate is modelled as an initial perturbation, hw , where hw = aw sin( k w x ) cos(k w y ) ; the wavelength associated with the wave number k w is 1cm and is taken from inspection of the experimental photograph in Figure 2, and the amplitude is 0.5cm, again taken from Figure 2.

4


Figure 2 is a photograph taken from the At=0.26 experiment. On the right at the mid-plane is the splitter plate. The air/helium streams move from right to left, and the buoyancy driven mix is seen to develop downstream. On the far left of WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

264 Advanced Computational Methods in Heat Transfer IX the photograph (2m from the splitter plate) the mixing has expanded to the top and bottom of the channel (i.e. a total depth of 120cm). The photograph shows a well mixed region with a wide range of mixing length scales. Close inspection of the region close to the end of the splitter plate, reveals initial perturbations of wavelength about 1cm, and amplitude 0.5cm, associated with wake shedding.

Figure 2:

Photograph from At=0.26 experiment.

Figure 3 below shows plots of initial density interface (on the left) used for the At=0.035 simulations (and is the same for all the simulations), and it is evident that there are both short and longer wavelength disturbances corresponding to conditions at the end of the splitter plate. The right side of Figure 3 shows the mix edges interfaces at f=0.01 (lower surface) and f=0.99 (upper surface), and corresponds to a distance downstream of 175cm (near the left side of the photograph in Figure 2). Comparison of the computed and experimental disturbances in Figure 2 and the right of Figure 3 shows more fine scale in the experiment, and this is because of computational grid resolution.

Figure 3:

Initial conditions and late time (t=2.92s) mixing edges for At=0.035.



265

Calibration of digital photographs provided time averaged non-dimensional density profiles (the scalar fraction, f) and these are shown in Figures 4 and 5, with the corresponding results from the simulations using MILES. Comparison in Figure 4 for an At=0.035 of the experimental density widths and profiles with the corresponding simulations is good, and in particular the almost linear density profile across the mixing region is well captured by the MILES simulations. Also of note is that the experimental mix is symmetric around the centreline, indicating that the Boussinesq approximation is valid for an At=0.035. Similarly, in Figure 5, comparison of experiment with simulation for the At=0.26 show they also agree quite well. However, the experiments and the simulations show a slight asymmetry, associated with the higher At=0.26, that perhaps best seen by inspecting the penetration at the edges of the mix in Figure 5.

At=0.035 Experiment Figure 4:

At=0.035 Simulation

Comparison of mix profiles for At=0.035. h2

At=0.26 Experiment Figure 5:

h1

At=0.26 Computation

Comparison of mix profiles for At=0.26.

To explore higher At (i.e. large density differences) two additional simulations have been performed for At=0.5 and At=0.9. To facilitate comparison, the product At*gz was held constant at the value for the At=0.035 WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)

266 Advanced Computational Methods in Heat Transfer IX case, so At*gz=-0.34335 ms-2, and the value of gz then computed for each value of At, so for At=0.5 a value of gz=-0.6867 ms-2 was assigned, and for At=0.9 the value was gz=-0.3815 ms-2. This scaling comes from consideration of the gravitational term in the governing equations. To gauge asymmetry the bubble penetration, h1 , and spike penetration, h2 , were computed as the distance from the centreline to the values of f = 0.99 and f = 0.01 respectively (see Figure 5 for a graphical representation at t=1.25s), and their ratio is plotted in Figure 6. Inspection of Figure 6 reveals that there is practically no asymmetry for At=0.035, however, asymmetry is clearly seen for At=0.5, with the spike/bubble penetration being about 1.2 near the end of the calculation at t=2s. Indeed, there is a strong asymmetry for At=0.9 with h2/h1=1.7 by t=2s. Figure 7 plots the edges of the mix region, in a similar fashion to Figure 5, for each of the cases at t=2s. The At*gz scaling ensures a comparison at similar development of the mix. Comparison of the mix edges for At=0.035 with At=0.5 in Figure 7 shows little difference in bubble and spike penetration. However, for At=0.9 the asymmetry is clear, with bubbles clearly shown at the top, and finger shaped spikes at the bottom.

Figure 6:

Spike to bubble penetration (h2/h1) asymmetry for increasing At

Results from the simulations show that significant departures from the Boussinesq approximation occur by At=0.5, but are not particularly evident at At=0.26. This suggests that the Boussinesq approximation is reasonable for pure buoyancy flows at least for At

Advanced Computational Methods in Heat Transfer IX

Advanced Computational Methods and Experiments in Heat Transfer X

Heat Transfer Design Methods

Advances in Heat Transfer, Volume 39 (Advances in Heat Transfer)

Computational Fluid Dynamics and Heat Transfer: Emerging Topics (Developments in Heat Transfer) (Developments in Heat Transfer Objectives)

Advances in Heat Transfer, Volume 35 (Advances in Heat Transfer)

Advances in Heat Transfer, Volume 34 (Advances in Heat Transfer)

Advances in Heat Transfer, Volume 37 (Advances in Heat Transfer)

Computational Fluid Mechanics and Heat Transfer

Computational Fluid Mechanics and Heat Transfer,

Computational Fluid Mechanics And Heat Transfer

Heat transfer

Heat transfer

Heat Transfer

Heat Transfer

Computational fluid mechanics and heat transfer

Heat Transfer

Advances in Numerical Heat Transfer

HEAT TRANSFER IN INDUSTRIAL COMBUSTION

Advanced Computational Methods In Science And Engineering

HEAT TRANSFER IN INDUSTRIAL COMBUSTION

Heat Transfer in Process Engineering

Heat Transfer in Industrial Combustion

Heat Transfer in Process Engineering

Gene Transfer to Animal Cells (Advanced Methods)

Building Heat Transfer

Convective Heat Transfer

Handbook of Heat Transfer

Engineering Heat Transfer

Process Heat Transfer

Handbook of Heat Transfer

Advanced Computational Methods in Heat Transfer IX

Advanced Computational Methods and Experiments in Heat Transfer X

Heat Transfer Design Methods

Advances in Heat Transfer, Volume 39 (Advances in Heat Transfer)

Computational Fluid Dynamics and Heat Transfer: Emerging Topics (Developments in Heat Transfer) (Developments in Heat Transfer Objectives)

Advances in Heat Transfer, Volume 35 (Advances in Heat Transfer)

Advances in Heat Transfer, Volume 34 (Advances in Heat Transfer)

Advances in Heat Transfer, Volume 37 (Advances in Heat Transfer)

Computational Fluid Mechanics and Heat Transfer

Computational Fluid Mechanics and Heat Transfer,

Computational Fluid Mechanics And Heat Transfer

Heat transfer

Heat transfer

Heat Transfer

Heat Transfer

Computational fluid mechanics and heat transfer

Heat Transfer

Advances in Numerical Heat Transfer

HEAT TRANSFER IN INDUSTRIAL COMBUSTION

Advanced Computational Methods In Science And Engineering

HEAT TRANSFER IN INDUSTRIAL COMBUSTION

Heat Transfer in Process Engineering

Heat Transfer in Industrial Combustion

Heat Transfer in Process Engineering

Gene Transfer to Animal Cells (Advanced Methods)

Building Heat Transfer

Convective Heat Transfer

Handbook of Heat Transfer

Engineering Heat Transfer

Process Heat Transfer

Handbook of Heat Transfer

Recommend Documents