PIV and water waves

COASTA ADVANCES \i ID OCEAN E PIV AND wflra wflucs John Grue Philip L.-F. Liu & Ceir K. Pcdersen VOLUME World Scient...

Author: John Grue | Philip L. F. Liu | Geir K. Pedersen | Philip L-F Liu

104 downloads 2298 Views 6MB 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

COASTA

ADVANCES \i ID OCEAN E

PIV AND wflra wflucs John Grue Philip L.-F. Liu & Ceir K. Pcdersen

VOLUME

World Scientific

PIV O D WATER. WOJES

This page is intentionally left blank

ADVANCES IN COASTAL AND OCEAN E N G I N E E R I N G

PIV 01D WflTft WflUES VOLUME 9

€ditors

John Grue University of Oslo, Norway

Philip L.-F. Liu Cornell University, USfl

Geir K. Pedersen University of Oslo, Norway

\[p World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONGKONG • TAIPEI • CHENNAI

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

PIV AND WATER WAVES Copyright © 2004 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-238-914-8 ISBN 981-238-949-0 (pbk)

Printed in Singapore.

Preface

Particle Image Velocimetry (PIV) denotes a group of laboratory techniques that extract velocity fields from particle displacements in subsequent digitized images of two-dimensional fluid sections. The velocities are found from cross-correlations or tracing of individual particles. The state of the art PIV employs digital camera technology, lighting by lasers and data processing on rapid computers. The advantage of PIV over competing techniques, as the Laser Doppler Velocimetry, is the simultaneous measurement of complete velocity fields that enables studies of complex flows, including vortical structures, flow separation, Reynolds stresses, turbulence characteristics and two-dimensional divergence at interfaces. Powerful experimental methods, like PIV, represent tools for advancement of theoretical computations, used for validation of models or as guidance in developing new theories where existing models either have shortcomings or do not exist. PIV and Water Waves grew out of a conference that was held on the subject in Peterhouse, Cambridge, UK, during 17-19 April 2002. The conference marked the end of a 4^ years long Strategic Program at the University of Oslo, entitled 'General analysis of Realistic Ocean Waves (GROW)'. In the project we developed a PIV system. Experiments on free surface and internal waves were performed and supported by theoretical and numerical modeling. With the wish to exchange results with other groups working with similar focus, we announced an open invitation to participate in this conference. As a result, more than twenty scientific papers on the subject were presented at the meeting. These papers are included as extended abstracts in part II, Chapters seven through nine, of this volume. Two additional papers that were presented at the meeting were later extended. They appear as Chapters three and five in part I of the book together with four invited review articles. The idea with this part is to review the status of the PIV technique and its various applications to free surface flows, interfaces, boundary layers, and internal waves. Chapter one gives

VI

Preface

an introduction to image based laboratory techniques with special focus on the use of image processing, digital data analysis and pattern matching in PIV. Various details of the method are discussed: resolution, displacement estimation, how to avoid peak-locking, true and false correlation peaks. The low-intensity mode of PIV, Particle Tracking Velocimetry, and how to extract higher order quantities from velocity fields, are described and illustrated by several examples. Chapter two describes a submersible PIV system for the measurement of turbulent dissipation, the Reynolds stresses and potential vortical structures in the bottom boundary layer of the coastal ocean. The system has the capacity to operate continuously for hours, and has been deployed at several sites in the coastal ocean. Wave induced, oscillatory boundary layer flow over a fixed ripple bed is discussed in Chapter three. PIV measurements of flow separation and vortex generation processes may serve as guidance in the choice of appropriate closure model of the turbulence equations, employed in simulations. Chapter four reviews PIV applied to flows in boundary layers and wakes of ships. Propulsion hydrodynamics, cavitation and free surface waves interacting with moving bodies are discussed. PIV measurements in the interface between water and air indicates regions of strong upwelling behind crests of microbreaking waves and regions of convergence in front of the crests (Chapter five). These processes are important for scalar exchange, e.g., green house gasses, between air and water. Field observations, laboratory measurements and theories are reviewed. Internal wave motion is the subject of Chapter six. Large amplitude solitary wave motion in experimental two-layer or stratified fluids compare excellent with fully nonlinear theory when the motion is smooth, but departs from theory when convective breaking or shear instability sets in. Solitary waves interacting with a submerged ridges or slopes may exhibit strong local deformation and cause strong currents at the sea floor. It is the hope that this volume will stimulate to enhanced interest in the PIV technique, and to further progress within the field of fluid mechanics, exploring the powerful combination between PIV and theory. We are indebted to the external referees and to Mrs. Dina Haraldsson for her willingness and proficient help in preparing parts of the text. The Strategic University Program was funded by the Research Council of Norway. The support is gratefully acknowledged. December 2003, The editors

Contents

Preface Part I

v Review Chapters

Chapter 1 Quantitative Imaging Techniques and Their Application to Wavy Flows J. Kristian Sveen and Edwin A. Cowen Chapter 2 PIV Measurements in the Bottom Boundary Layer of the Coastal Ocean W. A. M. Nimmo Smith, T. R. Osborn and J. Katz Chapter 3 Water Wave Induced Boundary Layer Flows Above a Ripple Bed Philip L.-F. Liu, Khaled A. Al-Banaa and Edwin A. Cowen

1

51

81

Chapter 4 Ship Velocity Fields Joe Longo, Lichuan Gui and Fred Stern

119

Chapter 5 The Air-Water Interface: Turbulence and Scalar Exchange S. Banerjee and S. Maclntyre

181

Chapter 6 Internal Wave Fields Analyzed by Imaging Velocimetry John Grue

239

Part II

Extended Abstracts

Chapter 7 Wave Breaking, Surface Motion, Surf Zone, Air-Sea Interaction and Wind Waves vii

279

viii

Contents

7.1. PIV Measurements of Accelerations in Water Waves A. Jensen, M. Huseby, D. Clamond, G. Pedersen and J. Grue

279

7.2. PIV Measurements of the Velocity Field in Steep Water Wave Events D. Clamond, J. Grue, M. Huseby and A. Jensen

282

7.3. LDA Laboratory Measurements in the Surf Zone for the RANS Depth-Averaged Shallow-Water Equations T. Feng and P. Stansby

285

7.4. Wave Acceleration Measurements Using PTV Technique Kuang-An Chang, Edwin A. Cowen and Philip L.-F. Liu 7.5. Two-Color LIF-PIV for Gas Exchange at Air-Water Interface Nobuhito Mori 7.6. Run-Up of Solitary Waves Geir Pedersen, Stefan Mayer and Atle Jensen 7.7. DPIV Measurements for Unsteady Deep-Water Wave Breaking on Following Currents Chin H. Wu, Aifeng Yao and Kuang-An Chang Chapter 8 PIV Methods, Boundary Layer Flows and Turbulence 8.1. A Digital High Speed Camera System Applied to PIV Measurements in Water Waves J. Kristian Sveen and Atle Jensen 8.2. Three-Dimensional Stereoscopic Particle Tracking Velocimetry for a Large Domain (3DSPTV-LD) and Its Application Yasunori Watanabe

288

291 294

297

301

301

304

8.3. Enhanced PIV Technique to Investigate Bottom Boundary Layer Dynamics Ahmed S. M. Ahmed and Shinji Sato

307

8.4. The Turbulent Boundary Layer Structure Beneath a Gravity Wave and Its Proper Decomposition Edwin A. Cowen and Stephen G. Monismith

311

Contents

8.5. Intermittency in Surf Zone Turbulence and Observations of Large-Scale Eddies Using PIV D. T. Cox and S. L. Anderson 8.6. Turbulence Distributions and Flow Structure in the Coastal Ocean from PIV Data W. A. M. Nimmo Smith, L. Luznik, W. Zhu, J. Katz and T. R. Osborn Chapter 9 Breakwaters and Internal Waves 9.1. Blocking Effects in Supercritical Flows Over Topography V. I. Bukreev, A. V. Gusev and V. Yu. Liapidevskii

ix

314

317

321 321

9.2. Study of the Flow Pattern in a Perforated Breakwater Using PIV Measurements Jean-Marc Rousset and Tom Bruce

324

9.3. Pseudo Image Velocimetry of a Regular Wave Flow Near a Submerged Breakwater Francisco Taveira-Pinto

327

9.4. Laboratory Modelling of the Motion of an Internal Solitary Wave Over a Ridge in a Stratified Fluid Y. Guo, P. A. Davies, J. Kristian Sveen, J. Grue and P. Dong

330

9.5. Measurements of Velocities During Run-Up of Long Internal Waves on a Gently Sloping Beach J. Kristian Sveen and John Grue

334

9.6. Comparison of Modelling of Strong Internal Wave Events with PIV Studies of the Phenomenon Deborah J. Wood, J. Kristian Sveen and John Grue

337

CHAPTER 1 QUANTITATIVE IMAGING TECHNIQUES A N D THEIR APPLICATION TO WAVY FLOWS

J. Kristian Sveen and Edwin A. Cowen Mechanics Division, Department of Mathematics, University of Oslo Box 1053 Blindern, N-0316 Oslo, Norway E-mail: [email protected] School of Civil and Environmental Engineering, Cornell University Ithaca, NY 14853-3501, USA E-mail: [email protected]

Quantitative imaging (QI) techniques are a general class of optically based laboratory measurement techniques used in the field of experimental fluid mechanics, which have seen rapid growth over the last two decades. They are particularly well suited for the study of wavy fluid flows which are characterized by unsteady free surfaces and internal motions. This paper presents an overview of QI techniques in general, with a particular focus on particle image velocimetry (PIV). We present QI methods in the context of the broader fields of pattern recognition and image processing techniques, which are currently used in a wide range of fields. In this review QI methods and their fundamentals are described in detail and recent developments, targeted at increasing accuracy and resolution, are described and put into perspective. More specifically we identify QI techniques as a digital data analysis (through software) set of issues built upon general principles of pattern matching. Throughout the paper we address the aspects that are particular to wavy flows although these issues can be argued to be important for any unsteady fluid flow of interest, e.g. turbulence. This review article thus serves as a general reference for the neophyte and experienced fluid mechanics experimentalist. 1. I n t r o d u c t i o n Perhaps the greatest challenge of making measurements in wavy free surface flows is measuring flow field characteristics near the dynamically moving free surface - i.e., between the trough and crest. Quantitative imaging (QI) techniques are a robust solution t o this problem, as demonstrated in Fig. 1, and hence QI techniques are becoming the methods of choice when 1

2

J. K. Sveen and E. A.

x (cm)

Cowen

x (cm)

Fig. 1. Incipient shoaling breaking wave. Left image with every 4 t h determined vector shown in each coordinate direction; right image is a magnified subregion of left image with every 2 n d determined vector in each coordinate direction.

attempting to interrogate laboratory wavy free surface flows. An important aspect to the emerging dominance of QI techniques is their ability to capture whole field properties - e.g., u(x,z) (the velocity field), u?y(x,z) (the vorticity field), u'w'(x,z) (the Reynolds stress field), and e(x,z) (the turbulent dissipation field). While many wavy free surface flows are periodic the reality is that due to reflections and to the difference between the phase velocity and the energy propagation velocity (group velocity) flows are often quasi-periodic at best. Researchers interested in spatial gradients have traditionally attempted to employ single-point measurement technologies at two or more spatial locations but on different experimental runs. Variability among experimental runs will lead to variability of induced rms velocities which is often on the order of the turbulence intensity itself making it extremely challenging to employ data from different runs for the determination of gradients. While QI techniques do not eliminate wave-to-wave variability they do capture a spatial field under the identical free surface conditions allowing accurate instantaneous gradients to be determined. Presently the most cited reason to work with single-point measurement technologies is a need for improved temporal resolution relative to QI techniques. However, over the last decade computers and image capture technologies have progressed sufficiently that QI techniques are now capable of reasonably high temporal as well as spatial resolutions, further

Quantitative

Imaging Techniques and Their Application

to Wavy Flows

3

propelling the popularity of QI techniques. A challenge for these techniques is optical access, particularly for existing facilities that were not designed with PIV in mind. Recent advances with horoscopes and index-of-refraction material matching, however, demonstrate that optical access issues can usually be overcome with a bit of ingenuity. In order to visualize some of the methods presented in this paper, we have chosen to use the free, open-source PIV program MatPIV 1 and two example images taken from the paper by Jensen et al? This paper is organized as follows. Section 2 presents a brief introduction to some of the more commonly used QI techniques in the perspective of pattern matching. Section 3 is written to provide a basic overview of the fundamentals of Particle Image Velocimetry (PIV), while section 4 targets the more fundamental aspects of the technique. Section 5 gives a brief overview of Particle Tracking Velocimetry (PTV) and section 6 focuses on higher order measurements from velocity fields. Finally, Section 7 contains a short conclusion, and also provides an overview of a few other areas where the same principles of pattern matching are applied. 2. Quantitative Imaging Techniques QI techniques can be broken immediately into several fundamentally different types of techniques - flows seeded with discrete particles, flows seeded with continuous tracers (e.g., fluorescent dyes) and unseeded flows (to look at density differences). The former are generally employed for the determination of velocity while the second and third are generally used to determine a scalar field quantity (e.g., concentration or temperature). The literature is rife with various acronyms for these types of techniques and we will briefly introduce some of the more popular QI nomenclature here. 2.1. Particle

Based QI

Techniques

• Particle Streak Velocimetry (PSV) -A general class of techniques where the image exposure time is long relative to the time a particle occupies a point in space. The result is images of particle streaks. The length of the streak can be calculated to determine the velocity based on the known exposure time. • Laser Speckle Velocimetry (LSV) -A general class of techniques where the seeding density is sufficiently high that an image captures predominantly overlapping and interfering particle images, which can be thought of as an intensity texture or speckle field.

4


Cowen

Essentially no discrete particle images are seen. The velocity is extracted by correlating the speckle pattern in a small subregion with that in another subregion, either optically (Young's fringe analysis) or digitally (auto or cross correlation analysis). • Particle Image Velocimetry (PIV) -This term is sometimes taken to mean the entire broad class of discrete particle based techniques, however, its preferred definition is a general class of techniques where the seeding density is moderate such that the nearest neighbor distance of particle images is on the order of a few to perhaps ten times the particle diameter ensuring that all small subregions have several distinct discretely imaged particles within them and relatively few particle images overlap. The velocity field can be extracted in a number of ways, the most popular of which are digitally via auto or cross correlation analysis. • Particle Tracking Velocimetry (PTV) A general class of techniques where the seeding density is sufficiently low that an image captures predominantly non-overlapping or interfering particle images and the velocity can be extracted by tracking the motion of individual particles over known times.

Other common names for particle based techniques include: pulsed light velocimetry (PLV), particle image displacement velocimetry (PIDV), particle displacement velocimetry (PDV), digital particle image velocimetry (DPIV), digital particle tracking velocimetry (DPTV), correlation image velocimetry (CIV), spatial correlation velocimetry (SCV) and large-scale particle image velocimetry (LSPIV). We note that three dimensional QI techniques have seen rapid growth recently and are based on one of four fundamental approaches: holography 3 , stereoscopic imaging with multiple cameras 4 , depth-of-field with a single camera 5 , and scanning a light sheet through a volume 6 . We will restrict our review of QI techniques to two-dimensional implementations. Ron Adrian, in a survey through 19957, showed that the number of publications per year on particle based QI techniques grew exponentially. All indications are that the trend continues. The common principle to all particle based techniques is that the instantaneous fluid velocities can be measured by recording the position of images produced by small tracer particles, suspended in the fluid, at successive instants in time. The techniques listed above, as well as others, fall into two broad categories, each with different development paths: PIV and PTV.

Quantitative


to Wavy Flows

5

LSV and PIV are different operating modes of the same technique. The velocity field is generally determined on an interrogation grid and each velocity vector is the average velocity over many tracers contained in a small volume of fluid. These techniques have their roots in solid mechanics. They were originally used to determine the in-plane displacement and strain of solids with diffusively scattering surfaces. PTV and PSV can also be thought of as different operating modes of the same technique. In contrast to PIV, velocity vectors are determined from the individual particle images or streaks produced by a single particle at random locations. These techniques have their roots in the field of flow visualization; particle streak photography and stroboscopic photography. Prandtl 8 was an early developer of particle tracking techniques, although not the first. For the remainder of this manuscript we will use PIV to indicate the class of particle based QI techniques where the velocity is extracted by looking at the movement of an ensemble of particle images. PTV will be used to indicate the class of particle based QI techniques where the velocity is extracted by looking at the movement of a single particle image.

2.2. Tracerless

QI

Techniques

This class of techniques relies on measuring either the concentration or displacement of a chemical tracer substance added to the flow. • Laser induced fluorescence (LIF) -A general class of techniques where a flow is seeded with a dilute fluorescent chemical tracer that will fluoresce proportionally with its local concentration. The local concentration field is often the objective of LIF measurements but careful choice of dye can allow the measurement of temperature, and pH as well as other scalar properties of the flow. • Scalar image velocimetry (SIV) -Really a specific analysis applied to LIF images where the gradient in intensity information recorded in images is treated like speckle in LSV and correlation based analysis is used to extract velocity field information. Another common name for chemical tracer based QI techniques is planar laser induced fluorescence (PLIF). As with the particle based techniques we note that three dimensional scalar QI techniques have seen growth recently and are generally based on scanning a laser light sheet through a volume 9 ' 10 .

6

2.3. Other QI


Cowen

Techniques

Other QI techniques do exist that do not rely on the use of passive tracers. In fact some of the oldest flow visualization 11 techniques do not, such as shadowgraphy, schlieren imaging and the Mach-Zendner Interferometer. Today many researchers utilize so-called Synthetic Schlieren or Quantitative Shadowgraph techniques using digital cameras and the very same principles of pattern matching as are used in for example Particle Image Velocimetry. We shall briefly return to this aspect in section 7.

2.4. QI Techniques — Image Processing Recognition by a Different Name

and

Pattern

Many newcomers to PIV in particular, and QI techniques in general, are often confused by the relatively large number of details that need to be addressed in order to apply these techniques to real fluid flow experiments. One of our primary goals in this review is to focus on the foundations upon which these techniques are built. PIV in particular has received a lot of attention within the last 10 to 15 years. As implied in section 2.1, authors often use different nomenclature for what are essentially identical approaches, perhaps with subtle implementation differences. We would like to stress that PIV relies on image processing and pattern recognition analysis and as such it should more properly be viewed as an interdisciplinary field between the experimental fluid mechanics research community and the image processing and pattern recognition (IP&PR) research communities. QI techniques are widely known and used by IP&PR researchers and in fact, experimental fluid mechanicians often re-invent analysis techniques as they ignore the previous efforts documented in the IP&PR literature. The basics for understanding PIV and pattern recognition may actually be found in most introductory books on image processing12. The details of applying QI techniques to fluid flows essentially consists of imaging a temporally and spatially varying pattern within the flow, generally by adding discrete tracer particles although we should not feel restricted to this particular case, illuminating the flow in a nearly two-dimensional slice as the majority of imaging devices capture two-dimensional information, and using IP&PR algorithms to extract the displacement in a known time of the imaged tracers. The entire QI measurement process can be divided into two fundamental components: (1) a hardware problem of experimental techniques, including illumination, seeding and image recording, and

Quantitative


to Wavy Flows

7

(2) an analysis (software) problem of applying IP&PR techniques to extract displacement or other information of interest from the images. The latter point should be recognized as the fundamental application of QI methods, while the former is merely an experimental problem (which in many cases is the hardest part as the majority of QI experimentalists will agree the challenge is getting 'good' images - with 'good' images in hand it is simply a matter of finding the right IP&PR technique to extract the information one desires!). Hence we contend that QI analysis in general, and PIV as a particular example, is simply a sub-specialty within the broader fields of IP&PR. For our purposes QI/PIV are defined as extracting information of interest to fluid mechanicians by image processing and pattern recognition means. There are subtleties that we avoid by this generalization, but as a starting point this view is effective and allows us to break down a large problem into smaller, solvable pieces. We can view the more recent developments in QI and PIV analysis techniques as emerging to solve problems that arise in terms of accuracy and resolution in displacement estimates - to date the fundamental quantity that the vast majority of QI velocimetry techniques seek to extract from images. The reader should note that applications of pattern matching is currently an area of active research. For example, looking at the motion picture industry, the MPEG standards 13 for sound and video compression, storage and transfer are built on many of the same ideas we use in PIV. As an example we can consider the DVD standard (MPEG2), where local motion estimates are used in order to limit the amount of storage required. Instead of saving every single frame in a movie, only a few (typically every 8-16 frames) are stored as full frames and only the local motion is stored for the remaining images. There are many advances from this industry that have yet to be applied to PIV. For example, we should be able to perform motion estimation in the frequency domain within the PIV framework, much like is done in image processing14. In this way we should be able to avoid one fast Fourier transform (FFT), and hence save about one-third of the calculation time. In section 7 we present a brief overview of other research fields that use pattern matching approaches. 3. P I V — A General Overview Let us begin the description of the basics of PIV with a fluid flow that is seeded with particles that can be considered passive tracers, perhaps because they are very small or if they are not so small they are near neutrally

8


Cowen

buoyant with respect to the flowing fluid. A light source, generally a laser light source, is shaped with optical components into a light sheet to illuminate the particles. The light sheet is quasi-two-dimensional in the sense that it is thin in the direction orthogonal to the plane of motion that contains the two components of velocity we are interested in measuring while it is broad in the other two directions. An imaging device, in this case a digital camera, is equipped with appropriate optics (e.g., a lens) to collect images of the particles as they pass through the light sheet. In general, computer controlled timing signals are sent to the digital camera and laser light source (or the optical components that shape the light source into a light sheet) to synchronize the light source to the camera such that discrete images of particles (e.g., short time exposure images) are captured at desired times within each collected image. There are, of course, myriad ways in which particle images can be collected, which fall into two general categories: the single exposure of multiple images and the multi-exposure of single images. Again there are myriad ways that the mean displacement of particles in any sub-region (henceforth referred to as a subwindow) can be extracted from the images but the fundamental technique used in multi-exposed single images is autocorrelation analysis while the fundamental technique used in single-exposed multiple images is cross-correlation analysis. As cross-correlation analysis is more straightforward as well as more accurate, we will restrict our discussion of PIV basics to cross-correlation analysis of image pairs - that is the exposure of two sequential images, each individually, specifically for analysis by cross-correlation. Let's assume we have collected an image pair where the second image was captured a known time, At, after the collection time of the first image. The most straightforward approach to cross-correlation analysis is to define a square subwindow with side length N = 2" where n is an integer. N is typically taken as a power of 2 to take advantage of determining the crosscorrelation in the frequency domain via the fast Fourier transform (FFT). As will be described later N need not be restricted to these discrete values but as this restriction is frequently employed we will assume it for now. Let us assume a value of N = 32 and an image size of NR x NC pixels (an acronym for "picture element" that describes the smallest discrete unit of scattered light intensity measured by a digital camera, sometimes also referred to as a "pel"), where NR and NC are the number of rows and columns in the digital image, respectively. The simplest algorithm is to divide each image of the pair into non-overlapping N x N subwindows and to then perform the two-dimensional cross-correlation of each subwindow

Quantitative


to Wavy Flows

9

pair in the image pair. The cross-correlation function is denned as N-1N-1

R S

(^ =^ E E

f

y ( M ' ) ^ ( < + *. 3+t)

(1)

where R is the cyclic cross-correlation between subwindows / , J in the first image of the image pair (F') and the second image of the image pair (F"), i,j is the pixel location within subwindow I, J, and s,t is the 2-D cyclic lag at which the cross-correlation is being computed. As indicated above R is often calculated in the spectral domain and hence equation (1) can be found as: R(S,t) = T~1 [^{F'ÂiJ^îF'/î

+ s, j + t)}]

(2)

where T and T~x are the Fourier and inverse Fourier transform operators and the star denotes complex conjugate. This basic feature is known as the correlation theorem and can be found in most introductory level books on image processing 12 . For the purpose of visualization we consider the example images in Fig. 2a-b, which shows the image pair F' and F" along with the non-overlapping subwindow interrogation grid. Fig. 2c and 2d shows subwindow J = 10, J = 15 for each image in the pair along with panel e, which shows the resultant cyclic cross-correlation of this subwindow pair. The correlation plane contains a peak which has a maximum at (s, t) = (26,25). The displacement is measured from the center of the correlation plane to this peak. The integer displacement in our example is estimated to be dx = —6 and dy = —7. Our simple introduction here raises several issues. First we note that the maximum unambiguous displacement that can be resolved is N/2 pixels. If the displacement is larger than N/2 pixels (but less than TV pixels, so at least a few of the particles in the first image subwindow remain in the second image subwindow) the correlation peak will alias to the location — (N — £) where £ is the actual displacement. If the displacement is larger than N pixels then R represents the cross-correlation of two uncorrelated subwindows and the returned displacement estimate will be the result of a random noise peak (i.e., the lag where a maximum number of particles randomly align themselves between the two subwindows). Secondly, there is an implicit assumption that the particles are being translated without rotation or shear. If the particles undergo rotation and/or shear over the time At between the capture of the two images then we must be concerned about the effect of this rotation and/or shear on the existence of a usable correlation peak. Thirdly, if the flow is not two-dimensional where the outof-plane motion is identically zero then there is a finite non-zero probability

10

J. K. Sveen and E. A. Cowen

of a particle appearing in the first subwindow moving out of the light sheet (and hence the image) by the time the second image is captured. This outof-plane motion can clearly lead to a measurement bias if the out-of-plane motion is correlated to the velocity itself. And lastly, if N = 32 our maximum resolvable displacement is just 16 pixels. If we cannot extend this or resolve the displacements to sub-pixel accuracy (e.g., estimate the location of the correlation peak to fractional pixel values) then our maximum accuracy is greater than 3% and our typical accuracy is considerably higher than this. The above example also raises many other questions. With the above primer in PIV as our starting point we will now turn to the details of PIV. 4. P I V — The Fundamentals There are essentially three typical implementations of PIV: single exposure multiple image (cross-correlation based), double-exposure single image (auto-correlation based) and multi-exposure single image (auto-correlation based). There are several excellent fundamental references on QI techniques in general and PIV techniques in particular and the reader should explore these references for further details and perspectives on PIV 1 5 ' 1 6 ' 1 7 ' 1 8 , 1 9 . 4.1. Displacement

Estimates

— Correlation

Approaches

PIV techniques often rely on either auto or cross-correlation of subregions to extract the mean displacement of particles contained within the subregion. This methodology thus relies on estimating the auto or cross-covariance function between the subregions to extract the displacement estimate. For details on auto-correlation see Adrian 20 and Keane and Adrian 21 . The covariance can be determined in a number of ways but it is generally determined by either Fourier transforms 22 or the direct determination of the two-dimensional covariance function23 (also known as the correlation coefficient12). The expected value of the cross-covariance R(s, t) (see equation 1) is shown by Westerweel17 to be E{R(s,t)}=(l-^(l-lf)R(s,t)

(3)

and it is seen immediately to be biased (note the bias vanishes as N —> oo). The bias occurs because the shift over (s, t) results in only a part of F' correlating with F". Adrian 24 reported this as the result of in-plane loss-of-pairs. In the presence of strong gradients the interpolation region contains more particle pairs

Quantitative


to Wavy Flows

11

b)

:

a; 3F«ttKtt#BKW»K«0##*&«VKKB8E8i8f««»»i#lffaft4 «£###»£

200 400 600

l£B8M»Kg8Btt«i(>*#t a«ssss*iK!»issa»B£sssiaK«i iBB8&tt8Sg8!«8!SS##8W*ft#««:8!£ Kg&BBBKM&WH*RM8Wai!

SSS«SffiKBiB!«lSSSSSSaf*«S(ftfSi KBBHBlBlBttSRMttWKSIIIKBi BRKfllBWMaKKiiltattBKMKI BKMttBBRMMHCaSBHSBI

800

BgKgttBB&£!8BttttB«!##i jasat -KBttgBBJgffBKttttttRaSii SSfSSi

1000 Hi

2()i

„ , , . . .._J«Httg88&»tttt$£&a*ttM«MK8stt&«ie«4ftt»K*«:l!iBE &»#g»aK«K££ggttttttB^8Hatt«#BHtttt&SS»K:iiagE#i«tt3£ te&S888«KStt«8im&ttHH#88»K8BKK88K&MBKI«!8:K8£ttttK f«Bg|«KWK8J«gKBK8KBBiSKS«SttgB«&»S:KKi«BBi8K8»&tt!g &M#H8»RHKKa«KK*aâBMMg«£»Ma#*fe«tttt£##Hg&g; fwfigBttaMKJttKKBH* "— —-————

!KKB!BBBSSf*8f*SS*SSEWi lSfWW*»*«*cSSSBSK< isxsaHBasvKaMBiSiaisi

!HHK0MS8SH«»i8!S8SHIKWI lg:£MS3l»BaSKBS!g&M»#K&&*g :sasf*»BSSsspi«f!9i*saa»sfsiSiS9is :»«ssaiswsajiss as asssiasssssfsissass

200 400 600 80010001200

200 400 600 80010001300

d)

c)

e)

Fig. 2. Images with superimposed non-overlapping grid. Black cells marks the subwindows shown in c) and d). Correlation plane shown in e). Upper arrow shows the displacement. Lower arrows denote the displacements along each axis. Black vertical line marks the center of the plane.

with small displacements, therefore, this is an under bias. Prom the above equation it is apparent that the bias grows linearly with the shift size. Westerweel shows that two conditions arise from looking at the variance of the expected value of R. He finds that the noise in R due to random correlations

12


Cowen

is approximately a uniform random field and goes as 1/N2. The probability that the noise peak is greater than the correlation peak grows for increasing displacement (s,£). 4.1.1. Removing Effects of Correlation with the Mean Background As shown by Adrian 24 , the correlation between two images may be split into 3 contributions: (1) correlation of the mean background intensities, Rc(s,t), (2) correlation of the mean and the fluctuating intensities, Rf(s,t), (3) correlation of the fluctuating intensities, Rd(s^t).

and

In this way the correlation may be written as R = Rc + Rf + Rj. The latter part of the correlation plane will contain the displacement peak, and to avoid the other parts of the correlation, the mean is normally subtracted from each image prior to computing the correlation. The process is shown in Fig. 3 and also included in equation (4).

R

Rc A

Fig. 3.

Rf „

Rd

± 28>29>30>31. piD methods are described in more detail in section 4.7.3.

4.3. Subwindow

Size

An important consideration is the size of the subwindow to be used for PIV analysis as this leads directly to issues of resolution. Prasad et al.32

Quantitative


to Wavy Flows

15

260

280

300

320

340

260

280

300

320

340

260

280

300

320

340

260

280

300

320

340

260

280

300

320

340

260

280

300

320

340

Fig. 4. Schematic view of the three most basic dynamic subwindow location techniques. The upper pair shows fixed interrogation windows. The middle pair shows first order window shifting 2 2 , 2 5 and the lower pair shows second order accurate window shifting 26 .

investigated a range of subwindow sizes from N = 32 to N = 256 finding that a reduction of subwindow size from the traditional N — 256 to N = 128 resulted in no appreciable degradation of accuracy. Westerweel17 explored in detail the requisite subwindow size to obtain the desired information,

16


Cowen

namely the location of the correlation peak to sub-pixel accuracy. Westerweel17 sought the effective sampling rate needed to yield a good representation of an original QI image. A signal is band limited if it can be constructed from only signals with frequency components less than some maximum, i.e., its Fourier transform is non-zero over only a finite portion of the frequency domain. Therefore, F(F)(s,t)

= 0, for |*| > S, \t\ > T.

(5)

The bandwidth is defined as the maximum of S and T. A band limited continuous signal can be reconstructed exactly (given infinite samples) provided that the sampling rate is at least twice the bandwidth. This rate, known as the Nyquist rate, is max(2Sr, 2T). Goodman 33 has shown that the bandwidth of the image intensity for a thin spherical lens image system (aperture D, focal length / , coherent light with wave length A) is given by:

Xz0

\f(M + 1)

w

where ZQ = f(M + 1) is the image distance and M — ZQ/ZQ is the magnification (Zo is the object distance). Now, if an image is obtained with illumination wavelength A = 0.5 /xm, f/D = 8, and M = 1, then it follows that W = 125 m m - 1 and therefore that the Nyquist rate, 2W, is 250 m m - 1 , explaining the traditional subwindow size of 256 pixels and a subwindow area of 1mm2 early in the history of PIV. However, this is a significant hurdle for CCD based image acquisition. Most CCD's pixels are at least 10 /xm x 10 itm, therefore 2W = 100 m m - 1 is the best that can be done. However, the reality is that the goal of PIV interrogation is not to reconstruct the image exactly, but only to obtain the position of the displacement covariance peak. What is required is not the exact details of the particle shapes (edges) but just the details of their positions (lower wave number information). The covariance function has a spectral density function that is "nearly band limited" meaning its value vanishes for sufficiently large (s, t) but may not be exactly zero. Therefore F(s,t) ~ 0 for \s\ > S, \s\ > T. Parzen 34 showed that for a 1-D signal the bandwidth, Wp with a circularly symmetric spectral density function is defined as the width of a cylinder with the same volume, therefore Wp = (irs(0,0)y1/2-

(7)

It can be shown that if a square region is sampled instead of a circular region and the covariance of particle images with diameter d are of interest, the

Quantitative


to Wavy Flows

17

covariance width, ah, is ah = ds/2/(2A4ir) which leads to Wp = l/(27rcr/,). Thus, given the same optical parameters as above and d = 20/xm (ah — 3.7 /xm), 2WP = 86 m - 1 , suggesting that 64 x 64 pixels over a bit less than 1 mm 2 is sufficient. Note that larger particle image diameters reduce the required sampling rate, so that 30/im images —>~ 32 x 32 pixels/mm 2 . Thus the dominant subwindow sizes in currently used PIV algorithms are N — 64 and N = 32. For more details on band limited signals see Westerweel17.

4.4. Sub-Pixel Displacement

Estimation

It can be shown that for any non-zero particle image diameter the width of the correlation peak will be greater than one pixel. By including values adjacent to the maximum in R, the center of the peak can be estimated to sub-pixel accuracy. We decompose the displacement (s,t) as s = sQ + es,

t = t0 + et

(8)

where So,io is the integer displacement and es,et is the fractional part. Therefore -0.5 < es < 0.5

and

- 0.5 < et < 0.5.

(9)

In the absence of an estimate for es and e* the error is ± A / 2 where A is one pixel. For N = 32 a displacement of eight pixels has an uncertainty of ±0.5/8 ~ 6%. If we digitize the same subwindow into 256 pixels (N = 256) the uncertainty is now ±0.5/64 ~ 1%. Therefore to work at small N, we need subpixel estimates of the center of the covariance peak. It can be shown that the covariance cov{e r , e*} = 0 (i.e., eêi = 0), therefore, we can work with the 1-D problem without loss of generality. For narrow correlation peaks the covariance width is small enough that only the nearest pixels to the peak contain significant information. Since QI techniques generally satisfy the conditions for narrow covariance peaks, three-point sub-pixel estimators are generally sufficient. As an example we will consider the correlation peak shown in Fig. 2e. Fig. 5a shows a close up of this peak, while Fig. 5b shows the highest value, RQ, of the correlation along with the two nearest pixels in the vertical direction, i?_i and R+\. The three most commonly used estimators are Center-of-Mass, Parabolic fit and Gaussian fit.


Fig. 5.

Cowen

Close-up of the largest peak in the correlation plane in Fig. 2e.

Center-of-Ma88

(COM)

The sub-pixel estimation of the Center-of-Mass is calculated from .Y ._..

ii|-i "- R-i ,1Qx R i -I"- R-o "V it-fi This method ignores the need for a peak since R® can be less than i2__i or J?+x. Fig. 5 shows an example of a correlation peak and the positions of jfiL-i, R-\-i and RQ. Parabolic Another option is to assume that the peak has a certain shape, for example a parabola. We can then evaluate __ R-i -~ R+i , . €p 2(H__ 1 -2Ho + -R + i)" ' This formula actually requires a peak since |J?o| must be greater than \R i| and

|JR+I|.

Gaussian The most common assumption in PIV is to use a Gaussian peak-fit. This is mainly due to -the fact that spherical particles image as Airy functions and the central lobe of an Airy function is well approximated by a Gaussian curve. The log of a Gaussian curve is parabolic and therefore l n i i - i — lni? + i (12) €G 2(ln # _ i - 2 In R0 + In j?, +1 )'

Quantitative


to Wavy Flows

19

Note, again that Ro must be a peak and R4 > 0 for i = —1,0,1. Clearly all three are a balance of R-\ and i?+i with a normalizing value in the denominator. Note that ec will always yield a result and ec has the most restrictions. Therefore we expect ec to be the most robust and ea the least. On the other hand, ec does not even acknowledge that we have a peak while ec acknowledges and uses shape information. Therefore we may expect ea to be the best performer and ec the worst. What is the expectation of these estimators? We begin with the bias by ignoring the fractional displacement for a moment. Therefore R-\ = R+i = ai-Ro where 0 < a\ < 1 and a\ is proportional to the width of the correlation peak. Westerweel17 shows ai

l + 2ai

m=

for ec

a\ for ep 4(1 - d )

AT-TOQ

(13)

1 for ec I 41n(l/ f l l ) Clearly the terms on the right hand side of the bracket are simply constants, but note that the first term is a function of the displacement mo (the integer pixel displacement). This is a negative bias, resulting from the probability of smaller velocities leading to higher pairing. If PIV is to operate in an unbiased manner this bias must be corrected. The correction is: R*{s,t) =

R(s,t) Fi(s,t)

(14)

where Fi

Hi N

1*1

(15)

N

We note that this could have been predicted looking at equation (3). We now consider the fractional part of the displacement. A particle will have a small imbalance in i?_i and -R+i; therefore we will model the particle image as: R.

,1+e

Ro = a\ and _R+i

,1-e

(16)


20

Note that this is not Gaussian. Westerweel17 shows that in this case sinh(elnai) e c

=

W~i

\ i i €-1'

(n>

cosh(emai) + | a a sinh(elnai)/2 piling m a i ^ / z , cosh(elnai) — a\~

«-5^-

(19)

The Gaussian sub-pixel fit is independent of a\, but COM and parabolic are functions of a±. If we look at the RMS error, f_1,2(error)2de, we find that the Gaussian performs the best under all particle diameters. We note that other peak fitting functions have been applied in the literature, for example various versions of spline-curve fits 23 . These, however, have the drawback that they are considerably more computationally intensive without significant gain in accuracy. As will be discussed in the next section, there are alternative methods to improve the sub-pixel fit. 4.5. Peak-Locking 17

and

Solutions

Westerweel has argued that the requisite bandwidth for a PIV image is constrained only by the need to locate the correlation peak. Hence, if this bandwidth criteria is met, we should be able to locate the correlation peak to subpixel accuracy in an unbiased manner. However, as demonstrated in section 4.4, we find in general a strong bias of subpixel fit estimators toward integer pixel locations. Looking at figure 6 we note that for actual subpixel displacements of—0.5 < e < 0 the bias error is positive - hence the determined displacement is biased towards a subpixel displacement of zero. We see the same effect for 0 < e < 0.5, namely the bias error is negative resulting again in a bias toward zero pixel displacements. This effect is often referred to as peak-locking as correlation peaks have a tendency to 'lock' onto integer-pixel displacements. The easiest way to identify the presence of peak-locking in a data set is to plot the histogram of the un-calibrated displacements (i.e., the displacements in pixels) as shown in figure 7. Depending on the quantity of interest peak-locking may not be a significant problem. For example, if one is interested in the mean velocity or even the variance, peak-locking contributes little to the error as long as the histogram spans at least two integer pixels of displacement. If the probability density function (PDF) of the underlying velocity field is of known shape then the histogram could be corrected by redistributing the displacements to have the assumed form of the PDF.

Quantitative


to Wavy Flows

21

ccnter-of-itiosE parabolic fit Gaumian fit

I

i

i

i

i

L.

-^

Fig. 6. Sub-pixel bias. Comparison of different sub-pixel peak fits. Figure reprinted with permission from Westerweel 17 .

However, if the displacements range over less than two integer pixels the possibility of a strong bias in all calculated statistics exists. Further, if one is interested in derivative quantities (e.g., vorticity, acceleration) or other quantities involving the difference of two displacement calculations (e.g., turbulent structural functions) then errors induced by peak-locking effects may be considerable and must be monitored. There has been research on minimizing or even eliminating the effects of peak-locking. As there is no peak-locking effect if the subpixel displacement is truly zero the majority of efforts have focused on iterative solutions where the image subwindows are displaced dynamically in fractional steps (a process that is known as continuous window shifting as opposed to the more conventional discrete window shifting where subwindows are shifted an integer number of pixels). Continuous window shifting is generally achieved by interpolating the original image at subpixel locations. The idea is to dynamically shift the subwindow(s) until the subpixel displacement is identically zero. Gui and Wereley35 have found that continuous

22


2x10"

1.8 1.8 1.4 1.2

vector 1 count 0.8 0.6 0.4 0.2 0.

u (pixels/cm) Fig. 7.

Example of velocity histogram showing the peaklocking effect.

window shifting using simple bilinear interpolation and as few as four iterations provides a significant minimization of peak-locking effects. Fincham and Delerce29 employ a slightly different approach to continuous window shifting as they use a spline-thin-shell interpolator to analytically model the first image subwindow and then utilize a PIV result already calculated for the image pair to translate and deform this analytic function based on the velocity field estimate. This analytic function is then compared to the subwindow in the second image in a least-squares sense and iteratively adjusted to minimize the difference. They demonstrate that the final result reduces peak locking and gradient bias while enhancing the robustness of PIV in the presence of locally large shear. It must be pointed out that their approach is computational expensive without any obvious improvement relative to simple bilinear interpolation. As an alternative to interpolation in the spatial domain, Liao and Cowen36 have shown that interpolations for continuous window shifting can be performed more efficiently in the spectral domain taking advantage of the shift property of the Fourier transform. They demonstrate the elimination of peak locking effects at minimal computational cost, given the Nyquist particle image sampling criteria is met. We also note that bi-linear subpixel interpolation has been used earlier in other contexts of pattern matching which we will return to in section 7.2. Finally we note that the solutions to peak-locking problems are closely

Quantitative


to Wavy Flows

23

connected to the evaluation methods in question, and readers may find several relevant publications cited in section 4.7.3. In one of the more recent works, Scarano and Riethmuller 30 give a general review and comparison of several iterative PIV methods. 4.6. Spurious

Vector

Detection

There are three fundamental approaches to test the validity of an individual vector: signal quality, spatial consistency (smoothness in space), and temporal consistency (smoothness in time). Signal quality based tests look at a specific property of the images, such as the magnitude of the correlation peak relative to the magnitude of the background noise in the correlation plane, to test the validity of a vector. Spatial consistency tests compare each vector with certain properties calculated from the local neighborhood of vectors (local filtering), or the entire vector field (global filtering). Temporal consistency tests are similar to spatial consistency tests with the exception that the vector whose quality is being tested is compared to either a statistic developed locally in time (e.g., acceleration) or globally in time (e.g., its distance from the mean relative to the standard deviation 22 ). Filtering using spatial statistics is the standard approach, however, and has the added advantage that it works on individual instantaneous vector fields. That said, the recent advances in camera speed have lead to temporally resolved data. As single-point measurement techniques can only rely on the temporal history of data to remove spurious points there is a well developed literature on the removal of spurious data points from temporal records (e.g., spike detection in acoustic Doppler velocimetry, which is essentially a local gradient (the acceleration) threshold in the time domain). The time domain is a relatively unexploited method for spurious vector detection in PIV which should be developed. The detection of spurious vectors is an essential part of any PIV measurement, and the topic is well covered in the literature 17 ' 37 ' 38 . Below is a review of the fundamental approaches currently in use. 4.6.1. Detectability It would be nice if we could simply look at a correlation-peak (PIV) and decide, based on some criteria, whether or not we think it is valid. Keane and Adrian 39 proposed the detectability, DQ, as such a measure. They define the detectability as the ratio of the peak value of the first correlation peak to the second and suggest that if DQ is greater than 1.2 — 1.5 that the vector is valid. The detectability criterion is often referred to as signal-to-

24


Cowen

noise ratio (snr) filtering. Westerweel37 demonstrates that any of the three methods based on the statistics of the measured vectors in a field perform better. These methods are known as global mean, local mean and local median filters.

4.6.2. The Global Mean Often during PIV measurements we observe that the data contains a few vectors that are very different from the whole ensemble of vectors. These vectors are typically the result of either i?_i or R+\ being close to zero, see equation (12). If Vij is the observed vector at i, j then the global mean v = jj J2r s Vr,sThe variance of this statistics will be a^ +cr^ where a% is the variance of the actual velocity field and a\ is the variance of the error. So, one possibility is to try to select the allowable variance of the signal and remove outliers (invalid vectors). However, of -C a% in practice so this is hard, hence, a user determined threshold is generally applied.

4.6.3. The Local Mean Let us now extend this idea by looking at individual vectors and comparing them with their closest neighbors. As opposed to looking at individual vectors compared to the global properties, this filter should be able to remove local "large" vector to vector variations. To do this we evaluate

NM

kJ^M

typically using a 3 x 3 region (NM = 8) around each vector, where M is the set of points in the neighborhood. The common approach is now to say that a vector is invalid if it is "very" different from the mean of its neighbors (for example twice as large). We immediately notice that in the cases where the neighborhood actually contains one or more outliers, our criterion for filtering is very difficult to determine, and actually spurious vectors are smoothed out with this method. It turns out we need to know a priori which vectors are bad to make reasonable decision criteria. If we can, perhaps using global mean, this performs fairly well.

Quantitative


to Wavy Flows

25

4.6.4. The Local Median The obvious extension of the local filter is to use the local median value instead of the mean. The median value is the nth value in a 2n + 1 long sequence (data sorted in ascending order) and is well known to be robust to outliers for estimating the mean of data. It should be relatively obvious that spurious data normally will be sorted to either side in this sequence. The advantage is therefore that the median is much more robust to outliers and eliminates the need for a priori information on which vectors are bad. Westerweel37 finds that this performs the best. 4.6.5. Adaptive Gaussian Filtering (Temporal Filtering) Given the underlying stochastic nature of turbulence it is possible to exploit knowledge of the PDF of the velocity component statistics to remove spurious vectors. Cowen and Monismith 22 demonstrate such a technique for both PIV and PIV stray vector removal. The idea is to choose a robust estimator of the center of the distribution (accumulated in time as the temporal ensemble at a point), such as the median, and then iteratively test the data for lying within the assumed bounds of the PDF (in the case of Cowen and Monismith, a Gaussian) given the number of data points in the ensemble. The data that is retained is used to recalculate the center of the data, where the mean may be used after the first iteration in general, and the retained data is again tested for lying within the bounds of the assumed PDF. This processes is iterated until no data is removed and hence is an adaptive filter. 4.6.6. Filtering Example Fig. 8 shows a velocity field as measured by conventional FFT-based PIV and filtered using a global filter, a local median filter and a signal-to-noise ratio filter. The images in question are identical to the images shown in Fig. 2 and Fig. 11. Fig. 8a-c shows a part of the velocity field as measured from the example images. The three sub-panels depict the velocity field after first a snr-filter has been applied, followed subsequently by a global filter and a local filter. First the vector field has been filtered by a signal-to-noise ratio filter, discarding all vectors where the ratio of the tallest peak in the correlation plane to the second tallest is less than 1.3. Secondly the remaining vectors are filtered using a global filter with the criterion defined as Vij sg mean(v r>s ) ± k • std(iv iS ),

(20)

26


Cowen

where A; is a constant denned by the user (in the present example k = 3) and r = 1,...,R and s = 1 , . . . , S and (R, S) is the number of velocity vectors. Finally a local median filter is applied, and in this case the criterion used for validation is v(i,j) ^ median[i;(r, s)] ± k • std[v(r, s)],

(21)

where again A; is a user defined threshold (in the present example k = 3) and r — i — l,i,i+ 1 and s = j — l,j,j + 1 are the indexes of the 3 x 3 neighboring vectors.

4.6.7. Correlation Based Correction Another approach to vector validation was suggested by Hart 40 , and in fact his proposal is more a general processing technique than a filtering technique. The idea is to take a correlation plane and compare with one or more adjacent correlation planes. The comparison is performed via an element by element multiplication, and thus represents a zero-dimensional correlation (therefore the name "Correlation Based Correction" or CBC). The point here is that each correlation plane typically will consist of a peak that signifies the displacement plus several additional peaks which are due to noise. The noise-peaks are randomly distributed in the correlation plane, but the displacement peaks are not. Therefore the multiplication of two or more correlation planes cancels the noise peaks, while the displacement peak remains. The principle is shown in Fig. 9. In this way we are able to calculate the displacement of a vector centered between the adjacent regions used in the calculation. There are basically two ways to use CBC processing. The first is to use the signal to noise ratio in the combined correlation plane as an indicator, discarding vectors where this ratio is smaller than a certain threshold. The second method, suggested by Hart 40 , is to compare the peak in the combined correlation plane with the peaks in each of the correlation planes that produced it. If the peak in the combined plane exists as a peak in at least one of the other planes, it is assumed to represent the local displacement. Hart 40 concludes that the method improves sub-pixel accuracy and eliminates spurious vectors, reduces bias errors and improves vector yields. However, he provides no evidence of the reduction in bias errors with regard to peak locking and, in fact, tests performed in the preparation of this manuscript suggest that CBC processing may be sensitive to peak locking.

Quantitative

a)


to Wavy Flows

27

100 P5

_~

150 200

—^—^"V^-

s>£ .

:_;_:.

" z-~—Z-~.

_v:

300 350 400 450 500

550

600

650

700

750

800

850

900

950

1000

800

850

900

950

1000

800

850

900

950

1000

i b)

100 150 200 250

J 300 350

450 t 500

550

600

650

700

750

i

c) 1°° 150 200 250 3

300 350

500 500

550

600

650

700

750

i Fig. 8. Example of PIV velocity filtering, a) Velocity-field - black vectors removed by the snr-filter. Gray vectors are the remaining, b) shows the results after the global filter gray, valid, black, outliers, c) black vectors, outliers identified by the local median filter.

4.7. Alternative

Displacement

Estimation

Approaches

We have until now only discussed the "original" approach to PIV, namely by evaluation of the cross-correlation function via the use of FFTs. Further-


28

640

660

680

700

Cowen

640

660

680

700

Fig. 9. a) ana b), soiici square corresponds to panel c), dash-dot led sûaAc U> panel d) and dashed square to panel e). Multiplication of correlation tables c), d) and e) produces the combined correlation plane f).

more we have seen that in order to normalize the correlation function, we must divide it by the covariance function. The latter is hard to implement using FFTs and therefore a first order approximation is applied by dividing the correlation plane with the standard deviations of the original images. in the following sections we will review a few different approaches that avoid the use of FFTs. Typically their downside is the additional computational time needed, but on the other hand often at improved accuracy. The cross-correlation function is used in many applications for pattern matching. By definition it is a measure of the "likeness" of two images (or subwindows). In the cases where there is a certain degree of match between the images, it will produce a plane which will contain a peak at the position of the highest match. In contrast we could have chosen to look for the position where the images are the least unlike. This can be done by looking at the minimum of the squared difference between the images. 4.7.1. MQD The use of the minimum squared distance (MQD) for PIV evaluations was first proposed by Gui and Merzkirch 41 and subsequently expanded in Gui

Quantitative


to Wavy Flows

29

and Merzkirch 42 . The MQD method evaluates the function M-1N-1

where the location of the minimum value of R indicates the displacement between the windows. The method is reported to increase the accuracy compared to the cross-correlation function, and it is claimed that the reason for this is that "... the MQD method contains a term which accounts for a non-uniform distribution of the particle images and for a non-uniformly distributed illumination intensity...". If we turn to the basic mathematics of the method 43 , it should be fairly obvious that M-1N-1

E T,lF'(hJ)-F"(i + s, j + t)}2 = i=0 j = 0 M-liV-1

E

E [*"(«'J')' " *F'&3)F"{i + s,j + t) + F"(i + s,j+1)\

i=0

j=0

(22)

which in turn means that the MQD method equals the (zeroth order) autocorrelation of the first (sub-)image, plus the autocorrelation of the second (sub-)image (or a part of it, that is), minus two times the cross-correlation of image one and two. If the images now contain a nonuniform seeding or illumination, this will show up in both the auto-correlations and in the crosscorrelation, but will cancel each other out in the MQD function, roughly speaking. If, on the other hand, the images are uniformly seeded and the illumination is uniform, there is no need to evaluate the full MQD-function. The evaluation of the auto-correlation function can be understood as a measure of the non-uniformities within the overlapping regions. As an example we may consider each interrogation region as a wave-field, where the particles show up as small waves. A non-uniformity in the seeding density, or a background illumination gradient, will appear as a lower frequency wave in this field. Depending on the amplitude of this lower frequency, our PIV measurement may in some cases be overshadowed by it. We may argue that the comparison between MQD and cross-correlation is a little unfair since the MQD plane is not normalized. A proper normalization of the cross-correlation plane can only be achieved in the spatial domain, and this is the basis of the next evaluation method we shall look at.

30


Cowen

4.7.2. CIV The cross-correlation function, as defined in equation (1), may be used for pattern matching, but the reader may notice that the product will be sensitive to changes in the amplitude in the image matrices. If we double the values in one of the images, the correlation values will double. For this reason the function known as the correlation coefficient (also termed the covariance function earlier in this paper), defined as J?(.] { , )

1 S , S j [F'(i, 3) ~ F>) [F"(i + s,j+t)F"\ 2 2 N Ei Ej([F'(i,J) ~ F'] [F"(i + s,j + t)- F"P)i/2

^ '

12

is often used in pattern matching (with M = N). Here F" is the average value of image 2, which is evaluated only once, and F" is the mean value of image 1 which needs to be evaluated for every (s, t). As we already have seen, the denominator may be approximated by iV 2 std(F')std(F") in the cases where we have uniform illumination or seeding. If this is not the case, the standard deviation of the region in image 1, which overlaps image 2, will actually change as the values of (s, t) change. The method of direct calculation of correlations in PIV, has been suggested by a few authors, most notably Huang et al.27, but also Fincham and Spedding 23 . The latter authors suggest that there should be two main reasons for calculating the correlations in the spatial domain. Firstly they point out the argument about normalization that we have already presented. Secondly they claim that the use of FFTs place a limitation on the size of the interrogation windows. Traditionally interrogation windows needed to be of size 2™, but recent developments in FFTs have made this demand less stringent 44 . We note that the benefits of calculating the full correlation coefficient is well known within the image processing community 12 ' 43 and this fact can be found in many introductory level books on image processing. 4.7.3. PIP-Matching and PID The concept of Particle Image Pattern (PIP) was first introduced by Huang et al.27 and is basically a direct calculation of the full correlation coefficient, see equation (23). The authors vary the interrogation window sizes in order to maintain all particles in both frames. The works of Huang et al.27'28 also contain an approach to dealing with large gradients in flows, known as particle image deformation, or PID. Large gradients may lead to interrogation regions with a spatial variation in velocities, again leading to loss of correlation. To adjust for this they

Quantitative


to Wavy Flows

31

distort the interrogation windows in an iterative manner where the rotation and shear is calculated from the velocity field and subsequently used to distort the images. These distorted windows are then used to calculate a new velocity, and the process is iterated upon until some predefined criteria is met. The window distortion is calculated from the shear components (du/dy, —dv/dx) and strain components (du/dx, dv/dy), neglecting the translation. A schematic drawing is shown in Fig. 10.

Fig. 10.

Schematic drawing of the interrogation window distortion in the PID method.

Scarano and Riethmuller 30 extended the PID method to also use interrogation window offsets, as well as distortion (Originally the PID method used larger interrogation regions in the second window to maintain a high correlation). Additionally they apply progressive grid refinements to maximize the spatial resolution, and in this respect they combine the results of Huang et al.27'28 with those of Westerweel et al.25 They report a decreased sensitivity with respect to peak-locking effects. Several authors 46 ' 47 ' 48 ' 49 have applied the PID concept or similar approaches and readers are advised to study these works individually. For the present review, the results by Lin and Perlin 47 are of particular interest since it deals with PIV measurements in water waves. More specifically they combine the PID methods with a particle tracking approach 50 in order to measure velocities in the boundary layer of gravity-capillary waves. 4.7.4. Other Methods for Displacement

Estimation

Some researchers have suggested the use of the phase correlation functions for pattern matching. This method is detailed in Kuglin and Hines 51 , and it is based on using the phase information in the images rather than the amplitude information. Finally we would like to mention that within the field of Optic (or Opti-

32


cal) Flow, another approach to motion estimation is often used. Under the assumption that the image intensity, E, is conserved and that the displacements between frames are small, one may solve Et(x,y,t)+v-VE(x,y,t) = 0,

(24)

where subscript denotes partial differentiation with respect to time, v is the velocity vector and V is the gradient operator. Interested readers may find a good starting point in the paper by Beghdadi et al.52 4.8. Comments Correlation

on the Normalization

of the

Cross

In the cases where the interrogation windows contain an unevenly distributed pattern or where we perhaps have non-uniformities in our illumination, the cross correlation may contain errors. In these cases the calculation of the full correlation coefficient may be more accurate. Since this is a time consuming task to calculate, an approximation is often used for the denominator. In PIV we normally use the standard deviations of the two sub-windows. For example in wavy flows, we often find a thick band of light at the fluid surface, where the light sheet leaves the fluid. For demonstrational purposes we will consider the work by Jensen et al.2, where errors in the PIV calculations are observed close to the free surface. Two images from those experiments are investigated (figure 4 b in that paper), and we focus our attention at this problematic region. Fig. l l a - b shows the original images taken with a time separation of At = 0.0012s. Fig. l l c - d shows two corresponding 64 x 64 pixel subregions, close to the free surface. The reader may notice that there is a gradient in the background illumination in these two sub-windows. Correlation via FFT produces the correlation plane shown in Fig. l i e and, although difficult to spot visually, the central peak is actually higher than the true displacement peak. The central peak corresponds to a correlation of the gradient in the background illumination within the sub-windows, and since this does not move between exposures, it shows up as a zero-displacement peak. If we, on the other hand, use direct calculation of the cross-correlation function, we get a correlation plane as shown in Fig. l i e . Here the true correlation peak is slightly taller than the false (zero-displacement) peak. We observe that the full correlation calculation is less sensitive to the background illumination gradient. This brief example has an interesting aspect if we compare with the results from the MQD algorithm. Fig. 12 shows the corresponding plane as calculated using the MQD method. Interestingly we can observe that due to the nature of

Quantitative


to Wavy Flows

33

the algorithm, the false correlation peak has opposite sign of the true displacement peak and the true displacement may easily be identified as the minimum value (black color) in this figure. This indicates that the MQD method may be favorable in many cases where the image quality is not ideal. In other areas of pattern matching several approaches for calculating the normalized correlation plane are used. For example one method was developed for the motion picture Forest Gump43, which relies on using precomputed integrals of the image and image energy over the interrogation region.

5. P T V Particle Tracking Velocimetry is probably the oldest of the particle based QI techniques. In its simplest form it can be thought of as low particle image density PIV, where low particle image density means that the concentration of tracers is sufficiently low that the maximum displacement of a particle in some time, At, will always be less than the mean nearest neighbour spacing between particle images. In PTV's early incarnations particle images were manually digitized and tracked, but with the aid of computers the process is now automated. However, due to the difficulty in determining a particle's pair when the particle's mean spacing is on the order of, or less than the maximum expected displacement, the technique has remained, until the 1990's 22 ' 50 ' 53 ' 54 , a low seeding density technique. That decade saw multiple researchers working with a variety of computer algorithms for particle pairing - the process of identifying which particle image, in any particular image, resulted from the identical particle that produced a particle image in one or more other images (singly exposed multiple images) or a different particle image in the same image (multiply exposed single images). During the 1990's the restriction on displacements relative to the mean particle separation distance was relaxed. The specific seeding density limit depends on the physical parameters of the experiment and the procedure used for particle pairing, and ultimately Cowen and Monismith 22 demonstrated that the limit on seeding density was essentially set only by the requirement of having the majority of the particles produce non-overlapping images. While the low seeding density restriction on PTV was an early criticism of the technique, it was overcome. A second significant criticism of PTV is the resultant randomly distributed velocity vectors, which is still an important area of research. The traditional solution to dealing with randomly located PTV data is to interpolate it onto a regular grid. This interpola-


:u

400

800

Cowen

1280

True correlation peak False correlation peak Fig. 11. a)—b) original images, black squares indicate the position of subwindows shown in c)—d). e) FFT-based correlation plane, f) Direct calculation of cross correlation.

Quantitative


20

40

60

80

100

to Wavy Flows

35

120

Fig. 12. Figure showing result of MQD-interrogation of the subwindows in Fig. 11 c)-d). Note t h a t the displacement is found as the minimum value (black color) in this plane.

tion has an associated error, that while not a major problem relative to the mean velocity field, is problematic when considering the variance and the co-variance as well as the calculation of any velocity gradients. Several research efforts at determining optimal interpolation methods exist (e.g., Agui and Jiminez 57 , Spedding and Rignot 58 ) and Cowen and Monismith 22 demonstrate the success of a zeroth order interpolation scheme - simple data binning over small spatial regions. While the above criticism is generally restricted to PTV, the reality is that the majority of PIV techniques produce quasi-randomly distributed velocity vectors due to the perturbation to the position of the PIV interrogation grid as a result of using first-order displacement differences to estimate the velocity. As discussed, Wereley and Meinhart 26 have resolved this issue through the use of second-order accurate displacement differences that are centered on the interrogation grid node. It should be noted that in the limit of small subwindow size PIV converges to PTV which points to a remaining perturbation to the location of the second-order accurate displacement differences proposed by Wereley and Meinhart - the true location of the estimated average displacement between two subwindows must account for the location of particles in each subwindow (e.g., a particle weighting of the information in each subwindow) and as the subwindow gets small the probability that the center-of-mass of the particle location information is not at the center of the subwindow, as implicitly assumed in all subwindow techniques, increases.

36


Cowen

The fundamental PTV approach to the determination of the velocity is to measure the displacement of an individual particle, located at a random point in space and time, either in doubly-exposed single images or successive singly-exposed images. The velocity is calculated as: U

^'V' '

=

Ax(x,y,t) At

^ ^

Clearly in the limit, when At —> 0, we recover the fundamental definition of velocity. However, we must keep in mind that we are approximating the Eulerian velocity as the second-order difference (or in some cases even the first-order difference) of the Lagrangian track of a particle. In fact, it is important to remember that PTV essentially is a Lagrangian measurement technique. Dalziel 53 ' 54 developed an effective and relatively successful particle tracking algorithm, which was commercialized through the code Dig/mage and applied in many publications 59,60 ' 61 . In the Diglmage code, for example, the Lagrangian velocity is calculated by a least squares fit to more than two points in the particle path. By fitting a quadratic function we can obtain the Lagrangian acceleration directly, a concept also utilized by Chang et al.62 to obtain acceleration from a single-camera configuration.

5.1. Particle

Detection

Algorithms

All PTV based techniques employ some form of particle detection algorithm. This usually involves thresholding the image, leading to a binary image. The threshold may be set manually or determined automatically. In either case the ideal image will have a histogram that looks like the one shown in Fig. 13. If in fact two separate peaks exist (one for the background and one for the particles) it is a simple process to pick a threshold value between the two peaks. However, as discussed shortly, it is more likely that the histogram will decay monotonically (a negative exponential probability density function), making the threshold choice somewhat unclear. In the latter case a user defined threshold is often the simplest solution. Preconditioning of the images may, generally speaking, serve to enhance the particle images from the background. Such preprocessing may be accomplished by convolving the images with a top-hat function, which will serve to amplify the particle images. Subsequently thresholding is applied, followed by binarization. Once an image has been binarized, particles are generally defined in one of two ways, either as horizontally and vertically adjacent "hot" pixels or as horizontally, vertically and diagonally adjacent "hot" pixels.

Quantitative


to Wavy Flows

37

m

su 60

count

40 30

20

to

0

••———

0

—

30

iMllM^^tetfM^tf^

™—

100

— —~—™

™»™™™»™™™™™™™,™™™.

150

200

250

Fixe] values, px Fig. 13.

Histogram of image intensities. Notice the small peak between 20 < px < 40.

Particle tracking may occur over multiple exposed or sequential singly pulsed images. Multiply exposed particle image based techniques can lead to higher accuracy since the probability of incorrectly drawing a vector decreases with each additional exposure. However, these techniques suffer from lower vector density as the probability of tracers remaining in an illuminated region is reduced the longer they are tracked. Multiply exposed image based techniques- can also be used to make higher order estimates of the velocity field (by retaining higher order terms in the Taylor Series expansion implied by equation 25) and the estimate of the acceleration field, which will be discussed further in section 6.4. 5.2. Locating

a Particle's

Center

Spherical particles tend to scatter light with an intensity pattern given by a rotated Airy wave. The central lobe of an Airy wave is well approximated as Gaussian. As discussed in section 4.4 as long as the particle image diameter is small ( 2 - 4 pixels) then a three-point Gaussian sub-pixel estimate of the particle's location is optimal. Unlike correlation peaks, however, we have no guarantee that the particle image diameter will be greater than one pixel. It is important in implementing the image acquisition and particle seeding that proper choices are made to insure that particle images will have diameters greater than one pixel and preferably less than four pixels 22 . Further, the same considerations about peak-locking as discussed in section 4.5 exist. In many cases, however, particle images may overlap (or nearly so) and in these cases the Gaussian fit may not work as well. In this case


38

Cowen

the particle image intensity centroid may be used as an estimate of the particle's position. 5.3. Particle

Pairing

After the individual particles are located at two different time steps, usually the next step will be to perform a pairing or matching of them. That is, we want to know which particle image in the second frame results from a particular particle that was imaged in the first frame. Hybrid PTV techniques have been developed that employ PIV based algorithms as predictors of the velocity field allowing particle pairing searches to be conducted in small search regions relative to the region to where a particle might have moved 22 ' 50 . This predictor approach overcomes the low seeding density limitation on PTV allowing particles to be tracked in flows where the mean spacing between particles is considerably less than the expected maximum displacements. Ultimately it must be remembered that these PIV predictor techniques suffer from the limitations of the PIV algorithms used to develop the predictive velocity field. The algorithm in Dig/mage 53 ' 54 is based on Operations Research. Particle pairing is accomplished via the use of a cost function. This cost function is dependent on individual particle factors, such as size, intensity, velocity history and shape. Particles are paired after a minimization process, where the cost of pairing particles in one frame with the particles in the next is minimized. The algorithm in Diglmage is similar to the so called transportation algorithm, although not quite the same since particles are allowed to leave and re-enter the light-sheet. There are many other algorithms that have been proposed for particle tracking that include fuzzy logic55 and simulated annealing 56 . An intriguing capability of several of these techniques is their ability to extract higherorder flow quantities, such as the local strain rate and rotation rate, directly from the analysis. 6. Higher Order Measurements from Velocity Fields Obtained by QI-Techniques In a great number of applications of Ql-techniques the velocity field information is only part of a complete description of the physical processes under study. For example many flows of interest are turbulent and derivative quantities of the velocity field may be of fundamental importance to describe the turbulence. In both turbulent and viscous flows the vorticity field is often an important descriptor of the flow physics. In stratified flows

Quantitative


to Wavy Flows

39

it may be necessary to simultaneously measure the density field. In wavestructure interaction flows knowledge of the acceleration or pressure field may be of vital interest. In the present chapter we will turn our attention to the use of the velocity fields, obtained from Ql-imaging techniques, to determine higher order quantities. Such quantities may include shear, strain, vorticity, streamlines, acceleration, pressure and estimates of turbulence characteristics (e.g., the Reynolds stresses, turbulence dissipation, and spectra). 6.1. Vorticity,

Strain Rate and

Divergence

The calculation of vorticity, strain rate and divergence is often referred to as the calculation of differential quantities. These quantities are defined as dv

du

1 fdv

s

= A^

.„ . du\

+

-dy}

_ .

(27)

dw _ dv du dz dy dx' where u>, s and —dw/dz denote the (in-plane) vorticity, strain rate and two-dimensional divergence, respectively. The differentiation of the velocity is a straightforward calculation, using for example a second-order accurate central difference numerical scheme. However, QI velocity measurements contain errors which can lead to strong noise in the differential quantity. Furthermore the data is discrete, which also may introduce noise. In these cases many scientists apply higher order numerical schemes 63 , others apply smoothing 64 functions to deal with this noise, and Cowen and Monismith 22 argue for the careful choice of the differentiation length scale to maximize resolution and minimize noise amplification. Readers are advised to consult Raffel et al.lg, Nogueira et al.38 and Lourengo and Krothapalli 65 for further details. We also note that it is possible to measure differential properties directly from the images using image distortion techniques 66 ' 67 and other non-correlation based displacement extraction techniques 56 . 6.2. Streamlines

and Pressure

Estimation

Defining the streamfunction, ip, can be very useful in two-dimensional applications. The streamfunction is constant along a streamline, where the streamline is tangent to the velocity field at every point.

40


Cowen

Consider a two-dimensional incompressible flow, thus V • v = 0. The streamfunction can be obtained from dtp = u dy — v dx where u = dip/dy and v = —dtp/dx. This implies that the unknown scalar function tp at the location P is defined as

fP i>p = ipo + /

{udy — vdx),

(29)

Jo where tpo is the streamfunction at the origin O (typically set to zero 68 ). The path of the numerical integration can be freely chosen, but in order to minimize noise accumulation, the integrations are often performed on a staggered grid starting from the center of the velocity field and proceeding out towards the edges 68 . In order to estimate the pressure gradient we have to know the acceleration field (see section 6.4). If we make the assumptions that the flow is twodimensional, steady and incompressible, it is possible to estimate the pressure field from the 2D Navier-Stokes equations 68 or similar approaches 69 using the measured velocity field. 6.3.

Turbulence

As many fluid flows of interest are turbulent it is not surprising that QI techniques have a development history that reflects a desire to extract information on the turbulent statistics and turbulent structures of the flows. The application of QI techniques to turbulent flows presents unique challenges due to the range in space and time scales, which leads not only to a large requisit dynamic range in velocity to characterize the flow, but also the potential for strong strain rates, which historically are problematic for traditional PIV type techniques (e.g., 22 ). 6.3.1. Dissipation The determination of the dissipation of turbulence, e, is an important but challenging experimental goal. In theory it should be particularly straightforward using QI techniques based on its definition e = 2i/(sijSij), where Sij is the fluctuating strain rate tensor defined as

(30)

Quantitative


to Wavy Flows

41

Hence, the in-plane fluctuating strain rate components can be directly calculated by second-order central differencing and the out-of-plane fluctuating strain rate components can be modeled by an understanding of the anisotropy ratio to the in-plane components 22 . However, presumably due to the noise level present in many velocity field calculations, relatively few researchers have reported dissipation results. Cowen and Monismith 22 report a dynamic algorithm that adjusts the length scale over which the velocity data is differentiated, arguing that the signal-to-noise ratio can be maximized by optimally choosing the length scale of the gradient calculation. Doron et al.74 discuss optimal assumptions about the anisotropy of the out-of-plane gradients impact on the total dissipation. Cowen et al.71 demonstrate several impacts of understating the dissipation structure in the swash zone. They show that the dissipation is in balance with the turbulent kinetic energy and that the turbulence is decaying as free turbulence during the swash uprush. They use the dissipation to estimate the friction velocity and the friction coefficient of the bed. The single point measurement fields with sufficient temporal resolution to capture temporal frequency based spectra (whose Ql-based calculation is described in the next section) have a well developed history of using the inertial subrange spectral energy measurements to estimate the turbulence dissipation. For temporally resolved QI measurements this is another approach to estimating the dissipation. For an example see Liao and Cowen75 and for a discussion of the errors with respect to direct measurements see Doron et al.74

6.3.2. Temporal and Spatial Spectra In unsteady flows in general and turbulent flows in particular, it is often of interest to look at flow statistics in the spectral domain. For single-point measurements this is only possible in the temporal frequency based sense. However, a significant advantage of QI techniques is the ability to determine instantaneous spatial information and hence spatial frequency based spectra. With typical QI hardware, say 30 frame per second cameras with 1024 x 1024 pixel imagers, we often find ourselves in the situation where the high spectral frequency content can only be determined in the spatial domain while the low spectral frequency content must be determined in the temporal domain. Typically a characteristic advective velocity is used to make the Galilean transformation from time to space and the temporal and spatial spectra are combined. In turbulent flows this transformation is known as Taylor's frozen turbulence hypothesis 70 and the characteristic

42


Cowen

velocity scale is simply the mean velocity. However, in turbulent flows one must be careful as Taylor's hypothesis requires (u' ) 1//2

•:i.

o r! CO

V " = T

i"

•

fi

•'.

m

0 +s T) CD r

••H

+3 CS

+3

=O : ' ft /: 8

-M>

siti

." :.:

1 00

bb

s .3

ovi

PIV Measurements

in the Bottom Boundary

Layer of the Coastal Ocean

65

tally friendly, food-grade hydraulic oil. A remotely controlled manifold at the top of the cylinder was used to switch operation between the hydraulic turntable motor and the cylinder, reducing the required number of lines linking the platform to the surface. The submersible system also contains a Sea-Bird Electronics, SeaCat 19-03 CTD, optical transmission and dissolved oxygen content sensors, a ParoScientific Digiquartz, Model 6100A, precision pressure transducer (for measuring surface waves), an Applied Geomechanics, Model 900a biaxial clinometer, and a KVH C100 digital compass.

3. D e p l o y m e n t s The initial measurements using the original, single camera PIV system were performed in the New York Bight near the Mud Dump Site, 7 miles east of Sandy Hook, New Jersey, in June 199821. Data were collected at six different elevations ranging from 10cm to about 1.4m above the sea floor. Limited by storage capacity at that time, the data for each elevation consisted of 130s of image pairs recorded at 1Hz. The experiments using the two-camera PIV system took place during May, 2000 and September, 2001 at two sites on the east coast of the USA, one in the vicinity of the Longterm Ecosystem Observatory (LEO15), about 5km off the New Jersey coast, the other inside the wall of the harbor of refuge near the mouth of Delaware Bay 22 . The system was deployed from the RV Cape Henlopen (one of the UNOLS ships managed by the University of Delaware). As in the initial deployments, the ship was held in a fixed position using a three point mooring system. A ship-board ADCP provided us with data on the mean velocity distribution in the water column for the entire duration of the tests. We also acquired water column profiles of temperature and salinity and collected samples of particles at different elevations and from the seabed itself. During May, 2000 we recorded three data sets, two at the first site and one at the second. The exact position of the first site was 39°27'00"N and 74°14'15" W, i.e. about 0.5 nautical miles to the south-east of Node B (39°27'25"N, 74°14'45"W), and about 1.25 nautical miles to the south-east of Node A (39°27'41"N, 74°15'43"W) of LEO-15. The seabed consists of coarse sand ripples of approximately 50cm wavelength and 10cm height. The currents in the region are generally moderate, however the site is exposed to the oceanic swell. On the night of May 15-16 we recorded data at 0.5Hz for 9hrs continuously. On the night of May 16-17 we recorded data at 0.5Hz for 13hrs continuously. Thus, we obtained PIV images that will

66

W. A. M. Nimmo Smith,

T. R. Osborn and J. Katz

enable us to study variations in flow structure and turbulence during an entire tidal cycle. During these tests the cameras were configured as illustrated in Fig. 3. Both were focused on the same vertical plane, had the same field of view (51cm), and their centers were located lm apart. The data was acquired at three different elevations, 9.5-60.5cm, 64-115cm and 118.5-169.5cm above the bottom. At each elevation we aligned the system to the mean flow direction (using the video images of the vane) and then acquired data, typically for thirty minutes, i.e. 2x900, 2Kx2K, 12bit PIV images. The elevation was then changed and the next image set was acquired. Since the flow direction changed with time and with elevation we had to re-align the platform for each data set. In a few cases the flow direction changed significantly during the run and we had to stop it early. Samples of the raw image data were examined at the end of each data series to check the data quality and if necessary the laser heads were re-aligned. On the night of May 19-20, 2000 we acquired a data set behind the wall of the harbor of refuge of Delaware Bay. Here, the flow is characterized by strong tidal currents (with surface velocities in excess of 1.5ms _1 ) and little wave motion. The seabed consists of fairly smooth sandy mud, with none of the sediment ripples observed at the LEO-15 site. This time the two sample planes were perpendicular to each other and the data was acquired at 3.3Hz. We recorded five data sets at different times and elevations, each consisting of 1000 image pairs. The latest version of the PIV system was deployed in May 2001 for testing, and in September 2001 for data collection. For the September 1-10 experiments, we again deployed the system near LEO-15, 5 nautical miles south-east of our previous sampling location at 39°23'37"N, 74°9'32"W, in 21m deep water. Here the water column was strongly stratified, with a sharp thermocline situated about 7m above the seabed. Data were collected for 20min periods at a time, sampling at either 2 or 3.3Hz. A total of 700GB of image data were collected at various elevations, up to 8m above the seabed and under differing conditions of mean flow and wave induced motion. Combined, we recorded over 1.2TB of images. 4. Analysis Techniques and Sample Results 4.1. Flow

Structure

Time series of the sample-area mean horizontal velocity ([/), obtained by averaging the velocity over the entire sample area of one camera, which are typical of the conditions encountered in Delaware Bay and near LEO-15 are shown in Fig. 9. At the Delaware Bay site (Fig. 9a), the mean flow is

PIV Measurements

in the Bottom Boundary

Layer of the Coastal Ocean

67

strong, with only relatively weak orbital wave motion. In this 5min, 3Hz time series, the mean velocity is 35cms _ 1 and the orbital wave motion has a maximum amplitude of about 7cms _ 1 and a period of about 9s. Smaller amplitude (about 4cms _1 ) and longer period (about 150s) variations in the mean flow, or "beating", are also visible in the horizontal velocity record. Figures 9b and c show characteristic 20min time series, from the LEO-15 site. In Fig. 9b, the mean velocity is about 9cms _ 1 , with the orbital motion caused by surface waves, traveling in groups, superimposed. It is evident that the amplitude of the streamwise wave-induced flow, generated by the predominantly 10s period surface gravity waves, is comparable to the mean velocity. Thus, at the extreme, the flow direction is actually reversed. Fig. 9c shows a period with very little flow, consisting mostly of oscillatory wave motion.

300

1200

600

1200

Time (s) Fig. 9. Time series of the sample area mean horizontal velocity for each of the sampling periods, a Delaware Bay, Z=35cm, b LEO-15, Z=50.5cm, c LEO-15, Z=50.5cm.

The corresponding vertical distribution of mean (ensemble averaged over all measurements and over all the data at the same elevation) horizontal velocity for the period shown in Fig. 9a, and that of the adjacent 5min periods, is shown in Fig. 10. It can be seen that they have a nearly loga-

68

W. A. M. Nimmo Smith,

T. R. Osborn and J. Katz

rithmic profile. We also observed a log layer in the New York Bight 27 under similar conditions of relatively high mean flow and low orbital wave motion. The vertical distributions of mean horizontal velocity measured at LEO-15, under conditions similar to the period shown in Fig. 9b, are presented in Fig. 11. Here, the vector maps forming each mean profile are conditionally sampled based on the phase of the wave-induced horizontal velocity. At no point is a logarithmic profile in evidence, consistent with the conclusions but not with the profile shape obtained in laboratory tests of an oscillatory boundary layer 28 . We do not have an explanation for this phenomenon. It may be related to the low turbulent Reynolds number of the LEO-15 data 23 , where the Taylor microscale Reynolds number is less than 100.

Fig. 10. Vertical distribution of mean horizontal velocity measured at the Delaware Bay site (from Ref 2 2 ) . The discontinuity in the profile is a result of performing measurements at different times and the "short" period of measurements at each of the three elevations.

To investigate the small-scale structure of the flow, Fig. 12 shows characteristic distributions of velocity and vorticity. During periods of no mean current and only oscillatory wave motion (Fig. 9c), the flow is very smooth, or "quiescent", with only very small turbulent fluctuations as shown in Fig. 12a. The quiescent periods also exist for some of the time when there is a moderate mean current (Fig. 9b). However, in between these quiescent periods, the character of the flow structure changes markedly. The more common pattern consists of powerful vortical structures with high core vorticity, of about 2cm in diameter, being advected through the sample area,

PIV Measurements

in the Bottom Boundary 1

20

r

U(z) 5/16/2000

•18 -16

•

....

'ID

E

Layer of the Coastal Ocean •

.

•

•

i

i

MaxU MinU Accel. Decel. j /

?f

j f

ô, -14 3

i

69

.-•/"

**^

-. >~

V'

/

i? •12

o

^ ^ > - ^

O

*

>^ ^ o

-10 .2

a E -8

^ • ^

.+

w £•.

. ^

120 100

^

'•""

(cm/s) (cm/s) (cm/s) (cm/s)

Fig. 5. The phase-averaged velocity field in CaseA at phase (a). Four vertical cross sections of the horizontal velocity component are also plotted at the same phase (see Fig. 3 for free surface location) with their respective 95% confidence intervals.

As the lee vortex moves from the ripple crest to the ripple trough, a thin viscous boundary layer develops between the vortex and the ripple bed surface. To illustrate this process, the vertical profile of the horizontal velocity at u)t = 0.557T (see Fig. 7) and x = 1.48 cm is replotted in Fig. 11. It is quite clear that the horizontal velocity approaches the ambient velocity at z> 4.0 cm, while the profile reflects the existence of the lee vortex.

Water Wave Induced Boundary

X = 0.60(cm) 4 _3 E

2.2

-20

Fig. 6.

t

0 (cm/s)

Layer Flows Above a Ripple Bed

X=1.22(cm)

X = 1.84(cm)

X = 2.47(cm)

4

4

4

3

3

3

2

2

2

1

1

0 20-20

^ 0 (cm/s)

n 20 -20

( 0 (cm/s)

91

1 n 20 -20

/ 0 20 (cm/s)

The phase-averaged velocity field in CaseA at phase (b). See caption for Fig. 5.

Underneath the vortex a very thin inner boundary layer, with the thickness of the order of 0.40 cm, is observed. This inner boundary layer has a characteristic length scale of order of {vT)1/2 (see table 1), which is equivalent to the Stoke's viscous layer thickness (Horikawa and Ikeda 11 ). Inside the inner boundary layer the direction of the horizontal velocity is the same as the free stream velocity. To ensure that this observation is not the result of the lack of resolution in data analysis, the averaging process has been repeated with smaller bins, 1.26 mm x 0.40 mm in size. The results with the refined bin size are very close to those obtained by the coarse bin size as shown in Fig. 11, confirming the boundary layer measurements.

3.1.1. Wave-Induced Steady Streaming Because of the nonlinearity in the wave field above the ripple bed, the period-averaged velocity does not vanish. The residual velocity, known as the steady streaming, is defined as:

(Ui)

1 ft+T (ui)dt;i -TJt '

=1,2

(3)

92

P. L.-F. Liu, K. A. Al-Banaa

 (cm/s)

Fig. 7.

 (cm/s)

and E. A.

 (cm/s)

Cowen

 (cm/s)

The phase-averaged velocity field in CaseA at phase (c). See caption for Fig. 5.

where T and (it*) are defined in equation 2. Thus, (ttj) = («») + (tij) and (ttj)' = 0. The magnitude and the direction of the steady streaming velocity have direct influence on near bed transport, such as sediment transport and the stability of the ripple bed. In Fig. 12, the spatial distribution of the steady streaming velocity vectors is presented. Two circulation cells with opposite direction appear above two adjacent ripple crests for CaseA. The circulation cells provide a mechanism for transporting sediment from the ripple trough to the crest and hence likely contribute to ripple growth. At the outer edge of the boundary layer, the vertical steady streaming vanishes as expected, while the horizontal component has a negative value; i.e., the steady streaming is in the direction opposite to that of wave propagation. Many researchers have also observed this negative steady streaming in laboratory wave facilities (e.g. Kemp and Simons 23 ; Nepf et al. 24 ), which is generally assumed to be the result of mass conservation in a closed wave tank. The circulation cells for the steady streaming are not symmetric with respect to the ripple trough. The asymmetry is partly the result of the negative wave-induced streaming just outside of the boundary layer. Consequently, the ripple-length-averaged steady streaming velocities (over one



93

X(cm) X = 0.60(cm)

4 _3 32

r

X = 1.84(cm) 4

3

3

2

2

1

1

0

c u > (cm/s)

Fig. 8.

X = 1.22(cm) 4

(

I

0

s u > (cm/s)

X = 2.47(cm) 4 3 2

}

1 n

 (cm/s)

< (cm/s)

The phase-aver aged velocity field in CaseA at phase (d). See caption for Fig. 5.

ripple wavelength, L), which can be defined as: Ui

--t

(ujdx,

1,2

(4)

do not vanish in the boundary layer. Fig. 13 plots the horizontal component of the ripple-length averaged streaming velocities above the ripple crest level for CaseA. In this case, the steady boundary layer has been established within the measurement area since the steady streaming velocity becomes essentially independent of the distance above the bed at « 20 5 from the ripple crest, where S is the laminar oscillatory boundary layer length scale as denned in Table 1. The velocity has been normalized by the maximum orbital velocity at the outer edge of the boundary layer, U0. Since the ripple-length-averaged steady streaming is positive in the vicinity of the ripple crest, the ripple bed would have been migrating in the direction of wave propagation, if the ripple bed were made of fine sediments. We also note that the steady streaming at the outer edge of the boundary layer, U/U0, is approximately -0.055 for CaseA and -0.080 for CaseB, which is in the same as order of magnitude of the wave slope (see table 1). As shown in Fig. 13, the steady streaming has a negative slope above the ripple bed, which has impacts on the asymmetric behavior in the phase-averaged vortex and hence on the vortex decay rate.


94

x = o 60(cm)

X = 1.22(cm) 4

1

3 2

I

-20

Fig. 9.

0 (cm/s)

3 2

\ 0 (cm/s)

n 20 -20

I

I

0 (cm/s)

Cowen

X = 2 47(cm)

X = 1.84(cm) 4

1

1

n 20 -20

and E. A.

4 3 2 1 n 20 -20

( 0 20 (cm/s)

X=2.47(cm)

X=1.84(cm) 4

4

3

3

2

2

1

1

n

/ (cm/s)

Fig. 20. The spatial variations of the phase-averaged turbulence intensity field in CaseA at phase (a). Four vertical profiles of turbulence intensity are also plotted at the same phase (see Fig. 3 for free surface location) with their respective 95% confidence interval.

cm < z < 2.14 cm. The reason for selecting these locations is to illustrate the effects of the flow separation due to high shear on the estimation of p and e. In Fig. 36 the temporal evolution of p, e and the differences between them are plotted (see Fig. 3 for free surface profile) for the CB close to the bed. It is clear that the two mechanisms are balanced since the CB is close to the ripple bed and the viscous terms are important. Thus, the common assumption that the p and e are balanced is applicable (Le32 and Pope 30 ). We observe that both terms peak after the wave crest past the measurement area at ult = 0.837T. In the shear layer (the upper CB), Fig. 37, the p is relatively higher than the e. We also observe that the difference between p and e peaks at the same phase as that of p, i.e., ut = 0.557T. Le 32 suggested that this discrepancy could be due to the velocity-pressurefluctuation term in the TKEB equation. However, we observed that the advection by the mean velocity is also very important at the upper CB. In fact, we found that the advective term is twice the p at oot = 0.357T and 22% at ut = 0.557T. It is also interesting to point out that the p peak occurs at the same moment that the wave crest reaches the measurement area (see Fig. 3 for free surface location) and the flow starts to separate above the ripple crest (see Fig. 7). This separation corresponds to an increase in the



105

0 10 -10 0 10 -10 0 10 -10 0 10 (cm/s) (cm/s)

Fig. 21. The spatial variations of the phase-averaged turbulence intensity field in CaseA at phase (b). The rest of figure caption remains the same as in Fig. 20.

0 10 -10 0 10 -10 0 10 -10 0 10 (cm/s) (cm/s)

Fig. 22. The spatial variations of the phase-averaged turbulence intensity field in CaseA at phase (c). The rest of figure caption remains the same as in Fig. 20.

106


and E. A.

Cowen

0 10 -10 0 10 -10 0 10 -10 0 10 (cm/s) (cm/s)

Fig. 23. The spatial variations of the phase-averaged turbulence intensity field in CaseA at phase (d). The rest of figure caption remains the same as in Fig. 20.

0 10 -10 0 10 -10 0 10 -10 0 10 (cm/s) (cm/s)

(cm/s)

107

{cm/s ) -(cm /s ) -(cm /s )

Fig. 31. The spatial variations of the phase-averaged Reynolds stress field in CaseA at phase (e). The rest of figure caption is the same as figure 27.

112


and E. A. Cowen

4 3.5 3 2.5

f N

2

1.5 1 0.5 0 0.5

1

1.5

X=0.60(cm)

2 X(cm)

X=1.22(cm)

2.5

3

X=1.84(cm)

3.5 X=2.47(cm)

4 3

_ 3 E S.2

-10

0

10 - 1 0

-(cm 2 /s 2 )

0

10 -10

-(cm 2 /s z )

0

10 - 1 0

-(cm 2 /s 2 )

0

10

-(cm 2 /s 2 )

Fig. 32. The spatial variations of the phase-averaged Reynolds stress field in CaseA at phase (f). The rest of figure caption is the same as figure 27.

vortex. Interestingly we observe, as shown in Fig. 38, that the difference of the positive and negative vorticity peak's decay rate, as determined by a first order exponential curve fit, is twice the mean background vorticity induced at second-order, e.g., AC = 2^-, where AC is the difference in the determined positive and negative vortices decay rates. While we expected the background shear to play a role in setting the decay rate it appears that the effect of viscous and turbulent diffusion is adjusting itself to be an order one process. Following Sleath 25 ' 26 and Jensen 27 we determined that the two cases, A and B, are fully developed turbulent boundary layers. With these results we extended the curve presented by Sleath and Jensen to smaller values of the roughness parameter, A/KS. Also, we find that the maximum turbulent intensity is roughly 40% of the maximum phase average velocity, which is high compared to other flows. The Reynolds stress was found to peak as the wave crest approaches the measurement area as a result of the ripple crest-induced flow separation. As the flow decelerates, the Reynolds stress becomes relatively weak. The spatial distribution of the phase dependant p and e in the TKEB equation are estimated based on the DPTV results. We observed two funda-



3

1

f

2

T

•

p 2 n U" J

)

X(cm)

3

p 2 T, \X?

1

p 2 TI \T>

3

V [ J

\ p2llU"J

" 1 1 ^

•

2

3

«2>\U„

3

ja 2 TI U ^

X(cm)

113

*J2TlU"J

3

*>2nU;J

^ u l T

3

Fig. 33. The spatial variations of the p and e in CaseA at phases (a &; b). Four vertical cross sections of the p and e are also plotted at the same phase (see Fig. 3 for free surface location). The contours are normalized by U^/2TJ. The "solid-line" represents the positive contour and the "dash-dot-line" represents the negative contour. The left column is e and the right is p.

1

)

3

e 21! U" 3

X(cm)

l

2

3

3

e 2 TI U" 3

3

e 2 n IT3

e 2 n IT3

3

P __£«

a-

^.f®:Sp>-

1

'

-iSSS.

d

)

X(cm)

2

3

3

' E2TIU"J

)

3

e2TiU"J

3

*> 2 Ti U" d

2

X(cm)

3

I I '

p 2 r| LT3

P 2 T| U ^

3

| p 2 TI U" 3

Fig. 34. The spatial variations of the p and e in CaseA at phases (c & d). See caption for Fig. 33.


114

and E. A.

Cowen

3

h

- ^

T40te

&

1 1 0

1

*

-0.0C-:

j£>

»

X(cm) 2

3 im—.

-0.1S

0

eZnU - 3

0.1S-0.15

0

e2nU-»

0.15-0.15

0

e2T]U^

0.15-0.15

0

'2:: ~ vr>.-\"*

0.005

•/ - v'v"-.!:-:""""'"-"* Dim

•QMS

-0.005

™JLa-ULu»i—l^-^JU^^Â^^.J^. G015 0 020 0.025 0 030 0.035 0.040

(a)

i • t . • , tJU. 0.015 0.020

_

0.025 0.039 J *

0.035

0.040

(b)

Fig. 25. Mean crossplane vectors (a) and w' /Ushi turbulence (b) distributions at x/L=QM90 and F^=0.30 from Roth et al96'97. Both'(a) and (b) also show the mean free surface shape measured from PIV images. Roth et al.96,97 apply single-exposed, digital PIV with a 27 mJ Nd:YAG laser and the cross-correlation algorithm to measure the flow structure and turbulence within a 7.01-m ship model (model no. 5422). The model has a bow sonar dome and transom stem. The experimental setup and procedures are very similar with those of Dong et al.94,95 in terms of facility,' particles, light delivery to the measurement region, and triggering of data acquisition. Differences include model, Fr, laser, and camera. The tests were done at one Fr=0.30 (/te=1.6xl07). Data is acquired at multiple axial stations between a location upstream of the bow stem and jt/L=0.1, however, the focus is on results at a single crossplane (x/L=O.069O). The dataset includes 92 instantaneous vector fields for each crossplane from which mean and turbulent statistics are computed. The PIV results reveal the detailed'structures of the bow wave at various axial and transverse locations for Fr=0.30 including free-

Ship Velocity Fields

157

surface shape and the mean and turbulent flowfield. Fig. 25 illustrates sample results for mean velocity vectors and the vertical normal stress as well as the mean free surface shape at a single crossplane. As x/L increases, vector maps show a growing region of negative vorticity or shear layer originating at the toe of the wave and then penetrating into the wave before curling upward and remaining close to the forward face. Coincident with this region is concentrated distributions of normal and shear Reynolds stresses whose magnitudes are consistent with theoretical predictions. This experiment expands on the scope of Dong et al.94'95 through investigation of bow-wave turbulence, turbulence production, and Reynolds number effect. Significant efforts were also made to investigate convergence of mean and turbulence variables and effects of interrogation window size on turbulence structure and statistics. 5.4. Wave hydrodynamics Coakley and Duncan98 apply film-based PIV with a 200 mJ Nd:YAG laser and the auto correlation evaluation method to investigate the wave-breaking region induced by a towed, submerged hydrofoil in close proximity to the free surface. The tests are performed in a 1.2(W)xl(D)xl5(L)-m3 wave tank with a stationary PIV measurement system. Although a range of chord lengths and foil depths are tested, results are presented for one foil chord length (c=15 cm) and one foil depth (d/c=l.02). The foil towing speed for the 15-cm version corresponds to Fr=0.571. The dataset includes 10 instantaneous vector fields from which average values are computed. The results clearly reveal a thin shear layer above the underlying flow at the forward face of the breaking wave. The mean shear layer is thin at the toe of the wave and increases with thickness with increasing distance downstream. The authors have presented less than 10% of the data so the remainder may have enough information to derive turbulence statistics and increase understanding of the fundamental physics of breaking waves. Dabiri and Gharib" apply single-exposed, digital PIV with the cross correlation evaluation method to investigate flowfield structures in a 2D spilling breaking wave. The tests are conducted in the 0.152(W)x0.152(D)x0.61(L)-m3 Caltech water tunnel with a stationary PIV measurement system. A schematic diagram (Fig. 26) of the experimental setup highlights the water tunnel, PIV system hardware, light-delivery method, and measurement regions. Two cases are studied: (1) the first focuses on a breaking wave with small capillary waves upstream of the breaking region; and (2) the second focuses on a capillary-gravity wave with no breaking. The Froude and Reynolds numbers based on breaker and wave heights, respectively, are (Fr=2.04, 7370) and (Re=l.62, 1050). The PIV measurements are acquired in three and two overlapping sections for piecewise reconstruction of the area of interest for cases one and two, respectively. Measurements resolve the vector and vorticity fields near

158

J. Longo et al

the free surface for both cases and reveal that the dominant source of vorticity flux is from decelerating free-surface fluid in the breaking region. The capillary waves are not a source of significant vorticity flux. Sample vector and vorticity results from section 25 case 1 are shown in Fig. 27. The vectors identify a mixing layer accompanied by strong positive vorticity that transitions into a shear layer with increasing distance downstream.

Fig. 26. Schematic of experimental setup from Dabiri and Gharib . J 40 cm s*"1

I

x (cm)

'

|

'

x (cm)

(a) (b) Fig. 27. Vector and vorticity fields from section 2 and case 1 from Dabiri .and Gharib". Chang and Liu100'101 apply digital PIV with a 200 mJ Nd:YAG laser to investigate the mean and turbulent structures in a 2D spilling breaking wave. The tests are done in the Cornell University 0.6(W)x0.9(D)x30(L)»m3 wave tank with a stationary PIV measurement system. The breaking-wave height, wavelength, and phase velocity are 14.5 cm, 1.21 m, and 1.21 m/s, respectively. A schematic diagram (Fig. 28) of the experimental setup highlights the wave tank and wavemaker, PIV system hardware, and light-delivery method. Ensemble average velocities are computed from 20 multiple tests with a vertical lightsheet centered on the breaking region and a field-of-view that captures roughly one-third of a wavelength. Sample results are shown in Figs. 29 and 30 for mean velocity and turbulence intensity, respectively, at two discrete times during the wave-breaking event. The mean results show that the velocity beneath the aerated breaking region

159

Ship Velocity Fields IJaht sheet optics

To wavamak&r control psnel

—a^-A FOV

U

Ttmir^b«x_ Tolassrs

•^VêcsH "H-' Mirror coniroiter

frame grabber

Fig. 28. Schematic of experimental setup from Chang and Liu100,101

80 85 90 95 100 105 110 115

80 85 90 95 100105 110 115

x (cm)

x(cm)

(a)

(b)

Fig. 29. Mean velocity of the third breaking wave at (a) 3.290 s and (b) 3.398 from Chang and Liu100'101.

(a) (b) Fig. 30. Turbulence intensity I (cm/s) of the third breaking wave at (a) 3.290 s and 100,101 (b) 3.398fromChang and Liu; is less than one-half of the phase velocity and matches closely with numerical results of Lin and Liu102. Turbulence intensity results indicate that in the trough region, the maximum ensemble-averaged value is roughly 10% of the phase velocity. Variation in the

160

J. Longo et al.

turbulence intensity is reasonably small for sample numbers greater than 16. The authors use the transport equation for turbulent kinetic energy to assess the transport processes under breaking waves. They find that under the trough level in a rectangular control volume, the diffusive transport term is an order-of-magnitude smaller than the advection and dissipation terms. They also find that the turbulence dissipation is greater than the turbulence production for all of the measurements. The authors publish a related wave-breaking study (Chang and Liu101) with the same facility and equipment to study bias errors on mean and turbulence measurements resulting from small ratio of particle image diameter to pixel size and correction procedures. Qiao and Duncan103 apply film-based PIV with a 200 mJ Nd:YAG laser and the auto correlation evaluation method to investigate the flowfield at the crests of gentle 2D spilling breakers. The tests are performed in a 1.22(W)xl(D)xl4.8(L)-m 3 wave tank with a stationary PIV measurement system. A schematic diagram (Fig. 31) of the experimental setup highlights the wave tank and wavemaker, PIV system hardware, and light-delivery method. Camera for surface profile pictures

Light sheet

Optics and the underwater mirror Wavemajcer Underwater mirror \ . Light sheet Mirror and cylindrical lerts Nd: YAG lasers

^

'

Cameras \ Two-dimensional wave structures

Plan View

Fig. 31. Schematic of experimental setup from Qiao and Duncan" The breaking-wave frequencies, wavelengths, and heights are (1.42, 1.26, 1.15 Hz), (77.4, 98.3,118.1 cm), and (3.83, 4.87, 5.87, 5.92 cm), respectively. In order to map the breaking wave structures, PIV recordings are acquired sequentially along the length of the wave tank by movement of the camera for successive waves. The dataset includes instantaneous vector fields from which vorticity is computed and maximum horizontal and wave-toe velocities are identified. PIV measurements track the evolution of the spilling breaker including initial formation of a bulge on the forward face of the wave. At this stage, differences in the wave-crest flowfield as a function of wave frequency are in the noise of the measurements. The next phase of the breaking process occurs when the leading edge of the bulge (toe) moves

161


down the wave face. Fig. 32 shows sample results with identification of the toe region and a thin layer of vorticity distributed from the toe to the crest. Progressive extension of this region from the toe toward the crest suggests that vorticity enters the flow through tie free surface at a location near the toe. Although coarse PIV resolution precludes measurement of the thin shear layer on the forward face of the breaker, measurements near the free surface between the toe and wave crest are as high as 1.4 times the linear wave phase speed. JV

JU

s20 10 b

10

20

30

40"

X(mm)

(a)

"50

"0

10

20

30

40

50

X(mm)

(b)

Fig. 32. Instantaneous vectors (a) and vorticity contours (b) from Qiao and Duncan103. Sarpkaya and Merrill104 apply digital PIV to investigate the breakup of a bow sheet (i.e., liquid wall jet) into ligaments and droplets over smooth and sandroughened walls to gain insight into the underlying physics of bow-sheet breakup and spray from naval vessels. The tests are conducted in a 0.5(W)x0.5(D)x6(L)-m water tunnel with a stationary PIV measurement system at several Fr= 15-30, J?e=2.4xlO4-8.5xl04, and Weber number= 1500-7500. The emphasis of the study is flow visualization of free-surface structures (ligaments and drops) but PIV measurements of the underlying wall-bounded Equid jet provide evidence that the Reynolds stress normal to the wall increases with distance from the wall and gives rise to eruptions of slender ligaments through the free surface. Most ligaments lean backwards as they travel forward with the wall jet because the majority of ejections occur in the second quadrant, i.e., for (-K', + V'). The results are being used for numerical modeEng of spray sheets. §S« Propidsor hydrodynamics Jiang et al.105 apply multiple-exposed, film-based PIV with a 15-W copper vapor laser and the auto correlation evaluation method to investigate ring-vortex creation from propeller crashback with propeller no. 4381. The tests are done in the David Taylor Model Basin 24-inch water tunnel with a stationary PIV measurement system at two advance coefficients (J=-0.472, -0.732). The region of interest is confined to

162

J. Longo et at

a local area near the Made tip. The dataset includes instantaneous vector, fields at three time instances after crashback for both advance coefficients from which vorticity is computed. The data captures the detailed ring-vortex flow structures and their locations for propeller inflow, reverse flow, and during ring-vortex formation. The tip vortices are stronger for the case with higher propeller loading. Cotroni et al.79 apply digital PIV with a 200 mJ Nd:YAG laser and the cross correlation evaluation method to study propeller wake flow from a 0.274-rn diameter, four-bladed propeller with a uniform pitch (pitch/diameter=L49). The experiments are conducted in the 0.6(W)x0.6(D)x2.6(L)~m3 Italian Navy Cavitation Tunnel (CEQMM). A schematic diagram (Fig. 33) of the experimental setup highlights the cavitation tunnel test section, PIV system hardware, synchronizing hardware, light-delivery method, and propeller-wake geometry.

(a)

(b)

Fig. 33. Schematic diagram of experimental setup (a) and wake geometry and measurement positions (b) from Cotroni et al.79. The propeller inflow and revolution speed is 7.5 m/s and 25 rev/s, respectively, for an advance ratio of 1.1. The Reynolds number is 2.0xl0 6 based on propeller diameter. Instantaneous and phase-averaged (from 30 PIV image pairs in' 5° increments) data is acquired in a 15.2x15.2 cm2 window centered on the propeller shaft axis and positioned at the aft edge of the propeller blades. Sample instantaneous results are shown in Fig. 34. Contours of vorticity clearly ilustrate and locate the cross sections of the tip-vortex system captured in the field of view and the complex interaction between the vortex generated at the blade-hub juncture and the recirculation zone behind the blunt end of the propeller shaft. A secondary tip-vortex system is also measured but is an order-of-magnitode weaker than the primary tip-vortex system. The core of the primary tip-vortex system is mapped in xy-space with error bars, for the y position. The test procedures such as phaseaveraging, window offsetting (Westerweel56), and adaptive correlation (Hart58) are advanced and prove that PIV can be used to measure complex unsteady flows with improved accuracy..

163


xm

xm

(a) (b) Fig. 34. Contours of instantaneous vorticity (a) and horizontal component of velocity (b) for a phase'angle of 10° (Cotroni et al.79). Di Felice et al.80 apply, digital PIV with the same facility and measurement system as Cotroni et al.79 (Fig. 33) to study the propeller wake flow of a smaller (D=0.227m; pitch/diameter=l.l) four-bladed marine propeler. The propeller inflow and revolution speed are 4.25 m/s and 25 rev/s, respectively, for an advance ratio of 0.748. The Eeynolds number is 1.12xl06 based on propeller chord length. Instantaneous and phase-averaged (from 65 PIV image pairs in 5° increments) data axe acquired in a 10x10 cm2 window centered on the propeller shaft axis and positioned at three overlapping positions aft of the propeller hub. The authors report uncertainties for the measurements (0.6%) away from the blade wake and tip vortex, two regions where strong centrifugal forces repel seed particles, create voids in the image recordings, and degrade the accuracy of the data. Phase-averaged PIV results for U and V are presented for six crossings of the tip vortex downstream of the propeller. Qualitatively, distributions of U and V are similar after rotation of the latter counterclockwise by 90°. The viscous blade wake is evident as a velocity defect mat is released from the blade and mostly dissipates within one propeller diameter downstream. Considerable deformation of the streamlines in the vicinities of the.tip and hub vortices is presented. The vorticity released from the Made trailing .edge weakens after one propeller diameter downstream but the tip vortex systems retains it strength through the measurement area. The measurements are made with sufficient spatial resolution to track the tip vortex system which highlights the slipstream contraction. Turbulence intensities are computed from 65 realizations at discrete phase angles. Results illustrate the tip-vortex system breakdown as a general diffusion of the turbulence patterns as they are converted downstream. Judge et al.29'30 apply digital PIV with an 800 ml Nd:YAG laser and the cross correlation evaluation method to investigate tip leakage vortices from a 0.8504»m diameter, three-bladed, ducted rotor (propeller no. 5206). The rotor has a blade

J. Longo et al

164

chord-length C=0.381 m across the span and a tip gap of 6.67 mm. The experiments are conducted in the Naval Surface Warfare Center 36-inch variable pressure water tunnel. A schematic diagram (Fig. 35) of the experimental setup highlights the water tunnel test section, ducted rotor, and light-delivery method. The propeller inflow and revolution speed are 6.89 m/s and 8.33 rev/s, respectively, for an advance ratio of 0.972. The Reynolds number is 7xl0 6 based on the propeller chord length. Instantaneous and phase-averaged (from 532 PIV image pairs) data are acquired in a 3x2.4 cm2 window centered on the tip leakage vortex. PIV and LDV data in the region of the tip vortex are in good agreement.

Fig. 35. Schematic diagram of experimental setup from Judge et al."1"''30. Instantaneous PIV recordings at the same Made positions show variability in location (2%-3% vortex core radii), strength (22% of average circulation), and vortex-core size (31% of average vortex core size). Fig. 36 presents tip-leakage vortex data for two blade phase angles separated by roughly 25° and reveals how the core structure wanders in space and reorganizes in time. As the tip-leakage vortex is convected downstream, its size and circulation strength increase smoothly and the average and maximum tangential velocities

(a)

(b)

Fig. 36. Velocity vectors and contours of vorticity for the tip-leakage vortex at two phase angles: S=6R/C=0.0296 (a) and S=0.5259 (b) from Judge et al.2930.


165

decrease smoothly. The test section Met pressure is controlled to suppress cavitation but further tests at lower inlet pressure are examining the effect of the tipleakage vortex variability on cavitation inception. Calcagno et al.26 apply digital stereoscopic PIV with a 200 mJ Nd:YAG laser to investigate the propeller wake flow from a 6.096»m Series 60 Q=0.60 container/cargo ship model with a single 0.222-m, 5»bladed MAU. The experiments are conducted in the 3.6(W)x2.25(D)xl0(L)~m3 INSEAN Circulating Water channel. A schematic diagram (Fig. 8) highlights the water channel test section, propeller and hull geometries, asymmetric two-camera configuration for 3D stereoscopic PIV, and light-delivery method. The propeller inflow and revolution speed are 1.22 m/s (Fn=0.16) and 6.7 rev/s, respectively, for an advance ratio of 0.82. The Reynolds number is 2.7xl05 based on the propeller diameter. Instantaneous and phase-averaged (129 PIV image pairs from 0°-69° in increments of 3°) data are acquired in a 20x20 cm2 window at three separate measurement planes orthogonal to the propeller shaft and downstream of the propeller disk (Fig. 37;x£=0.9997,1.0,1.0187). Typical phase-averaged results (Fig. 38) reveal the absence of propeller wake symmetry inherent to isolated propeller studies. f f i l

26

Fig. 37. Ship model, propeller, and measurement locations from Calcagno et al.

''..."

1"

K i:

f

A

i

/44^"

^ l l § § a ' - "*••

1

" -**3r4ve

-&)

o

sa

i*»

ISC

(a) 0>) Fig. 38. Average velocity vectors and contours of vorticity at i=0° from Calcagno et al;26

166

J. Longo et al.

The largest longitudinal velocities are in the lower part of the propeller disk at radial positions of r/R=0.6-0.7. The tip vortex system is captured in the measurements at r/R=0.9 where strong velocity gradients create peaks in the vorticity contours. Turbulence statistics from 129 realizations of the flow are computed and indicate maximum values are produced at the tip and hub vortex cores. The turbulent wake of the hull emerges in the turbulence statistics and strong turbulence is created by the passage of the blades through the hull wake. The wake evolves longitudinally with a noticeable slipstream contraction between the first and second planes and strong diffusion and dissipation of the blade wakes. The hub and tip vortices are attenuated at the third measurement plane but still very apparent. The authors highlight the advantages of a 3D PIV system including time efficiency for data acquisition and capability of measuring three velocity components for highly threedimensional flows that have strong velocity gradients. This study is complementary to previous work by Toda et al. 106 where the velocity field was measured with a multi-hole probe. Lee et al.42 apply digital PIV with a 125 mJ Nd:YAG laser and the particle tracking evaluation metiiod to study propeller wake flow from a five-bladed propeller with a diameter of 54 mm. The experiments are conducted in a 0.3(W)x0.2(D)xl.2(L)-m 3 circulating water channel. The propeller inflow and advance ratios are 0.33 m/s and 7=0.59, 0.72, 0.88, respectively. The Reynolds number is 1.8xl05 based on propeller diameter. Instantaneous and phase-averaged (400 PIV image pairs from 0°-54° in increments of 18°) data are acquired in a 11.8x11.8 cm2 window that is centered on the propeller axis shaft. The field of view allows data acquisition two propeller diameters downstream of the blade trailing edges. The authors apply an adaptive hybrid 2-frame particle tracking velocimetry (PTV) evaluation method to improve spatial resolution and measurement accuracy while reducing computational time. The data is presented as a function of propeller loading. In general, as the propeller loading increases, phase-averaged axial velocity, vorticity, axial and vertical turbulence intensity, and Reynolds shear stress increase within the propeller slipstream. 5.6. Appendages Liu and Chang107 apply multiple-exposed, film-based PIV with a 15-W copper vapor laser and the auto-correlation evaluation method to investigate the structure of vortices from a towed submarine sailplane with a half-span of 70.4 cm, base chord length of 65.3 cm, tip chord length of 43.7 cm, and maximum thickness of 12.1 cm. The experiments are performed in the 6.4(W)x4.9(D)x514(L)-m3 David Taylor Model Basin high-speed basin with a 2D, stationary PIV system at one Fr=0.81 (Re=l.3xl06) based on the sailplane base chord length. Data is acquired at three angles of attack (5°, 10°, 15°) at ten discrete times corresponding to a range of 0.05-


167

48 chord lengths downstream of the sailplane. The dataset includes instantaneous vector fields from which vorticity and circulation are presented. The PIV data captures the shear layer and tip vortex roll-up at the trailing -edge of the sailplane. Computed circulation strengths increase three-fold as the sailplane angle of attack is increased from 5° to 15°. Yeung et al.37 apply digital PIV with a 10-W argon laser and the cross correlation evaluation method to investigate vorticity fields from a 2.54-cm thick acrylic flat plate in forced roE motion, where the experimental setup approximates the flow generated by ship bilge keels. The experiments are conducted in the 2.44(W)xl.52(D)x61(L)-m3 University of CaMfomia, Berkeley towing tank with a 2D stationary PIV system. Draft, roll amplitude, and roll period are tested in the following ranges, (L=31.75 cm, 16.51 cm), (C4=5°-15°), and 0p=1.8 sec-4 sec), respectively, where the latter two are chosen to closely follow -typical roll resonances of ships. The field of view covers a local region at the bottom edge of the plate and the surrounding area where instantaneous vector fields are acquired at several instants of time. Vorticity contours are computed from the instantaneous vector fields. The Keulegan-Carpenter number {KC=2nmJJtp) is used to identify asymmetric and symmetric flow regimes. For KC>13, an asymmetric period steady state is reached in the flowfield with a vortex pair shed toward the side of initial plate swing. For KC 500

Sc1'2^]1/2

6a

Banerjee 4

Surface divergence (no shear) Interfacial shear (liquid side) Interfacial shear (gas side) Eddy resolving analytical Surface processes

C 3 u[0.3(2.83Ref / 4 -2.14Re?/3)]i/4Re-1/2

8

Banerjee 4

10

Banerjee 4

11

Banerjee

a l

"Met,™

0-07 < i c t , G S c - V 3

Csanady 1 9

C

4Pluliict,w

C5P2Uiict!W(l x(l +

+

Rf/RfCr)V*

Soloviev and Schluessel 20

Ke/KeCr)-1/2

turbulence near free surfaces, but we will not consider these further here. To proceed, the —5/3 power range at the low wave numbers suggests quasi two-dimensionality of turbulence in the near-surface region. There is up cascading of energy from the wave number range at which the spectrum splits to the small wave numbers. Physically, this means that the smaller structures merge to form larger structures, which is what is seen in Fig. 4. These conclusions regarding quasi two dimensionality of the surface turbulence are supported by the direct numerical simulations of Pan and Banerjee. In fact, the two-dimensionality can be seen from the contours of the vortical structures hanging down below the free surface in Fig. 8. Several aspects need to be discussed related to these experiments. First, particles were scattered by Kumar et al. on the free surface in order to conduct the DPIV. As McKenna et al. have shown, this may give rise to some surfactant effects, even though such effects would be expected to be less apparent in open-channel flow than in grid-stirred tanks. Kumar et al.

The Air- Water Interface:

Turbulence and Scalar Exchange

195

were aware of this problem, however, and used rather low particle coverage of the free surface to minimize it. Details may be found in their paper. Nonetheless, these effects cannot be discounted. Second, the direct numerical simulations of Pan and Banerjee were in the "rigid lid55 approximation, i.e., the free surface was replaced by a slip surface. The interface had no give in the vertical direction in the simulations, and this might not be a good assumption. Third, while the comparison in the shapes of the energy spectra between the experiments and simulations is remarkable, nonetheless the coincidence in actual values is not significant, as the integrals are normalized to the same values over the same range of wave numbers, i.e. it is only the shapes that are similar. Finally, the experiments under consideration did not have significant interfacial waves, and the Reynolds numbers were quite low, though turbulent. The low Reynolds numbers for the flow could lead to an overestimate of the importance of surface renewal motions that originate at the bottom wall. At high Reynolds numbers, the channel flow might be quite homogeneous and isotropic near the free surface and the surface-region turbulence structure might be more like that for a grid-stirred system.

Fig. 4.

Merger of two vortices at the free surface of an open-channel flow. Top view.

For details of the DPIV technique used by Kumar et al. 17 , reference should be made to the original paper. As free surface turbulence involves motions with a wide dynamic range, some special image processing considerations are involved, as discussed briefly in what follows. The methodology - based on multi-grid image processing algorithms for rigid body motion analysis, estimates the displacement vectors at discrete particle locations. The essence of this technique is to estimate large-scale motions from image intensity patterns of low spatial frequencies and small-scale motions from

196

S. Banerjee and S.

Maelntyre

Fig. 5. Annihilation of a vortex at the free surface by an upwelling. The dark area in panel "d" is an upwelling where the visualizing particles have been swept outward.

intensity patterns of high spatial frequencies. Cross-correlation between a pair of time-separated particle images is implemented by the hierarchical computational scheme of Burt 31 . Each image is convolved with a series of band-pass filters and subsampled to obtain a set of images progressively decreasing in resolution and size. A coarse estimate of the displacement field obtained from pairs of lower resolution images is used to obtain more accurate estimates at the next (finer) level. Processing starts at the level of lowest resolution and stops at the highest resolution level, which contains the original image pair. Due to subsampling of low-resolution images, the match template size can be kept constant for all stages of computation, thus eliminating the dependence of the largest resolvable displacement on the size of match template. In the present work, the search area at each level is kept constant at 3 x 3 pixels and the match template size at 5 x 5 pixels for all levels of computation. The algorithm has been implemented using simple thresholding based on the confidence level of an estimated displacement vector, as suggested by Anandan 32 . However, the confidencelevel-based smoothing technique for rigid body motions (continuous velocity fields) could not be applied to displacement estimates obtained at discrete points, i.e., the particle locations. Instead, smoothing was performed over the area covered by each particle. The algorithm has been tested against



197

10° 10''

10*

10"

10

o """

1

(a

10

ioo

attik,

10°

•,

10-

ID"*

10

s

10

(b)

^\.«

„

0

\ \ ^ 1

• DNS HDPIV [Exp k" k"

10""

-

^^

-

1

10

100

2*hk,

Fig. 6. Energy spectra for the streamwise velocity component at the free surface (x streamwise direction, y — spanwise direction).

direct numerical simulations of turbulent flows when the flow field is known and particle images have been generated from these with the addition of noise. Both the accuracy of motion estimation and the computation time are seen to improve as compared to conventional PIV methods. Before moving on to scalar transfer results in open-channel flows, the effects of waves interacting with a turbulent stream will be considered. Early studies of mechanically-generated waves interacting with a turbulent current were those of van Hoften and Karaki 33 , Iwagaki and Asaro 34 , Kemp and Simons 35,36 , Simons et al. 37 , and more recently, Rashidi et al. 38 , Supharatid et al. 39 , Nan et al. 40 , and Nan 41 . The most extensive work is that of Nan, in which 3D laser Doppler anemometry (LDA) measurements were made as a function of flow depth for a wide range of wave-to-current conditions, albeit at relatively low depth-based Reynolds numbers - though still resulting in fully developed turbulence. Nan's data, over a range of

198

S. Banerjee and S.

Maclntyre

10°

10''

"s-

^ V . 10*

-

\

• DNS 1 HOPIV (Expl - - - k "4

--

10"

k

\ •

1

10 s 0

*

1

10

(C)

100

fcthk,

10°

10''

-

10 z

-

• '

r

^ < ! ! \

1

" ^ v

\ 10"4

" (d)

0

\

• DNS HDPIV (Exp k" — - k"'

1

10

100

2nhk,

Fig. 7. Energy spectra for the spanwise velocity component at the free surface (x streamwise direction, y — spanwise direction).

Reynolds numbers and wave strength parameters, indicates turbulence enhancement in the near-surface regions, the effect increasing with the wave strength parameter Ub/U which is the ratio of the wave-induced Stokes drift velocity to the mean velocity. This is shown for the streamwise, wall-normal and spanwise components in Figs. 9, 10, and 11. While there is some variation in the friction velocities measured at the bottom of the channel with and without waves, these changes are not large, so the figures do indicate the enhancement in turbulence intensities caused by the waves, even though the intensities, with and without waves, are nondimensionalized by the respective friction velocities. Note that the fluctuating velocities due to the wave motions themselves have been removed by ensemble averaging over many identical waves. Therefore, what is left are the turbulent fluctuations. Nan took data with essentially 2D waves, so that the spanwise fluctuations are not at all contaminated by the wave mo-



199

Fig. 8. Vortical structures hanging down from a free surface on which the instantaneous streamlines are shown (Pan and Banerjee 3 0 ). The surface is viewed from the side of a channel flow, looking up from below. The streamwise direction is 4wh and the spanwise direction is 2irh in length for a liquid layer thickness of 2/i in these calculations.

tion. (In. fact, the spanwise wave components of motion are negligible.) Nan attributed the increase in turbulence to- a wave-induced Reynolds stress, which was significant when waves interacted with the turbulent current. While this suggests that waves could have substantial effects on mass transfer, we are not aware of any experiments that have directly measured the effect of such mechanically generated waves on scalar exchange. However, there have been experiments on gas transfer in laboratory flumes and circular wind-wave facilities, as well as combinations of mechanically generated waves with wind to simulate such effects. These do indicate increases in gas transfer velocities when waves are present. We turn now to consideration of scalar transfer at the free surface of open-channel flow. A pioneering study in this regard was that of Komori et al. 28 They not only measured gas transfer rates (for C(>2)5 but also made simultaneous measurements of surface renewal events which emanated as "bursts" from the bottom of the channel. The data cover a range of water velocities of 5.9 to 23.5 cm/s and depth-based Re of about 2800 to 10,000. They found a remarkable correlation between the frequency of surface renewal events (f8) and the gas transfer velocity, ft. However, the measured

200

S. Banerjee and S.

$~* 3°

o (0

3

iyu

BJ+

I

0.33 0.30 0.25 0.29 0.42 0.57 0.54

0.24 0.27 0.37 0.35 0.35 0.36 0.44

-

§3 2 S

1

-

-

—

'

1

'

I

'

4

~ -

//

-

Run 1 3 5 6

8

.yy

i l l

*•

Maclntyre

i

3

T^r^-^

1

ill

E

3

n

0.0

i

i

0.2

0.4

,

t

06

,

i

0.8

1.0

y/h Fig. 9. Enhancement of streamwise turbulence intensity due to waves as a function of depth. The average position of the free surface is at y/h = 1.0. Here Uf,/U and Gi+ are parameters that are measures of wave strength and frequency. The mean depth of the flow is h, and y is measured from the bottom of the channel. The subscript w denotes channel flow with superimposed mechanically generated waves, and the subscript c denotes channel flow without waves, u* is the friction velocity of the channel flow. Depth based Reynolds number ~ 5000.

gas transfer velocity is about a third of what would be predicted by surface renewal theory, as pointed out by Banerjee4, i.e. (3 = 0.35 (Dfg)1/2. The discrepancy may have arisen because Komori et al. did not directly measure surface divergence, but rather viewed the structures that were ejected from the bottom of the channel, not all of which may have impinged on the free surface. Knowlton et al. 42 attempted to validate the surface divergence theory by calculating surface divergence directly from the velocity field measured by Kumar et al. 17 and then using it to solve the 3D concentration field equation from which they calculated the mass transfer coefficient. The only case for which Kumar et al.'s data coincided with one of Komori's cases was for depth-based Re = 2800. Knowlton et al. found remarkable agreement with Komori et al.'s data for this case. However, when Knowlton et al. calculated the gas transfer data based on a "rigid lid" direct numerical simulation, they found gas transfer velocities that were about 2-3 times higher. These results are also shown in Fig. 12. In Table 2, we show the predictions of Eq. (8) with C = 0.20 (which was the value for C that agreed with the stirred vessel gas transfer data of McKenna et al.) It is evident that


1

t

-

XTs 3

E

—

Turbulence and Scalar

1

1

ub/U

TO+

Run

0.33 0.30 0.25 0.29 0.42 0.57 0.54

0.24 0.27 0.37 0.35 0.35 0.36 0.44

1 3 5 6 7

i

,

201

Exchange

.

I.

;

.

/

/ . / / / /

-

8 9

-

-

"

•

0.0

I

I

0.2

0.4

i

,

r

0.6

0.8

1.0

y/h Fig. 10. Enhancement of the vertical turbulence intensity due to waves as a function of depth. Same notation as in Fig. 9.

1

-

0.33 0.30 0.25 0.29 2 0.42 - — — 0.57 - — 0.54

0.24 0.27 0.37 0.35 0.35 0.36 0.44

1

'

1

/ / / /

/

.,

-

——

'—

;

i

Run 1 3 5 6 7 8 9

-

0.0

i

i

1

0.2

0.4

0.6

.

1

0.8

1.0

y/h Fig. 11. Enhancement of the turbulence intensity in the spanwise direction due to waves, as a function of depth. Same notation as in Fig. 9.

the agreement with these predictions is quite good. The velocity scale for Eq. (8) is taken to be the wall friction velocity, and the length scale was the depth. While these scales are reasonable, it is likely that the length scale is a weak function of the depth-based Reynolds number, i.e., (A/depth) varies

202

S. Banerjee and S.

Maclntyre

as Re~ ' , which is what would be expected in the core region of pipe flow. Making such an assumption would improve the agreement between Eq. (8) and the experimental data, which is rather over predicted at low Reynolds numbers and under predicted at high Reynolds numbers. Knowlton et al. concluded that a rigid lid approximation was poor for mass transfer calculations and that the "give" in the surface significantly reduced surface divergence at the air-water interface. This is also consistent with C ~ 0.20 in Eq. (8). These data have been taken in relatively shallow channels, and ejection events from the bottom boundary maintain their coherence up to the interface. For much higher Reynolds numbers and greater water depths, the bulk turbulence structure is expected to more closely approximate homogeneous, isotropic turbulence, where the surface divergence model given in Eq. (8) might again be expected to predict gas transfer velocity. In all this work the effect of heat loss from the liquid was not investigated. As demonstrated in the next section, such effects are important in the field - particularly at low wind velocities. We move on now to field experiments at relatively low wind velocities.

FULL DNS: 0.0750 Sc"° 5 DNS/DPIV (2h=1.5cm) EXPERIM: 0.0273 S c ~ 0 5

Schmidt Number

Fig. 12. Comparison of DNS with rigid slip surface as the interface in channel flow with experiment and calculations based on experimentally determined surface divergence field. Note that the rigid surface gives higher values for the gas transfer velocity.

The Air- Water Interface: Table 2. C = 0.2. Run No. I II III IV V VI VII VIII IX


203

Comparison of Komori et al.'s 2 8 experimental data with Eq. (8), with u* cm/s

S (cm) 1.1 2.9 3.1 5.0 5.1 6.4 7.0 10.0 11.2

3.2. Field

(cm/s) 23.5 9.7 18.3 5.9 11.9 19.9 13.8 10.5 10.9

1.48 0.61 1.06 0.37 0.69 1.01 0.75 0.58 0.59

experimental P X 10 5 (m/s) 1.65 0.75 1.60 0.45 0.90 1.2 1.3 0.7 0.8

Eq. (8) prediction /3 x 10 5 (m/s) 1.97 0.81 1.22 0.49 0.78 1.01 0.8 0.56 0.55

Studies

Field studies using tracers to determine gas transfer coefficients have shown considerable scatter at low wind speeds (see summary in Maclntyre et al. 5 , Cole and Caraco 43 , Crucius and Wanninkhof 44 ). While shear will make a contribution to turbulence at low wind speeds, the contribution from heat loss and rain is significant (Shay and Gregg 45 , Anis et al. 46 , Maclntyre et al. 5 , Ho et al. 47 ). Eddy sizes and rates of dissipation of turbulent kinetic energy e in the upper water column including the air-water interface can be determined with temperature-gradient microstructure profiling. Such profilers ascend through the water column in free fall mode sampling temperature, temperature gradients, conductivity, fluorescence, and photosynthetically available radiation at 100 Hz. With an ascent rate of 0.1 m/s, mm scale resolution is obtained. While estimates of e have been obtained in numerous oceanographic investigations, the ship's wake confounds the values in the upper 10 m. Data obtained during deployments from small boats while the profiler is rising illustrate the turbulence at the air-water interface. Examples of such profiles in lakes include Maclntyre 48 ' 49 ' 50 , Robarts et al. 51 , Maclntyre et al. 52 , Maclntyre and Jellison 53 . Typically during low winds and when incoming radiation exceeds heat losses, the upper water column is stably stratified to the air-water interface or has small eddies (1 mm 20 cm scale) at the air-water interface. When heat loss exceeds incoming radiation, eddies can be much larger, typically meter scale, and, depending on depth of the water body, can extend to the bottom or at least through the weakly stratified upper mixed layer. In those cases, temperature profiles have a characteristic shape with low temperatures near the air-water interface, slightly higher temperatures just below and then a decrease of temperatures (Anis and Mourn 46 ). Rates of dissipation of turbulent kinetic

204

S. Banerjee and S.

Maclntyre

energy below the air-water interface during periods of heat loss depend on buoyancy flux (Shay and Gregg 45 , Anis and Mourn 46 ). Profiles illustrating dissipation rates at low speeds are included in (Maclntyre 48 ' 49 ' 50 ). During times when wind speeds were less than 3 m s _ 1 , values ranged from detection (10~ 10 m 2 s - 3 ) up to 10~ 7 m 2 s - 3 . At Mono Lake, CA, when winds were low but sometimes reached 4 m s _ 1 , dissipation rates occasionally exceeded 10~ 6 m 2 s - 3 (Maclntyre and Jellison 53 ). A recent study in the Soppensee, a small (25 ha) sheltered lake in Switzerland, employed thermistor chains, a microstructure profiler, measurements of surface meteorology, and eddy covariance techniques to determine key processes driving gas flux (Eugster et al. 8 ). Wind speeds were always less than 3 m s _ 1 . During times when heat inputs exceeded heat losses, the variance in temperature gradient in 15 minute bins, a proxy for turbulent mixing, at 4.4 m depth was negligible. In contrast, the variance was higher at times when heat losses exceeded heat gains. Similarly, bin averaged temperature gradients between 4.5 and 7 m depth increased at times of heat loss. At these times, gas flux was elevated 30% over stratified periods. Eddy covariance measurements at Toolik Lake, AK, showed gas fluxes were three times higher during periods of heat loss than during stratified periods (Maclntyre et al. 54 , Eugster et al. 8 ). Schladow et al. 55 describe the formation of thermal plumes and show the dependence of the gas transfer coefficient for oxygen on heat loss. These data clearly illustrate the importance of convective motions due to heat loss for gas exchange at low wind speeds and indicate that use of a surface renewal model to calculate gas transfer coefficients would more accurately represent the turbulence at the air-water interface than a model based on wind speed alone.

4. Sheared Air-Water Interface In Sec. 3, turbulence structure and scalar exchange were discussed for situations in which turbulence was generated by motions in the water column in the absence of significant wind shear. Field studies indicated the importance of heat loss from the water as a mechanism for enhancing turbulent mixing and hence gas transfer at the surface. The structures in the interfacial region therefore arose from interactions with far-field turbulence and thermal plumes generated due to cooling of the surface layer. Such mixed convective flows near the air-water interface have not, however, been studied in the laboratory or in simulations. In this section we turn to phenomena that occur when wind shear dominates in the generation of near-surface turbulence.

The Air-Water

4.1. Laboratory

Interface:

Studies


and

205

Exchange

Simulations

A useful starting point is to consider a simple flow situation as shown in Fig. 13, which is a schematic of pressure-driven air and water streams between two horizontal plates. ^^r

Side view

yiiituruniiit

to pulsed current source

, , , , , , , , , , 1 1 1 } > J 1>I>!

1 >J 1 I i { I

. 25 Jim I t wire (vertical)

if

air interface

nuriamtal wire

. .

^j

W plauvkw

.

#

microtabWe tracer — lines

* • water

to pulsed current source

^ V S ' - V V U T O ; ivv'.i'H. v . v vvs. Vv s vv.v ^ v u v v u liorizoiilnl (spanwiK) Ptwirc

^

-microbubbte tracer lines

vertical wire _.î

)A\\\\wlî^^wiv^^v\.vvvv^^srgc7'

N* Fig. 13. Schematic of a co-current or countercurrent gas-liquid flow between two horizontal plates. This configuration defines a canonical experiment for investigations of turbulence and scalar exchange at gas-liquid interfaces. The wires are used in experiments to electrochemically generate microbubble lines that act as flow tracers - a device widely used to visualize turbulent flow structures.

Flat sheared interface First, let us consider the simplest situation where the air flow exerts shear on the liquid but not enough to form significant interfacial waves. Now consider what happens to the liquid near the bottom boundary, as this will serve as a reference. If the flow is visualized with lines of micro-bubble tracers which are generated by passing a pulsed current through a spanwise line, the streaky structures shown in Fig. 14 are seen to form close to the bottom boundary.

206

S. Banerjee and S.

Maclntyre

Fig. 14. Visualization of turbulence structure near the bottom boundary of turbulent channel flow. The flow tracers are microbubbles electrochemically generated by pulsing current through a spanwise (horizontal) wire lying a small distance above the channel floor. The flow structure consists of high-speed regions with alternating low-speed, "streaky", regions where the microbubbles accumulate. These regions form and reform and move around but their essential streaky character is always there.

These structures have been known since the pioneering experiments of Kline et al. 56 , who also observed that the low-speed regions would periodically be ejected in bursts, as shown in Fig. 15, which is a side view of the flow with the microbubble-generating wire placed vertically. It is known that these bursts or ejections, and the related sweeps that replace the ejected fluid, strongly influence heat and mass transfer at a wall. Bursts in boundary layers appear to be large-scale motions that contain within them several ejections away from the boundary with interspersed sweeps that bring fluid toward the boundary. A question that arises is whether such structures also occur near gas-liquid interfaces on the liquid side. They can be expected on the gas side, since the liquid surface, to a flowing gas, looks much like a wall. However, to the liquid, the interface is almost a slip surface, with a mean shear being impressed due to the gas flow. The boundary conditions are very different from those at solid walls, and it is not clear what is to be expected. To clarify this, Rashidi and Banerjee 57 carried out experiments in which they visualized the flow structure in a flowing stream of liquid, with shear imposed by the air flow. Fig. 16, upper two panels, shows structures close to the gas-liquid interface, on the liquid side, as the shear rate due to the gas motion is increased. One goes from a patchy structure at low shear to lowspeed, high-speed streaky structures that are qualitatively similar to those observed near a wall. Furthermore, if the flow is viewed from the side, one sees burst-like structures emanating from the gas-liquid interface, much like



207

Fi&. J.'i. Rule VH'-W of £». i-liannol flow, showing development of a burst which consbtf-; of an "active" period in which the low speed region shown in Fig. 14 appear to be lilted up in "ejections" followed by downdrafts of fluid called "sweeps" that bring high speed bulk fluid to the wall region. Flow is from right to left and the panels should be viewed starting on the right.

those seen in wall turbulence (see Fig. 16, lower panel). This led Rashidi and Banerjee 57 to conclude that shear rate was the primary determinant of structure in such situations, and the details of the boundary condition were secondary. There are, as discussed later, important quantitative differences in the interface low field on the liquid side from wall turbulence, though the qualitative features are similar. The criterion for transition from the patchy structures seen at low shear to the streaky structures was clarified through direct numerical simulations by Lam and Banerjee 58 . They showed that the criterion for formation of streaky regions near a boundary depends on a mean shear rate (non dimensionalized in such a. way that it is effectively a turbulence production to dissipation ratio)'being greater than one. Lombard! et al. 59 did a direct numerical simulation where the gas and liquid was coupled through continuity of velocity and stress boundary conditions at the interface. They artificially raised-the surface tension to maintain a fiat interface, as waves would introduce additional complexity. The interfacial plane itself shows regions of high shear stress and low shear stress, with low shear stress regions corresponding to the low-speed regions and the high shear stress to the high-speed regions. The low shear stress regions were streaky in nature with high shear stress islands. Lombard! et al. showed that the high shear stress regions occur below the sweeps on the gas side, i.e., the motions that bring high-speed fluid from the outer regions to the interfaces on the gas side. Conversely, ejections on the gas side, which take low-speed fluid away from the interface into the outer flow, strongly correlate with low shear stress regions. The liquid does not behave in this way and the ejections and sweeps do not correlate with shear stress on the interface. The difference between the gas and the liquid phases in the nearinterface region is further clarified by observing the velocity fluctuations

208

S. Banerjee and S.

Maclntyre

H$K;

Fig. 16, Upper panels: Structures at a gas-liquid interface with decreasing wind shear (a) to (d). Lower panel: Side view of a channel l o w with the l o o r at the bottom and the interface at the top of each panel. The top series of panels are for air flow countercurrent to the liquid. Qualitatively similar burst-like structures are seen at the interface, as well m at the channel l o o r . The bottom series of panels indicates bursts formed in cocurrent gas-liquid flow. Bursts are formed at the interface due to wind shear. Flow is from right to left, and time increases in the panels from right to left.



Exchange

209

on each side of the interface as shown in Fig. 17. The left panel of the figure is for the gas, whereas the right panel is for the liquid. Gas-side turbulence, as is evident, behaves much like flow over a solid wall, i.e., the fluctuations are almost identical to that at a solid boundary in all directions - streamwise, spanwise, and wall-normal. If the distance is measured from the wavy interface, then there is little effect of waves on intensity provided that everything is nondimensionalized with the friction velocity calculated by subtracting the form drag from the total drag. On the other hand, the liquid, as evident from the bottom figure, has the largest fluctuations in the streamwise and spanwise direction right at the interface itself. It sees the interface virtually as a free slip boundary, except for the mean shear. There is a somewhat greater effect of waves, but still rather small in the nondimensionalized form shown. These calculations are done with density for the gas and the liquid typical of air and water, i.e., ~ 1 : 1000. The cases with waves are at conditions where there is no microbreaking.

Fig. 17. Turbulence intensities on the gas side (left panel) and liquid side (right panel) as a function of nondimensional distance z+ from the interface, velocity U\-streamwise, U2spanwise, C/3-interface-normal. Case Do-flat interface; Di-wavy interface. All quantities nondimensionalized by the friction velocity, i.e., the shear velocity calculated on the basis of frictional drag(excluding form drag) at the interface.

Certain other aspects of turbulence on each side of a gas-liquid interface are worth considering. In the wall region, it is known that ejections and sweeps have a spacing in time that lies between a non-dimensional time of 30-100, with 100 being the upper limit. This last period corresponds to the time between assemblages of ejections and sweeps, i.e., bursts. This is shown in Fig. 18 for wall turbulence (i.e., at a solid boundary) as a function of the shear Reynolds number. Rashidi and Banerjee 57 found that on the liquid side of a sheared interface, this type of parameterization held. In Fig. 18

210

S. Banerjee and S.

Maclntyre

the time between interfacial bursts is also plotted as a function of the shear Reynolds number, and the behavior at a macroscopic level is very similar to that at a solid boundary. Thus, the qualitative behavior is similar, though within the bursts themselves there are substantial differences between the gas and liquid sides. I

' '•'

i " ""•

1

' • ' '"I

"

"I

100

•oo °

80 T

o

°

;

60

40 4n

"9

• O

intvriact tux K» wall bursts lit w i n I

2500

i* i

In

iini

5000 Re L

t

i i n* i ,». mmli

T500

10000

•

12500

Fig. 18. Scaling of interfacial and wall bursts. The time between bursts Tg is scaled with the local shear stress and kinematic viscosity.

Turbulence near nonbreaking wavy surfaces The previous discussion focused on turbulence phenomena for coupled gasliquid flows with fiat interfaces - a situation close to the experiments of Rashidi and Banerjee 57 . To understand the effect of waves on the gas side, flow over a surface with two dimensional waves (with spanwise crests and troughs) was investigated by direct numerical simulation (De Angelis et al. 60 ). The waves introduced a streamwise length scale, as indicated by the vortex structure shown in Fig. 19. However, when the phase average velocities are removed from the instantaneous field, then, typically, streaky regions of high speed-low-speed flow are seen, as indicated in Fig. 20. The streak spacing is ~ 100 in units nondimensionalized with the frictional drag (removing the form drag component from the total drag). This suggests that turbulence phenomena near the wavy surface may scale with frictional drag, rather than total drag. On examining the time between sweeps and bursts, a time non-dimensionalized



Exchange

211

a) Fig. 19. Quasi-streamwise vortical structures (shaded) over a wavy boundary with flow viewed from the side. The different shadings indicate different directions of rotation.

Fig. 20. Streaky structures over a solid wavy boundary with different wave lengths. The top left-hand and right-hand panels have the mean velocity removed and clearly show the streakiness of the high and low velocity regions very near the surface. The second row shows the velocities with the mean velocity included, and the third row shows the local shear stresses.

with the frictional drag, i.e. t+ — tDf/fi, is seen to be between 30 and 90, in agreement with what happens at a flat wall. Here Df is the frictional stress. These simulations are of interest for gas-side turbulence phenomena. However, liquid-side behavior may be expected to be substantially different, as discussed for the flat-interface case. De Angelis and Banerjee 61 have reported some details of DNS with a nonbreaking deformable interface between turbulent air and water streams. Further details are available in De Angelis62. The turbulence intensities and other qualitative features, e.g.

212

S. Banerjee and S.

Maclntyre

streak spacing and burst frequency, on both gas and liquid sides of the interface, were found to scale with upict and v, the kinematic viscosity. Fig. 21 shows an instantaneous snapshot of the interface configuration, together with the high-speed/low-speed streaky structures close to the interface on each side. It should be emphasized that these results only hold for situations where the interface does not break.

Fig. 21. Instantaneous configuration of the gas-liquid interface showing spanwise wave crests. The colors indicate velocity just above the interface (top panel). Red indicates high velocity, blue indicates low velocity, with yellow and green in between. The typical streaky structures seen in experiments are evident in these simulations as well. The bottom panel is just below the interface at the same instant and shows some differences in the local velocities.

Turbulence in microbreaking When Lîo exceeds 3 to 5 m/s, relatively short waves steepen and start to microbreak. This is illustrated in Fig. 22, which shows a microbreaking wave viewed from the side. If microbreaking waves are viewed from the top, they are easily visualized by following the trajectories of particles scattered on the water surface. It is found that the liquid does not cross the crest of microbreaking waves, which consequently forms regions of convergence where particles gather. Therefore, an operational way of identifying microbreaking is to observe particles gathering at, and being convected with, the wave crests (see Fig. 23). Non-breaking waves do not show this behavior particles on the liquid surface pass through wave crests. Regions of high surface divergence form behind microbreaking waves, as illustrated in Fig. 23. This may be expected to significantly impact scalar exchange, as indicated in the "surface divergence" Eq. (6a). The subsurface motions associated with the convergence zones at the wave crests and the

The Air- Water Interface: Turbulence and Scalar Exchange

213

ffortzuntal Ptnftfcui (an) Fig. 22. A side Yiew of a microbreakiiig waYe showing typical dimensions.

divergence zones behind remain to be clarified. At this point, we have seen plunging motions associated with microbreaking that appear much larger than those associated with bursts that develop due to wind shear in the absence of microbreaking (Leifer et al. 63 ). This is illustrated in Fig. 24, which shows that the plunging motions associated with microbreaking are almost an order of magnitude larger than motions associated with the "usual" bursts. The impact of microbreaking structures on scalar exchange has yet to be studied. 4*2. Laboratory

Studies

and Simulations

— Scalar

Exchange

Simulation As simulations proved useful in clarifying aspects of turbulence structure near wavy (nonbreaking) air-water interfaces, they have been extended to studies of scalar exchange (De Angelis 62 ). The calculations for Pr or Sc ~ 0(1) are straightforward once the velocity field has been calculated. However, for higher Sc, the spacing of collocation points-normal to the interface must be rescaled, i.e., interface-normal mesh spacing must be reduced as a function of Sc to resolve concentration fluctuations (see De Angelis62 and De Angelis et al. 18 ). Contours of the instantaneous scalar fluxes at interfaces from such DNS are shown in Figs. 25 and 26, and compared with the shear stress at the interface for the gas and the liquid side, respectively. It is immediately

214

S. Banerjee and S.

0

10

20

Madntyre

30

40

Downwind Position (cm)

1 HI r 1

0

10

20

30

40

Downwind Position (cm) Fig. 23. Top view of a microbreaking wave, visualized by particles scattered in the water surface. The particles are swept along by the wave crests, which are clearly discernible in the top panel. The velocity vectors calculated for particle motion (by DP1V) are shown in the bottom panel. Note the convergence zone at the crest and the divergence zone behind.

evident that the gas-side luxes correlate well with the shear stress. This suggests that sweeps give rise to higher scalar exchange rates, as they also produce regions of high shear stress. On the other hand, the flux field on the liquid side shows a much finer structure (see Fig. 26) and no such correlation exists. It is of interest, therefore, to understand what processes control the liquid-side fluxes. To clarify what happens on the liquid side, recall the earlier discussion that liquid-side bursts (sweeps and ejections) do not affect interfacial shear stress, which is mainly determined by gas-phase motions.. Therefore, we must directly examine the relationship between liquid-side motions and



215

Fig. 24. Bursts formed by microbreaking shown in bottom 6 panels G-L, and the usual bursts in top 6 panels A-F. Scales are the same. Microbreaking leads to "giant" bursts. The vertical dimensions in the pictures are about 8 cm, and the liquid layer is 10 cm deep (the bottom is not shown).

liquid-side scalar transfer rates. The procedure is illustrated in Fig. 27. Here we plot the instantaneous interface-normal velocity fluctuations, together with the instantaneous Reynolds stress (t^t^), aDLd the instantaneous scalar flux (ui is the velocity fluctuation in the streamwise direction and tig is in the interface normal direction). It is clear from the figure that the scalar flux increases sharply when there is a high velocity (t4) towards the interface of

216

S. Banerjee and S.

Maclntyre

(a)

Fig. 25. (a): The mass l u x field on the gas side of the interface. Note close correlation of regions of high mass transfer with high shear stress. (b):The non-dimensional shear stress field on the liquid side of the interface. The time at which the field is shown is not exactly the same as in Fig. 25(a).

high velocity bulk fluid (high u^). This is a sweep. The figure illustrates that the sweep, which leads to a high surface divergence event, is strongly correlated with high liquid-side scalar flux, which decays like what might be expected of transient absorption into a batch of stagnant fluid between sweeps. This suggests a basis for the "surface renewal" model with sweeps providing the renewal events - as developed previously to parameterize the liquid-side scalar flux in Eq. (10). Scalar flux parameterization The previous discussion suggests a physical basis for the surface renewal theory used to parameterize scalar transfer rates on the liquid side. It is of interest to see how Eq. (10) compares with data and simulations. The

The Air- Water Interface;


217

(a)

Fig. 26. (a): The shear stress field on the gas-side of the interface. The shear is nondimensionalized by pu*2f where u* is the average friction velocity. The streamwise dimension is 4wh and the spanwis© dimension is 2wh, where the gas-layer thickness is 2h. (b): The mass flux field on the liquid side of the interface. Notice the lack of correlation with the shear stress field. The mass flux field is much more fine grained. The time at which the field is shown is not exactly the same as in Fig. 25(b).

Schmidt number dependencies in Eqs. (10) and (11) are compared with, simulation results for two different Schmidt numbers in Fig. 28. It is clear that the Schmidt number dependence is correctly predicted at high Schmidt numbers, but there is some deviation at low Schmidt numbers. Also, the numerical value of the RHS of the equations is roughly correct. Eq. (10) is also compared with wind-wave tank data for SFg and OO2 transfer rates from Wanninkhof and Bliven64 and Ocampo-Torres et al. 65 in Figs. 29 and 30, respectively. The agreement is quite good, though the prediction always lies somewhat below the data. This could be due to a small effect related to turbulence from the channel bottom reaching the interface, which enhances the local turbulence. Turning now to the gas side,

S. Banerjee and S.

218

Maclntyre

ax

? D*"

-10.0 0,0

100.0

r

200.0

300.0

Fig. 27. Panels showing that a sweep, with high u i « 3 gives rise to a high mass transfer rate /3'L on the liquid side. Note t h a t the surface renewal model predicts the instantaneous and integrated value of 0'L quite well if the sweep is considered as a "renewing" event.

id-

ler 10' LIQUID SIDE Fig. 28. Gas transfer velocities nondimensionalized by the frictional velocities versus Schmidt number from direct numerical simulations with a flat interface. The solid lines are from Eqs. (11) and (10) for the gas and liquid sides, respectively. Left panel-gas side, right panel-liquid side.

the gas sees the liquid much like a solid surface, as discussed earlier. So, the surface renewal theory has to be modified somewhat for such applications, as in Eq. (11). This also leads to a different dependence on the Schmidt number. While this is encouraging, the issue of estimating the friction velocity



Exchange

219

60.0 •

4O.0

20.0 •

100.0 U., (ems') Fig. 29. Gas transfer velocity for SF6 desorption in the Delft wind-wave tank versus shear velocity (gas-side), from Wanninkhof and Bliven 6 4 . The open triangles may have been data taken in breaking conditions. The solid lines are from Eq. (10) and the same equation +50%.

10'

o

o •

o ° #~

o

M

10"

10"

10'

10'

10"

Re.

10'

10'

Fig. 30. Gas transfer velocity, non-dimensionalized by the shear velocity plotted against the shear Reynolds number (based on liquid depth and artificial shear velocity) from Ocampo-Torres et al. 6 5 The lines are Eq. (10) and the same equation +50%.

M

frict remains. This is a difficult estimate to make for field data. Form drag may be expected to increase with wave amplitude and steepness, therefore

220

S. Banerjee and S.

Maclntyre

estimating u* from total drag and using it in the proposed parameterization will significantly overestimate the scalar exchange rates. The effect of microbreaking has to be factored in as it is not clear whether these parameterizations will be applicable in such circumstances. Finally, as discussed in the next section, evaporation, and hence heat loss, from the water increases with wind speed, giving rise to natural convective effects that enhance windinduced turbulence. Such effects are important in field experiments, which are now discussed.

4.3. Field

Studies

In field settings, as wind speeds increase that increase shear, evaporation rates also increase, and at least two mechanisms for generating turbulence co-occur. Shear on the water side of the air-water interface is parameterized by the friction velocity u*w, where r = pwu2w = PO.CDU2 and pa and pw are density of air and water respectively, CD is the drag coefficient, and U is wind speed. A similar parameter exists for heat loss, the convective velocity scale w*, and is calculated from heat gains and losses into the actively mixing surface layer (Imberger 66 ). Only when heat gains exceed heat losses will the surface of the water body be influenced by shear alone. More often, heat loss and shear co-occur at the air-water interface of lakes, wetlands and oceans and contribute to turbulence production. Examples of profiles showing thermal structure and dissipation rates near the airwater interface for winds up to 12 m s _ 1 are provided in Maclntyre 48 ' 49 ' 50 , Robarts et al. 51 , Maclntyre et al. 52 , Maclntyre and Jellison 53 . A full surface energy budget was not completed for those studies, but dissipation rates in Mono Lake in 1995 reached 10~~5 m 2 s - 3 when winds were 12 m s"1 and significant white capping occurred. Surface meteorological data obtained from a meteorological station (http://ecosystems.mbl.edu/ARC/data_doc/lanwater/landwater default.htm) on Toolik Lake, Alaska, a 1.5 km 2 kettle lake in the foothills of the Brooks Range, illustrate the variability in surface forcing over aquatic ecosystems (Figs. 31-33). Data are from 12 and 14 July 2000 (day of year 194 to 197). Five minute averaged wind speeds were less than 4 m s _ 1 at night and increased in the afternoon to highs between 6 and 8 m s _ 1 , a range where microbreaking and shear are both important (Fig. 31 A). Solar insolation, which acts to stratify the lake, was barely detectable for a few hours at midnight and had daily maxima of 800 W m 2 (Fig. 31B). Intermittent cloudiness caused large fluctuations in irradiance. A surface energy budget was calculated as in Maclntyre et al. 67 Surface heat fluxes, which are



Exchange

221

the sum of heat fluxes from sensible, latent, and long wave radiation, were always negative, ranging from —100 to —400 W m - 2 (Fig. 31Ca). The effective heat flux (Fig. 31Cb) represents the actual heating of the upper surface layer. The surface layer is defined as the actively mixing layer near the airwater interface and operationally as having temperatures within 0.01 °C of those measured within the upper 5 cm of the lake. When the effective heat flux is greater than zero, the uppermost part of the water column will stratify; when it is less than zero, cooling occurs and turbulence can be induced not only by shear or microbreaking waves, but also by heat loss. Heat losses occur in the surface layer for nearly 12 hours of each 24-hour period and for an even longer period on cloudy days. Wind consistently induces some shear at the air-water interface; the shear velocity, u*, varied from 0.002 to 0.01 m s _ 1 (Fig. 32A). The convective velocity scale w„ is more intermittent due to solar heating. Values of w* are comparable to the friction velocity when winds are high and solar radiation low (e.g. day 194.8) and are zero when heat inputs exceed losses. At night, when wind speeds are low, tu* is often twice as high as it*, indicating heat losses will be the major source of turbulence at the air-water interface. Surface layer depths were calculated using data from Brancker TR1050 thermistors on a taut line mooring with loggers located at 0.05, 2.55, 3.3, 3.55, 3.8, 4.05, 4.3, 4.55, 4.8, 5.05, 5.3, 5.55, 5.8, 6.55, 7.55, and 8.55 m below the air-water interface (Fig. 32B). Due to insufficient thermistors in the upper water column, the surface layer was generally assumed to be 2 m deep during times of heating. The depth of the surface layer varied due to cooling and, during periods of high wind (e.g. Day 195.7), is confounded due to tilting of the thermocline. At these times, the surface layer deepens downwind; shallows upwind. The flux of energy into the mixed layer (Fig. 32C) was calculated following Imberger 66 as Fq — ql/2 — (wl + 1.333u»)/2, an approach that does not explicitly include the effects of microwave breaking, q represents the turbulent velocity scale due to both shear and heat loss. The rate of dissipation of turbulent kinetic energy (e) for the surface layer is estimated by dividing Fq by the depth of the actively mixing surface layer. Based on this approach, e rang ed from 1 0 - 1 0 to l O - 7 m s (Fig. 33A). Dissipation rates are underestimates during heating periods and when the thermocline is strongly tilted due to overestimates of the depth of the surface layer. Comparison of the flux of energy into the surface layer when winds dropped to 1 m s - 1 (u*w = 0.002 m s _ 1 ) but the lake was either cooling (Day 196.1, w* = 0.006 m s - 1 ) or gaining heat (Day 196.35, w* = 0) illustrates the contribution of heat loss to turbulence in the surface layer when shear is low. When cooling occurred (Day 196.35), Fq for the

222

S. Banerjee and S.

194

194.5

195

Maclntyre

195.5

196

196.5

197

196.5

197

196 5

197

1000

194.5

195.5

400

HeatFrluxes f

a-

200 0 -200 -400 194

194.5

195

195.5

196

Day of Year 2000 Fig. 31. (A): Five minute averaged wind speeds (m s _ 1 ) , B) solar insolation (W m - 2 ) and C) surface heat flux (a.) and effective heat flux (W m - 2 ) (b.) into Toolik Lake from July 12-15 (day of year 194 - 197) 2000 (unpublished data, Sally Maclntyre and George Kling).



Exchange

223

surface layer was 10~ 7 m 3 s - 3 and e was 10~ 8 m 2 s~ 3 , whereas when the surface layer was gaming heat (Day 196.1), Fq was 1 0 - 9 m 3 s - 3 . e was less than 10~ 9 m 2 s~ 3 . Generally, winds drop at night when cooling occurs, so the heat loss leads to continued production of turbulence and often to entrainment of deeper waters into the upper mixed layer. Temperature-gradient microstructure profiles illustrate the thermal structure, rates of dissipation of turbulent kinetic energy, and eddy sizes during periods with 7 m s _ 1 and considerable convective cooling (Fig. 34, 35) and during a period when winds were 4 m s " 1 but heating was occurring (Fig. 36). The temperature profile at 2014 h 12 July 2000 showed slightly cooler temperatures near the surface, a 0.02 °C increase in temperature by 4.5 m, and a gradual decrease in temperature to the base of the mixed layer at ca. 6.1 m (Fig. 34). Such a pattern is typical of conditions when convection due to heat loss occurs (Shay and Gregg 45 , Anis and Mourn 46 ). Temperature fluctuations of 0.01 °C and smaller were ubiquitous. Below the mixed layer, temperatures decreased rapidly. Eddy sizes are computed by calculating the displacement scale, the depth to which cooler water would descend to find water of similar density. In this case, the upper 50 cm may be considered an eddy, and dissipation rates are slightly higher within it (10~ 7 m 2 s~3) than in the water immediately below. A second eddy occurred between 0.5 and 3 m depth, and dissipation rates within it were ca. 3 x 10~ 8 m 2 s - 3 . A third eddy extended from 3 to 5 m and had similar dissipation rates. Despite this structure, the near uniformity in e and the overall pattern in temperature indicates this is a convectively mixed layer. A profile taken 15 minutes earlier had a similar temperature profile and average dissipation rate, but no accentuation of dissipation in near surface waters (not shown). While the profile at 2043 h (Fig. 35) was also obtained during a cooling period, the 6 m deep upper mixed layer does not have the classic structure of a convecting layer, but instead consists of 3 layers. In the uppermost one, temperature ranged from 11.97 to 12.01 °C and dissipation rates were 10~ 6 m 2 s - 3 at the surface to and decreased to 3 x 10~ 8 m 2 s - 3 at its base at 1.5 m. The higher values at the surface may have been due to the combination of shear, heat loss, and microbreaking. While the middle layer may have been an overturning eddy, the step changes associated with the top and bottom of the layers suggests lateral advection was occurring; the enhanced dissipation rates at the top and bottom of the lower layer support this interpretation. Dissipation rates estimated from the surface energy budget for the entire 6 m upper mixed layer were 10~ 7 m 2 s~ 3 (Fig. 33), which is

224

S. Banerjee and S.

Maclntyre

0.015 u. w»

A.

$ y

M.

•a

« 0.005

v* • .. K , , UJJBL 194.5 194

' f*V-

II

•

•

•

•

1

195

1195.5 II

V : fe ? . , . , • , ,

196

:«A

, , ,

196.5

197

194

194.5

195

195.5

196

196.5

197

194

194.5

195

195.5

196

196.5

197

Day of Year 2000 Fig. 32. (A): Friction velocity in water u» and convective velocity wm, (B): depth of the surface layer, and (C): flux of energy into the surface layer for days of year 194-197, Toolik Lake, AK. (unpublished data, Sally Maclntyre and George Kling).


'«


Exchange

225

10

197

195

195.5 Day of Year 2000

196

196.5

197

Fig. 33. (A): Estimates of rates of dissipation of turbulent kinetic energy (e) based on surface energy budgets and depth of the surface layer derived from thermistor chains. Arrows indicate revised estimates of e when the depth of the surface layer was determined from high resolution temperature profiles in Figs. 35, 36(B).) Gas transfer coefficient calculated from surface divergence model (Eq. 8) ( • • • ) , and two empirical wind based models (Cole and Caraco 4 3 , gray, and Kling et al. 6 8 , black). Data for these models were derived from the meteorological data from Toolik Lake (unpublished data Sally Maclntyre and George Kling); calculations are described in the text. Gas transfer velocity is augmented over wind based models when heat losses are significant and reduced when surface waters gain heat (see Fig. 32A).

226

S. Banerjee and S.

Maclntyre

Temperature f C)

Fig. 34. Temperature profile and rates of dissipation of turbulent kinetic energy (histogram) at 2014 h on 12 July 2000, Toolik Lake, AK., show the structure within the 6.2 m deep upper mixed layer at times when the mixed layer is losing heat and winds reached 7 m s _ 1 . e is higher near the air-water interface and typically between 1 0 - 8 m 2 s - 3 and 1 0 - 7 m 2 s - 3 in the rest of the surface layer. Dissipation rates calculated as in Maclntyre et al. 5 2 .

in reasonable agreement with the measured values. However, recalculating e from the surface energy budget based on a surface layer depth of 0.80 m gives dissipation rates of 10~ 6 m 2 s - 3 which is representative of conditions at the surface (Fig. 33A). Four microstructure casts were obtained in the period from 1978 h to 2051 h 12 July, and the measured dissipation rates in surface waters ranged from 10~ 8 m 2 s - 3 to 10~ 6 m 2 s - 3 . This range reflects the dynamic nature of the processes at the air-water interface. The microstructure cast taken at 1618 h on 14 July was obtained while the 4.7 m upper mixed layer was gaining heat (Fig. 36). Surface temperatures had increased by a degree since noon; winds were 4 m s - 1 . While turbulence occurred throughout the 4.7 m upper mixed layer, dissipation rates were 10~ 6 m 2 s~ 3 in the uppermost 0.8 m. Eddies below it ranged in size from a few centimeters to 50 cm indicating the upper mixed layer did



227

Exchange

Temperature f C) 11.7 Qj

11.74 11.78 11.82 11.86 •

•

•

•

•

11.9 •

11.94 11.98 12.02 12.06 ^pqsss-| I

12.1

I

Fig. 35. Temperature profile and rates of dissipation of turbulent kinetic energy at 2043 h on 12 July 2000.

not mix as an entity. Dissipation rates were at least an order of magnitude lower than in the surface layer. Eddies embedded in stably stratified waters indicate shear induced turbulence. Estimates of dissipation from the surface energy budget (Fig. 33A) using a surface mixing layer of 0.25 m, based on the step change in temperature at that depth, were slightly less than the value of 1 0 - 6 m 2 s - 3 obtained while profiling. In summary, the distribution of turbulence in the upper mixed layer differs during times with and without cooling. In nearly all cases, turbulence is highest in a shallow layer near the surface. When heating occurs and winds are moderate, the turbulence is more accentuated in surface waters and decreases considerably by the base of the mixed layer. When cooling co-occurs with wind forcing, the turbulence in the upper mixed layer tends to be more homogeneous and eddies are larger. However, even that pattern is variable, possibly due to horizontal advection of different water masses.

228

S. Banerjee and S.

Maclntyre

Temperature f C) 11.14 11.28 11.42 11.56

11.7

11.84 11.98 12.12 12.26

io" £ (m 2 s 3 p '

12.4

10

Fig. 36. Temperature profile and rates of dissipation of turbulent kinetic energy at 1618 h on 14 July 2000 show the structure within the 4.7 m deep upper mixed layer, e is higher in the upper 0.8 m; shear induced eddies occur below.

Ho et al. 69 have calculated the flux of kinetic energy into the mixed layer due to rainfall ranging from 13.6 to 115.2 mm h~x. The fluxes ranged from 0.09 to 1.27 W m~ 2 . Assuming that the energy was initially introduced to a depth of 0.2 m, as supported by measurements of temperature change and SF6 distribution in experiments with rainfall of 68 and 76 mm ft.-1 (David Ho, personal communication), and converting to W k g - 1 (equivalent to m 2 s~3) by assuming a density of 1000 kg m - 3 , gives dissipation estimates on the order of 5 x 1 0 - 4 m 2 s - 3 . These values are an order of magnitude higher than has been observed with microstructure profiling at low to moderate winds and indicates that intense rainstorms could dominate the surface energy budget under those conditions. Dissipation rates under breaking waves in Lake Ontario ranged from 10" 5 to 10" 2 m 2 s" 3 (Agrawal et al. 70 , Terray et al. 71 ). That e did not follow law of the wall scaling in the near surface layer was attributed to the more intense turbulence during white capping and wave breaking.



229

5. Calculation of Gas Transfer Coefficients During Low and Moderate Wind Speeds Most attempts to relate gas transfer velocities to surface meteorology have used wind as the independent parameter (Liss and Merlivat 1 , Kling et al. 68 , Cole and Caraco 43 , Wanninkhof and McGillis2, Crusius and Wanninkhof 44 ). However, the surface renewal model takes into account the wide variety of processes occurring at the air-water interface (Crill et al. 72 , Soloviev and Schluessel20, Maclntyre et al. 5 , Eugster et al. 8 ). As we have seen above, dissipation rates in near-surface waters calculated from surface meteorological data and depth of the surface layer obtained from high resolution profiles are in good agreement, or are less than, those obtained from temperaturegradient microstructure profiles. Estimates might be improved were the contribution of microwave breaking to energy flux into the surface layer included. Gas transfer velocities (keoo) were calculated using the small eddy, SE, (Eq. 5, 5a this paper) the surface divergence, SD, (Eq. 8 this paper), and the Soloviev and Schluessel20, SS, versions of the surface renewal model (see Table 1) using five minute averaged meteorological data from Toolik Lake, AK. The 600 implies the values are normalized to CO2 at 20 °C. For the small eddy version of the surface renewal model, the gas transfer coefficient k = a\Dll2{e/v)xlA. D is molecular diffusivity of the gas. e is the rate of dissipation of turbulent kinetic energy and nu is kinematic viscosity. When written in terms of the Schmidt number and turbulent Reynolds number, Ret = ul/u, the expression becomes kSc1^2 = c\ uRe 4 . e is related to u and / through the expression e = u3/l (Taylor 73 ), e and I can be obtained from temperature-gradient microstructure profiles in which e is estimated from a least squares fit of the power spectral densities of the temperaturegradient signal to the Batchelor spectrum and I is the overturning scale. Here e is calculated from surface energy budgets (Maclntyre et al. 67 ) in which the energy flux into the surface layer Fq = (w3 + 1.33 3 u 3 )/2 and e = 0.82Fq/l and I is the depth of the surface layer (Imberger 66 , Maclntyre et al. 5 ). We let c\ = 0.56 as in Crill et al. 72 but recognize that this coefficient was obtained in laboratory experiments and has not been validated in the field. To correct for the somewhat lower estimates of e from surface energy budgets relative to microstructure profiles due to lack of sufficient selfcontained temperature loggers, e was increased five fold (e.g., Fig. 33A); mixed layer depths during low winds (< 3 m s _ 1 ) and heating (iv* < 0.003 m s _ 1 ) were assumed to be 0.3 m.

230

S. Banerjee and S.

Maclntyre

Following Soloviev and Schluessel20, k = 1 . 8 5 , 4 ^ * Sc~1/2(1 + Rf0/RfCI)lA(l

+

Ke/Kea)-1/2

where Ao, based on cool skin data, is 13.3, Rfo is a surface Richardson number, and RfCI is a critical Richardson number determined to be 1.5 x 10~ 4 . H is surface heat flux obtained from summing latent and sensible heat fluxes and long wave radiation flux (e.g. Fig. 31C, curve a.). Ke is the Keulegan number, v%/gv whose critical value is 0.18. a is the coefficient of thermal expansion, g is gravity, u is kinematic viscosity, and cp is specific heat. The spread in calculated gas transfer coefficients at each wind speed and for each method, as evidenced by the 9% confidence intervals (Fig. 37, Maclntyre, Shaw and Kling, to be submitted), is due in part to the enhancement of turbulence when heat loss is included and its decrease when stratification decreases heat loss (Fig. 32A). Curves for gas transfer velocities as a function of wind speed alone developed from field studies (Cole and Caraco (CC) 43 , Crusius and Wanninkhof (CW) 44 , Kling et al. (K) 68 ) are plotted for comparison. CC and SE give similar estimates at all wind speeds and are low in comparison to the other approaches at moderate winds. CW was forced to zero at low winds but is similar to SS and K at moderate winds. The data on which CC and CW are based only includes winds up to 5 and 6 m s _ 1 , respectively. Gas transfer coefficients obtained from experiments (Clark et al. 74 , Crusius and Wanninkhof44, Maclntyre et al. 5 ) are plotted with the five curves in Fig. 38. Considerable scatter is evident in the experimental data at both low and moderate wind speeds. At wind speeds below ca. 4 m s _ 1 , the majority of the estimates of k are less than 3 cm hr _ 1 . In a recent laboratory study of gas flux due to heat loss, gas transfer coefficients, normalized to CO2 at 20°C using a Schmidt number of 530 for oxygen, were 0.4 and 0.9 cm h~~l for outgoing heat fluxes of 400 and 570 W m~ 2 (Schladow et al. 55 ). These low values of k would support Crusius and Wanninkhof's 44 assumption of minimal flux at wind speeds less than 1 m s _ 1 . However, even at winds between 1 and 3 m s _ 1 , some of the data from Lake 302N and that from Sutherland Pond have values of k that exceed those predicted from CW and are within the range of estimates from SS. Field data of Cole and Caraco (not shown) indicate k ranges between 1 and 4 cm h r _ 1 at winds between 1 and 2 m s _ 1 . The low values in Schladow et al. 55 were obtained with an undisturbed surface; hence even the minor disturbances at low winds may contribute to vorticity that enhances gas flux (e.g. Maclntyre 75 ). At winds between 3 and 8 m s _ 1 , the gas transfer coefficients

The Air-Water

Interface:


231

from Lake 302N, Sutherland Pond, Mono Lake, Pyramid Lake and Crowley Lake are in the range predicted from SS, SD, CW, and K. That the field data, which was not collected over time intervals defined to identify times of convection, microbreaking, or high winds, tends to follow these four models at moderate winds suggests these models better include the greater diversity of processes contributing to gas flux than do SE or CC. Clark et al.'s 74 results from Sutherland Pond further indicate that estimates of gas transfer coefficients based on the typical averaging schemes in tracer studies may be too low. During two of his sampling periods, Clark's estimates of k were obtained during steady winds, 3.2 and 3.4 m s _ 1 . These k values are better predicted by SS, SD, CW, and K than by CC or SE. Experimental results from a wide variety of lakes are well predicted by two approaches using surface renewal, SS and SD, and results with these models are similar in the mean to those obtained following Kling et al. 68 , who primarily used other data to develop their regression. To illustrate the importance of including a surface renewal approach for estimating gas flux, k is calculated over 3 diurnal cycles using the meteorological and time series temperature data from Toolik Lake (Fig. 33B). K and SD provide comparable estimates of k during heating periods, but K underestimates k when heat loss occurs. For this data set, CC underestimates k by ca. a factor of 2 under all conditions. These comparisons indicate the potential greater accuracy of models for predicting k that include the various mechanisms that induce turbulence at the air-water interface. Frequently, gas flux estimates from lakes used in regional carbon budgets are based on k values of 1 to 3 cm h r - 1 (Kling et al. 76 , Cole et al. 77 , Richey et al. 78 ). While many of the sites have low wind speeds, estimates are likely to be too low by a factor of at least 2. In tropical waters, k estimated by SE was two times higher than CC (Maclntyre et al. 54 ). If the trends observed here apply to the tropical data, gas flux estimates from those regions may be underestimated by a factor of four. Ho et al. 69 show gas transfer coefficients ranging from 16 to 70 cm h - 1 for rainfall ranging from 13 to 115 mm h _ 1 . Rainfall of such intensity is most likely in tropical environments and would only occur intermittently, but gas fluxes would be comparable to those anticipated when winds are high enough to induce considerable wave breaking. Clearly, consideration of the wide diversity of processes that lead to enhanced gas fluxes will have considerable impact on estimates of global carbon budgets. These approaches will also provide better estimates of k for biological studies estimating net community productivity (e.g. Cole et al. 79 ). These results strongly indicate that further work evaluating the surface renewal model in field situations is warranted, with special attention to heat loss, rain, and microbreaking.

232

S. Banerjee and S.

Maclntyre

Fig. 37. Estimates of gas transfer velocity obtained from 5 minute averaged meteorological data from Toolik Lake, Alaska, using three different surface renewal models: small eddy (SE, Eq. 5a this chapter calculated as in Crill 7 2 ), surface divergence (SD, Eq. 8 this chapter) and following Soloviev and Schluessel (SS, 1994 20 ). 95% confidence intervals are indicated. Gas transfer coefficients calculated from wind based expressions (Cole and Caraco 4 3 , CC, Crusius and Wanninkhof 44 , CW, Kling et al. 6 8 , K) are shown for comparison. Curves illustrate two to three fold variations in calculations of flux are possible dependent upon model selection. The range in values for each wind speed using surface renewal is due to diurnal heating and cooling of surface waters.

6. Conclusions Turbulence phenomena and scalar exchange at the air-water interface have been discussed in terms of the models used for prediction, numerical simulations, laboratory experiments and field observations. The focus of the discussion has centered on evasion of sparingly soluble gases from water bodies. These processes are controlled by liquid-side turbulence, and the main resistance to transfer lies in a very thin region at the interface. As it is difficult to make measurements and do computations of phenomena in the vicinity of a deforming, and perhaps breaking, interface, our understanding is still at a much earlier stage than for such processes at solid boundaries. Nonetheless, considerable progress has been made in understanding the mechanisms that control scalar exchange, when the interface remains continuous (does not break). In the absence of shear, and when the far-field turbulence approximates the homogeneous isotropic case, it has been found that the Hunt-Graham 15


—

K CW

cc

. — 3D ,-.20

SE


• * o « n A

Exchange

233

Mono Lake Crowley Lake Pyramid Lake ^ Rockkmd Late Sutherland Pond Laka302N

I | 10

i

Fig. 38. Data from tracer experiments (see text for description) with curves of Fig. 37 for comparison show that gas transfer coefficients at moderate wind speeds are well predicted by Soloviev and Schluessel 20 which explicitly includes waves. Similarities of mean values for each wind speed of SS and SD with K and CW and field data support these surface renewal models and the need for experimental work to define gas fluxes as a function of all processes inducing turbulence at the air-water interface.

blocking theory applies quite well, and predictions of the near-interface damping of the normal component of turbulence, and enhancement of the tangential components, are well predicted. Banerjee4 has applied the HuntGraham theory to calculate the surface divergence and developed the socalled surface divergence (SD) model, which predicts gas transfer across unsheared interfaces rather well. It has also been shown that this model predicts at low turbulent Reynolds number the same behavior as the Fortescue and Pearson 12 large-eddy (LE) model, and the Banerjee et al. 13 small-eddy (SE) model at high turbulent Reynolds numbers. In situations in which the shear rate imposed by the wind is high, turbulence is generated in the vicinity of the interface, much like near solid boundaries. For situations in which the air-water interface does not break, models for gas transfer based on scaling of active periods (such as sweeps and ejections) with interfacial frictional shear, have proved to be successful in predicting laboratory data. However, microbreaking (breaking of waves of amplitude ~ 1 cm. and length ~ 10 cm) starts to occur at L710 of about 3-5 meters/sec, and very little is currently understood about the effect of these phenomena on turbulence generation and scalar exchange. Labora-

234

S. Banerjee and S.

Maclntyre

tory experiments indicate strong regions of convergence in the surface fluid motion near wave crests and regions of divergence behind, suggesting that such waves would markedly increase the r.m.s. surface divergence and hence gas transfer. Clearly, considerable work still needs to be done to characterize such microbreakers, both in laboratory experiments and in the field, and to better understand their effects on turbulence and scalar exchange. Field observations also suggest that natural convective motions due to cooling of the interfacial layer play a significant role by affecting both turbulence and scalar exchange over a wide range of wind velocities. This suggests that parameterizations based on wind speed alone are inadequate for calculations of mass transfer. The field observations also suggest that surface divergence approaches are quite fruitful, as they can accommodate a variety of processes that may lead to turbulence generation. The SD model is found to predict field data rather well even at high wind speeds. This is surprising because the model was originally developed for a no-shear interface based on Hunt-Graham theory. On the other hand, the SE model significantly underpredicts gas transfer coefficients at moderate wind speeds - possibly because energy dissipation due to microbreaking was underestimated. These discrepancies indicate better estimates of energy dissipation and length scales of the turbulence from the near-surface regions are needed in the presence of microbreakers. These analyses illustrate the utility of extending the surface divergence/surface renewal approaches, which were derived on the basis of theory and laboratory experiments, to field settings where turbulence is induced by a variety of processes. Much work, however, remains to be done. In particular, the effect of natural convective motions observed in the field and microbreaking need investigation. To this end, more extensive and accurate field observations are of the highest priority, particularly in well-characterized conditions. Acknow

ledgement

We are grateful for continued support, extending from 1985, for SB's research program in this area by USDOE Basic Energy Sciences Contract #DE-FG03-85ER13314. References 1. P. S. Liss and L. Merlivat, in The Role of Air-Sea Exchange in Geochemical Cycling, Ed. P. Buat-Menard, 113-129, D. Reidel, Norwell, Mass. (1986). 2. R. Wanninkhof and W. R. McGillis, Geophys. Res. Letters 26, 1889 (1999). 3. M. Donelan and R. H. Wanninkhof, in Gas Transfer at Water Surfaces, Geophysical Monograph, 127, AGU (2002).


235

4. S. Banerjee, Proc. Ninth Int. Heat Transfer Conf., Keynote Lectures, 1, 395 (1990). 5. S. Maclntyre, R. Wanninkhof and J. Chanton, in Biogenic Trace Gases: Measuring Emissions from Soil and Water, Eds P. Matson and Ft. Harriss, Blackwell, 52 (1995). 6. B. Jaehne and H. Haussecker, Ann. Rev. Fluid Mech. 30, 443 (1998). 7. M. A. Donelan, W. M. Drennan, E. S. Saltzman and R. Wanninkhof, Eds., Gas Transfer at Water Surfaces, (Geophysical Monograph 127, AGU, 2002) 8. W. G. Eugster, G. W. Kling, T. Jonas, J. P. McFadden, A. Wuest, S. Maclntyre and F. S. Chapin III, J. Geophys. Res. (Atmospheres). 108 (D12), 4362 (2003). 9. W. K. Lewis and W. G. Whitman, Ind. Eng. Chem. 16, 1215 (1924). 10. R. Higbie, Trans. Am. Inst. Chem. Eng. 31, 365 (1935). 11. P. V. Danckwerts, Ind. Eng. Chem. 43, 1460 (1951). 12. G. E. Fortescue and J. R. A. Pearson, Chem. Engr. Science 22, 1163 (1967). 13. S. Banerjee, E. Rhodes and D. S. Scott, Ind. Eng. ChE Fundamentals 7, 22 (1968). 14. T. J. Theofanous, R. N. Houze and L. K. Brumfield, Int. J. Heat Mass Transfer 6, 13 (1976). 15. J. C. R. Hunt and J. M. R. Graham, J. Fluid. Mech. 84, 209 (1978). 16. M. J. McCready, E. Vassilliadou and T. J. Hanratty, AIChE J. 32, 1108 (1986). 17. S. Kumar, R. Gupta and S. Banerjee, Phys. Fluids 10, 437 (1998). 18. V. De Angelis, P. Lombardi, P. Andreussi and S. Banerjee, "Micro-physics of scalar transfer at air-water interfaces", Proceedings of the IMA Conference: Wind-over-Wave Couplings, Perspectives and Prospects, Ed. S. G. Sajjadi, N. H. Thomas and J. C. R. Hunt, Oxford Press, 257 (1999). 19. G. T. Csanady, J. Geophys. Res. 95, 749 (1990). 20. A. V. Soloviev and P. Schluessel, J. Phys. Oceanogr. 24, 1319 (1994). 21. B. Caussade, J. George and L. Masbemat, AIChE Journal 36, 265 (1990). 22. M. Coantic, J. Geophys. Res. 9 1 , 3925 (1986). 23. B. H. Brumley and G. H. Jirka, J. Fluid Mech. 183, 235 (1987). 24. D. R. Chu and G. H. Jirka, Int. J. Heat Mass Transfer 35, 1957 (1992). 25. S. P. McKenna, W. R. McGillis and E. J. Bock, in TSFP-1, Eds S. Banerjee and J. K. Eaton, Begell House, NY, 455 (1999). 26. S. Komori, H. U-F. Ogino and T. Mizurhina, Int. J. Heat Mass Transfer 25, 513 (1982). 27. M. Rashidi and S. Banerjee, Phys. Fluids 31 (9), 2491 (1988). 28. S. Komori, Y. Murakami and H. Ueda, J. Fluid Mech. 203, 103 (1989). 29. J. C. Lamont and D. S. Scott, AIChE J. 15, 1215 (1970). 30. Y. Pan and S. Banerjee, Phys. Fluids 7, 1649 (1995). 31. P. J. Burt, Comput. Graphics Image Proc. 16, 20 (1981). 32. P. Anandan, Int. J. Comput, Vision 2, 283 (1989). 33. J. D. A. van Hoften and S. Karaki, 15th ICCE, Ch. 23, 404 (1976). 34. Y. Iwagaki and T. Asano, Coastal Eng. Jpn. 23, 1 (1980). 35. P. H. Kemp and R. R. Simons, J. Fluid Mech. 116, 227 (1982). 36. P. H. Kemp and R. R. Simons, J. Fluid Mech. 130, 73 (1983).

236

S. Banerjee and S. Maclntyre

37. R. R. Simons, A. J. Grass and A. Kyriacou, in Proc. of 21st ICCE, 363 (1988). 38. M. Rashidi, G. Hetsroni and S. Banerjee, Phys. Fluids 4 (12), 2727 (1992). 39. S. Supharatid, H. Tanaka and N. Shuto, Coastal Engineering in Japan 35 (1992). 40. X. Nan, D. Kaftori and S. Banerjee, in Proc. of FEDSM '98, ASME Fluids Engineering. Division Summer Meeting, Washington, D.C. (1998). 41. X. Nan, Ph. D. Thesis, Department of Mechanical Engineering, University of California, Santa Barbara (2003). 42. B. Knowlton, R. Gupta and S. Banerjee, TSFP-1, S. Banerjee, J. K. Eaton, eds., Begell House, NY, 461 (1999). 43. J. J. Cole and N. F. Caraco, Limnol. Oceanogr. 43, 647 (1998). 44. J. Crusius and R. Wanninkhof, Limnol. Oceanogr. 48, 1010 (2003). 45. T. J. Shay and M. C. Gregg, J. Phys. Oceanogr. 16, 1777 (1986). 46. A. Anis and J. N. Mourn, J. Phys. Oceanogr. 24, 2142 (1994). 47. D. T. Ho, R. Bliven, R. Wanninkhof and P. Schlosser, Tellus 49, 149 (1997). 48. S. Maclntyre, Limnol. Oceanogr. 38, 798 (1993). 49. S. Maclntyre, Jap. J. Limnol. 57, 395 (1996). 50. S. Maclntyre, in Physical Processes in Lakes and Oceans, Coastal and Estuarine Studies, AGU, Ed. J. Imberger, 539 (1998). 51. R. D. Robarts, M. Waiser, O. Hadas, T. Zohary and S. Maclntyre, Limnol. Oceanogr. 43, 1023 (1998). 52. S. Maclntyre, K. M. Flynn, R. Jellison and J. R. Romero, Limnol. Oceanogr. 44, 512 (1999). 53. S. Maclntyre and R. Jellison, Hydrobiologia 466, 13 (2001). 54. S. Maclntyre, W. Eugster and G. W. Kling, in Gas Transfer at Water Surfaces, Eds M. A. Donelan, W. M. Drennan, E. S. Saltzman, and R. Wanninkhof, AGU, (2001). 55. S. G. Schladow, M. Lee, B. E. Hurzeler and P. B. Kelly, Limnol. Oceanogr. 47, 1394 (2002). 56. S. J. Kline, W. C. Reynolds, F. A. Schraub and P. W. Runstadler, J. Fluid Mech. 70, 741 (1978). 57. M. Rashidi and S. Banerjee, Phys. Fluids 2, 1827 (1990). 58. K. Lam and S. Banerjee, Phys. Fluids 4 (2), 306 (1992). 59. P. Lombardi, V. De Angelis and S. Banerjee, Phys. Fluids 8 (6), 1643 (1996). 60. V. De Angelis, P. Lombardi and S. Banerjee, Phys. Fluids 9, 8 (1997). 61. V. De Angelis and S. Banerjee, "Heat and transfer mechanicsms at wavy gas-liquid interfaces", TSFP-1, S. Banerjee, J. K. Eaton, eds, Begell House, New York, 1249 (1999). 62. V. De Angelis, Ph. D. Dissertation, Department of Chemical Engineering, University of California, Santa Barbara, (1998). 63. I. Leifer, B. D. Piorek, W. C. Smith and S. Banerjee, in TSFP-3, Ed. N. Kasagi et al., Sendai, Japan, 2, 705 (2003). 64. R. H. Wanninkhof and L. F. Bliven, J. Geophys. Res. 96, 2785 (1991). 65. F. J. Ocampo-Torres, M. A. Donelan, N. Merzi and F. Fija, Tellus46B, 16 (1994). 66. J. Imberger, Limnol. Oceanogr. 30, 737 (1985).


237

67. S. Maclntyre, J. R. Romero and G. W. Kling, Limnology and Oceanography 47, 656 (2002). 68. G. W. Kling, G. W. Kipphut and M. C. Miller, Hydrobiologia 240, 23 (1992). 69. D. T. Ho, W. E. Asher, L. F. Bliven, P. Schlosser and E. L. Gordan, J. Geophys. Res. 105, 24045 (2000). 70. Y. C. Agrawal, E. A. Terray, M. A. Donelan, P. A. Huang, A. J. Williams III, W. M. Drennan, K. K. Kahma and S. A. Kitaigorodskii, Nature 359, 219 (1992). 71. E. A. Terray, M. A. Donelan, Y. C. Agrawal, W. M. Drennan, K. K. Hahma, A. J. Williams III, P. A. Huang, and S. A. Kitaigorodskii, J. Phys. Oceanogr. 26, 792 (1996). 72. P. M. Crill, K. B. Bartlett, J. O. Wilson, D. I. Sebacher, R. C. Harriss, J. M. Melack, S. Maclntyre, L. Lesack and L. Smith-Morrill, J. Geophys. Res. 93, 1564 (1988). 73. G. I. Taylor, Proc. Roy. Soc. Lond. A151, 421 (1935) 74. J. F. Clark, P. Schlosser, R. Wanninkhof, J. H. Simpson, W. S. F. Schuster and D. T. Ho, Geophys. Res. Letters 22, 93 (1995). 75. S. Maclntyre, "Current fluctuations in the surface waters of small lakes", 125-131, W. H. Brutsaert and G. H. Jirka (eds.), Gas Transfer at Water Surfaces, D. Reidel Publishing Co., (1984). 76. G. W. Kling, G. W. Kipphut and M. C. Miller, Science 251, 298 (1991). 77. J. J. Cole, N. F. Caraco, G. W. Kling and T. K. Kratz, Science 265, 1568 (1994). 78. J. E. Richey, J. M. Melack, A. K. Aufdenkampe, V. M. Ballester and L. L. Hess, Nature 416, 617 (2002). 79. J. J. Cole, S. R. Carpenter, J. F. Kitchell and M. L. Pace, Limnol. Oceanogr. 47, 1664 (2002).

This page is intentionally left blank

CHAPTER 6 I N T E R N A L WAVE FIELDS ANALYZED BY IMAGING VELOCIMETRY

John Grue Mechanics Divsion, Department of Mathematics, University of Oslo P.O. Box 1053 Blindern, 0316 Oslo, Norway E-mail: [email protected]

We review recent laboratory measurents of internal waves using Particle Tracking Velocimetry and Particle Image Velocimetry. The methods enable accurate recordings of the wave induced velocity and vorticity fields. Global properties like wave length and wave speed are obtained from the measurements. Solitary waves and dispersive wave trains in twolayer fluids with an upper layer that is either homogeneous or linearly stratified, and a lower layer that is homogeneous, are investigated. The measurements provide references for mathematical models and are used to judge their applicability. Particular focus is paid to solitary waves that exhibit convective breaking, shear instability, how wave breaking changes the global wave properties, how solitary waves break at a submerged ridge or shelf, and how the wave breaking induces transport of suspended particles. Run-up of very long internal waves at a shelf-slope is found to introduce strong fluid velocities where the pycnocline meets the slope. The experiments are used to interpret observations in large scale. 1. I n t r o d u c t i o n A physical laboratory simulator represents a tool for investigating complex flow phenomenae where theoretical models have shortcomings or do not exist. L a b o r a t o r y experiments of internal waves (and other flows) m a y be used t o explain observations in the sea and may serve as guidance in developing theories of internal wave propagation where a breaking of the flow eventually is inherent. T h e experimental wave t a n k may be used t o study waves with very large amplitudes, waves t h a t propagate along thin or broad pycnoclines, conditions t h a t lead to breaking of t h e waves, properties of internal waves t h a t break and internal waves t h a t interact with a sub-sea ridge or shelf-slope. Laboratory experiments represent an important complement to 239

240

J. Grue

theoretical and numerical modelling of internal waves. Measurements may be used to confirm mathematical solutions and are useful to judge the range of validity and possible limitations of simulation models. Our main interest in studying internal waves was motivated by requests from industry, including the magnitude of the fluid velocities induced by internal waves, the vertical motion of pycnoclines in the deep ocean, and interpretation of sudden temperature or current spikes observed in the ocean close to the sea floor. Internal wave loading on pipelines, or on a submerged floating tunnel that was proposed as connection across a Norwegian fjord, represented other questions from industry. Marine biologists observed large vertical periodic motion of fish and navigation of fish along density contours in the ocean, and questioned if this could be explained by internal wave motion. Further, understanding and description of currents and waves in fjords, including the motion of interfaces in the sea, are important for a proper management of the ecosystem. The questions supported our general curiosity on the phenomenon of internal waves, the experimental measurements and the mathematical modelling of the waves. Descriptions of internal waves in large scale may be found in, e.g., Osborne, Burch and Scarlet 1 , Ostrovsky and Stepanyants 2 , Huthnance 3 . Both internal solitary waves of large amplitude (Pingree and Mardell 4 , Apel et al. 5 , Stanton and Ostrovsky 6 ) and dispersive wave trains (Gjevik and H0st 7 ) are observed. Mathematical description of the waves in terms of weakly nonlinear Korteweg-de Vries (KdV) and Benjamin-Ono equations may be employed for small and moderate wave amplitudes. Fully nonlinear theories are in general required for a proper modelling when the excursion of the isopycnals become comparable to their average depth (Holyer8, Meiron and Saffman9, Amick and Turner 10 , Turner and Vanden-Broeck 11 ' 12 , Evans and Ford 13 , Grue et al. 14 - 15 - 16 , Lamb 17 ). While several papers describe the mathematical modelling of the waves (solitary waves, periodic or unsteady wave trains), comparatively few experimental papers on the subject are published. The most important experimental investigations on nonlinear internal waves are summarized here. The amplitude-wave length relationship of interfacial solitary waves with small to moderate amplitude was experimentally determined by Koop and Butler 18 , Segur and Hammack 19 , Kao, Pan and Renouard 20 . Comparisons with weakly nonlinear theories were made. In the latter work the fluid velocity was measured using hot-film probes. Michallet and Barthelemy 21 measured profiles and speeds of solitary waves in a two-fluid system, with the amplitude ranging from small to relatively large values. Their attempts to investigate wave amplitudes close to the maximal one were prevented by

Internal

Wave Fields Analyzed by Imaging

Velocimetry

241

breaking during the wave generation process. Internal solitary waves of mode two propagating along the pycnocline were experimentally investigated by Davis and Acrivos 22 , Maxworthy 23 and Stamp and Jacka 24 . The small scale experiments in the latter work exhibited wave amplitudes that were up to five times the half-width of the pycnocline. Internal waves interacting with a shelf-slope show formation of turbulent boluses travelling up a slope, intense breaking where the pycnocline intersects the slope and partial reflection of the incoming waves (Cacchione and Southard 25 , Wallace and Wilkinson 26 , Kao et al. 20 , Helfrich27). Small particles may be transported along the pycnocline in the direction away from the slope as a consequence of the breaking (Michallet and Ivey 28 ). Internal waves interacting with a submerged ridge have been experimentally studied by Wessels and Hutter 29 , Vlasenko and Hutter 30 , Sveen et al. 31 , Guo et al. 32 . Up to now, measurements of internal waves were usually carried out using resistance wave gauges, ultrasonic probes or hot-film probes. We shall in this paper review recent measurements using the optical methods Particle Tracking Velocimetry (PTV) and Particle Image Velocimetry (PIV). Efficient processing of the relatively large data sets are enabled by powerful computers. It is indeed the computer efficiency and rapid storage techniques gained during the last decade that have made PTV and PIV tractable. Velocity profiles, local vorticity and global properties like propagation speed and wave length are derived from the velocimetry. Details of breaking processes are quantified by the methods. Comparisons with mathematical and numerical computations are performed whenever possible. The paper is organized as follows: The experimental wave tank and the PTV and PIV methods in use are summarized in section 2. An estimate of how close the particles follow the fluid flow is given. Wave propagation along a thin pycnocline is discussed in section 3. Measurements of solitary waves of depression with amplitudes ranging from a small value up to the maximal theoretical wave amplitude are discussed. Experiments with dispersive wave trains are included. Breaking and broadening of solitary waves propagating in a two-fluid system with a linear stratification and a homogeneous fluid in the upper and lower layer, respectively, are discussed in section 4. Features of waves breaking at a submerged ridge are reported in section 5. The motion of internal waves, the induced breaking and the runup at steep or mild-sloping beaches are discussed in section 6. Vertically induced velocities in the ocean identified by fish motion are described in section 7. Section 8 contains a description of the fully nonlinear mathematical/numerical reference models, while section 9 is a conclusion.

242

J. Grue

2. The experimental wave tank 2.1. The wave tank and

stratification

We shall mainly report experiments that were carried out in the Hydrodynamic Laboratory at the University of Oslo. Sections of a wave tank 0.5 m wide, 1 m deep and 6.2, 12.3 or 21.4 m long were used. Each experiment began by calibrating a two-fluid system where a lower layer of brine had density pi = 1.022...g/cm3 and depth hi. The top layer was either of fresh (homogeneous) water or had a linear stratification with density />2=0.999... g/cm 3 at the upper surface of the layer. The depth of the upper layer was /i2- The density profiles were recorded by a Yokogawa SC12 meter which determines the density from the local conductivity of the fluid. Reference measurements were taken at some selected locations using a Mettler-Toledo DA-300M density meter which determines the density to four decimal places. Some of the runs (described in sections 3.2, 6.2) were performed in a smaller wave tank with dimensions 0 . 3 m x 0 . 7 m x 7 m (width, depth, length). 2.2. Particle

tracking

and particle

image

velocimetry

For the benefit of the reader we briefly summarize the most important points of the PTV and PIV methods used in the experiments (Grue et al. 1 4 ' 1 5 , Sveen et al. 31 ). The PTV method traces individual particles in sequences of images. The method is ideal when the local fluid acceleration is small which is true for the slowly progressing internal waves in focus. Vertical sections of the wave tank were illuminated for subsequent recording of the wave motion. The light sheets were parallel to the side of the tank, 5 cm thick and had a distance of 10 cm from the glass wall. Crunched particles of pliolite VTAC with diameter in the range 0.8-1 mm were in the viewing sections seeded to the fluid. The density of the pliolite is approximately 1.0228 g/cm 3 . Particles were treated using a wetting agent to obtain an effectively neutral buoyancy in the entire range of the density profile. The viewing sections were typically 50 cm x 40 cm large. Monochrome COHU 4912 CCD cameras with a resolution 575x560 of pixels were used in the flow recordings. A particle at rest was normally covered by four pixels in the CCD-chip. The effective shutter speed (using a mechanical shutter) was usually 1/100 s. (In some initial experiments the shutter speed was 3/100 s.) The time between each frame is 1/25 s. The video recordings were digitized by a frame grabber card for subsequent analysis. In the experiments presented here we typically identified 800-4000 particles in each frame. Particles were traced during five frames us-

Internal


Velocimetry

243

ing the Diglmage program developed and described by Dalziel 33 . A perspex plate with reference coordinates was submerged in the fluid and recorded after each set of runs. Reference points were mounted to the wave tank. The mapping coordinates between the recording section and the camera is linear. In some of the waves the acceleration was not always small. This is true when breaking of the flow was observed. We could not trace particles long enough for the PTV algoritm to be effective, for example. Data processing using PIV was then used to estimate the velocity field. For the PIV algorithm we implemented the method outlined by Willert and Gharib 34 using an interrogation window shifting procedure of Westerweel, Dabiri and Gharib 35 . The images were interrogated in three steps where the two first steps were used to estimate the window shift with integer accuracy. In the final step the displacement was estimated to sub-pixel accuracy using a three point Gaussian peak fit. Images were interrogated using windows of 32x32 pixels. In some experiments 64x64 pixels were used due to insufficient particle seeding. A signal to noise ratio filter was used to validate the final velocity vectors. A local median filter effectively removed vectors deviating significantly from their neighbours. The implementation is documented in full in Sveen36. 2.3. Relative

accuracy

of the optical

method

The difference AV between the fluid velocity and the velocity of a seeded particle may be analysed using the equation of momentum for the particle, assuming that the Stokes drag is much less than the time derivative of the particle momentum, i.e. \mV\ > > |67rpî?AV|. Here a dot denotes time derivative, v the kinematic viscosity and R, V, m = (4ir/3)pR3 the radius, velocity and mass of the particle (assuming neutral boyuancy), respectively, p denotes fluid density. This gives |AV| < < a|V| where a — m/Girpi'R = 2R2/9u. The acceleration is estimated by |V| ~ | V m | / T where | V m | denotes a typical maximal velocity and T a typical period of the flow. Characteristic values of | V m | and T in the present experiments are 10 cm/s and 5-10 s, respectively. With R = 0.5 mm and v = 10~ 6 m 2 /s we find IAVI L | «

2.R2 _ -0.005-0.01.

(1)

The corresponding particle Reynolds number Re = 2R\AV\/i> is less than unity. The seeded particles thus quite closely follow the fluid motion, with an accuracy indicated by (1).

244

J. Grue

The accuracy of the digital tracing of a particle may be estimated as follows: We assume that the position and displacement of each particle is determined with an accuracy better than the length of a pixel which corresponds to about 1 mm in physical space in our case. (Dalziel33 found in his experiments that that the position of a particle was determined with an accuracy of about 0.2 times the pixel length.) A particle with a typical velocity of 10 cm/s, traced during a sequence of five frames, i.e. a time interval of 0.2 s, moves a distance of 2 cm. This means that the relative error in the particle displacement and the velocity is at most lmm/2cm=5%. This is a much larger figure than the error in (1). We have found that a relative error of the measured velocities is approximately in agreement with the error analysis just described. We find, for example, that the measured fluid velocity induced by a solitary wave, relative to the linear long wave speed Co of the two-layer fluid, has a deviation of about 7-8% at most, from a line fitted to all experiments that were made, see Fig. 2c. Our results include analysis using both PTV and PIV, preferring the method giving the best accuracy for the actual run. We note that in all practical applications it is possible to obtain the same accuracy using PTV and PIV. 3. Motion along a thin pycnocline 3.1. Solitary

waves

Investigations of large amplitude solitary waves propagating in a two-layer fluid represented a main focus of the laboratory studies that were undertaken. The waves were measured in the wave tank (dimensions are previously given) using the procedure described above: A two-layer fluid with a lower layer of brine and an upper layer of homogeneous fresh water was calibrated. In most of the runs the lower layer depth was hi = 62 cm and the upper layer depth /12 = 15 cm which means a depth ratio of /11//12 = 4.13. The thickness of the pycnocline was 1-2 cm. Vertical sections of the wave tank were illuminated for subsequent recording of the wave motion. Solitary waves were produced by trapping a volume of light fluid behind a gate that was lowered in one end of the wave tank. Upon removal of the gate the initial depression rather quickly was transformed into a solitary wave propagating along the tank. A single solitary wave was in each experiment produced by a combined adjustment of the position of the gate and the added volume (see Fig. la). The wave amplitude ranged from a rather small value to about the maximal possible one, as indicated by the measurements presented in Fig. 2c. The video cameras placed along the tank, at positions 4.5 m and 10.4 m from the end

Internal

Wave Fields Analyzed by Imaging Velocimetry

245

of the tank where the waves were generated, recorded the wave induced motion of the particles that was seeded into the fluid. From the recordings we determined the induced velocity profiles, speed and shape of the waves. The velocity measurements of the individual runs were scaled by the linear long wave speed of the two-fluid system given theoretically by 2 _ ghih2{p\ - P2) 0 P2/11 + Pih2

(2)

where g denotes the acceleration of gravity. The effect on the wave and fluid velocities due to small variations of the relative density jump in the different runs was then ruled out. The wave speed c was estimated by the elapsed time of the wave propagation between the two viewing sections (at 4.5 and 10.4 m, including accurate reference points) divided by distance. The confined pycnocline is very visible due to a higher concentration of particles there than elsewhere. The pycnocline motion was determined by visual inspection of video images. The wave speed scaled by the theoretical linear wave speed determined from eq. (2) is visualized in Fig. 2a. The wave speed is seen to grow at a rate that is decreasing with amplitude. At maximal amplitude the growth is zero. The experimental wave amplitudes are found to undergo a small reduction during the propagation due to the effect of viscosity. The decrease of the amplitude between the recording stations is indicated by the horizontal bars in the figure. (The largest amplitude refers to the first recording station.) Wave profiles were obtained from the video images by first tracing the vertical motion of the middle position of the pycnocline, as function of time. The recorded motion was then plotted versus time multiplied by the computed (and measured) wave speed c. Supplementary estimates of the pycnocline motion were obtained by tracing the jump in the velocity profile (across the pycnocline). The two methods give the same result, see Fig. lb. The main output from the velocimetry is the velocity field induced by the wave motion. The velocities in any position of the wave were determined. An example of the velocity profile at the wave crest is visualized in Fig. 2b. Results from a total of 10 different runs with slightly different pycnocline thicknesses are included. The consistent results illustrate the robustness and accuracy of the experiments. We were also interested in measuring the induced fluid velocities in the layers as function of wave amplitude. Results for the horizontal fluid velocity in the upper layer are obtained for the whole range of possible wave amplitudes (Fig. 2c). The maximal theoretical

246

J. Grue

h2

""-

-^^

I

!

i _=pLJu——

-0.25 i r]_ -0.75 h2 -1.25h -1.75. -25

D

an

\

c) i

-15

-5

i

15

25

-ct/h2 Fig. 1. Solitary waves propagating along a thin pycnocline. Sketch of experiment (upper). Profiles r)(ct) of waves with amplitude a/hi = 1.23 (mid) and a/h.2 = 1.51 (lower). Squares: measured pycnocline. Triangles: measured jump in velocity. Solid line: fully nonlinear theory. Dashed line: KdV theory. h\/h,2 = 4.13. From Grue et al. 1 5 . Reproduced with permission by J. Fluid Mech.

amplitude in these experiments is a/h2 = 1.55 (accounting accurately for the slightly small difference between the densities p\ and p2). The corresponding theoretical maximal fluid velocity is U/CQ = 0.77. In the figure the function {u/CQ)/{a/h2) is plotted. For very small a/h2 the function is close

Internal


Velocimetry

247

to unity. At maximal amplitude the value of (u/co)/(a//i2) is about 0.5 for the actual depth ratio between the layers. The experimental solitary waves may be computed employing a theoretical two-layer model where in the latter the thin pycnocline is replaced by an interface separating the motion in the upper and lower layer. An interesting aspect of the campaign is that the fully nonlinear model may be used to check the PTV method. The high precision computations provide a theoretical reference when the experimental conditions fit with the assumptions of the theory. Vice-versa, a fit between theory and experiment documents the applicability of the theoretical interface model. Fully nonlinear computations of solitary waves were performed using the theoretical-numerical two-layer interface model developed by Grue et a j 14,15 rp n e r e s m t s exhibit a surprisingly good agreement between measurements and theory for the whole range of wave amplitudes. The agreement is good even for a pycnocline which is not very sharp. The range up to the highest waves is covered. We were particularly interested in investigating experimental solitary waves with amplitudes as close as possible to theoretical maximum, which in the Boussinesq limit (Ap/p < < 1) means that the pycnocline moves to a position that is mid-way in the fluid. (This implies the simple formula amaxjhi = \{h\/h2 — 1) for the maximal wave amplitude, with numerical value 1.56 when h\jhi = 4.13.) We observe a good correspondence between theory and experiment for the profile of the maximal wave, in the leading part of the wave (Fig. lc). In this experiment the minimal value of the Richardson number, estimated from velocimetry, was 0.07. Local breaking of the flow due to a shear instability was then expected. Indeed, breaking is observed in the tail of the wave. The shear instability is very visible. Instability and subsequent breaking of the flow cannot be predicted by the theoretical two-layer model. The results in Fig. 2c indicate a relative accuracy of the measurements of about 7-8%. This represents the maximal deviation of U/CQ from the theoretical line and from a fitted line of the recordings. We have here studied the motion of solitary depression waves. The polarity of the solitary waves is determined according to the sign of the coefficient h\pi — h\p2- The waves are depression or elevation waves when this factor is negative or positive, respectively. Korteweg-de Vries theory provides a supplementary reference when the wave amplitude is small. The assumption of a weak nonlinearity is violated for moderate and large wave amplitudes, however. The amplitude range where KdV theory is valid is found to be rather small (a/h2 less than about 0.4).

248

J. Grue

a)

0.8

1.2

a/h2

b)

c)

Fig. 2. Solitary waves, a) Excess speed vs. wave amplitude, b) velocity profile above and below crest (a//i2 = 0.78), c) nondimensional velocity in upper layer (at level y = — /12/2) vs. nondimensional amplitude a/h?. /11//12 = 4.13. From Grue et al. 1 5 . Reproduced with permission by J. Fluid Mech.

Internal


Velocimetry

249

b)

x/h2 Fig. 3. Fully nonlinear wave profile (square), KdV (dots) and Intermediate-long-wave equations (solid line), a) 0//12 = 0.15, b) a//i2 = 0.7. /11//12 = 4, P2/P1 = 0.9. From Grue et al. 1 4 . Reproduced with permission by J. Fluid Mech.

It may be questioned if the Intermediate-long-wave-equations 19 may be more relevant than KdV theory in the present examples, where the wave length L//12 ~ 10—20 is not much longer than the depth ratio /11//12 = 4.13. The Intermediate-long-wave-equations have very limited range of validity and represent a poorer approximation than KdV theory, however (Fig. 3). This is true even if the wave length is comparable to the total depth of the fluid. Extensions based on the KdV and the BO equations, valid for solitary waves with large amplitude, are discussed by Ostrovsky and Grue 37 .

3.2. Dispersive

wave

trains

Experiments were also performed with a portion of brine trapped by the gate. This corresponds to an initial condition where a part of the interface has an elevation. The resulting wave system is a dispersive wave train propagating along the tank with the longer waves travelling faster than the shorter ones. The mechanical energy per wave length is relatively much smaller than the wave energy transported by a single solitary wave. The induced fluid velocities are correspondingly smaller (Fig. 4). The experimental waves are adequately modelled by the (linear) Cauchy-Poisson problem modified to interfacial flows (Carlin 38 , Hald 39 ). The experiments and theory

J. Grue

250

document wave systems that may be generated upon release of a volume of heavy water into a two-layer system. Such wave trains are observed in large scale in Skagerak off the southern coast of Norway. During summer a thermocline is developed in the sea with an average depth of 15 m, a relative jump in density of 3 . 7 x l 0 - 3 , with corresponding linear long wave speed Co — 70 cm/s. Wave groups show leading waves that typically are 1.5 km long, see Gjevik and H0st 7 . The wavenumber at any position of the wave train is implicitly determined by: d/dk[(x/t)k — uj(k)] = 0, where LO2 = g'k/(cothkhi + cothfc/12), k the wavenumber, x horizontal position, t time, u! wave frequency, g' = gAp/p, g acceleration of gravity, Ap/p relative jump in density. The laboratory results and the mathematical model predictions give an indication of the magnitude of the fluid velocities that are induced by the waves in large scale. (Nondimensional velocity in the upper layer, U/CQ ~ 0.1 corresponds to nondimensional interfacial vertical excursion, a/h2 ~ 0.15.)

4. Solitary waves that break Heating from sun may result in a linear stratification of the upper part of the ocean. Such a stratification is season-dependent and is observed at many locations. The thickness of the linearly stratified layer typically ranges from some tens of meters to a few hundreds. Examples are represented by the Sulu Sea (Fig. 5) and the Knight Inlet in British Columbia (Farmer and Smith 40 ). Another example is off the northern coast of Norway, see section 7. Internal wave propagation along such a stratification may be studied in the laboratory wave tank by preparing a fluid where an upper part of the undisturbed water column has a constant Brunt-Vaisala frequency and a lower part is homogeneous. The properties of internal solitary waves propagating along such a stratification were investigated experimentally and theoretically by Grue et al. 16 . The waves were generated using the same procedure as for the two-layer fluid with constant density in the layers (sections 2, 3). In all experiments reported here the thickness h\ of the lower homogeneous fluid is 4.13 times the thickness h2 of the upper fluid, where the latter has a linear stratification (and the density is continuous in the transition between the layers). The recorded velocities in the experiments were scaled by the linear long wave speed of the two-fluid system determined by 16 N0h2 Co

cot

N0h2 Co

h2 1- 7— = 0, hi

(3)

Internal


Velocimetry

251

191199 251199--A&B 291199 -A&B, 031299-A&B 071299-A&B 0.15

-

0.1

/.'' I

0.05

•''i'.l * i]

0.5

; 1.5

a//i2 Fig. 6. Propagation speed c vs. wave amplitude. Symbols: measurements. Solid line: theory. From Grue et al. 1 6 . Reproduced with permission by J. Fluid Mech.

not indicate that a saturation of the wave, in the form of a maximal value of the amplitude, exists. This is unlike what was observed for a two-layer fluid with constant density in each layer. Another important difference from the previous two-layer fluid is that an abrupt velocity transition across the thin pycnocline is replaced by a smooth velocity field. The maximal fluid velocity is now occurring at the top of the water column and is significantly larger than with the other two-fluid system, for otherwise same wave amplitude. The variation of the horizontal velocity profile during the passage of a solitary wave is visualized in Fig. 7. The induced fluid velocity in the upper part of the upper layer is rather pronounced despite the relatively modest nondimensional wave amplitude of a/h2 = 0.4. There is again a good correspondence between experiment and theory. The wave motion introduces a nonzero vorticity field in the stratified fluid. Vorticity profiles are obtained from the particle imaging measurements using a finite difference operator, i.e., df/dxi = (2/j+2 + fi+i —

254

J. Grue

o

o

o

o

Fig. 7. Time series of passage of wave. Horizontal velocity U/CQ as function vertical coordinate. Symbols: measurements. Solid line: theory. From Grue et al. 1 6 . Reproduced with permission by J. Fluid Mech.

Internal


255

Velocimetry

/i_i — 2/i_2)/(10AX). The experimental vorticity estimates are supported by the fully nonlinear computations for nonbreaking waves (Fig. 8a). Breaking of the waves is manifested by pronounced spikes in the vorticity field. A pronounced negative vorticity is observed at a level of J///12 = —0.3 in Fig. 8b, for example. The spike corresponds to a vortex seen in the velocity field and is caused by convective breaking. The maximal value of the vorticity in the series of experiments reported here is |w|/(co//i2) — 3, a result that is true for the largest waves. With NQH^/CO = 1.711.., the maximal value of the vorticity relative to the Brunt-Vaisala frequency becomes, about, OJ/NQ ~ 1.7.

%*-

b )

M.

c0/h2

c0/h2

Fig. 8. Vorticity (UJ) at crest of wave vs. vertical coordinate, a) Without breaking, b) with breaking. Solid line (theory, a//i2 = 0.65), other lines (experiments). From Grue et al. 1 6 . Reproduced with permission by J. Fluid Mech.

For increasingly larger solitary waves, both the wave speed c and the induced fluid velocity relative to c grow until a saturation is reached. The limiting mechanism of the waves is found to be convecting breaking taking place when the induced fluid velocity attempts to exceed the wave speed. The convective breaking is observed in a region in the centre of the wave in the upper fluid. In the breaking region the fluid velocity has the form v = ci + v ' , where ci denotes the wave velocity and v' a velocity field where |v'| < < c. (The waves transport a body of mass.) The vector i denotes the horizontal unity vector along the direction of the wave propagation. The horizontal velocity profile above wave crest is characterized by u/c being

256

J. Grue

unity in a vertical range corresponding to almost the thickness of the upper layer (Fig. 9a-b). The experimental waves exhibit broadening when breaking occurs (Fig. 9c). The nondimensional wave amplitude then exceeds about 0.80.9. (The theory predicts u/c = 1 for a//i2 = 0.855.) The broadening effect found here is entirely different from the mechanism taking place in a two-fluid system with constant density in the layers. Then the limiting amplitude and limiting wave speed, and thereby the broadening of the waves, are determined by a finite total depth of the fluid (Amick and Turner 10 , Turner and Vanden-Broeck 12 ), a result which is experimentally confirmed by the results in section 3. We note that in the present two-layer fluid a solitary wave (of mode one) always exhibits an excursion out of the layer with linear stratification. Observations in the wave tank, and computations, confirm this result (results not shown). The polarity of the solitary waves may be determined analytically from the sign of the coefficient of the quadratic nonlinear term in weakly nonlinear KdV- or Intermediate-long-wave theory. This shows that the polarity is always the same for a two-layer fluid where one layer is linearly stratified and the other layer is homogeneous (and the stratification is continuous), see also Grue et al. 16 eq. (8.1). The results exhibit good correspondence between experiment and theory up to breaking. The fully nonlinear inviscid theory does neither explain the breaking nor the broadening of the waves observed in the experiments, however.

5. Wave-breaking at a submerged ridge A solitary wave that interacts with a submerged ridge undergoes a local distortion that causes a scattering of the wave. Mechanical energy may be lost if breaking occurs. The case of a solitary wave incident normal to a long-crested ridge was experimentally investigated using PTV and PIV by Sveen et al. 31 . A stably stratified two-fluid system had homogeneous fluids in each layer. Conditions leading to breaking at the ridge and transmission of the solitary waves represented the main objectives. We summarize the features that is characteristic for strong interaction between wave and ridge. The incoming internal solitary waves break during the deformation if the amplitude relative to the local depth at the top of the ridge exceeds a certain level. The breaking process is illustrated in Fig. 10. The incoming wave is in this case interacting with a relatively wide ridge. More specifically, the parameters in the experiment were: lower layer

Internal


257

Velocimetry

te//l2

b)

o

12.5

o

-

-

10

c)

o X

o-g.

0

o 7.5

O

~^*~-^*_ =^t^^

"

X

Fig. 9. a) Wave induced horizontal velocity at the free surface as funtion of time (initial volume 100 dm 3 , measurements at several positions along the wave tank), b) horizontal velocity profile at crest in breaking waves, c) wave length A//12 = ( l / " m o i ) f^° u dx //12 vs. Umax/c obtained in experiments (symbols) and in theory (solid and dashed lines). From Grue et al. 1 6 . Reproduced with permission by J. Fluid Mech.

258

J. Grue

depth outside ridge h\ = 45 cm, upper layer depth /12 = 10 cm, incoming (nondimensional) wave amplitude a//i2 = 0.59, width of ridge 6.2 m, height of ridge above tank floor 30.5 cm, constant ridge slope ±0.1 and top of ridge positioned 7.83 m from the endwall of the wave tank where the wave was generated. (There is a smooth transition at the top of the ridge between the two slopes.) The length of wave tank was 21.5 m. The top of the ridge was in all experiments below the pycnocline at rest, and the upper layer was always thinner than the lower layer. The incoming waves were solitary waves of depression. During a strong interaction between wave and ridge the excursion of the pycnocline typically became lower than the mid depth of the water column. A significant distortion of the wave was observed as it encountered the turning point, i.e. the horizontal position (on the slope) where the pycnocline is half-way between the bottom and top of the water column. The incident depression wave was then transformed into a shorter leading depression and a subsequent dispersive wave train, where the first of the waves typically has a relatively high elevation. This process is the same as the one that occurs to a solitary wave of depression travelling up a sloping beach, beyond the turning point, described theoretically in Djordjevic and Redekopp 43 , Knickerbocker and Newell44, Malomed and Shrira 45 and experimentally by Helfrich, Melville and Miles46. The process on the ridge develops a breaking of the wave in the form of mixing between salt and fresh water at the pycnocline, in the rear of the leading depression, and in the leading part of the subsequent wave of elevation. The mixing is visible as a fog in the upper picture in Fig. 10. A second breaking event occurs when the tail of the wave collapses due to the local strong steepness there, see the lower picture in Fig. 10. Estimates indicate a magnitude of the accelerations of about 0.55' during the collapse. This means that a Rayleigh-Taylor instability is not the cause of the breaking. Dispersion has a weak or strong effect on the incoming wave, depending on the slope of the ridge. If the slope is relatively small, dispersion has a stronger effect than indicated in Fig. 10, on the incoming depression solitary wave, as it arrives at the turning point. The wave is in the small slope case transformed into a relatively short leading depression that is decreasing in time, and a subsequent dispersive wave train, with a number of waves that gains in time. The dispersive wave train represents the origin of boluses or vortex blobs - travelling up the slope, see the forthcoming section 6.2. If the ridge slope is relatively steep, and the waves have sufficiently large amplitude, wave breaking in the form of overturning is observed at the obstacle.

Internal

t


Velocimetry

259

-0.6

X

h2

b)

£ ^0.5

X

h2 Fig. 10. Waves breaking at a ridge. Time difference 2.04 seconds. From Sveen et a!.3 Reproduced with permission by J. Fluid Mech.

260

J. Grue

For a closer examination of the local process at the top of the ridge, prior to wave breaking, vertically averaged horizontal velocities in the upper and lower layer, u2 and u\, respectively, are measured. Local thicknesses of the layers at the top of the ridge are defined by h\^, where the sum h\ + h,2 equals the total depth of the fluid at the top of the ridge, see Fig. 11a. The values of hi and h2 are obtained using u\h\ 4- u2h2 = 0, given h\ + h2. The local thicknesses of the layers define the local nonlinear shallow-water speed at the top of the ridge, serving as a reference speed, i.e. Co = [g'hih2/(hi + h.2)]1/2, where g' = gAp/p, and the maximal value of h,2 and the minimal value of hi are used. The measured M2 above the ridge show good correspondence with the fluid velocity in the upper layer due to the undisturbed incoming wave (results not shown). On the other hand, due the thin local depth of the lower layer below the wave crest, at the top of the ridge, the magnitude of the velocity ui in the lower layer may become large. This is readily seen from the equation of continuity, i.e., U\hi+U2h2 = 0, which means that if î//i2 becomes small, |ui/«2| becomes correspondingly large. This is characteristic for waves with large negative excursions. An example is displayed in Fig. l i b . Measurements of ui^/co show that breaking of incoming solitary waves of depression always occur when |ui|/co exceeds the value of 0.7, i.e. somewhat less than critical flows (Fig. l i e ) . This is a result from a total of 56 runs where the parameters in the experiments were: wave amplitude in the range 0.2-1.9 times the depth of the upper layer, ratio between the lower and upper layer depth in the range 3-8.5, and ridge slope in the range 0.10.33. Incipient wave breaking happens either in the form of local spilling or violent overturning of the waves. Shear instability, which plays no role in the initial part of the wave breaking process, is observed during the later stages in some of the breaking events. In all cases of breaking, vortices are shed on the up-slope side of the ridge. The (almost critical) lower layer flow at the slope is attached close to the top of the ridge, but separates at some point, as indicated in the photos in Fig. 10. The initial separation point is on the up-slope side. As a result of the separation, tracer particles are distributed vertically across the lower layer. The transmitted waves exhibit a leading solitary wave with a subsequent dispersive wave train. The amplitude of the transmitted solitary wave corresponds in general to that of the incoming wave if breaking does not occur at the ridge. In the case of wave breaking the solitary wave undergoes a significant reduction in amplitude, however. The breaking process leads locally to a mixing between fresh and salt water at the ridge. Portions of mixed fluid that locally widens the pycnocline may subsequently generate

Internal


Velocimetry

261

solitary waves of mode two propagating along the pycnocline, see Fig. 12. The induced velocity within the pycnocline, due to the mode two wave, is as large as 30 % of the horizontal fluid velocity due to the leading solitary wave of mode one, with nondimensional amplitude a/h,2 = 0.35 in the actual example. Generation of waves of mode two at a ridge has also been studied by Vlasenko and Hutter 30 . A corresponding interaction between an incoming solitary wave and a ridge is true also if the undisturbed upper layer has a linear stratification and the lower layer is homogeneous. A strong interaction between wave and ridge exhibits a pronounced increase of the lower layer velocity at the top of the ridge, a separation of the flow on the upwave side of the ridge and subsequent breaking of the wave. The reduction of the solitary wave amplitude due to the wave-ridge encounter is somewhat less pronounced if the upper layer is linearly stratified as compared to when the fluid is homogeneous (Guo et al. 32 ). 6. Wave breaking and run-up at a shelf-slope 6.1. Steep slope. Strong local breaking and

mixing

Internal solitary waves that interact with a relatively steep shelf-slope (continuing through the entire water column) represented one of the focuses in the work by Kao et al. 20 . They ascribed the breaking of the waves at the shelf, observed in their experiments, to shear instability. Michallet and Ivey 28 studied solitary waves of large amplitude that broke as they encountered the shelf-slope. In the latter work PIV was used to quantify the velocity fields. The mixing efficiency at the shelf and the magnitude of a reflected solitary wave travelling away from the beach were measured for wave slopes in the range 0.07-0.2. Michallet and Ivey suggested that the mechanism leading to breaking of the waves at the shelf was due to gravitational instability (this is not supported by our measurements), following the separation of the flow in the lower layer. The mixing efficiency was estimated from the difference between the initial density profiles and the density profiles at the shelf-slope after breaking. The main part of the mixing was found to occur in the lower layer. Their and our results for solitary depression waves interacting with a submerged ridge, presented in the previous section, are conform on this point. The reflected to incident wave energy ratio (ER/EO) as function of the wave length to slope length ratio (Lw/Ls) was quantified. The results showed that ER/EO — Lw/Ls for Lw/Ls < 0.6 and that a maximum value of ER/EQ ~ 0.65 was reached for larger ratios of Lw/Ls- A strong

J. Grue

a)

b)

10

15

20

tc0/h2

1.25 •

T~

0.75 •

0.3

0.4

0.5

0.6

0.7

0.8

(a + h2)/(h1 + h2 - hr) Fig. 11. Waves breaking at a ridge, a) Sketch, b) horizontal velocities at top of the ridge, c) in plot: breaking (filled squares), strong local deformation (open circles), weak interaction (dots). From Sveen et al. 3 1 . Reproduced with permission by J. Fluid Mech.

Internal


263

Velocimetry

1

0.5

h2 -0.5

-1

-0.05

0

0.05

0.1

U/CQ

Fig. 12. Velocity profile in wave of mode two propagating along the pycnocline. From Sveen et al. 3 1 . Reproduced with permission by J. Fluid Mech.

flow up the slope in the form of a bolus could emerge from the interaction between the incident wave and the slope. After breaking, an off shelf flow was observed in the lower layer. Suspended particles were transported along the pycnocline. Particles that initially were located in the breaking region (at the slope) were spread far offshore.

6.2. Run-up

at a

shelf-slope

Relatively short incoming periodic waves or solitary waves of depression interacting with a weakly sloping beach show a transformation into boluses propagating up-slope (Cacchione and Southard 25 , Wallace and Wilkinson 26 , Helfrich27). A solitary wave of depression may split into a shorter leading depression with a subsequent dispersive wave system, each individual wave developing into a bolus. This results in a sequence of boluses propagating up the slope (Fig. 15). We shall here extend previous work, reporting experiments where the initial elevation or depression of the interface is very long. Our particular focus is strong run-up and strong currents that are generated where the pycnocline intersects the slope. The experiments are motivated by strong current events that have been observed at the sea floor at the offshore gas

264

J. Grue

field Ormen Lange. This is located in water 850 m deep on the slope of the Norwegian shelf. A thermocline at an average depth of about 500 m represents a characteristic feature of the ocean at the location. The thermocline separates the warm northward Norwegian Atlantic Current with temperature about 8 C° from cold water masses in the deep ocean with temperature -0.5 C°. Outside the shelf-slope, the lower cold layer has a thickness that is 1-2 times the depth of the upper warm water. (At the location of Ormen Lange the upper layer is thicker than the lower.) The relative density jump across the thermocline is 0.5 x 1 0 - 3 . Fig. 13 shows a recording of a strong landward current event at the sea floor at the actual location. The recorded current speed has a maximum of 0.5 m/s (Fig. 13a). The flow direction is up the shelf-slope. There is only a very small velocity component along the shelf-slope. Prior to the current event the temperature at the sea floor exhibited a slow build-up phase reaching a maximum of 3 C° (Fig. 13c). This indicates that the thermocline had moved downward the slope prior to the event, beyond the measurement position. The sudden return of the temperature coincides with the rapid increase of a current at the sea floor. The duration of the event is about 16-24 hours. It is convenient to put the observed current on nondimensional form. As reference speed we use the linear long wave speed of the two-fluid system determined by CQ = [g'hifi2/(hi + /12)]1/2 ~ 1-3 m/s where g' = gAp/p and we have put h2 = 500 m for the upper layer depth and the lower layer depth is determined by h\/li2 = 2. The maximum recorded speed of 0.5 m/s corresponds to about 40 % of the linear long wave speed. Further, the duration of the event corresponds to a nondimensional time ico//i2 in the range 150-224. Observations of the thermocline motion in the ocean indicate that a strong current event may occur due to very long wave motion along the thermocline. Typical periods may be in the range 1-2 days. A corresponding estimated wave length is 100-200 km which means 200-400 times the upper layer thickness. A set of model experiments in scale 1 : 5000 were undertaken in a small wave tank of length 7 m to shed light on the wave events observed at Ormen Lange (Sveen and Grue 47 ). The upper layer depth in the wave tank was h2 = 10 cm, the lower layer depth in the deep water part h\ = 20 cm and the slope factor was 0.3 m / 7 m corresponding to 2.5 degrees. The lower layer of brine and upper layer of fresh water was separated by the pycnocline being about 1 cm thick. With a relative jump in density of 0.02 the linear long wave speed in the laboratory was 13 cm/s.

Internal


Velocimetry

265

Currents and temperature at position II 5m asb

20

21

22

23

24

25

26

27

28

29

1

1

1

b)

40 Component normal to max velocity 20

0

1

0 -_

20

1

"\IM

l>' '

^

20 20

ufll [•Hs

kj,: TVU JWT—-"^ ^

I

I

I

23

24

25

: . . . ~sr^.:

23

1

1 n>JlAAywS

p^"

1

nAJ^wft,

^ : W

^J"

1

I

29

:\

24 25 26 Dates November 1999

jT.... V J U A < X W - :

28

-

29

30

_

30

Fig. 13. Strong current event. Measurements 5 m above sea floor. Reproduced with permission by Norsk Hydro ASA.

Waves were generated by trapping a volume either of brine or fresh water behind the gate, elevating or depressing, respectively, the pycnocline relative to the level elsewhere in the tank (Fig. 14a). In the interest of investigating the response due to very long initial disturbances, the gate was placed 2 m from the end of the wave-generating part, corresponding to 20 times the upper layer thickness. Both an initial elevation and depression leads to the formation of a long bore at the point where the pycnocline intersects the shelf-slope. The bore propagates up the slope in the form of

266

J. Grue

g a i e linear stratification

a)

•>r"Mk3Slli

b)

c)

Fig. 14. Internal run-up on slope, a) Sketch of experiment, b) velocity field during internal wave run-up, c) induced horizontal velocities in lower (u%) and upper (U2) layer.

Internal


Velocimetry

267

an internal run-up, with a leading part of the bore resembling the head of a gravity current (Simpson and Britter 41 , Simpson 42 ), see Fig. 14b. The induced fluid velocity in the head of the lower layer flow shows a maximal value of about 40 % of the linear long wave speed (U/CQ ~ 0.4), i.e. the same nondimensional speed as observed in the field (Fig. 14c). In the figure is plotted vertically averaged horizontal velocities in the layers. The duration of the wave event in laboratory of tco/h2 = 40 nondimensional periods is about 3 times shorter than in large scale. This difference is probably due to the relative difference of the input wave length in laboratory and large scale. The most important point with the two-dimensional laboratory investigation was to estimate the magnitude of the fluid velocities that may be introduced during run-up of very long internal waves. The laboratory measurements may be used, e.g., as reference for simplified ocean model simulations in two dimensions with the rotation (of the earth) terms switched off. Three-dimensional and rotation effects should be included in full simulations of internal run-up.

7. Fish as markers of internal waves In-situ regsistrations of internal wave motion were made by the Norwegian Institute of Marine Research in a small campaign off the Norwegian coast during August 1998. While the main objective was to register fish schools by means of echo-sounders, the data logs showed more than just the appearance of fish. A pronounced oscillatory motion of the schools became evident and were observed from both ship and buoy. The observations were made in the Norwegian Sea north of S0r0ya in Finnmark at position 71°N 22°10'E. A part of the record from the buoy is shown in Fig. 16 (God0 48 ). It was evident that the fish was eating plankton that was drifting with the flow in the ocean. The motion of the individual fishes were traced by the data processing routines and are visible in the plot by small streaks. It can be concluded that the fishes serve as markers of an oscillatory vertical fluid motion in the sea, with an average level at 100 m depth, amplitude of 20 m at a depth of the sea of 146 m. Hydrographical data from the area show that a roughly linear stratification in the upper part of the sea during late summer and autumn is developed annually. The thermocline is 200 m thick and has a relative density jump of 1 0 - 3 . The deep part of the sea has a constant density. Syntetic Aperture Radar (SAR) images from satellite were obtained during autumn 2000 and revealed occurrence of internal wave motion in the actual area in

268

J. Grue

the form of dispersive wave trains propagating from shallow to deep water. Wave lengths (from several images) were in the range 300-500 m. We interpret the set of observations on basis of the experiments and theories of waves moving along the linearly stratified upper layer and the homogeneous lower fluid. Relevant for the actual observation we assume that the homogeneous lower layer is relatively thin compared to the thickness of the upper layer. The linear long wave speed is estimated using eq. (20) (see section 8.2 below) with h2 » hi, giving c0 ~ N0h2/Tr ~ 30 cm/s. The Brunt-Vaisala frequency is determined by N$ = gAp/(ph2) with h2 = 146 m. The equations (18)-(20) show that a wave length of A = 300 m corresponds to a wave speed of 30 cm/s. The wave induced oscillatory vertical velocity at level 100 m becomes 12 cm/s at maximum, i.e. 40 % of the wave speed. The linear formulae predicts a horizontal surface velocity that somewhat exceeds the wave speed. The estimated wave period becomes 16 minutes (frequency 0.39 m i n - 1 ) which is significantly longer than indicated by the raw data from the echo-sounder (period 5 min, frequency 1.26 m i n - 1 ) . (The buoy was drifting with the unmonitored surface current.) Recent measurements 49 in the area (70° 27,6'N, 17° 13,9'E) reveal a maximal current speed of 65 cm/s, contributing to the frequency of encounter by an amount Uk = 0.82 m i n - 1 . This may explain the difference between the observed and estimated wave periods. The wave trains are most truly tidally driven. The Atlas of the tides of the shelves of the Norwegian and Barents Seas (Gjevik, N0st and Straume 50 ) exhibts dominating M2 and S2 modes with maximal currents at the location of about 25 cm/s and 10 cm/s, respectively. At spring tide the tidal current has a maximal speed of 35 cm/s. This corresponds to the magnitude of the linear long wave speed of the stratified fluid at the location. It is widely documented that a pronounced internal wave making may take place when the flow is close to critical, i.e. the bulk flow velocity is close to the linear long wave speed of the stratified fluid, see e.g., Grue at al. 14 . 8. Fully nonlinear reference models For completeness were include a brief documentation of the fully nonlinear solitary wave models used in the reference computations. 8.1. Interface

model

Several works on modelling fully nonlinear internal solitary waves are published (e.g. Amick and Turner 10 , Turner and Vanden-Broeck11, Evans and

Internal


Velocimetry

269

Ford 13 ). Our method is based on the fully nonlinear transient interfacial model Grue et al. 14 , solving the Laplace equation in the layers, integrating the prognostic equations resulting from the kinematic and dynamic boundary conditions at the (moving) interface. The model accounts for full nonlinearity, full dispersion, any density or depth ratio, and puts no assumption on the wave length. The model was developed with the primary goal to compute transient waves. A version of the method to obtain waves of permanent form is derived (Grue et al. 15 , section 3.1). The coded method is fully documented and may be downloaded, see Rusas 51 . The main steps of the method which is used in the present computations are outlined. Solitary waves of permanent form are modelled in the frame of reference moving with the wave speed c. In this frame of reference the interface is "frozen". There is a horizontal current in the far field with velocity —c. A coordinate system O — xy is introduced with the x-axis at the level of the interface in the far field and the y-axis pointing upwards. We apply complex analysis and introduce the complex variable z = x+iy and complex velocity 52(2) = u2(x,y) — c — iv2(x,y) in the upper layer and qi(z) = u\{x,y) — c — ivi(x,y) in the lower. The components Uk — c and Vk (k = 1,2) determine the horizontal and vertical components of the velocity field, respectively. At the top and bottom of the fluid layer the vertical velocity is zero, qi + c and q2 + c, being analytic functions, are found using Cauchy's integral theorem, giving (the fluid in each layer is assumed incompressible, homogeneous and inviscid, and the motion irrotational)

Js

z'-z

Jsz*

+ 2ih2-z>

\

27ri(g2 + c), z' e f i 2 ( 4 ) -7ri(g2 + c ) , z' € S

in the upper layer, where q'2 = q2(z'), and PV f

qi z)

^

+ C

dz 4- f

Q1{Z)

+ C

dz* - f

27d(

PIV and water waves

Linear Water Waves

Linear water waves

Linear Water Waves

Numerical modeling of water waves

Water waves and ship hydrodynamics: An introduction

Numerical Modeling of Water Waves, Second Edition

Linear water waves: a mathematical approach

Water waves. The mathematical theory with applications

Water Waves and Ship Hydrodynamics: An Introduction, Second Edition

Water Resources and Water Management

Magnetization Oscillations and Waves

Dielectrics and waves

Vibrations and Waves

Waves and Mean Flows

Lenses and Waves

Waves and compressible flow

Vibrations and Waves

Electromagnetic Fields And Waves

Waves and compressible flow

Vibrations and Waves

Shock waves and explosions

Vibrations and Waves

Waves and Mean Flows

Shock and denotation waves

Tsunami and nonlinear waves

Electromagnetic Waves and Antennas

Vibrations and Waves

Vibrations and Waves

Tsunami and Nonlinear Waves

Magnetization Oscillations and Waves

PIV and water waves

Linear Water Waves

Linear water waves

Linear Water Waves

Numerical modeling of water waves

Water waves and ship hydrodynamics: An introduction

Numerical Modeling of Water Waves, Second Edition

Linear water waves: a mathematical approach

Water waves. The mathematical theory with applications

Water Waves and Ship Hydrodynamics: An Introduction, Second Edition

Water Resources and Water Management

Magnetization Oscillations and Waves

Dielectrics and waves

Vibrations and Waves

Waves and Mean Flows

Lenses and Waves

Waves and compressible flow

Vibrations and Waves

Electromagnetic Fields And Waves

Waves and compressible flow

Vibrations and Waves

Shock waves and explosions

Vibrations and Waves

Waves and Mean Flows

Shock and denotation waves

Tsunami and nonlinear waves

Electromagnetic Waves and Antennas

Vibrations and Waves

Vibrations and Waves

Tsunami and Nonlinear Waves

Magnetization Oscillations and Waves

Recommend Documents