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An Investigation of Two-Dimensional Flow Separation with Reattachment Djilali, Nedjib 1987

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An Investigation of Two-Dimensional Flow Separation with Reattachment By  Nedjib Djilali B.Sc.(Hons), M.Sc,  University  D.I.C.,  A  THESIS THE  Hatfield  of London,  Imperial  SUBMITTED  College,  IN PARTIAL  REQUIREMENTS DOCTOR  Polytechnic,  1977 1979  1979  FULFILLMENT  OF  FOR T H E DEGREE OF  OF  PHILOSOPHY  in THE  FACULTY  OF GRADUATE  Department of Mechanical  STUDIES  Engineering  We accept this thesis as conforming to the required standard  T H E  U N I V E R S I T Y  O F  BRITISH  J u l y 1987 © Nedjib D j i l a l i , 1987  C O L U M B I A  In  presenting  degree at  this  the  thesis in  University of  partial  fulfilment  of  of  department  this or  publication of  thesis for by  his  or  her  iep*-  representatives.  i M g c U Q . m <-^vf  4  ,  for  an advanced  Library shall make  it  agree that permission for extensive  It  this thesis for financial gain shall not  The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date  that the  scholarly purposes may be  permission.  Department of  requirements  British Columbia, I agree  freely available for reference and study. I further copying  the  is  granted  by the  understood  that  head of copying  my or  be allowed without my written  A bstract T h i s thesis presents an experimental study and n u m e r i c a l predictions of the separated-reattaching flow around a bluff rectangular section.  T h i s laboratory con-  figuration, chosen for its geometric simplicity, exhibits a l l m a i n features of twodimensional flow separation w i t h reattachment. Detailed turbulent flow measurements of the mean and fluctuating flow field are reported.  T h e measurement techniques used are: hot-wire anemometry, pulsed-  wire anemometry and pulsed-wire surface shear stress probes. T h e separated shear layer appears to behave like a conventional m i x i n g layer over the first half of the separation bubble, b u t exhibits a lower g r o w t h rate a n d higher turbulent intensities i n the second half.  In the reattachment region, the flow is found to be highly  turbulent a n d unsteady. A finite difference m e t h o d is used, i n conjunction w i t h a modified version of the T E A C H code, to predict the mean  flow  field.  T w o discretization schemes  are used: the h y b r i d - u p w i n d differencing ( H D ) scheme, a n d the bounded-skewh y b r i d differencing ( B S H D ) scheme. L a m i n a r flow computations are performed for Reynolds numbers i n the range 100 to 325. T h e H D computations underpredict the separation-bubble length by up to 35% as a result of false diffusion.  The B S H D  predictions, on the other h a n d , are i n excellent agreement w i t h the experimental results reported i n the literature. T u r b u l e n t flow computations using the k — e turbulence m o d e l and the B S H D scheme result i n a reattachment length about 30% shorter t h a n the present measured value. W h e n a curvature correction is incorporated into the m o d e l , a reattachment length of 4.3.D, compared to the experimental value of 4.7D, is predicted. T h e predicted mean flow, turbulent kinetic energy field and pressure d i s t r i b u t i o n are i n good agreement w i t h experimental observations. A n alternative m e t h o d of analysis, based on the m o m e n t u m integral technique, is presented.  T h e m e t h o d is not applied to the blunt-rectangular plate problem,  b u t its use is i l l u s t r a t e d for the simpler case of the flow i n a sudden expansion, and promising results are obtained.  ii  C ontents  Abstract  ii  L i s t of Figures  vi  L i s t of Tables  ix  Nomenclature  x  Acknowledgements 1  Introduction  1  1.1  Schematic of the Flowfield  3  1.2  L i t e r a t u r e Review  5  1.2.1  E x p e r i m e n t a l Studies  5  1.2.2  T h e o r e t i c a l studies  9  Scope of the Present Investigation  12  1.3 2  3  xiii  Experimental Arrangement  a n d M e a s u r e m e n t Techniques  14  2.1  E x p e r i m e n t a l Facility and E q u i p m e n t  14  2.2  Measurement Techniques  18  2.2.1  Pressure Measurements  18  2.2.2  Velocity and Turbulence Measurements  18  2.2.3  W a l l Shear Stress Measurements  21  E x p e r i m e n t a l Results  26  3.1  26  3.2  Two-dimensionality Structure of the Separation B u b b l e  29  3.2.1  30  Reattachment L e n g t h and W a l l Measurements iii  CONTENTS  3.2.2 3.3  3.4 4  5  M e a n F l o w Properties  34  Structure of the Shear Layer  45  3.3.1  G r o w t h of the Shear Layer  45  3.3.2  Frequency Spectra and Autocorrelations  47  Effect of A n g l e of Separation  53  Mathematical Model  58  4.1  Background  59  4.2  Conservation Equations and T i m e - A v e r a g i n g  59  4.3  k — e Turbulence M o d e l  61  4.4  General Transport E q u a t i o n  63  4.5  B o u n d a r y Conditions  66  Computational Procedure  69  5.1  Finite Volume Formulation  70  5.1.1  G r i d L a y o u t and Variables L o c a t i o n  70  5.1.2  Integration of General Transport E q u a t i o n  70  5.2  5.3  6  iv  F i n i t e Difference Discretization  73  5.2.1  H y b r i d Differencing  73  5.2.2  False Diffusion  76  5.2.3  Skew Differencing  78  Treatment of B o u n d a r y Conditions  81  5.3.1  Types of Boundaries  81  5.3.2  Special Case: Corner Nodes  86  5.4  S o l u t i o n Procedure  86  5.5  P r e l i m i n a r y computations  88  5.5.1  L o c a t i o n of Inlet and Outlet Boundaries  88  5.5.2  N o n - u n i f o r m G r i d Arrangement  89  5.5.3  Convergence C r i t e r i o n  92  N u m e r i c a l Results  93  6.1  Laminar Flow  94  6.1.1  G r i d Independence  94  6.1.2  Effect of Reynolds N u m b e r and C o m p a r i s o n w i t h E x p e r i ments  96  CONTENTS  6.2  6.3 7  v  Turbulent F l o w  104  6.2.1  Preamble and Effect of G r i d Refinement  104  6.2.2  Predictions and C o m p a r i s o n w i t h E x p e r i m e n t a l D a t a  6.2.3  Effect of Solid Blockage  C o m p u t a t i o n a l costs  Conclusions  References  . . . .  109 120  :  122 124 130  APPENDICES A  Tabulated D a t a  136  B  Potential Flow Analysis  142  B.l  Theory  142  B. 2  Results  146  C  M o m e n t u m Integral A n a l y s i s  148  C. l  Formulation  149  C.2  Turbulent F l o w  152  C.3  Results  154  C.4  Closing Remarks  157  D  M o d i f i e d k-e M o d e l  160  E  Wall Function Treatment  163  List of F i g u r e s 1.1  Configurations w h i c h exhibit two-dimensional flow separation w i t h reattachment  1.2  2  Schematic of the flow around a blunt plate  4  2.1  B o u n d a r y layer w i n d tunnel  16  2.2  M o d e l of blunt rectangular plate used i n w i n d tunnel experiments.  2.3  Pulsed-wire anemometer probe  20  2.4  W a l l shear stress probe  22  3.1  Surface flow v i s u a l i z a t i o n patterns  28  3.2  D i s t r i b u t i o n of surface forward flow fraction  31  3.3  Reattachment lengths found for various solid blockage ratios  31  3.4  D i s t r i b u t i o n of surface pressure coefficient  33  3.5  D i s t r i b u t i o n of mean and fluctuating w a l l shear stress coefficients.  3.6  M e a n l o n g i t u d i n a l velocity profiles  35  3.7  M e a n streamlines deduced from pulsed-wire measurements  37  3.8  F l u c t u a t i n g (r.m.s) longitudinal velocity profiles  39  3.9  F o r w a r d flow fraction profiles  42  .  .  16  33  3.10 Skewness and flatness distributions of fluctuating w a l l shear stress. .  44  3.11 Representative shear layer positions  46  3.12 G r o w t h of the shear layer  46  3.13 Frequency spectra of velocity  fluctuations  along shear layer edge y . e  3.14 D o m i n a n t frequencies of velocity spectra 3.15 A u t o c o r r e l a t i o n of velocity  fluctuations  49 50  . .  51  3.16 Integral time scales deduced from autocorrelation measurements. . .  52  3.17 Effect of separation angle on surface pressure d i s t r i b u t i o n  54  3.18 R e d u c e d pressure distributions  54 vi  along shear layer edge y . e  LIST  OF FIGURES  vii  3.19 V a r i a t i o n of reattachment  length and base pressure w i t h angle of  separation  57  3.20 Effect of t r i p p i n g the boundary layer on reduced pressure d i s t r i b u tion  57  4.1  Flow domain  66  5.1  G r i d layout  71  5.2  (a) L o c a t i o n of scalar a n d velocity cells; (b) T y p i c a l control volume.  72  5.3  F i n i t e difference nodes  76  5.4  (a) F i n i t e difference c o m p u t a t i o n a l star; (b,c) Interpolation ranges for skew u p w i n d differencing scheme  80  5.5  Inflow b o u n d a r y cells  82  5.6  Outflow b o u n d a r y cells  83  5.7  S y m m e t r y axis cells  85  5.8  Solid wall b o u n d a r y cells  85  5.9  Cells near plate corner  87  5.10 C o m p u t a t i o n a l d o m a i n  89  5.11 T y p i c a l g r i d d i s t r i b u t i o n  91  6.1  Effect of g r i d refinement on reattachment length  95  6.2  V a r i a t i o n of reattachment length w i t h Reynolds number  97  6.3  Streamlines for laminar flow  99  6.4  V e l o c i t y field for l a m i n a r flow  6.5  Pressure d i s t r i b u t i o n along top surface of the plate (laminar flow).  6.6  Turbulent flow: effect of grid refinement on c o m p u t e d reattachment  101  length 6.7  106  Effect of g r i d refinement on computed velocity profile at x/D  — 2  ( P D M computation) 6.8  107  Effect of g r i d refinement on computed wall shear stress along top surface of plate ( P D M computation)  6.9  . 103  108  L o c a t i o n of grid points adjacent to the w a l l i n terms of the wall coordinate y  +  ( P D M computation)  108  6.10 M e a n l o n g i t u d i n a l velocity profiles ( B S H D c o m p u t a t i o n and P W A measurements  110  LIST  OF FIGURES  viii  6.11 M e a n l o n g i t u d i n a l velocity profiles ( P D M c o m p u t a t i o n a n d P W A measurements  Ill  6.12 Contours of constant effective viscosity  113  6.13 Turbulent kinetic energy profiles  114  6.14 Turbulent kinetic energy profiles  115  6.15 Contours of pressure ( P D M computation)  118  6.16 C o m p a r i s o n of computed and experimental pressure distributions.  . 118  6.17 C o m p a r i s o n of computed and experimental w a l l shear stress d i s t r i butions  119  6.18 P r e d i c t e d streamline pattern ( P D M computation)  119  6.19 Effect of blockage on turbulent flow reattachment lengths  121  6.20 P r e d i c t e d pressure distributions for various blockage ratios  121  6.21 C o m p u t i n g time on V A X 11/750 computer as a function of c o m p u t a t i o n a l array size (Turbulent  flow)  123  B.l  Wake source model  143  B.2  P h y s i c a l and transform planes  143  B.3  C o m p a r i s o n of calculated separation streamlines w i t h experiment.  B. 4  C a l c u l a t e d pressure distributions for different specified base pressures  . 147 147  C. l  M o d e l l i n g of 2-D sudden expansion  C.2  V a r i a t i o n of reattachment length w i t h Reynolds number  155  C.3  Effect of expansion ratio on reattachment length  155  C.4  Pressure d i s t r i b u t i o n for a 5:2 expansion ratio  156  C.5  Effect of expansion ratio on reattachment length for turbulent flow. . 158  C.6  C o m p a r i s o n of calculated and experimental pressure d i s t r i b u t i o n for turbulent  E.l  Near w a l l control volume  flow  flow  150  158 164  List of Tables 2.1  Measurement uncertainties  25  4.1  k — e m o d e l constants  63  4.2  Diffusion coefficients and source terms  65  5.1  L i n e a r i z e d source terms  75  A.l  Experimental data  137  C.l  Integral constants for cubic velocity profile  159  ix  Nomenclature AR  Aspect ratio (=  a  Coefficient i n finite difference equation.  BR  Solid blockage ratio (=  C  Convective flux coefficient.  n  Ci,Ci,  C,j.  S/D).  D/H).  Turbulence model constants.  Cf  W a l l shear stress coefficient (=  C  Pressure coefficient (= (p — P o o ) / f p ^ < L ) -  p  T^/^pU^).  C *  R e d u c e d pressure coefficient (= ( C — C )/(l  D  P l a t e thickness; also diffusive flux coefficient.  E  Integration constant i n logarithmic law of the w a l l .  ER  C h a n n e l expansion ratio.  p  p  E ,E  G r i d expansion ratio.  F  T o t a l flux coefficient.  x  F  y  T  Pmin  —  Flatness factor of T.  f  Frequency.  G  Generation of turbulent kinetic energy.  H  W i n d - t u n n e l / c h a n n e l height.  h  Step height.  k  Turbulent kinetic energy; also interpolation factor in skew differencing.  L , Ld u  Distance from recirculation zone to upstream a n d downstream boundaries.  P(r)  P r o b a b i l i t y density function of r.  Pe  C e l l Peclet number.  p  Static pressure.  p*  Pressure (= p +  Re  R e y n o l d s number (=  R  A u t o c o r r e l a t i o n of u .  S  uu  \pk). TJ^Dju).  S p a n of w i n d - t u n n e l m o d e l . x  C )). Pmin  NOMENCLATURE  Source t e r m i n transport equation. Coefficient of linearized source t e r m . S  T  Skewness factor of r . Integral timescale.  t  Time. M e a n velocity i n x,-, x , a n d y directions.  Ui,U,V  u,v  C a r t e s i a n coordinates.  •^i j i y x  T i m e - m e a n reattachment length.  y  W a l l coordinate (=  z = x + iy  C o m p l e x physical plane.  a  A n g l e of separation.  Ax, Ay  C o m p u t a t i o n a l cell dimensions.  r  +  yU /u). T  Kronecker delta (= 1 for i = j a n d = 0 otherwise). Sx, 6y  G r i d spacing.  e  D i s s i p a t i o n rate of turbulent kinetic energy.  4>  General variable.  r  Diffusion coefficient for <f> variable.  7  F o r w a r d flow fraction,  AC  von K a r m a n constant.  A  Pressure gradient parameter.  A*  D y n a m i c viscosity.  v  K i n e m a t i c viscosity. Shear stress; also time lag.  T  Non-dimensional stream function (= / Jo F l u i d density.  P  Constants i n k and e equations.  f = i + in  C o m p l e x transform plane.  (U/Uoo)  diy/D)).  xii  NOMENCLATURE  S u b / superscripts  eff  Effective value.  i,j  Tensor indices; grid points.  max  M a x i m u m value.  min  M i n i m u m value.  P,N,S,E,W,NE, NW, SE, SW  G r i d points.  n,s,e,w  C e l l faces.  s  V a l u e at separation.  t  Turbulent value.  w  W a l l value.  oo  Free-stream value.  ( )'  R . m . s . value of fluctuating quantity.  ( )  Time-averaged value.  A cknow ledgements I w o u l d like to express m y gratitude to my supervisor, Professor I. S. Gartshore, who remained throughout the course of this study a s t i m u l a t i n g source of guidance a n d encouragement. W o r k i n g w i t h h i m has been a pleasure a n d a privilege. N e x t , I w o u l d like to express m y sincere appreciation to Professor M . Salcudean for the interest shown i n this work. Her enthusiasm and valuable suggestions cont r i b u t e d m u c h to the progress of this research. I w o u l d also like to thank Professor G . V . P a r k i n s o n for his helpful suggestions, comments a n d discussions. T h a n k s are due to the technicians of the Department of M e c h a n i c a l Engineering. W i t h o u t their expertise and ingenuity, many aspects of the experimental part of this work w o u l d not have been possible. L a s t , b u t most of a l l , I w o u l d like to thank A n n e for her loving support and her patience.  xiii  Chapter 1  Intro d u c t i o n  Separated-reattaching flows, t y p i c a l of flows a r o u n d bluff bodies, occur i n a large variety of environmental and engineering situations. T h e recirculating flow regions— k n o w n as separation bubbles—encountered i n these flows have a significant impact on the performance of, for example, airfoils at higher angles of attack,  turbine  blades, diffusers a n d combustors. Separated flows determine, to a large extent, the drag of r o a d vehicles a n d are the dominant feature of atmospheric flows over b u i l d ings, fences a n d hills. T h e y are also a critical factor i n the design of structures, such as bridges, susceptible to potentially disastrous w i n d induced oscillations. W h i l e the importance of separated flows w i t h reattachment has long been recognized, a n d much progress has been accomplished over the last twenty years, many aspects of these  flows—particularly  i n the turbulent flow regime, r e m a i n poorly  understood, because they are difficult to measure or predict. In an effort to isolate those flow features of fundamental importance, a n u m ber of laboratory geometries have been devised to generate two-dimensional separated reattaching flows.  Some of these geometries are shown i n Figure 1.1. 1  The  CHAPTER  1.  2  INTRODUCTION  ^777777777777, ^7?  A  '777//  Backward-facing step  Forward-facing step  Elunt / / / / / / / / / / / / / / / /  rectangular  plate  F i g u r e 1.1 (a-g). Configurations which exhibit two-dimensional flow separation w i t h reattachment  CHAPTER  1.  INTRODUCTION  3  flow around the blunt rectangular section (Figure 1.1 g) is one of the simplest twodimensional recirculating flows, yet it exhibits all the i m p o r t a n t characteristics of separated reattaching  flows.  It combines several of the advantages of the other  geometries: fixed separation p o i n t , single p r i m a r y recirculation zone and simple u p s t r e a m b o u n d a r y conditions w h i c h make it ideal as a test case for numerical methods. In a d d i t i o n it is the simplest shape geometrically. T h e present work is a detailed experimental and c o m p u t a t i o n a l study of the flow a r o u n d a blunt rectangular section. T h e aims are first to provide a description of the structure of a separated reattaching flow, and secondly to model this flow using a numerical method.  1.1  Schematic of the Flowfield  T h e general features of the flow around a bluff rectangular plate are described in this section. T h e information is largely d r a w n from the relevant literature which is reviewed i n the next section. T h e m a i n characteristics of the flow around a rectangular plate are perhaps best described b y breaking up the flowfield into several zones, each h a v i n g distinct dominant features. A schematic view of the flow is shown i n F i g u r e 1.2. W h e n the oncoming i r r o t a t i o n a l flow (I) impinges on the front face of the plate, a boundary layer (II) develops on either side of the stagnation point. D u e to highly favourable pressure gradients, this boundary layer remains t h i n up to the sharp corner where it separates a n d forms a free shear layer (III) w i t h a large streamline curvature. T h e separated shear layer is i n i t i a l l y highly curved. In a first phase, the separated shear layer, as it proceeds downstream, grows under relatively constant pressure by  CHAPTER  1.  INTRODUCTION  4 Mean separation streamline  ©  v)  I  Irrotatlonal flow k  /—(VI)  IV Recirculating flow region  II Boundary layer  V  III Free shear layer  VI Relaxing shear layer and redeveloping boundary layer  Reattachment region  F i g u r e 1.2. Schematic of the flow around a blunt plate.  entraining fluid from b o t h the "outer" i r r o t a t i o n a l flow a n d the "inner" recirculating flow regions. In the reattachment zone ( V ) , the shear layer curves towards and interacts strongly w i t h the w a l l to w h i c h it eventually reattaches. P a r t of the flow at reattachment is deflected upstream into the recirculation zone ( I V ) , to compensate for the fluid d r a w n out b y entrainment; the rest is deflected downstream into the recovery zone (VI) where a new b o u n d a r y layer develops a n d merges w i t h the outer shear layer.  T h e flow i n the reattachment zone is characterized by large pressure gradients, low m e a n velocities, very large local turbulent intensities a n d instantaneous reversals.  A n i m p o r t a n t length scale of this flow is the reattachment length  flow x. r  T h i s length, w h i c h is a measure of the extent of the separation bubble, is defined as the distance from separation to the point of zero mean w a l l shear stress.  CHAPTER  1.  INTRODUCTION  5  F r o m this schematic description, it is clear that the flow a r o u n d a bluff plate, t h o u g h it is one of the simplest separated-reattaching flows, is quite complex. T h e h i g h levels of turbulent intensities combined w i t h fluctuations i n flow direction make measurements i n the recirculating flow region difficult, a n d conventional measurement techniques, such as hot-wire anemometry, are of l i m i t e d use there.  1.2  Literature Review  1.2.1  Experimental Studies  In the last decade, experimental research i n separated-reattaching flows has been greatly s t i m u l a t e d by the development of instruments suitable for measurements in r e c i r c u l a t i n g flows, especially the laser-Doppler a n d pulsed-wire anemometers. G e n e r a l reviews of the literature have been undertaken b y B r a d s h a w & W o n g (1972) for earlier experimental w o r k , a n d by W e s t p h a l et al. (1984) and Simpson (1981; 1985) for more recent developments. T h e latter reference is a comprehensive survey of measurement techniques, experimental studies, as well as c a l c u l a t i o n methods. T h e following is an overview of the previous experimental studies directly relevant to this work. One of the earliest studies of the flow past a blunt rectangular section is due to Roshko &: L a u (1965), w h o also considered, i n the same paper, the flow around various forebody shapes w i t h splitter plates. A l t h o u g h their study d i d not involve any flowfield measurements, it gave i m p o r t a n t insight i n the pressure recovery process i n reattaching  flows.  A n i m p o r t a n t finding of Roshko Sz L a u was that the  CHAPTER  1.  INTRODUCTION  6  pressure distributions for a l l cases considered collapsed to a single curve when the pressure was normalized by the pressure at separation a n d the streamwise distance was n o r m a l i z e d by the reattachment length. T h i s suggested that some features of separated-reattaching flows are universal, a n d that the reattachment length is an i m p o r t a n t characteristic length scale of these flows. Extensive measurements were performed by O t a a n d co-workers i n a series of experiments ( O t a & K o n 1974; O t a & Itasaka 1976; O t a & N a r i t a 1978), and i m p o r t a n t observations were made: • In the separation bubble the pressure i n the cross-stream direction remains nearly constant • M a x i m u m backflow velocities of about 25-30% of the free stream velocity occur i n the middle of the bubble. • Peak turbulent intensities are of the order of 30% of the free stream velocity a n d occur a r o u n d reattachment. • Reattachment occurs 4 to 5D downstream from separation. It was also noted that w h e n the plate is heated, m a x i m u m heat transfer rates occur at reattachment. W h i l e these measurements give a good description of the gross features of the flow, their accuracy is dubious as a result of the following experimental procedures. F i r s t , the mean and turbulent velocities were measured using a P i t o t tube and a hot-wire anemometer respectively. B o t h techniques are unreliable i n this k i n d of flow, a n d this is reflected b y some inconsistencies reported i n the near wall profiles.  Second, an aspect ratio (tunnel w i d t h / p l a t e thickness) of about 5 was used  CHAPTER  1.  INTRODUCTION  in the experiments.  7  Three-dimensional effects are, therefore, likely to have been  i m p o r t a n t . F i n a l l y the procedure for measuring the reattachment length was not explained i n sufficient detail. K i y a et al.  (1981) also used a hot-wire. T h o u g h they present  measurements  only outside the recirculation bubble, some of these measurements fall w i t h i n the highly turbulent flow region a n d should therefore be viewed w i t h c a u t i o n . A comprehensive set of measurements were made by K i y a & Sasaki(1983). T h e y used directionally sensitive split-film sensors i n conjunction w i t h hot-wire anemometry to measure mean velocities, fluctuating velocities a n d forward flow fraction. T h e y also presented a few measurements of the turbulent shear stresses a r o u n d reattachment. I m p o r t a n t aspects of the unsteady nature of the flow were also reported in this paper a n d are reviewed later. T h e effect of R e y n o l d s number was investigated by O t a et al. (1981) who also considered the effect of separation angle. U s i n g flow v i s u a l i z a t i o n (water w i t h alum i n i u m powder), they observed three flow regimes:  i) T h e l a m i n a r separation-laminar reattachment regime i n w h i c h the reattachment length increases w i t h R e y n o l d s number.  A m a x i m u m reattachment  length of about 4.5D was reported to occur at Re ~ 270. These observations are i n qualitative agreement w i t h those made b y L a n e & Loehrke (1980). Lane & Loehrke found, however, a m a x i m u m reattachment length of about  6.5D  at Re = 325. T h i s difference is probably due to the larger aspect ratio used by L a n e &; Loehrke (11.5 as opposed to 4.55).  ii) T h e l a m i n a r separation-turbulent reattachment regime characterized by the appearance of instabilities in the shear layer near separation and transition  CHAPTER  1.  INTRODUCTION  8  to turbulence before reattachment.  In this regime, the reattachment length  decreases w i t h R e y n o l d s number.  iii) T h e turbulent separation-turbulent reattachment regime (Re ~ 2 x l 0 ) where 4  the separated shear layer becomes turbulent very soon after separation. T h e Reynolds number is found to have no effect on the reattachment length i n this flow regime.  T h i s Reynolds-number-independent regime was also observed b y H i l l i e r & C h e r r y (1981a). T h e y noted that the flow is essentially R e y n o l d s number independent i n the range 3.4 x l O  4  < Re < 8.0 x l O , w i t h a weak elongation appearing only when 4  Re < 8.0 x 10 . In the same paper, Hillier & C h e r r y showed that the flow is very 4  sensitive to grid-generated free-stream-turbulence levels. For example a shortening of the bubble from 4 . 8 8 D to 2.12D was reported w h e n the free stream turbulence intensity was increased from about 0.1% to 6.5%. T h e effect of free stream turbulence was also investigated by K i y a &: Sasaki (1983b). T h e y used a r o d upstream of the plate to generate the turbulence and obtained results similar to those of H i l l i e r & Cherry. D z i o m b a (1985) used wires on the front face of the plate to t r i p the b o u n d a r y layer just before separation. He found this to have the same qualitative effect as an increase i n free stream turbulence. He argued, however, that the shortening of the bubble was mostly due to an effective change i n the angle of separation. T h e unsteady structure of the separation bubble has been the subject of a series of t h o r o u g h studies b y H i l l i e r & C h e r r y (1981b), C h e r r y et al. (1983;1984) and K i y a & Sasaki (1983a). These studies, using a c o m b i n a t i o n of flow visualization, measurements of fluctuating surface pressures a n d a judicious use of pressure-velocity  CHAPTER  1.  INTRODUCTION  9  correlations, clearly demonstrated that: • T h e shear layer near separation exhibits a low-frequency flapping m o t i o n . T h e mechanisms for this low frequency unsteadiness are not clearly understood. • D o w n s t r e a m from separation, large scale vortices are shed i n pseudoperiodic bursts. In an extension of these studies, K i y a & Sasaki (1985a) used c o n d i t i o n a l s a m p l i n g of the velocity field to deduce the structure of the large scale vortices. T h e y concluded that these vortices have a h a i r p i n structure. T h e unsteadiness of the reattachment process seems to be an inherent feature of separation bubbles i n a l l two dimensional geometries. F o r example, it has been observed i n the b a c k w a r d facing step flow ( E a t o n & J o h n s t o n 1982) and i n the flow around the flat p l a t e / s p l i t t e r plate c o m b i n a t i o n (Gartshore &; S a v i l l , 1982). C h e r r y et al. (1984) suggested that it is the large-scale shedding of vorticity that causes the m a x i m u m shear layer turbulent stresses a n d pressure  1.2.2  fluctuations  to occur i n the v i c i n i t y of reattachment.  Theoretical studies  T h e theoretical analysis of separated-reattaching flows poses many difficulties as a result of shear layer curvature, strong pressure gradients, and flow recirculation. A t higher R e y n o l d s numbers, an a d d i t i o n a l difficulty is the varied a n d complex nature of the turbulence field. T h e prediction of these flows can be attempted by using either zonal m o d e l l i n g or global modelling; b o t h approaches rely on numerical solution methods. In zonal modelling, one recognizes that different regions of the flow have different dominant features and a c o m p u t a t i o n a l procedure is developed accordingly. A n example of this approach is the viscous-inviscid interaction  CHAPTER  1.  INTRODUCTION  10  p r o c e d u r e . G l o b a l models, on the other h a n d , use the same set of equations for the 1  entire flowfield. A m o n g s t these, computationally intensive finite difference methods w h i c h solve the Navier-Stokes equations (in their time-averaged form i n the case of turbulent flows) have gained a wider acceptance over the last two decades as a result of the availability of more powerful computers.  These methods have been  used, w i t h v a r y i n g degrees of success, to predict a number of recirculating flows. T h e l a m i n a r flow a r o u n d the blunt flat plate was c o m p u t e d by G h i a &: Davis (1974) who used a finite difference method to solve the Navier-Stokes equations i n their streamfunction-vorticity form. T h e i r results were subsequently compared b y L a n e & Loehrke(1980) to their o w n experimental data and showed a large discrepancy. T h e possibility that the numerical solutions h a d not attained true convergence was put forward as a possible explanation for this discrepancy. A related case, the l a m i n a r flow through a cascade formed by a stack of flat plates, was considered recently by M e i &: P l o t k i n s (1986). T h e i r formulation also used the streamfunction-vorticity formulation of the Navier-Stokes equations. B u t second order u p w i n d differencing was used instead of the first order scheme of G h i a & Davis.  T h o u g h their results cannot be compared directly to the experimental  data of L a n e & Loehrke, it is interesting to note that they reported similar trends: flow separation was first found to occur at Re ~ 110 a n d the reattachment length was found to v a r y linearly w i t h Reynolds number up to Re ~ 300. T h i s was the largest R e y n o l d s number for w h i c h convergence could be obtained. T h e y noted that the first order differencing scheme resulted i n shorter reattachment lengths than the second order scheme. T h i s was due to false diffusion . 2  A brief account of this method for separated-reattaching flows is given in Appendix C . False diffusion, or numerical diffusion, is the truncation error associated with the use of upwind differencing in a discretization scheme. X  2  CHAPTER  1.  INTRODUCTION  11  U s i n g the discrete vortex m e t h o d , K i y a & Sasaki (1982) carried out an inviscid s i m u l a t i o n of the turbulent flow on a blunt flat plate. T h e s i m u l a t i o n consisted of t r a c k i n g elemental vortices w h i c h were shed downstream from the separation corner. In order to represent the viscous/turbulent dissipation of vorticity, the circulation of elemental vortices was reduced as a function of their age. T h e model required some empirical i n p u t (pressure at separation a n d mean reattachment length) to determine some free parameters. In general, the predictions of the mean velocity and surface pressure were reasonable, except i n the reattachment region. Remarkably, the unsteadiness of the flow was fairly well represented. Furthermore the  fluctuating  component of the surface pressure, a quantity w h i c h cannot be obtained at all w i t h the steady state finite difference m e t h o d , was i n tolerable agreement  with  experiments. N o finite difference predictions of the turbulent flow around a flat plate are reported i n the literature. M a n y such predictions have, however, been attempted for other geometries, a favourite being the downward facing step. A comprehensive a n d c r i t i c a l review of m a n y of these predictions, a l l based on the solution of the timeaveraged Navier-Stokes equations i n conjunction w i t h a turbulence closure m o d e l , can be found i n a recent article b y Nallassamy (1987). In the context of this study, it is of special interest to note that, i n most computations, the equations were discretized using u p s t r e a m differencing. T h e gross features of the recirculation zone were i n general underpredicted, a n d discrepancies of up to 30% i n the reattachment length were reported.  T h i s was p a r t l y due to  false diffusion w h i c h is inherent to upstream differencing. M a n y authors, however, a t t r i b u t e d the discrepancies to inadequate turbulence modelling. Since errors due to m o d e l l i n g cannot usually be dissociated from n u m e r i c a l errors, it was pointed  CHAPTER  1.  INTRODUCTION  12  out by C a s t r o (1977), among others, that prolonged arguments about turbulence model deficiencies were somewhat pointless unless false diffusion was reduced to negligible levels. T h i s c o u l d be achieved, i n principle, by refining the g r i d .  But  this is often i m p r a c t i c a l because c o m p u t i n g costs increase r a p i d l y w i t h the n u m ber of c o m p u t a t i o n a l nodes. A n alternative approach is the use of "higher order" differencing schemes such as the "skew-upwind differencing" of R a i t h b y (1976b) or the "quadratic upstream interpolation" of L e o n a r d (1979). T h e a p p l i c a t i o n of these schemes to turbulent recirculating flows has, i n general, resulted i n improved predictions (e.g. Leschziner & R o d i 1981).  1.3  Scope of the Present Investigation  T h e literature survey has shown that despite the extensive information available on the large-scale unsteadiness of the flow around the blunt rectangular plate, there is relatively little reliable quantitative d a t a on the flow w i t h i n the separation bubble. In particular, all available measurements i n this region were made w i t h instruments w h i c h are prone to measurement errors i n turbulent recirculating flows. O n the theoretical side, the only turbulent flow c o m p u t a t i o n available i n the literature is a discrete vortex s i m u l a t i o n , and though the gross features of the flow are reasonably well reproduced, the predictions of various mean flow quantities are only fair, i n d i c a t i n g that other approaches are w o r t h exploring.  CHAPTER  1.  INTRODUCTION  13  T h e objectives of the experimental part of this investigation were (i) to gain further insight into the structure of a two-dimensional separation bubble, and (ii) to provide dependable d a t a for comparison and evaluation of numerical predictions. T o this end, detailed  flowfield  and surface measurements were performed using  pulsed-wire anemometry as well as conventional hot-wire anemometry. T h e theoretical study h a d two objectives: (i) to devise and test a simple calculation procedure based on a m o m e n t u m integral technique, and (ii) to compute the flow using a finite difference method i n conjunction w i t h a two-equation turbulence model (A; — e). T h e first of these objectives was met only partially. Encouraging results were, however, obtained for the simpler case of a sudden expansion flow. T h e c o m p u t a t i o n a l study was carried out successfully for b o t h laminar and turbulent flow a r o u n d a blunt rectangular plate.  Chapter 2  Experimental  A r r a n g e m e n t and  Measurement Techniques  In this chapter, the w i n d tunnel a n d w i n d tunnel model used for the turbulent flow experiments reported i n the next chapter are described. T h e measurement techniques a n d related c a l i b r a t i o n procedures are discussed.  2.1  Experimental Facility and Equipment  Wind Tunnel A l l experiments were performed in the U . B . C . low speed, blower type b o u n d a r y layer w i n d tunnel shown schematically i n Figure 2.1. T h i s tunnel has a 2.4m wide, 1.6m h i g h , 24.5m long test section, a n d a m a x i m u m design w i n d speed of 25 m / s . In the velocity range used for the present experiments (4-15 m / s ) , the velocity d i s t r i b u t i o n in the e m p t y test section was uniform w i t h i n 1%, w i t h turbulent intensities i n the range 0.25-0.4%. 14  CHAPTER2.  EXPERIMENTAL  ARRANGEMENT  AND  TECHNIQUES  15  Wind Tunnel Model T h e m o d e l w h i c h was constructed for the tests consisted of a rectangular base section w i t h endplates a n d side extensions. T h i s configuration, shown i n F i g u r e 2.2, was selected after p r e l i m i n a r y tests, carried out by D z i o m b a (1985) a n d discussed i n the next chapter, indicated that the use of endplates was crucial to the twodimensionality of the flow. T h e section h a d a chord of 800 m m , a thickness D of 89.9 m m ( 3 j " ) — c o r r e s p o n d i n g to a solid blockage ratio D/H between end plates of 1000 m m , giving an aspect ratio S/D  of 5.6%—and a span of 11.1. W i t h the side  extensions mounted, the model spanned 2.2 m across the w i n d t u n n e l . In a d d i t i o n , a t a i l was attached to the t r a i l i n g edge of the model to suppress any periodic vortex shedding w h i c h might otherwise have "contaminated" the flow i n the separation bubble. T h e m o d e l was mounted at a zero angle of incidence, w i t h its front end located about 1.5 m downstream of the nozzle exit. It was held b y eight 0.7 m m diameter piano wires, w h i c h were fastened to the roof a n d floor of the w i n d t u n n e l .  The  s y m m e t r y of the flow was ensured by adjusting the position of the model u n t i l the difference i n pressure coefficients at equivalent positions on the top a n d b o t t o m surfaces were w i t h i n 1% of the d y n a m i c head. T h e base section was made of a l u m i n i u m , and h a d a removable plexiglass top. T h e b o t t o m surface of the model h a d a series of pressure taps at 20 m m intervals along the center line (x-direction), and at 100 m m intervals i n the spanwise (z) direction. T h e angle a at w h i c h the shear layer separates from the plate could be altered (from 45 to 90°) by changing the shape of the leading edge of the plate. T h i s was achieved by adding triangular front-pieces to the front face of the model.  CHAPTER2.  EXPERIMENTAL  4  ARRANGEMENT  AND  TECHNIQUES  1 honeycomb and screens in 4 « 4 m sellling section  Figure 2.1. B o u n d a r y layer w i n d t u n n e l .  F i g u r e 2.2. M o d e l of blunt rectangular plate used i n w i n d tunnel experiments.  CHAPTER2.  EXPERIMENTAL  ARRANGEMENT  AND  TECHNIQUES  17  Traverse Mechanism Measurements i n the flowfield, using various probes, were carried out using a traverse mechanism designed a n d b u i l t specifically for the project. T h e traverse was mounted on t r a n s l a t i o n bearings a n d guided b y two rails for h o r i z o n t a l traversing. A u t o m a t i c v e r t i c a l traversing was obtained by a lead screw mechanism w h i c h was driven by a microcomputer controlled stepper motor. T h e positioning accuracy of the traverse mechanism was 1.5 m m i n the horizontal direction, a n d 0.08 m m i n the vertical direction. T h e b u l k of the traverse mechanism was placed under the floor of the w i n d t u n n e l . T h e only parts p r o t r u d i n g into the flowfield were the probe a n d its support; their interference w i t h the flow was m i n i m a l : changes i n pressure d i s t r i b u t i o n were less t h a n 0.01 i n C , a n d the reattachment length, as measured by surface flow p  v i s u a l i z a t i o n remained unchanged when the traverse mechanism was introduced. T h e rigidity of the mechanism was also checked, a n d no flow induced vibrations were detected.  Data Acquisition System A C B M S u p e r P E T m i c r o c o m p u t e r was used for sampling and processing of velocity a n d shear stress data, as well as for the control of the traverse mechanism.  The  pulsed-wire anemometer unit was interfaced directly to the computer parallel user port, a n d a 12-bit analogue to digital converter was used to sample the analogue signal from the hot-wire anemometer.  A drive was connected to the parallel user  port for direct c o n t r o l of the the traverse mechanism stepper motor.  CHAPTER2.  2.2 2.2.1  EXPERIMENTAL  ARRANGEMENT  AND  TECHNIQUES  18  Measurement Techniques Pressure Measurements  T h e static pressure d i s t r i b u t i o n on the surface of the bluff plate was measured using a Barocel differential pressure transducer and a 48-port Scanivalve system. T h e Scanivalve was mounted inside the model a n d connected to the 0.5 m m diameter pressure taps by short lengths of T y g o n t u b i n g . Reference static a n d d y n a m i c pressures were measured w i t h a P i t o t - s t a t i c probe located at a distance 10D upstream of the front face of the bluff body. T h e pressure transducer h a d a linear response a n d d i d not require any calibrat i o n . However, the zero level (i.e. the output voltage for zero differential pressure) was found to oscillate slightly (with an amplitude of about 1.5% of the full scale reading), a n d therefore required frequent zero level checks. T h e output voltage from the transducer was measured using an integrating voltmeter; integration times of 10 seconds were used for averaging. T h e uncertainty i n the pressure measurements is estimated to be ± 0.03 m m water, corresponding to an uncertainty of ± 0.007 i n the pressure coefficient C . p  2.2.2  Velocity and Turbulence Measurements  T h e velocity field measurements were made b y traversing hot-wire a n d pulsed-wire probes at 10 streamwise stations; each traverse consisted of 20 to 34 points. T h e repeatability of the measurements was usually w i t h i n 1% for the mean velocities a n d w i t h i n 2.5% for the fluctuating velocities.  CHAPTER2.  EXPERIMENTAL  ARRANGEMENT  AND  TECHNIQUES  19  Hot-Wire Anemometer ( H W A ) O u t s i d e the recirculating flow region, streamwise mean and  fluctuating  velocities,  correlations, a n d frequency spectra were measured using a hot-wire probe a n d a D I S A constant temperature anemometer system. T h e probes were s t a n d a r d D I S A single wire probes, w i t h 5 n diameter, 1.25 m m length platinum-coated tungsten wires. T h e hot-wire anemometer bridge was operated at a 1.6 overheat ratio, and the signal was low-pass filtered (10 K H z cut-off frequency). O n line c a l i b r a t i o n , using K i n g ' s law w i t h an exponent of 0.45, was performed against a P i t o t - s t a t i c probe in low turbulence conditions (u/U was used for a l l measurements.  < 0.4%), a n d a digital s a m p l i n g rate of 4 K H z  C o r r e l a t i o n functions and frequency spectra were  obtained using an analogue P A R correlator and a frequency s p e c t r u m analyser. T h e error estimates for the hot wire-anemometer measurements are given i n table 2.1.  Pulsed-Wire Anemometer ( P W A ) T h e high turbulence intensities a n d reversed flows encountered i n recirculating flow regions require the use of special instrumentation. A pulsed-wire anemometer system, manufactured b y P E L A F l o w Instruments, was used i n the present study. T h e general principle of the instrument, originally developed by B r a d b u r y & C a s t r o (1971), is based on the measurement of the flow velocity by t i m i n g the passage of a heat tracer between two points. T h e pulsed-wire probe, shown in Figure 2.3, consists of two sensor or "receiving" wires, a n d a t h i r d pulsed or " t r a n s m i t t i n g " wire located between the two sensor wires. T h e central wire is heated periodically by short d u r a t i o n voltage pulses; the air passing the wire at that t i m e is heated and  CHAPTER2.  EXPERIMENTAL  ARRANGEMENT  AND  TECHNIQUES  20  Sensor wires  Pulsed wire  Figure 2.3. Pulsed-wire anemometer probe. convected w i t h the local instantaneous flow velocity. T h e time taken b y the heated air to reach one of the two sensor wires, w h i c h operate as resistance thermometers, is a direct measure of the magnitude of the instantaneous velocity; the direction of this velocity is determined by the sensor wire w h i c h detects the tracer. T h e instrument has two drawbacks. F i r s t it is relatively large a n d therefore measurements close to solid walls are not possible. Secondly s m a l l scale turbulence m a y influence the probe response.  T h i s was not likely to be a p r o b l e m i n the present investigation  since a relatively large scale experimental facility was used, as recommended by B r a d b u r y & C a s t r o (1971). T h e response of the instrument is not linear due to t h e r m a l diffusion and viscous wake effects. T o take this into account, B r a d b u r y & C a s t r o recommend the use of an e m p i r i c a l c a l i b r a t i o n fit of the form  (2-1)  CHAPTER2.  EXPERIMENTAL  ARRANGEMENT  AND  TECHNIQUES  21  where U is the flow velocity, A and B are c a l i b r a t i o n constants, and T is the time of flight. T h e pulsed-wire probes were calibrated, using P E L A software, i n a low turbulence flow against a Pitot-static probe.  T h e c a l i b r a t i o n constants were  determined by a least square fit procedure, w i t h a resulting s t a n d a r d deviation of less t h a n 2%. T h e c a l i b r a t i o n was frequently checked a n d found to be very stable. Measurements were performed at a sampling rate of 50 H z , w i t h the number of samples taken 5000 i n the outer region a n d 7500 i n the shear-layer/recirculating flow region. T h e accuracy of the pulsed-wire anemometer has been assessed quite thoroughly ( B r a d b u r y 1976; C a s t r o & C h e u n 1982; Westphal et al 1984), a n d its performance was found to be comparable to the hot-wire anemometer i n regions where b o t h instruments are applicable. It should be noted i n this context that, due to electronic noise, there is a lower l i m i t of 2% to the turbulent intensities that can be measured w i t h the instrument ( B r a d b u r y 1976). T h e estimated uncertainties quoted i n Table 2.1 are those given by Westphal et al (1984).  2.2.3  Wall Shear Stress Measurements  T h e d i s t r i b u t i o n of the mean and fluctuating wall shear stresses, a n d of the forwardflow-fraction  were obtained w i t h a pulsed wall-probe w h i c h was mounted on a sup-  p o r t i n g block. T h e block fitted flush w i t h the surface of the m o d e l , a n d could slide along a groove w h i c h was cut out along the centerline of the m o d e l . T h e groove was 600 m m long a n d allowed positioning of the probe anywhere between x/D 1.8.  = 0.2 to  CHAPTER2.  EXPERIMENTAL  ARRANGEMENT  Sensor wires  AND  TECHNIQUES  22  Pulsed wire  F i g u r e 2.4. W a l l shear stress probe.  T h e pulsed wall-probe—also a time of flight type of instrument—is a recent development (Wesphal et al 1981; C a s t r o & D i a n a t 1983) w h i c h has evolved from the P W A . T h e probe, shown i n F i g u r e 2.4, consists of an array of three wires mounted about 0.05 m m above a p l u g . T h e spacing between the wires is 0.75 m m ; the sensor wires are 2 m m i n length w i t h a 2.5 /z diameter, a n d the pulsed-wire is 3 m m i n length w i t h a 9.0 fi diameter. T h e measurement principle is based on the assumption that, i n the p r o x i m i t y of a w a l l , the instantaneous velocity profiles r e m a i n s i m i l a r a n d scale on the w a l l shear stress. T h e electronics used for the P W A was also used for the wall probe, and a c a l i b r a t i o n function of the same form as equation (3.1) was used. T h e c a l i b r a t i o n procedure posed a few problems. Because of the non-linearity of the c a l i b r a t i o n function, c a l i b r a t i o n of the probe i n a turbulent b o u n d a r y layer w o u l d lead to errors, even for moderate turbulent intensities.  Nevertheless turbulent flow c a l i b r a t i o n  CHAPTER2.  EXPERIMENTAL  ARRANGEMENT  facilities are easier to set u p . Westphal et al.  AND  TECHNIQUES  23  (1981) calibrated their probe i n a  turbulent flow apparatus a n d devised a correction procedure to compensate for the "nonlinear averaging error".  A similar procedure was adopted i n a recent study  by R u d e r i c h &; Fernholz (1986). T h e nonlinear averaging errors can be avoided by c a l i b r a t i n g the probe i n a l a m i n a r flow, w h i c h is the procedure used by C a s t r o & D i a n a t (1983) a n d W e s t p h a l et al (1984), w h o h a d high aspect ratio l a m i n a r channel facilities.  Since no such facility was available, the c a l i b r a t i o n for the present measurements was c a r r i e d out i n a two-dimensional l a m i n a r b o u n d a r y layer generated on a flat plate.  A slightly favourable pressure gradient was found to be necessary to pre-  vent early t r a n s i t i o n to turbulence. T h e near w a l l velocity profiles were measured using a hot-wire anemometer, a n d the w a l l shear stress was deduced from the resulting gradient dU/dy  at the w a l l . T h i s c a l i b r a t i o n procedure became increasingly  u n c e r t a i n w i t h higher w a l l shear stress because of the decreasing thickness of the b o u n d a r y layer at the highest velocities. Consequently, the c a l i b r a t i o n of the probe was done i n the range 0.0-0.10 N / m . T h i s range covers adequately the m a x i m u m 2  mean shear stresses encountered i n the separation bubble (~ 0.09 N / m ) , however 2  instantaneous shear stresses larger t h a n the upper c a l i b r a t i o n l i m i t are encountered because of the the highly turbulent nature of the flow. These were determined by assuming that the c a l i b r a t i o n curve extrapolated correctly to these values. It is w o r t h n o t i n g that C a s t r o Sz D i a n a t recommend a c a l i b r a t i o n function of the form T = A/T  + B/T  2  + C/T  s  for a better fit to values of r i n excess of 0.3 N / m . Since 2  this is higher t h a n the largest value used for the c a l i b r a t i o n , the original calibration  CHAPTER2.  EXPERIMENTAL  ARRANGEMENT  AND  TECHNIQUES  24  function was retained. Considering the large uncertainties, due m o s t l y to a narrow c a l i b r a t i o n range a n d estimated to be ± 25% i n r and ± 30% i n r ' , the shear stress measurements presented here should be viewed w i t h c a u t i o n . T h e measurements of the surface forward-flow-fraction 7 were also made using the w a l l probe.  A direct measure of 7 is given by the ratio of positive samples  to the t o t a l number of samples. T h e number of positive samples was obtained by appropriately setting the measurement range to include positive samples only and -7 was t h e n evaluated b y N° of -t-ve samples T o t a l n ° of samples - n ° of "zero" samples where the number of "zero" samples corresponds to heat tracers w h i c h fail to trigger either sensor wire.  CHAPTER2.  EXPERIMENTAL  ARRANGEMENT  AND  TECHNIQUES  Table 2.1. Measurement uncertainties Measured C  p  quantity  (Pressure transducer) U/Uoo  (Hot-wire)  U/Uoo  (Pulsed-wire)  Vut/Un  (Hot-wire)  y/vf/Uoo  (Pulsed-wire)  7  ( W a l l probe)  C  f  yfcf  ( W a l l probe) ( W a l l probe) x  Uncertainty  estimate  ±0.007 ± 2 % (for u/U  < 0.3)  ± 2 to 5% ± 4 % (for u/U ± 2 to 5% (for u/U ±0.02 ±25% ±30%  r  ±0.12}  X  ±0.0152)  y  ±0.0012?  < 0.3) > 0.05)  Chapter 3  E x p e r i m e n t a l Results  T h e results of turbulent flow experiments are presented a n d discussed i n this chapter. F i r s t , the results of surface flow visualization experiments carried out to examine the two-dimensionality of the flow for different model configurations are presented. In the second section, w a l l and flowfield measurements of various mean and fluctuating quantities are presented, comparisons are made w i t h d a t a available i n the literature, a n d some conclusions about the structure of the separation bubble are d r a w n . F u r t h e r information about the structure of the free shear layer, deduced from frequency spectra and velocity autocorrelation measurements, are presented next. F i n a l l y , the results of experiments to determine the effect of v a r y i n g the angle of separation are discussed.  3.1  Two-dimensionality  E x p e r i m e n t a l l y , a reattaching flow region is difficult to make convincingly twodimensional: for example, most studies reviewed i n C h a p t e r one report a curved reattachment line w h i c h is s y m p t o m a t i c of three-dimensional effects. 26  The main  CHAPTER  3.  EXPERIMENTAL  27  RESULTS  factors affecting two-dimensionality are aspect ratio (AR  = S/D)  a n d end condi-  tions. Reattachment length data c o m p i l e d from various sources by Cherry, Hillier &; L a t o u r (1984), hereafter referred to as C H L , shows that the reattachment length increases gradually w i t h aspect ratio.  T h i s effect becomes negligible for aspect  ratios greater t h a n about ten, which is the m i n i m u m aspect ratio recommended by de Brederode (1975), i n conjunction w i t h the use of end plates, to avoid threedimensional effects. T h o u g h the use of endplates is advocated by most workers (e.g. C H L ) their use is by no means universal, because their usefulness is not clear. For instance, K i y a a n d Sasaki (1983), hereafter referred to as K S , d i d not use endplates i n their experiments. In order to assess the influence of endplates  a n d end-conditions on two-  dimensionality, D z i o m b a ( l 9 8 5 ) carried out a series of tests i n the U B C aerodynamics laboratory. U s i n g pressure measurements and the surface flow visualization technique of Langstone & B o y l e (1982), D z i o m b a investigated the following configurations for the degree of two-dimensionality:  i) Base model (AR = 11.1).  ii) M o d e l w i t h endplates (AR = 1 1 . l ) .  iii) M o d e l w i t h endplates a n d side extensions (AR — 11.1). iv) M o d e l w i t h endplates mounted on side extensions (AR = 24.4).  T h e surface streamline pattern for configuration 1 is shown i n F i g u r e 3.1(a) and reveals a complex three dimensional structure. D z i o m b a observed a highly curved reattachment line associated w i t h two leading edge corner vortices. Large spanwise  CHAPTER  3.  EXPERIMENTAL  X/D  28  RESULTS  (a)  (b) X  /  D  (c) X/D  Figure 3.1. Surface flow visualization patterns obtained by D z i o m b a (1985) for three configurations:  (a) base model; (b) model w i t h endplates; (c) model w i t h endplates  and side extensions.  CHAPTER  3.  EXPERIMENTAL  RESULTS  29  variations i n the pressure d i s t r i b u t i o n were also reported. T h i s configuration yielded the longest reattachment length (x = r  5.7D).  T h e a d d i t i o n of endplates (configuration 2) reduced the centerline reattachment length to x  T  x  r  = 5.2D. Three-dimensional effects c a n still be seen (20% v a r i a t i o n in  i n the spanwise direction) though they are much less pronounced because the  corner vortex systems are weaker a n d the ventilation process w h i c h takes place at the extremities of the model is i n h i b i t e d b y the endplates. A m u c h i m p r o v e d s i t u a t i o n results from the a d d i t i o n of side extensions (configur a t i o n 3). G o o d spanwise uniformity and a straight reattachment line are obtained over the largest part of the m o d e l . T h e reattachment length is reduced further to i  r  = 4 . 7 D . Increasing the aspect ratio from 11.1 to 24.5 resulted i n very similar  flow characteristics. Configuration 3 was therefore retained for the present investigation since it resulted i n a nominally two-dimensional separation region extending ± 3 . 5 Z ) , or about ± 0 . 7 5 x , either side of the m i d s p a n . r  3.2  Structure of the Separation Bubble  P r e l i m i n a r y experiments indicated that b o t h reattachment length a n d pressure dist r i b u t i o n remained unchanged over the R e y n o l d s number range 2.5 x 10 to 9.0 x 10 4  4  confirming the observation of Hillier & C h e r r y (1981a) that the flow is essentially R e y n o l d s number independent for Re ~ 2.7 X 1 0 . A l l present experiments were 4  conducted at a R e y n o l d s number of 5.0 x 10 ± 0.1 x 10 . 4  4  CHAPTER  3.2.1  3.  EXPERIMENTAL  RESULTS  30  Reattachment Length and Wall Measurements  T h e mean reattachment point, defined as the location where the time-averaged wall shear stress flow-fraction  1  vanishes, can be determined from the location where the forward7 is equal to 0.5 as was shown by W e s t p h a l et al. (1981). T h e surface  probe was used to measure the forward-flow-fraction d i s t r i b u t i o n ( 7 vs.  x) and  the results are shown i n F i g u r e 3.2. T h e reattachment length deduced from these measurements, x /D r  = 4.7, is i n very good agreement w i t h the value obtained from  surface flow v i s u a l i z a t i o n . It is interesting to note that 7 is never equal to zero; that is the flow is never fully reversed even i n the m i d d l e of the separation bubble. Furthermore there is a broad region, extending about 2.5.D downstream of the mean reattachment p o i n t , where the near-wall flow can be m o v i n g instantaneously i n either downstream or upstream direction. T h i s suggests that the instantaneous  reattachment point wanders up and  downstream i n a region surrounding the time-averaged reattachment point; this view is consistent w i t h the large scale unsteadiness of the flow observed by K S and C H L . If the reattachment zone is defined, somewhat arbitrarily, as a region where 0.1 < 7 < 0.9, its w i d t h is then about 2AD or 0.5a; . r  A comparison of the reattachment length obtained i n the present experiments a n d i n various other studies is shown i n F i g u r e 3.3; only data obtained i n low free stream turbulence is included. T h e reattachment length is seen to decrease w i t h increasing solid blockage, w i t h the exception of the value reported by K S which is •"^This is the fraction of the time the flow moves in the downstream direction; it is equal to the ratio of the number of positive samples to the sum of all samples, i.e r+00 1=  /  r + 00  P(r )dT / w  JO  /  J-co  where P(T ) is the probability density of r . W  w  w  P{T )CLT W  W  CHAPTER  3.  EXPERIMENTAL  RESULTS  31  X/Xr 0  0.25  0.50  1  0.75  1.25  0.75  / •  •  /  •  1.50  2  1.75  7 ^  / •  / • /  0.25  • «  1  \  :  •  '  •  I  0  '  • /  /•  I  2  '  1  4  '  6  1  I  8  1  10  X/D Figure 3.2. D i s t r i b u t i o n of surface forward flow fraction.  Q  A 5<>  O  C  03  Legend O Kiya & Sasaki 1983 A Latour & Hllllar 1980 o  E o 3-  "5  Cherry el al. 1984  V  Chang 1982  •  Roshko & Lau 1965  9  Ota & Itasaka 1976  •  Present Experiments  ffi de Brederode 1975  1 0  1  1 2  1 3  1 4  (  5  6  Percentage Solid Blockage BR F i g u r e 3.3.  R e a t t a c h m e n t lengths found for various solid blockage ratios (data  c o m p i l e d by C h e r r y et al. 1984).  CHAPTER  3.  EXPERIMENTAL  RESULTS  32  low compared to the t r e n d of the remainder of the data. T h i s may be attributed to slight differences i n free stream turbulence, but it should also be pointed out that the n o m i n a l 0% blockage of K S was obtained by fitting the w i n d t u n n e l w i t h false boundaries w h i c h might not have eliminated a l l w a l l effects. T h e mean pressure coefficient d i s t r i b u t i o n is shown i n F i g u r e 3.4 a n d is found to compare well w i t h the C values reported by C H L . B o t h distributions are uncorp  rected for blockage w h i c h accounts for the s m a l l differences (BR = 3.79% for C H L as opposed to 5.6% i n the present experiments). We note that after a s m a l l i n i t i a l dip of about 0.05 i n C  p  the pressure remains approximately constant up to about  x = 0 . 5 x , a r a p i d recovery takes place thereafter. T h i s recovery continues beyond r  the reattachment point up to x ~ 1.4x . r  Figure 3.5 shows the d i s t r i b u t i o n of the mean a n d fluctuating shear stress coefficients.  T h e p o s i t i o n at w h i c h the mean shear stress is equal to zero corresponds  to the p o s i t i o n where 7 = 0.5 as a n t i c i p a t e d . 2  Cj attains a m i n i m u m value of  —2.4 x 1 0 ~ at 1 = 0.63x a n d rises r a p i d l y afterwards. 3  r  T h e r.m.s. value of the  shear stress coefficient increases steadily throughout the reversed flow region and attains a plateau (Cj ~ 2.4 x 10~ ) i n the reattachment zone. T h e magnitudes of both 3  mean a n d fluctuating shear stress coefficients are higher b y a factor of (about) two t h a n those encountered i n the b a c k w a r d facing step geometry (see, e.g., Westphal et al. 1984).  We note that the equivalence of T = 0 and 7 = 0.5 is not always true. Examination of the data of Ruderich and Fernholz (1986) for the flat plate/splitter plate geometry reveals that the forwardflow-fraction is equal to about 0 . 7 at the zero shear stress location associated with a secondary reseparation . 2  W  CHAPTER  3.  EXPERIMENTAL  RESULTS  -0.2  <•  33  •  _^ •  j  -0.4  o  7 -0.6-  • -0.8  -1  0  0.25  0.50  1  0.75  1.25  1.50  1.75  2  X/Xr F i g u r e 3.4. D i s t r i b u t i o n of surface pressure coefficient: # , present measurements; V , measurements of C h e r r y et al. (1984).  X/Xr 0  0.25  0.50  1  0.75  1.25  4.0  O  1.50  1.75  2  • • n 0 O  2.0  o o  ••  a"°"  o °  M—  •  0.0  •  •  -2.0-  ••  •  •  o°o °o o  •  o  o  •  -4.0  0  2  4  6  8  10  X/D F i g u r e 3.5. D i s t r i b u t i o n of wall shear stress coefficients: •  , mean; o > r.m.s.  CHAPTER  3.2.2  3.  EXPERIMENTAL  RESULTS  34  Mean Flow Properties  T h e mean velocity profiles are presented i n F i g u r e 3.6(a) and(fc) for the reversed and reattached flow regions respectively. H o t - w i r e data are presented only i n regions where 7 > 0.9 or 7 < 0.1. A l l velocity data are normalized by the free stream velocity UQO, a n d the o r i g i n for y is the plate surface. T h e mean separation streamline is also plotted for reference. Backflow velocities as large as OSU^ are encountered i n the reversed flow region. T h i s value is about 50% larger t h a n that i n the backward facing step geometry (Westphal et al.  1984) w h i c h accounts p a r t l y for the higher w a l l shear stresses  presented earlier. We also note that the backflow velocities r e m a i n relatively high in the near-wall region. L o c a l m a x i m a i n the velocity profiles are also apparent in the first half of the bubble. T h e velocities further away from the w a l l r e m a i n higher t h a n the free stream as a result of blockage effects. T h e recovery region profiles show an inflexion point as a result of the merging of a new b o u n d a r y layer w i t h the reattaching shear layer. We note also a r a p i d increase of the near-wall velocities w i t h downstream distance. T h e mean streamline pattern obtained by integration of the velocity profiles is shown i n F i g u r e 3.7. T h e dimensionless streamfunction is defined as  ry/D *=/  Jo  (U/U^/diy/D)  T h e time-averaged streamlines are somewhat deceptive i n that they disguise the h i g h l y unsteady nature of the reattachment process, illustrated earlier by the forwardflow-fraction  distribution.  CHAPTER  3.  EXPERIMENTAL  RESULTS  F i g u r e 3.8 shows the r.m.s.  38  l o n g i t u d i n a l velocity profiles.  T h e consistently  higher values of the P W A data are due to electronic noise w h i c h makes the i n strument unsuitable for measurements of turbulent intensities smaller t h a n about 2% ( B r a d b u r y 1976). T h e H W A data on the other h a n d appear reasonable only w h e n the forward-flow-fraction 7 is outside the range 0.1 < 7 < 0.9 a n d when the local intensity is below 20%. T h e results show the same overall trends as the splitfilm sensor results of K S , w i t h the notable exception that the turbulent intensities obtained i n the present measurements are generally higher b y 15 to 20% . 3  T h e peak turbulence intensity ujU^  is observed to occur upstream of reattach-  ment, p r o b a b l y as a result of the higher velocity difference across the shear layer. After a gradual decrease i n the reattachment region, the m a x i m u m turbulent intensities fall r a p i d l y i n the recovery region. W e observe that the turbulent intensities decay fairly r a p i d l y i n the outer region of the shear layer, but r e m a i n h i g h next to the w a l l . These near-wall intensities decrease much more slowly w i t h streamwise distance t h a n the m a x i m u m intensities. It is interesting to note that the peak turbulence intensity of about 30% measured i n this flow is substantially higher t h a n the value of about 18% found i n the plane m i x i n g layer ( W y g n a n s k y & Fiedler 1970). A more appropriate comparison, however, can be made b y using the velocity difference across the shear layer to normalize the present data. about 22% at x/D  T h e m a x i m u m value of u/(U  = 3 to about 25% at x/D  max  — cT  min  ) varies from  = 5. T h i s is still higher by 20 to 40%  t h a n the plane m i x i n g layer value.  Young (quoted by Simpson 1985) indicates that fluctuating quantities that are up to 3 0 % too low can be obtained with hot film probes; this would account for the discrepancy between the present measurements and those of KSA. 3  Figure  3.8(a).  F l u c t u a t i n g (r.m.s.)  pulsed-wire data; • , hot-wire data;  longitudinal  velocity  profiles:  O  ,  , separation streamline ( * = 0);  to CO  Figure 3.8(b). F l u c t u a t i n g (r.m.s.) longitudinal velocity profiles:  O , pulsed-wire  data; • , hot-wire data. O  CHAPTER  3.  EXPERIMENTAL  RESULTS  41  T h e forward-flow-fraction profiles, measured using the P W A , are shown i n F i g u r e 3.9 together w i t h the loci of 7 = 0.1,0.5 and 0.9. T h e data points at y/D from the surface probe measurements.  = 0 are  Similarly to the w a l l data , the p o s i t i o n of  7 = 0.5 is found to correspond to U = 0. A t x/D  = 1 a n d 2, 7 reaches a m i n i m u m  slightly away from the w a l l , but further downstream the m i n i m u m is reached at the w a l l ; this is consistent w i t h the location of the m a x i m u m backflow velocities in F i g u r e 3.6.  In fact the shapes of the mean velocity and forward-flow-fraction  profiles are s t r i k i n g l y s i m i l a r . F u r t h e r insight into the structure of the separation bubble can be gained by e x a m i n i n g the skewness, S , T  a n d the flatness factor, F , of the w a l l shear stress. T  These are the n o r m a l i z e d t h i r d a n d fourth moments of the p r o b a b i l i t y density, P , T  a n d are defined by  T h e skewness a n d flatness factor distributions are shown i n F i g u r e 3.10, a n d they exhibit the same trends as reported by R u d e r i c h and Fernholz (1986) for the flat p l a t e / s p l i t t e r plate geometry. B o t h skewness a n d flatness factors, are found to depart m a r k e d l y from their respective G a u s s i a n values of 0 a n d 3. H i g h values of the flatness factor are usually indicative of a burst-type or large amplitude intermittent signal (Townsend 1976). Such behaviour of the fluctuating shear stress and near-wall velocity could be associated w i t h the large scale unsteadiness and bubble bursting phenomenon reported by b o t h K S and C H L . T h e inrush of h i g h velocity fluid resulting from this phenomenon  CHAPTER  3.  EXPERIMENTAL  RESULTS  Q / A  CHAPTER  3.  EXPERIMENTAL  RESULTS  d  d  in  CM  T -  d  d  o d  m cn  m od  in  - cn  d  m od  y -3  O O O O 0 0 O OXOXCPOOLltlllUEKb 00  o  in  O  in  --i  n  n  n  n n n m  1111 iiiiimiiiiHimnit  O CD  m d  m id  3 o  o o o o o ojDOQxniiinffl  in  m  iri  d  m CM  m  d  qA  CHAPTER  3. EXPERIMENTAL  RESULTS  44  X/Xr 0.5  1  1.5  1.50-  • n o D  .  0.75  •  •  D  • •  •  "  H  L  l_  CO  6  6f  Gau s s i a n - ^ 7  to CD  •  0-  c  3  •  CD  • -0.75  • •  ^ 00 00  -0  •  c o  • -1.50  | 1  J  0  ,  .  '  1  1  1 8  -3  1  10  X/D F i g u r e 3.10. Skewness a n d flatness distributions of fluctuating w a l l shear stress: •  , S; T  • , F. T  w o u l d cause large amplitude fluctuations leading to p r o b a b i l i t y distributions having a higher flatness factor. T h e observed increase of F  T  i n the reattachment region is  compatible w i t h this e x p l a n a t i o n , since the activity associated w i t h the large scale unsteadiness is most vigorous there. T h e skewness is negative i n most of the reversed flow region, presumably because of the intermittent large amplitude negative  fluctuations.  After a local m i n i m u m i n  the m i d d l e of the separation bubble, the skewness rises steadily, crosses zero close to the mean reattachment p o i n t , and continues to rise thereafter due to increasing occurrences of large amplitude positive  fluctuations.  CHAPTER  3.3  EXPERIMENTAL  3.  45  RESULTS  Structure of the Shear Layer  3.3.1  Growth of the Shear Layer  The position y where the r.m.s. velocity u attains a m a x i m u m can be interpreted as c  representing the centre of the shear layer b o u n d i n g the separation bubble ( K S ) since it corresponds closely to the position where the velocity profiles have an inflexion point. F i g u r e 3.11 shows that the position of y changes very little w i t h x even i n c  the reattachment region (the shaded plot reflects the uncertainty i n locating u  m a x  due to scatter i n the data) . F i g u r e 3.11 also shows the streamwise v a r i a t i o n of an edge of the shear layer y defined as the position of 2.5% local turbulent intensity. e  T h i s corresponds closely to the position where an intermittent signal first appears. A n i m p o r t a n t parameter characterizing the shear layer is the growth rate which can  be deduced from the shear layer w i d t h .  O n e measure of this w i d t h is the  v o r t i c i t y thickness 6 defined b y ( B r o w n & Roshko 1974) U  ^ u  Umax (dU/dy)  Ui m  n  max  The streamwise v a r i a t i o n of the v o r t i c i t y thickness, plotted i n F i g u r e 3.12, shows that, initially, the shear layer grows i n a linear fashion. T h e growth rate  d6 /dx w  is found to be equal to 0.147, w h i c h is essentially identical to the value of 0.148 reported b y R u d e r i c h & Fernholz for the normal-plate/splitter-plate geometry. T h i s is w i t h i n the range of values (0.145-0.22) reported by B r o w n & R o s h k o for the conventional single stream m i x i n g layer. T h e present results indicate that at about x/x  r  ~ 0.65 there is a sudden decrease i n the growth rate (d6 /dx u  — 0.097), a  feature not shown by the data of R u d e r i c h & Fernholz. T h e sharp change in the slope after x/x  r  ~  1.1 is due to the smaller {dU/dy)  max  gradients.  C H L used  CHAPTER  3.  EXPERIMENTAL  46  RESULTS  X/Xr 0  0.4  0.8  1.2  1.6  2  X/D F i g u r e 3.11. Representative shear layer positions: A , y (location of u/U = 2.5%); e  zzzzz , y (location of c  0  {u/U^naz).  0.25  0.50  0.75  1  1.25  X/Xr F i g u r e 3.12. G r o w t h of the shear layer. V o r t i c i t y thickness 6 : - O , present; •  ,  R u d e r i c h &; Fernholz (1986). M a x i m u m slope thickness 6 :  ,  U  ms  C h e r r y et al. (1984).  A , present;  CHAPTER  EXPERIMENTAL  3.  47  RESULTS  the m a x i m u m slope thickness 6  ma  (defined i n the same way as S , b u t w i t h o u t u  i n c o r p o r a t i n g a mean reversed flow velocity U i ) m n  to represent the g r o w t h of the  shear layer. T h e i r streamwise coordinate was corrected for the displacement of the v i r t u a l o r i g i n , therefore only the mean slope line of their d a t a is shown i n F i g u r e 3.12.  3.3.2  Frequency Spectra and Autocorrelations  Frequency spectra a n d autocorrelation functions of the l o n g i t u d i n a l fluctuating velocity were measured along y . e  A t this p o s i t i o n , the velocity  fluctuations  are ir-  r o t a t i o n a l most of the t i m e , w i t h very short high-frequency bursts o c c u r r i n g very occasionally, and most of the information contained i n these fluctuations is therefore related to the large scale structure of the shear layer. T h e spectra at various streamwise positions along y , presented i n F i g u r e 3.13, e  show the same features as the measurements of C H L . A t x/x  r  = 0.01 there is a  distinct h i g h frequency c o n t r i b u t i o n w i t h a peak at a reduced frequency of about 34.  /x /^co r  A s we progress further downstream from separation, a progressive  fall i n the dominant frequency takes place. T h i s fall ceases at x/x  r  = 0.64, where  the spectra become dominated b y a broader b a n d c o n t r i b u t i o n centered around fxr/Uoo  cn 0.6.  T h i s streamwise v a r i a t i o n is illustrated i n F i g u r e 3.14 (the fre-  quencies were taken from the corresponding peaks i n F i g u r e 3.13) w h i c h also shows the measurements of D z i o m b a (1985) a n d C H L . A l l measurements show the same trends, a n d two regions can be observed. In the first region, extending from separat i o n to about 60% of the separation bubble length, the frequency decreases linearly w i t h x, i n d i c a t i n g that the large scale structures grow linearly just as i n a conven-  CHAPTER  3.  EXPERIMENTAL  48  RESULTS  t i o n a l free shear layer. In the second region, w h i c h overlaps the first, starting at about 50% of the separation bubble length, the characteristic frequency remains essentially constant (decreasing values shown by the d a t a of D z i o m b a for x > 1.2 are due to the c o n t a m i n a t i o n from the t r a i l i n g edge of the m o d e l w h i c h was not equipped w i t h a t a i l ) . T h i s characteristic frequency is associated w i t h the pseudoperiodic shedding of v o r t i c i t y from the bubble, a phenomenon observed by K S a n d C H L amongst others. F i g u r e 3.15 shows the autocorrelation R dimensional time lag TUoo/x . r  uu  plotted as a function of the non-  Close to separation, at x/x  = 0.01, the autocorrela-  r  t i o n exhibits a long t a i l . A s i m i l a r observation was made by K S w h o a t t r i b u t e d this t a i l to the flapping of the shear layer caused by the large scale unsteadiness of the b u b b l e . T h e h i g h frequency waviness of the t a i l is due to the contributions from the large scale structures of the shear layer. T h i s waviness can s t i l l be detected, w i t h correspondingly lower frequencies, at x/x  r  — 0.108 a n d 0.216. A t x/x  r  = 0.86  we note the reappearance of negative correlations w h i c h become even larger around reattachment. T h e streamwise d i s t r i b u t i o n of the integral timescales T obtained by integrating u  the corresponding autocorrelation to the first zero crossing, are shown i n F i g u r e 3.16. Consistently w i t h earlier observations, the timescales increase i n a linear fashion w i t h x up to x/x  r  =: 0.85 a n d stabilize thereafter at a value of T Uoo/x u  r  ~ 0.2.  F i g u r e 3.13. Frequency spectra of velocity  fluctuations  along shear layer edge y . c  CHAPTER  3.  EXPERIMENTAL  RESULTS  CHAPTER  3.  EXPERIMENTAL  RESULTS  0$  TUoo/Xr  F i g u r e 3.15. A u t o c o r r e l a t i o n of velocity fluctuations along shear layer edge  CHAPTER  3.  EXPERIMENTAL  RESULTS  X/Xr 0  0.5  1.5  0.2  o -  ^  8  o  o  o  o  o  0.1  o o o 0.0  I  I  I  1  I  2  3  4  5  I  I  7  8  X/D  F i g u r e 3.16. Integral t i m e scales deduced from autocorrelation measurements.  CHAPTER  3.4  3.  EXPERIMENTAL  RESULTS  53  Effect of Angle of Separation  T h e angle a at w h i c h the shear layer separates from the front face of the blunt plate has a significant impact on the pressure d i s t r i b u t i o n , as shown i n F i g u r e 3.17. Decreasing the separation angle induces earlier recovery of the pressure, w i t h a shift of the pressure distributions towards the leading edge a n d a corresponding shortening of the separation bubble length. There is also a decrease of the base pressure coefficient Cp , and the pressure dip immediately downstream of separation b  becomes more pronounced. It is well k n o w n that separated reattaching flows have very similar pressure distributions, a n d that a reasonable collapse of the d a t a is obtained over a wide range of geometries w h e n the pressure is plotted, as suggested by R o s h k o & L a u (1965), i n terms of x/x  r  a n d the reduced pressure coefficient  C* p  1  -  c _ . . .  T h e pressure distributions of F i g u r e 3.17 were replotted using these reduced coordinates, a n d the result is shown i n F i g u r e 3.18, w h i c h also includes the flat p l a t e / s p l i t t e r plate data of Ruderich&; Fernholz (1986) for comparison. T h e collapse of the data onto a single curve, w i t h slight deviations of the flat plate/splitter plate data, is quite remarkable and confirms the assumption that the reattachment length x  r  is a basic length scale for separated-reattaching  flows.  CHAPTER  3.  EXPERIMENTAL  RESULTS  V ....V..  -0.2  A  54  x xp;^$^ a° o  A -0.4-  p  •  x A  CL  o  -©f  •  A X o -0.6  A ^  -0.8-  A.k  <r=J  *• O  *8  O  a=  •  a = 85  90  x a = 75 A a = 60  v a = 45 -1  10  X/D  Figure 3.17. Effect of separation angle on surface pressure d i s t r i b u t i o n . 0.5  0.4  "  A;  o  a=  a X  a = 85  0.3  4  Q. O  0.2 o  90  cx = 75  A a = 60  0.1  a = 45 •  0.0-  t-  0.5  I  1.5  X/Xr  R 4 F 1986  —t— 2.5  F i g u r e 3.18. Reduced pressure distributions of Figure 3.18. and comparison w i t h flat plate/splitter plate d a t a of R u d e r i c h & Fernholz (1986).  RESULTS  55  T h e recovery pressure rise coefficient Cp *,  w h i c h measures the difference be-  CHAPTER  3.  EXPERIMENTAL  m  tween the highest a n d the lowest pressure, was found to be equal to 0.4 as compared to an averaged value of 0.36 cited b y Roshko &; L a u for a variety of configurations. T a n i (quoted by Westphal et al. 1984) noted that the u l t i m a t e pressure recovery is higher for thinner b o u n d a r y layers at separation. In the present case the effective thickness of the b o u n d a r y layer is quite s m a l l as a result of the strongly favourable pressure gradients at separation ( C H L estimated the m o m e n t u m thickness 6  sep  to  be about 0 . 0 0 4 D ) . T h e v a r i a t i o n w i t h separation angle of base pressure (measured at x/D  = 0.17,  the location of the first tap) and reattachment length (measured using the wall probe) are shown i n F i g u r e 3.19. T h e reattachment length for a = 45° is about 50% shorter t h a n for the base model (a = 9 0 ° ) , while the base pressure coefficient Cp is about 10% lower. It is interesting to note that these trends are qualitatively h  similar to the effect of either t r i p p i n g the b o u n d a r y layer on the front face before it separates ( D z i o m b a 1985), or increasing the free-stream turbulence level (Hillier & C h e r r y 1981a; K i y a & Sasaki 19836; Hillier & D u l a i 1985). D z i o m b a reported reductions of up to 40% i n the reattachment length w i t h a corresponding 15% drop i n the base pressure coefficient. T h e shortening of the bubble was a t t r i b u t e d to an effective change i n the separation angle due to the formation of a s m a l l recirculation bubble between the t r i p wires and the sharp edge of the plate. T h e pressure distributions reported by D z i o m b a were replotted i n terms of reduced coordinates for the basic undisturbed flow and two t r i p p e d flows. T h e results i n F i g u r e 3.20 show that, though reasonable, the collapse is not as good as that obtained w i t h various separation angles.  In particular, C * increases from about Pm  0.4 to 0.43 when the b o u n d a r y layer is t r i p p e d , and the pressure recovery process  CHAPTER  3.  EXPERIMENTAL  RESULTS  56  starts earlier, resulting i n a shift of the d a t a towards the left. T h i s , together w i t h the p r o p o r t i o n a l l y higher decrease i n C  ph  noticed earlier, indicates that the effects  of the t r i p wire is perhaps p a r t l y due, but not confined to an effective change in separation angle. A d d i t i o n a l factors to be considered are possible changes i n the state of the separating b o u n d a r y layer and i n the growth rate of the shear layer. T h i s concludes the discussion of the experimental results. T h e modelling of the mean properties of this flow is examined next.  CHAPTER  3.  EXPERIMENTAL  RESULTS  57 0.90  -0.85  Q_ O I  x -0.80  0.75  40  50  a F i g u r e 3.19.  V a r i a t i o n of reattachment  length a n d base pressure w i t h angle of  separation. 0.5  \ ,0L  -  •  y  0.4-  —  —  • -  • -  .r.A.v  <  /  M  0.3  /•'/ i // / ; • 9  Q. O  0.2-  O Undi s t u r b e d //  ;  A 2.3mm 41 w i r e  H 0.0  1 0.5  • 1  ,  1.5  3 . 1mm <> f wi re 1  1  2.5  X/Xr  F i g u r e 3.20. Effect of t r i p p i n g the boundary layer on reduced pressure d i s t r i b u t i o n . D a t a taken from D z i o m b a (1985); t r i p wires were placed on front face of plate.  Chapter 4  Mathematical Model  T w o different approaches to the modelling of separated reattaching flows can be taken. T h e first, a zonal approach, consists of d i v i d i n g the flow field into several regions, each h a v i n g dominant features. T h e flow is then analyzed using the m e t h o d w h i c h is o p t i m u m for each of those regions. In the second approach the flow field is solved for globally using a set of equations w h i c h a p p l y throughout the d o m a i n . B o t h these approaches are examined i n this study. A solution of the bluff rectangular plate p r o b l e m is not completed w i t h the first approach, but examples of the use of zonal analyses are given i n Appendices C a n d D .  In this chapter, we present the background for the global modelling of the flow. T h e averaging procedure of the Navier-Stokes equations is then briefly described, a n d the turbulence model used in this study, the k — e m o d e l , is reviewed. Next, to set the stage for the numerical method presented i n the next chapter, the general form of the various transport equations to be solved is given. F i n a l l y , the boundary conditions for the blunt rectangular plate geometry are described. 58  CHAPTER  4.1  4.  MATHEMATICAL  MODEL  59  Background  T h e simplifying assumptions made i n the derivation of the b o u n d a r y layer equations are unfortunately not s t r i c t l y v a l i d for separated flows.  A n accurate description  of these flows requires the use of the exact equations expressing the principle of conservation of m o m e n t u m : the Navier-Stokes equations. T h e n u m e r i c a l solution of these equations for l a m i n a r flows, although not a t r i v i a l task, is always possible. However, for turbulent flows, a n u m e r i c a l solution of the full set of equations i n their three-dimensional time dependent form is not quite feasible at present—at least not for flows of p r a c t i c a l importance. T h i s is because the exceedingly refined grids required to resolve the smallest scales of turbulent m o t i o n present at realistic Reynolds numbers (Re ~ 10 ) w o u l d tax the storage 4  capacity a n d speed of present day computers. Consequently one has to resort to the ensemble or time-averaging procedure, first proposed b y Reynolds over a hundred years ago. However, because the equations are non-linear, the averaging procedure produces e x t r a u n k n o w n terms: the turbulent or Reynolds stresses. T h e net result is that one ends up w i t h more unknowns t h a n equations, a n d a d d i t i o n a l equations are required to "close" the p r o b l e m . These a d d i t i o n a l equations are provided by m o d e l l i n g the turbulent stresses.  4.2  Conservation Equations and Time-Averaging  For incompressible flow, the equations expressing the principle of conservation of mass a n d m o m e n t u m are, i n Cartesian tensor co-ordinates (see R e y n o l d s 1974),  dxj  = 0  (4.1)  CHAPTER  4. MATHEMATICAL  MODEL  60  and dUi  d ,  T T T T  ,  dp  d  f  (dUi  dUj\\  ,  these equations are v a l i d for laminar and turbulent flow, w i t h £7, p representing instantaneous  ,  = 1,2,3) and  velocities and static pressure respectively.  T h e instantaneous general variable (f> is decomposed into a mean, cf>, and a fluctuating component, <p , as follows  (4.3)  <j> = j> + <p  where the time-averaged value, <f>, is defined as  w i t h an averaging t i m e At  long enough compared w i t h the longest t i m e scales of  the turbulent m o t i o n . Introducing these definitions i n equations (4.1) a n d (4.2) to decompose Ui and p into mean a n d  fluctuating  components, a n d time-averaging, we o b t a i n , for a  statistically steady flow,  <«>  It =  0  axj  and d ,  x  dp  d  [  (dUi  dUA  )  ,  .  where the overbars for the mean variables have been dropped for convenience. E q u a t i o n (4.6) contains six new unknowns, the turbulent or R e y n o l d s stresses —pufuj, w h i c h arise from the averaging of the nonlinear convective terms i n equat i o n (4.2). Physically, the R e y n o l d s stresses represent diffusion of m o m e n t u m by  CHAPTER  4. MATHEMATICAL  MODEL  61  turbulent m o t i o n . In order to o b t a i n a closed set of equations, some assumptions are necessary to relate the Reynolds stresses to other existing variables. T h e various assumptions w h i c h can be made constitute the central theme of turbulence modelling.  4.3  k — e Turbulence Model  Turbulence modelling has been an active field of research for many years; although m u c h progress has been made (see R o d i 1983, for a critical state of the art review), the models currently available are necessarily approximate and still rely on empirical information. T h e k — e m o d e l (Launder & Spalding 1974) used i n this study, requires the solution of two a d d i t i o n a l transport equations: one for the turbulent kinetic energy, k, a n d another for its dissipation rate, e. T h i s model seems, at present, to offer the best compromise between generality a n d c o m p u t a t i o n a l economy. T h e framework of the k — e model is the eddy viscosity concept. T h i s concept is expressed by an equation of the form  -  pUiUj  =  fi  t  fdUi —  \OXj  +  dUA  -r—  OXi J  -  2 , .  -pkbij  6  .  (4.7)  where k is defined as * = ^fu!+ ^  + ^)  (4-8)  T h e t e r m involving the Kronecker delta on the r.h.s. of (4.7) ensures that the s u m of the n o r m a l stresses is equal to 2k. T h i s t e r m is a scalar quantity which can be incorporated i n the pressure gradient t e r m of the m o m e n t u m equation.  CHAPTER  4. MATHEMATICAL  MODEL  62  T h e next step is to determine the eddy or turbulent viscosity fi i n terms of t  definable quantities.  T h i s is done by first assuming fx to be p r o p o r t i o n a l to a t  characteristic velocity V, a n d a length scale £:  Ht cx VI  (4.9)  A physically meaningful scale characterizing turbulent velocity fluctuations is yfk\ using this scale i n (4.9), results i n the K o l m o g o r o v - P r a n d t l relation  oc  (4.10)  py/kl  k a n d I are related to the dissipation rate of turbulent kinetic energy, e, by dimensional analysis ( R o d i 1984) e oc —  (4.11)  C o m b i n i n g these two expressions, we o b t a i n lit = G > f c / e  (4.12)  2  where C is a n empirically determined constant of proportionality. M  T h e p r o b l e m of determining the turbulent stresses has thus been reduced to determining k a n d e. E x a c t transport equations for b o t h k a n d e c a n be obtained by m a n i p u l a t i n g equation (4.2); the resulting equations are, however, of little use because they contain higher order correlation terms w h i c h are u n k n o w n . To obtain a closed set, these terms must be modelled. I n the "standard" k — e model, the modelled transport equations for k a n d e, take the form (Launder & Spalding 1974):  k  dx  k  dx  k  1 cr dx J k  k  e f f  \dx  k  dx ) {  dx  k  CHAPTER  4.  MATHEMATICAL  MODEL  63  Table 4.1. k — e model constants c 0.09  1.44  2  1.92  1.0  0.4187  and  pU — dx k  k  = — ( I + Ci/x.// dxk \ o~ dx J k \dx e  k  dx{ J  k  dx  k  T h e effective viscosity, / i / / , is the s u m of the laminar a n d eddy viscosities e  T h e values of the empirical constants appearing i n equations (4.13) and (4.14) are listed i n table 4.1. T w o variants of the s t a n d a r d k — e model w h i c h incorporate modifications accounting for streamline curvature are presented i n A p p e n d i x D .  4.4  General Transport Equation  For steady two-dimensional flow, the equations presented i n the previous section reduce to: continuity equation: ATT dU  dx  PiVdV__  (4.15)  Q  dy  rc-momentum equation:  d (  au]  d  f  (du  av\\  ^H&l+^W'U" "^]} 1  , ( 4  -  i 6 )  CHAPTER  4. MATHEMATICAL  MODEL  64  where p* = p + y - m o m e n t u m equation:  /^-transport equation:  e-transport equation: d P  dx^^  P  dy  dx  (n de\  I <7  e  5  (  eff  dx )  (ix de\ eff  <9y V a  £  dy  „ e„ „ e + d - G - C p2  (4.19)  where G , w h i c h represents the generation of turbulent kinetic energy, is given by 2 G = /z | 2 t  / ~ - \ 21  fdU  TJ  dx  \dy  \dy  t  dV\  2  dx J  (4.20)  a n d the effective viscosity is obtained from the two a u x i l i a r y equations  Veff  =  V- + ft k  (4.21)  2  e  (4.22)  Note that the continuity and m o m e n t u m equations are also v a l i d for steady l a m i nar flow, E q u a t i o n s (4.15) to (4.19) can be conveniently represented for numerical purposes by the following general equation (4.23)  T is a general diffusivity coefficient a n d <S^ a general source t e r m . T h e particular values taken by T a n d  are given i n Table 4.2. for each of the transport equations.  CHAPTER  4.  MATHEMATICAL  MODEL  Table 4.2. Diffusion coefficients and source terms Conserved property  4>  T  s*  Mass  1  0  0  z-momentum  u  y-momentum  V  T.K.E.  k  Peff  §2± j_  a /„  au\ G-  , a pe  Ok  T . K . E dissipation  e  »eff  C-J-G - C p*— k k 2  ( a v \  CHAPTER  4.5  4.  MATHEMATICAL  MODEL  66  Boundary Conditions  T o completely specify the p r o b l e m , it is necessary to impose b o u n d a r y conditions o n a l l the boundaries of the flow d o m a i n . For the blunt rectangular plate geometry considered here, there are four types of boundaries. These are shown i n F i g u r e 4.1, a n d the corresponding b o u n d a r y conditions are given below.  / / / /  / \ \ /  /  /  /  /  /  /  /  /  /  /  /  /  /  /  /  /  /  /  \\ /  / /  x  p  -t->  OJ +J OJ  O  CM  B  F i g u r e 4.1. F l o w d o m a i n .  i) I n f l o w ( A B ) : Values of a l l the variables, pressure e x c e p t e d , are specified. 1  For the present flow configuration, u n i f o r m profiles for U, k and e are i m 2  posed; V is set to zero. T h e turbulent kinetic energy, k, is determined from the experimentally measured free stream turbulence intensity, whereas e has to be estimated from k and a characteristic length scale. The pressure need not be specified when the normal velocity is imposed. Conditions on k and e are only required for turbulent flow computations  1  2  CHAPTER  4.  MATHEMATICAL  MODEL  67  ii) O u t f l o w ( E F ) : Ideally, the outflow boundary should be located i n a region where the flow is fully developed; a zero streamwise gradient c o n d i t i o n across d() the outflow boundary, —— = 0, is then applicable to a l l variables. L o c a t i n g ox this boundary i n the fully developed flow region is, however, not practical because a very large c o m p u t a t i o n a l d o m a i n , extending far beyond the region of interest, w o u l d be required. However, because the flow after reattachment is everywhere i n the downstream direction it is not very sensitive to downstream conditions. It is therefore permissible to locate the outflow boundary closer to reattachment, i.e. i n a region where the flow is not fully developed, p r o v i d i n g this location is sufficiently far downstream from the recirculating flow region to ensure that the flow i n the upstream region is not affected by downstream conditions. T h e zero streamwise gradient c o n d i t i o n is then applied to a l l variables at this location. T h e penalty for this inexact treatment is a local distortion of the flow field near the outflow w h i c h , anyhow, is not a region of p r i m e interest. iii) S y m m e t r y A x i s ( B C ) : T h e n o r m a l velocity, V, is set to zero along the axis d( 1 of symmetry, and a zero cross-stream gradient c o n d i t i o n , —— = 0, is imposed oy for a l l variables. iv) S o l i d W a l l s ( A F , C D a n d D E ) : T h e no-slip boundary c o n d i t i o n is imposed at a l l solid boundaries. In a d d i t i o n , a special treatment of the near-wall region k = — , is often suffice 2  is required, because the turbulent Reynolds number, Re  t  ciently s m a l l i n the v i c i n i t y of solid boundaries for viscous effects to become important.  T h e k — e model, w h i c h was devised for h i g h Reynolds number  fully turbulent flows, neglects these effects; consequently, it is not valid in  CHAPTER  4.  MATHEMATICAL  MODEL  68  near-wall regions. A n alternative to devising a low Reynolds number model w h i c h w o u l d take viscous effects into account, is the use of w a l l functions, as proposed by L a u n d e r and Spalding (1974). T h e wall function treatment, presented i n A p p e n d i x E , connects the w a l l shear stress to the velocity just outside the viscous sublayer by assuming one dimensional Couette flow and local e q u i l i b r i u m . A bonus of this treatment is c o m p u t a t i o n a l economy : it is no longer necessary to have the h i g h concentration of grids n o r m a l l y required to resolve the very steep gradients prevailing i n the viscous sublayer.  Chapter 5  Computational Procedure  T h e governing equations presented i n the previous chapter were discretized using a finite volume m e t h o d , and the resulting set of algebraic equations were solved by a n iterative procedure using a modified version of the T E A C H - I I code developed by Benodekar, G o s s m a n & Issa (1983). In this chapter, an overview of the m e t h o d of solution is given. T h e staggered g r i d system used to discretize the solution d o m a i n is first described.  T h i s is fol-  lowed (Section 5.2) by a summarized account of the finite volume m e t h o d , h y b r i d differencing and the associated p r o b l e m of false diffusion; a remedy to this problem, a variant of the skewed differencing scheme of R a i t h b y (1976), is also introduced. In Section 5.3, the implementation of the b o u n d a r y conditions is discussed.  An  outline of the iterative solution procedure is given i n Section 5.4. T h e results of p r e l i m i n a r y computations, carried out to determine various parameters (extent of the solution d o m a i n , o p t i m u m g r i d d i s t r i b u t i o n , convergence criteria) are presented i n the last section. 69  CHAPTER  5.1 5.1.1  5.  COMPUTATIONAL  PROCEDURE  70  Finite Volume Formulation Grid Layout and Variables Location  T h e c o m p u t a t i o n a l d o m a i n is d i v i d e d into a number of adjacent control volumes (or cells), w i t h their centres located at the nodes of a C a r t e s i a n grid system as shown i n F i g u r e 5.1. T h e location of the nodes at w h i c h the variables are to be calculated differs, depending on the variable i n question as shown i n F i g u r e 5.2. Scalar variables (p, k, and e) are evaluated at the m a i n g r i d nodes (shown as • ) , whereas the velocities U and V are evaluated at staggered locations (shown by —• and 1") w i t h respect to the m a i n g r i d nodes . 1  T h e g r i d is arranged so that the faces of the scalar cells coincide w i t h the faces of the physical boundaries. A d d i t i o n a l "fictitious" cells are located on the periphery of the d o m a i n to facilitate the implementation of the b o u n d a r y conditions. T h e nodes surrounding a central node P are identified by (using the compass convention) N, S, E, W, NE,  NW,  SE and SW; the faces of a control volume are  identified b y n , s, e and w.  5.1.2  Integration of General Transport Equation  T h e finite volume f o r m of equation (4.47) is obtained b y integration over the control volume shown i n F i g u r e 5.2(b), i.e  dxdy  (5-1)  The staggered grid arrangement is used almost universally in finite difference calculations because it has the advantage of locating the velocities where they are required to evaluate the convective fluxes, i.e at the faces of the control volume. Additionally, the velocity-pressure staggering averts physically unrealistic "wiggles" in the solution domain (see Patankar 1980). 1  CHAPTER  5.  /////  '////.  COMPUTATIONAL  ' / / /  / / / /  PROCEDURE  / / / / /  71  / / / / y / / / / / / / / ' / / /  /  / / / / / / /  \  \  1  ]  1  ;  ;  !  ;  |  WW  WW  \ \ *\  \ \  i  \  F i g u r e 5.1. G r i d layout.  WW  W  W  W W  1  ^ \ \ \ \  /  CHAPTER  5.  COMPUTATIONAL  PROCEDURE  72  1 P(1.J+D i i L. 1 1 L _ 'P(M.J)  I V(1,j)  I  1  r - - l - 1 .i ;pd.j)  jiud.j) I  i 1  p(i+i,j)  (a)  p(i,j-D  6x  NW  1  N  A  NE  *1  n  r y  i  n  W  w  P  [e  L  SW  v/if  /  E  |  d.j)  i  s  . 5E  S  x  e  _|  u  X  Figure 5.2: (a) L o c a t i o n of scalar and velocity cells, (b) T y p i c a l control volume  CHAPTER  5.  COMPUTATIONAL  PROCEDURE  73  using G a u s s ' divergence theorem, the volume integrals can be transformed  into  surface integrals:  where F ,...,F e  e  represent the sums of convective and diffusive fluxes across the  faces e , . . . , s. For example: F = C + D e  e  e  with C = e  (5.3)  r{pU<t>) dy Xe  and (5.4)  5.2 5.2.1  Finite Difference Discretization Hybrid Differencing  So far, the terms i n the integral equation (5.2) are exact.  T h e first step i n re-  ducing the equations to an algebraic form is to express, by using finite difference approximations, the convective and diffusive fluxes (C ,D e  e  etc . . . ) i n terms of the  n o d a l values of the variable (f>. T h e h y b r i d differencing scheme ( H D ) , presented i n  CHAPTER  5.  COMPUTATIONAL  PROCEDURE  74  this subsection, is the c o m b i n a t i o n of two types of finite difference approximations: central difference ( C D ) and u p w i n d difference ( U D ) . T h e C D a p p r o x i m a t i o n assumes that <f> varies linearly between E and P.  For  example, a p p l y i n g central differencing to approximate C (equation 5.3) yields: e  C. = ,,V, p*±4=) A y ,  (5.5)  S i m i l a r l y for the diffusive flux (equation 5.4)  C e n t r a l differencing, w h i c h is formally second order accurate i n 6x, is quite satisfactory for problems where diffusion is the dominant feature.  However, for  higher R e y n o l d s number convection dominated flows, the use of central differencing for the convective terms leads to numerical instabilities when the cell Reynolds or Peclet number Pe = \pU6/T\  is larger t h a n 2. T h e most widely used method to  counteract this type of instability is u p w i n d differencing. T a k i n g the example of C  e  again, at e the u p w i n d value of <f> is assumed to prevail, i.e. C  e  =  p U <f>pAyj  for  U >0  C  t  =  p U (j> Ayj  for  U <0  e  t  e  t  E  e  e  T h i s amounts to replacing the linear v a r i a t i o n of <f> assumed previously i n central differencing by a stepwise variation; this leads to an a p p r o x i m a t i o n w h i c h is u n c o n d i t i o n a l l y stable, but only first order accurate i n 6x. T o take advantage of the superior accuracy of the central differencing scheme at low Peclet numbers,  the  C D and U D schemes are combined to form the h y b r i d central-upwind differencing scheme ( H D ) : C D is always used for the diffusive terms, while for the convective terms C D is used when \Pe\ < 2 and U D is used when \Pe\ > 2.  CHAPTER  5.  COMPUTATIONAL  PROCEDURE  75  Table 5.1. Linearized source terms Conserved property i-momentum  s} U  St  0  p*\?Ay  +  Ay  + (*//£) V  y-momentum  0  P*\> Ax+ n  |>*  (M.//iJ)>y  + ("«//1r) l > * T.K.E.  k  T . K . E dissipation  e  _£ £kp Jl  G Ax  AxA  - ^ A x A y  Ay  dj^GAxAy  T o complete the discretization procedure the source t e r m  is linearized as  follows (Patankar 1980):  fj  S+dx dy = SUP  + St  (5.8)  Sp and Si are derived using C D approximations; the various expressions are given in Table 5.1. S u b s t i t u t i n g for the source and flux terms into equation (5.2) yields the general finite volume e q u a t i o n  2  (see Patankar 1980):  (ap — Sp)<f>p = a,N<t>N + a>s<t>s + O>E4>E + aw4>w + <Sy  with a =^2ai P  i 2  Also referred to as finite difference equation.  (i =  N,S,E,W)  (5.9)  CHAPTER  5.  COMPUTATIONAL  PROCEDURE  76  and a  N  =  max(|C„/2|,D ) - C / 2  a  s  =  max{\C,/2\,D.)  a  E  =  max(|C,/2|,2? )-C /2  aw  =  m a x ( | C / 2 | , 1?^) +  n  n  + e  w  C./2 e  C /2 w  T h e algebraic equations (5.9) are solved b y the iterative procedure described i n Section 5.4.  5.2.2  False Diffusion  T h e simple u p w i n d i n g procedure just described improves numerical stability, but it does so at a cost.  Because it is only first order accurate, u p w i n d differencing  introduces a potentially damaging t r u n c a t i o n error c o m m o n l y k n o w n as artificial or false diffusion.  -1 6X1-1  6  x •}  F i g u r e 5.3. F i n i t e difference nodes.  T h e t r u n c a t i o n error of a finite difference a p p r o x i m a t i o n can be evaluated using a T a y l o r series expansion; hence  ^ = ^(-^-)g),^(-^) (0), !  +  CHAPTER  5.  COMPUTATIONAL  PROCED  URE  77  the convective t e r m U— c a n therefore be expressed as ox  + ... d4> T h e first t e r m on the R H S is the u p w i n d difference a p p r o x i m a t i o n of U—— (for ox U > 0); the second t e r m is the leading t r u n c a t i o n error, a n d has the form of a diffusion t e r m w i t h an effective diffusion coefficient 6x;i-l 2 T h e effect of false diffusion is to artificially increase the (physical) diffusion coefficients; this results i n smearing of the gradients i n the flowfield. T h e question of how i m p o r t a n t is false diffusion was addressed by R a i t h b y (1976) who showed that false diffusion is negligible i n situations where either the local flow direction is closely aligned w i t h g r i d lines or i n the absence of strong cross-stream gradients i n <f>.  However, i n the presence of both skewness a n d strong gradients,  R a i t h b y showed that artificial diffusion can become comparable to, or even larger t h a n p h y s i c a l diffusion, a n d can lead to significant errors. It is i m p o r t a n t to note that skewness and strong gradients are often prevailing conditions i n recirculating flows .in general a n d turbulent ones i n particular. F u r t h e r , i n turbulent flows, as a consequence of false diffusion, it is not possible to properly assess the performance of a given turbulence model because it is difficult to dissociate errors due to modelling deficiencies from those arising from false diffusion. In principle, the errors due to false diffusion can be reduced to an acceptable level b y increasing the number of c o m p u t a t i o n a l nodes i n c r i t i c a l regions of the  CHAPTER  5.  COMPUTATIONAL  PROCEDURE  78  flow. A s i d e from this "brute force", c o m p u t a t i o n a l l y expensive a n d often impractical prescription, two remedies to false diffusion have been proposed: the skew u p w i n d differencing ( S U D ) scheme of R a i t h b y (1976b), w h i c h uses flow oriented differencing, a n d the quadratic upstream weighted interpolation ( Q U I C K ) scheme of L e o n a r d (1977), w h i c h uses higher order differencing. However, b o t h schemes suffer from nonphysical oscillations or "wiggles" as well as solution undershoots a n d overshoots. These are a consequence of negative coefficients appearing i n the finite difference equations, a p r o b l e m referred to as unboundedness . 3  L a i & Goss-  m a n (1982) developed a variant of the skew scheme, the bounded skew h y b r i d differencing scheme ( B S H D ) , w h i c h eliminates negative coefficients and is therefore u n c o n d i t i o n a l l y stable . T h i s scheme was used i n this study, a n d is o u t l i n e d next.  5.2.3  Skew Differencing  T h e basic cause of false diffusion lies i n the practice of treating the flow across a control volume face as locally one-dimensional, w h i c h results i n errors in the d i s t r i b u t i o n of <j>. T h e skew u p w i n d differencing ( S U D ) scheme of R a i t h b y (1976b) reduces this error substantially b y t a k i n g into account the local flow direction. In this scheme, the value of <f> required to evaluate the convective t e r m is determ i n e d by back projecting the local velocity vector u n t i l it intersects a grid line, as illustrated i n F i g u r e 5.4(a); <f> is then obtained by either linear or stepwise interpolation between the two neighbouring nodes lying on the same g r i d line. E x a m p l e s for each type of interpolation follow. The problem of unboundedness which can result in numerical instability and poor convergence is discussed in some details by Lai (1982). 3  CHAPTER  5.  COMPUTATIONAL  PROCEDURE  79  W h e n the projection intersects w i t h a v e r t i c a l grid line, as shown i n F i g u r e 5.4(b), then the value of <j> at face w is given by:  4>w =  where k  w  (1 — kw)<t>w + k <f>sw w  is a linear interpolation factor.  W h e n the intersection lies on a horizontal grid line, as shown i n F i g u r e 5.4(c), a stepwise interpolation is used, and <$> takes the nearest nodal value, (f>sw in this w  case. T o account for a l l possible flow directions and the two types of interpolations, the interpolation factor is given by (for U  w  > 0 and V  w  > 0)  T h e S U D reduces false diffusion substantially, but can result i n negative coefficients i n the finite difference equations.  T o suppress these negative coefficients,  L a i & G o s s m a n (1982) developed a flux blending technique w h i c h results i n the bounded skew u p w i n d differencing ( B S U D ) scheme: the S U D and U D schemes are blended i n such a fashion as to maximize the c o n t r i b u t i o n from the more accurate S U D while m a i n t a i n i n g p o s i t i v i t y of a l l coefficients. T h i s is done through an o p t i m i z a t i o n procedure described by L a i (1982) (see Benodekar et al. 1985 for a s u m m a r i z e d account). F i n a l l y , the bounded skew h y b r i d differencing ( B S H D ) scheme is, as its name implies, a h y b r i d c o m b i n a t i o n of the B S U D and C D schemes: the B S U D is used to approximate the convective terms for \Pe\ > 2, and C D is invoked for \Pe\ < 2.  CHAPTER  5.  COMPUTATIONAL  PROCEDURE  80  6X,  Ax. m  NE  /  Ay,  --sw  SE  NW  NW  (1  1  <i  I  II  ii  r  W  k Sy. ,  r  \,  1 w  Iw  \/ i i  1  *  I  \  I l l /  i  IV  IV  III  i SW II  1  11  SW  /  •—*k  1  F i g u r e 5.4. F i n i t e difference computational star and illustration of linear and stepwise interpolation ranges for skew u p w i n d differencing scheme.  CHAPTER  5.  COMPUTATIONAL  PROCEDURE  81  D i s c r e t i z a t i o n of equation (5.2) using the B S H D results i n a finite volume equat i o n of the f o r m :  (ap — Sp)(j)p  =  a <j>N + cts<f>s + o-E<i>E + a-w<i>w + N  O-NE^NE + 0-Nw4>NW + O-SE^SE + ^SW^SW + S$  (5.10)  the expressions for the coefficients a,- can be found i n Benodekar et al. (1982).  5.3  Treatment of Boundary Conditions  T h e finite difference representation of the b o u n d a r y conditions (discussed previously i n Section 4.3.) is given i n this section.  5.3.1  Types of Boundaries  i) I n f l o w : A n example of a cell at the inflow b o u n d a r y is shown i n Figure 5.5. T h e profiles were assumed to be u n i f o r m for a l l variables. T h i s corresponds closely to experimentally observed conditions.  U  =  t/oo  V  -  0  w  k  w  = w  =  A-oo  «oo  Uoo was the experimentally measured free stream velocity.  T h e turbulent  kinetic energy was estimated, assuming isotropy, from the experimentally  CHAPTER  5.  COMPUTATIONAL  PROCEDURE  82  r  1 1  J k  kp » p e  » W E  w  INFLOW BOUNDARY Figure 5.5. Inflow b o u n d a r y cells.  measured free stream turbulent intensity V t ? ; it was taken as  fcoo = 9.375 X 1 0 £ / ^ _ 6  T h e dissipation rate, €<„ cannot be measured experimentally a n d has to be estimated.  T h e generally accepted practice of estimating e as a function of  the local value of k a n d a characteristic length scale \H  Coo =  was followed; i.e.  k^/{\H)  It s h o u l d be noted that a judicious choice of A is required to ensure a realistic d i s t r i b u t i o n of k upstream of the bluff body, since too s m a l l or too large a value of too w o u l d cause unrealistic growth or decay of the turbulent kinetic energy. A value of A = 0.09 was chosen to ensure that the turbulent kinetic energy was maintained at its free stream level at x/D  ~ —4 as is observed  CHAPTER  5.  COMPUTATIONAL  PROCEDURE  83  experimentally. T h e choice of A was otherwise not c r i t i c a l : changing A by a factor of five h a d v i r t u a l l y no effect on the results (less t h a n 0.2% change in X ). r  ii) O u t f l o w : T h e zero gradient c o n d i t i o n at the outflow b o u n d a r y is obtained  OUTFLOW BOUNDARY Figure 5.6.Outflow b o u n d a r y cells.  by setting  UP  Uw + Uipfc  V  E  V  k  E  kp  P  tE  UJNC  is an incremental value w h i c h ensures that continuity is satisfied at the  outlet after each iteration. A t convergence  UINC  becomes equal to zero.  CHAPTER  5.  COMPUTATIONAL  PROCEDURE  84  iii) S y m m e t r y A x i s : T h e s y m m e t r y c o n d i t i o n is implemented as follows  U  =  s  V k  s  s  e  s  U  P  =  0  —  k  =  e  P  P  In a d d i t i o n it is necessary to ensure that the fluxes across the s y m m e t r y axis are equal to zero; this is done by setting the appropriate a,- coefficients to zero:  as — 0 ,  for  Up,kp,ep  iv) S o l i d W a l l s : T h e impermeability c o n d i t i o n is s i m p l y obtained by p u t t i n g  V  s  = 0  T h e b o u n d a r y conditions for U,k and e are implemented by first cancelling the fluxes t h r o u g h the boundary side of the cells, i.e.  as = 0 ,  for  Up,kp,e , P  and then by evaluating the diffusive fluxes at the w a l l using w a l l functions as described i n A p p e n d i x D .  CHAPTER  5.  COMPUTATIONAL  PROCEDURE  1  F i g u r e 5.7. S y m m e t r y axis cells.  p' p  k  v  \ \ \ \  \  1  •  !  e  s \  k  s  \  , e  V  s  F i g u r e 5.8. Solid w a l l boundary cells.  CHAPTER  5.3.2  5.  COMPUTATIONAL  PROCEDURE  86  Special Case: Corner Nodes  T h e corner cells of the bluff b o d y warrant special attention because of the staggered g r i d arrangement. T h i s arrangement results i n corner cells w i t h only a half-face i n contact w i t h the w a l l as illustrated i n F i g u r e 5.9; this raises the question of how are the convective fluxes t h r o u g h these half-faces to be calculated. T h e convective fluxes across each of these two half-faces are evaluated by:  a) A s s u m i n g a n o r m a l velocity across these half faces equal to that at the outer edge of the half cell (i.e. V  s  is used for the n o r m a l velocity across the half-face  of the U-ce\\ a n d vice-versa)  b) T a k i n g an effective area for the flux calculations equal to A y / 2 x l a n d A x / 2 x 1 for U and V respectively.  T h e above treatment of the corner cells was found to have a drastic effect on the solution: H D turbulent flow calculations w i t h o u t this special treatment resulted i n a 40% shorter reattachment length (for a 80 x 40 g r i d ) .  5.4  Solution Procedure  T h e s o l u t i o n m e t h o d i n the T E A C H - I I code uses the P I S O (Pressure I m p l i c i t Split Operator) a l g o r i t h m , described i n detail by Benodekar et al. (1983). T h i s a l g o r i t h m consists of a two-stage predictor corrector procedure, w h i c h involves the s p l i t t i n g of operations to deal w i t h the c o u p l i n g between velocity a n d pressure variables so that at each stage of the solution procedure, a set of algebraic equations in terms of a single u n k n o w n variable is obtained.  CHAPTER  5.  COMPUTATIONAL  PROCEDURE  Figure 5.9. Cells near plate corner.  87  CHAPTER  5.  COMPUTATIONAL  PROCEDURE  88  In the P I S O procedure, the algebraic equations i n question are solved by a series of "guess a n d correct" operations. F i r s t , the velocities are calculated from the m o m e n t u m equation using the pressure field prevailing at the n  th  iteration. T h e  velocity a n d pressure fields are then adjusted through two corrector steps to ensure that mass conservation is satisfied. T h e procedure is repeated u n t i l convergence. For turbulent flow calculations, the algebraic equations for k a n d e are solved i n the iteration loop, just after the second corrector step. A t each stage of the above procedure, the set of algebraic equations is solved using a line by line m e t h o d i n conjunction w i t h a tri-diagonal m a t r i x solution algorithm.  5.5  Preliminary computations  In this section, we discuss the results of p r e l i m i n a r y runs made to determine: the effect of the location of the inlet a n d outlet boundaries, the o p t i m u m g r i d d i s t r i b u t i o n , a n d the appropriate convergence criteria.  5.5.1  Location of Inlet and Outlet Boundaries  Specifying the extent of the c o m p u t a t i o n a l d o m a i n is an important consideration. Too large a d o m a i n results i n unnecessarily large arrays; too s m a l l a d o m a i n can affect the accuracy of the solution i n the region of interest. Several computations were therefore carried out to determine the appropriate location of the inlet and outlet boundaries. T h e distance from the front face of the plate to the inlet boundary (Figure 5.10) was gradually reduced from L  u  = 15Z> to 7.5D.  T h e location of  this b o u n d a r y was found to have no noticeable effect on the flow i n the recircula-  CHAPTER  5.  COMPUTATIONAL  PROCED  89  URE  t i o n region, provided that Lu > 9D. For Lu < 9D a s m a l l increase (typically about 0.3% for Lu = 7.5J9) i n the separation bubble was observed. ////  /\\ / / / / / / / / / / / / / / / / / / /  \ \/  //^  F i g u r e 5.10. C o m p u t a t i o n a l d o m a i n .  T h e effect of the location of the downstream b o u n d a r y was noted for Ld = 15D to 8D. T h e effect was negligible for Ld ~ 11D. C o m p u t a t i o n s w i t h smaller Ld resulted in g r a d u a l l y longer bubbles (typically about 1.0% for Ld = 81?). A l l subsequent computations were therefore performed w i t h  5.5.2  12D  <  L  14D  <  L  u  d  <  10D  <  12D  Non-uniform Grid Arrangement  A number of g r i d distributions were investigated. N o n - u n i f o r m grids were found to be the most efficient way of o b t a i n i n g the fine g r i d arrangements required i n regions of steep gradients.  CHAPTER  5.  COMPUTATIONAL  PROCEDURE  90  A proper resolution of the region immediately upstream of and around the sharp corner was found to be p a r t i c u l a r l y critical to the accuracy of the solutions. Consequently, a cluster of fine equally spaced grids was located there. T h e g r i d spacings were expanded on either side of this sharp corner, i n b o t h x and y-directions, as shown i n F i g u r e 5.11. In a d d i t i o n the x—direction g r i d was refined locally around the reattachment region. T h i s local refinement was not critical to the c o m p u t a t i o n , but allowed a more precise location of the reattachment lengths, w h i c h were determ i n e d b y linear interpolation from the computed w a l l shear stress distributions. A p r e l i m i n a r y g r i d refinement study showed that the solutions are more sensitive to g r i d refinements i n the y - direction t h a n i n the x-direction, and that o p t i m u m array sizes are obtained when the mesh at the corner has a ratio A x / A y ~ 2.5. To m i n i m i z e the t r u n c a t i o n errors associated w i t h non-uniform g r i d distributions, the effect of the g r i d expansion factors E  x  — A x , / A x , _ i and E  v  = Ayy/Ay _i ;  was investigated. A comparison was made w i t h solutions obtained using uniform grids i n selected regions of the d o m a i n . It was found that an economy i n c o m p u t a t i o n a l cells (compared to a uniform distribution) can be obtained w i t h no noticeable loss i n accuracy, provided that the expansion factors are kept w i t h i n the following ranges: 0.8  <  E ,E x  y  <  1.2  for {  x < —D  and  x > x  r  y > D  0.9  <  E ,E x  y  <  1.1  for «  -D -0.5D  <  x <  y  <x  r  <D  A l l the computations presented i n the next chapter were performed w i t h expansion factors w i t h i n these ranges.  CHAPTER  5.  COMPUTATIONAL  PROCEDURE  CHAPTER  5.5.3  5.  COMPUTATIONAL  PROCED  URE  92  Convergence Criterion  Since an iterative solution procedure is used to solve the equations, it is necessary to establish a convergence criterion which measures the degree to w h i c h a computed solution satisfies the finite difference equations. In the present computations this convergence criterion was based on the values of the absolute residual errors of the continuity a n d m o m e n t u m equations. W h e n the sums of these residual errors were less t h a n 0.2% of the inlet mass flow and m o m e n t u m respectively, the solution was considered converged, a n d the iteration cycle was t e r m i n a t e d .  T h i s convergence  criterion is expressed by max  where X) \R<p\ is the s u m of the absolute residuals over the entire field. T h e residuals are defined, from equation (5.9), by  R<t> — i p a  T h e value of £ x ma  - Sp)<f>p - ^2 difa - Si  = 2 x 1 0 ~ was considered to be acceptable since a reduction 3  of this value b y a factor of five d i d not result i n any appreciable change i n the c o m p u t e d reattachment length (less t h a n 0.25%).  Chapter 6  Numerical  Results  T h e accuracy of a n u m e r i c a l m e t h o d , w h i c h is p a r t i c u l a r l y important when assessing a given turbulence m o d e l , is best established i n l a m i n a r flows. In this chapter we present the results of a systematic g r i d dependence study for the l a m i n a r flow past a bluff rectangular plate. T h e superiority of the B S H D scheme over the H D scheme is clearly demonstrated. T h e l a m i n a r flow computations, besides p r o v i d i n g a check o n the accuracy of the m e t h o d , yielded results w h i c h are interesting i n their own right, a n d a selection of these is presented a n d discussed. T h e second part of this chapter is devoted to the results of turbulent flow computations.  Some i m p o r t a n t numerical aspects of the solutions are first discussed,  a n d the results of a comprehensive grid dependence study are presented. T h e k — e turbulence m o d e l is found to perform rather p o o r l y i n its standard form, but results i n substantially improved predictions when the preferential dissipation modification is incorporated. D e t a i l e d comparisons of the predicted flow field (velocity, t u r b u lent kinetic energy, pressure, w a l l shear stress) are made w i t h the experimental data presented earlier i n C h a p t e r 3. A s an i l l u s t r a t i o n of the usefulness of the numerical m e t h o d for parametric studies, predictions of the effect of solid blockage are pre93  CHAPTER  6.  NUMERICAL  RESULTS  94  sented. In the last section, c o m p u t a t i o n a l costs and the relative performance of the various schemes are discussed.  6.1  Laminar Flow  T h e parameters for the laminar flow computations were chosen to correspond to the measurements of L a n e & Loehrke (1980) for comparison purposes . T h e solid 1  blockage ratio was equal to 8.36%. T h e b u l k of the computations were performed in the R e y n o l d s number range 100-325, since it was observed experimentally, and confirmed i n our p r e l i m i n a r y computations, that no separation occurs for Reynolds numbers below about 100, and that the flow becomes unsteady for values higher t h a n about 325.  6.1.1  Grid Independence  T h e effect of g r i d refinement on the solution was investigated systematically, using b o t h H D a n d B S H D schemes, for three Reynolds numbers: i?e = 125,225 and 325. T h e number of grids used ranged from 41 x 2 6 (corresponding to A z - / Z ? = 35% m t  and Ay /D min  = 12.5%) to 7 0 x 5 5 [Ax /D min  = 8%,Ay /D min  n  = 3.7%). A t y p i c a l  g r i d layout was shown i n Section 5.3. T h e reattachment length x was found to be T  a good measure of the sensitivity of the solution to g r i d spacing, therefore all grid refinement results are conveniently presented i n terms of the v a r i a t i o n of x . r  T h e results of the three series of tests, presented i n Figures 6.1(a) to (c), show a significant difference between the reattachment lengths predicted w i t h the two The only other measurements available (Ota et al. 1981), were not considered here because the experiments were carried out with a low aspect ratio of 4.55, and three-dimensional effects are therefore likely to have been important. 1  CHAPTER  6.  NUMERICAL  RESULTS  Laminar Flow, Re=125 X  1.5-  —o-  c c  -o-  I"  E .c o o "5  0.5-  O BSHD A HD  1  1000  1500  1  2000  ,  !  !  2500  3000  3500  4000  No. of C o m p u t a t i o n a l N o d e s NIxNJ  Laminar Flow, Re=225 x c C  E o p £  "o or  2-  O BSHD A HD  1000  i  i  I  I  I  1500  2000  2500  3000  3500  4000  No. of C o m p u t a t i o n a l N o d e s NIxNJ 7  2 1 1000  i  1  i  1  i  1500  2000  2500  3000  3500  4000  No. of C o m p u t a t i o n a l N o d e s NIxNJ  F i g u r e 6.1. Effect o f g r i d refinement on reattachment length.  CHAPTER  6.  NUMERICAL  discretization schemes. pendent for NIxNJ  RESULTS  96  A t Re — 125, the B S H D solution is essentially grid inde-  — 1728, whereas the H D solutions r e m a i n sensitive to grid  refinement even for the finest g r i d a n d tend a s y m p t o t i c a l l y — f r o m b e l o w — towards the g r i d independent B S H D solution. A t higher R e y n o l d s number, the B S H D solutions show more sensitivity to g r i d refinement, a n d more c o m p u t a t i o n a l nodes are required to reach g r i d independence.  T h e H D solutions r e m a i n sensitive to grid  spacing throughout the range a n d respond rather sluggishly to g r i d refinement. Increasing R e y n o l d s number causes the H D predictions to deteriorate further. C o n sidering, for example, the fine grid results, we see that while the H D reattachment length is 9% shorter t h a n the B S H D result at Re = 125, this discrepancy increases to 35% at i2e = 325. T h i s can be a t t r i b u t e d directly to the inherent false diffusion of the H D scheme w h i c h , for a given grid spacing, is expected to increase w i t h R e y n o l d s number.  6.1.2  Effect of Reynolds Number and Comparison with Experiments  In view of the results of the g r i d dependence study, further computations were only performed using the B S H D scheme. G r i d distributions of 70 x 43 to 78 x 43 were used,and the solutions can therefore be regarded as grid independent. T h e predicted v a r i a t i o n of the reattachment length w i t h R e y n o l d s number is plotted i n F i g u r e 6.2. Considering the experimental uncertainties (reported to be about 10% i n Re a n d 0 . 1 5 D i n x ) the agreement between predicted a n d measured r  values is excellent. In c o m m o n w i t h other reattaching flows, the reattachment length is seen to increase linearly w i t h R e y n o l d s number.  Separation is first observed  CHAPTER  6.  NUMERICAL  RESULTS  97  Reynolds Number Re F i g u r e 6.2. V a r i a t i o n of reattachment length w i t h R e y n o l d s number: —O— , B S H D computations; V , measurements of Lane &: Loehrke (1980).  at Re = 100 a n d , remarkably, it is predicted to occur slightly downstream of the corner i n agreement w i t h the experimental observations of L a n e & Loehrke. A s the R e y n o l d s number increases the separation point moves upstream a n d remains fixed at the sharp corner, w h i l e the bubble grows i n b o t h length a n d height as illustrated by the predicted streamline patterns shown i n F i g u r e 6.3. We also observe in these plots t h a t the eddy centre is always located upstream of the m i d d l e of the bubble, at about x / x = 0.35. T h o u g h separation actually occurs at the corner for Re > 100, r  this is not clearly shown i n the streamline plots as a result of the interpolation procedure used to determine the streamfunction contours.  CHAPTER  6.  NUMERICAL  RESULTS  98  (0)  REYNOLDS  NUMBER  = 100 .  (.D)  REYNOLDS  NUMBER  = 150 .  (C)  REYNOLDS  NUMBER  = 200 .  CHAPTER  6.  NUMERICAL  99  RESULTS  (d)  REYNOLDS  NUMBER  =250.  (e)  REYNOLDS  NUMBER  = 300 .  F i g u r e 6.3(a to d). Streamlines for laminar flow (For * > 0: A * = 0.125. For # < 0: (b) A * = 0.00075; (c) A * = 0.0017; (d) A # = 0.0026; (e) A * = 0.0034).  CHAPTER  6.  NUMERICAL  RESULTS  REYNOLDS  100  NUMBER  = 200 .  CHAPTER  6.  NUMERICAL  101  RESULTS  REYNOLDS  NUMBER  = 250 .  REYNOLDS  NUMBER  = 300 .  F i g u r e 6.4(a to d). Velocity field for laminar flow.  CHAPTER  6.  NUMERICAL  RESULTS  102  T h e velocity field plots (Figure 6.4) show that the largest backflow velocities encountered i n the recirculating flow region are very small, increasing from about 1% of the freestream velocity at J?e = 100 to about 10% at .Re = 300. After reattachment the b o u n d a r y layer recovers relatively quickly. A linear region is established near the w a l l , a n d the velocity profiles look very similar to those of a classical flat plate b o u n d a r y layer. It is also interesting to note that, i n the region surrounding the corner of the plate, the cross-stream velocities are of the same order as the streamwise velocities. T h i s results i n velocity vectors w h i c h are highly inclined, or skewed, w i t h respect to the grid-lines. It is the c o m b i n a t i o n of this skewness w i t h the i m p o r t a n t gradients also present i n this region that is responsible for the large false diffusion errors associated w i t h the H D scheme. Figure 6.5 shows the predicted pressure d i s t r i b u t i o n along the surface of the plate. T h e pressure coefficient at separation Cp is equal to about —2.4 at .Re = 100 t  a n d increases, as a result of a reduced curvature of the separating shear layer, to about —1.5 at Re = 300. In a l l cases, an immediate a n d r a p i d pressure recovery takes place. T h i s recovery process is slower, and is spread out over a larger region for the higher Reynolds number. A n o t h e r point of interest is the constant pressure gradient t h r o u g h most of the separation bubble at Re = 300 and the higher pressure coefficient after recovery. We note that the flow is confined b y solid walls.  The  b o u n d a r y layers w h i c h develop on these walls, i n c o m b i n a t i o n w i t h the reattached b o u n d a r y layers on the plate, result i n the favourable pressure gradients observed after  reattachment.  CHAPTER  6.  NUMERICAL  RESULTS  103  0  -0.5  -1  I fj  - 1 . 5 - If  J  Legend Re=100 R_e=200 _ Re=300  0  2.5  5  7.5  10  12.5  15  17.5  X/D F i g u r e 6.5. Pressure d i s t r i b u t i o n along top surface of the plate (laminar flow).  CHAPTER  6.2 6.2.1  6. NUMERICAL  RESULTS  104  Turbulent Flow Preamble and Effect of Grid Refinement  T h e turbulent flow calculations were carried out at a R e y n o l d s number of 5 x l 0 for a 4  blockage r a t i o of 5.6%; these conditions correspond to the measurements described i n C h a p t e r 3. A s i n the l a m i n a r flow case, b o t h H D a n d B S H D schemes were used. In the course of p r e l i m i n a r y computations using the B S H D scheme, the stagnation flow i n the region immediately upstream of the bluff plate was found to be inaccurately predicted, w i t h stagnation pressure coefficients reaching a m a x i m u m of about 0.92 instead of 1.0.  T h i s type of p r o b l e m has also been observed by L a i (1983) w h o  showed that the performance of skewed based schemes deteriorates i n the presence of strong adverse pressure gradients . To rectify the deficiency of the B S H D scheme 2  i n the stagnation region, the c o m p u t a t i o n a l d o m a i n was split, w i t h H D being used u p s t r e a m of the plate leading edge and B S H D downstream of it. T h i s split, though it m a y seem arbitrary, was i n fact based on L a i ' s conclusions that, i n situations of strong pressure gradients, the h y b r i d differencing scheme performs better than the skewed differencing scheme, because numerical diffusion is less significant under these circumstances. T h e use of a split d o m a i n produced differences of less t h a n 7% i n computed values of x , r  the split d o m a i n resulting i n the longest separation bubble and a  stagnation pressure coefficient of essentially unity. T h i s technique was used i n a l l subsequent B S H D calculations. A similar problem was reported by Castro et al. 1981 when using a vector differencing scheme for the flow over a normal flat plate. 2  CHAPTER  6.  NUMERICAL  RESULTS  105  T h e turbulence models used i n the present computations were the  standard  k — e m o d e l (see Section 4.3) and a variant of it w i t h a preferential dissipation modification ( P D M ) described i n A p p e n d i x D . A t h i r d m o d e l , involving a streamline curvature correction ( A p p e n d i x D ) , was also investigated.  Its use was, however, inconclusive because it led to numerical  instabilities (in the form of r a n d o m oscillations i n the c o m p u t e d flowfield) a n d a converged solution could not be obtained; the p r o b l e m persisted even w i t h severe under-relaxation.  Implementing the correction on a converged solution obtained  w i t h the s t a n d a r d k — e m o d e l proved equally fruitless.  S i m i l a r difficulties were  encountered by H a c k m a n (1982) for the backward facing step, i n d i c a t i n g a possible deficiency i n the modification, though it should be pointed out that Benodekar et al. (1985) d i d not report any difficulties i n using the modification to solve for the flow over a surface -mounted r i b . T h e results of the g r i d refinement studies are shown i n figure 6.6 for three cases: H D scheme w i t h standard k — e m o d e l , B S H D w i t h standard k — e m o d e l , a n d B S H D w i t h s t a n d a r d k — e model a n d preferential dissipation modification. These w i l l s i m p l y be referred to as H D , B S H D and P D M respectively. T h e g r i d distributions w h i c h were used ranged from 4 9 x 3 4 (Ax /D min  {Ax /D min  = 3.6%, Ay /D min  = 20%, A y  m i n  / Z > = 5.6%) to 82x66  = 0.9%).  T h e H D scheme was expected to yield a shorter reattachment length than the experimentally observed value, and while this is borne out by the results, the magnitude of the discrepancy (44% w i t h a fine 8 2 x 6 2 grid) was somewhat of a surprise. T h e discrepancies reported i n the literature for various other geometries are much smaller even though coarser grids were used. For example, D u r s t & Rastogi (1977) reported a 30% underprediction of x for the flow around a surface mounted block; r  CHAPTER  6.  NUMERICAL  RESULTS  106  X CT)  C  c  E o "o 1000  2000  3000  4000  5000  6000  No. of C o m p u t a t i o n a l N o d e s NIxNJ F i g u r e 6.6.  Turbulent flow: effect of g r i d refinement on computed reattachment  length; A , H D scheme w i t h k — c model; O , B S H D scheme w i t h k — e model; •  , B S H D scheme w i t h P D M .  they used a 50x27 g r i d . F o r the backward facing step flow, H a c k m a n (1982) reported a 12% underprediction w i t h a 5 0 x 5 0 g r i d . T h e B S H D scheme results i n a substantial lengthening of the predicted reattachment length as shown i n F i g u r e 6.6, but the discrepancy w i t h the experimental value is still large at 30%. In view of the good agreement found i n the laminar flow case this rather disappointing result can, w i t h confidence, be b l a m e d on turbulence m o d e l deficiencies. T h e use of the modified turbulence m o d e l , which was i n fact p r o m p t e d by these results, leads to an encouraging improvement of the predictions.  T h e P D M solutions appear to be independent of g r i d refinement for a  7 5 x 5 7 g r i d and a reattachment length of 4 . 3 D is produced—this is w i t h i n 10% of the experimentally observed value of 4.7D.  CHAPTER  6.  NUMERICAL  RESULTS  107  U/Uref F i g u r e 6.7. Effect of g r i d refinement on c o m p u t e d velocity profile at x/D  = 2( P D M  computation).  A s a further check of g r i d independence, F i g u r e 6.7 shows streamwise velocity profiles at x/D  — 2 predicted by the P D M using two different g r i d distributions.  These profiles are, for a l l intents and purposes, identical. It was however disturbing to find that this g r i d independence, demonstrated b o t h i n terms of the reattachment length a n d i n the flowfield, d i d not extend to the computed w a l l shear stress. F i g u r e 6.8 shows that the computed w a l l shear stresses change by as m u c h as 10% when refining the g r i d from 7 5 x 5 7 to 8 2 x 6 2 nodes. We recall that i n the w a l l function treatment ( A p p e n d i x E ) different assumptions are made for the near-wall velocity, depending on the value taken by the wall coordinate y  +  early (y  +  = pC^y/kyp/fj,.  T h e near-wall velocity is assumed to either vary l i n -  < 11.63) or according to the logarithmic law of the w a l l ( y  F i g u r e 6.9 shows that for the 8 2 x 6 2 grid y  +  +  > 11.63).  is always less t h a n 11.63 a n d therefore  CHAPTER  6.  NUMERICAL  RESULTS  108  4.0  2.0O  o  o.o  8 2 x 6 2 GRID  -2.0-  75x57 GRID  -4.0 2.5  5  7.5  10  12.5  15  17.5  X/D F i g u r e 6 . 8 . Effect of g r i d refinement on computed w a l l shear stress along top surface of plate ( P D M c o m p u t a t i o n ) . 20  / — Y+=11.63 15 — — — +  ,0  1  .  -—•  /  '  —  '  //  \ / 8 2 x 6 2 GRID 75x57 GRID  2.5  7.5  10  12.5  15  17.5  X/D F i g u r e 6 . 9 . L o c a t i o n of g r i d points adjacent to the wall i n terms of the w a l l coordinate y  +  ( P D M computation).  CHAPTER  6.  NUMERICAL  RESULTS  109  the c o m p u t a t i o n a l nodes adjacent to the w a l l are w i t h i n the viscous sublayer. For the 7 5 x 5 7 g r i d , however, y  +  becomes larger t h a n 11.63 for x/D  ~ 2 a n d there-  fore the logarithmic law is invoked to calculate the near-wall flow. It was initially thought that the change i n c o m p u t e d shear stress was due to the inadequacy of the logarithmic law of the wall i n this  flow . 3  F u r t h e r computations showed, however,  that even when the nodes adjacent to the w a l l are w i t h i n the viscous sublayer, further g r i d refinement results i n changes of the computed w a l l shear stress. T h e reasons for these changes i n wall shear stress are not clear, but they i n d i cate possible inconsistencies i n the w a l l function treatment. A l t h o u g h the present computations suggest that the flow field is not sensitive to the details of the nearwall flow—and this is substantiated b y the results of H a c k m a n (1982) w h o found different w a l l function treatments to have little impact on the b a c k w a r d facing step flow—this  6.2.2  matter clearly deserves further investigation.  Predictions and Comparison with Experimental Data  In the following we shall concentrate on the results obtained using the  BSHD  scheme. Complete results of the computations w i t h the modified k — e model are presented, together w i t h a selection of the results obtained w i t h the s t a n d a r d k — e model to illustrate the effect of the turbulence model modification. Figure 6.10 a n d 6.11 show a comparison of the computed velocity profiles w i t h experimental measurements at various streamwise locations.  T h e B S H D results  show that size of the separation bubble is underpredicted not only i n length but also in height. A l t h o u g h the gross features of the flow are reasonably well reproduced, There is some experimental evidence suggesting that the law of the wall is not valid in reattaching flows (Adams et al. 1984; Ruderich & Fernholz 1986). 3  CHAPTER  6.  NUMERICAL  110  RESULTS  U/Uref 0  Figure  0.5  0  6.10.  putation; computed;  O  Mean ,  PWA  0.5  0  longitudinal  0.5  0  0.5  velocity profiles:  measurements. , experimental.  Separation  0  0  0.5  1.0  , BSHD streamlines: —  15  com-  —- —  ,  CHAPTER  Figure  6.  6.11.  putation;  0 ,  computed; —  NUMERICAL  Mean  111  RESULTS  l o n g i t u d i n a l velocity profiles:  P W A measurements. , experimental.  Separation  , streamlines:  P D M com,  CHAPTER  6.  NUMERICAL  quantitative agreement is poor. predicted to be about value of about  0.26J7oo  0.32C/OQ  112  RESULTS  For example, the m a x i m u m backfiow velocity is  and occurs at x/D = 1.0 as opposed to an experimental  occurring at x/D = 2.0. T h e P D M predictions, on the other  h a n d , are i n very good agreement w i t h the experimental d a t a i n the recirculating flow region.  After reattachment, however, they start to deteriorate as a result  of a slower rate of recovery of the reattached b o u n d a r y layer.  It is interesting  to note the fortuitious agreement of the B S H D predictions w i t h experiments  at  x/D = 5.Q and 6 . 0 . T h i s is due to the earlier reattachment w h i c h provides a longer distance for the b o u n d a r y layer to recover.  T h e deterioration of the predicted  velocity profiles i n the recovery region is a c o m m o n feature of many  reattaching  flow calculations (see N a l l a s a m y 1 9 8 7 ) and is perhaps not surprising i n view of the complex and unsteady nature of the reattachment process. We note i n conjunction w i t h the l a m i n a r flow results presented earlier that the backflow velocities are much higher i n the turbulent flow case, and the near-wall velocities r e m a i n relatively high compared to l a m i n a r flow. F i g u r e 6 . 1 2 shows the effective viscosity contours.  C o m p a r e d to the standard  model the P D M reduces the eddy viscosity not only upstream of the plate but also along a region corresponding roughly to the centre of the separated shear-layer. T h e improved velocity field predictions are a direct result of the reduced eddy viscosities w h i c h contribute to a slower growth rate of the separated shear layer and hence a longer separation bubble.  We observe that the highest eddy viscosities occur i n  b o t h cases slightly downstream of the reattachment point. T h e predicted turbulent kinetic energy profiles are presented i n Figures 6 . 1 3 and 6.14.  Since no experimental measurements of the turbulent kinetic energy k were  available, this was estimated from the measured l o n g i t u d i n a l r.m.s. velocity using  CHAPTER  6.  NUMERICAL  RESULTS  113  (a)  F i g u r e 6.12. Contours of constant effective viscosity: (a) B S H D c o m p u t a t i o n ; (b) P D M computation.  CHAPTER  6.  NUMERICAL  RESULTS  114  k/uref*uref 0.05 j.  0 4,  0.05 1  0 i,  0.05 1  F i g u r e 6.13. T u r b u l e n t kinetic energy profiles: O , estimated from P W A measurements.  0.10 L  0.05  0.10  , computed (BSHD);  CHAPTER  6.  NUMERICAL  RESULTS  F i g u r e 6.14. Turbulent kinetic energy profiles: estimated from P W A measurements.  115  , computed ( P D M ) ; Q ,  CHAPTER  6.  NUMERICAL  the relation k = u /a, 2  RESULTS  116  where a is a measure of the degree of anisotropy of the flow.  A n average value of a = 1.1 was estimated from the flat p l a t e / s p l i t t e r plate data of R u d e r i c h & Fernholz (1986). T h i s is of course a rather crude estimate, but a useful one i n the present context. T h e T . K . E . profiles predicted using B S H D show higher peaks t h a n the estimated values up to x/D  = 2 a n d lower ones thereafter;  these peaks are observed to occur closer to the w a l l . T h e unrealistically high T . K . E . observed near separation contribute to the poor performance of the s t a n d a r d k — e model i n this p a r t i c u l a r flow. In p a r t i c u l a r we note that an excessive value of k results i n an overestimated eddy viscosity (fi oc t  k /e). 2  T h e P D M profiles show a remarkable s i m i l a r i t y w i t h the estimated values both in r e c i r c u l a t i n g flow a n d recovery regions. We note i n p a r t i c u l a r that the highest T . K . E . is predicted to occur at the same streamwise location as the estimated value (x/D — 3). T h e preferential dissipation modification, by increasing the dissipation rate of T . K . E . i n the h i g h streamline curvature region close to separation, results i n more realistic values of k. There is one i m p o r t a n t feature of the estimated T . K . E . w h i c h is not w e l l reproduced, that is the spread of T . K . E . into the outer region. T h i s indicates a possible inadequacy of the modelled diffusion t e r m of the  fc-equation.  T h e remainder of this discussion is confined to the P D M results. T h e predicted pressure field shown i n F i g u r e 6.15 brings to light some interesting points.  The  pressure is observed to r e m a i n fairly uniform w i t h i n the first half of the separation bubble. We also observe that large cross-stream gradients are present along a region corresponding roughly to the outer part of the shear layer. These gradients are more pronounced near separation, probably as a result of the higher streamline curvature. T h e corresponding pressure d i s t r i b u t i o n along the surface of the plate is shown i n F i g u r e 6.16. T h i s pressure d i s t r i b u t i o n — w h i c h , incidentally, is markedly  CHAPTER  6.  NUMERICAL  RESULTS  117  different t h a n that i n the l a m i n a r flow case—reproduces fairly w e l l the trends of the experimental data.  In p a r t i c u l a r we note the b r o a d region of relatively con-  stant pressure w i t h a s m a l l dip before recovery. T h e m i n i m u m pressure coefficient, excluding the separation value, is underpredicted b y about 10%, p r o b a b l y as a result of the shorter bubble. T h e unrealistically low pressure coefficient predicted at separation is due to the singularity associated w i t h the sharp corner. In the actual flow, this corner is slightly rounded, since i n practice it is not possible to machine a m o d e l w i t h a perfectly sharp corner. F i g u r e 6.17 shows the w a l l shear stress d i s t r i b u t i o n . A g a i n the experimental trends are well reproduced a n d the m a x i m a a n d m i n i m a compare well considering the large uncertainty i n the measured values. A n interesting feature of the w a l l shear stress is the h u m p w h i c h occurs near separation, outside the measurements  range.  A possible interpretation of this h u m p is a s m a l l secondary recirculating flow region (with one or several vortices) just downstream of separation. T h i s interpretation is supported b y the streamline pattern shown i n F i g u r e 6.18.  CHAPTER  6.  NUMERICAL  RESULTS  118  F i g u r e 6.16. C o m p a r i s o n of computed and experimental pressure distributions on top surface of the the plate:  , c o m p u t e d ( P D M ) ; O , measured.  CHAPTER  6.  NUMERICAL  RESULTS  119  X/D F i g u r e 6.17. distributions:  F i g u r e 6.18.  C o m p a r i s o n of c o m p u t e d  and  experimental w a l l  shear  stress  , c o m p u t e d ( P D M ) ; O , measured.  P r e d i c t e d streamline p a t t e r n  A * = 0.143; for # < 0, A * = 0.01.  ( P D M computation):  for #  >  0,  CHAPTER  6.2.3  6.  NUMERICAL  RESULTS  120  Effect of Solid Blockage  F i n a l P D M computations were performed for blockage ratios i n the range 1.5% to 20%. T h e predicted reattachment lengths are compared i n Figure 6.19 to measured values reported i n the literature. T h e experimental trends are well reproduced and the c o m p u t e d values are w i t h i n 10% of the measured ones.  V a r y i n g departures  from strict two-dimensionality i n the measured case c o u l d account, i n part, for the generally higher experimental values of x . r  It should also be noted that a l l calcula-  tions were carried out assuming the boundary layers originated at the inlet of the c o m p u t a t i o n a l d o m a i n , a n d i n i t i a l profiles for k a n d e were kept unchanged. These assumptions do not necessarily correspond to the actual experimental conditions, but the resulting errors are expected to be small—at least for s m a l l blockage and low free-stream turbulence conditions. T h e predicted pressure distributions are shown i n F i g u r e 6.20 for a few blockage ratios. A t higher blockage ratios, there is a substantial decrease i n the m i n i m u m pressure coefficient w i t h increasing blockage, a n d the pressure starts recovering sooner. T h i s is consistent w i t h the shorter separation bubbles predicted at higher blockage ratios. T h e m a x i m u m pressure coefficient after recovery is lower at higher blockage ratios as w o u l d be expected from a simple application of B e r n o u l l i ' s equat i o n w h i c h gives: C  P m a x  -1  1  (1 -  BR)  2  T h e predicted pressure recovery is of course lower t h a n this "ideal" value not only as a a result of head loss, but also because the effective blockage is increased due to the growth of the b o u n d a r y layers on the solid w a l l a n d on the blunt plate.  CHAPTER  6.  NUMERICAL  RESULTS  x  c 0)  c  Q)  o  5  7.5  10  12.5  15  17.5  Percentage Solid Blockage BR F i g u r e 6.19. Effect of blockage on turbulent flow reattachment lengths.  CL O  -1.25  F i g u r e 6.20. P r e d i c t e d pressure distributions for various blockage ratios.  CHAPTER  6.3  6.  NUMERICAL  122  RESULTS  Computational costs  M o s t of the present computations were performed on a V A X 11/750 computer. For l a m i n a r flow calculations the c o m p u t a t i o n a l effort ranged from 28 iterations a n d 19 minutes of C P U time, for a 4 1 x 2 6 g r i d , to 63 iterations a n d 131 minutes of C P U time, for a 7 8 x 4 5 g r i d . Turbulent flow calculations converged much more slowly. T h e y required from 45 minutes of C P U time a n d 83 iterations, for a 49x34 g r i d , to 545 minutes a n d 329 iterations, for a 8 2 x 6 2 g r i d . A comparison of c o m p u t i n g costs for the H D , B S H D a n d P D M is shown i n F i g u r e 6.21. T h e P D M scheme is the most expensive of the three schemes as a result of its slower convergence rate. For a l l schemes, the costs rise r a p i d l y w i t h increasing size of the c o m p u t a t i o n a l array. T h e generally large C P U time requirements are due to the relatively s m a l l size a n d slow execution speed of the V A X 11/750.  F o r comparison purposes,  some  test runs were made on the U B C C o m p u t i n g Centre mainframe computer ( A m d a h l 5850). T h e execution time was found to be shorter by a factor of about 25. A turbulent flow c o m p u t a t i o n using the P D M scheme w i t h a 7 5 x 5 7 g r i d w o u l d therefore require about 18 minutes of C P U time on the A m d a h l (c/. 439 minutes on V A X ) .  CHAPTER  6.  NUMERICAL  123  RESULTS  600  -l  1000  1  2000  1  3000  1  4000  1  5000  No. of C o m p u t a t i o n a l Nodes NIxNJ  F i g u r e 6.21. C o m p u t i n g time on V A X 11/750 computer as a function of computat i o n a l array size (Turbulent flow).  Chapter 7  C onclusions  E x p e r i m e n t a l l y , it is difficult to establish an accurately two-dimensional mean flow field i n a separated reattaching flow region. Considerable effort has been expended i n this s t u d y to approximate these ideal conditions over the central part of the bluff rectangular plate. A mean reattachment length of 4.7D is obtained from b o t h wall shear stress measurements a n d surface streamline v i s u a l i z a t i o n . W i t h i n the separation bubble, the mean wall shear stress is found to be of the same order of magnitude as i n the reattached flow, a n d backflow velocities of the order of 30% of the free-stream velocity are encountered. O v e r the first 60% or so of the b u b b l e , the separated shear-layer b o u n d i n g the reversed flow region has similar characteristics to those of a plane m i x i n g layer. T h e g r o w t h rate of the shear-layer is linear, a n d b o t h characteristic frequencies and integral scales v a r y linearly w i t h streamwise distance. A t about x ~ 0 . 6 x , the shear-layer appears to undergo a fairly abrupt change r  in structure: the shear-layer growth rate becomes significantly lower a n d the m a x i m u m turbulent intensities become substantially higher t h a n the corresponding plane 124  CHAPTER  7.  CONCLUSIONS  125  mixing-layer values. In the reattachment region, the linear v a r i a t i o n of the characteristic frequencies a n d integral scales ceases. Consistently w i t h the observations of K i y a & Sasaki (1983) and C h e r r y et al. (1984), the shear layer is characterized by a low-frequency unsteadiness. T h i s unsteadiness of the reattachment process is further i l l u s t r a t e d by the forward-flow-fraction measurements w h i c h suggest that the instantaneous  reattachment point moves around the time-averaged reattachment  point over a distance of about 0 . 5 x . r  In a d d i t i o n to the carefully established two-dimensionality of the flow, a number of precautions have been taken to ensure the reliability of the present data and its suitability as a test case for n u m e r i c a l predictions. T h e measurements have been made i n a large scale facility, a n d probe interference effects have been assessed and m i n i m i z e d . W h e r e possible, measurements of the same quantities have been made using different experimental methods.  A complete pulsed-wire anemometer  data  set is given i n A p p e n d i x A for reference. In the n u m e r i c a l predictions, there is some uncertainty i n the i n i t i a l conditions for the b o u n d a r y layers developing on the w i n d tunnel walls.  T h e errors arising  from this uncertainty are expected to be s m a l l for the low blockage ratios considered i n this study. Otherwise, the boundary conditions for the flow a r o u n d the blunt rectangular plate are clearly defined. L a m i n a r flow predictions show that the h y b r i d differencing scheme ( H D ) leads to significantly underpredicted reattachment lengths as a result of false diffusion. T h e skewed differencing scheme ( B S H D ) yields markedly improved predictions. These predictions are i n excellent agreement w i t h experimental observations, indicating that errors due to false diffusion are effectively eliminated w i t h the B S H D scheme.  CHAPTER  7.  CONCLUSIONS  126  A t higher R e y n o l d s numbers, however, the performance of the B S H D scheme is found to deteriorate i n the stagnation region.  T h i s p r o b l e m , w h i c h has also  been noted elsewhere, is due to a deficiency of the B S H D scheme i n the presence of strong adverse pressure gradients. In the present turbulent flow computations, this deficiency is "corrected" by reverting to lower order h y b r i d differencing i n the stagnation region. T h e use of H D differencing i n conjunction w i t h the standard k — e model is inadequate for this flow. E v e n w i t h fine grids (82 x 62), this c o m b i n a t i o n leads to a separation bubble 44% shorter t h a n observed experimentally. T h i s discrepancy is m u c h larger t h a n the values reported i n the literature for other types of separated reattaching  flows.  T h e results of the the B S H D computations show that the s t a n d a r d k — e model fails to represent accurately the effects of turbulence i n the region around separat i o n . T h e use of the preferential dissipation modification ( P D M ) leads to significantly i m p r o v e d predictions. A reattachment length of 4 . 3 D is obtained. D e t a i l e d predictions i n the separation bubble compare well w i t h experiments, a n d are much more satisfactory t h a n those obtained w i t h the discrete v o r t e x m e t h o d ( K i y a et al.  1982). D o w n s t r e a m of reattachment, however, a gradual deterioration of the  predictions is observed as the wall b o u n d a r y layer redevelops. A d i s t u r b i n g feature of the predictions is that although the flowfield results are found to be essentially g r i d independent for a 75 x 57 g r i d , this g r i d independence does not extend to the c o m p u t e d w a l l shear stress. T h e w a l l shear stress is found to change by as m u c h as 10% w i t h subsequent g r i d refinements. These changes indicate possible inconsistencies i n the near-wall treatment. T h e i m p a c t of these changes on  CHAPTER  7.  CONCLUSIONS  127  the predictions seems to be m i n i m a l i n the present case because the flowfield appears to be insensitive to the details of the near-wall flow. T h i s p r o b l e m , however, could be significant i n other flow situations. A n alternative m e t h o d of analysis, based on a novel use of the m o m e n t u m integral technique i n flows w i t h separation a n d reattachment has been proposed. T h e m e t h o d leads to encouraging results for the case of a sudden expansion flow, but problems of convergence exist w h e n the m e t h o d is applied to external flows using a direct viscous-inviscid interaction procedure. A few suggested areas of future research a n d some recommendations follow:  1. E x p e r i m e n t a l information to elucidate the structure of the flow i n the nearw a l l region is needed to guide the m o d e l l i n g of this region. A step i n this direction has been taken by A d a m s et al. (1984) for the turbulent flow over a b a c k w a r d facing step. T h e i r results suggest that the near-wall flow has a laminar-like structure, but more information is required p a r t i c u l a r l y for high R e y n o l d s number flows. Because the scale of the w a l l region is very s m a l l compared w i t h the overall scale of the separation bubble, experiments should be conducted on large scale models a n d m i n i a t u r i z e d pulsed-wire probes a n d / o r laser-Doppler anemometry should be used.'  2. M o r e definitive w a l l shear stress measurements should be made.  T h e large  uncertainty i n the present measurements is due to the c a l i b r a t i o n procedure. A better—though not ideal (see C a s t r o & D i a n a t 1985)—procedure would be to calibrate the probe i n an ordinary turbulent b o u n d a r y layer against a P r e s t o n tube.  CHAPTER  7.  CONCLUSIONS  128  3. Measurements of the turbulent stresses uv, a n d of the r.m.s. velocities v a n d w should be made. These measurements w o u l d not only give further insight into the structure of the flow, but w o u l d also enable a more t h o r o u g h assessment of turbulence models.  4. T h e extension of the m o m e n t u m integral analysis to external separated-reattaching flows is w o r t h pursuing. T h e use of an inverse or semi-inverse interaction procedure of the type used by W i l l i a m s (1985) s h o u l d be explored for the m a t c h i n g of the viscous flow to the external inviscid flow.  5. A criterion to assess the suitability of the B S H D scheme i n regions of strong pressure gradients should be devised.  B a s e d on this criterion, a "switch-  ing" technique from bounded-skew-upwind differencing to u p w i n d differencing should be considered. A t a more fundamental level, consideration should be given to the development of a skewed scheme w h i c h allows for strong pressure gradients.  6. T h e present results show that the blunt rectangular plate flow constitutes a severe test for discretization schemes as well as turbulence models. In b o t h , inadequacies appear to be magnified as a result of the h i g h streamline curvatures, large gradients, and the stagnation flow region. These features, as well as the simple u n i f o r m flow u p s t r e a m boundary conditions, suggest the a d o p t i o n of this flow configuration as a benchmark test for n u m e r i c a l methods a n d turbulence models.  CHAPTER  7.  CONCLUSIONS  129  F i n a l l y , the present predictions of the mean flow field using the modified k — e model are quite remarkable, considering the unsteadiness of the actual flow and the complexity of its turbulence structure i n the reattachment region.  A time-  dependent numerical solution w h i c h takes into account this large-scale unsteadiness w o u l d be a logical a n d , undoubtedly, challenging extension of this w o r k . Such an approach should also prove useful as a first step towards the c o m p u t a t i o n of the high R e y n o l d s number unsteady flows about prismatic sections—a class of flows w h i c h is of great p r a c t i c a l importance to many w i n d engineering applications.  References A C R I V O S , A . & S C H R A D E R , M . L . (1982). Steady flow i n a sudden expansion at h i g h R e y n o l d s number. Phys. Fluids 25, 923-930. A D A M S , E . W . , J O H N S T O N , J . P. & E A T O N , J . K . (1984). 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W E S T P H A L , R . V . , E A T O N , J . K . & J O H N S T O N , J . P . (1981). A new probe for measurement of velocity and wall shear stress i n unsteady reversing flow. J. Fluids Engng 103, 478-482. W E S T P H A L , R . V . , J O H N S T O N , J . P . & E A T O N , J . K . (1984). E x p e r i m e n tal study of flow reattachment i n a single sided sudden expansion. N A S A C o n t r a c t o r R e p o r t 3765. W I L L I A M S , B . R . (1985). T h e prediction of separated flow using a viscousinviscid interaction method. Aero. J. 89(885), 185-197. W Y G N A N S K I , I. & F I E D L E R , H . E . (1970). T h e two-dimensional m i x i n g layer. J. Fluid Mech. 41,327-361.  Appendix A  Tabulated  Data  A complete set of mean velocity (U), r.m.s. velocity ( V r ) , a n d forward flow fraction ( 7 ) profiles are given for reference. A l l measurements were made w i t h a pulsed-wire anemometer at a R e y n o l d s number of 5 x 1 0 . T h e tabulated values are presented 4  in non-dimensional form; the reference quantities are the free-stream velocity [T^, (= Uref)  and the plate thickness D (= 89.9 m m ) .  136  Table A . l ( a to j ) . Experimental data.  FILE NU.PW.JAN85A STATION X/D= 0.000 Uref= 8.38 M/SEC Y/D 0. 0759 0. 0843 0. 1 1 24 0. 1687 0. 2249 0. 2812 0. 3374 0. 3937 0. 4499 0. 5624 0. 6749 0. 7874 0. 8998 1. 0123 1. 1811 1. 3498 1. 5185 1. 6872 1. 8560 2. 0247  Y/Xr 0 .0163 0 .0182 0 .0242 0 .0364 0 .0485 0 .0607 0 .0728 0 .0850 0 .0971 0 .1214 0 . 1457 0 . 1700 0 . 1 943 0 .2186 0 .2550 0 .291 5 0 .3279 0 . 3644 0 .4008 0 . 4373  X/Xr=  FILE NU.PW.JAN85B STATION X/D= 1.000 Uref= 8.36 M/SEC  0.000  U/Uref  u/Uref  GAMMA  0.9508 0.9479 0.9473 0.9584 0.9654 0.9797 0.9878 0.9907 0.9932 1.0046 1.0109 1.0176 1.0258 1.0291 1.0231 1.0287 1.0353 1.0414 1.0483 1.0539  0. 0276 0. 0251 0. 0207 0. 0203 0. 0209 0. 0204 0. 0198 0. 0196 0. 0196 0. 0197 0. 0191 0. 0196 0. 01 97 0. 0198 0. 0205 0. 0198 0. 0202 0. 0212 0. 021 1 0. 0201  1 .0000 1 .0000 1 .0000 1 .0000 1 .0000 1 .0000 1 .0000 1 .0000 1 .0000 1 .0000 1 .0000 1 .0000 1 .0000 1.0000 1 .0000 1 .0000 1 .0000 1 .0000 1 .0000 1.0000  Y/D 0 .0758 0 .0843 0 .1124 0 .1406 0 . 1 687 0 .1968 0 .2249 0 .2530 0 .281 2 0 .3093 0 .3374 0 .3655 0 .3937 0 .4218 0 .4499 0 .4780 0 .5061 0 .5343 0 .5624 0 .6186 0 .6749 0 .7311 0 .7874 0 .8998 1 .0123 1 .1248 1 .2373 1 .3498 1 .5185 1 .6872 1 .8560 2 .0247  Y/Xr 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.  0163 0182 0242 0303 0364 0425 0485 0544 0607 0668 0728 0789 0850 091 1 0971 1032 1093 1 1 54 1214 1336 1457 1 579 1700 1943 2186 2429 2672 291 5 3275 3644 4008 4373  X/Xr=  0.216  U/Uref -0 .2404 -0 .2421 -0 .2296 -0 .2225 -0 .2006 -0 .1823 -0 .1516 -0 .1099 -0 .0404 0 .0385 0 . 1 535 0 .2853 0 .4087 0 .5505 0 .7148 0 .8664 1 .0319 1 . 1 623 1 .2434 1 .2919 1.2828 1 .2662 1 .2541 1.2266 1 .2069 1 . 1 949 1 .1823 1 .1693 1 .1517 1 .1417 1 .1395 1 . 1 332  u/Uref  GAMMA  0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.  0.0250 0.0240 0.0390 0.0480 0.0680 0.1050 0. 1730 0.2760 0.4220 0.5760 0.7590 0.8770 0.9560 0.9870 0.9970 1.0000 1 .0000 1 .0000 1 .0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000  1 543 1614 1 604 1717 1846 1895 2091 2133 2297 2355 2388 2453 2525 2606 2728 2664 2478 2041 1578 0792 0555 0456 0424 0369 0338 0324 0295 0266 0246 0244 0249 0246  FILE NU.PW.JAN85C STATION X/D= 2 . 0 0 0  Uref=  8.32  FILE NU.PW.JAN85D STATION X/D= 3 . 0 0 0  0.432  M/SEC  Y/Xr  Y/D 0 .0759 0 .0843 0 . 11 24 0 .1406 0 . 1 687 0 .1968 0 .2249 0 .2530 0 .2812 0 .3093 0 .3374 0 .3655 0 .3937 0 .4218 0 .4499 0 .4780 0 .5061 0 .5343 0 .5624 0 .6186 0 .6749 0 .731 1 0 .7874 0 .8998 1.0123 1.1248 1.2373 1.3498 1.5185 1.6872 1.8560 2 .0247  X/Xr=  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  .0163 .0182 .0242 .0303 .0364 .0425 .0485 .0546 .0607 .0668 .0728 .0789 .0850 .091 1 .0971 .1032 .1093 . 11 54 .1214 . 1 336 .1457 . 1 579 .1700 .1943 .2186 .2429 .2672 .2915 .3279 .3644 .4008 .4373  Uref=  U/Uref - 0 . 3258 - 0 . 3239 - 0 . 2963 - 0 . 2670 - 0 . 2342 - 0 . 1 994 - 0 . 1 508 - 0 . 1016 - 0 . 0545 - 0 . 0051 0. 0679 0 . 1 302 0 . 221 1 0. 2982 0 . 381 1 0. 4612 0 . 5531 0. 6640 0. 7422 0. 9478 1 . 1 147 1. 2 3 2 2 1. 2 8 8 2 1. 2 9 2 5 1. 2 7 3 7 1. 2 5 5 1 1. 2 4 1 3 1. 2 2 9 3 1. 2 0 4 1 1. 1 8 4 3 1. 1 7 7 6 1. 1 7 0 2  u/Uref  GAMMA  0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.  0 .0390 0 .0370 0 .0710 0 .0960 0 . 1 420 0 . 1 920 0 .2480 0 .3250 0 .4060 0 .5150 0 .6240 0 .6970 0 .7860 0 .8540 0 .9090 0 .9420 0 .9710 0 .9860 0 .9940 0 .9990 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000  1731 1774 1915 2073 2217 2376 2521 2633 271 1 2838 2864 2888 2886 2879 2877 2888 2956 2931 2915 2762 2341 1804 1 223 0686 0493 0431 0397 0368 0334 0277 0261 0260  8.34  Y/D 0 .0759 0 .0843 0 . 1 124 0 . 1 406 0 . 1 687 0 .1968 0 . 2249 0 .2530 0 .2812 0 .3093 0 .3374 0 .3655 0 .3937 0 .4218 0 .4499 0 .4780 0 .5061 0 . 5343 0 .5624 0 .6186 0 .6749 0 .7311 0 .7874 0 .8436 0 .8998 0 .9561 1.0123 1.1248 1.2373 1.3498 1.5185 1.6872 1.8560 2 .0247  X/Xr=  0.648  M/SEC  Y/Xr 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.  0163 0182 0242 0303 0364 0425 0485 0546 0607 0668 0728 0789 0850 091 1 0971 1 032 1093 1 1 54 1 241 1336 1 457 1 579 1700 1822 1943 2065 2186 2429 2672 2915 3279 3644 4008 4373  U/Uref  u/Uref  GAMMA  -0.2663 -0.2580 -0.2154 -0.1845 -0.1413 -0.0992 -0.0569 -0.0091 0.0614 0.1157 0.1777 0.2441 0.3041 0.3765 0.4489 0.5122 0.5976 0.6580 0.7348 0.8839 1 .0053 1 . 1058 1 . 1730 1.2020 1 .2217 1 .2170 1.2219 1.2093 1.2003 1.1931 1.1808 1.1691 1.1619 1 . 1582  0.2120 0.2146 0.2375 0.2480 0.2688 0.2776 0.2939 0.2978 0.3041 0.3166 0.3150 0.3146 0.3139 0.3139 0.3091 0.3102 0.3040 0.3021 0.2997 0.2796 0.2472 0.2107 0 . 1672 0.1342 0.0978 0.0819 0.0666 0.0464 0.0379 0.0334 0.0298 0.0286 0.0272 0.0268  0 . 1020 0 . 1 1 20 0 . 1570 0 .2290 0 .2850 0 . 3450 0 .4290 0 .4810 0 .5730 0 .6520 0 .7110 0 .7900 0 .8410 0 .8940 0 .9320 0 .9550 0 .9700 0 .9850 0 .9890 0 .9980 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000  FILE NU.PW.JAN85E STATION X/D= 4.000 Uref= 8.31 M/SEC  FILE NU.PW.JAN85F STATION X/D= 4.500 Uref= 8.34 M/SEC  X/Xr= 0.864  y/D  Y/Xr  U/Uref  u/Uref  GAMMA  0.0759 0.0843 0.1124 0.1406 0.1687 0.1968 0.2249 0.2530 0.281 2 0.3093 0.3374 0.3655 0.3937 0.4218 0.4499 0.47B0 0.5061 0.5343 0.5624 0.6186 0.6749 0.731 1 0.7874 0.8436 0.8998 0.9561 1.0123 1 . 1248 1 .2373 1 . 3498 1.5185 1.6872 1.8560 2.0247  0.0163 0.0182 0.0242 0.0303 0.0364 0.0425 0.0485 0.0546 0.0607 0.0668 0.0728 0.0789 0.0850 0.091 1 0.0971 0.1032 0.1093 0.1154 0.1214 0. 1336 0. 1457 0.1579 0.1700 0.1822 0.1943 0.2065 0.2186 0.2429 0.2672 0.2915 0.3279 0.3644 0.4008 0.4373  -0.0193 -0.0001 0.0239 0.0597 0.0898 0.1251 0.1647 0.2120 0.2590 0.2942 0.3350 0.3890 0.4392 0.4934 0.5408 0.5932 0.6454 0.6918 0.7443 0.8341 0.9308 0.9951 1.0569 1 .0908 1.1163  0.2450 0.2516 0.2547 0.2621 0.2706 0.2799 0.2828 0.2879 0.2850 0.2888 0.2959 0.2964 0.3015 0.3003 0.2966 0.2960 0.2894 0.2882 0.2783 0.2631 0.2366 0.2125 0.1791 0.1550 0.1247 0.1013 0.0813 0.0588 0.0484 0.0416 0.0357 0.0313 0.0309 0.0284  0.4690 0.4870 0.5110 0.5850 0.6270 0.6820 0.7210 0.7710 0.8160 0.8510 0.8980 0.9120 0.9380 0.9510 0.9720 0.9810 0.9860 0.9890 0.9930 0.9970 0.9980 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000  1.1265  1 . 1326 1 . 1346 1 . 1349 1.1319 1.1242 1.1199 1.1169 1.1215  y/D  Y/Xr  0.0759 0.0843 0.1124 0.1406 0. 1687 0. 1968 0.2249 0.2530 0.2812 0.3093 0.3374 0.3655 0.3937 0.4218 0.4499 0.4780 0.5061 0.5343 0.5624 0.6186 0.6749 0.731 1 0.7874 0.8436 0.8998 0.9561 1.0123 1 .1248 1 .2373 1.3498 1.5185 1.6872 1.8560 2.0247  0.0163 0.0182 0.0242 0.0303 0.0364 0.0425 0.0485 0.0546 0.0607 0.0668 0.0726 0.0789 0.0850 0.0911 0.0971 0.1032 0.1093 0.1154 0.1214 0.1336 0.1456 0. 1579 0.1700 0. 1822 0.1943 0.2065 0.2186 0.2429 0.2672 0.2915 0.3279 0.3644 0.4008 0.4373  X/Xr=  0.972  U/Uref  u/Uref  GAMMA  0.1413 0.1518 0.1718 0.2034 0.2296 0.2557 0.2757 0.3224 0.3537 0.3892 0.4171 0.4652 0.5005 0.5372 0.581 1 0.6405 0.6722 0.7159 0.7515 0.8334 0.8979 0.9601 1.0099 1 .0503 1.0700 1.0812 1 .0902 1.1008 1.1001 1.0907 1.0935 1 .0880 1.0878 1 .0944  0.2395 0.2413 0.2415 0.2470 0.2506 0.2570 0.2619 0.2685 0.2721 0.2739 0.2742 0.2762 0.2774 0.2764 0.2792 0.2802 0.2799 0.2733 0.2665 0.2523 0.2312 0.2061 0.1839 0.1499 0.1304 0.1084 0.0872 0.0648 0.0532 0.0446 0.0381 0.0333 0.0315 0.0313  0.7340 0.7600 0.7790 0.8070 0.8330 0.8420 0.8760 0.8980  0.9140  0.9330 0.9500 0.9620 0.9700 0.9760 0.9820 0.9870 0.9920 0.9940 0.9960 0.9970 0.9990 1.0000 1.0000 1.0000 1.0000 1.0000 1 .0000 1.0000 1 .0000 1 .0000 1 .0000 1 .0000 1 .0000 1 .0000  FILE NU.PW.JAN8 5G STATION X/D= 5.000 U r e £ = 8.35 M/SEC Y/D  0 .0759 0 .0843 0 . 1 124 0 . 1 406 0 .1687 0 .1968 0 .2249 0 .2530 0 .2812 0 .3093 0 .3374 0 .3655 0 .3937 0 .4218 0 .4499 0 .4780 0 .5061 0 .5343 0 .5624 0 .6186 0 .6749 0 .731 1 0 .7874 0 .8436 0 .8998 0 .9561 1 .0123 1 . 1 248 1 .2373 1 .3498 1 .5185 1.6872 1.8560 2 .0247  Y/Xr 0 .0163 0 .0182 0 .0242 0 .0303 0 .0364 0 .0425 0 .0485 0 .0546 0 .0607 0 .0668 0 .0728 0 .0789 0 .0850 0 .091 1 0 .0971 0 .1032 0 .1093 0 . 1 1 54 0 .1214 0 .1336 0 .1457 0 .1579 0 .1700 0 .1822 0 .1943 0 .2065 0 .2186 0 .2429 0 .2672 0 .2915 0 .3279 0 .3644 0 .4008 0 .4373  X/Xr=  FILE NU.PW.JAN85H STATION X/D= 6.000 Uref= 8.39 M/SEC  1.080  U/Uref  u/Uref  GAMMA  0.2831 0.2927 0.3058 0.3302 0.3505 0.3762 0.3971 0.4300 0.4540 0.4804 0.5132 0.5534 0.5864 0.6231 0.6573 0.6863 0.721 3 0.7671 0.7981 0.8591 0.9097 0.9551 1.0018 1.0257 1.0531 1.0648 1.0743 1.0863 1.0857 1.0862 1.0921 1.0885 1.0877 1.091 3  0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.  0. 9240 0. 9310 0. 9400 0. 9520 0. 9530 0. 9610 0. 9680 0. 9710 0. 9730 0. 9740 0. 9850 0. 9880 0. 9900 0. 9900 0. 9940 0. 9960 0. 9970 0. 9980 0. 9980 0. 9990 1. 0000 1 .0000 1 .0000 1 .0000 1. 0000 1 .0000 1 .0000 1 .0000 1 .0000 1 .0000 1. 0000 1 .0000 1 .0000 1. 0000  2191 2199 2197 2240 2282 2292 2358 2448 2482 2515 2595 2624 2655 2646 2670 2618 2564 2537 2466 2362 2210 1 948 1725 1 523 1 281 1024 091 1 0628 0536 0422 0355 0303 0289 0276  Y/D 0 .0675 0 .0758 0 .0842 0 . 1 1 24 0 . 1 406 0 .1685 0 . 1 967 0 .2248 0 .2530 0 .2812 0 .3092 0 .3373 0 .3654 0 .3937 0 .4217 0 .4499 0 .4780 0 .5061 0 .5342 0 .5624 0 .5904 0 .6185 0 .6749 0 .7310 0 .7874 0 .8435 0 .8998 0 .9560 1 .0123 1 . 1 248 1 .2373 1 .3498 1 .5185 1 .6871 1 .8560 2 .0246  Y/Xr 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.  0145 0163 0182 0241 0302 0364 0425 0485 0545 0607 0668 0728 0789 0850 0910 0971 1 032 1 091 1 154 1214 1 275 1 336 1 456 1 578 1700 1 821 1942 2064 2185 2428 2672 2915 3278 3643 4007 4371  X/Xr=  0.296  U/Uref  u/Uref  GAMMA  0.4623 0.4707 0.4753 0.4976 0.5081 0.5243 0.5423 0.5614 0.5778 0.5958 0.6183 0.6383 0.6573 0.6864 0.7070 0.7274 0.7459 0.7730 0.7983 0.8202 0.8378 0.8621 0.9035 0.9372 0.965B 0.9906 1.0155 1 .0320 1 .0428 1.0555 1.0627 1.0661 1.0660 1.0673 1.0657 1.0656  0 .2018 0 .2031 0 .2007 0 .2032 0 .2073 0 .2091 0 .2113 0 .2175 0 .2214 0 .2255 0 .2301 0 .2332 0 .2353 0 .2385 0 .2383 0 .2346 0 .2312 0 .2305 0 .2281 0 .2242 0 .2228 0 .2171 0 .2031 0 .1879 0 .1698 0 .1519 0 . 1 322 0 .1161 0 .1038 0 .0717 0 .0565 0 .0502 0 .0374 0 .0318 0 .0281 0 .0253  0 . 9960 0 .9970 0 .9970 0 .9980 0 .9980 0 .9980 0 .9980 0 .9980 0 . 9990 0 .9980 0 .9980 0 .9990 0 .9990 0 .9990 0 .9990 1 .0000 1 .0000 .9990 1 .0000 .9990 1 .0000 1 .0000 1 .0000 1 .0000 1 .0000 1 .0000 1 .0000 1 .0000 1 .0000 1 .0000 1 .0000 1 .0000 1 .0000 1 .0000 1 .0000 1 .0000  FILE NU.PW.JAN85I STATION X/D= 7.000 Uref= 8.42 M/SEC Y/D 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.  0673 0758 0842  Y/Xr  0. 0. 0. 1 1 240. 1 406 0. 1 686 0. 1 967 0. 2248 0. 2530 0. 2812 0. 3373 0. 3937 0. 4499 0. 5061 0. 5624 0. 6185 0. 6749 0. 7310 0. 7874 0. 8435 0. 8998 0. 0. 1 0123 . 0. 1 1248 . 1 .2373 0. 1 .3498 0. 0. 1 51B5 . 0. 1 6871 . 1. 8560 0. 2. 0247 0.  0145 0163 0182 024 1 0302 0364 0425 0485 0545 0607 0728 0850 0971 1 091 1214 1336 1456 1578 1700 1821 1942 2185 2428 2672 2915 3278 3643 4007 4371  X/Xr=  F I L E NU.PW.JAN85J STATION X/D= 8.500 Uref= 8.40 M/SEC  1.512  U/Uref  u/Uref  GAMMA  0. 5762 0. 5769 0. 5848 0. 6022 0. 6095 0. 6250 0. 6409 0. 6544 0. 6628 0. 6836 0. 7074 0. 7375 0. 7642 0. 7965 0. 8284 0. 8674 0. 8958 0. 9146 0. 9437 0. 9587 0. 9806 1 0087 . 1 .0241 1 .0373 1 .0497 1 .0529 1 .0529 1 .0479 1 .0539  0. 1885 0. 1881 0. 1884 0. 1895 0. 1898 0. 1913 0. 1972 0. 1 982 0. 2004 0. 2003 0. 2030 0. 2071 0. 2116 0. 207 1 0. 2051 0. 1955 0. 1876 0. 1794 0. 1654 0. 1 566 0. 1412 0. 1 136 0. 0936 0. 0697 0. 0538 0. 0426 0. 0340 0. 0312 0. 0268  0.9990 1.0000 1 .0000 0.9990 1 .0000 1.0000 0.9990 1.0000 1.0000 1 .0000 1 .0000 1.0000 1.0000 1.0000 1.0000 1.0000 1 .0000 1.0000 1.0000 1.0000 1.0000 1 .0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000  Y/D 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 2  0759 0843 1 124 1 406 1 687 1968 2249 2530 2812 3093 3374 3937 4499 5061 5624 6186 6749 7874 8998 0123 1248 2373 3498 5185 6872 8560 0247  X/Xr=  1.836  Y/Xr  U/Uref  u/Uref  GAMMA  0.0163 0.0182 0.0242 0.0303 0.0364 0.0425 0.0485 0.0546 0.0607 0.0668 0.0728 0.0850 0.0971 0.1093 0.1214 0.1336 0.1457 0.1700 0.1943 0.2186 0.2429 0.2672 0.2915 0.3279 0.3644 0.4008 0.4373  0.6784 0.6836 0.7030 0.7207 0.7343 0.7414 0.7515 0.7568 0.7735 0.7740 0.7795 0.8052  0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.  1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1 .0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1 .0000 1.0000 1.0000 1.0000 1.0000 1 .0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000  0.8199 0.8455 0.8617 0.8808 0.9049 0.9431 0.9820 1.0141 1.0292 1.0408 1.0486 1.0514 1.0522 1.0508 1.0499  1679 1701 1693 1716 1724 1733 1757 1 761 1746 1743 1756 1780 1784 1757 1753 1716 1675 1546 1403 1 198 0965 0782 0692 0509 0399 0324 0279  Appendix B  Potential Flow Analysis  B.l  Theory  T h e simple model outlined i n this section is inspired directly from Parkinson's wake source model (Parkinson and J a n d a l i 1970). T h e separated shear layer is, i n a first a p p r o x i m a t i o n , replaced by a streamline b o u n d i n g an external irrotational flow region on its upper side (Figure B . l ) . T h e irrotational flow can be modelled by introducing a surface point source, the strength and location of w h i c h are determined by a p p l y i n g two conditions: i) Specified base pressure at A. ii) Tangential  separation.  Since the b o d y is a polygon, the flowfield i n the physical plane (2-plane) can be m a p p e d conformally onto the upper half of the transform plane (f-plane) using a Schwarz-Christoffel transformation.  T h e solution of the flow is then readily ob-  tained, once the free parameters of the p r o b l e m are determined from the imposed conditions. 142  APPENDIX  B.  POTENTIAL  FLOW  143  ANALYSIS  U.  /  v V  Q  1  F i g u r e B . l . Wake source model.  ©  y.  1 U-  A ij  6  T  h  A'  Figure B . 2 . P h y s i c a l and transform planes.  APPENDIX  B.  POTENTIAL  FLOW  ANALYSIS  144  T h e tangential separation c o n d i t i o n , however, presents some difficulties.  But  these difficulties can be avoided by the i n t r o d u c t i o n of a s m a l l teat i n the physical plane at p o i n t A, as shown i n figure B . 2 . A p p l y i n g the Schwarz-Christoffel transformation: dz  K(  ,  A t the c r i t i c a l point A , dz/d$ has a simple zero; angles are therefore doubled through the transformation. Hence, to a stagnation streamline at A i n the f- plane corresponds a tangential separation streamline at A i n the 2-plane. T h e relative height of the teat 8/h c a n be made suitably s m a l l by appropriately choosing e. In the f-plane, we have a u n i f o r m flow past a source; the resulting complex p o t e n t i a l is: Voof + I;  ln(£ - &)  (B.2)  a n d 2-planes are given, respectively, b y  a n d the velocity potential i n the  W(t)  = ^  (B.3)  and  In the outer flow region bounded by the separation streamline, B e r n o u l l i ' s equation applies: p+lp\W(z)\  = p  2  0O  +  -pU  1  or ir  1  oo  oo  2 oo  (B.5)  APPENDIX  B.  POTENTIAL  FLOW  ANALYSIS  145  A p p l y i n g the first c o n d i t i o n (i.e. given base pressure pj, at A) to equation (B.8), we obtain: W{z)  = (i-c yi  2  Pb  evaluating W(z)  =k  (B.7)  at A gives the following relation:  (B.8)  a n d r e q u i r i n g A to be a stagnation point i n the f-plane (2  condition) leads to  Q  (B.9)  k(l - e)  2hU  G i v e n k a n d e, the source strength, Q, a n d its p o s i t i o n ,  are determined by  (B.8) a n d (B.9). T h e pressure d i s t r i b u t i o n along the streamline can be evaluated using ( B . 7 ) , a n d the shape of the separating streamline is obtained b y solving for the f co-ordinates satisfying (B.10)  9[F(f)] = 0  the corresponding physical plane co-ordinates are then found using  z h - In and  2 h  7T  [  2 >  / ( + l ) ( - 6 ) + ( + 1) + (f - e) f  f  — t a n y/e . 1  1 - e  w h i c h are obtained by integrating equation ( B . l ) .  ?  + i (B.ll) (B.12)  APPENDIX  B.2  B.  POTENTIAL  FLOW  ANALYSIS  146  Results  F i g u r e B . 3 shows the calculated position of the separation streamline for three different base pressures, chosen i n a range corresponding to the experimentally measured base pressure coefficient of about -0.8. T h e calculated a n d measured positions are in reasonable agreement over some distance downstream from separation.  This  agreement is of course not expected to extend much beyond i / Z ? ~ 2.0, where the reattachment process starts t a k i n g place. C a l c u l a t i o n s w i t h a s m a l l value of e result i n a streamline w h i c h is too close to the w a l l . In order to o b t a i n a streamline w h i c h is more representative of the actual flow, it is necessary to use large values for e (i.e. close to 1). A l t h o u g h this corresponds to large teats, this is of little consequence because the region below the b o u n d i n g streamline is outside the scope of the p o t e n t i a l flow m o d e l . T h e results presented here are for e = 0.99. T h e calculated pressure d i s t r i b u t i o n along the separated streamline is shown in F i g u r e B . 4 . A sharp dip i n pressure associated w i t h the high i n i t i a l curvature of the separating streamline , occurs immediately after separation, a n d is followed by a r a p i d recovery. These calculated distributions bear little resemblance to the measured surface pressure distributions w h i c h exhibit a m u c h broader region of almost constant pressure extending from separation to about x/D  = 2.0.  This  indicates that the diffusive effects, necessarily ignored by the potential flow analysis, are very i m p o r t a n t i n establishing the pressure i n the actual turbulent flow, even at separation where the shear layer is very t h i n .  APPENDIX  B.  POTENTIAL  FLOW  ANALYSIS  147  F i g u r e B . 4 . C a l c u l a t e d pressure distributions for different specified base pressures.  Appendix C  Momentum  Integral A naly sis  In this appendix, we consider the modelling of a separated shear layer using the b o u n d a r y layer equations i n their m o m e n t u m integral form . T h e m e t h o d is assessed for the fully developed flow i n a sudden expansion (Figure C . l ) , w h i c h is, in this context, a simpler flow t h a n the bluff plate flow: this flow does not have a potential flow core, a n d therefore does not require a viscous-inviscid m a t c h i n g procedure. Before discussing the assumptions w h i c h were made i n this analysis, it should be pointed out that, strictly, the validity of the b o u n d a r y layer equations is based on the c o n d i t i o n that d6 jdx <C 1. T h i s c o n d i t i o n is not met in reattaching flows as pointed out by, amongst others, B r a d s h a w & W o n g (1972). It appears, nevertheless, that for small separation regions, the b o u n d a r y layer or t h i n shear layer a p p r o x i m a t i o n retains its validity: for instance, the boundary layer calculations of G h i a et al. (1975) and B r i l e y & M c D o n a l d (1975) were found to compare well w i t h numerical solutions using the full Navier-Stokes equations. One of the m a i n attractions of the b o u n d a r y layer equations is the relative simplicity w i t h w h i c h they can be solved numerically. Because they are parabolic, c o m p u t a t i o n a l l y inexpensive forward marching algorithms can be used. However, 148  APPENDIX  C. MOMENTUM  INTEGRAL  ANALYSIS  149  w h e n m a r c h i n g against the streamwise velocity, w h i c h is the case i n the backflow region, it is no longer possible to use a simple forward m a r c h i n g technique, and an iterative procedure is required. T h a t is unless the culprit for this added complicat i o n , namely the streamwise convective t e r m i n the inner region, is removed. T h i s step, w h i c h may seem r a d i c a l at first sight, was first proposed by Reyhner &; F K i g g e - L o t z (1969) and is k n o w n as the F L A R E a p p r o x i m a t i o n . T h e implications of this a p p r o x i m a t i o n are examined i n some detail by M c D o n a l d & B r i l e y (1983) who conclude that the a p p r o x i m a t i o n is, i n general, only acceptable w h e n the magnitude of the reversed flow velocities do not exceed O . I Z / O Q . T h i s , based on experimental observations, implies that the a p p r o x i m a t i o n is v a l i d for l a m i n a r flows. T h i s conclusion is borne out by the results of K w o n et al.  (1984), w h o com-  p u t e d a l a m i n a r sudden expansion flow, using the b o u n d a r y layer equations and the F L A R E a p p r o x i m a t i o n . T h e y obtained results w h i c h are i n good agreement w i t h Navier-Stokes predictions as well as experimental observations. A major difference of the present formulation w i t h that of K w o n et al, is the use of an integral method as opposed to a finite difference method.  C.l  Formulation  T h e variables for the model are illustrated i n F i g u r e C . l . T h e flow is fully developed at separation, a n d , for the purpose of the analysis, is d i v i d e d into two regions. T h e b o u n d a r y layer equation i n its m o m e n t u m integral form is used for both regions, but w i t h different assumptions. In the "outer" region, corresponding to U > 0, the velocity profiles are assumed to be self-similar; whereas i n the backflow region, the w a l l shear stress and the fluid m o m e n t u m are neglected ( F L A R E  approximation).  APPENDIX  C. MOMENTUM  INTEGRAL  ANALYSIS  150  ' / / / / / / / / / / / / / / / / / / / / / / / /  / / /  / / / /  / / / / A 6(0)  ER  = JJ/6(0)  u(x,y) -  vA  /  - "  T  ®  6(X)  -o x  F i g u r e C . l . M o d e l l i n g of 2-D sudden expansion  flow.  T h e separation streamline is therefore identical to the line of zero velocity. A brief outline of the formulation is given below. In the case of steady two-dimensional l a m i n a r flow, the b o u n d a r y layer equations take the f o r m (Schlichting 1968)  ox  oy  dU  dV  dx  dy  =  ldp  dU  p dx  dy  2  2  0  (C.l) (C.2)  E q u a t i o n ( C . l ) is integrated i n the outer region between y = 0 and y =  6(x),  yielding ± [ * ax Jo  )  V  i  l  l  —  f  s £ ax  +  *  n  p  (C.3)  T h e outer region velocity profiles are assumed to be of the form  ^  = /(r?) + A (r ) ff  ?  (C.4)  APPENDIX  C. MOMENTUM  where /(??) a n d g(n) ordinate n = y/6(x),  INTEGRAL  ANALYSIS  151  are p o l y n o m i a l functions i n t e r m of the dimensionless coand A is a dimensionless pressure gradient parameter defined  by '  1  2  pvU  D  0  "  (C.5)  dx  S u b s t i t u t i n g for U using (C.4) a n d (C.5), equation (C.3) yields, for a cubic polynom i a l profile,  (Cx + 2 A C + A C ) [ U l 6 ) > + 2 ( C + AC )t/ <5A' = 2  2  - (J + ±)  2  3  2  3  0  ^  ( .6) C  where primes denote differentiation w.r.t. i a n d C\, C and C are constants deter2  3  m i n e d by the chosen velocity profile. These constants are listed at the end of this appendix. Similarly, integration of the continuity equation (C.2) through the outer region, gives  d /"*(*)  -T dx Jo  Udy = 0  or, replacing for U {C + KC ){U S)' 4  5  0  + C U A' 5  0  = 0  (C.7)  F i n a l l y , using the F L A R E a p p r o x i m a t i o n a n d neglecting the w a l l shear stress, i n tegration of equation ( C . l ) through the inner region yields  pdx  p  using (C.5) a n d s u b s t i t u t i n g for r$, this gives  A (4j  - 3) = - 6  (C.8)  APPENDIX  C.  MOMENTUM  INTEGRAL  152  ANALYSIS  T h e set of equations (C.6),(C.7) a n d (C.8) were solved numerically for UQ, 6 a n d A , using a R u n g e - K u t t a routine, subject to the i n i t i a l conditions  U {0)  =  U.  6{0)  =  H - h  0  (C.9)  A similar set of equations is obtained when a quartic p o l y n o m i a l instead of a cubic is chosen for the velocity profile.  C.2  Turbulent Flow  T h e formulation of the turbulent flow p r o b l e m proceeds i n essentially the same fashion, u s i n g the t i m e averaged form of the b o u n d a r y layer equations . 1  U ^ + V— dx dy dU  -  - - ^ pdx  - — ( — p dy ^ dy  uv) P>  (CIO)  uv  dV  <te + a ?  , =  < - >  0  c  n  T h e a d d i t i o n a l turbulent or R e y n o l d s stress terms, —puv w h i c h appears i n equat i o n (C.10) is modelled using the well k n o w n eddy viscosity concept. T h i s assumes that the mean turbulent stress is, i n analogy to l a m i n a r stresses, p r o p o r t i o n a l to the m e a n velocity gradient, i.e.  - puv - pu  t  dU  (C12)  ——  dy  where u , the apparent or eddy viscosity, is a coefficient for the turbulent diffusivity t  of m o m e n t u m . A d d i t i o n a l assumptions are required to determine u . t  The rationale for the time averaging procedure and turbulence modelling are discussed in Chapter 4. 1  APPENDIX  C. MOMENTUM  INTEGRAL  ANALYSIS  153  Since we observed i n section 3.5 that the separated shear layer appears to have a structure s i m i l a r to a 2-D free m i x i n g layer i n the region extending from separation to about half the length of the separation bubble, P r a n d t l ' s free shear layer model for u was used over the first half of bubble. T h e m o d e l , w h i c h assumes u to be t  t  constant across the shear layer, takes the form u = ab{Ui - U ) t  (C.13)  2  where a is an e m p i r i c a l constant, b a characteristic w i d t h , and U\ — U  2  is the  velocity difference across the layer. In the case of a two-dimensional m i x i n g layer, w i t h Ui = UQ a n d U = 0 , Schlichting (1968) defines a w i d t h 6 .i measured between 2  0  the positions where ( u / E / ) = 0.1 and (u/U) 2  = 0 . 9 , and gives a = 0.014. Hence  2  u = .0146 .i^o t  0  T h e rate of growth of the shear layer is given by dbn  —r ax  1  i  = .098  or, after integration, 6 .i = .098(x + x ) 0  0  Therefore u = .00137(x + x )U t  0  (C.14)  0  where the v i r t u a l o r i g i n , x , is determined from 0  „  _  Xn  —  M o ) .098  A simple way of accounting for the experimentally observed behaviour of the separated shear layer i n Section 3.5, is to apply equation (C.14) for 0 < x/x  r  < 0.5  APPENDIX  C. MOMENTUM  INTEGRAL  a n d then "freeze" the value of v for x/x t  r  154  ANALYSIS  > 0.5. Because x  r  is not k n o w n a  priori,  an iterative procedure is required. U s i n g a modest under-relaxation (0.9), a solution was obtained after 3 to 6 iterations, depending how good—or bad—the i n i t i a l guess for x was. A convergence r  criterion of less t h a n 0.5% change i n x between two successive iterations was used. T  C.3  Results  A c r i v o s & Schrader (1982) noted, for l a m i n a r flow, that the non-dimensionalisation of the b o u n d a r y layer equations by the Reynolds number leads to a reattachment length w h i c h w i l l always be p r o p o r t i o n a l to the Reynolds number. T h i s expected linear v a r i a t i o n is clearly demonstrated i n F i g u r e C . 2 , w h i c h shows the predictions of the present m e t h o d a n d other more detailed calculations. T h e m o m e n t u m integral m e t h o d gives, w i t h a fourth order p o l y n o m i a l , results w h i c h compare well w i t h those obtained from finite difference solutions. F i g u r e C . 3 shows the effect of varying the expansion ratio on reattachment length (note the non-dimensionalisation by R e ) . For ER > 2, the results show a decrease i n reattachment length w i t h increasing ER  i n agreement w i t h the predictions of  A c r i v o s & Schrader. T h e i r results, however, show a reversal of this trend for ER  <  2. T h e y a t t r i b u t e d this to larger w a l l shear stresses associated w i t h the smaller expansion ratios. T h e integral method cannot be expected to represent this effect since it assumes the w a l l shear stress to be negligible. T h e calculated pressure d i s t r i b u t i o n for a 5:2 expansion ratio is shown in F i g u r e C . 4 . A g a i n reasonable agreement is obtained, apart from the first part of the curve where the differences are probably due to the omission of the w a l l shear stress.  APPENDIX  C. MOMENTUM  INTEGRAL  155  ANALYSIS  30  25  A  20  15  10  200  F i g u r e C . 2 . V a r i a t i o n of reattachment length w i t h Reynolds number.  0.25  Legend  0.20  TIL  A  CUBIC  X  QUARTIC  •  A& S  0.15  •  R  0.10  0.05  0.00  ER  F i g u r e C . 3 . Effect of expansion ratio o n reattachment length.  APPENDIX  C. MOMENTUM  INTEGRAL  ANALYSIS  0.30 -i  X/Xr  F i g u r e C . 4 . Pressure d i s t r i b u t i o n for a 5:2 expansion ratio.  156  APPENDIX  C. MOMENTUM  INTEGRAL  ANALYSIS  157  Turbulent flow calculations were carried out over a range of expansion ratios, and the resulting reattachment lengths, w h i c h are independent of R e y n o l d s number, are shown i n F i g u r e C . 5 compared w i t h experimental results available i n the literature. T h e agreement is surprisingly good, considering the sweeping assumptions made i n the integral m e t h o d and the eddy viscosity model. Figure C . 6 shows the calculated pressure d i s t r i b u t i o n for ER = 1.25 compared to the experimental results of A d a m s et al. (1984). T h o u g h the pressures recover to a s i m i l a r value at reattachment, the discrepancy between the two distributions is rather large.  T h i s confirms the inadequacy of the F L A R E a p p r o x i m a t i o n for  turbulent flows w i t h large backflow velocities ( A d a m s et al. backflow velocities of the order of  C.4  reported m a x i m u m  0.2U}.  Closing Remarks  T h e a p p l i c a t i o n of the m o m e n t u m integral analysis to the sudden expansion flow was intended to be the first step towards the application of this method to the blunt plate p r o b l e m . For this type of external flow an iterative viscous-inviscid matching procedure is required. In the classical m a t c h i n g scheme, the displacement thickness is specified for an inviscid c a l c u l a t i o n , and the resulting pressure d i s t r i b u t i o n . T h i s scheme was applied to the b a c k w a r d facing step flow, but convergence could not be obtained. It was found subsequently that a similar problem has been encountered in m a n y studies involving separation, and that an inverse viscous-inviscid interaction procedure s h o u l d be adopted for such problems (see W i l l i a m s 1985). Because of time constraints, this procedure was not t r i e d .  APPENDIX  C. MOMENTUM  INTEGRAL  ANALYSIS  158  10  x  - e — e - -G  8-  6-  Legend  I  X  •  O PRESENT THEORY X DURST & TROPEA 1981 •  MOHSEN 1967  8  WAUSCHKUHN 4 RAM 1975  S ETHERIDGE k KEMP 1978  2-  X SMYTH 1979  1.2  1.4  1.6  •  WESTPHAL «t al. 1984  9  ADAMS •) al. 1984  1.8  —1 2.2  ER F i g u r e C . 5 . Effect of expansion ratio on reattachment length for turbulent  flow  ( D a t a of M o h s e n and W a u s c h k u h n & R a m compiled by D u r s t & Tropea (1981). 0.25-  0.4  0.6  X/Xr F i g u r e C . 6 . C o m p a r i s o n of calculated and experimental pressure d i s t r i b u t i o n for turbulent  flow.  APPENDIX  C. MOMENTUM  INTEGRAL  ANALYSIS  Table C . l . Integral constants for cubic velocity profile Constant  c  x  c  2  Integral  rpdrj  3  c  4  c  5  Value 17/35  1 / fgdn Jo  19/1680  f g dn 2  1/1680  Cfdr, Jo  5/8  l  C  term  Jo  Jo  9dn  1/48  159  Appendix D  Modified k - e M o d e l  T h e k-e m o d e l does not account for streamline curvature effects. Modifications to the m o d e l w h i c h attempt to take into account these effects have led to improved predictions of r e c i r c u l a t i n g flows i n some cases. T w o such modifications were used in this study, a n d are reviewed here.  Streamline Curvature Modification E x p e r i m e n t a l observations (see B r a d s h a w 1973) show that turbulent shear stresses a n d the degree of anisotropy between the n o r m a l stresses are very sensitive to streamline curvature. T h e k-e model does not, i n its "standard" form, reflect this sensitivity. Leschziner a n d R o d i (1981) developed an ad hoc modification, based on an algebraic stress m o d e l , w h i c h attempts to take these effects into account. isolated the effect on the coefficient C  M  They  of extra strain rates due to curvature and  arrived at the following correction:  (D.l)  160  APPENDIX  D.  MODIFIED  K-e  MODEL  161  where U is the velocity tangential to the streamline, n is the n o r m a l coordinate, s  and R  c  is the radius of curvature. A n a r b i t r a r y lower l i m i t of 0.025 on  posed to ensure, for physical realism, that C  M  was i m -  does not take very s m a l l or negative  values.  Preferential Dissipation Modification T h e second modification was originally proposed by Hanjalic a n d L a u n d e r (1979). It accounts for the preferential influence of n o r m a l stresses (over shear stresses) in p r o m o t i n g the transfer of turbulent energy from large to s m a l l eddies a n d thus the rate of dissipation. T h e modification replaces the t e r m representing "production of generation" i n the e transport equation, i.e.  'dJT  (dU_  dV^  dx,  \ dy  dx  (D.2)  by P! =  C[G - CiVt (  dU  dV  dy  dx J  s  Leschziner a n d R o d i noted that i n recirculating flows the v o r t i c i t y  (D.3) (dU/dy—dV/dx)  i n the above expression, bears no relation to the shear s t r a i n , and, therefore, the modification loses its intended purpose. T o overcome this p r o b l e m , they suggested the use of streamline coordinates to define the stresses. T h i s results i n a corrected t e r m w h i c h reads: P[ = i [ C G x  (D.4)  where C[ = 2.24 a n d C" — 0.8 (giving, consistently w i t h the s t a n d a r d k-e model,  APPENDIX  D. MODIFIED  K-e MODEL  162  C[ — C" = Ci). T h e shear s t r a i n i n the direction of the streamline, S , na  S  ns  = 0.5(S  yy  - S)  sin26 + S  xx  xy  cos 20  is given by  (D.5)  where 6 is the angle between the velocity vector a n d the z-axis, and  S„ = 2 f  (D.6)  d  Syy  =  x  dV 2 — W  (D.7) dV  e  n  ,  Appendix E  Wall Function  Treatment  T h e k — e turbulence m o d e l neglects low R e y n o l d s number effects w h i c h are important i n the v i c i n i t y of solid boundaries. A c o m p u t a t i o n a l l y efficient way of accounting for these effects is to use w a l l functions to determine the near w a l l turbulent kinetic energy a n d its rate of dissipation. T h e w a l l function m e t h o d , due to L a u n d e r & S p a l d i n g (1974), is based on two assumptions: first, the flow i n the v i c i n i t y of a solid b o u n d a r y behaves locally as a one dimensional Couette flow; secondly, the near w a l l turbulence characteristics are those associated w i t h the inertial sublayer. Following L a u n d e r a n d Spalding, we consider a g r i d point P adjacent to a wall (Figure E . I . ) , the point P is assumed to be sufficiently close to the wall for the shear stress to be approximately constant a n d equal to the wall shear stress. terms of the non-dimensional distance y , +  y where  V  +  this implies that: (E.l)  ~ 200  +  In  =  and 163  U y lu T  P  (E.2)  APPENDIX  E.  WALL  FUNCTION  TREATMENT  164  t P  VP \  W  \  \  \  \  \  \  \  \  \  \  \  \  \  F i g u r e E . l . Near w a l l control volume  V, = ^  (E.3)  For negligible pressure gradients, the m o m e n t u m equation reduces to: 1, . dU T - (M + Mt) = ~ p dy p  r  w  — —  ,  p  E.4  or, i n non-dimensional f o r m , (  »t\  where  U  +  dU  ,  +  = ^-  ,  (E.6)  T h e near w a l l region can be s u b d i v i d e d into a viscous sublayer, where molecular viscosity is dominant, and an inertial sublayer, where turbulent effects are dominant: — <  1  for  y  — >  1  for  y  Integrating (E.6), we o b t a i n , for y  +  +  +  <  11.63  >  11.63  < 11.63, U  +  = y  +  (E.7)  APPENDIX  When y  +  E.  WALL  FUNCTION  TREATMENT  165  > 11.63, the flow is fully turbulent, and the m i x i n g length argument  (Hinze 1959) gives: Ut = KpU y  (E.8)  T  R e p l a c i n g for fi , and integrating (E.5), we obtain the familiar logarithmic law of t  the w a l l : U  +  = -ln{Ey )  (E.9)  +  /C  where K is v o n K a r m a n ' s constant (/c = 0.4187), and E is a constant of integration (E = 9.0).  Wall shear stress In the inertial sublayer, the flow is assumed to be i n local e q u i l i b r i u m , i.e. the prod u c t i o n and the dissipation rate of turbulent kinetic energy are locally in balance. T h i s implies: dU - utJ—- ~ e dy  7 U J but  ^ p dy  therefore  f/  K  !• p  =  e  VI  =  ^ ~ e dy  2 T  ,„ . (E.10)  ( E  -  n )  (E.12) (E.13)  then m u l t i p l y i n g (E.13) by (E.12) we obtain:  Uj =  (E.14)  R e p l a c i n g for n by (equation 4.12) t  lh = C^p-  e  (E.15)  APPENDIX  E.  WALL  FUNCTION  TREATMENT  166  equation (E.14) gives: U  (E.16)  T  N o w , from equation (E.9) UU  T  -ln(Ey )  (E.17)  +  or, replacing for U and rearranging, T  r  w  = pKc)/*k U/]n{Ey ) 1/2  for  +  y  +  > 11.63  (E.18)  W h e n the node P lies i n the viscous sublayer, an expression for the shear stress is obtained directly from (E.7):  Tw = M — yp  for  y  +  < 11.63  (E.19)  E q u a t i o n s (E.18) and (E.19) are used to evaluate the near w a l l diffusive fluxes i n the m o m e n t u m equations.  Turbulent kinetic energy B o t h the source and generation terms of the /^-equation are modified for near wall computations. T h e w a l l shear stress is given by (see E.4) dU  or, using ( E . 8 ) ,  but r /p w  = U, 2  T = w  therefore  KpU y— dy  dU_U dy  (E.20)  T  L  Ky  (E.21)  APPENDIX  E.  WALL  FUNCTION  F i n a l l y , s u b s t i t u t i n g for dU/dy  TREATMENT  167  from (E.13), U?  (E.22)  Ky  W h e n calculating kp, it is necessary to assign a value for the average rate of energy dissipation over the control volume. Hence, using (E.22), "VP rvp  /  Jo  edy  U u  [VP  =  —dy  Jo  Ky  Jo  Ky Kyi  +  U — In y s  =  +  + constant  K  U ^Mn(£y ) /c s  *  +  N o w using (E.9) a n d (E.16), this expression can be re-written as [VP  /  tdy  c  3/4 3/2 H j t  f /  Jo  Therefore  II-  C  3/4fc3/2r  edxdy = -«  / +  Ax Ay  (E.23)  yp  this expression is used, i n c o m b i n a t i o n w i t h (E.7) or (E.9), to evaluate the source t e r m — pe i n the fc-equation (4.44).  {  -3/4,3/2  *  pc  pC  "* yp  k  \n(Ey )AxAy +  + +  y AxAy  for  y  for  y  +  +  > 11.63 " < 11.63  ( E  -  2 4 )  In a d d i t i o n , the t e r m representing the generation of k is modified to ensure that the shear stress T given b y the w a l l function is used instead of its finite difference W  counterpart. T h e generation t e r m reads (equation 4.46):  APPENDIX  E.  WALL  Near the w a l l dV/dx  FUNCTION  ~ 0, and fj, dU/dy t  TREATMENT  = r,  168  therefore  w  2  dU_ w  dy  (E.25)  Dissipation of T . K . E . F r o m equation E . 1 5 , e is linked to k by  substituting for ^< from equation (E.8), and m a k i n g use of (E.16) gives:  Ky  P  T h i s expression is used directly to evaluate e next to solid boundaries.  (E.26)  

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