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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), Hatfield Polytechnic, 1977 M.Sc, University of London, 1979 D.I.C., Imperial College, 1979 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F D O C T O R O F P H I L O S O P H Y in T H E F A C U L T Y O F G R A D U A T E S T U D I E S Department of Mechanical 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 B R I T I S H C O L U M B I A Ju ly 1987 © Nedjib Dj i l a l i , 1987 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of i M g c U Q . m <-^vf The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date i e p * - 4 , A bstract This thesis presents an experimental study and numerical predictions of the sepa-rated-reattaching flow around a bluff rectangular section. Th i s laboratory con-figuration, chosen for its geometric simplicity, exhibits a l l main features of two-dimensional flow separation w i th reattachment. Detai led turbulent flow measurements of the mean and fluctuating flow field are reported. The measurement techniques used are: hot-wire anemometry, pulsed-wire anemometry and pulsed-wire surface shear stress probes. The separated shear layer appears to behave like a conventional mix ing layer over the first half of the separation bubble, but exhibits a lower growth rate and higher turbulent intensities in the second half. In the reattachment region, the flow is found to be highly turbulent and unsteady. A finite difference method is used, in conjunction wi th 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 hybr id-upwind differencing (HD) scheme, and the bounded-skew-hybr id differencing ( B S H D ) scheme. Laminar flow computations are performed for Reynolds numbers in the range 100 to 325. The 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 hand, are in excellent agreement w i th the experimental results reported i n the literature. Turbulent flow computations using the k — e turbulence model and the B S H D scheme result in a reattachment length about 30% shorter than the present mea-sured value. W h e n a curvature correction is incorporated into the model , a reat-tachment length of 4.3.D, compared to the experimental value of 4.7D, is predicted. The predicted mean flow, turbulent kinetic energy field and pressure distr ibution are in good agreement w i th experimental observations. A n alternative method of analysis, based on the momentum integral technique, is presented. The method is not applied to the blunt-rectangular plate problem, but its use is i l lustrated for the simpler case of the flow in a sudden expansion, and promising results are obtained. i i C ontents Abstract ii List of Figures v i List of Tables ix Nomenclature x Acknowledgements xiii 1 Introduction 1 1.1 Schematic of the Flowfield 3 1.2 Literature Review 5 1.2.1 Exper imenta l Studies 5 1.2.2 Theoretical studies 9 1.3 Scope of the Present Investigation 12 2 Experimental Arrangement and Measurement Techniques 14 2.1 Exper imenta l Faci l i ty and Equipment 14 2.2 Measurement Techniques 18 2.2.1 Pressure Measurements 18 2.2.2 Veloci ty and Turbulence Measurements 18 2.2.3 Wal l Shear Stress Measurements 21 3 Experimental Results 26 3.1 Two-dimensionali ty 26 3.2 Structure of the Separation Bubble 29 3.2.1 Reattachment Length and W a l l Measurements 30 i i i CONTENTS iv 3.2.2 M e a n F low Properties 34 3.3 Structure of the Shear Layer 45 3.3.1 Growth of the Shear Layer 45 3.3.2 Frequency Spectra and Autocorrelat ions 47 3.4 Effect of Angle of Separation 53 4 Mathemat ica l M o d e l 58 4.1 Background 59 4.2 Conservation Equations and Time-Averaging 59 4.3 k — e Turbulence M o d e l 61 4.4 General Transport Equat ion 63 4.5 Boundary Condit ions 66 5 Computat ional Procedure 69 5.1 F in i t e Volume Formulat ion 70 5.1.1 G r i d Layout and Variables Loca t ion 70 5.1.2 Integration of General Transport Equat ion 70 5.2 F in i te Difference Discret izat ion 73 5.2.1 H y b r i d Differencing 73 5.2.2 False Diffusion 76 5.2.3 Skew Differencing 78 5.3 Treatment of Boundary Condit ions 81 5.3.1 Types of Boundaries 81 5.3.2 Special Case: Corner Nodes 86 5.4 Solut ion Procedure 86 5.5 Pre l iminary computations 88 5.5.1 Loca t ion of Inlet and Outlet Boundaries 88 5.5.2 Non-uniform G r i d Arrangement 89 5.5.3 Convergence Cr i ter ion 92 6 Numer ica l Results 93 6.1 Lamina r F low 94 6.1.1 G r i d Independence 94 6.1.2 Effect of Reynolds Number and Comparison w i t h Exper i -ments 96 CONTENTS v 6.2 Turbulent F low 104 6.2.1 Preamble and Effect of G r i d Refinement 104 6.2.2 Predictions and Comparison wi th Exper imenta l D a t a . . . . 109 6.2.3 Effect of Sol id Blockage 120 6.3 Computa t iona l costs : 122 7 Conclusions 124 References 130 A P P E N D I C E S 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 Analysis 148 C . l Formulat ion 149 C.2 Turbulent F low 152 C.3 Results 154 C.4 Clos ing Remarks 157 D Modif ied k-e M o d e l 160 E Wal l Funct ion Treatment 163 List of Figures 1.1 Configurations which exhibit two-dimensional flow separation w i t h reattachment 2 1.2 Schematic of the flow around a blunt plate 4 2.1 Boundary layer w i n d tunnel 16 2.2 M o d e l of blunt rectangular plate used in wind tunnel experiments. . 16 2.3 Pulsed-wire anemometer probe 20 2.4 W a l l shear stress probe 22 3.1 Surface flow visual izat ion patterns 28 3.2 Dis t r ibu t ion of surface forward flow fraction 31 3.3 Reattachment lengths found for various solid blockage ratios 31 3.4 Dis t r ibu t ion of surface pressure coefficient 33 3.5 Dis t r ibu t ion of mean and fluctuating wal l shear stress coefficients. . 33 3.6 M e a n longitudinal velocity profiles 35 3.7 M e a n streamlines deduced from pulsed-wire measurements 37 3.8 F luc tua t ing (r.m.s) longitudinal velocity profiles 39 3.9 Forward flow fraction profiles 42 3.10 Skewness and flatness distributions of fluctuating wa 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 ye. 49 3.14 Dominant frequencies of velocity spectra 50 3.15 Autocorre la t ion of velocity fluctuations along shear layer edge ye. . . 51 3.16 Integral t ime scales deduced from autocorrelation measurements. . . 52 3.17 Effect of separation angle on surface pressure dis tr ibut ion 54 3.18 Reduced pressure distributions 54 v i LIST OF FIGURES v i i 3.19 Var ia t ion of reattachment length and base pressure w i t h angle of separation 57 3.20 Effect of t r ipping the boundary layer on reduced pressure distr ibu-t ion 57 4.1 F low domain 66 5.1 G r i d layout 71 5.2 (a) Locat ion of scalar and velocity cells; (b) Typ ica l control volume. 72 5.3 F in i te difference nodes 76 5.4 (a) F in i te difference computational star; (b,c) Interpolation ranges for skew upwind differencing scheme 80 5.5 Inflow boundary cells 82 5.6 Outflow boundary cells 83 5.7 Symmetry axis cells 85 5.8 Solid wall boundary cells 85 5.9 Cells near plate corner 87 5.10 Computa t iona l domain 89 5.11 T y p i c a l gr id distr ibution 91 6.1 Effect of gr id refinement on reattachment length 95 6.2 Var ia t ion of reattachment length w i th Reynolds number 97 6.3 Streamlines for laminar flow 99 6.4 Veloci ty field for laminar flow 101 6.5 Pressure dis t r ibut ion along top surface of the plate (laminar flow). . 103 6.6 Turbulent flow: effect of grid refinement on computed reattachment length 106 6.7 Effect of gr id refinement on computed velocity profile at x/D — 2 ( P D M computation) 107 6.8 Effect of gr id refinement on computed wall shear stress along top surface of plate ( P D M computation) 108 6.9 Loca t ion of grid points adjacent to the wal l in terms of the wall coordinate y+ ( P D M computation) 108 6.10 M e a n longitudinal velocity profiles ( B S H D computat ion and P W A measurements 110 LIST OF FIGURES v i i i 6.11 M e a n longitudinal velocity profiles ( P D M computat ion and P W A measurements I l l 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 Comparison of computed and experimental pressure distributions. . 118 6.17 Comparison of computed and experimental wa l l shear stress dis t r i -butions 119 6.18 Predicted streamline pattern ( P D M computation) 119 6.19 Effect of blockage on turbulent flow reattachment lengths 121 6.20 Predicted pressure distributions for various blockage ratios 121 6.21 Comput ing time on V A X 11/750 computer as a function of compu-tat ional array size (Turbulent flow) 123 B . l Wake source model 143 B .2 Physica l and transform planes 143 B .3 Comparison of calculated separation streamlines wi th experiment. . 147 B . 4 Calculated pressure distributions for different specified base pres-sures 147 C . l Mode l l i ng of 2-D sudden expansion flow 150 C.2 Var ia t ion of reattachment length w i th Reynolds number 155 C.3 Effect of expansion ratio on reattachment length 155 C.4 Pressure dis t r ibut ion for a 5:2 expansion ratio 156 C.5 Effect of expansion ratio on reattachment length for turbulent flow. . 158 C.6 Compar ison of calculated and experimental pressure dis t r ibut ion for turbulent flow 158 E . l Near wal l control volume 164 List of Tables 2.1 Measurement uncertainties 25 4.1 k — e model constants 63 4.2 Diffusion coefficients and source terms 65 5.1 Linearized source terms 75 A . l Exper imenta l data 137 C . l Integral constants for cubic velocity profile 159 ix Nomenclature AR Aspect ratio (= S/D). an Coefficient in finite difference equation. BR Sol id blockage ratio (= D/H). C Convective flux coefficient. Ci,Ci, C,j. Turbulence model constants. Cf W a l l shear stress coefficient (= T^/^pU^). Cp Pressure coefficient (= (p — Poo ) / f p ^ < L ) -C p * Reduced pressure coefficient (= ( C p — CPmin)/(l — CPmin)). D Pla te thickness; also diffusive flux coefficient. E Integration constant in logarithmic law of the wal l . ER Channel expansion ratio. Ex,Ey G r i d expansion ratio. F Tota l flux coefficient. FT Flatness factor of T. f Frequency. G Generation of turbulent kinetic energy. H Wind- tunnel /channel height. h Step height. k Turbulent kinetic energy; also interpolation factor in skew differencing. Lu, Ld Distance from recirculation zone to upstream and downstream boundaries. P ( r ) Probabi l i ty density function of r. Pe Ce l l Peclet number. p Static pressure. p* Pressure (= p + \pk). Re Reynolds number (= TJ^Dju). Ruu Autocorre la t ion of u . S Span of wind-tunnel model . x NOMENCLATURE ST t Ui,U,V u,v •^i j i y xr y+ z = x + iy a A x , A y Sx, 6y e 4> r 7 AC A A* v T P f = i + in Source term in transport equation. Coefficient of linearized source term. Skewness factor of r . Integral timescale. T i m e . M e a n velocity in x,-, x , and y directions. Cartesian coordinates. Time-mean reattachment length. W a l l coordinate (= yUT/u). Complex physical plane. Angle of separation. Computa t iona l cell dimensions. Kronecker delta (= 1 for i = j and = 0 otherwise). G r i d spacing. Dissipat ion rate of turbulent kinetic energy. General variable. Diffusion coefficient for <f> variable. Forward flow fraction, von K a r m a n constant. Pressure gradient parameter. Dynamic viscosity. Kinemat ic viscosity. Shear stress; also time lag. Non-dimensional stream function (= / (U/Uoo) diy/D)). Jo F l u i d density. Constants in k and e equations. Complex transform plane. NOMENCLATURE x i i Sub / 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 Value at separation. t Turbulent value. w Wa l l value. oo Free-stream value. ( )' R.m.s . value of fluctuating quantity. ( ) Time-averaged value. A cknow ledgements I would like to express my gratitude to my supervisor, Professor I. S. Gartshore, who remained throughout the course of this study a st imulat ing source of guidance and encouragement. Working wi th h i m has been a pleasure and a privilege. Next , I would like to express my sincere appreciation to Professor M . Salcudean for the interest shown in this work. Her enthusiasm and valuable suggestions con-t r ibuted much to the progress of this research. I would also like to thank Professor G . V . Parkinson for his helpful suggestions, comments and discussions. Thanks are due to the technicians of the Department of Mechanical Engineering. Wi thou t their expertise and ingenuity, many aspects of the experimental part of this work would not have been possible. Las t , but most of a l l , I would like to thank Anne for her loving support and her patience. x i i i Chapter 1 Intro duction Separated-reattaching flows, typical of flows around bluff bodies, occur in a large va-riety of environmental and engineering situations. The recirculating flow regions— known as separation bubbles—encountered in these flows have a significant impact on the performance of, for example, airfoils at higher angles of attack, turbine blades, diffusers and combustors. Separated flows determine, to a large extent, the drag of road vehicles and are the dominant feature of atmospheric flows over bui ld-ings, fences and hi l ls . They are also a cr i t ical factor in 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 th reattachment has long been recog-nized, and much progress has been accomplished over the last twenty years, many aspects of these flows—particularly in the turbulent flow regime, remain poorly understood, because they are difficult to measure or predict. In an effort to isolate those flow features of fundamental importance, a num-ber of laboratory geometries have been devised to generate two-dimensional sepa-rated reattaching flows. Some of these geometries are shown in Figure 1.1. The 1 CHAPTER 1. INTRODUCTION 2 ^7? ^77 777777777, A '777// Backward-facing step Forward-facing step / / / / / / / / / / / / / / / / Elunt rectangular p late Figure 1.1 (a-g). Configurations which exhibit two-dimensional flow separation wi th reattachment CHAPTER 1. INTRODUCTION 3 flow around the blunt rectangular section (Figure 1.1 g) is one of the simplest two-dimensional recirculating flows, yet it exhibits all the important characteristics of separated reattaching flows. It combines several of the advantages of the other geometries: fixed separation point , single pr imary recirculation zone and simple upstream boundary conditions which make it ideal as a test case for numerical methods. In addi t ion it is the simplest shape geometrically. The present work is a detailed experimental and computat ional study of the flow around a blunt rectangular section. The 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 The general features of the flow around a bluff rectangular plate are described in this section. The information is largely drawn from the relevant literature which is reviewed in the next section. The ma in characteristics of the flow around a rectangular plate are perhaps best described by breaking up the flowfield into several zones, each having distinct dominant features. A schematic view of the flow is shown in Figure 1.2. When the oncoming irrotat ional flow (I) impinges on the front face of the plate, a boundary layer (II) develops on either side of the stagnation point. Due to highly favourable pressure gradients, this boundary layer remains th in up to the sharp corner where it separates and forms a free shear layer (III) w i th a large streamline curvature. The separated shear layer is in i t ia l ly 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) /—(VI) I Irrotatlonal flow k II Boundary layer III Free shear layer IV Recirculating flow region V Reattachment region VI Relaxing shear layer and redeveloping boundary layer Figure 1.2. Schematic of the flow around a blunt plate. entraining fluid from bo th the "outer" irrotat ional flow and the "inner" recirculating flow regions. In the reattachment zone ( V ) , the shear layer curves towards and interacts strongly w i t h the wal l to which it eventually reattaches. Par t of the flow at reattachment is deflected upstream into the recirculation zone ( IV) , to compensate for the fluid drawn out by entrainment; the rest is deflected downstream into the recovery zone (VI) where a new boundary layer develops and merges w i t h the outer shear layer. The flow in the reattachment zone is characterized by large pressure gradients, low mean velocities, very large local turbulent intensities and instantaneous flow reversals. A n important length scale of this flow is the reattachment length xr. This length, which is a measure of the extent of the separation bubble, is defined as the distance from separation to the point of zero mean wal l shear stress. CHAPTER 1. INTRODUCTION 5 F r o m this schematic description, it is clear that the flow around a bluff plate, though it is one of the simplest separated-reattaching flows, is quite complex. The high levels of turbulent intensities combined wi th fluctuations in flow direction make measurements in the recirculating flow region difficult, and conventional measure-ment techniques, such as hot-wire anemometry, are of l imi ted use there. 1.2 Literature Review 1.2.1 Experimental Studies In the last decade, experimental research in separated-reattaching flows has been greatly st imulated by the development of instruments suitable for measurements in recirculating flows, especially the laser-Doppler and pulsed-wire anemometers. General reviews of the literature have been undertaken by Bradshaw & Wong (1972) for earlier experimental work, and by Westphal et al. (1984) and Simpson (1981; 1985) for more recent developments. The latter reference is a comprehensive survey of measurement techniques, experimental studies, as well as calculation methods. The following is an overview of the previous experimental studies directly rele-vant to this work. One of the earliest studies of the flow past a blunt rectangular section is due to Roshko &: L a u (1965), who also considered, in the same paper, the flow around various forebody shapes w i th splitter plates. A l though their study d id not involve any flowfield measurements, it gave important insight in the pressure recovery pro-cess in reattaching flows. A n important finding of Roshko Sz L a u was that the CHAPTER 1. INTRODUCTION 6 pressure distributions for al l cases considered collapsed to a single curve when the pressure was normalized by the pressure at separation and the streamwise distance was normalized by the reattachment length. This suggested that some features of separated-reattaching flows are universal, and that the reattachment length is an important characteristic length scale of these flows. Extensive measurements were performed by O t a and co-workers in a series of experiments (Ota & K o n 1974; O t a & Itasaka 1976; O t a & N a r i t a 1978), and important observations were made: • In the separation bubble the pressure in 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 in the middle of the bubble. • Peak turbulent intensities are of the order of 30% of the free stream velocity and occur around reattachment. • Reattachment occurs 4 to 5D downstream from separation. It was also noted that when the plate is heated, m a x i m u m heat transfer rates occur at reat-tachment. Whi 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 to t tube and a hot-wire anemometer respectively. B o t h techniques are unreliable in this k ind of flow, and this is reflected by some inconsistencies reported in the near wall pro-files. Second, an aspect ratio (tunnel wid th /p la te thickness) of about 5 was used CHAPTER 1. INTRODUCTION 7 in the experiments. Three-dimensional effects are, therefore, l ikely to have been important . F ina l l y the procedure for measuring the reattachment length was not explained in sufficient detail . K i y a et al. (1981) also used a hot-wire. Though they present measurements only outside the recirculation bubble, some of these measurements fall w i th in the highly turbulent flow region and should therefore be viewed wi th caution. A com-prehensive set of measurements were made by K i y a & Sasaki(1983). They used directionally sensitive spli t-f i lm sensors in conjunction w i th hot-wire anemometry to measure mean velocities, fluctuating velocities and forward flow fraction. They also presented a few measurements of the turbulent shear stresses around reattach-ment. Important aspects of the unsteady nature of the flow were also reported in this paper and are reviewed later. The effect of Reynolds number was investigated by O t a et al. (1981) who also considered the effect of separation angle. Us ing flow visual izat ion (water w i th alu-m i n i u m powder), they observed three flow regimes: i) The laminar separation-laminar reattachment regime in which the reattach-ment length increases w i th Reynolds 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 in qualitative agreement w i th those made by Lane & Loehrke (1980). Lane & Loehrke found, however, a max imum reattachment length of about 6.5D at Re = 325. Th i s difference is probably due to the larger aspect ratio used by Lane &; Loehrke (11.5 as opposed to 4.55). ii) The laminar 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 th Reynolds number. iii) The turbulent separation-turbulent reattachment regime (Re ~ 2 x l 0 4 ) where the separated shear layer becomes turbulent very soon after separation. The Reynolds number is found to have no effect on the reattachment length in this flow regime. Th i s Reynolds-number-independent regime was also observed by Hi l l ie r & Cherry (1981a). They noted that the flow is essentially Reynolds number independent in the range 3.4 x l O 4 < Re < 8.0 x l O 4 , w i th a weak elongation appearing only when Re < 8.0 x 10 4 . In the same paper, Hi l l ie r & Cherry showed that the flow is very sensitive to grid-generated free-stream-turbulence levels. For example a shortening of the bubble from 4.88D to 2.12D was reported when the free stream turbulence intensity was increased from about 0.1% to 6.5%. The effect of free stream turbulence was also investigated by K i y a &: Sasaki (1983b). They used a rod upstream of the plate to generate the turbulence and obtained results s imilar to those of Hi l l ie r & Cherry. Dz iomba (1985) used wires on the front face of the plate to t r ip the boundary layer just before separation. He found this to have the same qualitative effect as an increase in free stream turbulence. He argued, however, that the shortening of the bubble was mostly due to an effective change in the angle of separation. The unsteady structure of the separation bubble has been the subject of a series of thorough studies by Hi l l ie r & Cherry (1981b), Cherry et al. (1983;1984) and K i y a & Sasaki (1983a). These studies, using a combination of flow visualization, mea-surements of fluctuating surface pressures and a judicious use of pressure-velocity CHAPTER 1. INTRODUCTION 9 correlations, clearly demonstrated that: • The shear layer near separation exhibits a low-frequency flapping mot ion. The mechanisms for this low frequency unsteadiness are not clearly understood. • Downstream from separation, large scale vortices are shed in pseudoperiodic bursts. In an extension of these studies, K i y a & Sasaki (1985a) used condit ional sampling of the velocity field to deduce the structure of the large scale vortices. They concluded that these vortices have a ha i rp in structure. The unsteadiness of the reattachment process seems to be an inherent feature of separation bubbles in al l two dimen-sional geometries. For example, it has been observed in the backward facing step flow (Eaton & Johnston 1982) and in the flow around the flat plate/spl i t ter plate combinat ion (Gartshore &; Savi l l , 1982). Cherry et al. (1984) suggested that it is the large-scale shedding of vort ici ty that causes the m a x i m u m shear layer turbulent stresses and pressure fluctuations to occur in the v ic in i ty of reattachment. 1.2.2 Theoretical studies The 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 Reynolds numbers, an addit ional difficulty is the varied and complex nature of the turbulence field. The prediction of these flows can be attempted by using either zonal modell ing or global modelling; both approaches rely on numer-ical solution methods. In zonal modell ing, one recognizes that different regions of the flow have different dominant features and a computat ional procedure is devel-oped accordingly. A n example of this approach is the viscous-inviscid interaction CHAPTER 1. INTRODUCTION 10 procedure 1 . G loba l models, on the other hand, use the same set of equations for the entire flowfield. Amongst these, computationally intensive finite difference methods which solve the Navier-Stokes equations (in their time-averaged form in the case of turbulent flows) have gained a wider acceptance over the last two decades as a result of the availabili ty of more powerful computers. These methods have been used, w i t h varying degrees of success, to predict a number of recirculating flows. The laminar flow around the blunt flat plate was computed by G h i a &: Davis (1974) who used a finite difference method to solve the Navier-Stokes equations in their streamfunction-vorticity form. Thei r results were subsequently compared by Lane & Loehrke(1980) to their own experimental data and showed a large discrep-ancy. The possibil i ty that the numerical solutions had not attained true convergence was put forward as a possible explanation for this discrepancy. A related case, the laminar flow through a cascade formed by a stack of flat plates, was considered recently by M e i &: Plotk ins (1986). The i r formulation also used the streamfunction-vorticity formulation of the Navier-Stokes equations. B u t second order upwind differencing was used instead of the first order scheme of G h i a & Davis . Though their results cannot be compared directly to the experimental data of Lane & Loehrke, it is interesting to note that they reported similar trends: flow separation was first found to occur at Re ~ 110 and the reattachment length was found to vary linearly w i th Reynolds number up to Re ~ 300. Th i s was the largest Reynolds number for which convergence could be obtained. They noted that the first order differencing scheme resulted in shorter reattachment lengths than the second order scheme. Th i s was due to false diffusion 2 . X A brief account of this method for separated-reattaching flows is given in Appendix C . 2False diffusion, or numerical diffusion, is the truncation error associated with the use of upwind differencing in a discretization scheme. CHAPTER 1. INTRODUCTION 11 Us ing the discrete vortex method, K i y a & Sasaki (1982) carried out an inviscid s imulat ion of the turbulent flow on a blunt flat plate. The simulat ion consisted of t racking elemental vortices which were shed downstream from the separation corner. In order to represent the viscous/turbulent dissipation of vorticity, the circulat ion of elemental vortices was reduced as a function of their age. The model required some empirical input (pressure at separation and mean reattachment length) to determine some free parameters. In general, the predictions of the mean velocity and surface pressure were reasonable, except in the reattachment region. Remarkably, the unsteadiness of the flow was fairly well represented. Furthermore the fluctuating component of the surface pressure, a quantity which cannot be obtained at all w i th the steady state finite difference method, was in tolerable agreement wi th experiments. N o finite difference predictions of the turbulent flow around a flat plate are reported in the literature. M a n y such predictions have, however, been attempted for other geometries, a favourite being the downward facing step. A comprehensive and cr i t ical review of many of these predictions, al l based on the solution of the time-averaged Navier-Stokes equations in conjunction wi th a turbulence closure model, can be found in a recent article by Nallassamy (1987). In the context of this study, it is of special interest to note that, in most com-putations, the equations were discretized using upstream differencing. The gross features of the recirculation zone were in general underpredicted, and discrepancies of up to 30% in the reattachment length were reported. This was par t ly due to false diffusion which is inherent to upstream differencing. M a n y authors, however, at t r ibuted the discrepancies to inadequate turbulence modell ing. Since errors due to modell ing cannot usually be dissociated from numerical errors, it was pointed CHAPTER 1. INTRODUCTION 12 out by Castro (1977), among others, that prolonged arguments about turbulence model deficiencies were somewhat pointless unless false diffusion was reduced to negligible levels. This could be achieved, in principle, by refining the gr id . Bu t this is often impract ical because computing costs increase rapidly w i th the num-ber of computat ional nodes. A n alternative approach is the use of "higher order" differencing schemes such as the "skew-upwind differencing" of Rai thby (1976b) or the "quadratic upstream interpolation" of Leonard (1979). The applicat ion of these schemes to turbulent recirculating flows has, in general, resulted in improved predictions (e.g. Leschziner & R o d i 1981). 1.3 Scope of the Present Investigation The 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 l i t t le reliable quantitative data on the flow wi th in the separation bubble. In part icular , al l available measurements in this region were made wi th instruments which are prone to measurement errors in turbulent recirculating flows. O n the theoretical side, the only turbulent flow computat ion available in the literature is a discrete vortex simulat ion, and though the gross features of the flow are reasonably well reproduced, the predictions of various mean flow quantities are only fair, indicat ing that other approaches are worth exploring. CHAPTER 1. INTRODUCTION 13 The 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 data for comparison and evaluation of numerical predictions. To this end, detailed flowfield and surface measurements were performed using pulsed-wire anemometry as well as conventional hot-wire anemometry. The theoretical study had two objectives: (i) to devise and test a simple calcu-lat ion procedure based on a momentum integral technique, and (ii) to compute the flow using a finite difference method in conjunction w i th a two-equation turbulence model (A; — e). The first of these objectives was met only partially. Encouraging results were, however, obtained for the simpler case of a sudden expansion flow. The computat ional study was carried out successfully for bo th laminar and turbulent flow around a blunt rectangular plate. Chapter 2 Exper imenta l Arrangement and Measurement Techniques In this chapter, the w i n d tunnel and wind tunnel model used for the turbulent flow experiments reported in the next chapter are described. The measurement techniques and related cal ibrat ion 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 boundary layer w i n d tunnel shown schematically in Figure 2.1. This tunnel has a 2.4m wide, 1.6m high, 24.5m long test section, and a max imum design wind speed of 25 m/s . In the velocity range used for the present experiments (4-15 m / s ) , the velocity distr ibution in the empty test section was uniform wi th in 1%, w i th turbulent intensities in the range 0.25-0.4%. 14 CHAPTER2. EXPERIMENTAL ARRANGEMENT AND TECHNIQUES 15 Wind Tunnel Model The model which was constructed for the tests consisted of a rectangular base section w i t h endplates and side extensions. This configuration, shown in Figure 2.2, was selected after prel iminary tests, carried out by Dz iomba (1985) and discussed in the next chapter, indicated that the use of endplates was crucial to the two-dimensionality of the flow. The section had a chord of 800 m m , a thickness D of 89.9 m m (3j")—corresponding to a solid blockage ratio D/H of 5.6%—and a span between end plates of 1000 m m , giving an aspect ratio S/D of 11.1. W i t h the side extensions mounted, the model spanned 2.2 m across the wind tunnel . In addi t ion, a t a i l was attached to the t ra i l ing edge of the model to suppress any periodic vortex shedding which might otherwise have "contaminated" the flow in the separation bubble. The model was mounted at a zero angle of incidence, w i th its front end located about 1.5 m downstream of the nozzle exit. It was held by eight 0.7 m m diameter piano wires, which were fastened to the roof and floor of the wind tunnel . The symmetry of the flow was ensured by adjusting the position of the model unt i l the difference in pressure coefficients at equivalent positions on the top and bot tom surfaces were wi th in 1% of the dynamic head. The base section was made of a luminium, and had a removable plexiglass top. The bo t tom surface of the model had a series of pressure taps at 20 m m intervals along the center line (x-direction), and at 100 m m intervals in the spanwise (z) direction. The angle a at which 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. This was achieved by adding triangular front-pieces to the front face of the model. CHAPTER2. EXPERIMENTAL ARRANGEMENT AND TECHNIQUES 1 honeycomb and 4 screens in 4 « 4 m sellling section Figure 2.1. Boundary layer w i n d tunnel . F igure 2.2. M o d e l of blunt rectangular plate used in w i n d tunnel experiments. CHAPTER2. EXPERIMENTAL ARRANGEMENT AND TECHNIQUES 17 Traverse Mechanism Measurements in the flowfield, using various probes, were carried out using a tra-verse mechanism designed and bui l t specifically for the project. The traverse was mounted on translat ion bearings and guided by two rails for horizontal traversing. Automat ic vert ical traversing was obtained by a lead screw mechanism which was driven by a microcomputer controlled stepper motor. The posit ioning accuracy of the traverse mechanism was 1.5 m m in the horizontal direction, and 0.08 m m in the vert ical direction. The bulk of the traverse mechanism was placed under the floor of the wind tunnel . The only parts protruding into the flowfield were the probe and its support; their interference w i t h the flow was min imal : changes in pressure dis t r ibut ion were less than 0.01 in C p , and the reattachment length, as measured by surface flow visual izat ion remained unchanged when the traverse mechanism was introduced. The r igidi ty of the mechanism was also checked, and no flow induced vibrations were detected. Data Acquisition System A C B M S u p e r P E T microcomputer was used for sampling and processing of velocity and 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, and 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 control of the the traverse mechanism stepper motor. CHAPTER2. EXPERIMENTAL ARRANGEMENT AND TECHNIQUES 18 2.2 Measurement Techniques 2.2.1 Pressure Measurements The static pressure dis t r ibut ion on the surface of the bluff plate was measured us-ing a Barocel differential pressure transducer and a 48-port Scanivalve system. The Scanivalve was mounted inside the model and connected to the 0.5 m m diameter pressure taps by short lengths of Tygon tubing. Reference static and dynamic pres-sures were measured w i th a Pitot-stat ic probe located at a distance 10D upstream of the front face of the bluff body. The pressure transducer had a linear response and d id not require any calibra-t ion . 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), and therefore required frequent zero level checks. The output voltage from the transducer was measured using an integrating voltmeter; integration times of 10 seconds were used for averaging. The uncertainty in the pressure measurements is estimated to be ± 0.03 m m water, corresponding to an uncertainty of ± 0.007 in the pressure coefficient C p . 2.2.2 Velocity and Turbulence Measurements The velocity field measurements were made by traversing hot-wire and pulsed-wire probes at 10 streamwise stations; each traverse consisted of 20 to 34 points. The repeatabili ty of the measurements was usually w i th in 1% for the mean velocities and wi th in 2.5% for the fluctuating velocities. CHAPTER2. EXPERIMENTAL ARRANGEMENT AND TECHNIQUES 19 H o t - W i r e Anemometer ( H W A ) Outside the recirculating flow region, streamwise mean and fluctuating velocities, correlations, and frequency spectra were measured using a hot-wire probe and a D I S A constant temperature anemometer system. The probes were standard D I S A single wire probes, w i t h 5 n diameter, 1.25 m m length platinum-coated tungsten wires. The 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 cal ibrat ion, using K i n g ' s law w i t h an exponent of 0.45, was performed against a Pitot-stat ic probe in low turbulence conditions (u/U < 0.4%), and a digi tal sampling rate of 4 K H z was used for al l measurements. Correlat ion functions and frequency spectra were obtained using an analogue P A R correlator and a frequency spectrum analyser. The error estimates for the hot wire-anemometer measurements are given in table 2.1. Pulsed-Wire Anemometer ( P W A ) The high turbulence intensities and reversed flows encountered in recirculating flow regions require the use of special instrumentation. A pulsed-wire anemometer sys-tem, manufactured by P E L A Flow Instruments, was used in the present study. The general principle of the instrument, originally developed by Bradbury & Castro (1971), is based on the measurement of the flow velocity by t iming the passage of a heat tracer between two points. The pulsed-wire probe, shown in Figure 2.3, consists of two sensor or "receiving" wires, and a th i rd pulsed or "transmitt ing" wire located between the two sensor wires. The central wire is heated periodically by short durat ion voltage pulses; the air passing the wire at that t ime is heated and CHAPTER2. EXPERIMENTAL ARRANGEMENT AND TECHNIQUES 20 convected w i th the local instantaneous flow velocity. The time taken by the heated air to reach one of the two sensor wires, which 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 which detects the tracer. The instrument has two drawbacks. Fi rs t it is relatively large and therefore measurements close to solid walls are not possible. Secondly small scale turbulence may influence the probe response. Th i s was not likely to be a problem in the present investigation since a relatively large scale experimental facili ty was used, as recommended by Bradbury & Castro (1971). The response of the instrument is not linear due to thermal diffusion and viscous wake effects. To take this into account, Bradbury & Castro recommend the use of an empir ical cal ibrat ion fit of the form Sensor wires Pulsed wire Figure 2.3. Pulsed-wire anemometer probe. (2-1) CHAPTER2. EXPERIMENTAL ARRANGEMENT AND TECHNIQUES 21 where U is the flow velocity, A and B are cal ibrat ion constants, and T is the time of flight. The pulsed-wire probes were calibrated, using P E L A software, in a low turbulence flow against a Pitot-stat ic probe. The cal ibrat ion constants were determined by a least square fit procedure, wi th a resulting standard deviation of less than 2%. The cal ibrat ion was frequently checked and found to be very stable. Measurements were performed at a sampling rate of 50 H z , w i th the number of samples taken 5000 in the outer region and 7500 in the shear-layer/recirculating flow region. The accuracy of the pulsed-wire anemometer has been assessed quite thoroughly (Bradbury 1976; Castro & Cheun 1982; Westphal et al 1984), and its performance was found to be comparable to the hot-wire anemometer in regions where both instruments are applicable. It should be noted in this context that, due to electronic noise, there is a lower l imi t of 2% to the turbulent intensities that can be measured w i t h the instrument (Bradbury 1976). The estimated uncertainties quoted in Table 2.1 are those given by Westphal et al (1984). 2.2.3 Wall Shear Stress Measurements The dis t r ibut ion of the mean and fluctuating wall shear stresses, and of the forward-flow-fraction were obtained w i th a pulsed wall-probe which was mounted on a sup-port ing block. The block fitted flush w i th the surface of the model , and could slide along a groove which was cut out along the centerline of the model . The groove was 600 m m long and allowed positioning of the probe anywhere between x/D = 0.2 to 1.8. CHAPTER2. EXPERIMENTAL ARRANGEMENT AND TECHNIQUES 22 The pulsed wall-probe—also a time of flight type of instrument—is a recent development (Wesphal et al 1981; Castro & Dianat 1983) which has evolved from the P W A . The probe, shown in Figure 2.4, consists of an array of three wires mounted about 0.05 m m above a plug. The spacing between the wires is 0.75 m m ; the sensor wires are 2 m m in length w i th a 2.5 /z diameter, and the pulsed-wire is 3 m m in length w i th a 9.0 fi diameter. The measurement principle is based on the assumption that, in the proximity of a wal l , the instantaneous velocity profiles remain s imilar and scale on the wal l shear stress. The electronics used for the P W A was also used for the wall probe, and a cal ibrat ion function of the same form as equation (3.1) was used. The calibration procedure posed a few problems. Because of the non-linearity of the calibration function, cal ibrat ion of the probe in a turbulent boundary layer would lead to errors, even for moderate turbulent intensities. Nevertheless turbulent flow calibration Sensor wires Pulsed wire Figure 2.4. Wa l l shear stress probe. CHAPTER2. EXPERIMENTAL ARRANGEMENT AND TECHNIQUES 23 facilities are easier to set up. Westphal et al. (1981) calibrated their probe in a turbulent flow apparatus and devised a correction procedure to compensate for the "nonlinear averaging error". A s imilar procedure was adopted in a recent study by Ruder ich &; Fernholz (1986). The nonlinear averaging errors can be avoided by cal ibrat ing the probe in a laminar flow, which is the procedure used by Castro & Dianat (1983) and Westphal et al (1984), who had high aspect ratio laminar channel facilities. Since no such facility was available, the cal ibrat ion for the present measurements was carried out in a two-dimensional laminar boundary layer generated on a flat plate. A slightly favourable pressure gradient was found to be necessary to pre-vent early t ransi t ion to turbulence. The near wal l velocity profiles were measured using a hot-wire anemometer, and the wal l shear stress was deduced from the re-sult ing gradient dU/dy at the wal l . Th i s cal ibrat ion procedure became increasingly uncertain w i t h higher wal l shear stress because of the decreasing thickness of the boundary layer at the highest velocities. Consequently, the cal ibrat ion of the probe was done in the range 0.0-0.10 N / m 2 . Th i s range covers adequately the max imum mean shear stresses encountered in the separation bubble (~ 0.09 N / m 2 ) , however instantaneous shear stresses larger than the upper cal ibrat ion l imi t are encountered because of the the highly turbulent nature of the flow. These were determined by assuming that the cal ibrat ion curve extrapolated correctly to these values. It is worth not ing that Castro Sz Dianat recommend a cal ibrat ion function of the form T = A/T + B/T2 + C/Ts for a better fit to values of r in excess of 0.3 N / m 2 . Since this is higher than the largest value used for the cal ibrat ion, the original calibration CHAPTER2. EXPERIMENTAL ARRANGEMENT AND TECHNIQUES 24 function was retained. Considering the large uncertainties, due mostly to a narrow cal ibrat ion range and estimated to be ± 25% in r and ± 30% in r ' , the shear stress measurements presented here should be viewed wi th caution. The measurements of the surface forward-flow-fraction 7 were also made using the wa l l probe. A direct measure of 7 is given by the ratio of positive samples to the total number of samples. The number of positive samples was obtained by appropriately setting the measurement range to include positive samples only and -7 was then evaluated by N° of -t-ve samples Tota l n° of samples - n° of "zero" samples where the number of "zero" samples corresponds to heat tracers which fail to trigger either sensor wire. CHAPTER2. EXPERIMENTAL ARRANGEMENT AND TECHNIQUES Table 2.1. Measurement uncertainties Measured quantity Uncertainty estimate Cp (Pressure transducer) ± 0 . 0 0 7 U/Uoo (Hot-wire) ± 2 % (for u/U < 0.3) U/Uoo (Pulsed-wire) ± 2 to 5% Vut/Un (Hot-wire) ± 4 % (for u/U < 0.3) y/vf/Uoo (Pulsed-wire) ± 2 to 5% (for u/U > 0.05) 7 (Wal l probe) ± 0 . 0 2 Cf (Wal l probe) ± 2 5 % yfcf (Wal l probe) ± 3 0 % xr ± 0 . 1 2 } X ± 0 . 0 1 5 2 ) y ±0 .0012? Chapter 3 Exper imenta l Results The results of turbulent flow experiments are presented and discussed in this chap-ter. F i rs t , the results of surface flow visual izat ion experiments carried out to ex-amine the two-dimensionality of the flow for different model configurations are pre-sented. In the second section, wal l and flowfield measurements of various mean and fluctuating quantities are presented, comparisons are made w i t h data available in the literature, and some conclusions about the structure of the separation bubble are drawn. Further information about the structure of the free shear layer, deduced from frequency spectra and velocity autocorrelation measurements, are presented next. F ina l ly , the results of experiments to determine the effect of varying the angle of separation are discussed. 3.1 Two-dimensionality Experimental ly , a reattaching flow region is difficult to make convincingly two-dimensional: for example, most studies reviewed in Chapter one report a curved reattachment line which is symptomatic of three-dimensional effects. The main 26 CHAPTER 3. EXPERIMENTAL RESULTS 27 factors affecting two-dimensionality are aspect ratio (AR = S/D) and end condi-tions. Reattachment length data compiled from various sources by Cherry, Hil l ier &; La tour (1984), hereafter referred to as C H L , shows that the reattachment length increases gradually w i th aspect ratio. Th i s effect becomes negligible for aspect ratios greater than about ten, which is the m i n i m u m aspect ratio recommended by de Brederode (1975), in conjunction wi th the use of end plates, to avoid three-dimensional effects. Though 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 and Sasaki (1983), hereafter referred to as K S , d id not use endplates i n their experiments. In order to assess the influence of endplates and end-conditions on two-dimensionality, Dziomba(l985) carried out a series of tests in the U B C aerody-namics laboratory. Us ing pressure measurements and the surface flow visualizat ion technique of Langstone & Boyle (1982), Dz iomba investigated the following config-urations for the degree of two-dimensionality: i) Base model (AR = 11.1). ii) M o d e l w i th endplates (AR = 11 . l ) . iii) M o d e l w i th endplates and side extensions (AR — 11.1). iv) M o d e l w i th endplates mounted on side extensions (AR = 24.4). The surface streamline pattern for configuration 1 is shown in Figure 3.1(a) and reveals a complex three dimensional structure. Dz iomba observed a highly curved reattachment line associated wi th two leading edge corner vortices. Large spanwise CHAPTER 3. EXPERIMENTAL RESULTS 28 X/D X / D X/D (a) (b) (c) Figure 3.1. Surface flow visualization patterns obtained by Dziomba (1985) for three configurations: (a) base model; (b) model wi th endplates; (c) model wi th endplates and side extensions. CHAPTER 3. EXPERIMENTAL RESULTS 29 variations in the pressure distr ibution were also reported. Th i s configuration yielded the longest reattachment length (xr = 5.7D). The addi t ion of endplates (configuration 2) reduced the centerline reattachment length to xT = 5.2D. Three-dimensional effects can s t i l l be seen (20% variation in xr in the spanwise direction) though they are much less pronounced because the corner vortex systems are weaker and the ventilation process which takes place at the extremities of the model is inhibi ted by the endplates. A much improved si tuat ion results from the addit ion of side extensions (configu-rat ion 3). G o o d spanwise uniformity and a straight reattachment line are obtained over the largest part of the model. The reattachment length is reduced further to i r = 4 .7D. Increasing the aspect ratio from 11.1 to 24.5 resulted in very similar flow characteristics. Configuration 3 was therefore retained for the present investi-gation since it resulted in a nominally two-dimensional separation region extending ±3 .5Z) , or about ± 0 . 7 5 x r , either side of the midspan. 3.2 Structure of the Separation Bubble Pre l iminary experiments indicated that both reattachment length and pressure dis-t r ibut ion remained unchanged over the Reynolds number range 2.5 x 10 4 to 9.0 x 10 4 confirming the observation of Hi l l ier & Cherry (1981a) that the flow is essentially Reynolds number independent for Re ~ 2.7 X 10 4 . A l l present experiments were conducted at a Reynolds number of 5.0 x 10 4 ± 0.1 x 10 4 . CHAPTER 3. EXPERIMENTAL RESULTS 30 3.2.1 Reattachment Length and Wall Measurements The mean reattachment point, defined as the location where the time-averaged wall shear stress vanishes, can be determined from the location where the forward-flow-fraction1 7 is equal to 0.5 as was shown by Westphal et al. (1981). The surface probe was used to measure the forward-flow-fraction dis tr ibut ion ( 7 vs. x) and the results are shown in Figure 3.2. The reattachment length deduced from these measurements, xr/D = 4.7, is in very good agreement w i th the value obtained from surface flow visual izat ion. It is interesting to note that 7 is never equal to zero; that is the flow is never fully reversed even in the middle of the separation bubble. Furthermore there is a broad region, extending about 2.5.D downstream of the mean reattachment point , where the near-wall flow can be moving instantaneously in either downstream or upstream direction. Th i s suggests that the instantaneous reattachment point wanders up and downstream in a region surrounding the time-averaged reattachment point; this view is consistent w i th 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 id th is then about 2AD or 0.5a; r. A comparison of the reattachment length obtained in the present experiments and in various other studies is shown in Figure 3.3; only data obtained in low free stream turbulence is included. The reattachment length is seen to decrease wi th increasing solid blockage, wi th 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 r + 00 1= / P(rw)dTw/ / P{TW)CLTW JO J-co where P(TW) is the probability density of rw. CHAPTER 3. EXPERIMENTAL RESULTS 31 X/Xr 0 0.25 0.50 0.75 1 1.25 1.50 1.75 2 0.75 0.25 • / • 7 ^ / • / • / • / • • \ • • : / • / « 1 ' I ' I ' 1 ' 1 I 1 0 2 4 6 8 10 X/D Figure 3.2. Dis t r ibut ion of surface forward flow fraction. Q C 03 E o "5 5<> 3-A Legend O Kiya & Sasaki 1983 A Latour & Hllllar 1980 o Cherry el al. 1984 V Chang 1982 • Roshko & Lau 1965 9 Ota & Itasaka 1976 • Present Experiments ffi de Brederode 1975 O 1 1 1 1 (  0 1 2 3 4 5 6 Percentage Solid Blockage BR Figure 3.3. Reattachment lengths found for various solid blockage ratios (data compiled by Cherry et al. 1984). CHAPTER 3. EXPERIMENTAL RESULTS 32 low compared to the trend of the remainder of the data. Th i s may be attr ibuted to slight differences in free stream turbulence, but it should also be pointed out that the nominal 0% blockage of K S was obtained by fitting the w i n d tunnel w i th false boundaries which might not have eliminated al l wal l effects. The mean pressure coefficient dis t r ibut ion is shown in Figure 3.4 and is found to compare well w i t h the Cp values reported by C H L . B o t h distributions are uncor-rected for blockage which accounts for the small differences (BR = 3.79% for C H L as opposed to 5.6% in the present experiments). We note that after a small in i t ia l dip of about 0.05 in Cp the pressure remains approximately constant up to about x = 0 .5x r , a rapid recovery takes place thereafter. This recovery continues beyond the reattachment point up to x ~ 1.4x r. Figure 3.5 shows the distr ibution of the mean and fluctuating shear stress coef-ficients. The posi t ion at which the mean shear stress is equal to zero corresponds to the posit ion where 7 = 0.5 as ant ic ipated 2 . Cj attains a m i n i m u m value of —2.4 x 10~ 3 at 1 = 0.63x r and rises rapidly afterwards. The r.m.s. value of the shear stress coefficient increases steadily throughout the reversed flow region and at-tains a plateau (Cj ~ 2.4 x 10~ 3) in the reattachment zone. The magnitudes of both mean and fluctuating shear stress coefficients are higher by a factor of (about) two than those encountered in the backward facing step geometry (see, e.g., Westphal et al. 1984). 2 We note that the equivalence of TW = 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 forward-flow-fraction is equal to about 0 . 7 at the zero shear stress location associated with a secondary reseparation . CHAPTER 3. EXPERIMENTAL RESULTS 33 o - 0 . 2 - 0 . 4 - 0 . 6 -- 0 . 8 -1 _^  j • • <• 7 • 0 0.25 0.50 0.75 1 1.25 1.50 1.75 2 X/Xr Figure 3.4. Dis t r ibut ion of surface pressure coefficient: # , present measurements; V , measurements of Cherry et al. (1984). X/Xr 0 0.25 0.50 0.75 1 1.25 1.50 1.75 2 4.0 O M — 2.0 0.0 - 2 . 0 -- 4 . 0 o o ° n 0 O a"°" o o ° o ° o o • • • • • o o • • • • • • • • 10 0 2 4 6 8 X/D Figure 3.5. Dis t r ibu t ion of wall shear stress coefficients: • , mean; o > r.m.s. CHAPTER 3. EXPERIMENTAL RESULTS 34 3.2.2 Mean Flow Properties The mean velocity profiles are presented in Figure 3.6(a) and(fc) for the reversed and reattached flow regions respectively. Hot-wire data are presented only in regions where 7 > 0.9 or 7 < 0.1. A l l velocity data are normalized by the free stream ve-locity UQO, and the or igin for y is the plate surface. The mean separation streamline is also plotted for reference. Backflow velocities as large as OSU^ are encountered in the reversed flow region. Th i s value is about 50% larger than that in the backward facing step geometry (Westphal et al. 1984) which accounts part ly for the higher wal l shear stresses presented earlier. We also note that the backflow velocities remain relatively high in the near-wall region. Loca l max ima in the velocity profiles are also apparent in the first half of the bubble. The velocities further away from the wal l remain higher than the free stream as a result of blockage effects. The recovery region profiles show an inflexion point as a result of the merging of a new boundary layer w i th the reattaching shear layer. We note also a rapid increase of the near-wall velocities w i th downstream distance. The mean streamline pattern obtained by integration of the velocity profiles is shown in Figure 3.7. The dimensionless streamfunction is defined as ry/D * = / (U/U^/diy/D) Jo The time-averaged streamlines are somewhat deceptive in that they disguise the highly unsteady nature of the reattachment process, i l lustrated earlier by the forward-flow-fraction dis tr ibut ion. CHAPTER 3. EXPERIMENTAL RESULTS 38 Figure 3.8 shows the r.m.s. longitudinal velocity profiles. The consistently higher values of the P W A data are due to electronic noise which makes the in-strument unsuitable for measurements of turbulent intensities smaller than about 2% (Bradbury 1976). The H W A data on the other hand appear reasonable only when the forward-flow-fraction 7 is outside the range 0.1 < 7 < 0.9 and when the local intensity is below 20%. The results show the same overall trends as the split-f i lm sensor results of K S , w i t h the notable exception that the turbulent intensities obtained in the present measurements are generally higher by 15 to 20% 3 . The peak turbulence intensity ujU^ is observed to occur upstream of reattach-ment, probably as a result of the higher velocity difference across the shear layer. After a gradual decrease in the reattachment region, the m a x i m u m turbulent inten-sities fall rapidly i n the recovery region. We observe that the turbulent intensities decay fairly rapid ly in the outer region of the shear layer, but remain high next to the wal l . These near-wall intensities decrease much more slowly w i th streamwise distance than the m a x i m u m intensities. It is interesting to note that the peak turbulence intensity of about 30% mea-sured i n this flow is substantially higher than the value of about 18% found in the plane mix ing layer (Wygnansky & Fiedler 1970). A more appropriate comparison, however, can be made by using the velocity difference across the shear layer to normalize the present data. The max imum value of u/(Umax — c T m i n ) varies from about 22% at x/D = 3 to about 25% at x/D = 5. This is s t i l l higher by 20 to 40% than the plane mix ing layer value. 3Young (quoted by Simpson 1985) indicates that fluctuating quantities that are up to 30% too low can be obtained with hot film probes; this would account for the discrepancy between the present measurements and those of KSA. Figure 3.8(a). Fluctuat ing (r.m.s.) longitudinal velocity profiles: O , pulsed-wire data; • , hot-wire data; , separation streamline ( * = 0); to CO Figure 3.8(b). Fluctuat ing (r.m.s.) longitudinal velocity profiles: O , pulsed-wire data; • , hot-wire data. O CHAPTER 3. EXPERIMENTAL RESULTS 41 The forward-flow-fraction profiles, measured using the P W A , are shown in Figure 3.9 together w i th the loci of 7 = 0.1,0.5 and 0.9. The data points at y/D = 0 are from the surface probe measurements. Similar ly to the wal l data , the posit ion of 7 = 0.5 is found to correspond to U = 0. A t x/D = 1 and 2, 7 reaches a m i n i m u m slightly away from the wa l l , but further downstream the m i n i m u m is reached at the wal l ; this is consistent w i t h the location of the m a x i m u m backflow velocities in Figure 3.6. In fact the shapes of the mean velocity and forward-flow-fraction profiles are s t r ikingly s imilar . Further insight into the structure of the separation bubble can be gained by examining the skewness, ST, and the flatness factor, FT, of the wa l l shear stress. These are the normalized th i rd and fourth moments of the probabil i ty density, PT, and are defined by The skewness and flatness factor distributions are shown in Figure 3.10, and they exhibit the same trends as reported by Ruder ich and Fernholz (1986) for the flat plate/spl i t ter plate geometry. B o t h skewness and flatness factors, are found to depart markedly from their respective Gaussian values of 0 and 3. High 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 associ-ated wi th the large scale unsteadiness and bubble bursting phenomenon reported by both K S and C H L . The inrush of high velocity fluid resulting from this phenomenon CHAPTER 3. EXPERIMENTAL RESULTS Q / A CHAPTER 3. EXPERIMENTAL RESULTS d d CM T -d d o d in in d y -3 O O O O 0 0 O OXOXCPOOLltlllUEKb 00 in m d in d m od - cn m od o O - - i n n n n n n m 1111 i i i i imii i iHimnit O CD 3 o o o o o o ojDOQxniiinffl m cn in m id m iri m CM m d qA CHAPTER 3. EXPERIMENTAL RESULTS 44 0.5 X / X r 1 1.5 1.50-0.75 CO to CD c CD 0 -- 0 . 7 5 -1.50 J • . n o • • D • LD • • " H l_ Gau • ssian -^7 • • • • • 1 • | , . ' 1 1 1 1 6 6 f 3 ^ 00 00 c - 0 o - 3 0 8 10 X / D Figure 3.10. Skewness and flatness distributions of fluctuating wal l shear stress: • , ST; • , FT. would cause large amplitude fluctuations leading to probabil i ty distributions having a higher flatness factor. The observed increase of FT in the reattachment region is compatible w i th this explanation, since the act ivi ty associated w i th the large scale unsteadiness is most vigorous there. The skewness is negative in most of the reversed flow region, presumably because of the intermittent large amplitude negative fluctuations. After a local min imum in the middle of the separation bubble, the skewness rises steadily, crosses zero close to the mean reattachment point , and continues to rise thereafter due to increasing occurrences of large amplitude positive fluctuations. CHAPTER 3. EXPERIMENTAL RESULTS 45 3.3 Structure of the Shear Layer 3.3.1 Growth of the Shear Layer The posit ion yc where the r.m.s. velocity u attains a max imum can be interpreted as representing the centre of the shear layer bounding the separation bubble (KS) since it corresponds closely to the position where the velocity profiles have an inflexion point . F igure 3.11 shows that the posit ion of yc changes very litt le w i th x even in the reattachment region (the shaded plot reflects the uncertainty in locating u m a x due to scatter in the data) . Figure 3.11 also shows the streamwise variat ion of an edge of the shear layer ye defined as the position of 2.5% local turbulent intensity. Th i s corresponds closely to the posit ion where an intermittent signal first appears. A n important parameter characterizing the shear layer is the growth rate which can be deduced from the shear layer wid th . One measure of this w id th is the vort ic i ty thickness 6U defined by (Brown & Roshko 1974) ^ Umax Umin u (dU/dy) max The streamwise variat ion of the vort ic i ty thickness, plotted in Figure 3.12, shows that, in i t ia l ly , the shear layer grows in a linear fashion. The growth rate d6w/dx is found to be equal to 0.147, which is essentially identical to the value of 0.148 reported by Ruder ich & Fernholz for the normal-plate/split ter-plate geometry. This is w i th in the range of values (0.145-0.22) reported by Brown & Roshko for the conventional single stream mix ing layer. The present results indicate that at about x/xr ~ 0.65 there is a sudden decrease in the growth rate (d6u/dx — 0.097), a feature not shown by the data of Ruderich & Fernholz. The sharp change in the slope after x/xr ~ 1.1 is due to the smaller {dU/dy)max gradients. C H L used CHAPTER 3. EXPERIMENTAL RESULTS 46 X/Xr 0 0.4 0.8 1.2 1.6 2 X / D Figure 3.11. Representative shear layer positions: A , ye (location of u/U = 2.5%); zzzzz , yc (location of {u/U^naz). 0 0.25 0.50 0.75 1 1.25 X / X r Figure 3.12. Growth of the shear layer. Vor t ic i ty thickness 6U: - O , present; • , Ruder ich &; Fernholz (1986). M a x i m u m slope thickness 6ms: A , present; , Cherry et al. (1984). CHAPTER 3. EXPERIMENTAL RESULTS 47 the m a x i m u m slope thickness 6ma (defined in the same way as Su, but without incorporating a mean reversed flow velocity Umin) to represent the growth of the shear layer. The i r streamwise coordinate was corrected for the displacement of the v i r tua l or igin, therefore only the mean slope line of their data is shown in Figure 3.12. 3.3.2 Frequency Spectra and Autocorrelations Frequency spectra and autocorrelation functions of the longi tudinal fluctuating ve-locity were measured along ye. A t this posit ion, the velocity fluctuations are ir-rotat ional most of the t ime, wi th very short high-frequency bursts occurring very occasionally, and most of the information contained in these fluctuations is therefore related to the large scale structure of the shear layer. The spectra at various streamwise positions along ye, presented in Figure 3.13, show the same features as the measurements of C H L . A t x/xr = 0.01 there is a distinct h igh frequency contribution w i th a peak at a reduced frequency / x r / ^ c o of about 34. A s we progress further downstream from separation, a progressive fall in the dominant frequency takes place. This fall ceases at x/xr = 0.64, where the spectra become dominated by a broader band contr ibut ion centered around fxr/Uoo cn 0.6. Th i s streamwise variat ion is i l lustrated in Figure 3.14 (the fre-quencies were taken from the corresponding peaks in Figure 3.13) which also shows the measurements of Dz iomba (1985) and C H L . A l l measurements show the same trends, and two regions can be observed. In the first region, extending from separa-t ion to about 60% of the separation bubble length, the frequency decreases linearly w i th x, indicat ing that the large scale structures grow linearly just as in a conven-CHAPTER 3. EXPERIMENTAL RESULTS 48 t ional free shear layer. In the second region, which overlaps the first, starting at about 50% of the separation bubble length, the characteristic frequency remains essentially constant (decreasing values shown by the data of Dz iomba for x > 1.2 are due to the contamination from the t ra i l ing edge of the model which was not equipped wi th a tai l ) . This characteristic frequency is associated w i th the pseudo-periodic shedding of vort ic i ty from the bubble, a phenomenon observed by K S a n d C H L amongst others. Figure 3.15 shows the autocorrelation Ruu plotted as a function of the non-dimensional t ime lag TUoo/xr. Close to separation, at x/xr = 0.01, the autocorrela-t ion exhibits a long t a i l . A similar observation was made by K S who at tr ibuted this t a i l to the flapping of the shear layer caused by the large scale unsteadiness of the bubble. The high frequency waviness of the ta i l is due to the contributions from the large scale structures of the shear layer. This waviness can s t i l l be detected, w i t h correspondingly lower frequencies, at x/xr — 0.108 and 0.216. A t x/xr = 0.86 we note the reappearance of negative correlations which become even larger around reattachment. The streamwise dis tr ibut ion of the integral timescales Tu obtained by integrating the corresponding autocorrelation to the first zero crossing, are shown in Figure 3.16. Consistently w i t h earlier observations, the timescales increase in a linear fashion wi th x up to x/xr =: 0.85 and stabilize thereafter at a value of TuUoo/xr ~ 0.2. Figure 3.13. Frequency spectra of velocity fluctuations along shear layer edge yc. CHAPTER 3. EXPERIMENTAL RESULTS CHAPTER 3. EXPERIMENTAL RESULTS 0$ TUoo/Xr Figure 3.15. Autocorre la t ion of velocity fluctuations along shear layer edge CHAPTER 3. EXPERIMENTAL RESULTS 0 0.2 8 ^ 0.1 0.0 X / X r 0.5 2 3 4 5 X / D 1.5 o o o o o -o o o o I I I 1 I I I 7 8 Figure 3.16. Integral t ime scales deduced from autocorrelation measurements. CHAPTER 3. EXPERIMENTAL RESULTS 53 3.4 Effect of Angle of Separation The angle a at which the shear layer separates from the front face of the blunt plate has a significant impact on the pressure dis tr ibut ion, as shown in Figure 3.17. Decreasing the separation angle induces earlier recovery of the pressure, wi th a shift of the pressure distributions towards the leading edge and a corresponding shortening of the separation bubble length. There is also a decrease of the base pressure coefficient Cpb, and the pressure dip immediately downstream of separation becomes more pronounced. It is well known that separated reattaching flows have very similar pressure distr ibutions, and that a reasonable collapse of the data is obtained over a wide range of geometries when the pressure is plotted, as suggested by Roshko & L a u (1965), in terms of x/xr and the reduced pressure coefficient Cp* 1 - c _ . . . The pressure distributions of Figure 3.17 were replotted using these reduced co-ordinates, and the result is shown in Figure 3.18, which also includes the flat plate/spl i t ter plate data of Ruderich&; Fernholz (1986) for comparison. The col-lapse of the data onto a single curve, wi th slight deviations of the flat plate/split ter plate data, is quite remarkable and confirms the assumption that the reattachment length xr is a basic length scale for separated-reattaching flows. CHAPTER 3. EXPERIMENTAL RESULTS 54 CL o -0.2 -0.4--0.6 -0.8--1 V A ....V.. A • p x -©f A • A X o A.k A * • ^ O *8 x a ° o xp;^$^ X/D < r = J O a = 90 • a = 85 x a = 75 A a = 60 v a = 45 10 Figure 3.17. Effect of separation angle on surface pressure distr ibut ion. 0.5 0.4 0.3 Q. O 0.2 0.1 0.0-4 o t-0.5 " A; o a = 90 a a = 85 X cx = 75 A a = 60 a = 45 • R 4 F 1986 I 1.5 X/Xr —t— 2.5 Figure 3.18. Reduced pressure distributions of Figure 3.18. and comparison wi th flat plate/spli t ter plate data of Ruderich & Fernholz (1986). CHAPTER 3. EXPERIMENTAL RESULTS 55 The recovery pressure rise coefficient Cpm*, which measures the difference be-tween the highest and the lowest pressure, was found to be equal to 0.4 as compared to an averaged value of 0.36 cited by Roshko &; L a u for a variety of configurations. Tan i (quoted by Westphal et al. 1984) noted that the ult imate pressure recovery is higher for thinner boundary layers at separation. In the present case the effective thickness of the boundary layer is quite small as a result of the strongly favourable pressure gradients at separation ( C H L estimated the momentum thickness 6sep to be about 0.004D). The var iat ion 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 in Figure 3.19. The reattachment length for a = 45° is about 50% shorter than for the base model (a = 90°) , while the base pressure coefficient Cph is about 10% lower. It is interesting to note that these trends are qualitatively s imilar to the effect of either t r ipping the boundary layer on the front face before it separates (Dziomba 1985), or increasing the free-stream turbulence level (Hill ier & Cherry 1981a; K i y a & Sasaki 19836; Hi l l ie r & D u l a i 1985). Dz iomba reported reductions of up to 40% in the reattachment length w i t h a corresponding 15% drop in the base pressure coefficient. The shortening of the bubble was at tr ibuted to an effective change in the separation angle due to the formation of a smal l recirculation bubble between the t r ip wires and the sharp edge of the plate. The pressure distributions reported by Dz iomba were replotted in terms of re-duced coordinates for the basic undisturbed flow and two tr ipped flows. The results in Figure 3.20 show that, though reasonable, the collapse is not as good as that ob-tained w i th various separation angles. In particular, CPm* increases from about 0.4 to 0.43 when the boundary layer is t r ipped, and the pressure recovery process CHAPTER 3. EXPERIMENTAL RESULTS 56 starts earlier, resulting in a shift of the data towards the left. Th i s , together wi th the proport ional ly higher decrease in Cph noticed earlier, indicates that the effects of the t r ip wire is perhaps part ly due, but not confined to an effective change in separation angle. Add i t i ona l factors to be considered are possible changes in the state of the separating boundary layer and in the growth rate of the shear layer. Th i s concludes the discussion of the experimental results. The modell ing of the mean properties of this flow is examined next. CHAPTER 3. EXPERIMENTAL RESULTS 57 x 0 . 9 0 - 0 . 8 5 Q_ O I - 0 . 8 0 0 . 7 5 4 0 5 0 a Figure 3.19. Var ia t ion of reattachment length and base pressure w i th angle of separation. 0.5 0.4-0.3 Q. O 0.2-0.0 \y,0L - • — — • -.r.A.v • -/ < M /•'/ i // / ; • 9 / / ; O Undi s turbed A 2.3mm 41 w i re H • 3 . 1mm <f> wi re 1 1 , 1 1 0.5 1.5 X/Xr 2.5 Figure 3.20. Effect of t r ipping the boundary layer on reduced pressure distr ibution. D a t a taken from Dz iomba (1985); t r ip wires were placed on front face of plate. Chapter 4 M a t h e m a t i c a l M o d e l T w o different approaches to the modelling of separated reattaching flows can be taken. The first, a zonal approach, consists of dividing the flow field into several regions, each having dominant features. The flow is then analyzed using the method which is op t imum for each of those regions. In the second approach the flow field is solved for globally using a set of equations which apply throughout the domain. B o t h these approaches are examined in this study. A solution of the bluff rectan-gular plate problem is not completed w i th the first approach, but examples of the use of zonal analyses are given in Appendices C and D . In this chapter, we present the background for the global modell ing of the flow. The averaging procedure of the Navier-Stokes equations is then briefly described, and the turbulence model used in this study, the k — e model, is reviewed. Next, to set the stage for the numerical method presented in the next chapter, the general form of the various transport equations to be solved is given. F ina l ly , the boundary conditions for the blunt rectangular plate geometry are described. 58 CHAPTER 4. MATHEMATICAL MODEL 59 4.1 Background The simplifying assumptions made in the derivation of the boundary layer equations are unfortunately not s tr ict ly val id for separated flows. A n accurate description of these flows requires the use of the exact equations expressing the principle of conservation of momentum: the Navier-Stokes equations. The numerical solution of these equations for laminar flows, although not a t r iv ia l task, is always possible. However, for turbulent flows, a numerical solution of the full set of equations in their three-dimensional t ime dependent form is not quite feasible at present—at least not for flows of pract ical importance. Th i s is because the exceedingly refined grids required to resolve the smallest scales of turbulent mot ion present at realistic Reynolds numbers (Re ~ 10 4) would tax the storage capacity and speed of present day computers. Consequently one has to resort to the ensemble or time-averaging procedure, first proposed by Reynolds over a hundred years ago. However, because the equations are non-linear, the averaging procedure produces extra unknown terms: the turbulent or Reynolds stresses. The net result is that one ends up w i t h more unknowns than equations, and addi t ional equations are required to "close" the problem. These addit ional equations are provided by modell ing the turbulent stresses. 4.2 Conservation Equations and Time-Averaging For incompressible flow, the equations expressing the principle of conservation of mass and momentum are, in Cartesian tensor co-ordinates (see Reynolds 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 val id for laminar and turbulent flow, w i th £7, = 1,2,3) and p representing instantaneous velocities and static pressure respectively. The instantaneous general variable (f> is decomposed into a mean, cf>, and a fluctuating component, <p , as follows <j> = j> + <p (4.3) where the time-averaged value, <f>, is defined as w i th an averaging t ime At long enough compared wi th the longest t ime scales of the turbulent mot ion . Introducing these definitions in equations (4.1) and (4.2) to decompose Ui and p into mean and fluctuating components, and time-averaging, we obtain, for a statist ically steady flow, It =0 <«> axj and d , x dp d [ (dUi dUA ) , . where the overbars for the mean variables have been dropped for convenience. Equat ion (4.6) contains six new unknowns, the turbulent or Reynolds stresses —pufuj, which arise from the averaging of the nonlinear convective terms in equa-t ion (4.2). Physical ly, the Reynolds stresses represent diffusion of momentum by CHAPTER 4. MATHEMATICAL MODEL 61 turbulent mot ion . In order to obtain a closed set of equations, some assumptions are necessary to relate the Reynolds stresses to other existing variables. The var-ious assumptions which can be made constitute the central theme of turbulence modell ing. 4.3 k — e Turbulence Model Turbulence modell ing has been an active field of research for many years; although much progress has been made (see R o d i 1983, for a cr i t ical state of the art review), the models currently available are necessarily approximate and s t i l l rely on empirical information. The k — e model (Launder & Spalding 1974) used in this study, requires the solution of two addit ional transport equations: one for the turbulent kinetic energy, k, and another for its dissipation rate, e. This model seems, at present, to offer the best compromise between generality and computational economy. The framework of the k — e model is the eddy viscosity concept. This concept is expressed by an equation of the form fdUi dUA 2 , . . - pUiUj = fit — + -r— - -pkbij (4.7) \OXj OXi J 6 where k is defined as * = ^ f u ! + ^ + ^ ) (4-8) The term involving the Kronecker delta on the r.h.s. of (4.7) ensures that the sum of the normal stresses is equal to 2k. This term is a scalar quantity which can be incorporated in the pressure gradient term of the momentum equation. CHAPTER 4. MATHEMATICAL MODEL 62 The next step is to determine the eddy or turbulent viscosity fit in terms of definable quantities. Th i s is done by first assuming fxt to be proport ional to a characteristic velocity V, and a length scale £: Ht cx VI (4.9) A physically meaningful scale characterizing turbulent velocity fluctuations is yfk\ using this scale in (4.9), results in the Kolmogorov-Prandt l relation oc py/kl (4.10) k and I are related to the dissipation rate of turbulent kinetic energy, e, by dimen-sional analysis (Rodi 1984) e oc — (4.11) Combin ing these two expressions, we obtain lit = G>fc 2 / e (4.12) where C M is an empirical ly determined constant of proportionality. The problem of determining the turbulent stresses has thus been reduced to determining k and e. Exac t transport equations for both k and e can be obtained by manipula t ing equation (4.2); the resulting equations are, however, of litt le use because they contain higher order correlation terms which are unknown. To obtain a closed set, these terms must be modelled. In the "standard" k — e model, the modelled transport equations for k and e, take the form (Launder & Spalding 1974): kdxk dxk 1 crk dxkJ e f f \dxk dx{) dxk CHAPTER 4. MATHEMATICAL MODEL 63 Table 4.1. k — e model constants c 2 0.09 1.44 1.92 1.0 0.4187 and pUk— = — ( I + C i / x . / / dxk dxk \ o~e dxk J k \dxk dx{ J dxk The effective viscosity, / i e / / , is the sum of the laminar and eddy viscosities The values of the empirical constants appearing in equations (4.13) and (4.14) are listed in table 4.1. T w o variants of the standard k — e model which incorporate modifications ac-counting for streamline curvature are presented in Append ix D . 4.4 General Transport Equation For steady two-dimensional flow, the equations presented in the previous section reduce to: continuity equation: ATT PiV (4.15) dU dV__Q dx 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-momentum equation: /^-transport equation: e-transport equation: d (neffde\ ( 5 (ixeffde\ „ e„ „ e P d x ^ ^ P dy dx I <7e dx ) <9y V a £ dy + d - G - C 2 p - (4.19) where G , which represents the generation of turbulent kinetic energy, is given by 2 / ~ T J - \ 21 dx \dyt G = /z t | 2 and the effective viscosity is obtained from the two auxi l iary equations fdU dV\2 \dy dx J (4.20) Veff = V- + ft k2 e (4.21) (4.22) Note that the continuity and momentum equations are also va l id for steady lami-nar flow, Equations (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 and <S^  a general source term. The particular values taken by T and are given in 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 Peff §2± j_ a /„ au\ , a ( a v \ T . K . E . k Ok G- pe T . K . E dissipation e »eff C-J-G - C2p*— k k CHAPTER 4. MATHEMATICAL MODEL 66 4.5 Boundary Conditions To completely specify the problem, it is necessary to impose boundary conditions on a l l the boundaries of the flow domain. For the blunt rectangular plate geometry considered here, there are four types of boundaries. These are shown in Figure 4.1, and the corresponding boundary conditions are given below. / / / / / \ \ / / / / / / / / / / / / / / / / / / / \ \ / / / x p OJ B CM -t-> OJ +J O Figure 4.1. F low domain. i) I n f l o w ( A B ) : Values of a l l the variables, pressure excepted 1 , are specified. For the present flow configuration, uniform profiles for U, k and e2 are im-posed; V is set to zero. The 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. 1The pressure need not be specified when the normal velocity is imposed. 2Conditions on k and e are only required for turbulent flow computations CHAPTER 4. MATHEMATICAL MODEL 67 ii) O u t f l o w ( E F ) : Ideally, the outflow boundary should be located in a region where the flow is fully developed; a zero streamwise gradient condit ion across d() the outflow boundary, —— = 0, is then applicable to a l l variables. Locat ing ox this boundary in the fully developed flow region is, however, not practical because a very large computat ional domain, extending far beyond the region of interest, would be required. However, because the flow after reattachment is everywhere in the downstream direction it is not very sensitive to downstream conditions. It is therefore per-missible to locate the outflow boundary closer to reattachment, i.e. in a region where the flow is not fully developed, providing this location is sufficiently far downstream from the recirculating flow region to ensure that the flow in the upstream region is not affected by downstream conditions. The zero stream-wise gradient condit ion is then applied to a l l variables at this location. The penalty for this inexact treatment is a local distort ion of the flow field near the outflow which, anyhow, is not a region of prime interest. iii) S y m m e t r y A x i s ( B C ) : The normal velocity, V, is set to zero along the axis d( 1 of symmetry, and a zero cross-stream gradient condit ion, —— = 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 ) : The no-slip boundary condit ion is imposed at a l l solid boundaries. In addit ion, a special treatment of the near-wall region k2 is required, because the turbulent Reynolds number, Ret = — , is often suffi-ce ciently small in the v ic in i ty of solid boundaries for viscous effects to become important . The k — e model, which was devised for high Reynolds number fully turbulent flows, neglects these effects; consequently, it is not val id in CHAPTER 4. MATHEMATICAL MODEL 68 near-wall regions. A n alternative to devising a low Reynolds number model which would take viscous effects into account, is the use of wal l functions, as proposed by Launder and Spalding (1974). The wall function treatment, presented in Append ix E , connects the wa l l shear stress to the velocity just outside the viscous sublayer by assuming one dimensional Couette flow and local equi l ibr ium. A bonus of this treatment is computat ional economy : it is no longer necessary to have the high concentration of grids normally required to resolve the very steep gradients prevailing in the viscous sublayer. Chapter 5 C o m p u t a t i o n a l Procedure The governing equations presented in the previous chapter were discretized using a finite volume method, and the resulting set of algebraic equations were solved by an iterative procedure using a modified version of the T E A C H - I I code developed by Benodekar, Gossman & Issa (1983). In this chapter, an overview of the method of solution is given. The staggered gr id system used to discretize the solution domain is first described. Th i s is fol-lowed (Section 5.2) by a summarized account of the finite volume method, hybrid differencing and the associated problem of false diffusion; a remedy to this problem, a variant of the skewed differencing scheme of Ra i thby (1976), is also introduced. In Section 5.3, the implementation of the boundary conditions is discussed. A n outline of the iterative solution procedure is given in Section 5.4. The results of prel iminary computations, carried out to determine various parameters (extent of the solution domain, op t imum grid distr ibution, convergence criteria) are presented in the last section. 69 CHAPTER 5. COMPUTATIONAL PROCEDURE 70 5.1 Finite Volume Formulation 5.1.1 Grid Layout and Variables Location The computat ional domain is divided into a number of adjacent control volumes (or cells), w i th their centres located at the nodes of a Cartesian grid system as shown in Figure 5.1. The location of the nodes at which the variables are to be calculated differs, depending on the variable in question as shown in Figure 5.2. Scalar variables (p, k, and e) are evaluated at the main gr id nodes (shown as • ) , whereas the velocities U and V are evaluated at staggered locations (shown by —• and 1") w i th respect to the ma in gr id nodes 1 . The gr id is arranged so that the faces of the scalar cells coincide w i th the faces of the physical boundaries. Add i t iona l "fictitious" cells are located on the periphery of the domain to facilitate the implementation of the boundary conditions. The 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 by n , s, e and w. 5.1.2 Integration of General Transport Equation The finite volume form of equation (4.47) is obtained by integration over the control volume shown in Figure 5.2(b), i.e dxdy (5-1) 1The staggered grid arrangement is used almost universally in finite difference calculations be-cause 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). CHAPTER 5. COMPUTATIONAL PROCEDURE 71 ///// '////. ' / / / / / / / / / / / / / / / / y / / / / / / / / ' / / / / / / / / / / / / \ \ ] 1 1 ; ; ! ; WW WW WW W W W W 1 ^ \ \ \ \ | \ \ * \ \ \ i \ Figure 5.1. G r i d layout. CHAPTER 5. COMPUTATIONAL PROCEDURE 72 1 P(1.J+D i i L. 1 1 L _ . I V(1 , j ) I 1 r - - l - 1 i j i u d . j ) i ' P ( M . J ) ; p d . j ) I 1 p ( i , j - D p(i+i,j) ( a ) 6x 1 NE NW N A*1 E n yn W w r i P [e SW L d . j ) | i 5E s S . v / i f / u x e _| X Figure 5.2: (a) Loca t ion of scalar and velocity cells, (b) Typ ica l control volume CHAPTER 5. COMPUTATIONAL PROCEDURE 73 using Gauss ' divergence theorem, the volume integrals can be transformed into surface integrals: where Fe,...,Fe represent the sums of convective and diffusive fluxes across the faces e , . . . , s. For example: Fe = Ce + De w i t h C e = r{pU<t>)Xedy and 5.2 Finite Difference Discretization 5.2.1 Hybrid Differencing So far, the terms in the integral equation (5.2) are exact. The first step in re-ducing the equations to an algebraic form is to express, by using finite difference approximations, the convective and diffusive fluxes (Ce,De etc . . . ) in terms of the nodal values of the variable (f>. The hybr id differencing scheme ( H D ) , presented in (5.3) (5.4) CHAPTER 5. COMPUTATIONAL PROCEDURE 74 this subsection, is the combination of two types of finite difference approximations: central difference (CD) and upwind difference ( U D ) . The C D approximation assumes that <f> varies linearly between E and P. For example, applying central differencing to approximate Ce (equation 5.3) yields: C. = ,,V, p*±4=) A y , (5.5) Similar ly for the diffusive flux (equation 5.4) Centra l differencing, which is formally second order accurate in 6x, is quite satisfactory for problems where diffusion is the dominant feature. However, for higher Reynolds 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 than 2. The most widely used method to counteract this type of instabil i ty is upwind differencing. Taking the example of Ce again, at e the upwind value of <f> is assumed to prevail , i.e. Ce = peUe<f>pAyj for Ue>0 Ct = ptUt(j>EAyj for Ue<0 This amounts to replacing the linear variat ion of <f> assumed previously in central differencing by a stepwise variat ion; this leads to an approximation which is un-condit ional ly stable, but only first order accurate in 6x. To 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 hybr id 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 s} St i -momen tum U 0 p*\?Ay + A y + ( * / / £ ) | > * y-momentum V 0 P*\>nAx+ ( M . / / i J ) > y + ("«//1r) l>* T . K . E . k _£Jl£kpAxA G Ax Ay T . K . E dissipation e - ^ A x A y dj^GAxAy To complete the discretization procedure the source te rm 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. Subst i tut ing for the source and flux terms into equation (5.2) yields the general finite volume equation 2 (see Patankar 1980): (ap — Sp)<f>p = a,N<t>N + a>s<t>s + O>E4>E + aw4>w + <Sy (5.9) w i t h aP=^2ai (i = N,S,E,W) i 2Also referred to as finite difference equation. CHAPTER 5. COMPUTATIONAL PROCEDURE 76 and aN = m a x ( | C „ / 2 | , D n ) - C n / 2 as = max{\C,/2\,D.) + C./2 aE = m a x ( | C , / 2 | , 2 ? e ) - C e / 2 aw = m a x ( | C w / 2 | , 1?^) + Cw/2 The algebraic equations (5.9) are solved by the iterative procedure described in Section 5.4. 5.2.2 False Diffusion The simple upwinding procedure just described improves numerical stability, but it does so at a cost. Because it is only first order accurate, upwind differencing introduces a potentially damaging truncation error commonly known as artificial or false diffusion. -1 6 X 1 - 1 6 x •} Figure 5.3. F in i te difference nodes. The t runcat ion error of a finite difference approximation can be evaluated using a Taylor series expansion; hence ^ = ^ ( - ^ - ) g ) , ^ ( - ^ ) ! ( 0 ) , + -CHAPTER 5. COMPUTATIONAL PROCED URE 77 the convective term U— can therefore be expressed as ox + ... d4> The first term on the R H S is the upwind difference approximation of U—— (for ox U > 0); the second term is the leading t runcat ion error, and has the form of a diffusion te rm w i t h an effective diffusion coefficient The effect of false diffusion is to artificially increase the (physical) diffusion coeffi-cients; this results in smearing of the gradients in the flowfield. The question of how important is false diffusion was addressed by Rai thby (1976) who showed that false diffusion is negligible in situations where either the local flow direction is closely aligned wi th gr id lines or in the absence of strong cross-stream gradients in <f>. However, in the presence of both skewness and strong gradients, Rai thby showed that artificial diffusion can become comparable to, or even larger than physical diffusion, and can lead to significant errors. It is important to note that skewness and strong gradients are often prevail-ing conditions in recirculating flows .in general and turbulent ones in particular. Further, in 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 diffi-cult 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 by increasing the number of computational nodes in cr i t ical regions of the 6x; i-l 2 CHAPTER 5. COMPUTATIONAL PROCEDURE 78 flow. Aside from this "brute force", computat ionally expensive and often imprac-t ical prescription, two remedies to false diffusion have been proposed: the skew upwind differencing (SUD) scheme of Ra i thby (1976b), which uses flow oriented differencing, and the quadratic upstream weighted interpolation ( Q U I C K ) scheme of Leonard (1977), which uses higher order differencing. However, bo th schemes suffer from nonphysical oscillations or "wiggles" as well as solution undershoots and overshoots. These are a consequence of negative coefficients appearing in the finite difference equations, a problem referred to as unboundedness3. L a i & Goss-man (1982) developed a variant of the skew scheme, the bounded skew hybr id differencing scheme ( B S H D ) , which eliminates negative coefficients and is therefore uncondit ionally stable . This scheme was used in this study, and is outl ined next. 5.2.3 Skew Differencing The basic cause of false diffusion lies in the practice of treating the flow across a control volume face as locally one-dimensional, which results in errors in the dis t r ibut ion of <j>. The skew upwind differencing (SUD) scheme of Rai thby (1976b) reduces this error substantially by taking into account the local flow direction. In this scheme, the value of <f> required to evaluate the convective term is deter-mined by back projecting the local velocity vector unt i l it intersects a grid line, as i l lustrated in Figure 5.4(a); <f> is then obtained by either linear or stepwise interpo-lation between the two neighbouring nodes lying on the same gr id line. Examples for each type of interpolation follow. 3 The problem of unboundedness which can result in numerical instability and poor convergence is discussed in some details by Lai (1982). CHAPTER 5. COMPUTATIONAL PROCEDURE 79 W h e n the projection intersects wi th a vert ical grid line, as shown in Figure 5.4(b), then the value of <j> at face w is given by: where kw is a linear interpolation factor. W h e n the intersection lies on a horizontal grid line, as shown in Figure 5.4(c), a stepwise interpolation is used, and <$>w takes the nearest nodal value, (f>sw in this case. To account for a l l possible flow directions and the two types of interpolations, the interpolation factor is given by (for Uw > 0 and Vw > 0) The S U D reduces false diffusion substantially, but can result in negative coef-ficients in the finite difference equations. To suppress these negative coefficients, L a i & Gossman (1982) developed a flux blending technique which results in the bounded skew upwind differencing ( B S U D ) scheme: the S U D and U D schemes are blended in such a fashion as to maximize the contribution from the more accu-rate S U D while maintaining posi t ivi ty of a l l coefficients. Th i s is done through an opt imizat ion procedure described by L a i (1982) (see Benodekar et al. 1985 for a summarized account). F ina l ly , the bounded skew hybr id differencing ( B S H D ) scheme is, as its name implies, a hybr id combination 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. 4>w = (1 — kw)<t>w + kw<f>sw CHAPTER 5. COMPUTATIONAL PROCEDURE m --sw Ax. / 6 X , NE Ay, SE 80 NW NW k Sy. , (1 1 II 1 W w I r \, \/ 1 * I I I i SW II 1 i i IV 11 <i i i I r Iw \ I l l / i IV SW / •—*k 1 Figure 5.4. F in i t e difference computat ional star and i l lustrat ion of linear and step-wise interpolation ranges for skew upwind differencing scheme. CHAPTER 5. COMPUTATIONAL PROCEDURE 81 Discret izat ion of equation (5.2) using the B S H D results in a finite volume equa-t ion of the form: (ap — Sp)(j)p = aN<j>N + cts<f>s + o-E<i>E + a-w<i>w + O-NE^NE + 0-Nw4>NW + O-SE^SE + ^SW^SW + S$ (5.10) the expressions for the coefficients a,- can be found in Benodekar et al. (1982). 5.3 Treatment of Boundary Conditions The finite difference representation of the boundary conditions (discussed previously in Section 4.3.) is given in this section. 5.3.1 Types of Boundaries i) I n f l o w : A n example of a cell at the inflow boundary is shown in Figure 5.5. The profiles were assumed to be uniform for a l l variables. This corresponds closely to experimentally observed conditions. Uw = t /oo Vw - 0 kw = A-oo = «oo Uoo was the experimentally measured free stream velocity. The turbulent kinetic energy was estimated, assuming isotropy, from the experimentally CHAPTER 5. COMPUTATIONAL PROCEDURE 82 r 1 J 1 k w» EW kp »e p INFLOW BOUNDARY Figure 5.5. Inflow boundary cells. measured free stream turbulent intensity V t ? ; it was taken as fcoo = 9.375 X 1 0 _ 6 £ / ^ The dissipation rate, €<„ cannot be measured experimentally and has to be estimated. The generally accepted practice of estimating e as a function of the local value of k and a characteristic length scale \H was followed; i.e. Coo = k^/{\H) It should be noted that a judicious choice of A is required to ensure a realistic dis tr ibut ion of k upstream of the bluff body, since too smal l or too large a value of too would 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. The choice of A was otherwise not cr i t ical : changing A by a factor of five had v i r tua l ly no effect on the results (less than 0.2% change in Xr). ii) O u t f l o w : The zero gradient condit ion at the outflow boundary is obtained OUTFLOW BOUNDARY Figure 5.6.Outflow boundary cells. by setting UP VE kE tE Uw + Uipfc VP kp UJNC is an incremental value which 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 : The symmetry condit ion is implemented as follows Us = UP Vs = 0 ks — kP es = eP In addi t ion it is necessary to ensure that the fluxes across the symmetry 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 : The impermeabili ty condit ion is s imply obtained by put t ing Vs = 0 The boundary conditions for U,k and e are implemented by first cancelling the fluxes through the boundary side of the cells, i.e. as = 0 , for Up,kp,eP, and then by evaluating the diffusive fluxes at the wa l l using wa l l functions as described in Append ix D . CHAPTER 5. COMPUTATIONAL PROCEDURE 1 Figure 5.7. Symmetry axis cells. k p ' e p v s \ \ \ \ \ \ \ V 1 • ! k s , e s Figure 5.8. Sol id wal l boundary cells. CHAPTER 5. COMPUTATIONAL PROCEDURE 86 5.3.2 Special Case: Corner Nodes The corner cells of the bluff body warrant special attention because of the staggered gr id arrangement. This arrangement results in corner cells w i th only a half-face in contact w i th the wa l l as i l lustrated in Figure 5.9; this raises the question of how are the convective fluxes through these half-faces to be calculated. The convective fluxes across each of these two half-faces are evaluated by: a) Assuming a normal velocity across these half faces equal to that at the outer edge of the half cell (i.e. Vs is used for the normal velocity across the half-face of the U-ce\\ and vice-versa) b) Taking an effective area for the flux calculations equal to A y / 2 x l and A x / 2 x 1 for U and V respectively. The above treatment of the corner cells was found to have a drastic effect on the solution: H D turbulent flow calculations without this special treatment resulted in a 40% shorter reattachment length (for a 80 x 40 grid) . 5.4 Solution Procedure The solution method in the T E A C H - I I code uses the P I S O (Pressure Implici t Split Operator) a lgori thm, described in detail by Benodekar et al. (1983). This algori thm consists of a two-stage predictor corrector procedure, which involves the spl i t t ing of operations to deal wi th the coupling between velocity and pressure variables so that at each stage of the solution procedure, a set of algebraic equations in terms of a single unknown variable is obtained. CHAPTER 5. COMPUTATIONAL PROCEDURE 87 Figure 5.9. Cells near plate corner. CHAPTER 5. COMPUTATIONAL PROCEDURE 88 In the P I S O procedure, the algebraic equations in question are solved by a series of "guess and correct" operations. F i r s t , the velocities are calculated from the momentum equation using the pressure field prevailing at the nth i teration. The velocity and pressure fields are then adjusted through two corrector steps to ensure that mass conservation is satisfied. The procedure is repeated unt i l convergence. For turbulent flow calculations, the algebraic equations for k and e are solved in the i teration 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 method in conjunction w i th a tri-diagonal mat r ix solution algo-r i t hm. 5.5 Preliminary computations In this section, we discuss the results of prel iminary runs made to determine: the effect of the location of the inlet and outlet boundaries, the op t imum grid distr ibu-t ion, and the appropriate convergence cri teria. 5.5.1 Location of Inlet and Outlet Boundaries Specifying the extent of the computational domain is an important consideration. Too large a domain results in unnecessarily large arrays; too smal l a domain can affect the accuracy of the solution in the region of interest. Several computations were therefore carried out to determine the appropriate location of the inlet and outlet boundaries. The distance from the front face of the plate to the inlet bound-ary (Figure 5.10) was gradually reduced from Lu = 15Z> to 7.5D. The location of this boundary was found to have no noticeable effect on the flow in the recircula-CHAPTER 5. COMPUTATIONAL PROCED URE 89 t ion region, provided that Lu > 9D. For Lu < 9D a small increase (typically about 0.3% for Lu = 7.5J9) in the separation bubble was observed. //// / \ \ / / / / / / / / / / / / / / / / / / / \ \/ //^ Figure 5.10. Computa t ional domain. The effect of the location of the downstream boundary was noted for Ld = 15D to 8D. The effect was negligible for Ld ~ 11D. Computat ions w i th smaller Ld resulted in gradually longer bubbles (typically about 1.0% for Ld = 81?). A l l subsequent computations were therefore performed wi th 12D < Lu < 10D 14D < Ld < 12D 5.5.2 Non-uniform Grid Arrangement A number of gr id distributions were investigated. Non-uniform grids were found to be the most efficient way of obtaining the fine gr id arrangements required in 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 par t icular ly cr i t ical to the accuracy of the solutions. Conse-quently, a cluster of fine equally spaced grids was located there. The gr id spacings were expanded on either side of this sharp corner, in both x and y-directions, as shown i n Figure 5.11. In addit ion the x—direction gr id was refined locally around the reattachment region. This local refinement was not cr i t ical to the computat ion, but allowed a more precise location of the reattachment lengths, which were deter-mined by linear interpolation from the computed wal l shear stress distributions. A prel iminary gr id refinement study showed that the solutions are more sensitive to gr id refinements in the y- direction than in the x-direction, and that op t imum array sizes are obtained when the mesh at the corner has a ratio A x / A y ~ 2.5. To minimize the t runcat ion errors associated w i th non-uniform gr id distr ibu-tions, the effect of the gr id expansion factors Ex — A x , / A x , _ i and Ev = A y y / A y ; _ i was investigated. A comparison was made w i th solutions obtained using uniform grids in selected regions of the domain. It was found that an economy in computa-t ional cells (compared to a uniform distribution) can be obtained w i th no noticeable loss in accuracy, provided that the expansion factors are kept w i th in the following ranges: x < —D and x > x r 0.8 < Ex,Ey < 1.2 for { 0.9 < Ex,Ey < 1.1 for « y > D -D < x <xr -0.5D < y <D A l l the computations presented in the next chapter were performed wi th expansion factors w i th in these ranges. CHAPTER 5. COMPUTATIONAL PROCEDURE CHAPTER 5. COMPUTATIONAL PROCED URE 92 5.5.3 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 which a computed solution satisfies the finite difference equations. In the present computations this convergence cri terion was based on the values of the absolute residual errors of the continuity and momentum equations. When the sums of these residual errors were less than 0.2% of the inlet mass flow and momentum respectively, the solution was considered converged, and the iteration cycle was terminated. Th i s convergence cri terion is expressed by where X) \R<p\ is the sum of the absolute residuals over the entire field. The residuals are defined, from equation (5.9), by The value of £max = 2 x 10~ 3 was considered to be acceptable since a reduction of this value by a factor of five d id not result in any appreciable change in the computed reattachment length (less than 0.25%). max R<t> — iap - Sp)<f>p - ^2 difa - Si Chapter 6 N u m e r i c a l Results The accuracy of a numerical method, which is part icular ly important when assessing a given turbulence model , is best established in laminar flows. In this chapter we present the results of a systematic gr id dependence study for the laminar flow past a bluff rectangular plate. The superiority of the B S H D scheme over the H D scheme is clearly demonstrated. The laminar flow computations, besides providing a check on the accuracy of the method, yielded results which are interesting in their own right, and a selection of these is presented and discussed. The second part of this chapter is devoted to the results of turbulent flow com-putations. Some important numerical aspects of the solutions are first discussed, and the results of a comprehensive grid dependence study are presented. The k — e turbulence model is found to perform rather poorly in its standard form, but results in substantially improved predictions when the preferential dissipation modification is incorporated. Detai led comparisons of the predicted flow field (velocity, turbu-lent kinetic energy, pressure, wal l shear stress) are made wi th the experimental data presented earlier in Chapter 3. A s an i l lustrat ion of the usefulness of the numerical method for parametric studies, predictions of the effect of solid blockage are pre-93 CHAPTER 6. NUMERICAL RESULTS 94 sented. In the last section, computat ional costs and the relative performance of the various schemes are discussed. 6.1 Laminar Flow The parameters for the laminar flow computations were chosen to correspond to the measurements of Lane & Loehrke (1980) for comparison purposes 1 . The solid blockage ratio was equal to 8.36%. The bulk of the computations were performed in the Reynolds number range 100-325, since it was observed experimentally, and confirmed in our prel iminary computations, that no separation occurs for Reynolds numbers below about 100, and that the flow becomes unsteady for values higher than about 325. 6.1.1 Grid Independence The effect of gr id refinement on the solution was investigated systematically, using both H D and B S H D schemes, for three Reynolds numbers: i?e = 125,225 and 325. The number of grids used ranged from 41 x 2 6 (corresponding to A z m t - n / Z ? = 35% and Aymin/D = 12.5%) to 7 0 x 5 5 [Axmin/D = 8%,Aymin/D = 3.7%). A typical gr id layout was shown in Section 5.3. The reattachment length xT was found to be a good measure of the sensitivity of the solution to gr id spacing, therefore al l grid refinement results are conveniently presented in terms of the variat ion of xr. The results of the three series of tests, presented in Figures 6.1(a) to (c), show a significant difference between the reattachment lengths predicted w i th the two 1The 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. CHAPTER 6. NUMERICAL RESULTS 1.5-X c c I" E .c o o "5 0.5-Laminar Flow, Re=125 —o- -o-O BSHD A HD 1 1 , ! ! 1000 1500 2000 2500 3000 3500 4000 No. of Computa t iona l Nodes NIxNJ x c C E o p £ 2-"o or Laminar Flow, Re=225 O BSHD A HD i i I I I 1000 1500 2000 2500 3000 3500 4000 No. of Computa t iona l Nodes NIxNJ 7 2 1 i 1 i 1 i 1000 1500 2000 2500 3000 3500 4000 No. of Computa t iona l Nodes NIxNJ Figure 6 .1. Effect o f gr id refinement on reattachment length. CHAPTER 6. NUMERICAL RESULTS 96 discretization schemes. A t Re — 125, the B S H D solution is essentially grid inde-pendent for NIxNJ — 1728, whereas the H D solutions remain sensitive to grid refinement even for the finest gr id and tend asymptotical ly—from below— towards the gr id independent B S H D solution. A t higher Reynolds number, the B S H D solu-tions show more sensitivity to gr id refinement, and more computat ional nodes are required to reach gr id independence. The H D solutions remain sensitive to grid spacing throughout the range and respond rather sluggishly to gr id refinement. Increasing Reynolds number causes the H D predictions to deteriorate further. Con-sidering, for example, the fine grid results, we see that while the H D reattachment length is 9% shorter than the B S H D result at Re = 125, this discrepancy increases to 35% at i2e = 325. Th i s can be at tr ibuted directly to the inherent false diffusion of the H D scheme which, for a given grid spacing, is expected to increase wi th Reynolds number. 6.1.2 Effect of Reynolds Number and Comparison with Experiments In view of the results of the gr id 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. The predicted variat ion of the reattachment length w i th Reynolds number is plotted in Figure 6.2. Considering the experimental uncertainties (reported to be about 10% in Re and 0.15D in xr) the agreement between predicted and measured values is excellent. In common wi th other reattaching flows, the reattachment length is seen to increase linearly w i th Reynolds number. Separation is first observed CHAPTER 6. NUMERICAL RESULTS 97 Reynolds Number Re Figure 6.2. Var ia t ion of reattachment length w i th Reynolds number: —O— , B S H D computations; V , measurements of Lane &: Loehrke (1980). at Re = 100 and, remarkably, it is predicted to occur slightly downstream of the corner in agreement w i th the experimental observations of Lane & Loehrke. A s the Reynolds number increases the separation point moves upstream and remains fixed at the sharp corner, while the bubble grows in both length and height as illustrated by the predicted streamline patterns shown in Figure 6.3. We also observe in these plots that the eddy centre is always located upstream of the middle of the bubble, at about x / x r = 0.35. Though separation actually occurs at the corner for Re > 100, this is not clearly shown in the streamline plots as a result of the interpolation procedure used to determine the streamfunction contours. CHAPTER 6. NUMERICAL RESULTS 98 ( 0 ) R E Y N O L D S NUMBER = 100 . (.D) R E Y N O L D S NUMBER = 150 . ( C ) R E Y N O L D S NUMBER = 2 0 0 . CHAPTER 6. NUMERICAL RESULTS 99 (d ) R E Y N O L D S NUMBER = 2 5 0 . ( e ) R E Y N O L D S NUMBER = 3 0 0 . Figure 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 100 R E Y N O L D S NUMBER = 2 0 0 . CHAPTER 6. NUMERICAL RESULTS 101 R E Y N O L D S NUMBER = 2 5 0 . R E Y N O L D S NUMBER = 3 0 0 . Figure 6.4(a to d). Velocity field for laminar flow. CHAPTER 6. NUMERICAL RESULTS 102 The velocity field plots (Figure 6.4) show that the largest backflow velocities encountered in 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 reattach-ment the boundary layer recovers relatively quickly. A linear region is established near the wa l l , and the velocity profiles look very similar to those of a classical flat plate boundary layer. It is also interesting to note that, in the region surrounding the corner of the plate, the cross-stream velocities are of the same order as the streamwise velocities. Th i s results in velocity vectors which are highly inclined, or skewed, w i th respect to the grid-lines. It is the combination of this skewness w i th the important gradients also present in this region that is responsible for the large false diffusion errors associated w i th the H D scheme. Figure 6.5 shows the predicted pressure distr ibution along the surface of the plate. The pressure coefficient at separation Cpt is equal to about —2.4 at .Re = 100 and 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 and rapid pressure recovery takes place. Th i s recovery process is slower, and is spread out over a larger region for the higher Reynolds number. Another point of interest is the constant pressure gradient through most of the separation bubble at Re = 300 and the higher pressure coefficient after recovery. We note that the flow is confined by solid walls. The boundary layers which develop on these walls, in combination wi th the reattached boundary layers on the plate, result in the favourable pressure gradients observed after reattachment. CHAPTER 6. NUMERICAL RESULTS 103 0 - 0 . 5 -1 - 1 . 5 -fj I If J Legend Re=100 R_e=200 _ Re=300 0 2.5 5 7.5 10 12.5 15 17.5 X / D Figure 6.5. Pressure distr ibution along top surface of the plate (laminar flow). CHAPTER 6. NUMERICAL RESULTS 104 6.2 Turbulent Flow 6.2.1 Preamble and Effect of Grid Refinement The turbulent flow calculations were carried out at a Reynolds number of 5 x l 0 4 for a blockage rat io of 5.6%; these conditions correspond to the measurements described in Chapter 3. A s in the laminar flow case, bo th H D and B S H D schemes were used. In the course of prel iminary computations using the B S H D scheme, the stagnation flow in 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. Th i s type of problem has also been observed by L a i (1983) who showed that the performance of skewed based schemes deteriorates in the presence of strong adverse pressure gradients 2 . To rectify the deficiency of the B S H D scheme in the stagnation region, the computat ional domain was spli t , w i t h H D being used upstream of the plate leading edge and B S H D downstream of i t . Th i s split , though it may seem arbitrary, was in fact based on Lai ' s conclusions that, in situations of strong pressure gradients, the hybr id differencing scheme performs better than the skewed differencing scheme, because numerical diffusion is less significant under these circumstances. The use of a split domain produced differences of less than 7% in computed values of xr, the split domain resulting in the longest separation bubble and a stagnation pressure coefficient of essentially unity. This technique was used in al l subsequent B S H D calculations. 2 A similar problem was reported by Castro et al. 1981 when using a vector differencing scheme for the flow over a normal flat plate. CHAPTER 6. NUMERICAL RESULTS 105 The turbulence models used in the present computations were the standard k — e model (see Section 4.3) and a variant of it w i th a preferential dissipation modification ( P D M ) described in Append ix D . A th i rd model , involving a streamline curvature correction (Appendix D ) , was also investigated. Its use was, however, inconclusive because it led to numerical instabilities (in the form of random oscillations in the computed flowfield) and a converged solution could not be obtained; the problem persisted even wi th severe under-relaxation. Implementing the correction on a converged solution obtained w i t h the standard k — e model proved equally fruitless. Similar difficulties were encountered by Hackman (1982) for the backward facing step, indicat ing a possible deficiency in the modification, though it should be pointed out that Benodekar et al. (1985) d id not report any difficulties in using the modification to solve for the flow over a surface -mounted r ib . The results of the gr id refinement studies are shown in figure 6.6 for three cases: H D scheme wi th standard k — e model , B S H D wi th standard k — e model , and B S H D wi th standard k — e model and preferential dissipation modification. These wi l l s imply be referred to as H D , B S H D and P D M respectively. The gr id distributions which were used ranged from 49x34 (Axmin/D = 20%, A y m i n / Z > = 5.6%) to 82x66 {Axmin/D = 3.6%, Aymin/D = 0.9%). The 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 mag-nitude of the discrepancy (44% wi th a fine 82x62 grid) was somewhat of a surprise. The discrepancies reported in the literature for various other geometries are much smaller even though coarser grids were used. For example, Durs t & Rastogi (1977) reported a 30% underprediction of xr for the flow around a surface mounted block; CHAPTER 6. NUMERICAL RESULTS 106 X CT) C c E o "o 1000 2000 3000 4000 5000 6000 No. of Computa t iona l Nodes NIxNJ Figure 6.6. Turbulent flow: effect of gr id refinement on computed reattachment length; A , H D scheme wi th k — c model; O , B S H D scheme wi th k — e model; • , B S H D scheme w i t h P D M . they used a 50x27 gr id . For the backward facing step flow, Hackman (1982) reported a 12% underprediction w i th a 5 0 x 5 0 gr id . The B S H D scheme results in a substantial lengthening of the predicted reat-tachment length as shown in Figure 6.6, but the discrepancy w i th the experimental value is s t i l l large at 30%. In view of the good agreement found in the laminar flow case this rather disappointing result can, w i th confidence, be blamed on tur-bulence model deficiencies. The use of the modified turbulence model , which was in fact prompted by these results, leads to an encouraging improvement of the pre-dictions. The P D M solutions appear to be independent of gr id refinement for a 7 5 x 5 7 gr id and a reattachment length of 4 .3D is produced—this is w i th in 10% of the experimentally observed value of 4.7D. CHAPTER 6. NUMERICAL RESULTS 107 U/Uref Figure 6.7. Effect of gr id refinement on computed velocity profile at x/D = 2 ( P D M computat ion) . A s a further check of gr id independence, Figure 6.7 shows streamwise velocity profiles at x/D — 2 predicted by the P D M using two different gr id distributions. These profiles are, for al l intents and purposes, identical. It was however disturbing to find that this gr id independence, demonstrated both in terms of the reattachment length and in the flowfield, d id not extend to the computed wal l shear stress. Figure 6.8 shows that the computed wal l shear stresses change by as much as 10% when refining the gr id from 7 5 x 5 7 to 8 2 x 6 2 nodes. We recall that in the wal l function treatment (Appendix E) different assump-tions are made for the near-wall velocity, depending on the value taken by the wall coordinate y+ = pC^y/kyp/fj,. The near-wall velocity is assumed to either vary l i n -early (y+ < 11.63) or according to the logarithmic law of the wal l ( y + > 11.63). Figure 6.9 shows that for the 82x62 grid y+ is always less than 11.63 and therefore CHAPTER 6. NUMERICAL RESULTS 108 4.0 2.0-O o o.o - 2 . 0 -- 4 . 0 82x62 GRID 75x57 GRID 2.5 5 7.5 10 12.5 15 17.5 X/D Figure 6 .8 . Effect of gr id refinement on computed wa l l shear stress along top surface of plate ( P D M computat ion) . 20 15 + ,0 / — Y+=11.63 — — — / ' — 1 . -—• ' / / \ / 82x62 GRID 75x57 GRID 2.5 7.5 10 X/D 12.5 15 17.5 Figure 6 .9 . Loca t ion of gr id points adjacent to the wall in terms of the wal l coordi-nate y + ( P D M computat ion). CHAPTER 6. NUMERICAL RESULTS 109 the computat ional nodes adjacent to the wa l l are w i th in the viscous sublayer. For the 7 5 x 5 7 gr id , however, y+ becomes larger than 11.63 for x/D ~ 2 and there-fore the logarithmic law is invoked to calculate the near-wall flow. It was ini t ia l ly thought that the change in computed shear stress was due to the inadequacy of the logarithmic law of the wall in this flow3. Further computations showed, however, that even when the nodes adjacent to the wal l are w i th in the viscous sublayer, further gr id refinement results in changes of the computed wal l shear stress. The reasons for these changes in wall shear stress are not clear, but they ind i -cate possible inconsistencies in the wal l function treatment. A l though the present computations suggest that the flow field is not sensitive to the details of the near-wall flow—and this is substantiated by the results of Hackman (1982) who found different wal l function treatments to have litt le impact on the backward facing step flow—this matter clearly deserves further investigation. 6.2.2 Predictions and Comparison with Experimental Data In the following we shall concentrate on the results obtained using the B S H D scheme. Complete results of the computations w i th the modified k — e model are presented, together w i th a selection of the results obtained w i th the standard k — e model to illustrate the effect of the turbulence model modification. Figure 6.10 and 6.11 show a comparison of the computed velocity profiles wi th experimental measurements at various streamwise locations. The B S H D results show that size of the separation bubble is underpredicted not only in length but also in height. A l though the gross features of the flow are reasonably well reproduced, 3There is some experimental evidence suggesting that the law of the wall is not valid in reattaching flows (Adams et al. 1984; Ruderich & Fernholz 1986). CHAPTER 6. NUMERICAL RESULTS 110 U/Uref 0 0.5 0 0.5 0 0.5 0 0.5 0 0 0.5 1.0 15 Figure 6.10. M e a n longitudinal velocity profiles: , B S H D com-putat ion; O , P W A measurements. Separation streamlines: — — - — , computed; , experimental. CHAPTER 6. NUMERICAL RESULTS 111 Figure 6.11. putat ion; 0 computed; — M e a n longitudinal velocity profiles: , P D M com-, P W A measurements. Separation streamlines: , , experimental. CHAPTER 6. NUMERICAL RESULTS 1 1 2 quantitative agreement is poor. For example, the max imum backfiow velocity is predicted to be about 0 . 2 6 J 7 o o and occurs at x/D = 1.0 as opposed to an experimental value of about 0 . 3 2 C / O Q occurring at x/D = 2.0. The P D M predictions, on the other hand, are in very good agreement w i th the experimental data in the recirculating flow region. After reattachment, however, they start to deteriorate as a result of a slower rate of recovery of the reattached boundary layer. It is interesting to note the fortuitious agreement of the B S H D predictions w i th experiments at x/D = 5.Q and 6 . 0 . This is due to the earlier reattachment which provides a longer distance for the boundary layer to recover. The deterioration of the predicted velocity profiles in the recovery region is a common feature of many reattaching flow calculations (see Nallasamy 1 9 8 7 ) and is perhaps not surprising in view of the complex and unsteady nature of the reattachment process. We note in conjunction w i t h the laminar flow results presented earlier that the backflow velocities are much higher in the turbulent flow case, and the near-wall velocities remain relatively high compared to laminar flow. Figure 6 . 1 2 shows the effective viscosity contours. Compared 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. The improved velocity field predictions are a direct result of the reduced eddy viscosities which 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 in both cases slightly downstream of the reattachment point . The predicted turbulent kinetic energy profiles are presented in Figures 6 . 1 3 and 6 . 1 4 . Since no experimental measurements of the turbulent kinetic energy k were available, this was estimated from the measured longitudinal r.m.s. velocity using CHAPTER 6. NUMERICAL RESULTS 113 ( a ) Figure 6.12. Contours of constant effective viscosity: (a) B S H D computat ion; (b) P D M computat ion. CHAPTER 6. NUMERICAL RESULTS 114 0.05 j . k/uref*uref 0 0.05 0 0.05 0.10 4, 1 i, 1 L 0.05 0.10 Figure 6.13. Turbulent kinetic energy profiles: O , estimated from P W A measurements. , computed ( B S H D ) ; CHAPTER 6. NUMERICAL RESULTS 115 Figure 6.14. Turbulent kinetic energy profiles: estimated from P W A measurements. , computed ( P D M ) ; Q , CHAPTER 6. NUMERICAL RESULTS 116 the relation k = u2/a, 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 plate/spl i t ter plate data of Ruder ich & Fernholz (1986). This is of course a rather crude estimate, but a useful one in the present context. The T . K . E . profiles predicted using B S H D show higher peaks than the estimated values up to x/D = 2 and lower ones thereafter; these peaks are observed to occur closer to the wal l . The unrealistically high T . K . E . observed near separation contribute to the poor performance of the standard k — e model in this part icular flow. In part icular we note that an excessive value of k results in an overestimated eddy viscosity (fit oc k2/e). The P D M profiles show a remarkable s imilar i ty wi th the estimated values both in recirculating flow and recovery regions. We note in part icular that the highest T . K . E . is predicted to occur at the same streamwise location as the estimated value (x/D — 3). The preferential dissipation modification, by increasing the dissipation rate of T . K . E . in the high streamline curvature region close to separation, results in more realistic values of k. There is one important feature of the estimated T . K . E . which is not wel l reproduced, that is the spread of T . K . E . into the outer region. This indicates a possible inadequacy of the modelled diffusion term of the fc-equation. The remainder of this discussion is confined to the P D M results. The predicted pressure field shown in Figure 6.15 brings to light some interesting points. The pressure is observed to remain fairly uniform wi th in the first half of the separation bubble. We also observe that large cross-stream gradients are present along a re-gion 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. The corresponding pressure dis t r ibut ion along the surface of the plate is shown in Figure 6.16. This pressure dis tr ibut ion—which, incidentally, is markedly CHAPTER 6. NUMERICAL RESULTS 117 different than that in the laminar flow case—reproduces fairly wel l the trends of the experimental data. In part icular we note the broad region of relatively con-stant pressure w i t h a smal l dip before recovery. The m i n i m u m pressure coefficient, excluding the separation value, is underpredicted by about 10%, probably as a re-sult of the shorter bubble. The unrealistically low pressure coefficient predicted at separation is due to the singularity associated w i th the sharp corner. In the actual flow, this corner is slightly rounded, since in practice it is not possible to machine a model w i th a perfectly sharp corner. Figure 6.17 shows the wal l shear stress dis tr ibut ion. A g a i n the experimental trends are well reproduced and the max ima and min ima compare well considering the large uncertainty in the measured values. A n interesting feature of the wal l shear stress is the hump which occurs near separation, outside the measurements range. A possible interpretation of this hump is a small secondary recirculating flow region (with one or several vortices) just downstream of separation. Th i s interpretation is supported by the streamline pattern shown in Figure 6.18. CHAPTER 6. NUMERICAL RESULTS 118 Figure 6.16. Comparison of computed and experimental pressure distributions on top surface of the the plate: , computed ( P D M ) ; O , measured. CHAPTER 6. NUMERICAL RESULTS 119 X / D Figure 6.17. Compar ison of computed and experimental wal l shear stress distributions: , computed ( P D M ) ; O , measured. Figure 6.18. Predicted streamline pattern ( P D M computation): for # > 0, A * = 0.143; for # < 0, A * = 0.01. CHAPTER 6. NUMERICAL RESULTS 120 6.2.3 Effect of Solid Blockage F i n a l P D M computations were performed for blockage ratios in the range 1.5% to 20%. The predicted reattachment lengths are compared in Figure 6.19 to measured values reported in the literature. The experimental trends are well reproduced and the computed values are wi th in 10% of the measured ones. Va ry ing departures from strict two-dimensionality in the measured case could account, in part , for the generally higher experimental values of xr. It should also be noted that a l l calcula-tions were carried out assuming the boundary layers originated at the inlet of the computat ional domain, and in i t ia l profiles for k and 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 smal l blockage and low free-stream turbulence conditions. The predicted pressure distributions are shown in Figure 6.20 for a few blockage ratios. A t higher blockage ratios, there is a substantial decrease in the m i n i m u m pressure coefficient w i th increasing blockage, and the pressure starts recovering sooner. Th i s is consistent w i th the shorter separation bubbles predicted at higher blockage ratios. The m a x i m u m pressure coefficient after recovery is lower at higher blockage ratios as would be expected from a simple application of Bernoul l i ' s equa-t ion wh ich gives: C - 1 1  P m a x (1 - BR)2 The predicted pressure recovery is of course lower than 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 boundary layers on the solid wa l l and 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 Figure 6.19. Effect of blockage on turbulent flow reattachment lengths. CL O -1.25 Figure 6.20. Predicted pressure distributions for various blockage ratios. CHAPTER 6. NUMERICAL RESULTS 122 6.3 Computational costs Most of the present computations were performed on a V A X 11/750 computer. For laminar flow calculations the computational effort ranged from 28 iterations and 19 minutes of C P U time, for a 4 1 x 2 6 gr id , to 63 iterations and 131 minutes of C P U time, for a 7 8 x 4 5 gr id . Turbulent flow calculations converged much more slowly. They required from 45 minutes of C P U t ime and 83 iterations, for a 49x34 gr id , to 545 minutes and 329 iterations, for a 8 2 x 6 2 gr id . A comparison of computing costs for the H D , B S H D and P D M is shown in Figure 6.21. The 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 rapidly w i th increasing size of the computat ional array. The generally large C P U time requirements are due to the relatively small size and slow execution speed of the V A X 11/750. For comparison purposes, some test runs were made on the U B C Comput ing Centre mainframe computer (Amdah l 5850). The execution time was found to be shorter by a factor of about 25. A tur-bulent flow computat ion using the P D M scheme wi th a 75x57 gr id would 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 RESULTS 123 6 0 0 - l 1 1 1 1 1000 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0 No. of Computat ional Nodes NIxNJ Figure 6.21. Comput ing time on V A X 11/750 computer as a function of computa-t ional array size (Turbulent flow). Chapter 7 C onclusions Experimental ly , it is difficult to establish an accurately two-dimensional mean flow field in a separated reattaching flow region. Considerable effort has been expended in this study to approximate these ideal conditions over the central part of the bluff rectangular plate. A mean reattachment length of 4.7D is obtained from both wall shear stress measurements and surface streamline visual izat ion. 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 in the reattached flow, and backflow velocities of the order of 30% of the free-stream velocity are encountered. Over the first 60% or so of the bubble, the separated shear-layer bounding the reversed flow region has similar characteristics to those of a plane mix ing layer. The growth rate of the shear-layer is linear, and both characteristic frequencies and integral scales vary linearly wi th streamwise distance. A t about x ~ 0 .6x r , the shear-layer appears to undergo a fairly abrupt change in structure: the shear-layer growth rate becomes significantly lower and the maxi -m u m turbulent intensities become substantially higher than the corresponding plane 124 CHAPTER 7. CONCLUSIONS 125 mixing-layer values. In the reattachment region, the linear variat ion of the charac-teristic frequencies and integral scales ceases. Consistently w i t h the observations of K i y a & Sasaki (1983) and Cherry et al. (1984), the shear layer is characterized by a low-frequency unsteadiness. This unsteadiness of the reattachment process is fur-ther i l lustrated by the forward-flow-fraction measurements which suggest that the instantaneous reattachment point moves around the time-averaged reattachment point over a distance of about 0 .5x r . In addi t ion to the carefully established two-dimensionality of the flow, a number of precautions have been taken to ensure the rel iabil i ty of the present data and its sui tabil i ty as a test case for numerical predictions. The measurements have been made in a large scale facility, and probe interference effects have been assessed and minimized. Where possible, measurements of the same quantities have been made using different experimental methods. A complete pulsed-wire anemometer data set is given in Append ix A for reference. In the numerical predictions, there is some uncertainty in the in i t i a l conditions for the boundary layers developing on the w i n d tunnel walls. The errors arising from this uncertainty are expected to be small for the low blockage ratios considered in this study. Otherwise, the boundary conditions for the flow around the blunt rectangular plate are clearly defined. Lamina r flow predictions show that the hybr id differencing scheme (HD) leads to significantly underpredicted reattachment lengths as a result of false diffusion. The skewed differencing scheme ( B S H D ) yields markedly improved predictions. These predictions are in excellent agreement w i th 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 Reynolds numbers, however, the performance of the B S H D scheme is found to deteriorate in the stagnation region. This problem, which has also been noted elsewhere, is due to a deficiency of the B S H D scheme in the presence of strong adverse pressure gradients. In the present turbulent flow computations, this deficiency is "corrected" by reverting to lower order hybr id differencing in the stagnation region. The 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 combination leads to a separation bubble 44% shorter than observed experimentally. This discrepancy is much larger than the values reported in the literature for other types of separated reattaching flows. The results of the the B S H D computations show that the standard k — e model fails to represent accurately the effects of turbulence in the region around separa-t ion . The use of the preferential dissipation modification ( P D M ) leads to signifi-cantly improved predictions. A reattachment length of 4 .3D is obtained. Detailed predictions in the separation bubble compare well w i th experiments, and are much more satisfactory than those obtained w i th the discrete vortex method ( K i y a et al. 1982). Downstream of reattachment, however, a gradual deterioration of the predictions is observed as the wall boundary layer redevelops. A disturbing feature of the predictions is that although the flowfield results are found to be essentially gr id independent for a 75 x 57 gr id , this gr id independence does not extend to the computed wal l shear stress. The wal l shear stress is found to change by as much as 10% wi th subsequent gr id refinements. These changes indicate possible inconsistencies in the near-wall treatment. The impact of these changes on CHAPTER 7. CONCLUSIONS 127 the predictions seems to be min imal in the present case because the flowfield appears to be insensitive to the details of the near-wall flow. Th i s problem, however, could be significant in other flow situations. A n alternative method of analysis, based on a novel use of the momentum inte-gral technique in flows wi th separation and reattachment has been proposed. The method leads to encouraging results for the case of a sudden expansion flow, but problems of convergence exist when the method is applied to external flows using a direct viscous-inviscid interaction procedure. A few suggested areas of future research and some recommendations follow: 1. Exper imenta l information to elucidate the structure of the flow in the near-wal l region is needed to guide the modell ing of this region. A step in this direction has been taken by Adams et al. (1984) for the turbulent flow over a backward facing step. The i r results suggest that the near-wall flow has a laminar-like structure, but more information is required part icular ly for high Reynolds number flows. Because the scale of the wal l region is very small com-pared w i th the overall scale of the separation bubble, experiments should be conducted on large scale models and miniaturized pulsed-wire probes and/or laser-Doppler anemometry should be used.' 2. M o r e definitive wal l shear stress measurements should be made. The large uncertainty in the present measurements is due to the cal ibrat ion procedure. A better—though not ideal (see Castro & Dianat 1985)—procedure would be to calibrate the probe in an ordinary turbulent boundary layer against a Preston tube. CHAPTER 7. CONCLUSIONS 128 3. Measurements of the turbulent stresses uv, and of the r.m.s. velocities v and w should be made. These measurements would not only give further insight into the structure of the flow, but would also enable a more thorough assessment of turbulence models. 4. The extension of the momentum integral analysis to external separated-reattaching flows is wor th pursuing. The use of an inverse or semi-inverse interaction procedure of the type used by Wi l l i ams (1985) should be explored for the matching of the viscous flow to the external inviscid flow. 5. A cri terion to assess the suitabil i ty of the B S H D scheme in regions of strong pressure gradients should be devised. Based on this cri terion, a "switch-ing" technique from bounded-skew-upwind differencing to upwind differencing should be considered. A t a more fundamental level, consideration should be given to the development of a skewed scheme which allows for strong pressure gradients. 6. The present results show that the blunt rectangular plate flow constitutes a severe test for discretization schemes as well as turbulence models. In both, inadequacies appear to be magnified as a result of the high streamline cur-vatures, large gradients, and the stagnation flow region. 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Fluid Mech. 137, 83-113. K I Y A , M . & S A S A K I , K . (1983b). Free-stream turbulence effects on a separation bubble. J. Wind Engng & Ind. Aero. 14, 375-386. REFERENCES 133 K I Y A , M . & S A S A K I , K . (1985a). Structure of large-scale vortices and unsteady reverse flow in the reattaching zone of a turbulent separation bubble. J. Fluid Mech. 154, 463-491. K I Y A , M . & S A S A K I , K . (1985b). Turbulence structure and unsteadiness in a separation- reattachment flow. Proc. 5th Symp. on Turbulent Shear Flows, Cornell Univ., Ithaca, 5.7-5.12. K W O N , O . K . , P L E T C H E R , R . H . & L E W I S , J . P. (1984). Predic t ion of sudden expansion flow using the boundary layer equations. J. Fluids Engng 106, 285-291. L A I , K . Y . M . (1983). Numer ica l analysis of fluid transport phenomena. P h . D . thesis, U n i v . London . L A I , K . Y . M . k G O S M A N , A . D . (1982). F in i te difference and other approxi-mations for the transport and Navier-Stokes solutions. Dept . M e c h Engng, Imperial College, Rept . F S / 8 2 / 1 6 . L A N E , J . C . & L O E H R K E , R . I. (1980). Leading edge separation from a blunt plate at low Reynolds number. J. Fluids Engng 102, 494-496. L A N G S T O N E , L . S. & B O Y L E , M . T . (1982). A new surface streamline flow visual izat ion technique. J. Fluid Mech. 125, 53-57. L A U N D E R , B . E . & S P A L D I N G , D . B . (1974). The numerical computat ion of turbulent flow. Comp. Meths. Appl. Mech. Engng 3, 269-289. L E O N A R D , B . P . (1977). News flash: upstream parabolic interpolation. Proc. 2nd GAMM Conference on Numerical Methods in Fluid Mechanics, Koln, p. 97. L E S C H Z I N E R , M . A . & R O D I , W . (1981). Calcula t ion of annular and twin par-rallel jets using various discretization schemes and turbulence-model varia-tions. J. Fluids Engng 103, 352-360. M C D O N A L D , H . & B R I L E Y , W . R . (1983). A survey of recent work on in-teracted boundary-layer theory for flow wi th separation. In Numerical and Physical Aspects of Aerodynamic Flows II (ed. T . Cebeci), pp . 141-162. Springer-Verlag. M A C A G N O , E . O . & H U N G , T . K . (1967). Pressure, Bernoul l i sum, and mo-mentum and energy relations in a laminar zone of separation. Phys. Fluids 10, 78-82. REFERENCES 134 M E I , R . W . & P L O T K I N , A . (1986). Navier-Stokes solutions for laminar incom-pressible flows in forward-facing step geometries. AIAA J. 24, 1106-1111. N A L L A S A M Y , M . (1987). Turbulence models and their applications to the pre-dict ion of internal flows: a review. Computers & Fluids, 15, 151-194. O T A , T . , A S A N O , Y . & O K A W A , J . (1981). Reattachment length and transit ion of the separated flow over blunt flat plates. Bull. JSME 24(192), 941-947. O T A , T . & I T A S A K A , M . (1976). A separated and reattached flow on a blunt flat plate. J. Fluids Engng 98, 79-84. O T A , T . & K O N , N . (1974). Heat transfer in the separated and reattached flow on a blunt flat plate. J. Heat Transfer 96, 459-462. O T A , T . &: M O T E G I , H . (1983). Measurements of spatial correlations and au-tocorrelations in a separated and reattached flow over a blunt flat plate. J . Wind Engng & Ind. Aero. 12, 297-312. O T A , T . & N A R I T A , M . (1978). Turbulence measurements in separated and reattached flow over a blunt flat plate. J. Fluids Engng 100, 224-228. P A R K I N S O N , G . V . & J A N D A L I , T . (1970). A wake source model for bluff body potential flow. J. Fluid Mech. 40, 577-594. P A T A N K A R , S. V . (1980). Numerical Heat Transfer and Fluid Flow. M c Graw-H i l l (Hemisphere), New York . R A I T H B Y , G . D . (1976a). A cr i t ical evaluation of upstream differencing applied to problems involving fluid flow. Comp. Meths Appl. Mech. Engng 9, 75-103. R A I T H B Y , G . D . (1976b). Skew upwind differencing schemes for problems in-volv ing fluid flow. Comp. Meths Appl. Mech. Engng 9, 153-164. R E Y H N E R , T . A . & F L T J G G E - L O T Z , I. (1968). The interaction of a shock wave w i t h a laminar boundary layer. Int. J. Non-linear Mechanics 3, 173-199. R E Y N O L D S , A . J . (1974). Turbulent Flows in Engineering. John Wi ley & Sons, London . R O D I , W . (1984). Turbulence models and their applicat ion in hydraul ics—a state of the art review. International Associat ion for Hydraul ics Research, Delft, the Netherlands. REFERENCES 135 R O S H K O , A . & L A U , J . C . (1965). Some observations on transi t ion and reat-tachment of a free shear layer in incompressible flow. Proc. Heat Transfer & Fluid Mech. Inst. 18 (ed. A . F . Charwat t ) , pp . 157-167. Stanford Universi ty Press. R U D E R I C H , R . & F E R N H O L Z , H . H . (1986). A n experimental investigation of a turbulent shear flow wi th separation, reverse flow, and reattachment. J. Fluid Mech. 163, 283-322. S C H L I C H T I N G , H . (1968). Boundary Layer Theory, 6 th edit ion. M c G r a w - H i l l , New York . S I M P S O N , R . L . (1981). A review of some phenomena in turbulent flow separa-t ion . J. Fluids Engng 103, 520-532. S I M P S O N , R . L . (1985). Two-dimensional turbulent separated flow. AGARDo-graph No. 287. S M Y T H , R . (1979). Turbulent flow over a plane symmetric sudden expansion. J. Fluids Engng 101, 348-353. T O W N S E N D , A . A . (1976). The Structure of Turbulent Shear Flow, 2nd edition. Cambridge Universi ty Press. 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 in 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). Exper imen-ta l study of flow reattachment in a single sided sudden expansion. N A S A Contractor Report 3765. W I L L I A M S , B . R . (1985). The prediction of separated flow using a viscous-inviscid 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). The two-dimensional mix ing layer. J. Fluid Mech. 41,327-361. Appendix A Tabulated D a t a A complete set of mean velocity (U), r.m.s. velocity ( V r ) , and forward flow fraction ( 7 ) profiles are given for reference. A l l measurements were made wi th a pulsed-wire anemometer at a Reynolds number of 5 x 10 4 . The tabulated values are presented 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 X/Xr= 0.000 Uref= 8.38 M/SEC Y/D Y / X r U/Uref u/Uref GAMMA 0. 0759 0 .0163 0.9508 0. 0276 1 .0000 0. 0843 0 .0182 0.9479 0. 0251 1 .0000 0. 1 1 24 0 .0242 0.9473 0. 0207 1 .0000 0. 1687 0 .0364 0.9584 0. 0203 1 .0000 0. 2249 0 .0485 0.9654 0. 0209 1 .0000 0. 2812 0 .0607 0.9797 0. 0204 1 .0000 0. 3374 0 .0728 0.9878 0. 0198 1 .0000 0. 3937 0 .0850 0.9907 0. 0196 1 .0000 0. 4499 0 .0971 0.9932 0. 0196 1 .0000 0. 5624 0 .1214 1.0046 0. 0197 1 .0000 0. 6749 0 . 1457 1.0109 0. 0191 1 .0000 0. 7874 0 . 1700 1.0176 0. 0196 1 .0000 0. 8998 0 . 1 943 1.0258 0. 01 97 1 .0000 1. 0123 0 .2186 1.0291 0. 0198 1.0000 1. 1811 0 .2550 1.0231 0. 0205 1 .0000 1. 3498 0 .291 5 1.0287 0. 0198 1 .0000 1. 5185 0 .3279 1.0353 0. 0202 1 .0000 1. 6872 0 . 3644 1.0414 0. 0212 1 .0000 1. 8560 0 .4008 1.0483 0. 021 1 1 .0000 2. 0247 0 . 4373 1.0539 0. 0201 1.0000 FILE NU.PW.JAN85B STATION X/D= 1.000 X/Xr= 0.216 Uref= 8.36 M/SEC Y/D Y / X r U/Uref u /Uref GAMMA 0 .0758 0. 0163 -0 .2404 0. 1 543 0.0250 0 .0843 0. 0182 -0 .2421 0. 1614 0.0240 0 .1124 0. 0242 -0 .2296 0. 1 604 0.0390 0 .1406 0. 0303 -0 .2225 0. 1717 0.0480 0 . 1 687 0. 0364 -0 .2006 0. 1846 0.0680 0 .1968 0. 0425 -0 .1823 0. 1895 0.1050 0 .2249 0. 0485 -0 .1516 0. 2091 0. 1730 0 .2530 0. 0544 -0 .1099 0. 2133 0.2760 0 .281 2 0. 0607 -0 .0404 0. 2297 0.4220 0 .3093 0. 0668 0 .0385 0. 2355 0.5760 0 .3374 0. 0728 0 . 1 535 0. 2388 0.7590 0 .3655 0. 0789 0 .2853 0. 2453 0.8770 0 .3937 0. 0850 0 .4087 0. 2525 0.9560 0 .4218 0. 091 1 0 .5505 0. 2606 0.9870 0 .4499 0. 0971 0 .7148 0. 2728 0.9970 0 .4780 0. 1032 0 .8664 0. 2664 1.0000 0 .5061 0. 1093 1 .0319 0. 2478 1 .0000 0 .5343 0. 1 1 54 1 . 1 623 0. 2041 1 .0000 0 .5624 0. 1214 1 .2434 0. 1578 1 .0000 0 .6186 0. 1336 1 .2919 0. 0792 1.0000 0 .6749 0. 1457 1 .2828 0. 0555 1.0000 0 .7311 0. 1 579 1 .2662 0. 0456 1.0000 0 .7874 0. 1700 1 .2541 0. 0424 1.0000 0 .8998 0. 1943 1 .2266 0. 0369 1.0000 1 .0123 0. 2186 1 .2069 0. 0338 1.0000 1 .1248 0. 2429 1 . 1 949 0. 0324 1.0000 1 .2373 0. 2672 1 .1823 0. 0295 1.0000 1 .3498 0. 291 5 1 .1693 0. 0266 1.0000 1 .5185 0. 3275 1 .1517 0. 0246 1.0000 1 .6872 0. 3644 1 .1417 0. 0244 1.0000 1 .8560 0. 4008 1 .1395 0. 0249 1.0000 2 .0247 0. 4373 1 . 1 332 0. 0246 1.0000 F I L E N U . P W . J A N 8 5 C S T A T I O N X/D= 2 . 0 0 0 X/Xr= 0 . 4 3 2 Uref= 8 . 3 2 M / S E C Y / D Y / X r U/Uref u /Uref GAMMA 0 . 0 7 5 9 0 . 0 1 6 3 - 0 . 3 2 5 8 0 . 1731 0 . 0 3 9 0 0 . 0 8 4 3 0 . 0 1 8 2 - 0 . 3 2 3 9 0 . 1774 0 . 0 3 7 0 0 . 11 24 0 . 0 2 4 2 - 0 . 2 9 6 3 0 . 1915 0 . 0 7 1 0 0 . 1 4 0 6 0 . 0 3 0 3 - 0 . 2 6 7 0 0 . 2 0 7 3 0 . 0 9 6 0 0 . 1 687 0 . 0 3 6 4 - 0 . 2342 0 . 2 2 1 7 0 . 1 420 0 . 1 9 6 8 0 . 0 4 2 5 - 0 . 1 994 0 . 2 3 7 6 0 . 1 920 0 . 2 2 4 9 0 . 0 4 8 5 - 0 . 1 508 0 . 2521 0 . 2 4 8 0 0 . 2 5 3 0 0 . 0 5 4 6 - 0 . 1016 0 . 2 6 3 3 0 . 3 2 5 0 0 . 2 8 1 2 0 . 0 6 0 7 - 0 . 0 5 4 5 0 . 271 1 0 . 4 0 6 0 0 . 3 0 9 3 0 . 0 6 6 8 - 0 . 0051 0 . 2 8 3 8 0 . 5 1 5 0 0 . 3 3 7 4 0 . 0 7 2 8 0 . 0 6 7 9 0 . 2 8 6 4 0 . 6 2 4 0 0 . 3 6 5 5 0 . 0 7 8 9 0 . 1 302 0 . 2 8 8 8 0 . 6 9 7 0 0 . 3 9 3 7 0 . 0 8 5 0 0 . 221 1 0 . 2 8 8 6 0 . 7 8 6 0 0 . 4 2 1 8 0 . 0 9 1 1 0 . 2982 0 . 2 8 7 9 0 . 8 5 4 0 0 . 4 4 9 9 0 . 0 9 7 1 0 . 381 1 0 . 2 8 7 7 0 . 9 0 9 0 0 . 4 7 8 0 0 . 1 0 3 2 0 . 4 6 1 2 0 . 2 8 8 8 0 . 9 4 2 0 0 . 5 0 6 1 0 . 1 0 9 3 0 . 5531 0 . 2 9 5 6 0 . 9 7 1 0 0 . 5 3 4 3 0 . 11 54 0 . 6 6 4 0 0 . 2931 0 . 9 8 6 0 0 . 5 6 2 4 0 . 1 2 1 4 0 . 7 4 2 2 0 . 2 9 1 5 0 . 9 9 4 0 0 . 6 1 8 6 0 . 1 336 0 . 9 4 7 8 0 . 2 7 6 2 0 . 9 9 9 0 0 . 6 7 4 9 0 . 1 4 5 7 1 . 1 147 0 . 2341 1 . 0 0 0 0 0 . 7 3 1 1 0 . 1 5 7 9 1. 2 3 2 2 0 . 1804 1 . 0 0 0 0 0 . 7 8 7 4 0 . 1 7 0 0 1. 2882 0 . 1 223 1 . 0 0 0 0 0 . 8 9 9 8 0 . 1 9 4 3 1. 2 9 2 5 0 . 0 6 8 6 1 . 0 0 0 0 1 . 0 1 2 3 0 . 2 1 8 6 1. 2 7 3 7 0 . 0 4 9 3 1 . 0 0 0 0 1 . 1 2 4 8 0 . 2 4 2 9 1. 2551 0 . 0431 1 . 0 0 0 0 1 . 2 3 7 3 0 . 2 6 7 2 1. 2 4 1 3 0 . 0 3 9 7 1 . 0 0 0 0 1 . 3 4 9 8 0 . 2 9 1 5 1. 2 2 9 3 0 . 0 3 6 8 1 . 0 0 0 0 1 . 5 1 8 5 0 . 3 2 7 9 1. 2041 0 . 0 3 3 4 1 . 0 0 0 0 1 . 6 8 7 2 0 . 3 6 4 4 1. 1843 0 . 0 2 7 7 1 . 0 0 0 0 1 . 8 5 6 0 0 . 4 0 0 8 1. 1776 0 . 0261 1 . 0 0 0 0 2 . 0 2 4 7 0 . 4 3 7 3 1. 1702 0 . 0 2 6 0 1 . 0 0 0 0 F I L E N U . P W . J A N 8 5 D S T A T I O N X/D= 3 . 0 0 0 X/Xr= 0 . 6 4 8 Uref= 8 . 3 4 M / S E C Y/D Y / X r U /Ure f u /Uref GAMMA 0 . 0 7 5 9 0 . 0 1 6 3 - 0 . 2 6 6 3 0 . 2 1 2 0 0 . 1020 0 . 0 8 4 3 0 . 0 1 8 2 - 0 . 2 5 8 0 0 . 2 1 4 6 0 . 1 1 20 0 . 1 124 0 . 0 2 4 2 - 0 . 2 1 5 4 0 . 2 3 7 5 0 . 1570 0 . 1 4 0 6 0 . 0 3 0 3 - 0 . 1 8 4 5 0 . 2 4 8 0 0 . 2 2 9 0 0 . 1 687 0 . 0 3 6 4 - 0 . 1 4 1 3 0 . 2 6 8 8 0 . 2 8 5 0 0 . 1 9 6 8 0 . 0 4 2 5 - 0 . 0 9 9 2 0 . 2 7 7 6 0 . 3 4 5 0 0 . 2 2 4 9 0 . 0 4 8 5 - 0 . 0 5 6 9 0 . 2 9 3 9 0 . 4 2 9 0 0 . 2 5 3 0 0 . 0 5 4 6 - 0 . 0 0 9 1 0 . 2 9 7 8 0 . 4 8 1 0 0 . 2 8 1 2 0 . 0 6 0 7 0 . 0 6 1 4 0 . 3 0 4 1 0 . 5 7 3 0 0 . 3 0 9 3 0 . 0 6 6 8 0 . 1 1 5 7 0 . 3 1 6 6 0 . 6 5 2 0 0 . 3 3 7 4 0 . 0 7 2 8 0 . 1 7 7 7 0 . 3 1 5 0 0 . 7 1 1 0 0 . 3 6 5 5 0 . 0 7 8 9 0 . 2 4 4 1 0 . 3 1 4 6 0 . 7 9 0 0 0 . 3 9 3 7 0 . 0 8 5 0 0 . 3 0 4 1 0 . 3 1 3 9 0 . 8 4 1 0 0 . 4 2 1 8 0 . 091 1 0 . 3 7 6 5 0 . 3 1 3 9 0 . 8 9 4 0 0 . 4 4 9 9 0 . 0971 0 . 4 4 8 9 0 . 3 0 9 1 0 . 9 3 2 0 0 . 4 7 8 0 0 . 1 0 3 2 0 . 5 1 2 2 0 . 3 1 0 2 0 . 9 5 5 0 0 . 5 0 6 1 0 . 1093 0 . 5 9 7 6 0 . 3 0 4 0 0 . 9 7 0 0 0 . 5343 0 . 1 1 54 0 . 6 5 8 0 0 . 3 0 2 1 0 . 9 8 5 0 0 . 5 6 2 4 0 . 1 241 0 . 7 3 4 8 0 . 2 9 9 7 0 . 9 8 9 0 0 . 6 1 8 6 0 . 1 3 3 6 0 . 8 8 3 9 0 . 2 7 9 6 0 . 9 9 8 0 0 . 6 7 4 9 0 . 1 4 5 7 1 . 0 0 5 3 0 . 2 4 7 2 1 . 0 0 0 0 0 . 7 3 1 1 0 . 1 5 7 9 1 . 1 0 5 8 0 . 2 1 0 7 1 . 0 0 0 0 0 . 7 8 7 4 0 . 1700 1 . 1 7 3 0 0 . 1672 1 . 0 0 0 0 0 . 8 4 3 6 0 . 1822 1 . 2 0 2 0 0 . 1 3 4 2 1 . 0 0 0 0 0 . 8 9 9 8 0 . 1943 1 . 2 2 1 7 0 . 0 9 7 8 1 . 0 0 0 0 0 . 9 5 6 1 0 . 2 0 6 5 1 . 2 1 7 0 0 . 0 8 1 9 1 . 0 0 0 0 1 . 0 1 2 3 0 . 2 1 8 6 1 . 2 2 1 9 0 . 0 6 6 6 1 . 0 0 0 0 1 . 1 2 4 8 0 . 2 4 2 9 1 . 2 0 9 3 0 . 0 4 6 4 1 . 0 0 0 0 1 . 2 3 7 3 0 . 2 6 7 2 1 . 2 0 0 3 0 . 0 3 7 9 1 . 0 0 0 0 1 . 3 4 9 8 0 . 2 9 1 5 1 . 1 9 3 1 0 . 0 3 3 4 1 . 0 0 0 0 1 . 5 1 8 5 0 . 3 2 7 9 1 . 1 8 0 8 0 . 0 2 9 8 1 . 0 0 0 0 1 . 6 8 7 2 0 . 3 6 4 4 1 . 1 6 9 1 0 . 0 2 8 6 1 . 0 0 0 0 1 . 8 5 6 0 0 . 4 0 0 8 1 . 1 6 1 9 0 . 0 2 7 2 1 . 0 0 0 0 2 . 0 2 4 7 0 . 4 3 7 3 1 . 1582 0 . 0 2 6 8 1 . 0 0 0 0 FILE NU.PW.JAN85E STATION X/D= 4.000 X/Xr= 0.864 Uref= 8.31 M/SEC FILE NU.PW.JAN85F STATION X/D= 4.500 X/Xr= 0.972 Uref= 8.34 M/SEC y/D Y / X r U/Uref u/Uref GAMMA y/D Y / X r U/Uref u /Uref GAMMA 0.0759 0.0163 -0.0193 0.2450 0.4690 0.0759 0.0163 0.1413 0.2395 0.7340 0.0843 0.0182 -0.0001 0.2516 0.4870 0.0843 0.0182 0.1518 0.2413 0.7600 0.1124 0.0242 0.0239 0.2547 0.5110 0.1124 0.0242 0.1718 0.2415 0.7790 0.1406 0.0303 0.0597 0.2621 0.5850 0.1406 0.0303 0.2034 0.2470 0.8070 0.1687 0.0364 0.0898 0.2706 0.6270 0. 1687 0.0364 0.2296 0.2506 0.8330 0.1968 0.0425 0.1251 0.2799 0.6820 0. 1968 0.0425 0.2557 0.2570 0.8420 0.2249 0.0485 0.1647 0.2828 0.7210 0.2249 0.0485 0.2757 0.2619 0.8760 0.2530 0.0546 0.2120 0.2879 0.7710 0.2530 0.0546 0.3224 0.2685 0.8980 0.281 2 0.0607 0.2590 0.2850 0.8160 0.2812 0.0607 0.3537 0.2721 0.9140 0.3093 0.0668 0.2942 0.2888 0.8510 0.3093 0.0668 0.3892 0.2739 0.9330 0.3374 0.0728 0.3350 0.2959 0.8980 0.3374 0.0726 0.4171 0.2742 0.9500 0.3655 0.0789 0.3890 0.2964 0.9120 0.3655 0.0789 0.4652 0.2762 0.9620 0.3937 0.0850 0.4392 0.3015 0.9380 0.3937 0.0850 0.5005 0.2774 0.9700 0.4218 0.091 1 0.4934 0.3003 0.9510 0.4218 0.0911 0.5372 0.2764 0.9760 0.4499 0.0971 0.5408 0.2966 0.9720 0.4499 0.0971 0.581 1 0.2792 0.9820 0.47B0 0.1032 0.5932 0.2960 0.9810 0.4780 0.1032 0.6405 0.2802 0.9870 0.5061 0.1093 0.6454 0.2894 0.9860 0.5061 0.1093 0.6722 0.2799 0.9920 0.5343 0.1154 0.6918 0.2882 0.9890 0.5343 0.1154 0.7159 0.2733 0.9940 0.5624 0.1214 0.7443 0.2783 0.9930 0.5624 0.1214 0.7515 0.2665 0.9960 0.6186 0. 1336 0.8341 0.2631 0.9970 0.6186 0.1336 0.8334 0.2523 0.9970 0.6749 0. 1457 0.9308 0.2366 0.9980 0.6749 0.1456 0.8979 0.2312 0.9990 0.731 1 0.1579 0.9951 0.2125 1.0000 0.731 1 0. 1579 0.9601 0.2061 1.0000 0.7874 0.1700 1.0569 0.1791 1.0000 0.7874 0.1700 1.0099 0.1839 1.0000 0.8436 0.1822 1 .0908 0.1550 1.0000 0.8436 0. 1822 1 .0503 0.1499 1.0000 0.8998 0.1943 1.1163 0.1247 1.0000 0.8998 0.1943 1.0700 0.1304 1.0000 0.9561 0.2065 1.1265 0.1013 1.0000 0.9561 0.2065 1.0812 0.1084 1.0000 1.0123 0.2186 1 . 1326 0.0813 1.0000 1.0123 0.2186 1 .0902 0.0872 1 .0000 1 . 1248 0.2429 1 . 1346 0.0588 1.0000 1 .1248 0.2429 1.1008 0.0648 1.0000 1 .2373 0.2672 1 . 1349 0.0484 1.0000 1 .2373 0.2672 1.1001 0.0532 1 .0000 1 . 3498 0.2915 1.1319 0.0416 1.0000 1.3498 0.2915 1.0907 0.0446 1 .0000 1.5185 0.3279 1.1242 0.0357 1.0000 1.5185 0.3279 1.0935 0.0381 1 .0000 1.6872 0.3644 1.1199 0.0313 1.0000 1.6872 0.3644 1 .0880 0.0333 1 .0000 1.8560 0.4008 1.1169 0.0309 1.0000 1.8560 0.4008 1.0878 0.0315 1 .0000 2.0247 0.4373 1.1215 0.0284 1.0000 2.0247 0.4373 1 .0944 0.0313 1 .0000 FILE NU.PW.JAN8 5G STATION X/D= 5.000 X/Xr= 1.080 U r e £ = 8.35 M/SEC Y / D Y / X r U/Uref u/Uref GAMMA 0 .0759 0 .0163 0.2831 0. 2191 0. 9240 0 .0843 0 .0182 0.2927 0. 2199 0. 9310 0 . 1 124 0 .0242 0.3058 0. 2197 0. 9400 0 . 1 406 0 .0303 0.3302 0. 2240 0. 9520 0 .1687 0 .0364 0.3505 0. 2282 0. 9530 0 .1968 0 .0425 0.3762 0. 2292 0. 9610 0 .2249 0 .0485 0.3971 0. 2358 0. 9680 0 .2530 0 .0546 0.4300 0. 2448 0. 9710 0 .2812 0 .0607 0.4540 0. 2482 0. 9730 0 .3093 0 .0668 0.4804 0. 2515 0. 9740 0 .3374 0 .0728 0.5132 0. 2595 0. 9850 0 .3655 0 .0789 0.5534 0. 2624 0. 9880 0 .3937 0 .0850 0.5864 0. 2655 0. 9900 0 .4218 0 .091 1 0.6231 0. 2646 0. 9900 0 .4499 0 .0971 0.6573 0. 2670 0. 9940 0 .4780 0 .1032 0.6863 0. 2618 0. 9960 0 .5061 0 .1093 0.721 3 0. 2564 0. 9970 0 .5343 0 . 1 1 54 0.7671 0. 2537 0. 9980 0 .5624 0 .1214 0.7981 0. 2466 0. 9980 0 .6186 0 .1336 0.8591 0. 2362 0. 9990 0 .6749 0 .1457 0.9097 0. 2210 1. 0000 0 .731 1 0 .1579 0.9551 0. 1 948 1 . 0000 0 .7874 0 .1700 1.0018 0. 1725 1 . 0000 0 .8436 0 .1822 1.0257 0. 1 523 1 . 0000 0 .8998 0 .1943 1.0531 0. 1 281 1. 0000 0 .9561 0 .2065 1.0648 0. 1024 1 . 0000 1 .0123 0 .2186 1.0743 0. 091 1 1 . 0000 1 . 1 248 0 .2429 1.0863 0. 0628 1 . 0000 1 .2373 0 .2672 1.0857 0. 0536 1 . 0000 1 .3498 0 .2915 1.0862 0. 0422 1 . 0000 1 .5185 0 .3279 1.0921 0. 0355 1. 0000 1 .6872 0 .3644 1.0885 0. 0303 1 . 0000 1 .8560 0 .4008 1.0877 0. 0289 1 . 0000 2 .0247 0 .4373 1.091 3 0. 0276 1. 0000 FILE NU.PW.JAN85H STATION X/D= 6.000 X/Xr= 0.296 Uref= 8.39 M/SEC Y/D Y / X r U/Uref u/Uref GAMMA 0 .0675 0. 0145 0.4623 0 .2018 0 . 9960 0 .0758 0. 0163 0.4707 0 .2031 0 .9970 0 .0842 0. 0182 0.4753 0 .2007 0 .9970 0 . 1 1 24 0. 0241 0.4976 0 .2032 0 .9980 0 . 1 406 0. 0302 0.5081 0 .2073 0 .9980 0 .1685 0. 0364 0.5243 0 .2091 0 .9980 0 . 1 967 0. 0425 0.5423 0 .2113 0 .9980 0 .2248 0. 0485 0.5614 0 .2175 0 .9980 0 .2530 0. 0545 0.5778 0 .2214 0 . 9990 0 .2812 0. 0607 0.5958 0 .2255 0 .9980 0 .3092 0. 0668 0.6183 0 .2301 0 .9980 0 .3373 0. 0728 0.6383 0 .2332 0 .9990 0 .3654 0. 0789 0.6573 0 .2353 0 .9990 0 .3937 0. 0850 0.6864 0 .2385 0 .9990 0 .4217 0. 0910 0.7070 0 .2383 0 .9990 0 .4499 0. 0971 0.7274 0 .2346 1 .0000 0 .4780 0. 1 032 0.7459 0 .2312 1 .0000 0 .5061 0. 1 091 0.7730 0 .2305 .9990 0 .5342 0. 1 154 0.7983 0 .2281 1 .0000 0 .5624 0. 1214 0.8202 0 .2242 .9990 0 .5904 0. 1 275 0.8378 0 .2228 1 .0000 0 .6185 0. 1 336 0.8621 0 .2171 1 .0000 0 .6749 0. 1 456 0.9035 0 .2031 1 .0000 0 .7310 0. 1 578 0.9372 0 .1879 1 .0000 0 .7874 0. 1700 0.965B 0 .1698 1 .0000 0 .8435 0. 1 821 0.9906 0 .1519 1 .0000 0 .8998 0. 1942 1.0155 0 . 1 322 1 .0000 0 .9560 0. 2064 1 .0320 0 .1161 1 .0000 1 .0123 0. 2185 1 .0428 0 .1038 1 .0000 1 . 1 248 0. 2428 1.0555 0 .0717 1 .0000 1 .2373 0. 2672 1.0627 0 .0565 1 .0000 1 .3498 0. 2915 1.0661 0 .0502 1 .0000 1 .5185 0. 3278 1.0660 0 .0374 1 .0000 1 .6871 0. 3643 1.0673 0 .0318 1 .0000 1 .8560 0. 4007 1.0657 0 .0281 1 .0000 2 .0246 0. 4371 1.0656 0 .0253 1 .0000 FILE NU.PW.JAN85I STATION X/D= 7.000 X/Xr= 1.512 Uref= 8.42 M/SEC Y/D Y / X r U/Uref u /Uref GAMMA 0. 0673 0. 0145 0. 5762 0. 1885 0.9990 0. 0758 0. 0163 0. 5769 0. 1881 1.0000 0. 0842 0. 0182 0. 5848 0. 1884 1 .0000 0. 1 1 24 0. 024 1 0. 6022 0. 1895 0.9990 0. 1 406 0. 0302 0. 6095 0. 1898 1 .0000 0. 1 686 0. 0364 0. 6250 0. 1913 1.0000 0. 1 967 0. 0425 0. 6409 0. 1972 0.9990 0. 2248 0. 0485 0. 6544 0. 1 982 1.0000 0. 2530 0. 0545 0. 6628 0. 2004 1.0000 0. 2812 0. 0607 0. 6836 0. 2003 1 .0000 0. 3373 0. 0728 0. 7074 0. 2030 1 .0000 0. 3937 0. 0850 0. 7375 0. 2071 1.0000 0. 4499 0. 0971 0. 7642 0. 2116 1.0000 0. 5061 0. 1 091 0. 7965 0. 207 1 1.0000 0. 5624 0. 1214 0. 8284 0. 2051 1.0000 0. 6185 0. 1336 0. 8674 0. 1955 1.0000 0. 6749 0. 1456 0. 8958 0. 1876 1 .0000 0. 7310 0. 1578 0. 9146 0. 1794 1.0000 0. 7874 0. 1700 0. 9437 0. 1654 1.0000 0. 8435 0. 1821 0. 9587 0. 1 566 1.0000 0. 8998 0. 1942 0. 9806 0. 1412 1.0000 1 . 0123 0. 2185 1 . 0087 0. 1 136 1 .0000 1 . 1248 0. 2428 1 . 0241 0. 0936 1.0000 1 . 2373 0. 2672 1 . 0373 0. 0697 1.0000 1 . 3498 0. 2915 1 . 0497 0. 0538 1.0000 1 . 51B5 0. 3278 1 . 0529 0. 0426 1.0000 1 . 6871 0. 3643 1 . 0529 0. 0340 1.0000 1. 8560 0. 4007 1 . 0479 0. 0312 1.0000 2. 0247 0. 4371 1 . 0539 0. 0268 1.0000 FILE NU.PW.JAN85J STATION X/D= 8.500 X/Xr= 1.836 Uref= 8.40 M/SEC Y/D Y / X r U/Uref u /Uref GAMMA 0 0759 0.0163 0.6784 0. 1679 1.0000 0 0843 0.0182 0.6836 0. 1701 1.0000 0 1 124 0.0242 0.7030 0. 1693 1.0000 0 1 406 0.0303 0.7207 0. 1716 1.0000 0 1 687 0.0364 0.7343 0. 1724 1.0000 0 1968 0.0425 0.7414 0. 1733 1.0000 0 2249 0.0485 0.7515 0. 1757 1 .0000 0 2530 0.0546 0.7568 0. 1 761 1.0000 0 2812 0.0607 0.7735 0. 1746 1.0000 0 3093 0.0668 0.7740 0. 1743 1.0000 0 3374 0.0728 0.7795 0. 1756 1.0000 0 3937 0.0850 0.8052 0. 1780 1.0000 0 4499 0.0971 0.8199 0. 1784 1.0000 0 5061 0.1093 0.8455 0. 1757 1 .0000 0 5624 0.1214 0.8617 0. 1753 1.0000 0 6186 0.1336 0.8808 0. 1716 1.0000 0 6749 0.1457 0.9049 0. 1675 1.0000 0 7874 0.1700 0.9431 0. 1546 1.0000 0 8998 0.1943 0.9820 0. 1403 1 .0000 1 0123 0.2186 1.0141 0. 1 198 1.0000 1 1248 0.2429 1.0292 0. 0965 1.0000 1 2373 0.2672 1.0408 0. 0782 1.0000 1 3498 0.2915 1.0486 0. 0692 1.0000 1 5185 0.3279 1.0514 0. 0509 1.0000 1 6872 0.3644 1.0522 0. 0399 1.0000 1 8560 0.4008 1.0508 0. 0324 1.0000 2 0247 0.4373 1.0499 0. 0279 1.0000 Appendix B Potential Flow Analys is B . l Theory The simple model outl ined in this section is inspired directly from Parkinson's wake source model (Parkinson and Jandal i 1970). The separated shear layer is, in a first approximation, replaced by a streamline bounding an external i rrotat ional flow region on its upper side (Figure B . l ) . The irrotational flow can be modelled by introducing a surface point source, the strength and location of which are determined by applying two conditions: i) Specified base pressure at A. ii) Tangential separation. Since the body is a polygon, the flowfield in the physical plane (2-plane) can be mapped conformally onto the upper half of the transform plane (f-plane) using a Schwarz-Christoffel transformation. The solution of the flow is then readily ob-tained, once the free parameters of the problem are determined from the imposed conditions. 142 APPENDIX B. POTENTIAL FLOW ANALYSIS 143 U. Q / v V 1  Figure B . l . Wake source model . U-y. 1 A ij 6 T h © A' Figure B . 2 . Physical and transform planes. APPENDIX B. POTENTIAL FLOW ANALYSIS 144 The tangential separation condit ion, however, presents some difficulties. B u t these difficulties can be avoided by the introduction of a small teat in the physical plane at point A, as shown in figure B . 2 . A p p l y i n g the Schwarz-Christoffel transformation: dz K( , A t the cr i t ical point A , dz/d$ has a simple zero; angles are therefore doubled through the transformation. Hence, to a stagnation streamline at A in the f- plane corre-sponds a tangential separation streamline at A in the 2-plane. The relative height of the teat 8/h can be made suitably small by appropriately choosing e. In the f-plane, we have a uniform flow past a source; the resulting complex potential is: Voof + I; ln(£ - &) (B.2) and the velocity potential in the and 2-planes are given, respectively, by W(t) = ^ (B .3 ) and In the outer flow region bounded by the separation streamline, Bernoul l i ' s equation applies: p+lp\W(z)\2 = p0O + 1-pU2oo (B.5) or ir1 oo oo APPENDIX B. POTENTIAL FLOW ANALYSIS 145 A p p l y i n g the first condit ion (i.e. given base pressure pj, at A) to equation (B.8) , we obtain: W{z) = (i-cPbyi2 = k (B.7) evaluating W(z) at A gives the following relation: (B.8) and requiring A to be a stagnation point in the f-plane (2 condition) leads to Q 2hU k(l - e) (B.9) G iven k and e, the source strength, Q, and its posit ion, are determined by (B.8) and (B.9). The pressure distr ibution along the streamline can be evaluated using (B.7) , and the shape of the separating streamline is obtained by solving for the f co-ordinates satisfying 9 [ F ( f ) ] = 0 (B.10) the corresponding physical plane co-ordinates are then found using and z h h - In [ 2 > / ( f + l ) ( f - 6 ) + ( ? + 1) + (f - e) 2 7T 1 - e — tan 1y/e . + i ( B . l l ) (B.12) which are obtained by integrating equation ( B . l ) . APPENDIX B. POTENTIAL FLOW ANALYSIS 146 B.2 Results Figure B .3 shows the calculated position of the separation streamline for three differ-ent base pressures, chosen in a range corresponding to the experimentally measured base pressure coefficient of about -0.8. The calculated and 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 taking place. Calculations w i th a smal l value of e re-sult in a streamline which is too close to the wal l . In order to obtain a streamline which is more representative of the actual flow, it is necessary to use large values for e (i.e. close to 1). Al though this corresponds to large teats, this is of lit t le consequence because the region below the bounding streamline is outside the scope of the potential flow model. The results presented here are for e = 0.99. The calculated pressure dis t r ibut ion along the separated streamline is shown in Figure B .4 . A sharp dip in pressure associated w i t h the high in i t ia l curvature of the separating streamline , occurs immediately after separation, and is followed by a rapid recovery. These calculated distributions bear lit t le resemblance to the measured surface pressure distributions which exhibit a much 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 important in establishing the pressure in the actual turbulent flow, even at separation where the shear layer is very th in . APPENDIX B. POTENTIAL FLOW ANALYSIS 147 Figure B .4 . Calculated pressure distributions for different specified base pressures. Appendix C M o m e n t u m Integral A naly sis In this appendix, we consider the modelling of a separated shear layer using the boundary layer equations in their momentum integral form . The method is assessed for the fully developed flow in a sudden expansion (Figure C . l ) , which is, in this context, a simpler flow than the bluff plate flow: this flow does not have a potential flow core, and therefore does not require a viscous-inviscid matching procedure. Before discussing the assumptions which were made in this analysis, it should be pointed out that, strictly, the val idi ty of the boundary layer equations is based on the condit ion that d6 jdx <C 1. This condit ion is not met in reattaching flows as pointed out by, amongst others, Bradshaw & Wong (1972). It appears, nevertheless, that for small separation regions, the boundary layer or th in shear layer approximation retains its val idi ty: for instance, the boundary layer calculations of G h i a et al. (1975) and Br i ley & M c D o n a l d (1975) were found to compare well w i th numerical solutions using the full Navier-Stokes equations. One of the main attractions of the boundary layer equations is the relative sim-plici ty w i th which they can be solved numerically. Because they are parabolic, computat ional ly inexpensive forward marching algorithms can be used. However, 148 APPENDIX C. MOMENTUM INTEGRAL ANALYSIS 149 when marching against the streamwise velocity, which is the case in the backflow region, it is no longer possible to use a simple forward marching technique, and an iterative procedure is required. Tha t is unless the culprit for this added complica-t ion , namely the streamwise convective term in the inner region, is removed. Th i s step, which may seem radical at first sight, was first proposed by Reyhner &; FKigge-Lotz (1969) and is known as the F L A R E approximation. The implications of this approximation are examined in some detail by M c D o n a l d & Br i l ey (1983) who conclude that the approximation is, in general, only acceptable when the magnitude of the reversed flow velocities do not exceed O . I Z / O Q . Th i s , based on experimental observations, implies that the approximation is val id for laminar flows. Th i s conclusion is borne out by the results of K w o n et al. (1984), who com-puted a laminar sudden expansion flow, using the boundary layer equations and the F L A R E approximation. They obtained results which are in good agreement w i th Navier-Stokes predictions as well as experimental observations. A major difference of the present formulation wi th 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 The variables for the model are i l lustrated in Figure C . l . The flow is fully developed at separation, and, for the purpose of the analysis, is divided into two regions. The boundary layer equation in its momentum integral form is used for both regions, but w i th different assumptions. In the "outer" region, corresponding to U > 0, the velocity profiles are assumed to be self-similar; whereas in the backflow region, the wal l shear stress and the fluid momentum are neglected ( F L A R E approximation). APPENDIX C. MOMENTUM INTEGRAL ANALYSIS 150 / / / / ' / / / / / / / / / / / / / / / / / / / / / / / / / / / / u(x,y) / / / / A - - " T 6(0) v A ® 6(X) ER = JJ/6(0) -o x Figure C . l . Mode l l ing of 2-D sudden expansion flow. The 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 laminar flow, the boundary layer equations take the form (Schlichting 1968) ox oy dU dV dx dy ldp d2U p dx dy2 = 0 ( C . l ) (C.2) Equa t ion ( C . l ) is integrated in the outer region between y = 0 and y = 6(x), yielding ± [ * ) V i l l — f s £ + * n (C.3) ax Jo ax p The outer region velocity profiles are assumed to be of the form ^ = /(r?) + A f f(r ?) (C.4) APPENDIX C. MOMENTUM INTEGRAL ANALYSIS 151 where /(??) and g(n) are polynomial functions in term of the dimensionless co-ordinate n = y/6(x), and A is a dimensionless pressure gradient parameter defined by ' 1 2 D " ( C . 5 ) pvU0 dx Subst i tut ing for U using (C .4) and (C.5), equation (C.3) yields, for a cubic polyno-m i a l profile, (Cx + 2 A C 2 + A 2 C 3 )[Ul6)> + 2 ( C 2 + AC 3)t/ 0 2<5A' = - (J + ±) ^ ( C .6 ) where primes denote differentiation w.r.t . i and C\, C 2 and C 3 are constants deter-mined by the chosen velocity profile. These constants are listed at the end of this appendix. Similar ly, integration of the continuity equation (C.2) through the outer region, gives d /"*(*) -T Udy = 0 dx Jo or, replacing for U {C4 + KC5){U0S)' + C5U0A' = 0 (C.7) F ina l ly , using the F L A R E approximation and neglecting the wa l l shear stress, in-tegration of equation ( C . l ) through the inner region yields pdx p using (C.5) and subst i tut ing for r$, this gives A (4j - 3) = - 6 (C.8) APPENDIX C. MOMENTUM INTEGRAL ANALYSIS 152 The set of equations (C.6),(C.7) and (C.8) were solved numerically for UQ, 6 and A , using a Runge -Kut t a routine, subject to the in i t ia l conditions U0{0) = U. 6{0) = H - h (C.9) A similar set of equations is obtained when a quartic polynomial instead of a cubic is chosen for the velocity profile. C.2 Turbulent Flow The formulation of the turbulent flow problem proceeds in essentially the same fashion, using the t ime averaged form of the boundary layer equations 1 . U ^ + V— - - - ^ - — ( — - uv) (CIO) dx dy pdx p dy ^ dy Puv> dU dV , <te + a ? = 0 < c - n > The addit ional turbulent or Reynolds stress terms, —puv which appears in equa-t ion (C.10) is modelled using the well known eddy viscosity concept. This assumes that the mean turbulent stress is, in analogy to laminar stresses, proport ional to the mean velocity gradient, i.e. dU - puv - put —— (C12) dy where ut, the apparent or eddy viscosity, is a coefficient for the turbulent diffusivity of momentum. Add i t i ona l assumptions are required to determine ut. 1The rationale for the time averaging procedure and turbulence modelling are discussed in Chap-ter 4. APPENDIX C. MOMENTUM INTEGRAL ANALYSIS 153 Since we observed in section 3.5 that the separated shear layer appears to have a structure s imilar to a 2-D free mix ing layer in the region extending from separation to about half the length of the separation bubble, Prandt l ' s free shear layer model for ut was used over the first half of bubble. The model , which assumes ut to be constant across the shear layer, takes the form ut = ab{Ui - U2) (C.13) where a is an empir ical constant, b a characteristic w id th , and U\ — U2 is the velocity difference across the layer. In the case of a two-dimensional mix ing layer, w i t h Ui = UQ and U2 = 0 , Schlichting (1968) defines a w id th 6 0 . i measured between the positions where (u /E / ) 2 = 0.1 and (u/U)2 = 0 .9, and gives a = 0.014. Hence ut = .0146 0 .i^o The rate of growth of the shear layer is given by or, after integration, Therefore dbn i —r1 = .098 ax 6 0 . i = .098(x + x 0 ) ut = .00137(x + x0)U0 (C.14) where the v i r tua l or igin, x0, is determined from „ _ M o ) Xn — .098 A simple way of accounting for the experimentally observed behaviour of the separated shear layer in Section 3.5, is to apply equation (C.14) for 0 < x/xr < 0.5 APPENDIX C. MOMENTUM INTEGRAL ANALYSIS 154 and then "freeze" the value of vt for x/xr > 0.5. Because xr is not known a priori, an iterative procedure is required. Us ing a modest under-relaxation (0.9), a solution was obtained after 3 to 6 iterations, depending how good—or bad—the in i t ia l guess for x r was. A convergence cri terion of less than 0.5% change in xT between two successive iterations was used. C.3 Results Acr ivos & Schrader (1982) noted, for laminar flow, that the non-dimensionalisation of the boundary layer equations by the Reynolds number leads to a reattachment length wh ich w i l l always be proport ional to the Reynolds number. Th i s expected linear variat ion is clearly demonstrated in Figure C.2 , which shows the predictions of the present method and other more detailed calculations. The momentum integral method gives, w i t h a fourth order polynomial , results which compare well wi th those obtained from finite difference solutions. Figure C.3 shows the effect of varying the expansion ratio on reattachment length (note the non-dimensionalisation by Re) . For ER > 2, the results show a decrease in reattachment length w i t h increasing ER in agreement w i th the predictions of Acr ivos & Schrader. The i r results, however, show a reversal of this trend for ER < 2. They at tr ibuted this to larger wa l l shear stresses associated w i th the smaller expansion ratios. The integral method cannot be expected to represent this effect since it assumes the wa l l shear stress to be negligible. The calculated pressure dis tr ibut ion for a 5:2 expansion ratio is shown in Figure 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 wal l shear stress. APPENDIX C. MOMENTUM INTEGRAL ANALYSIS 155 30 25 A 20 15 10 200 Figure C.2 . Var ia t ion of reattachment length w i th Reynolds number. 0.25 0.20 0.15 0.10 0.05 0.00 • R T I L Legend A CUBIC X QUARTIC • A & S ER Figure C .3 . Effect of expansion ratio on reattachment length. APPENDIX C. MOMENTUM INTEGRAL ANALYSIS 156 0.30 -i X/Xr Figure C.4 . Pressure distr ibution for a 5:2 expansion ratio. APPENDIX C. MOMENTUM INTEGRAL ANALYSIS 157 Turbulent flow calculations were carried out over a range of expansion ratios, and the resulting reattachment lengths, which are independent of Reynolds number, are shown in Figure C.5 compared w i th experimental results available in the literature. The agreement is surprisingly good, considering the sweeping assumptions made in the integral method and the eddy viscosity model. Figure C.6 shows the calculated pressure distr ibution for ER = 1.25 compared to the experimental results of Adams et al. (1984). Though the pressures recover to a s imilar value at reattachment, the discrepancy between the two distributions is rather large. Th i s confirms the inadequacy of the F L A R E approximation for turbulent flows wi th large backflow velocities (Adams et al. reported max imum backflow velocities of the order of 0.2U}. C.4 Closing Remarks The applicat ion of the momentum integral analysis to the sudden expansion flow was intended to be the first step towards the application of this method to the blunt plate problem. For this type of external flow an iterative viscous-inviscid matching procedure is required. In the classical matching scheme, the displacement thickness is specified for an inviscid calculat ion, and the resulting pressure dis tr ibut ion. This scheme was applied to the backward facing step flow, but convergence could not be obtained. It was found subsequently that a similar problem has been encountered in many studies involving separation, and that an inverse viscous-inviscid interaction procedure should be adopted for such problems (see Wi l l i ams 1985). Because of t ime constraints, this procedure was not tr ied. APPENDIX C. MOMENTUM INTEGRAL ANALYSIS 158 10 8-6-X I • 2--e—e-x -G Legend O PRESENT THEORY X DURST & TROPEA 1981 • MOHSEN 1967 8 WAUSCHKUHN 4 RAM 1975 S ETHERIDGE k KEMP 1978 X SMYTH 1979 • WESTPHAL «t al. 1984 9 ADAMS •) al. 1984 1.2 1.4 1.6 ER 1.8 —1 2.2 Figure C .5 . Effect of expansion ratio on reattachment length for turbulent flow (Data of Mohsen and Wauschkuhn & R a m compiled by Durs t & Tropea (1981). 0.25-0.4 0.6 X / X r Figure C.6 . Comparison of calculated and experimental pressure distr ibution for turbulent flow. APPENDIX C. MOMENTUM INTEGRAL ANALYSIS 159 Table C . l . Integral constants for cubic velocity profile Constant Integral term Value cx rpdrj 17/35 c2 1 / fgdn Jo 19/1680 C 3 flg2dn Jo Cfdr, Jo 1/1680 c4 5/8 c5 9dn Jo 1/48 Appendix D Modif ied k - e M o d e l The k-e model does not account for streamline curvature effects. Modifications to the model which attempt to take into account these effects have led to improved predictions of recirculating flows in some cases. T w o such modifications were used in this study, and are reviewed here. Streamline Curvature Modification Exper imenta l observations (see Bradshaw 1973) show that turbulent shear stresses and the degree of anisotropy between the normal stresses are very sensitive to streamline curvature. The k-e model does not, in its "standard" form, reflect this sensitivity. Leschziner and R o d i (1981) developed an ad hoc modification, based on an algebraic stress model , which attempts to take these effects into account. They isolated the effect on the coefficient C M of extra strain rates due to curvature and arrived at the following correction: (D.l) 160 APPENDIX D. MODIFIED K-e MODEL 161 where Us is the velocity tangential to the streamline, n is the normal coordinate, and Rc is the radius of curvature. A n arbitrary lower l imi t of 0.025 on was im-posed to ensure, for physical realism, that C M does not take very smal l or negative values. Preferential Dissipation Modification The second modification was originally proposed by Hanjalic and Launder (1979). It accounts for the preferential influence of normal stresses (over shear stresses) in promoting the transfer of turbulent energy from large to small eddies and thus the rate of dissipation. The modification replaces the term representing "production of generation" in the e transport equation, i.e. 'dJT dx, (dU_ dV^ \ dy dx (D.2) by P! = C[G - CiVt ( dU dVs (D.3) dy dx J Leschziner and R o d i noted that in recirculating flows the vort ic i ty (dU/dy—dV/dx) in the above expression, bears no relation to the shear strain, and, therefore, the modification loses its intended purpose. To overcome this problem, they suggested the use of streamline coordinates to define the stresses. Th i s results in a corrected te rm which reads: P[ = i [ C x G - (D.4) where C[ = 2.24 and C" — 0.8 (giving, consistently w i th the standard k-e model, APPENDIX D. MODIFIED K-e MODEL 162 C[ — C" = Ci). The shear strain in the direction of the streamline, Sna, is given by Sns = 0.5(Syy - Sxx) sin26 + Sxy cos 20 (D.5) where 6 is the angle between the velocity vector and the z-axis , and S „ = 2dfx (D.6) dV Syy = 2 — (D.7) W dV e n , Appendix E Wal l Funct ion Treatment The k — e turbulence model neglects low Reynolds number effects which are impor-tant in the v ic in i ty of solid boundaries. A computat ionally efficient way of account-ing for these effects is to use wa l l functions to determine the near wal l turbulent kinetic energy and its rate of dissipation. The wal l function method, due to Launder & Spalding (1974), is based on two assumptions: first, the flow in the vicini ty of a solid boundary behaves locally as a one dimensional Couette flow; secondly, the near wa l l turbulence characteristics are those associated w i t h the inertial sublayer. Following Launder and Spalding, we consider a gr id 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 and equal to the wall shear stress. In terms of the non-dimensional distance y+, this implies that: y+ ~ 200 ( E . l ) where V+ = UTyPlu (E.2) and 163 APPENDIX E. WALL FUNCTION TREATMENT 164 t P VP \ W \ \ \ \ \ \ \ \ \ \ \ \ \ F igure E . l . Near wal l control volume V, = ^ (E .3) For negligible pressure gradients, the momentum equation reduces to: 1, . dU T rw , - (M + Mt) = ~ — — E.4 p dy p p or, in non-dimensional form, ( »t\ dU+ , , where U+ = ^- (E.6) The near wal l region can be subdivided into a viscous sublayer, where molecular viscosity is dominant, and an inertial sublayer, where turbulent effects are dominant: — < 1 for y+ < 11.63 — > 1 for y+ > 11.63 Integrating (E.6), we obtain, for y+ < 11.63, U+ = y+ (E.7) APPENDIX E. WALL FUNCTION TREATMENT 165 When y+ > 11.63, the flow is fully turbulent, and the mix ing length argument (Hinze 1959) gives: Ut = KpUTy (E.8) Replacing for fit, and integrating (E.5), we obtain the familiar logarithmic law of the wal l : U+ = -ln{Ey+) (E.9) /C where K is von Karman ' s constant (/c = 0.4187), and E is a constant of integration (E = 9.0). Wall shear stress In the inert ial sublayer, the flow is assumed to be in local equi l ibr ium, i.e. the pro-duction and the dissipation rate of turbulent kinetic energy are locally in balance. Th i s implies: dU ,„ . - utJ—- ~ e (E.10) dy 7 U J = e ( E - n ) but ^ K ! • = VI (E.12) p dy p therefore f / T 2 ^ ~ e (E.13) dy then mul t ip ly ing (E.13) by (E.12) we obtain: Uj = (E.14) Replacing for nt by (equation 4.12) lh = C^p- (E.15) e APPENDIX E. WALL FUNCTION TREATMENT 166 equation (E.14) gives: UT (E.16) N o w , from equation (E.9) UUT -ln(Ey+) (E.17) or, replacing for UT and rearranging, rw = pKc)/*k1/2U/]n{Ey+) for y+ > 11.63 (E.18) W h e n the node P lies in the viscous sublayer, an expression for the shear stress is obtained directly from (E.7): Tw = M — for y+ < 11.63 (E.19) yp Equations (E.18) and (E.19) are used to evaluate the near wal l diffusive fluxes in the momentum equations. Turbulent kinetic energy B o t h the source and generation terms of the /^-equation are modified for near wall computations. The wa l l shear stress is given by (see E.4) dU (E.20) or, using (E.8) , Tw = KpUTy— dy dU_UL dy Ky but rw/p = U2, therefore (E.21) APPENDIX E. WALL FUNCTION TREATMENT 167 Fina l ly , subst i tut ing for dU/dy 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 [VP U rvp u/ edy = —dy Jo Jo Ky Jo Ky+ Kyi Us = — In y+ + constant K Us * ^ M n ( £ y + ) /c Now using (E.9) and (E.16), this expression can be re-written as [VP / tdy Jo c 3 / 4 j t 3 / 2 f / H Therefore C 3 / 4 f c 3 / 2 r / + edxdy = -« Ax Ay (E.23) yp II-this expression is used, in combination w i th (E.7) or (E.9), to evaluate the source te rm — pe in the fc-equation (4.44). { -3/4,3/2 pc* k \n(Ey+)AxAy for y+ > 11.63 + " ( E - 2 4 ) pC"yp* y+AxAy for y+ < 11.63 In addi t ion, the term representing the generation of k is modified to ensure that the shear stress TW given by the wal l function is used instead of its finite difference counterpart. The generation term reads (equation 4.46): APPENDIX E. WALL FUNCTION TREATMENT 168 Near the wa l l dV/dx ~ 0, and fj,t dU/dy = rw, therefore 2 w dU_ dy (E.25) Dissipation of T . K . E . F r o m equation E.15 , e is l inked to k by subst i tut ing for <^ from equation (E.8), and making use of (E.16) gives: KyP This expression is used directly to evaluate e next to solid boundaries. (E.26) 

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