A NUMERICAL STUDY OF BUOYANT TURBULENT FLOWS …

A NUMERICAL STUDY OF BUOYANT TURBULENT

FLOWS USING LOW-REYNOLDS NUMBER k-e MODEL

by

BUNG RYEOL SEO. B.E., M.E.

A DISSERTATION

IN

MECHANICAL ENGINEERING

Submitted to the Graduate Faculty of Texas Tech University m

Partial Fulfillment of the Requirements for

the Degree of

DOCTOR OF PHILOSOPHY

Approved

May, 2001

ACKNOWLEDGEMENTS

Most of all, I would like to thank my graduate advisor. Dr. S. Parameswaran for his

guidance, persistent support, priceless advice, patience, and encouragement for this

research and my life. My thanks are extended to my dissertation committee members. Dr.

Timothy T. Maxwell, Dr. J. Walter Oler, Dr. Darryl L. James, and Dr. Zhimin Zhang, for

their constructive criticism and valuable suggestions and comments. 1 would like to-

extend my thanks to Dr. Wijesuriya P. Dayawansa and Dr. Karlene A. Hoo, who gave

valuable recommendations for this dissertation.

Sincere gratitude is also expressed to my friends M the CFDLAB, Sivapalan

Senthooran, Suthaharan Nagendran, Saman Jayantha, Kumar Aravamudhan and K

Prabaharan, for the special time we had.

I especially thank my wife, Soo Young Ahn, for her enduring love and support, and I

would like to dedicate this work to my son and daughter, Andrew and Hannah.

This research was supported by the U.S. Department of energy (DOE), Cooperative

Aggreement No. DE-FC04-5AL85832, conducted through the Amarillo National

Resource Center for Plutonium.

11

TABLE OF CONTENTS

ABSTRACT

LIST OF TABLES

LIST OF FIGURES

NOMENCLATURE

CHAPTER

I. INTRODUCTION

1.1 Problem Statement

VI

Vlll

IX

Xlll

1

1

1.2 Objectives 2

1.3 Contents of Dissertation 3

II. BUOYANT FLOW 4

2.1 Introduction 4

2.2 Literature Review 4

2.3 Buoyant, Laminar Boundary Layer Flow g

2.4 Buoyant, Turbulent Flow \ j

III. MATHE^lATICAL FORMULATION 13

3.1 Governing Equation for Momentum Transport 13

3.2 Standard k-s Model 16

3.3 Wall Functions Ig

3.4 Low-Reynolds Number k-s Model 20

3.5 Governing Equation for Energy Transport 22

3.6 Turbulent Prandtl Number 23

IV. METHOD OF COMPUTATION 27

111

4.1 Governing Equations in a General Coordinate System 27

4.2 Discretization 29

4.3 Pressure Correction 29

4.4 QUICK Scheme 31

4.5 Normal Distance Calculation 33

4.6 Solution Procedure 33

V. TEST CASE 1 : TURBULENT CHANNEL FLOW 37

5.1 Problem Statement and Boundary Conditions 37

5.2 Grid arrangement and Grid Independent Study 38

5.3 Results without Buoyant Effects 39

5.3.1 U Velocity Profile 39

5.3.2 Turbulent Kinetic Energy Profile 40

5.3.3 Turbulent Kinetic Energy Dissipation Rate Profile 41

5.4 Results with Buoyant Effects 42


5.4.2 Temperature Profile 43


5.4.4 Turbulent Kinetic Energy Rate Profile 43

VI. TEST CASE 2 : BACKWARD-FACING STEP 62

6.1 Problem Statement and Boundary Conditions 62

6.2 Grid Arrangement and Grid Independent Study 63

6.3 Influence oflnlet Boundary Conditions 65

6.4 Influence of Discretization Schemes for the Convective Terms 66

IV

6.5 Results without Buoyant Effects 66


6.5.2 V Velocity Profile 67

6.5.3 Reattachment Length 68

6.5.4 Skin Friction Coefficient Comparison 68


6.5.6 Turbulent Kinetic Energy Dissipation Rate Profile 70

6.5.7 Turbulent Prandtl Number Profile 70

6.5.8 Stanton Number Comparison 72

6.6 Results with Buoyant Effects 73

6.6.1 Reattachment Length 73

6.6.2 Turbulent Scales 74

6.6.3 Skin Friction Coefficient Comparison 76

6.6.4 Temperature Contour Comparison 77

6.6.5 Stanton Nimiber Comparison 77

6.6.6 Vortex Shedding 79

VII. CONCLUSIONS AND RECOMMENDATIONS 116

7.1 Conclusions 116

7.2 Recommendations 118

REFERENCES 119

APPENDIX 127

ABSTRACT

Numerical computation has been performed to determine the influence of buoyant

eflects on convective flows with the standard k-e and the low-Reynolds number k-s

models. The present study was motivated by the need to overcome the shortcoming of the

standard k-e model in separating and reattaching flows because of the wall-function

approach employed in the model.

The low-Reynolds number k-e model is considered to be an appropriate model for

recirculating flows because the model does not employ wall functions. Results of the two

different models are compared against the available experimental data and direct

numerical simulation (DNS) data. In this work, Kolmogorov velocity, u^={ve)^'*, is

introduced instead of shear velocity, u^=^T,„/p, to avoid the singularity that appears at

the separating and reattaching point for both thurbulence models. QUICK differencing

scheme is employed for the convective terms. Eddy-diffusivity concept is used in

modeling buoyant term. Momentum equation for the velocity field and the energy

equation for the temperature field are solved altematively because of the strong coupling

that exists between temperature and the velocity fields in a buoyant flow. Turbulent

Prandtl nimiber was allowed to vary in the low-Reynolds number k-e model to mimic the

experimental data.

Buoyant effects have been investigated with various Richardson numbers for the

backward-facing step flow. The reattachment lengths are calculated, and the skin friction

coefficient and the Stanton numbers are examined. Various separation patterns as well as

vortex shedding were observed beyond a critical Richardson number. Strouhal numbers

VI

based on the frequencies of the vortex shedding are also calculated. In addition, the

required grid configration for accurate results for low-Reynolds number k-e model has

been discussed for the backward-facing step flows. Numerical solution obtained with the

standard k-e model showed that the buoyancy-induced vortex shedding enhanced heat

transfer.

vu

LIST OF TABLES

3.1 Constants for the k-e Group of Models 25

3.2 Functions of the k-e Group of Models 26

6.1 Svmimary of the Reattachment Lengths for Various Cases 68

Vlll

LIST OF FIGURES

3.1 Time History of Turbulent Instaneous Velocity Component 14

4.1 Control Volume for Node P and Neighboring Control Volumes 27

4.2 Two possible cases of normal distance calculation 35

4.3 Normal Distance Contour for complex geometries 35

4.4 Flow Chart for Overall Procedure 36

5.1 Computational Domain and Boundary Conditions for Channel Flow 44

5.2 Grid Formation for Channel Flow(52x42 Grids with r^ =1.2) 45

5.3 Grid Independent Solution for U Profile Using Standard k-e Model 46

5.4 Grid Independent Solution for Nondimensional U Profile Using Standard k-e Model 47

5.5 Grid Independent Solution for Nondimensional k Profile Using Standard k-e Model 48

5.6 Grid Independent Solution for Nondimensional e Profile Using Standard k-e Model 49

5.7 Grid Independent Solution for U Profile Using Low-Reynolds Number k-e Model 50

5.8 Grid Independent Solution for Nondimensional U Profile Using Low-Reynolds Number k-e Model 51

5.9 Grid Independent Solution for Nondimensional k Profile Using Low-Reynolds Number k-e Model 52

5.10 Grid Independent Solution for Nondimensional e Profile Using Low-Reynolds Number k-e Model 53

5.11 Comparison of the Fully Developed U Profiles 54

5.12 Comparison of the Fully Developed Nondimensional k Profiles 5 5

IX

5.13 Comparison of the Fully Developed Nondimensional e Profiles 56

5.14 Effect of Buoyancy on U profiles 57

5.15 Effect of Buoyancy on Nondimensional U profiles 58

5.16 Effect of Buoyancy on Dimensionless Temperature 59

5.17 Effect of Buoyancy on Nondimensional k Profiles 60

5.18 Effect of Buoyancy on Nondimensional e Profiles 61

6.1 Problem Description and Boundary Conditions for Turbulent Flow 82

6.2. Grid Configuration for the Turbulent Flow

(LKE Model, 122x62 Grid) 83

6.3. Partial View of the Grid Configuration near Step End 83

6.4 Grid Independent Solution for U velocity at x/H=5.5 Using Low-Reynolds Number k-e Model 84

6.5 Grid Independent Solution for V velocity at x/H=5.5

Using Low-Reynolds Number k-e Model 85

6.6 Profiles for Inlet Boundary Conditions 86

6.7 Influence of the Inlet Velocity Profile on Reattachment Length 87

6.8 Influence of the Schemes for Discretization of Convective Term

on Reattachment Length 88

6.9 U Profiles for Reference Conditions 89

6.10V Profiles for Reference Conditions 90

6.11 Comparison of Model Results for Skin Friction Coefficient to Vogel's Experiment (8 / / / = 1.1) 91

6.12 Mean Velocity Profiles in Recirculating Region(Re=38000, ER=1.67) 92

6.13 y+ variations along the south wall 93

6.14k Profiles for Reference Conditions 94

6.15 e Profiles for Reference Conditions 95

6.16 Contour for Turbulent Prandtl Number Using Eq. (3.28) 96

6.17 Range of Turbulent Prandtl Number Using Eq. (3.28) 96

6.18 Turbulent Prandtl Number Profiles Variation 97

6.19. Comparison of Stanton Number to Experimental

Results with Various Inlet Boundary Layer Thickness 98

6.20. Particle Trace for Mixed Turbulent Convection Using standard k-e Model 99

6.21. Particle Trace for a Mixed Turbulent Convection Using Low-Reynolds number k-e Model 100

6.22. Influence of the Hichardson Number on Reattachment Length 101

6.23. Contour Plot for the Kolmogorov Length Scale (Ri„ =0.1,10 contour between 4.62x 10" and 6.89x 10' ) 102

6.24. Contour Plot for the Kolmogorov Time Scale {Ri„=0.\, 10 contour between 5.97x10"^ and 1.22x10"°) 102

6.25. Influence of the Richardson Number on Skin Friction Coefficient 103

6.26. Buoyancy-affected Temperature Contour for Recirculating Region

Obtained by Standard k-e Model 104

6.27. Buoyancy-affected Temperature Contour for Recirculating Region

Obtained by Low-Reynolds Number k-e Model 105

6.28. Fig. 6. 28 Temperature Contour Plots at Exit Boundary for Two Turbulent Models: Standard k-e Model(Ri=0); (b) Low-Reynolds Number k-e Model(Ri=0); (c) Standard k-e Model(Ri=0.1); (d) Low-Reynolds Number k-e Model(Ri=0.1) 106

6.29. Influence of the Richardson Number on Stanton Number 107

6.30. Effects of Turbulent Prandtl Number on Stanton Number 108

XI

6.31. Effects of Richardson Number on Wall Normal Distance

for Low-Reynolds Number k-e Model 109

6.32. Vortex Shedding af er the Backward-Facing Step with Ri„ =0.25 using Standard k-e Model 110

6.33. Reattachment Variation of the Flow with Ri„ =0.25 along the time 111

6.34. Vortex Interaction Between Two Bubbles after the Backward-Facing Step with Riff =0.20 using Low-Reynolds Nimiber k-e Model 112

6.35. Effect of Richardson Numberon Strouhal Number of \ onex Shedding Using Standard k-e Model 113

6.36. Stanton Number Change during Vortex Shedding Using Standard k-e Model 114

6.37. Cf Change during Vortex Shedding Using Standard k-e Model 115

xu

NOMENCLATURE

English Letter Symbol

C,, C,, Cy, C , model constants of k-e model

Cf mean skin friction coefficient, T^J{pull2)

f frequency of the vortex shedding,//z

Gjj buoyancy production in k and e equation s, Nl{s-m^)

Gfff Grashof number based on step height, gfiH^{T„ -TQ)/V'

H height of step, m

g^ acceleration due to gravity, m/s^

NUfj local Nusselt number, iS'r/(Re„ Pr)

/» production ofthe kinetic energy, Nl{s-m^)

p mean static pressure, NIrn^

Pr laminar Prandtl number

Pr, turbulent Prandtll number

Re, turbulent Reynolds number, k^ I ve

Re„ Reynolds number based oh step height

Riff Richardson number, Grfj /Re^

Shff Strouhal number,/?//«

St Standton number, q„ / pC,,u„ {T„ - T„)

T temperature, K

xiii

T fluctuation of temperature, K

t time, s

M,v mean velocity in X-and y-directions, respectively, «7/.y

" y fluctuation ofthe velocity in x- and y-direction, respectively, mis

I'r friction velocity, ^ | r„ , | /p , mis

X Cartesian Coordinate in streamwise direction, m

A,. flow reattachment length, m

y dimensionless length in y direction

y cartesian coordinate in normal to streamwise direction, m

y* nondimensional distance from wall surface, u^ylv

Greek symbols

a,a, molecular and eddy diffusivity for heat, m^ Is

Tf, r grid expansion ratio in streamwise and normal to streamwise direction

P volumetric coefficient of thermal expansion, AT''

5 boundary layer thickness, m

K turbulent energy, uûjl, m^ Is^

e turbulence dissipation rate, Nl{s-m^)

e nondimensional fluid temperature, (7;, - T) l{T^ - T^)

V,V, kinematic and eddy viscosity, respectively, nlp, m' Is

p fluid density, kglrn^

XIV

cr,. ,<Tj model constants in turbulent diffusion terms of k-e model

r„, shear stress evaluated at the wall, NIm'

Subscripts

0 reference values at inlet position

c center of inlet position in y direction

P cell center ofthe control volume

p 'lie point where wall functions are applied to

w wall surface

^ streamwise direction in computational domain

Tj normal direction to s t reamwise direction in computat ional doma in

XV

CHAPTER I

INTRODUCTION

1.1 Problem Statement

The standard k-e model has been widely used for its simplicity and economic

computational cost to account for rotation, buoyancy, chemical reaction, etc. The-

standard k-e model has played an important role in interpreting simple turbulent flows,

such as, duct flow or pipe flow. However, this model shows a weakness for recirculating

flow. It is a well-known problem that the standard k-e model underpredicts the

reattachment length in recirculating flow. In addition, one of the most difficult tasks for

researchers who used the standard k-e model for their studies was to find proper wall

functions for complicated turbulent flow. Conventional wall functions assumed turbulent

flow to be fully developed while separating and reattaching flow does not satisfy the

description of fully developed flow.

In the wall fiinction method, the boimdary conditions for the velocity and

temperature are matched with the logarithmic profile that exists near a wall in a forced

convection boundary layer flow. It has been known for some time that the logarithmic

profile does not hold near a wall for a pure natural convection boundary layer. George

and Capp (1979) have developed a wall function for natural convection turbulent flow

using analytical techniques. Henkes and Hoogendoom (1990), however, found that these

wall functions did not show sufficient agreement with experimental results found by

other researches such as Tsuji and Nagano (1988), while the low-Reynolds number k-s

models by Jones and Launder (1972), Chien (1982) and Lam and Bremhorst (1981)

showed fairly good agreement with other's experimental results. In addition, after

studying wall functions numerically. To and Humphrey (1986) suggest that turbulent

natural convection calculations should be carried out up to the wall, including the viscous

sublayer, with a low-Reynolds number k-e model.

In order to overcome the aforementioned problems of the standard k-s model, the

low-Reynolds number k-e model is investigated for buoyant separating flow in this study.'

It is impossible to predict the temperature field properly without accurate flow field

prediction. Two turbulent models will be used to predict buoyant separating flow and the

results will be compared against available experimental data.

For the standard k-e model, a constant turbulent number is used throughout the

calculation field because of the assumption of a high turbulent Reynolds number.

Re, -k- lv£. The low-Reynolds number k-s model, however, needs varying turbulent

Prandtl number to account for viscous sublayer effects. Finally, a varying turbulent

Prandtl number based on empirical correlations proposed by Kays and Crawford (1993)

will be introduced.

1.2 Objectives

The overall objective of this study is to employ a better turbulent model to improve

the prediction of separating and reattaching buoyant turbulent flows. The specific

objectives of this study are:

1. To develop a computational code that can predict accurate resuhs in the presence

of separating and reattaching flow.

2. To examine buoyant effects on the mixed flow.

3. To compare the low-Reynolds number k-e model with the standard k-e model for

channel flow and backward-facing step flow.

4. To improve temperature field predictions by introducing varying turbulent Prandtl

number based on empirical correlations.

1.3 Contents of Dissertation

The following chapter contains an overview and literature review of buoyant flows

for laminar and turbulence flow fields. Definitions of parameters which are used in this

work to define the characteristics of the buoyant flows are introduced. In Chapter 111,

mathematical formulation for a closure problem and the descriptions for the standard k-e

model and the low-Reynolds number k-e model are covered. Computational methods

such as discretization of the governing equations, pressure correction, and higher-order

convective term discretization techniques are discussed in Chapter IV. Also in Chapter

IV, the normal distjmce calculation algorithm which needs to be found for the damping

functions of the low-Reynolds number k-e model is presented. Chapter V and VI are

devoted to the test cases of turbulent buoyant flows for a channel and a backward-facing

step. For each case, results without and with buoyant effects for two turbulent models are

compared against available experimental data. Chapter VII contains the conclusions and

recommendations for future works.

CHAPTER II

BUOYANT FLOW

2.1 Introduction

Buoyant flows are caused by gravity. In this section, the reader is introduced to the

important features of buoyant flows and the important non-dimensional numbers that

characterize buoyant flow. Buoyancy force is the driving mechanism for natural

convection. Therefore, when the effect of temperature on fluid density is taken into

account, lighter parts of a heated fluid flow upward and the heavier ones flow downward

relative to a reference state in equilibrium. The phenomena are seen in the heating of a

room from convectors or space heaters, in providing the draft in chimneys, in cooling

products in refrigerators or cooling houses, in the cooling of transistors and transformers,

as well as in human beings and animals standing in a quiescent atmosphere. The most

frequent buoyant flows are encountered in many industrial flows such as electronic

cooling, HVAC systems and material processing, large-scale fires, and wild fires.

2.2 Literature Review

Many researchers have conducted studies on buoyant flows for various geometries.

In this literature review, boundary layer flow, flow between two plates, and flow over a

backward-facing step are discussed. In addition, vortex shedding and wall-functions

related literature are discussed.

Tsuji and Nagano (1988) measured velocity and temperature profiles for air along a

vertical plate with a constant wall temperature. They found a curve-fit for the wall-shear

stress measuring velocity using a hot-wire technique. The data is considered to be one of

the best-measured and is widely used by many researchers to study the boundary layer

buoyant flow.

Yin et al. (1990) developed a model which was originally based on the low-

Reynolds number k-e model to predict turbulent natural convection boundary layers. In

the model they divided the velocity into two components, i.e., a forced convection

component and a buoyancy-influenced component. A two-equation model for energy

equation by Nagano and Kim (1988) has been used for the calculation of temperature

field. They foimd that the combination of the modified low-Reynolds number k-e and the

two-equation model for energy equation is the best way to predict natural convection.

In the mean time, the study to develop the wall functions for the buoyant flow had

been done by several researchers (George & Capp, 1979; Tsuji & Nagano, 1989; Henkes,

1990; Henkes & Hoogendoom, 1990; Yuan et al., 1993). Henkes and Hoogendoom

(1990) suggested that the calculation for natural convection should been done up to the

wall with the low-Reynolds number k-e model for turbulence. They listed two differences

between the wall functions for the forced and natural convection boundary layer. First,

the inertial sublayer is fully turbulent, whereas the turbulent viscosity is very small in the

buoyant sublayer. Second, the log-law in the inertial sublayer gives a good approximaion

until close to the outer edge, >;"' >1000, while the extension ofthe buoyant sublayer is

restricted to the velocity maximum.

Yuan et al. (1993) tried to find the wall function for natural convection to avoid a

large number of grid points in the near-wall to save the computing costs, especially for 3-

D calculations. They introduced heat flux temperature and heat flux velocity as a

temperature and velocity scale. However, they found that their wall functions still

required a very fine grid system for accuracy.

Blackwell and Armaly (1993) conducted laminar mixed convection flow over the

backward-facing step as a benchmark test for the codes used for this study. The results of

many contributors (Acharya et al., 1993; Blackwell & Pepper, 1992; Chopin, 1993;

Cochran et al.. 1993; Hong et al., 1993; Iglesias et al., 1993; Sanchez & Vradis, 1993)

indicated that Uie reattachment length had been reduced as the Grashof number was

increased. With a fixed Reynolds number of 100 and an expansion ratio of 2, the

reattachment length, after the step, was found to be 5 and 3 for Grashof numbers of 0 and

1000, resj)ectively. They found that the Nusselt number and friction coefficient had been

increased due to buoyancy effects when Gr=1000 is used. Iwai et.al. (1999) conducted

the numerical simulations for mixed convective laminar flow, Re=125, over a backward-

facing step in the duct. It was found that the locations ofthe reattachment and the peak of

Nusselt number moved upstream as the Richardson number was increased, which is

similar to the study by Blackwell and Armaly (1993).

Kishinami et al. (1998) studied the laminar mixed convection over the backward-

facing step, which can be adjusted to change its height. They found that as the step height

is increased, the position of peak Nusselt number moves downward. Further, unstable

self-excited oscillation was noticed near the reattachment point within the limiting range

ofthe Richardson number, 0.3<Ri<2.5. In the oscillating region, the unsteady fluctuating

behaviors ofthe streamline and temperature contours were observed.

The problem of laminar flow over a backward-facing step geometry in natural,

forced, and mixed convection has been investigated rather extensively in the past, both

numerically and experimentally. On the other hand, studies of turbulent flow over a

backward-facing step have dealt mainly with forced convection only. Inagaki (1995),

Abu-Mulaweh et al. (1996), and Abu-Mulaweh (1999) are the only researchers who have

reported some results on free convection over a backward-facing step.

Abu-Mulaweh et al. (1999) conducted an experiment of turbulent natural

convection over a backward-facing step whose step height can be adjusted. Temperature

difference between heated wall and ambient air was maintained at 30 °C. Using three

different step heights, the authors found that both the reattachment length and the heat'

transfer rate from the downstream heated wall increase with increasing step height.

Kwak and Song (1998) conducted an experimental and numerical study of buoyant

flow over a vertical plate with horizontal rectangular grooves. They found that if the

Rayleigh number is not large enough the heat transfer rate of the plate with grooves

would be less than that of a smooth plate. It was found that if the Rayleigh number is not

sufficiently large enough, small recirculation will block the main stream from flowing

into the bottom surface ofthe grooves.

Similar study to augment heat transfer for laminar mixed convection for flat plate

with rectangular grooves has been done numerically by Wu and Pemg (1998). They

found that the installation of an inclined plate, which creates upstream vortex shedding in

the vertical block-heated channel, could effectively augment the blocks' heat transfer

performance in the channel.

Singh et al. (1998) investigated the flow field and the temperature distribution

around a heated/cooled circular cylinder placed in an insulated vertical channel using a

finite volume method. With a fixed Reynolds number of 100, the frequencies of vortex

shedding were examined with various Richardson numbers. For Ri up to 0.1, the

shedding frequency steadily increased, but for Ri>0.15, the shedding frequency suddenly

became zero, an indication that the shedding was stopped.

Heindel et al. (1994) compared two different low-Reynolds number k-e models for

natural convection in an enclosure. They concluded that the predictions of variable

coefficients model, which vary with the value of local turbulence Reynolds number, is

better than that ofthe model with fixed coefficients.

2.3 Buoyant, Laminar Boundary Layer Flow

For laminar boundary layer flow in which gravity acts in the negative x direction,

the body force per unit volume is pg^. With gravitational body force, the steady state

momentum equation for boundary layer can be written as follows.

.dU .,dU dP d ( dU^ pU — +pV — = -— + -^—-pg^. • (2.1)

ox dy ox oy\ ay J

The pressure gradient term can be defined by the pressure outside of the boundary

layer where the hydrostatic pressure can be applied to, dPldx = -pQg^ , and the

momentum equation becomes

-EAP-PO)- (2.2)

If density is a function of temperature and pressure, density can be expanded using

pU + pV = —H p — dx dy dy\ dy ^

the Tayk r series as follows.

/ ' p ^ ^ P = Po +

dp (T-To)^^] iP-Po) + - (2.3)

In a most natural pressure correction term in right hand side is neglected compared

to temperature difference term,

Po-p = P(T-To). (2.4)

where p = -{dpldT),, denotes the coefficient of volumetric thermal expansion, p^ and

TQ are the density and temperature where convection would stop for uniform temperature.

Finally, the momentum equation for natural convection can be written as follows.

d.\ dy dy ;'|^]-^,Ar-r„). (2.5)

In natural convection, some dimensionless parameters are introduced to

characterize buoyant flow. One of the most significant nondimensional parameter is the

Grashof number which is often interpreted as the parameter describing the ratio of

buoyancy to viscous forces,

Gr^MP^lZ^. (2.6)

where / is the characteristic length. The Grashof number is independent of the velocity

and is sometimes written in the following modified form to reveal constant heat flux at

the wall.

Gr =Gr,Nu^=f^ql\ (2.7) kv

Another parameter, the Rayleigh number, can be used to account for the inherent

coupling between energy and momentum balances.

Ra = i ^ ^ = GrPr. (2.8) va

This Rayleigh number is the counterpart of natural convection to the Reynolds

number of forced convection. For a natural convection boundary layer, Incropera and

DeWitt (1990) found that transition from laminar to turbulence occurs at the position

where Ra ~ 10^, regardless ofthe value ofthe Prandtl number. This universal transition

criterion can be supported by many experimental evidences (Fujii, 1959; Humphreys &

Welty, 1975; Godaux & Gebhart, 1974).

For mixed convection, the Richardson number can be defined as accounting for the

ratio of buoyant force to the change in momentum.

_. gP6Tl Gr

U' Re^ ^ ^ ^

where Re = —2-, denotes Reynolds number and U^ , the free stream velocity. It is

known for external flow that if the Richardson number is much less than 1, buoyancy

effects can be neglected. However, the role of the Richardson number in separating flow

has not been studied by many researchers.

Combined forced and free convection heat transfer coupled with thermal

conduction and separated recirculation flow becan-;c important in recent years, both in

academic and practical fields (Kishinami et al., 1995).

In natural convection, velocity profile has peak values close to the wall while in

forced convection the velocity profile is parabolic in which the free stream velocity is the

maximum velocity. For natural convection, Eckert (1950) assumed the following shapes

for velocity and temperature profiles, with the assumption that the momentum and

thermal boundary layers are of equal thickness,

(2.10) ^.=zfi_zV U^ SI s 'J

^-^" =fi_zf (2.11)

In buoyant flow, the amount of heat transferred can be affected by the Grashof

numbers and Prandti number, Pr, of the fluid. Ede (1967) found the correlation of the

local Nusselt number as a function of Pr and Gr^,

Nu=-"" 4

2Pr

5(l + 2Pr"^ + 2Pr)

1/4 ,1/4 (Gr^Pr)"\ (2.12)

A correlating equation for laminar constant-heat-flux Nusselt number was found in

10

a similar way by Fujii and Fujii (1976) using a modified Grashof number to contain the

heat flux, q, explicitly,

Pr

4 + 9Pr"^ + 10Pr)

Kays and Crawford (1993) compared the two cases above and revealed that the

Nu,= ,1/5 (Gr Pr)"^ (2.13)

constant-heat-flux Nusselt number is about 15 percent higher in laminar free convection. '

2.4 Buoyant, Turbulent Flow

The Reynolds-averaged Navier-Stokes equation and energy equation contain two

new terms, uÛj and u^T' which represent turbulent momentum transport and turbulent

heat transport, respectively. These two terms need to be modeled in order to have a closed

set of equations.

In direct analogy to the turbulent momentum transport, the turbulent heat transport

is assumed to be related to the gradient ofthe transport quantity, temperature,

dT -UjT = a , ^ , (2.14)

dxj

where a, is the turbulent heat diffusity. This turbulent heat diffusivity can be modeled by

the analogy between heat transport and momentum transport,

a =— (2.15) ' Pr , '

where v and Pr, are eddy diffusivity and turbulent Prandtl number, respectively. It can

be noted that these two properties are not fluid properties but instead depend on the state

of the turbulence. The turbulent kinetic energy and dissipation transport equations for

buoyant flows contain the buoyant production/destruction term, G^ , which can be written

as follows,

11

G,=-gJuJ\ (2.16)

Using Equations 2.14 and 2.16 can be arranged as follows,

G,=-g^P^^. (2.17) Pr, c\

The buoyancy production/destruction term represents the exchange between

potential energy and turbulent kinetic energy. Heindel et al. (1994) defined G^ as a sink

term if the term is negative, and defined it as a turbulent kinetic energy source for a

positive case. This distmction/production term is added to the k and s equations to

a^^count for turbulent buoyant flow. A set of turbulent transport equations and the Navier-

Stokes equation will be discussed in the following Chapter.

Turbulent buoyant velocity and temperature profiles assumed by Eckert and

Jackson (1950) are, in a similar way, used for laminar free-convection flow.

JLJy. u^ [s

^ ^- = l - f - | (2.19) Pw T^_

12

CHAPTER III

MATHEMATICAL FORMULATION

In analyzing separating flow, more complex governing equations must be solved

compared with tliose for non-separating flows. Separating flow can occur by pressure

difference without a sudden geometry change but it primarily occurs by flow geometry

change, such as, backward-facing step, forward-facing step, cavity flow, disk or cylinder

in flow stream, etc. One of main differences between separating flow and non-separating

flow is the velocity profile. Non-separating flow has a monotonic velocity profile and

provides researchers with rules which can be applied to numerical simulation while

separating flows do not. Especially, when the flow field and temperature field affect each

other, flow profile can be much more complicated than other flows.

Backward-facing step has been attractive to many researchers, and it became the

representative geometry to study separating flow. Backward-facing step can be seen in

many engineering applications, such as sudden expansion in a pipe or a channel. This

sudden expansion can make the flow separate from the wall. Backward-facing step can be

classified into three types: two-dimensional symmetric step, two-dimensional single step,

and axisymmetric aimular step. In this study, a numerical method using the standard k-e

model and the low-Reynolds number k-s model were used to study the flow over the

backward step with buoyancy effects.

3.1 Governing Equation for Momentum Transport

The characteristics of turbulence are random, diffusive, dissipative, and three-

13

dimensional. Figure 3.1 shows the time history of turbulent velocity.

Figure 3.1 Time History of Turbulent Instaneous Velocity Component

In Figure 3.1, instantaneous velocity at any pairticular time can be a summation of

the mean velocity and the fluctuating velocity:

« ' = ( / + «, (3.1)

where U is the mean velocity,

u is the fluctuating velocity.

For constant p and p, incompressible Navier-Stokes (1827 & 1845) equations can

be expressed in the following manner.

Continuity Equation,

5M

CbC; -i- = 0. (3.2)

Momentum Equations

du , du 1 dp Q —'- + U —'- = + — dt ' dx. p dxi dXf

du' du • \ • + •

dx , dx, \ + gJ{T-T,) (3.3)

where /?* = P + /? is the instantaneous pressure.

In engineering application, however, a more practical approach to describing

14

turbulent flows would be to model the averaged turbulent transport quantities. Using

short-term time Reynolds averaging over At, which is short but still much greater than

any significant period of fluctuations, dt, in velocity u, governing equations for

turbulence can be arranged as follows.

Continuity Equation,

dU. - = 0.

dx,

Momentum Equations

dU^^^dU.

(3.4)

dl ' dx, \_dP_ j _ p dXj dx:

( dU, dU,

dx, dx. \ "".I

-u,Uj\ + gJ{T-T,) (3.5)

In Equation (3.5), the double velocity correlation term, -uû • has been added as a

result of Reynolds time averaging and is known as the Reynolds stress term or the

turbulent shear stress term. The Reynolds stress term makes the set of equations not

closed. Therefore the turbulent shear stress term has to be modeled in terms of known

quantities in order to close this equation set for turbulence.

Many models have been introduced to model the unknown Reynolds stress term

known as zero-equation model or the first-order closure model.

Boussinesq eddy viscosity model (1977) dU

Prandtl mixing length model

Von Karman mixing length model

- MV = V,

- O T = / ^

-l^ = k^

dy

dU dy

dU

dy

dU .dÛ •I

dU

dy

(3.6)

(3.7)

(3.8) dy dy^

where / is a length scale characteristic of the size of momentum-transfer eddies. The

Boussinesq eddy viscosity model (EVM) is used in a generalized form.

15

uÛj = r . - ' -S..k. ^dxj dx, ^ 3 "

(3.9)

Here, k denotes the turbulent kinetic energy and S^j denotes the Kronecker delta function.

For this EVM the eddy viscosity, v,, should be modeled. Substituting Equation (3.9), into

Equation (3.5) gives

dl ' cbc, p dx, dXj \y + v,) —^ + —-1.

I, 5jf, dx, J -^5„^^gJ{T-T,). (3.10)

Prandtl (1925) and Kolmogorov (1941) setup the relation among eddy viscosity,

turbulent kinetic energy, k, and its dissipation rate, s, by dimensional analysis as

^,=C^ (3.11)

Here, C^ is considered as constant for high Reynolds number. For a simple flow, such as

pipe flow or channel flow, the Reynolds stress term can be modeled by a simple Prandtl

mixing length model called zero-equation model. The zero-equation model can be

modeled directly without any additional differential equations. However, more general

and complex flow needs to have a more sophisticated model that includes k and s. This is

referred to as the second-order closure problem. One of the most popular turbulence

models with various combinations of the variables proposed is the k-s model by Launder

and Spalding (1974).

3.2 Standard k-s Model

As previously mentioned, kinetic energy and its dissipation rate are introduced to

calculate eddy viscosity. The standard k-s model has been applied to many different flows,

such as plane jets, mixing layer, boundary layer flows, and so forth. Despite the existence

16

of more advanced turbulent models, such as Direct Numerical Simulation (DNS), Large

Eddy Simulation (LES) or Reynolds Stress Model (RSM), the standard k-s model has

been studied by many researchers because of its easy adaptivity and smaller

computational cost. This model was developed on the assumption that the flow is fully

turbulent and there is a local equilibrium where rate of production of turbulent stress, /"^,

equals rate of turbulent dissipation rate, e , near the wall. The assumption of fially

turbulent flow requires the local turbulent Reynolds number. Re, =k^ Ive, to be high.

For this reason, the standard k-s model is sometimes called the high-Reynolds numbtx k-

e model, compared to the low-Reynolds number k-s model in which calculation is carried

out up to the wall including the viscous sublayer. It is necessary for standard k-s model to

have the first calculation point far away from the wall where the local Reynolds number

is large enough to satisfy the above assumption. Empirical wall functions are used to

bridge the gap between the wall and the first grid node adjacent to the wall. Those wall

functions will be discussed later in this chapter.

The turbulent model, including buoyancy term, originally developed by Launder

and Spalding (1974), is described by the following equations.

The k-s equation

Â,U^^ = M[v.^]f-\.P,-e.U,. (3.12) dt 'dXf dxA\ cyjdxj p

The s-equation

d£ ,, de d — + U

( v,\ds\ ^ £ ^ ^ E^ 1

dt ' dx, dx. v + -

£ ^ C,-P,-C,— + -C,jG,. (3.13) ax, I K K p k

where.

17

A=v, V ^ 7 ^^. J ^j dx,

y^dT^

Pr, dy

The empirical constants are assigned their usual values, these are

G,=-g,P-^—. Pr, dy

( C „ . C , , , C^„ a , , 0-,) = (0.09,1.44,1.92,1.0,1.3).

One more exti-a constant, C,, is necessary. Rodi(1984) suggested that Cj should

be close to 1 in the boundary layer which is in the direction of gravity and close to 0 in

the boundary layer which is in the perpendicular direction of gravity. Following Henkes

et al. (1991), an approximation that satisfied this condition is given by

Cy = tanh (3.14)

3.3 Wall Functions

The fluxes of momentum and heat to the wall are supposed to obey the wall

functions by Launder and Spalding (1974), as follows

(3.15)

(3.16)

"' C"'k';' - ^ in Eyp

= Pr,

(c;'

iln K

kr)'"' V

Eyp V

where

P = 9.0[(Pr/Pr, )°^^ -1][1 + 0.28exp(-0.007 Pr/Pr,)] .

The P in the right-hand side of Equation (3.16) has its origin in an analysis of

experimental data conducted by JayatiUaka (1969). Equations (3.15) and (3.16) are

18

modified forms of u* and T* to avoid a shear velocity, u^ = ^[rj~p , which changes its

sign in a recirculating region. However, k in Equations (3.15) and (3.16) still become

zero at the separating and reattaching points. Therefore, the Kolmogorov velocity,

Mj = (ve)"*, is introduced to avoid this shortcoming using the following local equilibrium

approximation for high turbulent Reynolds number. According to experimental data, e

has a finite non-zero value, even at the wall, while k becomes zero,

e,=[{C]rkr)lKy,\ (3.17)

Here subscript P in Equation (3.17) denotes the first calculation point where wall

functions are applied. Equations (3.15) and (3.16) can be rearranged using Equation

(3.17) and Kolmogorov velocity as follows:

"p 1/3 n *"^ 1 1 ^ -MAf Re =—In

^ K {rip),.

EK'" Re »4/3

\n {Tp-TJCppK"'Rc .1/3

= Pr, In K

EK'" Re-,4/.1

+ P

(3.18)

(3.19)

where

Re = ypu,

Using wall functions related parameters, the Nusselt number and friction

coefficient can be found

where

Nu.= hH qlH {T^-T,.)u,H

" k k{T^-TJ T;{T„-TJa

• \l2pU^ ii2u;ul

.1/3

(3.20)

(3.21)

19

H is the characteristic length,

T^ is the reference stream temperature,

( / , is the reference velocity,

u*p denotes the dimensionless velocity for the first calculation point next to the

wall,

Tp denotes the dimensionless temperature for the first calculation point next

to the wall.

3.4 Low-Reynolds Number k-e Model

The first low-Reynolds-number k-e model was proposed by Jones and Launder

(1972). Low-Reynolds-number k-e model was made for the modification of turbulence

closure models that would enable their use at low Reynolds numbers and to calculate the

flow close to the wall. This model includes damping functions based on the local

turbulent Reynolds mmiber. The attempt has been made to predict the appropriate values

ofthe eddy viscosity and the dissipation near the wall where turbulent Reynolds number

is small. Low-Reynolds-number k-e by Lam and Bremhorst (1981), Jones and Launder

(1972) and its modification by Launder and Sharma (1974) are still considered adequate

for modeling turbulent natural convection. Chien (1982) was one ofthe researchers who

improved the low-Reynolds number k-e model by applying the Taylor series expansion

technique to investigate the proper behavior of the turbulent shear stress and the kinetic

energy and its dissipation near a solid wall. Craft et al. (1993) proposed a non-linear low-

Reynolds-number model that has a nonlinear relation between strain-stress and vorticity

that includes quadratic and cubic terms. Nagano and Tagawa (1990) developed the low-

Reynolds number k-e model to satisfy the physical requirements of wall and free

20

turbulence. Abe et al. (1994) introduced Kolmogorov velocity scale, u^ = (v / / f ) ' ^^

instead of the friction velocity, u^, to account for the near-wall and low-Reynolds-

number effects in both attached and detached flows. For a flow over backward-facing

step, experimental data by Kim et al. (1987), Driver and Seegmiller (1985), and Vogel

and Eaton (1985) are frequently used to verify the models. Unlike experimental data,

DNS data are often used to develop new turbulence models because of its detailed

information of flow quantities even very close to the wall.

Chen (1998) summarized k-e group of models as shown in Tables 3.1 and 3.2. A

variety of low-Reynolds number k-e models can be written in general form

dk ,, dk d — + U, = — dt ' dx, dx.

v + -\

'kj

dk dx,

•dU, •u,u i

' ' dx. £-D, (3.22)

d£ ,, d£ d — + U dt ' dXi dXf

^

v + V, d£

\ ê)

•dU,

STf-'^'^'i-'-^^-^^^^T^^- ("^'

where.

r uv = v.

dU, dUj +

^ dXj dx. -f' '

y, =cj„--.

Rcy — — , ve

Re^ = 4ky

u, =

y ' = ^

21

y =::—£. V

,1/4 «, = {vsY

Here, D in Table 3.1 has been introduced to compensate for the boundary condition of

turbulent kinetic energy dissipation rate, e, because physically in the Table 3.1, £•„, is not

zero.

Chien's model (hereafter referred as the CH model) was developed basically for

simple flows, such as, channel flow or pipe flow, or in other words nonseparating flow.

Wall dissipation, 2vkly^, in the k-equation is equal to the molecular diffusion at y=0. In a

similar way, another wall dissipation term has been introduced in the e-equation.

Turbulent eddy viscosity has been modified to include the damping effect due to the

presence of the wall. This damping function usually has an exponential form so that

damping effects can be faded out away from the wall.

The major difference between the Abe-Kondoh-Nagano model (hereafter referred

to as the AKN model) and CH model is the velocity scale. In AKN model, Kolmogorov

velocity scale, u^ =(v£-)"*, is used while CH model used friction velocity, «_., to account

for the near-wall and low-Reynolds effects in both attached and detached flows. Wall

dissipation term is given as a boundary condition for turbulent kinetic dissipation rate. In

this study the AKN model with the same buoyancy terms defined in the standard k-s case

has been selected to solve the problem of recirculating flow.

3.5 Governing Equation for Energy Transport

Analogous to the momentum equation, after Reynolds time averaging, the energy

equation has an extra term due to turbulence, known as the Reynolds flux,

22

-^^"'-S^'a^l''-^-"''^]- (3.24)

Reynolds heat fluxes in Equation (3.24) also need to be modeled. In Chapter II, The

modeling of Reynolds fluxes were discussed. Using Equations (2.14) and (2.15), the

energy equation can be rearranged as follows

This energy equation is strongly coupled with the momentum equation so that all

the numerical calculations for the governing equations can be performed simultaneously.

3.6 Turbulent Prandtl Number

The turbulent Prandtl number cannot be considered as a constant if calculations for

the turbulent model are carried up to the wall. According to the experimental data for

fully developed flow by Blackwell et al. (1972) and Hollingsworth et al. (1989), the

turbulent Prandtl number is sharply increased in the viscous sublayer region. In addition,

if the flow is not fully developed a more precise turbulent Prandtl number calculation will

be needed. From Equations (2.15) and (2.14), and the definition ofthe eddy viscosity.

Equation (3.6), the turbulent Prandtl number can be expressed as follows

v^^-^dTldy

a, -vT dUldy

To define the turbulent Prandtl number, four quantities in Eq, (3.26) should be

defined. Kays and Crawford (1993) showed that for fluids with 0.7 and 5.9 of laminar

Prandtl number, Pr , such as air and water, the turbulent Prandtl number in the

logarithmic region is close to 1. The experimental data by many researchers (Reynolds et

23

al., 1958; Blackwell, 1972; Gibson & Verriopoulous, 1984; Hollingsworth, 1989) show

that the turbulent Prandtl number has a range from 0.7 to 0.9, with a preponderance of

data around 0.85. For this reason, 0.85 is used as the turbulent Prandtl number for

standard k-s model. Kays and Crawford (1993) fitted the experimental data for boundary

layer into the following equation

1 Pr,=

2Pr. ' +CPe,]^-{CPe,y

700 fPr, 7oO

1-exp 1

CPe,^, /CO /

(3.27)

where Pe, = {\; /v)Pr, Pr,„ is the value of Pr, far from the wall that is usually assigned a

value of 0.85. C is an experimental constant with a value of 0.3. It .should be noted that

Equation (3.27) and all the experimental data mentioned above have been performed for

boundary layer flow, which is attached flow.

24

Table 3.1 Constants for the k-e Group of Models

Model

Standard

Launder-

Sharma

Hassid-

Poreh

Chien

Lam-

Bremhorst

Lam-

Bremhorst

Nagano-

Hishida

Nagano-

Tagawa

Abe-

Kondoh-

Nagano

Code

STD

LS 2\'

HP

CH

LB

LBl

NH 2v

NT

AKN

D

0

'd^] < 'y J

2A \'"

2A

0

0

0

0

f..,-B.C.

Wall

0

0

0

d-k

'V ^ = 0 dy

i ' 2 v —

y

2v — y

C.

0.09

0.09

0.09

0.09

0.09

0.09

0.09

0.09

0.09

f,

1.44

1.44

1.45

1.35

1.44

1.44

1.45

1.40

1.40

^2

1.92

1.92

2.00

1.80

1.92

1.92

1.90

1.30

1.40

o-A

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.45

1.50

ê

1.30

1.30

1.30

•

1.30

1.30

1.30

1.30

1.90

1.90

25

Table 3.2 Functions ofthe k-e Group of Models

Code f J n

STD l.O

LS

exp

LB

- 3 . 4

HP l-exp(-0.0015Re,)

CH l-exp(-0.0115.v')

[l-exp(-0.0165Re,,)f

L B l [l-exp(-0.0165Re,)f

20.5' 1 + -Re,

N H [l-exp{-Re,/26.5)f

NT ^ 26

AKN

4'^^] l -exp(-^)

^ 14 ( / „ N2>'\

. 5 f fRe Y

/ . fl

1.0

1.0

1.0

I - 0.3exp(-ReJ)

1.0 l-0.3exp(-Rc;)

1.0 l-0.22exp(-(Re,/6)^)

l + (0.05//^,) l-exp(Re.J.)

l+(0.05//^,) l-exp(Re^.)

1.0

1.0

1.0

l-exp(-^^) ^ 6.0

1 - 0.3exp 6.5

0.0

2vv, ^d'-U^"

dy' )

-2v I f> J

-2v{£ly') X exp(-0.5_v)

0

l-0.3exp(-Re7) '» , ( • - / „ )

l - e x p ( - ^ 3.1

l-0.3exp r / „ ^2^^

Re, ~6l

0.0

0.0

26

are

CHAPTER IV

METHODS OF COMPUTATION

A typical control volume for a node P and its neighbor node N,S,E and W

shown in Figure 4.1. Governing equations are integrated over the control volume for

nodes. A nonstaggered grid system has been used. Therefore, velocity, pressure, and'

temperature are stored at the cell center P. For the pressure correction, a modified version

of the SIMPLE algorithm by Patankar (1980) is used. The momentum equation and

energy equation are solved altematively because ofthe strong coupling between velocity

and temperature. Stone's (1968) sti-ongly implicit scheme is used to solve the resulting

discretized equations.

Figure 4.1 Control Volimie for Node P and Neighboring Control Volumes

4.1 Governing Equations in a General Coordinate System

The governing equations. Equations (3.10), (3.12), (3.13), (3.22), (3.23) and (3.25)

in the physical domain (x,y), which has curvilinear coordinates, have to be transformed to

the equations in the transformed domain (^,r|) which allows for all the computations to be

27

done on fixed square grids. The transformed grid frees the computational simulation from

resti-iction to certain boundary shapes and allows general codes to be written in which the

boundary shape is specified.

If ^denotes a general scalar variable, the general form ofthe differential equations

can be expressed in the transformed coordinate system {t„r]) as follows:

d d{p^)^]_

e, jik^^^H^'^^^^ = l A f ^ a ^

_i__a_ ^ J dTj , J ^''drjj Jd^[ J ^''drjfjdni J ^ " ^ J

where G^= pUy^-pVx^,

(4.1)

+ 5 .

G,f = pVx^ - pUy^,

11 ^^l+y].

gn =-{Xf^x^+y^y^).

g22=yl +xl.

J = x^y^-x^y^.

Here, J and gjj are the Jacobian of the transformation and the covariant metric

components, respectively, p^^ and Sy are the associated viscosity and source function

for variable , respectively. The detailed derivation of Equation (4.1), and the source

terms and the associated viscosities for each variable are given in the appendix. This

work originated from Parameswaran et al. (1992), and the transformed equations for the

k-e equation and the energy equation are added.

28

4.2 Discretization

Equation (4.1) is integrated over the control volume for node P to yield the

following finite volume equation for <;>,

pp.o,'"" '/"'+[{Frr'). - (F.-r')..,+(F,V-' )„ - ( F , > - ' ). ] Ar

Av

+ '''«'3 <«'"'-«'"'-k -22 \

vo/ ( < ^ P -<PS ) (4.2)

• ^ | / ^ e #

>"e#

-12 ^

vol

-21

vo/

(C-O--12 >

/"îr vo/ (C-O

Ke-fnJ -2n

/".# vo/ ( C - O + Vvo/

where F^ and F^ are the convective mass fluxes across the east and the north cell faces

G"=(A>.)J+(Ax)J

G'2 = G^' = -[(Ax)^(Ax),, + {t.y)^{^y)rf

G'•'•={^y)\^{^x)\

vol = volume of a general cell.

In Equation (4.2) ^;*' , ^,7 ' , (Z>;*' , and (Z>;*' are calculated according to the

Quadratic Upstream Interpolation for Convective Kinematics (QUICK) scheme by

Leonard (1979). The QUICK scheme is discussed in section 4.4. The superscripts n and

n+1 denote the old and new time levels.

4.3 Pressure Correction

For incompressible flow, there is no separate equation for pressure. The pressure

29

has to be obtained from the continuity and momentum equations. In this study, a modified

version ofthe SIMPLE algorithm by Patankar (1980) is used to obtain the pressure field.

The main aim of the pressure correction is to find a way of improving the guessed

pressure, p', to satisfy the continuity equation. The corrected pressure is called the

pressure correction, p', which can be used to find the corresponding velocity corrections.

The convective fluxes, F, and F^, at the cell faces are computed from the velocity

projections along Uie coordinate direction at the cell faces. All velocity projections are

calculated from the velocities stored at the center of each cell. The velocity projection in

the procedure is described in detail in Parameswaran (1985). Here, only the final form of

the pressure correction equation is given as:

PP Pê , PQ». , pq» , PQ.V . pa ; . pal . pal • pa:

ê -^11. -^.1 A ,

where A^, A„, A„ and A^ are the averaged momentum coefficients at the cell faces

derived from the momentimi coefficients at the cell center, and s,, is the continuity error

associated with the cell P given by:

^p = F\,y, - Fi + Kv - Fi, (4.4)

where F^^, F,',,, F^, and Fl,, are convective fluxes based on velocity projections.

The pressure correction procedure is similar to that of Issa (1986). The following

steps are carried out at each time step,

1. Equation (4.2) is solved to yield the intermediate velocity field «* and v* with the

existing pressure field.

2. The new fluxes F^ and Fj are computed from the velocity projections at the cell

faces.

30

3. The continuity error, e,., is calculated for each cell from the fluxed in step 2 and the

pressure correction equation Equation (4.3) is assembled and solved to yield a new

pressure field p'.

4. New velocity fields for M" and v" are obtained from Equation (4.2) using the new

pressure field in step. 3.

5. A new set of fluxes, F^ and Fj, are calculated using the updated velocity field and

pressure field.

6. The pressure correction equation. Equation (4.3) is solved for an improved pressure

field, p".

7. Steps (4)-(6) are repeated until the momentum and the continuity equations are

satisfied within the preset tolerances.

4.4 QUICK Scheme

It is well known that high accuracy can be achieved either by use of higher-order

discretization or by use of a finer mesh. The higher-order convective term discretization

techniques are necessary to gain accurate, stable, and fast numerical simulation. CFD

code, HEAD2D, which is used for this study, originally employed first-order upwind

scheme to avoid the stability problem of central differencing in discretizing the

convective terms in the governing equations. However, this lower-order upwind scheme

creates artificial dissipation and requires more grid points to secure the accuracy. In order

to overcome these problems, higher order techniques have been proposed by researchers.

Raithby (1976) proposed Skew Upstream Differencing Scheme (SUDS) to improve the

upwind scheme, and Leonard (1979) has developed Quadratic Upstream Interpolation for

Convective Kinematics (QUICK) for steady problems, and for unsteady problems

31

QUICK with estimated Streaming Terms (QUICKEST) scheme to discretize convective

terms. Leonard et al. (1993) developed Uniformly Third-order, Polynomial Interpolation

Algorithm (UTOPIA) and concluded that the UTOPIA scheme is the best scheme in a 2D

problem. However, according to the work by Kiris (1994) there is no significant

difference between the QUICK scheme and the UTOPIA scheme. In this study, the

QUICK scheme has been employed and compared with the results ofthe upwind scheme.'

Using the upwind scheme and QUICK scheme the convective terms, "" , ,"* ,

^"'^', and "" ' in Equation (4.2) can be expressed as follows.

Upwind Scheme

.„.._|«t.r''/^.">o, ' Ur''/ F;' < 0,

i i i + i

K*=-ifw if F;'>O,

<i.r' // F: < 0,

„.,_j«t>r''/^2">o. ^ " I . «+i

„^,_j<t,r ' / /F,">0, (4 5)

^.j,;^' // Fl' < 0

QUICK Scheme

„ , _ u^'+i/2((t.r' -<t>r')-9/3(f;^' - 2 C +<t.r)«/ F - >O, *"''[,!,-'+i/2((t.r' -<i)r')-9/3((t.^' -i^'ir +v;') if ," < o,

, _j(t,r+i/2«i,r' - c ) -9 /3« i ) ' ^ -2<t.r ^r;') if F,- >O, * ' 'Ur'+i/2(<t.r -<t'r')-9/3((t>r' -2^7' +C)?/^ p" < o

32

„„ (<t.r'+i/2((|,r -<t>r)-9/3(,t,r' -ir;' ^r;') if F,- >o, l C + i/2«t)r'-<t)r)-g/3((|,;;;-2C+<t'r')'/ 2"<o,

^„., j<i.rM/2(C-<t)r')-9/3«t.;:;'-2,t)r'+<t>r)'/ K > O , Ur' + i/2«t)r -<t,7')-9/3(f;' -2,^;;' +(|,r),/ F" < o ^ ' ^

where q is assigned as 3/8 and the QUICK scheme has two extra terms compared to the

upwind scheme to secure better accuracy.

4.5 Normal Distance Calculation

Unlike the standard k-e model, the normal distance from the calculation point to the

shortest wall is needed to calculate the damping functions in Table 3.2. For simple

geometry, such as pipe or channel, it is not a difficult task to find the distance from the

walls or obstacles. However, for complex geometry, it is necessary to develop a method

to calculate the normal distance. In this study, the new algorithm to find the distance was

introduced. Two possible cases for the normal distance from the arbitrary point C to

piecewise wall AB are shown in Figure 4.2. For case (a) in Figure 4.2, the shorter

distance between CA and CB is assigned as the normal distance for the piecewise wall

^B. For case (b) in Figure 4.2, the perpendicular distance CJ is assigned as the normal

distance for the piecewise wall AB. After considering all the distance to all piecewise

wall, the minimum value ofthe distance is assigned as the normal distance for point C.

The same approach has been used for the obstacles. Figure 4.3 shows two test geometries

for the normal distance calculation mentioned above.

4.6 Solution Procedure

In buoyancy flow the momentum field is strongly coupled with the temperature

33

field so that all the governing equations should be solved at every time step. The overall

solution procedure implemented in this study is shown in Figure 4.4. The do-loop, which

contains the subroutines from 'store' to 'output', is represented as a box.

34

C{x,.y,) C{x..y,)

J(Xj.y^) A{Xi.y^) B{x,.y,) ^(x,.y,) J{x^.yj)

(a) (b)

Figure 4.2. Two possible cases of normal distance calculation

B{x,.y,)

L.

(a) The Geometry with Bluff Body Placed in Channel

i -p- \ >.'•}—

yj

l ^ ^ i

(b) The Geometry with Backward-facing Step and Forward-facing Step

Figure 4.3 Normal Distance Contour for complex geometries

35

RADFLO

uaii input

uaii inietu

uaii caicdist

i uaii store

uaii caicu

uaii caicv

uaii outbc

uaii caicp

uaii caice

uaii caicte

uaii caicea

uaii moapro

uaii output

E N D

Reads input data

Reads input boundary conditions

Calculates the normal distance

Stores the variables for next level

Calculates U velocity

Calculates V velocity

does the block correction for boundaries

carries the block correction for boundaries

Calculates pressure corrections

Calculates turbulent kinetic energy

Calculates dissipation rate

Calculates eddy viscosity

Prints output data

Figure 4.4 Flow Chart for Overall Procedure

36

CHAPTER V

TEST CASE 1 : TURBULENT CHANNEL FLOW

In this chapter, channel flow has been selected as a test case to examine the

buoyancy effects on the flow mechanism. The results by two models, standard k-e model

and low-Reynolds number k-e, have been compared with the budget of DNS. The Two-

dimensional computational fluid code, HEAD2D originally developed by Parameswaran

(1993), is equipped with the standard k-e model. The HEAD2D code has been modified

to predict buoyant turbulent flow. The low-Reynolds number k-e model also has been

added to carry out the calculation up to the wall because buoyancy affected velocity

profile changes close to the wall.

5.1 Problem Statement and Boundary Conditions

The height ofthe channel is selected as 2H and the length ofthe channel is set to be

20 times the channel half-height, H. The Reynolds number based on the mean velocity

and the channel height is 5600, Re,, = lUHIv .

Inlet profiles for the velocity, the turbulent kinetic energy and its dissipation rate

are considered as fully developed which have been acquired numerically by updating

inlet profiles using exit profiles. This correction has been done until the values remain

unchanged within a certain limit. The inlet temperature, however, is given as a constant,

T,„ , which is the same for the temperature of the north wall, Tf, while high wall

temperature, T^ , has been imposed on the south wall. For the exit, zero-gradient

boundary condition is used for the velocities and temperature. Gravitational force acts in

37

the opposite direction ofthe flow to have opposing flow for the heated south wall. Figure

5.1 shows the computational domain and the boundary conditions for the channel flow.

5.2 Grid Arrangement and Grid Independent Study

For the numerical study, a non-uniform grid has been used in the direction normal

to the sti-eam, y, while a uniform grid has been assigned in the streamwise direction, x.'

The grid distribution in y direction is controlled by 5y^^,l5yf =r^. Here /-„ is the

expansion ratio of the grid in the streamwise direction. Values less than 1 have- ''icn .

assigned to the expansion ratio so that more grids can be clustered near both walls. One

ofthe grid configuration used in this study is shown in Figure 5.2.

For the standard k-e model three different sets of meshes, 52x12, 52x22 and 52x32

in the x and y directions are used for grid independent study. The r^ is adjusted for each

case so as not to have a grid point in the viscous sublayer where y* is less than 13 for air.

For the standard k-e model, if the first calculation point close to the wall is in the viscous

sublayer, the numerical prediction tended to overpredict in the viscous sublayer region

while it imderpredicts in the logarithm region. To avoid this problem, r^ is assigned as

1.2, 1.05, 0.95 for 52x12, 52x22 and 52x32 grids, respectively. The predicted velocity

profiles by the standard k-e model using three different grid sets are shown in Figure 5.3

and Figure 5.4. The resuh of 52x12 shows deviation from the results ofthe other two. In

Figure 5.3, U„, denotes the mean velocity. Figure 5.4 shows the same result in semi-

logarithmic coordinates using the nondimensional velocity and the nondimensional

distance from the wall. Figures 5.5 and 5.6 show the nondimensional turbulent kinetic

energy and its dissipation rate, which are defined as follows,

38

k*=klu] (5.1)

£*=£\'lu'. (5.2)

Thus, 52x22 has been selected as a grid independent solution for further study of

standard k-e model.

For the low-Reynolds number k-e model three different sets of meshes, 52x22,,

52x42 and 52x52 in the x and y directions, respectively, are used for grid independent

study. The Low-Reynolds number k-e model requires more grids than the standard k-s

model to have enough grid points within the viscous sublayer region to carry out the

turbulent calculation up to the wall. The grid expansion ratio in the y direction, r^, was

set to be 1.2 for all three sets. The number of grid points in the y direction is decided to

have at least one or two grid points in the viscous sbulayer region. According to Figure

5.7, all three curves of U profiles from three different number of grids are pretty close.

However, when the same results are plotted in a non-dimensional semi-logarithmic

coordinates as shown in Figure 5.8, the U profile with 52x22 grid was clearly deviated

from the other two curves. Figures. 5.9 and 5.10 for non-dimensional turbulent energy

and its dissipation rate show the same grid independent solution for three different

meshes. Therefore, a grid independent solution is obtained with 52x42 mesh for low-

Reynolds number k-s model.

5.3 Results without Buoyant Effects

5.3.1 U Velocity Profile

The mean U profiles for the standard k-e model and the low-Reynolds number k-s

model against DNS data in semi-logarithmic coordinates for channel flow is shown in

39

Figure 5.11. The semi-logarithmic coordinates provide a convenient way to examine the

most important region, the inner region, where most of the velocity change takes place.

Two graphs using two different turbulence models were plotted against DNS data by Kim

(1990) and the law ofthe wall. The law ofthe wall used in this comparison is defined as

following Kays and Crawford (1993)

M* =2.44 1nv'+5.0. (5.3)'

The standard k-s model tends to overpredict velocity while the low-Reynolds

number k-e model (AKN model) tends to underpredict. The shear velocity, u^, for DNS

data is 395 while the shear velocity for current channel flow with Re=5600 is 345. As the

Reynolds number is increased, the outer edge ofthe boundary layer goes to higher.

5.3.2 Turbulent Kinetic Energy Profile

Turbulent kinetic energy has a finite value at the center of the line of the channel

flow. Although there is no production of turbulence at the centerline owing to the zero

velocity gradient, turbulence is continuously diffused towards the center of the channel

from the high turbulence region which is near the wall where a high velocity gradient

exists. Abe et al. (1994) evaluated the near-wall limiting behavior ofthe velocities as

follows,

U = A^y + A^y'^ + Ayy^ +....

V= B2y^ + Byy^ +....

W^C^y + Cjy^ +Cyy^ +....

u = a^y + ajy^ + Oyy +....

40

v= by-+ byy'+ ....

w = co' + c,y^ +Cyy^ +.... (54)

According to Equation (5.4), near-wall limiting behavior of the turbulent kinetic

energy is proportional to ,- In Equation (5.5), only the first term has been written, while

neglecting other higher terms,

,^!r^^J_al^pyl^ (5.5)

This near-wall behavior is valid for the region between the wall and the location of

peak value. In Figure 5.12 DNS data and the result using a low-Reynolds number k-e

model show the near-wall limiting behavior for the turbulent kinetic energy while the

standard k-e model excludes the near-wall behavior assuming that the first calculation

point is located in the logarithmic region. According to Figure 5.12, the standard k-e

model has good agreement with the DNS data if y^ is greater than 100. However, it will

be a hard task to satisfy the condition before one performs the calculation, especially

when the flow is a separating flow. The results using the low-Reynolds number k-s model

show good agreement with DNS data overall including near-wall region and the peak

position at about y*=20.

5.3.3 Turbulent Kinetic Energy Dissipation Rate Profile

In Figure 5.13, a comparison ofthe s profiles calculated by the two models against

the DNS data is shown. In the logarithmic region both models have good agreement with

DNS data. As mentioned earlier, the standard k-s model excludes near-wall behavior for

its calculation and has monotonic decay in the logarithmic region and the outer region.

However, the DNS data and the results using low-Reynolds number k-s model have near

41

wall behaviors and non-zero e at the wall. The dissipation rate, e, near the wall surface

and Equation (5.4) lead to non-zero e at the wall.

£ = v< du V

+ 1 — dy) [dy)

^dw^' • = v(f l , -+cf) + ... (5.6)

\^y)

It is highly debated about where e has its peak and what the value should be at the

wall, though most modelers believe that the peak value of e should be where k peaks.

5.4 Results with Buoyant Effects

Three Richardson numbers, 0, 0.1 and 0.2, are introduced to study the effects of

buoyancy on the flow field in the channel flow. For the case with Re=0, the gravitational

force has been set to 0 to eliminate buoyancy effects.


For the south wall where there is a high temperature, the buoyancy force aids the

convective motion near-wall region resulting in acceleration of the velocity. The

buoyancy effects on the velocity profiles for half of the channel are shown in Figure 5.14.

Velocity profiles using both models show clear peak velocities near the wall due to the

buoyancy force. As the magnitude ofthe buoyancy force increased the velocity gradients,

in the near-wall also increased. The low-Reynolds number k-e model tends to have

velocity peak values close to the wall as the Richardson number is increased. In Figure

5.15, it is found that the value of w at the outer region became smaller as the Richardson

number increased due to the increase of shear velocity at the wall as well as the decrease

ofthe core velocity at the center ofthe channel.

42

5.4.2 Temperature Profile

In Figure 5.16, the effect of buoyancy forces on dimensionless temperature is

shown. It is shown that the trends of the temperature profiles are highly affected by the

velocity profiles in Figures 5.14 and 5.15. Temperature profiles cross each other where

the velocity profiles do in both models.


Figure 5.17 shows that turbulent kinetic energy, k, decreased as the Richardson

number becomes high. According to Equation (5.4), k is the function of the velocity

fluctuations. Garg et al. (2000) performed direct numerical simulation (DNS) and found

that in buoyancy affected flows, the changes in the fluctuation component of U, u, are

primarily due to the decreased correlation between the streamwise and vertical velocity

fluctuations, which reduces the Reynolds shear stress and, thus the production. Since u is

the source of the fluctuation velocities v and w components, the behavior of v and w

tends to decrease in buoyancy affected flow. Nakajima et al. (1980) also mentioned the

decrease of turbulent fluctuation with increasing Richardson number. For both models, a

reduction of k was observed as the Richardson number increased. The peak values for the

low-Reynolds mmiber k-e model decreased with the increase of the Richardson number,

while the peak values ofthe standard k-e model are fixed by the wall function.


The decrease of production also suppresses the dissipation rate of turbulent energy,

e, as the Richardson number increases. Profiles of s shown in Figure 5.18 are closely

related with the turbulent kinetic energy behavior affected by buoyancy.

43

T=Tc North W&ll

T=Tm V=0 IHXy)

2H

gc 4

dU_dV_dT

dx dx dx

^ O l i h ^ l L=20H T=TH

X

Figure 5.1 Computational Domain and Boundary Conditions for Channel Flow

44

Figure 5.2 Grid Formation for Channel Flow

(52x42 Grids with r^ =1.2)

45

0.0 0.2 0.4 0.6 y/H

0.8 1.0

Figvire 5.3 Grid Independent Solution for U Profile Using Standard k-s Model

46

10.0 100.0 y+

1000.0

Figure 5.4 Grid Independent Solution for Nondimensional U Profile

Using Standard k-s Model

47

4.0

3.0

i 2.0

1.0

0.0 0.0 100.0 200.0

y+ 300.0

Figure 5.5 Grid Independent Solution for Nondimensional k Profile

Using Standard k-s Model

48

0.25 0.20

^0.15 «0.10

0.05 0.00

0.0 100.0 200.0 300.0

y+

Figure 5.6 Grid Independent Solution for Nondimensional e Profile

Using Standard k-e Model

49

0.0 0.3 0.5 y/H

0.8

Figure 5.7 Grid Independent Solution for U Profile

Using Low-Reynolds Number k-e Model

1.0

50

% 10.0

1.0 10.0 100.0 1000.0

y+

Figure 5.8 Grid Independent Solution for Nondimensional U Profile

Using Low-Reynolds Number k-s Model

51

5.0

4.0

3.0

'2.0

1.0

0.0 0.0 100.0 200.0

y+ 300.0

Figure 5.9 Grid Independent Solution for Nondimensional k Profile


52

0.20

0.15

+ 0.10

0.05

0.00

0.0 100.0 200.0 300.0

y+

Figure 5.10 Grid Independent Solution for Nondimensional e Profile


53

2Q0

1.0 IQO KBO

y^

100QO

Figure 5.11 Comparison ofthe Fully Developed U Profiles

54

5.0

4.0

30 + •^2.0

1.0

00 i

%

•_

0.0

- Û*

— - -

100.0

y ^ ^ -

—

y+

^ L K E HD-STKE

• DNS

^*"~"~^*'0.~.i i i .^____^ ^

200.0 m

Figure 5.12 Comparison ofthe Fully Developed Nondimensional k Profiles

55

0.0 100.0 200.0 300.0

y+

Figure 5.13 Comparison ofthe Fully Developed Nondimensional e Profiles

56

0.0 0.3 0.5

y/H

0.8 1.0

Figure 5.14 Effect of Buoyancy on U profiles

57

25 .

20 :.

15 .

10

0

,LKE(Ri=0) "

.LKE(Ri=0.1)

.LKE(Ri=0.2)

.STKE(Ri=0)

.STKE(Ri=0.1)

.STKE(Ri=0.2)

10 100 10001

y+

Figure 5.15 Effect of Buoyancy on Nondimensional U profiles

58

1.2

0.9

(0

0 0.6

0.3

0.0 0.0 0.3 0.5

y/H

LKE(Ri=0) LKE(Ri=ai) LKE(Ri=0.2) STKE(Ri=0) STKE(Ri=0.2)i STKE(Ri=0.2):

- t>

0.8 1.0

Figure 5.16 Effect of Buoyancy on Dimensionless Temperature

59

5.0

4.0

3.0

2.0

1.0

0.0 0

—-0_LKE(Ri=0) J_n-LKE(Ri=0.1)

-,A_LKE(Ri=0.2) ^^STKE(Ri=0) ^K_STKE(Ri=0.1)

.STKE(Ri=0.2)

100 200 300'

y+

Figure 5.17 Effect of Buoyancy on Nondimensional k Profiles

60

+ 0>

0.25

0.20

0.15

0 100 200 300

y+

Figure 5.18 Effect of Buoyancy on Nondimensional e Profiles

61

CHAPTER VI

TEST CASE 2: BACKWARD-FACING STEP

Quite often, the boundary layer separates in many flow processes of practical

interest. Model testing is often performed on a backward-facing step because of its

simpler geometry and the availability ofthe experimental data. In this chapter, backward-

facing step has been selected to study buoyant turbulent flow for separating flow.

6.1 Problem Statement and Boimdary Conditions

Numerical computation has been performed for turbulent flqw over a backward-

facing step studied by Vogel and Eaton (1985) which is frequently used in benchmarking

the performance of turbulence models for separating and reattaching flows. The

computational domain consists of two walls and an adiabatic step. The step height is

given as H and the inlet height and the height after the step are assigned as 4H and 5H,

respectively, which make an expansion ratio (ER) of 1.25. The computational domain

was set to 30H in the streamwise direction. A diagram showing the computational

domain is shown in Figure 6.1. For the measurements, the average inlet velocity was 11.3

m/s and the height ofthe step was 3.8 cm. The Reynolds number based on the step height

and a laminar Prandtl number for the experiment are given as 28,000 and 0.71,

respectively. At the inlet (x=-4H), a fully developed velocity profile was prescribed from

the channel flow test case described in Chapter V. The coordinate x is measured from the

step end. Constant heat flux through the south wall was set to 270 W / m \ corresponding

to a maximum temperature of 15 °C above ambient temperature, and the north wall and

62

the step are considered as adiabatic. Constant inlet temperature was imposed, and at the

exit, zero-gradient boundary conditions are assigned for temperature, velocities, and

turbulent quantities. Gravitational force is set to act in the opposite direction of the flow

stream. Numerical predictions with a low-Reynolds number k-e model are compared to

the work by Seo and Parameswaran (2000) which was performed with the standard k-e

model.

6.2 Grid Arrangement and Grid Independent Study

For this study, non-uniform grids are employed which have clustered near the step

end and in the recirculation region. One of the grid arrangements and the partial view of

the grids used in this study are shown in Figures. 6.2 and 6.3, respectively. The grid

distribution is controlled by the parameters r^ and r^. Here r^ and r^ are expansion

ratios of the grids in the streamwise and normal directions, respectively. The

computational domain in the streamwise direction was divided into two regions; the r^^

was assigned to a value of less than one for the region before the step (-4H < x < 0), and

the r„ was assigned to a value of greater than one for the region after the step (x>0). In

the same manner, r , and r^j' which control the grid expansion ratios in the normal

direction of grids for the region between 0 < y < H and H < y < 5H, respectively. Grids

are clustered near the walls in order to place some of grid points inside the viscous

sublayer.

In any numerical calculation, it is necessary to show that the results are

independent of the grid size. In order to achieve this, calculations have been performed

with three grid sizes.

63

An earlier work of Seo and Parameswaran (2000) employed a mesh size of 102x62

with the following expansion factors for the grid distribution:

(r„,r., ,rj,) = (1.02, 0.8, 1.04). (6.1)

In the spanwise direction, only one expansion factor, r,,, was used for 0 < y < 5H

while two expansion factors, r,,, and r,,,. were employed for the current study with the

low-Reynolds number k-e model to have a clustered grid near the step end.

Three mesh sizes employed for the grid independent study for the low-Reynolds

number k-e model are 85x43, 122x62, and 122x93. The grid sizes used for the low-

Reynolds number k-e model are close to those employed for the standard k-e model.

However, more attractions in a crosstream direction have been imposed to place the grids

within the viscous sublayer. The values for the expansion factors are as follows:

('•n.' V ' rî '•42) = (1 -28, 113, 0.65, 1.025) . (6.2)

These values were adjusted in order to maintain continuity of the grid distribution

at the junction of the grid blocks. A sudden expansion or contraction in the grid size can

cause numerical instabilities. The y* values for the cells adjacent to the wall for 85x43,

122x62, and 122x93 meshes are adjusted to be 18.5, 10.7, and 2.9 at the exit,

respectively.

The low-Reynolds number k-e model with 85x43 mesh predicts the reattachment

length at x/H=5.72, while the predictions of reattachment length with 122x62 and 122x93

meshes are x/H=5.85 and x/H=5.87, respectively. The average reattachment length for

the three test cases is x/H=5.81 which has a 12% increase compared to the value of

x/H=5.24 as predicted by the standard k-e model.

64

Numerical results from three sets of grids are compared at the vicinity of the

reattaching point. Accurate prediction of the reattachment location is one of the key

problems in evaluation of a turbulent model. Location ofthe reattachment point has been

calculated using a skin friction coefficient which changes its sign at the reattachment

point.

Figures 6.4 and 6.5 show the plots of U and V velocities for the three meshes at x/H-

= 5.0. The deviation is not significant because of close predictions of the reattachment

length using the three meshes. However, the U velocity profile of 62x38 mesh shows a

small deviation from the other two profiles in Figure 6.4. Considering velocity profile

deviation and the reattachment length predicted, the 122x62 mesh is selected for further

study.

6.3 Influence oflnlet Boundary Conditions

Inlet boundary velocity profile, specifically boundary-layer thickness, has an

influence on the reattachment length and the corresponding momentum and heat transfer

characteristics, such as the skin fiiction coefficient and Stanton number. Two inlet

velocity profiles are employed to examine its influence on the reattachment length. One is

the constant inlet velocity profile that was used in the grid independent study and the

other is the fully developed velocity profile generated by channel flow in Chapter V. The

fiiUy developed velocities and turbulent quantities for inlet boundary conditions are

shown in Figure 6.6, for a chaimel with height of 4H. Using two inlet velocity profiles,

the changes ofthe reattachment lengths are examined.

65

The low-Reynolds number k-e model increased the reattachment length from

x/H=5.84 to x/H=6.38, which is a 9.2% increase, by introducing the fully developed inlet

velocity profile. The velocity vectors for each case are shown in Figure 6.7.

6.4 Influence of Discretization Schemes for the Convective Term

In section 4.4, the QUICK scheme for discretization ofthe higher-order convective

term has been discussed. In this section the influence of the QUICK scheme on the

reattachment length will be discussed and compared to upwind differencing scheme for

two turbulence models. The low-Reynolds number k-s model has been shown to increase

the reattachment length from jc , IH =5.91 to x,, ///=6.38 which is a 6.9% increase, by

replacing the upwind scheme with the QUICK scheme. The velocity vector plots are

shown in Figure 6.8. Figure 6.7(b) is plotted again for better comparison with Figure 6.8.

6.5 Results without Buoyant Effects

In this section, forced convection has been studied to examine the difference in

results between the standard k-s model and the low-Reynolds number k-s model. From

the grid independent study in the previous section, 102x62 and 122x62 meshes with

different attraction factors are employed for the standard k-s model and the low-Reynolds

number k-e model, respectively. The QUICK scheme for the convective term and the

fully developed inlet velocity profiles for the inlet boimdary condition are used for this

study.

66

Velocities and temperature are plotted against the experimental measurements by

Vogel and Eaton (1985) and the turbulent quantities from two different turbulence

models are compared to each other.


Velocity profiles for the velocity component U are shown in Figure 6.9 and

compared against the experimental measurement by Vogel and Eaton (1985).

Comparison is made at four representative locations in the recirculation, reattachment and

developing regions. The corresponding velocity vectors are shown in Figure 6.8 (b). The

reference velocity is measured at the inlet and is 1.08 times the mean velocity. Velocity

profiles obtained using the two turbulence models follow the trend of the experimental

velocity profiles faithfully. It can be seen from the figure that the low-Reynolds number

k-s model shows low sensitivity to the velocity change near the wall compared to the

standard k-s model and the experimental data. The discrepancy between the experimental

data and the numerical predictions for the y/H>1.0 region is due to the difference of inlet

velocity profiles between experiment and numerical computations. The boundary layer

thickness ofthe experimental inlet velocity profile is 1.1 times the step height while the

inlet velocity profile used for computation is a fiilly developed profile.

6.5.2 V Velocity Profile

Comparison of the V velocity profiles has been done for the same locations of the

U velocity profiles. In Figures 6.10 (a) and (b) which are in the recirculation region, the

magnitude of the V velocity of the standard k-s model is greater than that of the low-

67

Reynolds number k-e model because the standard k-e model has smaller recirculating

bubbles than the low-Reynolds number k-e model.

6.5.3 Reattachment Length

The reattachment length changes with the inlet profiles, turbulence models, and

with the differencing schemes. Experiment by Vogel and Eaton (1985) indicates that thd

reattachment of the current step is 6 2/3 times the step height. Table 6.1 shows the

summary of the reattachment lengths and the errors against the experimental data.

Compared to the experimental data, the standard k-e model underpredicts the

reattachment length by about 14-20% while the low-Reynolds' number k-s model

underpredicts about 4-12% ofthe same quantity.

Table 6.1 Summary ofthe Reattachment Lengths for Various Cases Model STKE

LKE

Inlet Velocity Condition Fully Developed

Constant

Fully Developed

Con\ecti\« Term QUICK QUICK upwind QUICK

Xr1/H 5.24 5.84 5.97 6.38

Xr2/H 0

0.61 0.51 0.67

Error(%) 21.4 12.44 10.49 4.35

6.5.4 Skin Friction Coefficient Comparison

The predicted skin friction coefficients are compared with the experimental data in

Figure 6.11. The pattern of the skin friction coefficients is closely related to the

reattachment length. Experimental results show the effect ofthe secondary bubble, which

makes the friction coefficient positive at the comer of the step. Numerical results,

however, do not show the clear effects ofthe secondary bubbles, although Table 6.1

shows there are secondary bubbles. The computational models overpredict the absolute

skin friction inside and outside the recirculation bubble. This means that the magnitude of

68

the velocity near the wall is overpredicted. For the standard k-e model the overprediction

can be due to the wall functions employed. Experimental results by Adams and Johnston

(1988) for the velocity profiles inside the recirculating region is compared with the

standard wall function in Figure 6.12. From the figure, it is clear that the velocities

obtained from the wall-fimctions are much higher than the experimental values. Figure

6.13 shows the vj. variations used for the two turbulence models. It is clear that yj, fot

the standard k-e model is less than 12 within the recirculation region and near the

reattachment p' int Since the velocities were prescribed from the wall function for these

points, predicted velocities, in general, would be much higher than the actual ones for the

standard k-e model. Hence, the wall function is inadequate to predict recirculating and

developing flows.

A low-Reynolds number k-e model has been employed to overcome this problem.

In low-Reynolds number k-e model, yJ, can be located anywhere theoretically. However,

Chien (1982) found that the yp should be less than 1 for an accurate prediction in

channel flow. In addition, the grid independent study for channel flow in Chapter V

supports Chien's statement. However, for practical difficulty, the fine grid system which

has a point within yJ, <l was not employed for the backward-facing step. However, the

meshes for a low-Reynolds number k-e model are designed to have smaller yJ,, e.g., less

than 10 even at the exit, compared to the standard k-e model. The standard k-e model

shows fast recovery of skin friction coefficient compared to the low-Reynolds number k-

s model and experiment. This may be due to the overprediction of the velocity near the

wall by the wall function in the developing region after the reattachment point.

69


The disd-ibution ofthe turbulent kinetic energy k is shown in Figure 6.14. In Figure

6.14(a) the maximum value of k appears in the shear layer downstream of the step.

Kasaki et al. (1993) stated that the location ofthe maximum in the y-direction changes as

the flow goes downsU-eam; from the separation to just upstream of the reattachment, it

moves toward the wall, but gradually away from the wall after reattachment due to the

turbulent kinetic energy diffusion toward the outer edges ofthe shear layer. The turbulent

kinetic energy has a maximum near the reattachment point. The standard k-s model

shows faster difftision of turbulent kinetic energy than low-Reynolds number k-s model

within the recirculation region. After the reattachment point it is known that the peak

value ofthe turbulent kinetic energy decreases with the distance downstream.


The mechanics of the turbulent kinetic energy dissipation rate profiles can be

explained by understanding turbulent kinetic energy profiles mentioned in the previous

section. In addition, s has a high peak near the wall which was not evident in k profiles. It

is observed that this peak value of s near the wall increases within the recirculation

region and reaches a maximum near the reattachment point as shown in Figure 6.15.

6.5.7 Turbulent Prandti Number Profile

The turbulent Prandtl number plays an important role in the prediction of the heat

transfer. For the standard k-s model a value of 0.85 was used for the turbulent Prandtl

70

number. For the low-Reynolds number k-s model Equation (3.28) first proposed by Kays

and Crawford (1993) was used to calculate the turbulent Prandtl number. Figure 6.16

shows tile contour plot of the turbulent Prandtl number predicted by the low-Reynolds

number k-e model using Equation (3.28). Basically the turbulent Prandtl number is

increased near the wall along the step and the south wall because of the increase of the

eddy viscosity due to turbulent kinetic energy production. In the same sense the rise in'

Pr, has been noticed in the shear layer downstream of the step end. The range of the

turbulent Prandti number has been examined by changing Pe, - 0 ' , /v)Pr and the plot is

shown in Figure 6.17. Pe, is a turbulent Peclet number, and is conveniently expressed as

the product of the Prandtl number and the eddy-viscosity ratio. The turbulent Prandtl

number varies between 1.7 and 0.85 as Pe, increases from 0 to 100 which can cover most

of the range for the channel flow. Figure 6.18 shows the Pr, profiles for selected

locations near the step end.

According to the Figures 6.16 and 6.18, Pr, is 0.85 for most of the recirculation

region which contradicts the experimental measurement by Pak (1999) and numerical

calculation by Rhee and Sung (2000) and Abe et al. (1995). Pak (1999) measured tiie

turbulent Prandtl number within the recirculation region using Equation (3.27) and found

that the turbulent Prandtl number is over 1 close to the wall and approaches 0.25

asymptotically away from the wall in the recirculation region. This is much less than the

conventional turbulent Prandti number, 0.85, used for boundary layer flow due to a

respectively high decrease of momentum compared to the decrease in thermal diffusion.

Rhee and Sung (2000) conducted numerical calculation using a nonlinear low-Reynolds

number k-s model to flnd that Pr, increases considerably in the recirculating region

71

owing to the enhanced eddy diffusity. They found that the turbulent Prandtl number

varies from 0 to 2.0 according to the recirculating pattern within the recirculation region.

Abe et al. (1995) found that a calculated turbulent Prandtl number with the two-equation

heat-ti-ansfer model is much higher than the conventional value of Pr,=0,85. Although

the turbulent Prandtl number plays a important role in the calculation of the heat transfer

turbulent the qualification of Prandtl number in a recirculating region is still an open'

research issue.

However, it is certain that a reasonable prediction of heat transfer with separating

and reattaching flow is impossible with a constant turbulent Prandtl number. In addition,

the empirical correlation proposed by Kays and Crawford, Equation (3.28), is not also

suitable for the recirculating flow because the correlation does not include the effects of

thermal eddy diffusivity which is used for the original calculation ofthe turbulent Prandtl

number. Equation (3.27).

6.5.8 Stanton Number Comparison

Computed Stanton numbers are compared with experimental values for non-

buoyant flow in Figure 6.19. Experimental values by Vogel and Eaton (1985) are

provided for different boundary layer thicknesses, 6/H, at the inlet. As tiie boundary layer

thickness increases, the curves of tiie Stanton numbers move downward. Numerical

predictions witii the standard k-s model, with a fully developed inlet velocity boundary

condition, underpredicts the Stanton number. This underprediction of the standard k-s

model is due to the overprediction of the wall temperature by tiie wall function. On the

contrary, with a tiirbulent Prandtl number of 0.85, the Stanton numbers were over-

72

predicted witii tiie low-Reynolds number k-e model. These overpredictions by the low-

Reynolds number k-e model have also been noticed in the work of Abe et al. (1995) with

Pr, =0.9 and the work of Rhee and Sung (2000) with Pr, =1.0. This significant

overprediction indicates that a constant Prandtl number assumption is no longer valid in

separating and reattaching flows. However, as Vogel and Eaton (1985) stated the peak

heat ti-ansfer rate for die backward-facing step occurs slightly upstream of reattachment

jK)int.

6.6 Results with Buoyant Effects

In this section, several different Richardson numbers are introduced to test the flow

patterns and corresponding characteristic in heat transfer over the back-ward-facing step

using two different turbulent models.

6.6.1 Reattachment Length

Several different Richardson numbers are introduced to test the flow patterns over

the backward-facing step. Figure 6.20 shows three representative particle traces for

results obtained using the standard k-e model. The reattachment length decreases as the

Richardson number increases. In addition, the size of the secondary bubble at the comer

of the step increases with increasing the Richardson number. This occurs because the

flow near the step moves in the main flow direction when there is significant buoyancy.

When the Richardson number is increased to 0.2, the length of the secondary bubble has

almost reached that of the main bubble. Consequently, the main bubble is lifted by the

secondary bubble and is about to be separated from the surface. The Richardson number

73

from which the main bubble is separated from the surface is referred to as the 'critical

Richardson number', hereafter.

The same ti-end has been noticed when the low-Reynolds number k-e model is

used. Compared to the results with the standard k-e model, the size of the secondary

bubble at the comer of the step with /?/„ =0 is significant, which was not shown cleariy

for the results with standard k-e model. This secondary bubble is noticed in the

experimental resuhs by Vogel and Eaton (1985) as a corner eddy. The low-Reynolds

numht'r k-i: model has relatively bigger in size recirculation bubbles, than the standard k-

e model. However, the critical Richardson number for the low-Reynolds number k-s

model is smaller than that of the standard k-s model due to the fast enlarging of the

secondary bubble as the Richard number increases.

Figure 6.22 shows the influence of the Richardson number on the reattachment

length. In Figure 15, A>, denotes the reattachment length of the main bubble and Xr^

denotes that of the secondary bubble. Two reattachment lengths approach each other

indicating the possible values of the critical Richardson numbers for two turbulence

models. The flow pattern after the critical Richardson number will be discussed in section

6.6.4.

6.6.2 Turbulent Scales

Turbulent scales such as time and length scales are important in conducting

experiment for turbulent flow. This section is dedicated to provide the quantitative ideas

that how much the time and length scales are, for the flow over backward-facing step. For

74

velocity scale, Kolmogorov velocity scale is employed for the study. The dimensions of s

and V are

£ = {lengthy litimef,

v = {length)-1 time.

Based on tiie dimensions of s and v, Kolmogorov defined a length scale as

L=(^''fe)"- (6.3)

By performing dimensional analysis using Kolmogorov velocity scale and

Kolmogorov length scale the time scale can be calculated,

T,-={vl£y'\ (6.4)

All the turbulent scales are defined by the turbulent kinetic energy dissipation rate,

e. Contour plots for the length scale and the time scale with ^/^ =0.1 are shown in Figures

6.23 and 6.24. Length scale varies between 4.62x10"'* and 6.89x10""' m with smallest

value near shear layer which defines the boundary of the main recirculating bubble after

the step end. It is also shown that length scales are relatively small within recirculating

region compared to values outside ofthe recirculating bubble. The variation of time scale

lies between 5.97x10'^ and 1.22x10'° s. Referring to the particle trace in Figure 6.20, like

the length scale, the minimum time scale exists along the shear layer after the step end.

Without buoyant effects, the length scale varies between 4.61x10"'' and 9.38x10"^ m

while the time scale varies between 5.95x10"^ and 2.46x10"° s. Maximum turbulent scales

are decreased by increase of the buoyant effects. However, it was found that the

minimum turbulent scales which are important in experimental measurement are not

affected by buoyant effects. Once the turbulent time scales are highly affected by shear

75

layer, geometry and the Reynolds number are more important factors than buoyant

effects to define the turbulent scales.

6.6.3 Skin Friction Coefficient Comparison

Figure 6.25 shows the variation of the skin friction coefficient with different

Richardson numbers. With a high Richardson number the flow is accelerated near the hot-

wall. The acceleration in die flow creates a high friction coefficient along the south wall

with constant heat flux. The friction factor remains almost positive throughout the entire

length of the south wall indicating that the secondary bubble has almost reached the size

of tiie primary bubble at /?/„=0.2 and /?/„=0.175 for the standard k-s model and the low-

Reynolds number k-s model, respectively. In Figure 6.25(b), the c/ curve with

Ri^ =0.175 for the low-Reynolds number k-s model shows fluctuation within the

recirculation region. This means that the secondary bubble becomes unstable due to the

interaction writh the main recirculation bubble and is about to create the vortex shedding.

The most significant difference between the two results is the magnitude of the friction

coefficients predicted by the two different turbulence models. The cf predicted by the

standard k-e model is more than double that of the cf by the low-Reynolds number k-s

model. This could be explained by the wall function functions used in the standard k-s

model which overpredict the velocity within the recirculation region and results in fast

recovery of the velocity in the developing region after the reati:achment point. However,

the friction coefficient in buoyany-affected flow is closely related with heat transfer.

Therefore, it is too early to say which model predicted better results in the skin friction

coefficient without any reliable data.

76

6.6.4 Temperature Contour Comparison

The size and pattern of recirculation bubbles are important factor to characterize

the temperature pattern after the step end. The buoyancy-affected temperature contours

obtained by two turbulent models are plotted in Figures 6.26 and 6.27. Figure 6.26(a)

shows that temperature contour has been rotated in clockwise direction due to the main

recirculating bubble. In (b) and (c) of Figure 6.26, the secondary bubble moves the hot

temperature region away from the step end as Richardson number increases due to the

increase of the secondary bubble in magnitude. The same trend has been noticed in

Figure 6.27 obtained by the low-Reynolds number k-e model.

Zero-gradient temperature boundary condition has been imposed to the exit

boundary condition. If the thermal layer is developing even at the exit the boundary

condition may create numerical error for the calculation of the thermal field. Magnified

temperature contours at the exit are shown in Figure 6.28. Temperature gradient near exit,

x/H=22, was calculated to examine the validity of non-zero boundary condition for the

temperature at the exit. Temperature gradients for cases (a) and (c) in 6. 28 were found to

be 1.67x10'^ and 3.33x10'^, respectively. Although the temperature gradient had been

increased as the buoyant effect was increased the magnitude of the temperature gradient

were close to the given zero-gradient temperature boundary condition at the exit.

6.6.5 Stanton Number Comparison

In Figure 6.29, computed Stanton numbers for different Richardson numbers are

compared with each other. For the standard k-e model, Stanton numbers for each case are

77

highly dependent on the structure of tiie recirculation bubble within the recirculation

region. After tiie reattachment point, the Stanton numbers mainly depend on the

Richardson number. It is found that the flow with a higher Richardson number has a

higher Stanton number in the region after the main bubble.

However, the Stanton numbers predicted by the low-Reynolds number k-e model

decrease as Uie Richardson number increases. This is a different trend compared to the'

results shown by tiie standard k-e model. The Stanton number tends to increase slightly

with the increase of buoyant effects in turbulent flow (Osborne &. Incropera, 1985). This

nonphysical prediction can be due to two different reasons.

One ofthe reasons for the low prediction ofthe Stanton number with the increase

of the Richardson number is the constant turbulent Prandtl number discussed in section

6.5.7. As mentioned, the turbulent Prandtl number should be defined by the ratio of the

eddy viscosity to the thermal eddy diffusivity. However turbulent heat transfer modeling

with a constant turbulent Prandtl number adopts one representative turbulent Prandti

number for the calculation ofthe whole domain, including the recirculation region. If the

turbulent Prandtl number is fixed as a constant for a flow with recirculating flow, thermal

eddy diffusivity should be proportional to the eddy viscosity regardless of its physical

values. Figure 6.30 shows the effect of the constant turbulent Prandtl number on the

Stanton numbers.

The second reason for the low prediction ofthe Stanton number for buoyant flow is

due to the non-dimensional distance from the hot wall, y]>. Figure 6.31 shows the y].

variation along the hot wall with various Richardson numbers. A high Richardson

number provides a high y], away from the step end. As discussed in Chapter V for

78

channel flow, the low-Reynolds number k-e model requires that the yp should lie with

tiie viscous sublayer, if possible >';<1. If this condition is not satisfied, the low-Reynolds

number k-e model generates inaccurate predictions because of the collapse of turbulent

quantities near the wall, as shown in Chapter V. However, in practice, it is difficult to

keep all .i> within such a range along all the boundary walls and obstacle placed in the

sti-eam. Especially when tiiere is a buoyancy effect , the orthogonal grid system cannot

catch up witii the velocity development along the heated wall. Therefore, an adaptive grid

system will be necessary when tiie low-Reynolds number k-e model is employed to

predict the buoyant flow or developing flow in a complex geometry.

6.6.6 Vortex Shedding

In Figure 6.22, the influence of the Richardson number on the reattachment length

is shown discussed. As mentioned in the section 6.6.1, two recirculating bubbles interact

with each other and become unstable creating vortex shedding after the critical

Richardson number. For the standard k-e model the critical Richardson number was

found to be 0.225. For the range of such a large Richardson number, pattems of two

recirculating bubbles have changed their form along time showing unsteady flow patterns

while the flows with small Richardson numbers give only steady-state solution. As the

secondary recirculation bubble grows, the main bubble becomes unstable and is divided

into two small bubbles creating periodic vortex shedding. In the same marmer, the main

bubble breaks the secondary bubble into two, creating vortex shedding with small

magnitude. Figure 6.32 shows the bubble pattems for a flow with a Richardson number

of 0.25, which produces vortex shedding. The period of vortex shedding was computed as

79

5.35s, approximately, based on the reattachment length in Figure 6.32. Figure 6.32(e)

shows the instant moment at which the flow is not reattached on the hot wall at time

3.99s, which makes the friction coefficients positive for the whole range along the wall.

The variation of the reattachment of the two recirculation bubbles for a flow with

/?/„=0.25 is shown in Figure 6.33.

The prediction by low-Reynolds number k-e model, after the critical Richardson'

mmiber, is shown in Figure 6.34. Interaction between the two bubbles is relatively small

compared to tiie result using the standard k-s model. The main recirculation bubble is all

consistently separated from the heated surface by the secondary bubble, and the center of

the main bubble is lifted up and down creating a neglectable vortex'. Unlike the standard

k-s model, the vortex pattem predicted by the low-Reynolds number k-s model changes

randomly without a clear period due to minor interaction between the two bubbles.

Cetegen et al. (1998) and Jiang and Luo (2000) correlated the oscillation frequency

with the Richardson number for the puffing phenomenon of an axisymmetric thermal

plume. Investigation of the relation between the Richardson number and the Strouhal

number, Sh„ =flllu, for the resuhs predicted by the standard k-s model was conducted

for this study. Frequencies ofthe vortex shedding were evaluated for the range of 0.225 <

Ri < 0.6 where f is the frequency of the vortex shedding.

Eleven point three (11.3) m/s of velocity and 3.8xl0"^m of step height have been

used to calculate the Strouhal numbers. The result is shown in Figure 6.35 with the

correlation, Sh = 0.048 /°'™ • It is also found that friction coefficients and the Stanton

numbers grew larger with increasing Richardson numbers regardless of vortex shedding.

80

Finally, tiiere is a question as to whether the buoyant-induced vortex shedding is

desirable for heat ti-ansfer. Figure 6.36 shows the Stanton number variation for different

times during tiie vortex shedding. The Stanton number for flow with /?/„=0.2 has been

plotted together as a reference. It shows that the Stanton number has increased, in the

recirculation region, as well as after the recirculation region, while the other flows

without the vortex shedding shows that the Stanton number in the recirculation region is

decreased as tiie Richardson number is increased. Fluctuation of the Stanton number and

friction factor have been also noticed right after the recirculation region as shown in

Figures 6.36 and 6.37, respectively.

81

•

•

' lMJ(y) • V=0

• T=To 1

H • t

ck=0 >

, m 4H

qw=o

^dU dV dT

8K 8K dx

'

q^=CQnst

261 1

=0

Figure 6.1 Problem Description and Boundary Conditions for Turbulent Flow

82

Figure 6.2. Grid Configuration for the Turbulent Flow (LKE Model, 122x62 Grid)

.—\—t'^ti—^±~^1i "irf—"ttl" 1

x/H

Figure 6.3. Partial View ofthe Grid Configuration near Step End

83

-0.50

3-0

.LKE( 123*93)"'

.LKE(122*62)

.LKE(85*43) '

x/H=5.0

0.00 0.50

U/Um

1.00 1.50

Figure 6.4 Grid Independent Solution for U velocity at x/H=5.5 Using Low-Reynolds Number k-s Model

84

5C

.LKE(123*93)

.LKE(122*62)

.LKE(85*43)

-0.15 -0.12 -0.09 -0.06 -0.03 0.00

V/Um

Figure 6.5 Grid Independent Solution for V velocity at x/H=5.5 Using Low-Reynolds Number k-s Model

85

4.0

3.0

5 2.0

1.0

0.0

0.0 1.0 U/Um

(a) U Profile

n

2.0

4.0

-1.0E-03 O.OE+00 1.0E-03

V/Um

(b) V Profile

(c) k Profile (d) s Profile

Figure 6.6 Profiles for Inlet Boundary Conditions

86

O.S 1 1.5 2 2.5 i 3.5 4 4.5 5 5.5 6 6.5

(a) Low-Reynolds Number k-e Model with Constant Inlet Velocity Profile(.x,i IH =5.84)

f --

-

= ^ t n ,

\ . I c \ \ \ \ \ s >.' \' V > ,'' r ' / ! ^ ^^

4 l i m ^ ^ ^ ^ ^ ^

- . — ^ r ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^

1' ' '|f ' V^f~^T^'%^^^ '\ " h ^ 0.5 1 l.S 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5

(b) Low-Reynolds Number k-s Model with Fully Developed Inlet Velocity Profile(;c,., ///=6.38) Figure 6.7 Influence ofthe Inlet Velocity Profile on Reattachment Length

87

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5

(a) Low-Reynolds Number k-e Model with Upwind Scheme( .v , / H =5.97)

0-5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5

(a) Low-Reynolds Number k-s Model with QUICK Scheme(x„ ///=6.38)

Figure 6.8 Influence ofthe Schemes for Discretization of Convective Term on Reattachment Length

88

% Experiment(Vogel) .<!—LKE -A—STKE

x/H=2.20

(a)

« Expenmerit(Vogel) H>-LKE .^—STKE

-0.2

x/H=6.67

0.2 0.6 U/Uref

X

9 ExperlmenKVogel) j K3—LKE -A—STKE

-0.2 0.2 0.6 U/Uref

(b)

Experimeht(\7ogel j

.IKE

J

-0-i-»

x/H=8.87

-0.2 0.2 0.6 U/Uref

(0 ("> Figure 6.9 U Profiles for Reference Conditions

89

X

-0.20 -0.10 0.00 0.10 0.201

V/Um

-0.20

(a)

0.20

0.201

(b)

X

-0.20 0.20

(c) (d)

Figure 6.10 V Profiles for Reference Conditions

90

4-O

Figure 6.11 Comparison of Model Results for Skin Friction Coefficient to Vogel's Experiment (6 / i / = 1.1)

91

15

10

+ 3

0

/ • • • / •

/ • • ^ / •

/ • y/ \ y+=u+

y^ Law of the wall ^^^.^"•^ • Experlment(Adams) x/Xr =

•""""^ • Experiment(Adams) x/Xr = 1 , 1 1 X

•

0.67 0.45

10

y+ 100

Figure 6.12 Mean Velocity Profiles in Recirculating Region(Re=38000, ER=1.67)

92

Figure 6.13 y+ variations along the south wall

93

X "5»

0.00 0.02 0.04 0.06

k/U'

(a)

ref

X

0.00 0.02 0.04 0.06

2 ref k/U^

X

0.00 0.02 0.04 0.06i

k/U" ref

(b)

0.00 0.02 0.04 0.06

k/Uêf

(C) (d)

Figure 6.14 k Profiles for Reference Conditions

94

0.00 0.01 0.02 0.03 0.04'

e/U^ ref

(a)

0.00 0.01 0.02 0.03 0.04|

0.00 0.01 0.02 003 0.041

^'^ ref

(b)

0.00 0.01 0.02 0.03 0.04|

e/U^ref

(c) (d)

Figure 6.15 s Profiles for Reference Condhions

95

1.2S 1.5

Figure 6.16 Contour for Turbulent Prandtl Number Using Eq. (3.28)

0 50 100

Figure 6.17 Range of Turbulent Prandti Number Using Eq. (3.28)

96

0.85

Pr. 1.35

Pr. 1.85

(a) (b)

0.85 1.35

Pr.

1.85i

x/H=1.23

-A-A-0.85 1.35 1.85:

(c) (d)

Figure 6.18 Turbulent Prandtl Number Profiles Variation near the Step End Using Eq. (3.28)

97

CO

0.006

0.005

0.004

0.003

0.002 -

0.001

* ff

STKE(Prt=0.85) • LKE(Prt=0.85)

-X— Abe(Prt=0.9) A Rhee(Prt=1.0)

10 15

X/H

Figure 6.19. Comparison of Stanton Number to Experimental Results with Various Inlet Boundary Layer Thickness

98

x/H

a) /?/„=()

b) ^/^=0.1

c) Ri„=0.2

Figure 6.20. Particle Trace for Mixed Turbulent Convection Using standard k-s Model

99

a) Riff=0

x/H b) Riff =0.1

c) Riff =0.175

Figure 6.21. Particle Trace for a Mixed Turbulent Convection Using Low-Reynolds number k-s Model

100

Xr1(STKE)^ Xi2(STKE) Xrl(LKE) Xr2(LKE)

0 0.05 0.1 0.15 0.2 0.25

Ri

Figure 6.22. Influence ofthe Richardson Number on Reattachment Length

101

V .

•I" =F ^ 2 - 3 4 5 6 7 8

Figure 6.23. Contour Plot for tiie Kolmogorov Length Scale {Riff =0.1, 10 contour between 4.62x10"^ and 6.89x10'^)

10

J-3

- .2

1L^ -h + + + + ^

2 3 4 5 6 7 8 Figure 6.24. Contour Plot for the Kolmogorov Time Scale {Riff=0.\, 10 contour between 5.97x10"^ and 1.22x10"°)

10

102

o o

u

10

8

6

4

2

0

-2

-4

10

8

6

4

«- 2

0

-2

.Ri=0.0(STKE) • Ri=0.1(STKE) i • RI=0.2(STKE) Vogel

x/H

(a) Standard k-s Model

-i»-Ri=0.0(LKE) _o_Ri=0.1(LKE) -A-Ri=0.175(LKE)

• Vogel

(1 ^^«5ft(|&§*^ 10 15

x/H

(b) Low-Reynolds number k-s Model

Figure 6.25. Influence ofthe Richardson Number on Skin Friction Coefficient

103

^•N

\ .

aaep —~^^^ asfitea

3 x/H

(a)/?/V,=0

6 3 x/H

(b) i?//,=0.1

1 2 3 4 5 x/H

(c)i?/;,=0.2

Figure 6.26 Buoyancy-affected Temperature Contour for Recirculating Region Obtained by Standard k-s Model

6

104

• ^ " • ^ " " " ^ ^ ^ — - ^ - — . ^

\ \ \ \

3 x/H

(a)/?/„=0

\ \ \ \ \

i^M^

x/H

(b) Riff =0.1

1 2 3 4 5 x/H

(c) Riff =0.175

Figure 6.27 Buoyancy-affected Temperature Contour for Recirculating Region Obtained by Low-Reynolds Number k-s Model

105

T-T;,=O.OO

22 24 x/H (b)

26

22 26 24 26 22 24 x/H x/H (c) (d)

Fig. 6. 28 Temperature Contour Plots at Exit Boundary for Two Turbulent Models: (a) Standard k-s Model(Ri=0); (b) Low-Reynolds Number k-s Model(Ri=0); (c)

Standard k-s Model(Ri=0.1); (d) Low-Reynolds Number k-s Model(Ri=0.1)

106

4.E-03

3.E-03

S 2.E-03

1.E-03

O.E+00 1

:-g^B5^ •

# ^ ^ ^ ^ r ^ jsf"V-o- Ri=0.d(gTOr*'*«=9*,fca

/ l i __Ri=0.1(STKE) fSjr - «^Ri=0.2(STKE) W^ • Vogel a — —.—...—-—.

10 ^ 15

x/H

(a) Standard k-s Model

(b) Low-Reynolds Numer k-s Model

Figure 6.29 Influence ofthe Richardson Number on Stanton Numbner

107

10

X/H

15 20

{a) Riff =0.0

Prt=0.35

^Prt=0.45

Prt=0.65

Prt=0.85

(c)i?i;,=0.175

Figure 6.30 Effects of Turbulent Prandtl Number on Stanton Number

108

0 10 15 x/H

20 25

Figure 6.31 Effects of Richardson Number on Wall Normal Distance for Low-Reynolds Number k-s Model

109

1 2

(a) time=0

(c) time=2.8s

(f) time=5.35s

Figure 6.32 Vortex Shedding after the Backward-Facing Step with Riff =0.25 using Standard k-s Model

110

0

Xrl

Xr2

8

time(s)

Figure 6.33 Reattachment Variation ofthe Flow with Riff =0.25 along the time

111

1 2 3

(b) time=0.71s

1 2 3

(c) time= 1.24s

1 2 3

(d) time= 1.92s

2 3

(e) time=2.65s

2 3

(f)time=3.22s

Figure 6.34 Vortex Interaction Between Two Bubbles after the Backward-Facing Step with Riff =0.20 using Low-Reynolds Number k-s Model

112

(O

0.044

0.042

0.04

0.038

0.036

0 034 .

^^

—o^

-- — - -

o^y""^ y^— _ _.

<>

^ ^ Sh = 0.048 Rl°^^^^ —

: o Sh Power (Sh)

0.2 0.3 0.4 RI

0.5

Figure 6.35 Effect of Richardson Number on Strouhal Number of Vortex Shedding Using Standard k-s Model

0.6

113

I . .

(O

0.004

0.003

0.002

0.001

0.000

X/H

Figure 6.36 Stanton Number Change during Vortex Shedding Using Standard k-s Model

114

o o o

O

X/H

Figure 6.37 Cf Change during Vortex Shedding Using Standard k-s Model

115

CHAPTER VII

CONCLUSIONS AND RECOMMENDATIONS

In this study, the low-Reynolds number k-s model is employed to study turbulent

buoyant flow to overcome the shortcoming of the standard k-s model in predicting the

reattachment length of separating and reattaching flow.

A constant turbulent Prandtl number was used for the standard k-s model and the

varying turbulent model proposed by Kays and Crawford (1993) was used for the low-

Reynolds number k-s model. An algorithm was introduced to find the normal distance of

the damping functions of tiie low-Reynolds number k-s model. The'QUICK scheme was

also used as an alternative of the upwind scheme to enhance the discretization of the

convective terms of the governing equations. The results obtained by the standard k-s

model and the low-Reynolds number k-s model were presented for channel flow and the

backward-facing step flow against experimental data. In addition, the influence of

buoyancy effects was discussed for the channel flow and backward-facing step flow. The

required conditions for the low-Reynolds number k-s model for accurate results were

discussed.

7.1 Conclusions

From the present study, the following conclusions were made concerning the

usage of k-s models to predict separating buoyant flow:

1. The reattachment length for the backward-facing step has been improved by 17%

by replacing the turbulent models of the standard k-s model with tiie low-Reynolds

116

number k-e model. Conventional log-laws cannot be applied to complex flows with

separations. Therefore, it is recommended to use the low-Reynolds number k-s model for

a recirculating flow.

2. The correlation of the turbulent Prandtl number proposed by Kays and Crawford

(1993), which varies between 0.85 and 1.75, did not provide significant effects on

recirculating flow because the correlation was originally developed for channel flow,'

which is an attached flow. With tiie constant turbulent Prandtl number, the low-Reynolds

number k-s model overpredicted the Nusselt number, while the standard k-s model

underpredicted the Nusselt number for backward-facing step flow.

3. In backward-facing step flow, buoyant effects reduced the size of the main

recirculating bubble while increasing that of the secondary bubble. The changes in the

recirculating bubble affected the skin friction coefficient and the Stanton number along

the heated wall.

4. Vortex shedding occurred when the Richardson number exceeded the critical

Richardson number. With a standard k-s model vortex shedding enhanced heat transfer

characteristics due to the fluctuations.

5. A low-Reynolds number k-s model showed physically wrong results when buoyant

effects were included. It was found that grid configuration plays a significant role when

the low-Reynolds number k-s model is used. For channel flow, yl along the wall

boundary conditions should be less than 1 to obtain accurate results. The same rule

should be applied to backward-facing step flow.

117

7.2 Recommendations

The following recommendations are made for improving the accuracy of the

prediction for buoyant flow with the low-Reynolds number k-s model.

1. A varying turbulent Prandtl number, based on a two-equation model for thermal

field, needs to be employed rather than a constant turbulent Prandtl number or varying

turbulent Prandtl nimiber based on an empirical correlation, to solve eddy diffusivity

directiy for heat wdthout prescribing tiie turbulent Prandtl number.

2. An adaptive grid system should be developed to obtain y*^ along the walls, or the

obstacles, for the low-Reynolds number k-s model. In addition, the stability ofthe multi-

block grid system should be studied to secure stable convergence.

118

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126

APPENDIX

DERIVATION OF GOVERNING EQUATIONS

IN A GENERAL COORDINATE SYSTEM

127

A.I Transformation of Governing Equation

U-momentum equation is selected to be transformed from the equation in physical

domain into the equation in ti-ansformed domain.

d{pU) d , d

dp d St dx

(dU dU dy ^^<#

dV dU — + —

^ dx dy

(A.l) + pgp(r-ro)

l.I Transform Relations for Gradient

Transform relations are well defined by Thomson et al. (1985). For 2-dimensional

domain the gradient can be defined in conservative form and non-conservative form as

follows:

Conservative form

Non-conservative form

A =-kA->'4/nl J

/,=-j[-V^+Vn]

where the Jacobian ofthe transform is defined

J=Xi,y^-Xr,yi,

(A.2)

(A.3)

(A.4)

(A.5)

(A.6)

128

Using the ti-ansform relations each term ofthe U-momentum Equation (A.l) can be

transformed into the term for computational domain.

A. 1.2 Transient Term

Transient term will keep its original form if the time derivative is understood to be at

fixed point in the transformed space.

d{pU) _ d{pU)

dt dt (A.7)

A. 1.3 Convective Term

Convective terms can be transformed using Equations (A.2) and (A.3).

. ;5 1 ±{pU') + -l-{pUV) J dx dy =>

[y^pU^X -{y,pU'hj[-kpUv\A^,pUv\[ (A.8)

where G^ = pUy^ -pVx^,G^ = pVx^ -pUy^

A. 1.4 Pressure Term

-^=>-j[ynP^-y^p^ (A.9)

A. 1.5 Diffusive Term

dx

(dU dU => —

J

|(>'.^.^[7k^.-3^.^l4k^^->'^^nl

1 - I - —

J dx\ y^y-eff

j[y,U,-y,uy-j[y,U,-y,U,] (A.10)

129

d_

dy J

+ — J

(A. l l )

A. 1.6 Buoyancy term

pgîT-T,)=>pg^(T-T„) (A.12)

Final transformed equation can be arranged as follows.

d{pU) 1 d . *W'^^Hi,M-^^ Peff _ dU], \ d fPeff dU^ -8u + — I, J " d^J jdt] J " ^ - drj

^ J d^ d (Peff „ dU\, \ d (Peff dU

-^12-r- +-7^r-J dT]) J drj

where tiie covariant metric components for 2-D are defined

J "^'^ d^

(A.13)

+ 5,

^11 =Xr^+y, 1 •

^12 = - ( j f 4 ^ n + n > ' T i ) '

gn=yl+xl

The detailed description of covariant metric components, g ,y, are given in Thomson

etal . (1985).

A.2 Governing Equation in General Form of Differential Equations

in General Coordinate System

130

J ^"a^j a^

^ J dt]

Peff d6

_1__5_ /V^ a ^ 1 a r;/e# „ d(i> ^ J drj

\

J ' a? +5.

(A.14)

A.3 Source functions for each equation

A.3.1 U-Momentum Equation

5=--^ ^''de, -ânJ y a ^ l y

r 2 dji

dk y:-:^-yi-^r^

dU_ an

^}_d_\\^

J dk\ J

dV '-yî;{-'-y^^

dV^ 1 d \V;sf • + —

d\)\ J dx\\ J

dV^

2dU dU ^' an ^ " a

^7l^fxh^-^^^^^l^^^^«^-^«>

(A. 15)

A.3.2 V-Momentum Equation

5.. = — dp dp

I "a^ â7j ^ 1 a |/ê#^

y a ^ J

dU

1 a / e# J d4[ J

1 a f/^.#

2 aF " d^ ^ ^ dT]

dU^

dU dU dV\\ 1 d \Peff(

j a ^ y

aF a^ - ^ ^ ^ ' 7

ar 5^

(A.16)

A.3.3 k Equation

Sk=P.

+ Pt [Ll^û, .x,uy-[y/, -y,v,^-pe-gP^'-j[-^.T,.x,T,]

(A.17)

131

A.3.4 e Equation

k

-^c\-Pt k J

£ p, 1

- C — (A. 18)

A.3.5 Energy Equation

Sr=0 (A.19)

A.4 The Associated Voscosities for each Equation

A.4.1 U-Momentum Equation

Peff=P + Pt (A.20)

A.4.2 V-Momentum Equation

Peff=P + Pt (A.21)

A.4.3 k Equation

Peff=P^Pt'^k (A.22)

A.4.4 s Equation

Peff^P^^^tlÊ (A.23)

132

A.4.5 Energy Equation

p,ff=klCp^p,l?x, (A.24)

133

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A NUMERICAL STUDY OF BUOYANT TURBULENT FLOWS …