wall functions for numerical simulation

Diss ETH No 11073

Wall Functions for Numerical

Simulation of Natural Convection

along Vertical Surfaces

A dissertation submitted to the

SWISS FEDERAL INSTITUTE OF TECHNOLOGY

for the degree of Doctor of Technical Sciences

presented by

Xiaoxiong Yuan

Master of Sciences

born 26 January, 1957

citizen of China

accepted on the recommendation of

Professor Peter Suter, examiner

(Institute of Energy Technology)

Professor Bernhard Muller, co-examiner

(Institute of Fluid Dynamics)

1995

\Z7

Juris Druck+Verlag Dietikon

1995

To my parents, and

to Yongqing and Yi

Foreword

This study was carried out at Laboratory of Energy Systems, Swiss Federal Institute of

Technology Zurich under the supervisions of Professor Peter Suter and Dr. Alfred Moser. To

them I am very sincerely grateful for their great valuable guidance and offering of comfortable

research environment.

Many sincere thanks are given to Professor Bernhard Muller of the Institute of Fluid Dynamics,

Swiss Federal Institute of Technology Zurich, for being the co-examiner to this thesis, and for

his careful reading and insightful comments.

I am grateful to Professor Bu-Xuan Wang and Professor Youting Shen of Tsinghua University,

Beijing, and Professor Shuzo Murakami of University of Tokyo for their advice and

encouragement for my study.

Professor Torstein Fannelop, Professor Bernhard Muller, Dr. Jiirg Kiiffer, and Dr. Alois

Schalin of the Swiss Federal Institute of Technology Zurich, and Professor Qingyan Chen of

Massachusetts Institute of Technology are sincerely acknowledged for their helpful discussions.

Particular thanks are given to Professor Yasutaka Nagano and Professor Toshihiro Tsuji of

Nagoya Institute of Technology, Professor Patrick Le Quere, Dr. Shihe Xin at the National

Centre of Scientific Research, France (CNRS), Dr. Dominiqe Blay and Dr. Sophie Mergui of

the University of Poitiers for providing their experimental data or DNS data. Professor L. R.

Glicksman of Massachusetts Institute of Technology is sincerely acknowledged for providing

the permission to reproduce the figures 5.4a, 5.5a and 5.6a.

I express my appreciation to all staff member of the laboratory for offering friendly atmosphere

and help during the past four and half years.

Finally, I thank my wife for her understanding and support all along.

Xiaoxiong Yuan

Zurich, Spring 1995

Table of Contents

Table of Contents i

ABSTRACT iii

Zusamenfassung iv

1. INTRODUCTION 1

2. NUMERICAL SIMULATION OF FLOWS 2

2.1. Governing Equations and Turbulence Models 3

2.1.1. Navier-Stokes Equations 3

2.1.2. Reynolds Equations and Turbulence Models 5

2.1.3. Wall-Treatment Approach 9

2.2. Boundary and Initial Conditions 10

2.3. Discretization of the Equations and Solving 11

3. WALL FUNCTIONS 12

3.1. Review of Wall Functions for Forced Convection 12

3.1.1. Wall Functions of Velocity and Temperature 12

3.1.2. Wall Functions of the Turbulent Kinetic Energy and the Turbulent

Dissipation Rate 15

3.1.3. Discussion 16

3.2. Review of Wall Functions for Natural Convection 17

3.3. Wall Functions of Velocity and Temperature for Natural Convection 19

3.3.1. Temperature Wall Function 20

3.3.2. Velocity Wall Function 24

3.3.3. Comparison of Scales between Natural and Forced Convection 31

3.4. Wall Functions of the Turbulent Kinetic Energy and the Turbulent Dissipation

Rate for Natural Convection 32

3.5. Discussion 34

4. VALIDATION OF THE NEWWALL FUNCTIONS AND SENSmVITY

ANALYSIS 34

4.1. Natural Convection along Vertical Plates 35

4.1.1. Boundary Conditions Treatment 35

4.1.2. Validation of the New Wall Functions 38

4.1.3. Sensitivity Analysis 46

i

4.2. Natural Convection in Differentially Heated Cavities 53

4.2.1. Validation of the New Wall Functions 53

4.2.2. Sensitivity Analysis 63

5. APPLICATION OF THE NEW WALL FUNCTIONS IN THE PREDICTION OF

AIR FLOWS IN BUILDINGS 65

5.1. Air Flows in a Room 65

5.2. Air Flows in Glazed Spaces 73

6. CONCLUSION 77

7. NOMENCLATURE 79

8. APPENDIX 82

A. Numerical Solution Method 82

B. The Wall Functions Adopted by PHOENICS and TASCflow 86

C. Implementation of the New Wall Functions in CFD Codes 88

9. REFERENCES 91

CURRICULUM VITAE 97

u

ABSTRACT

This thesis presents new wall functions of velocity, temperature, the turbulent kinetic energy,and the turbulent dissipation rate for natural convection, based on dimensional analysis and

experimental data. Because only fluid properties and local parameters are included, the

proposed wall functions are suitable for numerical simulation. A wall function serves as a

bridge across the near-wall region to save computing time and memory.

The validation of the new wall functions in prediction of natural convection along two vertical

flat plates and in three differentially heated cavities shows that the numerical results are nearly

independent of the wall distance of the first near-wall grid cell in a range of one order of

magnitude of the wall distance. Compared with the available experimental data or direct

numerical simulation results in a range of 5xl09 < Ran ^ 1.7xlOn, the predicted profiles of

temperature and velocity as well as the streamwise distributions of the wall shear stress and the

wall heat flux are in good agreement with the data for flows at Ran S 1010. The prediction of air

flows in three spaces indicates that the new wall functions may serve practical engineeringcalculation. They extend the application of the standard k-e model to natural convection.

Moreover, a certain analogy of length, velocity, and temperature scales between natural and

forced convection has been found. The new defined heat-flux temperature and the heat-flux

velocity are appropriate temperature and velocity scales for natural convection. The correlation

of temperature data is independent of the wall shear stress, while the correlation of velocity data

depends both on the wall shear stress and the wall heat flux in natural convection.

In principle, the derived wall functions are valid for natural convection of air (or gases whose

Prandtl number is almost the same as that for air) along vertical surfaces with constant wall

temperature or constant wall heat flux. The new wall functions are proved to be also valid at

vertical surfaces in cavities.

in

Zusammenfassung

In dieser Arbeit werden neue Wandfunktionen fur Geschwindigkeit, Temperatur, turbulente

kinetische Energie und Dissipationsrate der turbulenten kinetischen Energie fur natiirliche

AuftriebsstrOmungen mit freier Konvektion vorgestellt. Die Herleitung basiert auf

Dimensionsanalyse und Vergleich mit experimentellen Daten. Da in den neuentwickelten

Wandfunktionen nur Fluideigenschaften und lokal bekannte Variablenwerte, nicht aber globale

Grossen, wie etwa die Rayleighzahl, vorkommen, eignen sich diese Funktionen besonders fur

die numerische Simulation. Eine Wandfunktion dient zur Uberbriickung des wandnahen

Gebietes, um Rechenzeit und Speicherbedarf einzusparen.

Die Giiltigkeit dieser neuen Wandfunktionen wurde anhand der Berechnung der natiirlichen

Stromung an zwei vertikalen flachen beheizten Platten und drei differentiell beheizten Kavitaten

iiberpruft. Die numerischen Ergebnisse sind praktisch unabhangig von der Wahl des

Wandabstandes der ersten Rechengitterzelle im Bereich einer Grossenordnung des

Wandabstandes. Im Vergleich mit publizierten experimentellen Daten oder Resultaten aus

direkter numerischer Simulation im Bereich 5xl09 < Ran ^ 1.7xlOn stimmen die Temperatur-

und Geschwindigkeitsprofile sowie die Verteilungen der Wandschubspannung und des

Warmeflusses entlang der Oberflache gut iiberein fur Stromungen mit Ra > 1010.

Die Anwendung auf die Berechnung der Luftstromung in drei verschiedenen Raumen zeigt,

dass die neuen Wandfunktionen den praktischen ingenieurtechnischen Anforderungen geniigen.Sie erweitern den Anwendungsbereich des bekannten Standard-k-e-Modelles auf natiirliche

Stromungen.

Weiter hat sich eine gewisse Analogie zwischen Langen-, Geschwindigkeits- und

Temperaturmass bei erzwungener und natiirlicher Stromung gezeigt. Eine sogenannte

"Warmefluss-Temperatur" und eine sogenannte "Warmefluss-Geschwindigkeit" sind geeignete

Masse fur die natiirliche Konvektionsstromung. Die Korrelation der Temperaturwerte ist

unabhangig von der Wandschubspannung, wahrend dagegen die Korrelation der

Geschwindigkeitswerte im Falle der natiirlichen Stromung sowohl von der

Wandschubspannung als auch vom Warmefluss abhangig ist.

Die neuen Wandfunktionen gelten entsprechend ihrer Herleitung fur natiirliche Stromung von

Luft (oder Gasen mit ahnlicher Prandtlzahl wie Luft) entlang senkrechter Wande mit konstanter

Temperatur oder konstantem Warmefluss. Die Giiltigkeit wurde ebenfalls fur senkrechte

beheizte Wande in Kavitaten aufgezeigt.

IV

1. INTRODUCTION

Natural convection is a flow driven by buoyancy, in contrast to forced convection, which is

driven by pressure difference. Buoyancy means power to float or keep things floating due to the

combined presence of a fluid density gradient and a body force that is proportional to density.The variation of density may result from temperature differences in the fluid, or from

differences in the concentrations of chemical species in the fluid, or from the presence of

multiple phases of the fluid.

People underwent a long history to try to understand the behaviour of buoyant flows, after

Archimedes discovered buoyancy about 2200 years ago. As a milestone in the development of

fluid dynamics, Navier (1822) and Stokes (1845) established the differential equations of

viscous flows. Reynolds (189S) split physical quantities into mean part and fluctuating part,

derived the Reynolds equations, and defined Reynolds stress. His work marks the

establishment of the theory of turbulence. As computer resources have become more and more

abundant, the development of computational fluid dynamics (CFD) has made it possible to

predict laminar and turbulent flows. The advantages of CFD are: low cost, short time needed,

providing detailed and complete information, easy to simulate realistic conditions and ideal

conditions. Since CFD has made considerable progress over the past 20 years, it plays an

important role in solving practical problems related to fluid dynamics and heat and mass

transfer. However, many problems and questions still remain in CFD technique, for example,

validation problems, multisolution problems, the conflict between accuracy and computing

time, the establishment of appropriate turbulence models, boundary condition treatments, etc.

Recently, accurate prediction of convective heat transfer has attracted attention for the

development ofCFD (Launder 1993, and Moser 1992).

This thesis will focus on the boundary condition treatments, specially the treatment at a solid

wall, for the numerical simulation of natural convection. Because, firstly, the gradients of flow

variables near a wall are very steep, and secondly the heat transfer through the wall determines

the buoyancy, which is the driving force in natural convection, the proper treatment of a solid

wall becomes extremely important in the numerical simulation of natural convection. There are

three different approaches: (1) applying wall functions, (2) using a low Reynolds number

turbulence model, and (3) applying a two-layer model (see Section 2.1 for details). The first

approach is most practical because of the economy of calculation, but the often used log-law

wall functions derived from forced convection fail in prediction of natural convection (and in

many forced convection flows). The second approach is considered to be valid for natural

convection, however it requires a very fine grid in the near-wall region. Because of the limit of

computer resources, this approach still cannot be applied to some real engineering projects,

e.g., three-dimensional air flows in buildings. The third approach was introduced recently and

1

needs further investigation to be applicable for natural convection. To derive wall functions

suitable for natural convection is the main aim of this thesis.

Natural convection abounds in nature and in our living environment. In this study, only natural

convection resulting from temperature difference between a vertical plate and the fluid will be

considered. Natural convection in a cavity with heated or cooled vertical walls is also included.

This kind of natural convection plays an important role in many engineering problems of

practical interest, e.g., climate conditioned or ventilated spaces, solar collectors, cooling of

electronic equipment and nuclear reactors, heating or sterilisation in food industry.

Before discussing wall functions, we will describe the CFD technique adopted in Chapter 2,

which includes the governing equations, the turbulence model, boundary conditions,

discretization and the solving algorithm for the equations. In Chapter 3 the wall functions for

natural convection will be derived after a review of wall functions for forced convection. The

validation of the new wall functions will be given in Chapter 4, where the results of sensitivity

analysis will also be presented. We will demonstrate the application of the new wall functions in

predicting air flows in large spaces. Finally the thesis draws the conclusions arising from the

present study.

2. NUMERICAL SIMULATION OF FLOWS

Numerical simulation of flows requires typically the following components (Scheuerer 1993):

Mathematical Models. These usually consist of the basic physical conservation laws for

mass, momentum and energy. In the case of turbulent flows, averaged forms of the

conservation laws are often used. These laws are usually non-linear, second-order, partial

differential equations. To obtain unique solutions initial and/or boundary conditions are also

required.

Numerical Solution Method. The task of this component is to discretize the model

equations and approximate the equations by sets of linear algebraic equations, which are solved

by suitable algorithms.

Pre- and Post-Processing Software. Pre-processing software includes methods for

geometry and grid generation and boundary and initial conditions specification. Post-processing

software serves to graphically display, analyse and understand the computed results and to

derive secondary quantities (e.g., heat transfer coefficient) from the primitive variables.

2

The details of the mathematical models will be discussed in the following sections, while the

numerical solution method is explained in Appendix A.

2.1. Governing Equations and Turbulence Models

2.1.1. Navier-Stokes Equations

A fluid in motion follows the conservation laws of mass, momentum and energy. In a Cartesian

coordinate system, the conservation equations for a single species Newtonian fluid are

presented as follows.

The conservation of mass is expressed by the continuity equation:

where Uj stands for the component of velocity in the Xj-coordinate directions, p and t denote

density and time. The summation convention notation is applied.

The conservation equation of momentum can be derived from Newton's second law:

3, ^ 3 / ^ 3p 3^ij , .^pu^CpujU^ Pgi

-

^-

-^(2.2)

unsteady term convection buoyancy pressure diffusion

gradient

This is usually called the Navier-Stokes equation of motion. In this equation gi is the

gravitational acceleration, p is the static pressure, and Xy is the viscous stress tensor, which can

be expressed in terms of velocity gradients using Stokes' law:

Tu-A^diri^ (2-3)

where \i is the dynamic viscosity of the fluid, and o,j is Kronecker's delta function.

The first law of thermodynamics describes the conservation of energy:

unsteady term convection diffusion dissipation compressibility effect

3

where h stands for the enthalpy of the fluid and qy is the heat flux due to conduction, which is

expressed in terms of the temperature gradient using Fourier's law:

*-* (2"5)

where X is the heat conductivity of the fluid, and T the temperature of the fluid.

In order to close the above equations, thermodynamic property equations have to be introduced.

According to the theory of thermodynamics, the thermodynamic state of a fluid is determined by

only two independent thermodynamic properties (e.g., p and T). Any third thermodynamic

property (e.g., p, h) is related to the two independent properties, e.g.,

^-= Kjdp - PdT (2.6)

pdh = (1 - PT)dp + pCpdT (2.7)

where Cp is the specific heat capacity at constant pressure, and Kt and P are the coefficients of

isothermal compressibility and thermal expansion:

The fluid properties (\i, X, and Cp) may be dependent on the thermodynamic state. They can be

determined by only two independent thermodynamic properties as a third thermodynamic

property can. Therefore Equations (2.1) (2.9) are closed equations, which govern the

motion of any single species Newtonian fluid.

For ideal gases, density is related to pressure and temperature via the ideal gas law:

p-0 WO,

where Ro is the universal gas constant (8.313 J/K/mole) and M is the mass of the gas per mole.

From this equation, we have :

KT=l/p (2.11)

P=l/T (2.12)

h = CpT (2.13)

4

For perfect gases Cp is constant. Since the Eckert number (u2/CpT) is small for naturalconvection, the term of dissipation in Equation (2.4) is negligible. Furthermore, air natural

convection may be considered as an incompressible flow, and the compressibility term is also

negligible. Then Equation (2.4) can be rewritten as:

|(PT) + ^T,. !-(!-) ,2.14)

where Pr is Prandtl number of the fluid:

Pr = iyt (2.15)

Equations (2.1) (2.3), (2.10) and (2.14) form the governing equations of flows for perfect

gases, if dissipation and compressibility in the energy equation are negligible.

These governing equations are often used to solve laminar flows, but recently they are also used

to solve turbulent flows by resolving turbulence on a very fine mesh in very short time steps.That is called a Direct Numerical Simulation (DNS) of turbulence. Nieuwstadt (1990) reviewed

this method. The two main advantages of DNS are accurate governing equations without anyturbulence model and detailed results, which are helpful for understanding the behaviour of

turbulence. However, a DNS requires very high computing time and a large memory. One of

approaches to reduce the required computing time and memory is using large eddy simulations

(LES), in which the small eddies are filtered from the instantaneous motions and are modelled

by some simple models. A LES needs to resolve the unsteady, three-dimensional evolution of

the large scale of a turbulent flow field. At present LES still requires too much computational

time to be useful for engineering computations (Chen and Jiang 1992). The "Reynolds

equations", which were derived by Reynolds through averaging the Navier-Stokes equations,are most often used for solving turbulent flows.

2.1.2. Reynolds Equations and Turbulence Models

For turbulent flows the instantaneous variable

where the time interval, At, is long compared to the time scale of the turbulent fluctuations but

small enough to capture the unsteady mean flow. With the decomposition substituted, the

governing equations are time-averaged. If the density fluctuations are neglected, the result is

3 ._. 3 /_ \ _ 3p 3 f 2 3uic_(pui)+ (pUjUi) = pgi____^_k

(2.18)

3x

3uj 3uj

l3xJ 3xiJ-&*&)

i^^)-^ Pr3xU

-H*n

(2.19)

(2.20)

As a result of the time averaging, two unknown terms have been introduced: the turbulent

momentum flux, or the Reynolds stress, -pu/Uj', and the turbulent heat flux, -pUj'T.

These terms must be related to known quantities via a turbulence model before a closed solution

of the above equation system becomes possible. This approach is called turbulence transport

modelling.

There are many different turbulence models. The standard k-e model is most widely used in

engineering calculations at present. In this model, the Reynolds stresses are related to the rate of

mean strain through the turbulent viscosity vt

-UjUj =v,3uj 3uj3xj 3xj

-5ij

3 lJk + Vt^

JPrt 3xj

k2Vt = Cn

(2.21)

(2.22)

(2.23)

where Prt is the turbulent Prandtl number, k and e are the kinetic energy of the turbulent

fluctuations and the turbulent dissipation rate, respectively, which are defined as:

k = u,'u1'/2

3u. 3u.e = v'-

3x, 3xj

(2.24)

(2.25)

6

The k and e equations can be derived from the Navier-Stokes equations using time-averaging

and certain model assumptions.

In numerical simulation of natural convection, the Boussinesq approximation (Boussinesq,

1903) is often adopted. This approximation treats the fluid properties (p, \i, and Pr) as

constants except for the density in the buoyancy term which is simulated as:

p = p0[l-p(T-T0)] (2.26)

where po and p* are the density and the coefficient of thermal expansion at a reference

temperature To, which is usually taken as the ambient temperature in a natural-convection

boundary layer or the average temperature in an enclosure. Under this approximation the

governing equations with the standard k-e model can be written as:

3u

9xj9uj

j_

= 0

1 3P

a+(.J,).^r.To).J P^odxj

3x(v + vt)

aui+auil9xj 3xj J

3T 3 / v 8

aT+ax-M=3x-

3k a / ,\ a

Pr PrJ 3x

cte _3at ax

(uie) = ^7

( vt)dkV + L

akJ3x

- e + Pv + Gv:J

V + -9e

3tJ3x;

+^(clpk -ce2 + celce3Gk)

(2.27)

(2.28)

(2.29)

(2.30)

(2.31)

with

V, = c

Pk=vt

Gk=-giP

aui+auiN^aXj aXj>

vt 3T

3^3x:

Prt3xi

(2.23)

(2.32)

(2.33)

7

Here the overbars indicating time-mean values of the variables are omitted for simplicity. In the2 r)lc

momentum equations, - is ignored, and P is a modified pressure:3 dx;

P = p - po&Xi (2.34)

The turbulence model coefficients are listed in Table 2.1.

Table 2.1 The turbulence model coefficients

Cu Cel c62 CE3 Ok oE Prt

0.09 1.44 1.92 1.0 1.0 1.3 0.9

If the coefficients, c^ cei and c^, are taken as functions of the local Reynolds number to model

the near-wall turbulence, the model is called a "low Reynolds number k-e model". Actually ofe,

aE and Prt are not constant. For example, Prt depends on the type of flow (wall flow, core

flow, or free flow), molecular Prandtl number, Reynolds number (or Rayleigh number for

natural convection) and position (Yuan et al. 1992b). cE3 is not a constant, either. Henkes

(1990) suggested to take the value of cE3 as 0 to 1 depending on the vertical component of

velocity, u (parallel to gravity), and horizontal component, v, as follow:

ce3 = tanh|u / v| (2.35)

In the vertical boundary layer, u v, and cE3 = 1, while in the horizontal boundary layer u

v, and cE3 = 0. We shall use Equation (2.35).

Now let us discuss the effects of the Boussinesq approximation. The properties of a real fluid

depend on the temperature and the pressure. For natural convection and air flows in buildings,

the effects of pressure on the properties can be neglected, but we have to take care of the effects

of temperature. Table 2.2 lists the relative errors of air properties due to the Boussinesq

approximation at the reference temperature To = 290 K at atmospheric pressure. The error, 8^,is calculated according to:

Table 2.2 The relative errors of air properties due to the Boussinesq approximation

(T0 = 290 K, pq = 101300 Pa)

AT[K] 8p 8Cp 8v 5* 8pr 5b

20 -0.035 0.000 0.059 0.029 -0.003 0.001

60 -0.104 0.001 0.171 0.095 -0.010 0.010

200 -0.346 0.007 0.405 0.227 -0.029 0.118

460 -0.795 0.027 0.635 0.389 -0.044 0.629

820 -1.421 0.064 0.777 0.513 -0.038 2.002

8

5 = & \*

Xr(2.36)

where %t and Xo indicate p, Cp, v, X, and Pr at T = T0 + AT/2 and at T = T0, respectively. The

error of the buoyancy term in momentum, 8b, is estimated via [pr - po(l - PAT/2)]/pr.

From the table it can be seen that the variations of Cp and Pr are small, while correct

representation of the buoyancy term is sensitive to temperature. Because of the complexity of

the governing equations we cannot theoretically analyse the influences of the Boussinesq

approximation on the predicted results. Kato and his collaborators (1993) applied k-e turbulence

models to calculate a natural-convection boundary layer at temperature differences (between the

plate surface and the ambient air) of 43 K and 404 K and found that the Boussinesq

approximation is acceptable when AT = 43 K, whereas the variation of density with temperature

and the density fluctuation should be considered when AT = 404 K. Henkes and Hoogendoorn

(1992) calculated flows in a differentially heated cavity using the standard k-e model and

concluded that the Boussinesq approximation is acceptable up to AT/To = 1.5.

2.1.3. Wall-Treatment Approach

The standard k-e model discussed above is not valid in the near wall-region, where there is a

viscosity-affected region with low Reynolds number and steep gradients. That causes difficultyfor numerical simulations. There are three wall-treatment approaches to cope with that problem:

i. crossing the near-wall region by means of wall functions. This approach has been widely

used in practice because of its economy of calculation. Since existing wall functions (so called

log-law wall functions) were derived under the assumptions of Couette flow, local equilibrium

(production equals dissipation) and constant stress layer, they failed in the prediction of some

complex flows, for example, separated flows and natural convection. In this situation, new wall

functions are expected.

ii. extending the models to be suitable for low Reynolds number flows and solving the near-

wall region with a fine grid. The extended models are called low Reynolds number turbulence

models (LRN), which were suggested by numerous researchers. Patel et al. (1985) reviewed

these models. Due to the steep gradients, especially the steep gradient of the turbulent

dissipation rate, in the near-wall region, a very fine grid system is required when low Reynolds

number turbulence models are applied. According to Rodi (Rodi 1991), typically 60 to 100 grid

lines across boundary layers are required for proper numerical resolution depending somewhat

on the version considered. Skovgaard and Nielsen (1991) suggested at least 10 and Chen and

Jiang (1992) suggested 20 to 30 grid lines in the near-wall region. For prediction of 3-

dimensional flows the computer resources required by this approach is still beyond what can be

offered at the present time, especially for the simulation of practical cases.

9

iii. applying two-layer models. Two-layer models (TLM) divide the computational domain into

two regions: away from walls and near walls. The standard k-e model (or a Reynolds stress

model) is used in the region away from the wall. In the near-wall region, some simple models,

e.g., one-equation models, are employed. This approach was introduced by Iacovides and

Launder (1984) and later adopted by Chen and Patel (1988), Rodi (1991), et al. TLM's may

reduce computing time because relatively coarse grid system, say 6 to 10 grid lines in the near-

wall region, is acceptable for simple models (Rodi 1991). The two-layer models have been

tested in some forced convection. For boundary layers, the predictions with TLM's are

generally as good as those obtained with LRN's, and for the situations with adverse-pressure

gradient TLM's are better. For steady separated flows, the calculated results with TLM's are

clearly improved over those obtained with the k-e model employing the wall functions (Rodi

1991). Applications of this new approach have not been found in predictions of natural

convection and air flows in buildings.

The present thesis will focus on new wall functions and try to predict natural convection using

the standard k-e model in conjunction with the new wall functions with reasonable accuracy and

computing time for practical engineering application.

2.2. Boundary and Initial Conditions

The governing equations discussed above generally require initial and boundary conditions.

Initial conditions are specified initial values of all variables at t = 0. For a steady flow, initial

conditions refer to initial guesses, from which the solution is obtained after a number of

successful iterations. Good initial guesses sometimes save computing time significantly. A

solution with a coarse grid is often helpfully used as the initial guess for a fine grid. For some

unsteady flows, the flow is sensitive to small disturbances, and different initial conditions will

result in different solutions. That is a multiple-solution problem (see Chen and Jiang 1992 for

details).

The boundary of the calculation domain may be either solid or fluid. Except for flows in

enclosures, the computation domain comprises usually only a part of the whole flow field. In

this situation the calculation domain has to be chosen carefully. The appropriate boundary

conditions should result in a unique solution. A small change of the boundary condition should

cause only a correspondingly small change in the solution. Kreiss and Lorenz (1989) described

some techniques for choosing correct boundary conditions. At present no adequate

mathematical theory is available to ensure a correct boundary condition for the full Navier-

Stokes equations in general. Therefore, in order to find appropriate boundary conditions, we

10

have first to rely on physical arguments, then on known mathematical results, and finally on

heuristic considerations (Cebeci and Bradshaw 1977).

The boundary conditions for flows in enclosures may be easily specified, whereas it is not so

easy for flows along vertical heated plates, which will be discussed in Chapter 4. For

engineering applications the following kinds of boundaries are often used: inlet, outlet, wall and

symmetry. The conditions for each kind of boundary are stated as follows.

Inlet. An inlet boundary condition specifies the fluid flows across the boundary surface into the

calculation domain. For mass and momentum equations, we may specify all velocity

components, or mass flow rate and direction, or total pressure. For other transport equations(T, k, e, etc.) the values of the dependent variables must be specified.

Outlet. An outlet boundary condition specifies the fluid flows across the boundary surface out

of the calculation domain. For mass and momentum equations, the velocity or the mass flow

rate can be specified if the exiting flow distributions are known. If the distributions are not

known, we can specify the static pressure. For other transport equations a zero gradient (instreamwise direction) boundary condition is used.

Wall. A solid wall boundary condition specifies that the fluid cannot flow across the boundarysurface. For mass and momentum equations, a no-slip condition is specified on the boundarysurface. For energy transport equation, we can specify either the wall temperature or the wall

heat flux. If the flow at the nodal point of the near-wall grid cell is not dominated by the

viscosity effect, so called wall functions are needed to calculate the wall shear stress and the

wall heat flux (or the wall temperature) when the wall temperature (or the wall heat flux) is

specified. The details of wall functions will be discussed in Chapter 3.

Symmetry. At a symmetry boundary, the component of the velocity normal to the surface is set

to zero, and the gradients normal to the boundary surface of all other variables such as p, T, k,6 and the tangential velocity component are specified as zero.

2.3. Discretization of the Equations and Solving

In order to solve the governing equations numerically, they have to be replaced by their discrete

counterparts by means of a discretization. A successful discretization should result in zero

-deviation between the discrete equations and the continuous equations when the number of grid

points is increased to infinity. There are many strategies for discretization, e.g., finite

differences, finite volumes and finite elements. Both PHOENICS and TASCflow are based on

the finite-volume method, which is described in Appendix A.

11

The discretized equations are algebraic equations, which can be efficiently solved by means of

Tri-Diagonal Matrix Algorithm. Since the equations are strongly coupled and they are only

approximations of the original continuous differential equations, an iterative procedure is

necessary. The chosen procedure of the iteration is briefly stated in Appendix A.

3. WALL FUNCTIONS

Wall functions are analytic descriptions of the near-wall profiles of flow variables. In non-

dimensional representation these functions correlate data of different experiments. In this

chapter we will derive the wall functions for numerical simulation of natural convection after a

review of the wall functions for forced convection.

3.1. Review of Wall Functions for Forced Convection

3.1.1. Wall Functions of Velocity and Temperature

The investigation of wall functions can be traced back to 1936. Based on the experimental data

in the near-wall region, Bakhmeteff (1936) found a logarithmic law for the velocity wall

function:

u+=c1log(y+) + c2 (3.1)

where u+ and y+ are defined as:

(3.2)

(3.3)

(3.4)

where uT is known as the friction velocity.

Patankar and Spalding (1970) derived the wall functions of velocity and temperature for the

Couette flow, in which the variations of the variables, u, T, etc., in streamwise direction can be

neglected. If the near wall region is divided into two the laminar region and the log-law

region, and the viscous effects are dominated in the laminar region and the Prandtl's mixing

length hypothesis is applied in the log-law region, the wall functions may be expressed by:

12

u+=y+ y+11.5

T+ = Pry+ y+11.5

(3.5)

(3.6)

where T+ is a dimensionless temperature:

T+_(Tw-T)

with

(3.7)

T =_q

Cpput(3.8)

TT is known as the friction temperature. In Equation (3.5), k is the Von Karman constant, E = 9

for a moderate pressure-gradient flow on a smooth surface. P can be determined by (Rosten and

Worrell 1988):

-

14

(3.16)1/2

uk=S1/4k

where

(3.15)

(3.14)

Prt(u++P)=Tt'

KUku+=Ii!3-ln(Ey+^)

follows:as

(3.9))to(3.5)Equationsfromdistinguishto(1988)WorrellandRostenbygiven"generalised"

ofnametheadopt(wefunctionswallgeneralisedthesuggested(1974)SpaldingandLaunder

(3.13)

(3.12)

y+>44.5for4.05+T+=2.131ny+

44.5

30

20-

Equation (3.5)

Equation (3.12)

30

1000

Figure 3.2 Velocity wall function

20

10-

Equation (3.6)

Equation (3.13)

1000

Figure 3.3 Temperature wall function

These wall functions reproduce identically the log-law wall functions (Equations (3.5) and (3.6)in the log-law region) when the near-wall region is a constant shear stress layer, and productionand dissipation of energy are in balance. The generalised wall functions are considered to be

valid in flows with streamline-reattachment points and are generally better than the log-law wall

functions (Rosten and Worrell 1988).

There are other versions of wall functions for forced convection.

3.1.2. Wall Functions for the Turbulent Kinetic Energy and the Turbulent Dissipation Rate

For the local equilibrium Couette flow with constant shear stress, we may derive the following

equations:

k+ = 3.33

'-7

(3.17)

(3.18)

where k+ =5- and e+ = j . These two equations are often used as wall functions for theuT uT

turbulent kinetic energy and the turbulent dissipation rate.

15

Patel et al. (1985) studied and presented experimental profiles of the turbulent kinetic energyand the turbulent dissipation rate in the near-wall region as shown in Figures 3.4 and 3.5,

which can be fitted by Equations (3.19) and (3.20), respectively (Yuan et al. 1992b):

k+ = min{3.33,0.05y+}+ 0.1 + 0.003y+

1 + 0.00125y+

(3.19)

(3.20)

These forms are only appropriate for local equilibrium flows.

k+'

experimental data

*- Equation (3.17)A Equation (3.19)

L20 40 60 80

Figure 3.4 Wall function of the turbulentkinetic energy

0.4-

0.0

-experimental data

Equation (3.18) Equation (3.20)

20

r

40

I

60

I

80

Figure 3.5 Wall function of theturbulent dissipation rate

3.1.3. Discussion

The log-law and the generalised wall functions have been widely applied in engineering

simulation and adopted in commercial CFD codes. TASCflow uses the generalised wall

functions, while PHOENICS offers both the log-law and the generalised wall functions, see

Appendix B for details. Although these wall functions succeed in predicting many flows, they

result in grid-dependent numerical solution of natural convection. Figures 3.6 and 3.7 show the

predicted results of natural convection along a heated vertical plate (RaH = 1.7xlOn) and in a

closed cavity (RaH = 5xl010) using the standard k-e model in conjunction with the log-law or

the generalised wall functions. From the figures it can be seen that the predicted total heat

transfer, Q, and the total wall shear force, S, are very sensitive to the choice of the distance, y1;

16

2000

Q[W]

1000

Plate, the log-law wall functions

Plate, the generalised wall functions

Cavity, the log-law wall functions

Cavity, the generalised wall functions

8*

10 20 30 yi [mm ]

Figure 3.6 The variation of predicted total wall heat transfer with thedistance y from the wall to the nodal point of the near-wall grid cell

0.04

S

[N]0.03

0.02

4

0.01 -

0.00

--Q Plate, the log-law wall functions

x Plate, the generalised wall functions

-b - Cavity, the log-law wall functions-x - Cavity, the generalised wall functions

i

--T-

10 20 30 yi [mm ]

Figure 3.7 The variation of predicted total wall shear force with thedistance y from the wall to the nodal point of the near-wall grid cell

from the wall to the nodal point of the near-wall grid cell. When yt varies from 1 mm to 30

mm, the values of Q and S change by a factor 3 to 20. (In the turbulent region y+ is about 8 and

6 for the plate case and the cavity case, respectively, corresponding to y = 2 mm.) Therefore

new wall functions for numerical simulation of natural convection are needed.

3.2. Review of Wall Functions for Natural Convection

The bases for deriving wall functions are reliable experimental data and appropriate scales to

correlate these data. Several researchers made contributions to the measurement and analysis of

turbulent natural convection along heated vertical surfaces.

17

Cheesewright (1968) reported his experimental data in the forms of 6 = f(T|), and u/(gPATx)0-5= f(r|), where r\ = yGrx01/x. And he mentioned that "T| = yGrx01/x as the independent variable

does not achieve good correlation of the results. T| = yGrx-4/x would give much better

correlation over the inner part of the boundary layer but would give the wrong behaviour near

the outer layer edge."

Fujii et al. (1970) proposed to use = yNux/x as length parameter, which was employed later

by Miyamoto and Okayama (1982) and To and Humphrey (1986) for correlation of the

temperature and velocity profiles of the whole boundary layer, and by Tsuji and Nagano (1989)

and Henkes (1990) for correlation of temperature in the inner layer.

George and Capp (1979) divided the boundary layer into two inner and outer regions.

For the correlation of the velocity and temperature profiles, they recommended uSj =

(gPF0a)1M, Tsi = F03/4/(gPa)1/4 and r) = a3/4/(gPF0)1/4 as the velocity, temperature and

length scales for the inner region, and uso = (gPF

temperature profiles respectively, and in the outer layer 8U was adopted for both profiles.

Recently, Henkes and Hoogendoorn (1994) recommended (2ky3)1/2 and y^,, as velocity scale

and length scale for flows near vertical surfaces in a cavity. Here yitm is the position of the

maximum turbulent kinetic energy (km).

By means of asymptotic method Gersten and Herwig (1992) derived wall functions for natural

convection, which are similar to those obtained by George and Capp (1979).

Although the previous investigations did not result in applications to numerical simulation, they

presented useful data and knowledge for deriving suitable wall functions. Moser (1992)reviewed the methods and problems in numerical simulation of air flows in buildings. He

emphasised the need for proper wall functions for numerical simulation of natural and mixed

convection.

3.3. Wall Functions for Velocity and Temperature in Natural Convection

The contents of this section are quoted from Yuan et al. (1993). Under the Boussinesq

approximation, the governing equations for a 2-dimensional steady natural-convection

boundary layer along a vertical smooth plate (cf. Figure 4.1) can be expressed by (Bejan 1984):

3U 3U 3 ( 3U -rr '

dx dy 9y

3T 3T=_3

dx 9y 8y

v - u' V

I dy+ gP(T-T0) (3.22)

u^L+v^r = T:laj--v"r'] (3-23>

Velocity, u, v, and temperature, T, depend on position, x and y, boundary condition, Tw - T0,and fluid properties, gP, a, and v, which are the only fluid property parameters occurring in

Equations (3.21) to (3.23), i.e.,

u = f(x,y,Tw-T0,gP,

-^-= f(Grx,Pr) (3.27)

where us is the velocity scale, for which as proposed by Tsuji and Nagano (1989) us =[gP(Tw -

T0)v]1/3 and by Cheesewright and Mirzai (1988) us =[gP(Tw - T0)x]1/2. In Equations (3.26)and (3.27) Grx is the x-Grashof number:

Grx = gP(T^-T0)x3 ^

The profiles (3.24) and (3.25) are not in a form suitable for wall functions in a numerical finite-

volume method because the streamwise distance, x, still appears. The idea now is to use the

empirical information in expressions (3.26) and (3.27) to eliminate x and (Tw - To) in (3.24)

and (3.25). Equations (3.26) and (3.27) can be rewritten as dimensional equations:

^ = f(x,Tw-T0,gp\a,v) (3.29)

j= f(x, Tw - T0, gp, a, v) (3.30)

which are equivalent to:

^(TWP'^a'v) (3-31)

Tw-T0 = f(y,^pgP,a,v) (3.32)

Substituting Equations (3.31) and (3.32) into Equations (3.24) and (3.25), we have:

u=(y'7'^'SP'a'v) (3-33)

Tw-T={y-7^'gP'a'v) ^34>

which can be considered as general velocity and temperature profiles in boundary layers alongvertical plates. Next we discuss how to deduce wall functions for velocity and temperature by

means of dimensional analysis.

3.3.1. Temperature Wall Function

20

Among the seven variables in Equation (3.34), there are three independent units, m, K, and s.

According to the II theorem (Sedov 1959), only four independent non-dimensional parameters

can be formed, we choose:

(3.35)

(3.36)

(3.37)

(3.38)

PrV

a

a

T*T -T

T

*

ya

Rs.sSUt

with

T s*q

q^ ^

Upa(pCp)'

1/4

(3.39)

1/4

Tq is called the heat flux temperature. George and Capp (1979) used Tq and u, as the

temperature and velocity scales for the inner region.

In general, Equation (3.34) can now be written as:

f( T*, y*, R, Pr) = 0 (3.41)

Equation (3.41) formally represents the wall function for temperature in non-dimensional terms.

The temperature profile depends on two parameters, R and Pr.

We expect that the following equation can correlate temperature profiles:

T" = f(y", Pr) (3.42)

where:

T** = T*Ra (3.43)

y"=y*Rb (3.44)

21

When a = b = -1, then T" = T+, y" = Pr y+, and Equation (3.42) becomes:

T+ = f(y+,Pr) (3.45)

which is the temperature wall function for forced convection.1

For natural convection, we have to determine the values of a and b and the actual shape of the

function based on experimental data. Since only experimental data of air are available, the

dependency on the Prandtl number cannot be considered and Pr is set constant in the present

thesis.

Nagano et al. (1990) systematically presented their experimental data of turbulent natural

convection of air along a vertical plate with constant wall temperature in the range of 1.55xl010

10"

o Nagano etal. (1990)

Equation (3.46)

Equation (3.47) Equation (3.48) Cheesewright&Mirzai(1988) Cheesewright(1968)

Miyamoto et al. (1983)

' '

10" 10' 10' 10 J

Figure 3.9 The temperature wall function for natural convection

T=y y 100

(3.46)

(3.47)

(3.48)

The available experimental data of Cheesewright and Mirzai (1988) (Grx = 3.94xl010,

2.05xl010), and Cheesewright (1968) (Grx = 5.72xl010) for air flows along vertical plateswith constant wall temperature, and Miyamoto et al. (1983) (Grx = 2.37xl010, 1.26x10",

1.90xlOn) for air flows along vertical plates with constant wall heat flux are also shown in

Figure 3.9 to compare the wall function. The function agrees well with all these experimental

data and is obviously applicable to situations of both constant wall temperature and constant

wall heat flux. This results from the fact that the constant wall temperature condition is identical

to the constant wall heat flux condition for turbulent natural-convection boundary layers. This

fact may be observed from both experiment (Tsuji and Nagano 1989) and asymptotic analysis

(Gersten and Herwig 1992), which show for constant wall temperature Nux = c3 Grx1/3, or qw= c4 (Tw - T0)4/3x (where c3 and c4 are constant). This means that the wall heat flux is

independent of x for constant wall temperature condition.

According to the definitions of T* and y*, Equation (3.46) can be rewritten as:

X(TW-T)qw=- y

At the outer edge of the boundary layer the temperature becomes T = T0, and the dimensionless

temperature approaches T0,

t- _Zw~Jo _*" _ Z,

Tl0'

/ , \l/4

Nur(3.50)

According to the following relationship proposed by Tsuji and Nagano (1989):

Nux=0.11Gr,1/3

we have:

Tn =

( Pr2 Y

vo.irJ= 4.4 for Pr= 0.71

(3.51)

(3.52)

which is consistent with Equation (3.48).

The agreement of the correlation of temperature profiles with measurement both in forced and

natural convection with appropriate values of the exponents a and b confirms that Equation

(3.42) is a proper formula for the temperature wall function.

3.3.2. Velocity Wall Function

Similarly, we can determine 4 independent dimensionless parameters from Equation (3 33) as

follows:

Pr^a

"q

*

ya

Rs

Then Equation (3.33) can be changed into dimensionless form, i.e.,

f( u\ y*. R, Pr) = 0

(3.35)

(3.53)

(3.37)

(3.38)

(3.54)

The same basic steps listed above for the temperature profile are applied again to obtain the

velocity wall function. The following equation is expected to correlate velocity profile well:

24

u" = f(y",Pr) (3.55)

where:

(3.56)

(3.57)

(3.58)

y =yR

When c = -d = 1, then u** = u+, y" = Pr y+, and Equation (3.55) becomes:

u+ = f(y+,Pr)

in which the velocity wall function for forced convection is contained.

For natural convection of air, c and d can be determined based on the experimental data of

Nagano et al. (1990). Figure 3.10 (a) shows that u+ = f(y+) cannot correlate the experimental

data. When we choose c = 4, the maximum of u** is independent of Grx, as shown in Figure

3.10 (b), which is the essential condition to correlate the velocity profile. Then we get a

dimensionless velocity parameter:

3

UqU

T

6.59)

Nagano et al (1990) Gr, = 1 80E11

Nagano et al (1990) Gr, = 8 44E10

Nagano et al (1990) Gr = 3 62E10

Nagano et al (1990) Gr = 1 SSE10

103 y+ 104

(a) in the form of u+ = fly*)

25

0.3 r

10

Nagano et al (1990) Ckx = 1 80E11

Nagano et al (1990) Grx = 8 44E10



10*10' 10' 10" y

(b) in the form of u = f(y+)

Figure 3.10 The experimental data of velocity in natural convection

Now we have to determine the value of d, but whatever the value of d is chosen, u" = f(y")can still not correlate the experimental data well in the whole boundary layer. We have to divide

the boundary layer into inner and outer regions as George and Capp (1979), Tsuji and Nagano

(1989), and Henkes (1990) did. The inner region is the layer from the wall to the maximum

velocity position, and the outer region the remaining part of the boundary layer beyond the inner

region. Figure 3.11 shows that u" = f(y**) correlates the experimental data well when d = 2

for the inner region and d = 6 for the outer region.

Nagano et al (1990) Grx 1 80E11


Nagano et al (1990) G^ = 3 62E10


(a) in the inner region

26

Nagano et al. (1990) Grx = 1.80E11

Nagano et al. (1990) Gr,, = 8.44E10

Nagano et al. (1990) Gr8 = 3.62E10

Nagano et al. (1990) Gr = 1.55E10

10"1 yVlO0(b) in the outer region

Figure 3.11 The correlation of the velocity profile in natural convection

By means of curve-fitting, as shown in Figure 3.12, we obtain the velocity wall function as

follows:

u** = min{fi, f0}

fi = 1.41y" -3.1 ly * + 2.38y * y * < 0.53

fi = 0.228 y**>0.53

f0 = 0.228 y" 0.005

J...**f0 = -0.458 - 0.2581ny 0 - 0.024251n'y 0

0.005

o Nagano et al. (1990)

Equation (3.61) Equation (3.62) Cheesewright & Mirzai (1988) Cheesewright (1968) Miyamoto et al. (1983)


10

O Nagano et al. (1990)

Equation (3.64)

Equation (3.63) Cheesewright & Mirzai (1988). Cheesewright (1968) Miyamoto et al. (1983)

lo- y;* io10"J 10"

(b) in the outer region

Figure 3.12 The velocity wall function for natural convection

Let us compare Equation (3.61) with the analytical velocity profile deduced by Yang and Nee

(1970) and Tsuji and Nagano (1988):

*w 1 gP(Tw-T0) 2 j_ gpq^ 3v

y +3! Xv yu"y-r

1 f 93uV4! I av3

,O}/ 'y = 0

/+... (3.68)

If the higher-order terms can be neglected,.Equation (3.68) can be rewritten as:

28

u =Avj +Byj +Cy; (3.69)

where:

A = PT1

lgpa2(Tw-T0)B=~2

VU,

c.i ?6 Pro,2

According to Equation (3.51) and the following relationship:

^ n.rn.i^J/11.9

p[gP(Tw-T0)v]2/3

= 0.6840^'

proposed by Tsuji and Nagano (1989), we have:

1B=~

2x0.113/4Pr1/2

C = -0.684Gr,

1/11.9

6x0.111/2

(3.70)

(3.71)

(3.72)

(3.73)

(3.74)

(3.75)

For air, A = 1.41, B = -3.11, and C = 0.3437Grx1/n-9. Figure 3.13 shows the curves of

C = 7.59 for Gr.-1.0E16C = 4.25 for Gr=1.0B13C = 2.38 for Grx=1.0E10

C=1.33forGrx=1.0E7C = 0.0

O Nagano et al. (1990)

10"J 10-i 10"' 10" 10Xy j 102

Figure3.13 The curves of Equation (3.69) with A = 1.41 andB = -3.11

29

Equation (3.69) with different values of C (corresponding to different magnitudes of Grx),

from which it can be seen that the last term in the right hand side of Equation (3.69) is

negligible in the region of y** < 0.1. Since both A and B are independent of Grx, therefore u**

and yj" are indeed the proper dimensionless parameters to correlate the velocity profile in the

near-wall region.

For yj** > 0.1, u" is sensitive to the value of C in Equation (3.69). When Grx = 1010, C =

2.38, Equation (3.69) is identical to Equation (3.61). That does not mean that Equation (3.61)

is only valid in Grx = 1010. Equation (3.61) is a fitted curve based on the experimental data

hence it is valid in the same region as the experimental data (1.55xl010 ^ Grx < 1.8xl0n).

For the scales in the outer region, most previous researchers suggested the maximum velocity,

um, and the velocity boundary-layer thickness, & (= jTu I umdy).1 This is a reasonable choice

since um and c\, are directly related to the flow in the outer region. However, um and 8u are not

suitable parameters in a wall function for the outer region, since both um and c\, are not available

when the nodal point of the first near-wall grid is within the outer region. Therefore um and 8u

should be replaced by some other parameters which are available during numerical iterations.

Based on measurement of natural convection along a vertical plate with constant wall

temperature, Tsuji and Nagano (1989) found:

^= f(Grx) (3.76)

^&-= f(Grx) (3.77)

where us =[gP(Tw - To)v]1/3. These equations can be rewritten as:

um = f(x, Tw - To, gp, a, v) (3.78)

ou = f(x,Tw-T0,gp,a,v) (3.79)

Substituting Equations (3.31) and (3.32) into Equations (3.78) and (3.79), we have:

^{j-WP'g*-a'v) (3-80)

**'{j''**") (3-81)

1 In forced convection, the friction velocity, u and the boundary thickness, 5, are recommended as the scales for

the outer region of a boundary layer (Cebeci and Bradshaw 1984).

30

These two equations mean that um and 8u can be represented by uT (equal to (tw/p)1/2) and uq(equal to (gPocqw/(pCp))1/4). If the velocity profiles can be correlated by um and 8u, they mustalso be correlated by uT and uq in an appropriate way. Since u/u** (= u^/u,,3) and y/y0" (=

ctut6/uq7) can replace Ui and uq, u/u** and y/y0** can also correlate the velocity data if um and o\,can. Therefore, u** and y0** are believed to be acceptable parameters for the outer region.

However this is true only under the condition of constant wall temperature or constant wall heat

flux, because Equations (3.31), (3.32), (3.76) and (3.77) were obtained under these

conditions.

3.3.3. Comparison of Scales between Natural and Forced Convection

A comparison of the dimensionless parameters between natural and forced convection. Table

3.1, shows the analogy between natural and forced convection. The near wall boundary layer in

forced convection is a constant stress layer where Uj (the friction velocity), Tx = qw/pCpUj, (thefriction temperature), and v/uT are proper velocity, temperature, and length scales, respectively,

while the inner region in natural convection is a constant heat flux layer where Tq (the heat flux

temperature), uQ = (Tw/p)2/gPaTq, (the heat flux velocity), and a/uq should be proper

temperature, velocity, and length (for temperature profile) scales, respectively. As also noticed

by Cheesewright and Mirzai (1988), Tsuji and Nagano (1989), and Henkes (1990), the length

scale for the velocity profile is different from that for the temperature profile in natural

convection, which we need further research to explain.

Table 3.1 The analogy of the non-dimensional parameters between natural convection and

forced convection

Forced convection

T+sTw-T_(Tw-T)uxT

pcpin the inner region:

+ y"t (^y s = y

v

vpv

in the outer region:y

s

1/2

Natural convection

T -TT* = -^ = (TW-T)

gPa(pCp)TW

.. u ugpaT

Uq

mfor the temperature profile:

( ^1/4

y*szui=v jPa^a \a3pCp

for the velocity profile:

yi s \ in the inner regionauTz

yu

yo* s L in the outer regionauT

31

From the scales of temperature and velocity it can be seen that Tq is independent of xw, while uqdependent both on xw and qw This means that the correlation of temperature data is independent

of the wall shear stress and the correlation of velocity data is dependent both on the wall shear

stress and the wall heat flux, which is consistent with the argument of Cheesewright and Mirzai

(1988). It is interesting to compare the correlation in forced convection where velocity data are

independent of the wall heat flux, while temperature data are dependent both on the wall shear

stress and the wall heat flux.

3.4. Wall Functions for the Turbulent Kinetic Energy and the Dissipation Rate

for Natural Convection

Again by using the data1 of Nagano et al. (1990), shown in Figure 3.14, we obtained the wall

function for turbulent kinetic energy in a similar way:

2

k = 0.6y, in the inner region and y,

0.03 r

0.02

0.01

0.00

10" 10"

o

B Nagano et al. Grx= 1.80E11

Nagano et al Grx = 8.44E10

+ Nagano et al. Grx = 3.62E10

O Nagano et aL Grx = 1.55E10

Equation (3.82)

Equation (3.83)

{f%9

til"*1*! I i mill10 10' yV 102


B Nagano et al. Grx = 1.80E11

Nagano et al. Grx = 8.44E10

+ Nagano et al. Grx = 3.62E10

Nagano et al. Grx = 1.55E10

Equation (3.84)

10""* 10"J 10"" 10"1 y" 10

(b) in the outer region

Figure 3.14 The wall function of the turbulent kinetic energy

. Spalart (1988) and Kim etal. (1987)

Equation (3.86) Equation (3.87) |K = 0.41)

-l i l ilJ

0 20 40 60 80 y+ 100

Figure 3.15 The wall function of the dissipation rate

33

The turbulent dissipation rate is the rate at which kinetic energy is converted into internal

thermal energy. This conversion is a direct action of viscous stresses and not of the buoyancy.

The buoyancy influences the velocity distribution directly and then influences the e distribution

indirectly. Therefore one might assume that the difference of the e distributions between forced

and natural convection is not as large as the difference of the velocity distributions. The

validation (in Chapter 4) shows that this wall function seems to be acceptable.

3.5. Discussion

The proposed wall functions were derived from data measured on vertical flat plates. In these

experiments the boundary layers develop from a sharp leading edge, at which the boundary

layer thickness is zero. In that case the streamwise distance x is a fundamental parameter that

appears, for instance, in the Grashof number, Grx. For the flow in a cavity, the boundary layer

may have a different history. And the "plate length", x, is normally not defined. Wall functions

for numerical computations must therefore depend on local quantities only. This is the case for

the proposed wall functions. A flow near a vertical wall in a cavity can be considered as a

boundary layer flow in a stratified environment (Henkes and Hoogendoorn 1994), or in an

environment with non-zero temperature gradient. The proposed new wall functions are expected

to be applicable also for the vertical walls in cavities, as the log-law wall functions, derived

originally for a boundary layer flow with zero pressure gradient, are often applied to a flow

with non-zero pressure gradient.

The new wall functions are suitable for numerical simulation since they do not contain

parameters such as maximum velocity or boundary layer thickness which cannot be obtained

accurately during the calculation unless a very fine grid system is applied.

Since only experimental data of air are adopted, the derived wall functions do not account for

variations of Prandtl number. That means, the functions are valid only in gases whose Prandtl

number is almost the same as air. In principle, like the wall functions for forced convection, the

application of the wall functions for natural convection should be restricted to the boundary

layers with constant wall temperature or constant wall heat flux.

4. VALIDATION OF THE NEW WALL FUNCTIONS AND SENSITIVITY

ANALYSIS

Buoyant flows along flat plates and in closed differentially heated cavities are two basic kinds of

natural convection, in which the new wall functions together with the standard k-E model will

34

be tested. To check the usefulness of the new wall functions we will examine whether the

numerical results are independent of the distance y j from the wall to the nodal point of the first

grid cell. In the validation six numerical grid systems with y! equal to 1,2,5,10,20, and 30

mm will be applied. A commercial CFD code, PHOENICS (version 1.6), is adopted for

calculation. This chapter is a revision of Yuan et al. (1994a and 1994b).

4.1. Natural Convection along Flat Plates

Turbulent natural convection along a vertical heated plate has been calculated by many

researchers, e.g., Mason and Seban (1974), Cebeci and Khattab (1975), Plumb and Kennedy

(1977), Lin and Churchill (1978), To and Humphrey (1986), Nagano et al. (1989), Henkes

(1990), Peeters and Henkes (1992), and Kato et al. (1993), using one-equation models, low

Reynolds number models or Reynolds stress models with fine grid systems. The present

attempt is to obtain nearly grid-independent numerical results, using the standard k-e model

with wall functions.

Before presenting the results, we discuss the boundary conditions for this kind of flow.

4.1.1. Boundary Condition

As mentioned in Section 2.2, it is not clear how to specify appropriate conditions on the

artificial boundaries (A-B-C-D) of the computational domain for a natural-convection boundary

layer, as shown in Figure 4.1. When a heated plate is immersed in a static fluid, because of

buoyancy, the plate sucks the fluid into the domain from the bottom (C-D) and edge (B-C)

boundaries, and discharges it through the top (A-B) boundary. The vertical component of

velocity on B-C and the horizontal velocity on C-D can be set to 0 (as Nagano et al. (1989) and

Henkes (1990) did), but the other component of velocity and the mass flow rate on B-C and C-

D are unknown. However, we can assume inviscid flow from infinity to the boundaries B-C

and C-D. If the fluid is assumed to be at rest and at uniform temperature at infinity and if no

heat is exchanged between stream tubes, the Bernoulli's equation can be applied, i.e., the total

pressure, Pt, at any position along the stream tube is constant. Since the stream tubes originate

from the same location at infinity, the total pressure on the artificial boundaries B-C and C-D

is uniform. In other words, we can set the constant total pressure condition on the boundaries

B-C and C-D. This boundary condition requires the edge boundary to be sufficiently far away

from the plate. In Section 4.1.3 we will test how far it should be. Since the absolute value of

reference pressure may be chosen arbitrarily in numerical simulation, we set Pt = 0 on the

boundaries B-C and C-D.

35

A B

f

*

To

Figure 4.1 The sketch of natural-convection boundary layer

When the total pressure is specified, we can set a zero gradient condition for the normal velocity

on the boundary, since the velocity is determined by buoyancy, which takes place inside of the

domain. For the temperature and the turbulence quantities we can set T = To, k = 0, and e = 0

on B-C and C-D.

The boundary A-D is a wall. We use the wall functions to represent the distributions of the

variables.

The boundary A-B is an outlet, on which, according to the order-of-magnirude analysis for the

boundary layer equations (Cebeci and Bradshaw 1984), we can specify:

P = Pn (4.1)

where Pb is the modified pressure (see Equation (2.34)) at point B, Pb = PtB -

p(uB2 + vB2)/2. Since PtB = 0, and p(uB2+vB2)/2 is negligible when the point B is

sufficiently far away from the plate, P = 0 can be specified on A-B. (It will be shown in Section

4.1.3.1 that p(uB2 + vB2 J / 2 is negligible.) For the other variables, zero gradient conditionscan be applied.

The boundary conditions have been specified now on all boundaries of the computational

domain. During the calculation we found that some fluid flows into the domain across the

boundary A-B, as shown in Figure 4.2. In order to remove this effect, we add an additional

36

domain on the top, as shown in Figure 4.3. On the additional boundary A-E, a symmetry

condition may be specified.B

lg

To

y c

Figure 4.2 The predicted Figure 4.3 The computational domain forstreamlines the natural-convection boundary layer

In summary, Table 4.1 lists the boundary conditions for a natural-convection boundary layer.

Table 4.1 The boundary conditions for a natural-convection boundary layer

Boundary Setting

A-B P = 0

^-[u,v,T,k,e] = 0ox

B-C Pt = 0

dy

u = 0

T = T0

k = e = 0

C-D Pt = 0

1-v = 0

T = T0

k = e = 0

A-E Symmetry

D-E Wall functions

37

4.1.2. Validation of the New Wall Functions

To validate the new wall functions, we will compare the numerical results with the experimental

data of air natural-convection boundary layers along vertical heated flat plates, which were

measured by Nagano et al. (1990) (called Case N for convenience) and Cheesewright and

Mirzai (1988) (called Case CM), respectively. The geometry and the temperature in the

measurements are listed in Table 4.2. The fluid properties are estimated at the mean temperature

((Tw + To)/2), except for the thermal expansion coefficient P which is defined as 1/Tq.

Table 4.2 The main parameters of the boundary layers in measurements

CaseN Case CM

Plate height H [m] 4 2.75

Surface temperature Tw [K] 333

Ambient temperature To [K] 289

Temperature difference AT [K] 44 64,50,

32, 19

Grashof number Grn 2.4xlOn 1.4xlOn

to

4.2xl010

In the simulations the heights of the domains are chosen to be 4.5 m and 3.3 m for Case N and

Case CM, respectively, on which 90 and 72 grid lines are specified. The width of the domains

is 0.5 m, about twice the boundary layer thickness. The number of grid lines in the horizontal

direction is between 23 and 30 depending on the wall distance yj. Figure 4.4 shows the grid

systems applied in the simulations. The influences of the width of the computational domains

and the grid systems on the calculated results will be given in Section 4.1.3.2.

The boundary conditions are listed in Table 4.1. In case CM the temperature difference is

chosen to be 64 K, and the ambient temperature is 293 K.

Figure 4.5 shows the variations of the predicted total heat transfer, Q, with different values of

the first-grid-node distance y^ When y! varies from 1 mm to 30 mm (corresponding to y+ = 4

to 120 in the turbulent region approximately), the value of Q based on the log-law wall

functions (Rosten and Worrell 1988) changes by a factor 3, whereas the value based on the new

ones varies only within 12% for Case CM, and compared with the measured value the maximal

error is 16% for Case N. From Figure 4.6 we can see that the predicted wall heat flux, qw,

based on the new wall functions agrees well with the measurements for different positions x

along the vertical plate. The points in Figure 4.6 (b) are calculated based on the equation

presented by Cheesewright and Mirzai (1988) for turbulent natural convection:

38

CaseN 0 5m x 4 5m 30x90 30x90 30x90

Case CM 0 5mx 3 3m 30x72 30x72 30x72

1 1 yi= 1 mm y!= 2mm yj= 5 mm

The gridis shown

on the

right side

(a) The sketch

25x90

25x72

y! = 10 mm

24x90 23x90

24x72 23x72

yx = 20 mm yl= 30 mm

(b) The grid system (only one sixth

of computational domain)

Figure 4 4 The sketch of the boundary layer and numerical grid systems

Nux = 0 1 lGrx33 (4 2)

Since we do not know the exact temperature and fluid properties m the measurement, we plot

the results in non-dimensional form for Case CM

The variations of the calculated total wall shear force, S, with the wall distance yj of the first

nodal point are shown in Figure 4 7 The calculated wall shear force based on the newwall

functions changes by about 30% when yl vanes from 1 mm to 30 mm, whereas the predicted

force based on the log-law wall functions is very sensitive to the choice of y j It can be seen

from Figure 4 8 that the predicted wall shear stress, xw, based on the new wall functions is in

fair agreement with the measurements

39

Q[W]

1600

1200

800

400

~e~~- Based on the log-law wall functions-A Based on the new wall functions

Tsuji & Nagano (1989)

_l

0

Q[W]

1600

1200

800

400

0

10 15 20 25 y, [mm]

(a) Case N

fl Based on the log-law wall functions

A Based on the new wall functions

^

_1 I I I 1_

0 io 15 20 25 yi [mm]

(b) Case CM

Figure 4.5 The variation of the predicted total wall heat transferwith the first-grid-node distance

3.0 x [m] 4.0

(a) Case N

40

(b) Case CM

Figure 4.6 The predicted local heat flux along the wall based onthe new wall functions

S[N]

0.03

0.02

0.01 "

0.00

~o Based on the log-law wall functions

-a Based on the new wall functions

Tsuji & Nagano (1989)

0 5 10 15 20 25 y, [mm]

(a) Case N

SIN]

0.02

0.01

0.00

-~B Based on the log-law wall functions

A Based on the new wall functions

0 5 10 15 20 25 y! [mm]

(b) Case CM

Figure 4.7 The variation of the predicted total v. all shear force withthe first-grid-node distance

41

2 3

(a) Case N

......... yt = 2 mm

..... y(=5mm"""

y, = 10mm

-*-

y( =20 mm

~ ~

y( - 30 mm

Cheesewright & Mirzai

a (W)' * * ' * ' ' * ' ' ' ' ' ' ' ' * ' ' '

0 3xl010 6xl010 9xl010 1.2xlOM Grx

(b) Case CM

Figure 4.8 The predicted local shear stress along the wall based onthe new wall functions

T-Tnr

(a) Case N, at Grx = 8.4xl010

42

T-T,yj = lmm

y,s2mm

y^lOn

yj=20mm*

y|=30mm Cheesewright ft Mirzai

(b) Case CM, at Gr. = 5.4x10'

Figure 4.9 The predicted temperature profile based on the newwall functions

u[m/s]r

(a) Case N, at Gr. = 8.4x10

(b) Case CM, at Gr. =7.2x10'

Figure 4.10 The predicted velocity profile based on the new wallfunctions

43

Figure 4.11 The predicted profile of the turbulent kinetic energy

at Gr, = 8.4x10 based on the new wall functions

uv

[mVl

0.006

0.004

0.002

0.000

-0.002

-0.004

'

10" 10" 10" 10" y[m]

Figure 4.12 The predicted profile of theReynolds stress

at Gr =8.4x10 based on the new wall functions

Figure 4.13 The predicted profile of the turbulent heat flux

at Gr. = 8.4xl010 based on the new wall functions

44

y=lmm y =2mm y =5mm y=10mm y=20mmy=30mm

Figure 4.14 The contours of the predicted eddy viscosity

**

0.16

0.14

0.12

0.10

0.08

0.06

0.04

0.0 0.2 0.4 0.6 0.8 x/H

Figure 4.15 The distribution of y; when yx = 1 mm

45

' i__i i i_j

Figures 4.9 and 4.10 show the profiles of temperature and velocity, from which we can see that

the predicted results are nearly y^ -independent and are in good agreement with the

measurements.

The predicted turbulence quantities for Case N are compared with the experimental data in

Figures 4.11 to 4.13, where uV and v" T are calculated according to Equations (2.21), (2.22)

and (2.23). From Figure 4.11 we can see that the turbulent kinetic energy at the first grid cell,

which is determined directly by the wall function, is well predicted. But the value of k near the

position of maximum velocity is too low. The reasons for that are, perhaps, (1) the production

of k (Pt) is underestimated since the velocity gradient is small in this region, and (2) the

production by buoyancy (G^) is not well estimated by Equation (2.33), the predictions are

expected to be improved by applying the generalised gradient diffusion hypothesis to the

turbulent heat flux i.e., U;'T =-U:'uk'- , as suggested by Ince and LaunderV 2Prte 9xkJ

(1989). In Figure 4.12 the measurement showed that the Reynolds stress, -u' v', is negative in

the whole boundary layer. That will results in a negative value of the eddy viscosity in the inner

region, where the velocity gradient is positive, according to Equation (2.21). That means the

Boussinesq eddy viscosity concept is not valid in the inner region of natural-convection

boundary layer. However, Kato et al. (1993) measured natural-convection boundary layer with

a 2-D Laser Doppler Anemometer (Nagano et al. (1990) adopted the hot-wire techniques) and

showed positive value of the Reynolds stress in the near-wall region. Further investigation is

required to explain this inconsistency.

Figure 4.14 shows the contours of the calculated eddy viscosity, vt, for Case N. The calculated

vt varies from about 0 to over 200v, which indicates that the standard k-e model with the new

wall functions is capable of predicting the transition from laminar flow to turbulent flow. We

can clearly see from Figure 4.6 (a) that the transition, which is indicated by a sudden increase of

the wall heat flux, is well predicted when yi < 5 mm, where the nodal point of the first grid cell

is within the inner region.

Figure 4.15 shows the distribution of yj along the vertical plate when yi = 1 mm. In the

turbulent region (x/H > 0.4), y** is about 0.06. Therefore the range of 1 mm < yi < 30 mm

corresponds to the range of 0.06 < y j < 1.8 approximately, which covers the inner and outer

regions, and contains the position of the velocity maximum.

According to the results discussed above, we can conclude that the standard k-e model in

conjunction with the new wall functions can provide nearly yi-independent results, which are in

good agreement with the experimental data of turbulent natural-convection boundary layers.

4.1.3. Sensitivity Analysis

46

In this section we will present the results of the sensitivity analysis concerning boundary

conditions, computational domain, grid system, and the turbulent Prandtl number. Since the

total wall heat transfer, Q, and the total wall shear force, S, are comprehensive parameters of a

flow, which engineers are most interested in, we use these two parameters in the sensitivity

analysis.

The sensitivity analysis will be carried out only for Case N. To study whether the sensitivity is

grid-dependent or not, we present numerical results with three grid systems: yi = 1,5, and 30

mm. For convenience we call the unchanged setting the standard case.

4.1.3.1. Sensitivity analysis of the boundary conditions

As mentioned in Section 4.1.1, we have neglected p(u| + v| J / 2 and specified P = 0 on the

outlet in the standard case. Now we set P = -p(u| + v|j/ 2 on the outlet, where ub is taken asthe average velocity on the edge boundary (B-C) in the standard case. This case is called Case

BND1. The predicted total wall heat transfer and the total wall shear force are listed in Tables

4.3 and 4.4, from which it can be seen that there is no difference between the standard case and

Case BND1. Therefore we are allowed to set P = 0 on the outlet.

In Case BND2 we set P = 0 on the boundaries B-C and C-D (Figure 4.3), while

P + p(u2 + v2) / 2 = 0 is specified in the standard case. This change results in increases ofQ by3.7% to 0.4% and S by 2.6% to 0.5% corresponding to yi = 1 to 30 mm, as shown in Tables

4.3 and 4.4. That means that it is also acceptable to set P = 0 on the boundaries B-C and C-D,

but the constant total pressure condition is better, as discussed in Section 4.1.1.

In the standard case the vertical velocity component, u, on the edge boundary (B-C) and the

horizontal component, v, on the bottom boundary (C-D) are specified to be 0. If we set the zero

gradient boundary condition, i.e., = 0 on B-C and - = 0 on C-D, which is Case BND3,dy dx

Table 4.3 The comparison of the predicted total wall heat transfer

(The measured value is 871.8 W)

Case yi = 1 mm yi =5 mm yi = 30 mm

Q

[Wl

AQ/Q

[%]

Q

[Wl

AQ/Q

[%]

Q

rw]

AQ/Q

[%]

Standard 1009 - 859.6 - 819.1 -

BND1 1009 0.0 859.6 0.0 819.1 0.0

BND2 1046 3.7 869.1 1.1 822.7 0.4

BND3 998.4 -1.1 849.3 -1.2 819.8 0.1

47

Table 4.4 The comparison of the predicted total wall shear force

(The measured value is 0.0207 N)

yi = 1 mm yi =5 mm yi = 30 mm

S

[N]

AS/S

[%]

S

[N]

AS/S

[%]

S

[N]

AS/S

[%]

Standard 0.0271 - 0.0214 - 0.0201 .

BND1 0.0271 0.0 0.0214 0.0 0.0201 0.0

BND2 0.0278 2.6 0.0216 0.9 0.0202 0.5

BND3 0.0272 0.4 0.0216 0.9 0.0203 1.0

the maximal difference is 1.2%, as shown in Tables 4.3 and 4.4.

These small differences among the cases investigated above imply that the boundary conditions

specified in the standard case are reliable.

4.1.3.2. Sensitivity analysis of computation domain and grid system

The constant total pressure boundary condition requires the edge boundary to be far enough

away from the heated plate. The width of the computational domain, W, should be sufficiently

large. In the standard case W is 0.5 m. Now we test Cases DMN1 and DMN2 with W = 1 m

and 50 m, respectively, where the computational meshes are the same as in the standard case in

the region y




S

[N]

AS/S S

[N]

AS/S S

[N]

AS/S

[%]

Standard 0.0271 . 0.0214 . 0.0201 .

DMN1 0.0271 0.0 0.0214 0.0 0.0202 0.5

DMN2 0.0271 0.0 0.0214 0.0 0.0202 0.5

DMN3 0.0266 -1.8 0.0210 -1.9 0.0199 -1.0

additional domain does not improve the predicted results significantly but it improves the

appearance of the flow field, as shown in Figure 4.16 (the additional domain is not shown in

Figure 4.16(b)).

(a)CaseDMN3 (b) The standard case

Figure 4.16 The predicted streamlines

Computational meshes have often effects on numerical results. It is necessary to check whether

the results are grid-dependent or not. By applying the Richardson extrapolation technique

(1910), the solution error (Eh), defined as the difference between the converged solution to the

discretized equations (h) and the exact solution of the continuum problem (), can be estimated

by (Ferziger 1993):

E), =

Eh =* (h - $2h)/3 fr a second order method (4.4)

where fa is the converged solution with grid size h. This estimate is accurate only when h is

small. From a practical point of view, the error should be less than about 10% (Ferziger 1993).

In the present study a first order method was used. According to Equation (4.3), we can

calculate the extrapolated solution (|>e = 2 fa - fa^ and estimate the exact solution = fa. Now

we adopt this approach to analyze our calculated results. In the standard case the numbers of

grid lines in horizontal and vertical directions, NY and NX, are 23 to 30 and 90, respectively.

In Case Coarser the numbers are reduced to half and in Case Finer the numbers are doubled (y i

is kept at the required value). When we use the results of Case Coarser and Standard Case, as

shown in Tables 4.7 and 4.8, the extrapolated heat transfer (Qe) is 951 W, and the solution

error is 6.1% in the case of yi = 1 mm. When the data of Standard Case and Case Finer are

used, the maximum solution error is 4.4%. This is acceptable for engineering calculations.

Table 4.7 Analysis of the solution error for the total wall heat transfer


Case 3fi = l mm yi = 5 mm yi = 30 mm

Q

[W]

Qe

[W]

AQ/Qe

[%]

Q

[W]

Qe

[W]

AQ/Qe

[%]

Q

[W]

Q

[W]

AQ/Qe

[%]

Coarser 1067 824.8 820.4

Standard 1009 951.0 -6.1 859.6 894.4 3.9 819.1 817.8 0.2

Finer 968.3 927.6 -4.4 871.2 882.8 1.3 817.5 815.9 0.2

Table 4.8 Analysis of the solution error for the total wall shear force



SxlO3

[N]

Sexl03

[N]

AS/Se

[%)

SxlO3

[N]

Sexl03

[N]

AS/Se

[%]

SxlO3

[N]

Sexl03

[N]

AS/Se

[%]

Coarser 27.18 21.20 20.40

Standard 27.10 27.02 -0.3 21.40 21.60 0.9 20.10 19.80 -1.5

Finer 26.68 26.26 -1.6 21.40 21.40 0.0 20.00 19.90 -0.5

4.1.3.3. Sensitivity analysis of the turbulent Prandtl number

In the standard k-e model, the turbulent Prandtl number, Prt, is taken as a constant. However,

as shown in Figure 3.1, Prt varies in the near wall region. In the standard case Prt is 0.9. We

change the value by 20% to see the influence. Prt = 1.08 and Prt = 0.72 are specified in Case

PRT1 and Case PRT2, respectively. From Tables 4.9 and 4.10 we can see that Q is sensitive to

50

Table 4.9 The comparison of the predicted total wall heat transfer



Q

[W]

AQ/Q

[%]

Q

[W]

AQ/Q

[%]

Q

[W]

AQ/Q

[%]

Standard 1009 - 859.6 - 819.1 .

PRT1 947.3 -6.1 835.9 -2.8 813.6 -0.7

PRT2 1092 8.2 888.5 3.4 825.4 0.8




S

[N]

AS/S

[%]

S

[N]

AS/S

[%]

S

[N]

AS/S

[%]

Standard 0.0271 - 0.0214 - 0.0201 .

PRT1 0.0270 -0.4 0.0212 -0.9 0.0201 0.0

PRT2 0.0272 0.4 0.0215 0.5 0.0202 0.5

Prt, while S is not. When Prt changes 20%, the total heat transfer changes by about 8% to 1%

corresponding to yi = 1 to 30 mm. The reasons why the influence of Prt is yi dependent are:

(1) the variation of Prt has not been taken into account in the new wall functions, which govern

the variables at the near-wall grid cells, and (2) Prt enters the energy equation direcdy, which is

applied to the interior domain (the computational domain except for the near-wall control

volumes). The larger yi is, the less the interior domain is, and the weaker the influence of Prt

is.

4.1.3.4. Sensitivity analysis of the wall function of the turbulent dissipation rate

The wall function of the turbulent dissipation rate, Equations (3.86) and (3.87), is derived from

the DNS data of forced convection. Buoyancy may enhance the dissipation rate, as indicated in

the differential equation of e, Equation (2.31). In order to investigate the sensitivity of the

results to the wall function of e, we consider a linear profile with a value of + = 0.4 instead of

0.2 at the wall, as shown in Figure 4.17:

e+ = 0.4 - 0.0164y+ y+ < 12.2 (4.5)

e+ = l/(icy+) y+>12.2 (3.87)

Figure 4.18 shows the comparison of predicted results between the different wall functions of

e. Since the results for yi > 5 mm based on the two wall functions are almost the same, only the

results based on Equation (4.5) for yi = 1 mm and 2 mm are plotted in the figure. Compared

51

Spalart (1988) and Kim et al. (1987)

Equation (3.86)

Equation (3-87) (k = 0.41)Equation (4.5)

Sir*.

0 20 40 60 80 y+ 100

Figure 4.17 The wall function of the dissipation rate

600 0-

qw

[W/m2]

400 0^

200.0 -

0.0-

0.0

0.000

y, = 1 mm, Eq. (3.86)

yi = 2 mm, Eq. (3.86)"""

yi = 5 mm, Eq. (3.86)

y, = 10 mm, Eq. (3.86)

yi=20 mm, Eq. (3.86)

yi =30 mm, Eq. (3.86) Nagano et al.

_

yi = 1 mm, Eq. (4.5)

ii&rX~JJJrjf. y, = 2 mm, Eq. (4.5)

2 0 3 0 x [m] 4 0

(a) the wall heat flux

yi = 1 mm, Eq. (3.86)

yi = 2 mm, Eq (3 86)

y! = 5 mm, Eq. (3.86)

y, = 10 mm, Eq. (3.86)

yi = 20 mm, Eq. (3.86)

y, = 30 mm, Eq. (3.86)

Nagano et al.

yi = 1 mm, Eq. (4.5)

yi = 2 mm, Eq. (4.5)

1 I

3 o x [m] 4 o

(b) the wall shear stress

Figure 4.18 The predicted results based on the two different wall functions of e

52

with the measured data, the calculated wall heat transfer and the wall shear stress based on

Equation (4.5) are slightly better than those based on Equations (3.86). The conclusion is that

there is ro

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