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ANRCP-1999-3January 1999

Amarillo National

Resource Center for PlutoniumA Higher Education Consortium of The Texas A&M University System,Texas Tech University, and The University of Texas System

Application of a New Time Scale

Based Low K- Model to NaturalConvection from a Semi-Infinite

Vertical Isothermal Plate

S. Senthooran and S. ParameswaranDepartment of Mechanical Engineering

Texas Tech University

Edited by

Angela L. WoodsTechnical Editor

600 South Tyler Suite 800 Amarillo, TX 79101(806) 376-5533 Fax: (806) 376-5561

http://www.pu.org

This report wasprepared with thesupport of the U.S.Department of Energy(DOE) CooperativeAgreement No. DE-

FC04-95AL85832.However, any opinions,findings, conclusions,or recommendationsexpressed herein arethose of the author(s)and do not necessarilyreflect the views ofDOE. This work wasconducted through theAmarillo NationalResource Center forPlutonium.

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ANRCP-1999-3

AMARILLO NATIONAL RESOURCE CENTER FOR PLUTONIUM/ A HIGHER EDUCATION CONSORTIUM

A Report on

Application of a New Time Scale Based Low K-

Model ToNatural Convection from a Semi-Infinite Vertical Isothermal Plate


Texas Tech UniversityLubbock, Texas 79409

Submitted for publication to

ANRC Nuclear Program

January 1999

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TABLE OF CONTENTS

1. INTRODUCTION .....................................................................................................................1

2. MODEL EQUATIONS ............................................................................................................3

3. NUMERICAL SOLUTIONS ...................................................................................................5

4. RESULTS AND DISCUSSIONS .............................................................................................7

5. CONCLUSIONS .......................................................................................................................9

NOMENCLATURE .....................................................................................................................11

REFERENCES .............................................................................................................................13

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LIST OF FIGURES

Figure 1 : Velocity Profile at Gr x= 10 12 (Variation of u/U with ) ..............................................15

Figure 2: Temperature Profile at Gr x= 1012

(Variation of (t-t )/(t w- t) with ) .........................15Figure 3 : Wall Shear Stress (Variation of w / Ub2 with Gr x) ......................................................16

Figure 4 : Wall Heat Transfer (Variation of Nu x with Gr x) ..........................................................16

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Application of a New Time Scale Based Low K- Model ToNatural Convection from a Semi-Infinite Vertical Isothermal Plate


Texas Tech UniversityLubbock, Texas 79409

Abstract

The low k- model proposed by Yangand Shih (1992) is applied to the calculationof the turbulent natural convective boundarylayer over a semi-infinite, vertical, isothermalsurface. Using k/ as the turbulent time scalewill introduce a singularity in the equation,near the wall. This model uses a modified

turbulent time scale near the wall to eliminatethis singularity. The constants in the equationfor damping function are modified to producebetter results for both, natural convection andforced convection. The results are compared

with available experimental data and theresults obtained from Chien's model and arefound to be in reasonable agreement.

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1. INTRODUCTIONHeat transfers by natural convection from

vertical isothermal surfaces have receivedconsiderable research interest since theseflows are encountered in many industrial

applications. The two equation k- model iswidely used to model turbulent flows. Here k represents the turbulent kinetic energy and represents the dissipation rate of turbulentkinetic energy. This model is usually appliedwith logarithmic wall functions for velocityand temperature, to avoid the application of the k- model in the laminar sub-layer nearthe wall. These wall functions only hold forforced convection flows; they do not hold fornatural convection flows. Proper wall

functions for natural convection flows havestill not been found. Therefore low k- modelshould be applied for natural convectionflows, in which the calculations are performedup to the wall. The first low k- model was

proposed by Jones and Launder, which wasthen followed by a number of other models.

Yang and Shih proposed a new time scalebased k- model for near wall turbulence. Inthis model, the eddy viscosity is characterized

by a turbulent velocity scale and a turbulenttime scale. It uses a modified time scale nearthe wall such that there is no singularity in thedissipation equation near the wall. The modelconstants are exactly the same as those in thestandard k- model that ensures theperformance of the model far from the wall.In this paper the constants in the dampingfunction are modified to provide better resultsfor both, natural convection and forcedconvection. The modified model is applied to

the calculation of the turbulent naturalconvective boundary layer over a vertical,isothermal surface. The results are comparedwith available experimental data and theresults obtained from Chien's model.

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2. MODEL EQUATIONSThe time averaged turbulent boundary

layer equations for two-dimensional,incompressible buoyancy induced flow are asfollows.

Continuity:

Momentum:

Energy:

It is assumed that the Boussinesq is valid and

the turbulent stresses are proportional to themean velocity gradients.

The transport equations for k and aregiven as follows.

k equation:

equation:

The time scale (T t) used in thestandard k- model is k/ . Using this timescale up to the wall will introduce asingularity in the equation due to vanishingk at the wall. Therefore near the wall this

time scale has to be modified to avoid thissingularity. Yang and Shih showed that thetime scale near the wall should be theKolmogorov time scale because viscousdissipation dominates near the wall.Therefore they proposed that the turbulenttime scale be given by k/ away from the walland by the Kolmogorov time scale near thewall. The time scale for the whole regioncould be written as

T t = k/ + T k (6)where T k is the Kolmogorov time scale and isgiven by

Tk = c k ( / ) 1/2 (7)

This time scale ensures theperformance of the standard k- model farfrom the wall since k/ is much larger than T k away from the wall and the model constants

are the same as those in the standard k- model. Since k/ vanishes near the wall dueto the boundary condition for k, the time scalenear the wall would be T k .

Using this time scale as the turbulenttime scale and k 1/2 as the turbulent velocityscale would give the below expression for theeddy viscosity.

t = c f kT t (8)

f is the damping function, which is used toaccount for the wall effect and is given by

f = [1-exp(-a 1Ry -a3Ry3 -a5Ry5)] 1/2 (9)

( ) ( ) )2(

1

+

+

+=

+

t t g yu

ydxdp

yu

v xu

u

t

)3(PrPr y

t y y

t v

xt

ut

t

+

=+

)4(

2

+

+

=

+

yu

yk

y yk

v xk

u t k

t

)5( / 2

2

1 t t

t

T c yu

c

y y yv

xu

+

+

=+

)1(0=+

yv

xu

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where

Ry = k 1/2 y/ (10)

and a 1 = 3.0*10 -4, a2 = 6.0*10 -5, a3 = 2.0*10 -6.The suggested boundary condition for

on the wall is:w = 2 (dk 1/2 /dy) 2 (11)

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3. NUMERICAL SOLUTIONThe free convection turbulent

boundary layer equations are solved for asemi-infinite isothermal flat plate. Air is thefluid considered for these calculations. The

EXPRESS code, which uses the standard k- model and the Reynolds stress model, ismodified with the above low k- model andused to for these calculations. The abovegoverning equations are transformed from thephysical coordinates (x, y) into the coordinatesystem (x, ) where the non-dimensional crossstream coordinate is defined as:

= y/ . (12)

Even though varies with x, thecomputational domain is constrained to lie inthe region:

0 1. (13)

The control volume approach is usedto discretize the governing equation. In thisapproach the governing partial differentialequations are converted into algebraicequations by integrating them over the cells.

Since for gases and liquids withmoderate Prandtl number transition to

turbulent flow occurs between Gr x = 10 9 and1010, the calculation was started at Gr x = 10

10.Profile shapes used by Eckert and Jackson areused for initial velocity and temperatureprofiles. These profiles are given by:

u/U = (y/ ) 1/7 (1 - y/ )4 (14)

(t - t )/(t w - t) = 1 - (y/ ) 1/7 (15)

where U is the characteristic velocity and isgiven by:

U = [g (tw - t)x] 1/2 . (16)

At the free stream, the velocity

components are set to zero, the turbulencequantities are set to a pre-set free streamvalues and the temperature is set to a constantvalue (t ). At the wall the velocitycomponents are set to zero, the temperature isset to a constant value (t w), k is set to zero and is given by Equation (11).

At each x station, the system of non-linear equations are solved using the Thomasalgorithm (TDMA) and then marcheddownstream to the next x station.

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4. RESULTS AND DISCUSSIONSThe results obtained from the above

model are compared with the availableexperimental results and the results obtainedby using Chien's low k- model. Calculationsare performed for Pr = 0.71 with 44 grids inthe y direction. The non-dimensionalturbulent velocity and temperature profiles atGrx = 10 12 are compared with the profilesobtained from Chien's model in Figure 1 andFigure 2 respectively. These profiles areplotted versus . The velocity is normalizedwith the characteristic velocity U. Theprofiles are agreeing well with those obtainedusing Chien's model.

The distribution of wall shear stress wis shown in Figure 3. From the experimentalresults Tsuji and Nagano found a correlationfor the turbulent shear stress. It is given by

w / Ub2 = 0.684 Gr x1/11.9 (17)

The shear stress distribution is compared withthe above correlation and the distribution

obtained from Chien's model and found to beagreeing well.

Figure 4 shows the wall heat transferas a function of Gr x. The calculated valuesare compared with the values obtained from

Chien's model and the best fit curve foundfrom the experimental values. The best fitcurve for Nu x for air at large Gr x is given by

Nu x = 0.106 Gr x1/3 (18)

Though the wall heat transfer calculated usingChien's model is close to the experimentalvalues at the beginning, it is becoming toohigh for large Gr x. The presented low k- model is under predicting the wall heat

transfer. But it is becoming close to theexperimental results with increasing Gr x. Thedifference at the beginning may be due to theinfluence of the prescribed initial conditions.At large Gr x, results become independent of the initial profiles and the model gives betterresults.

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5. CONCLUSIONSThe low k- model proposed by Yang

and Shih is modified and applied to calculateturbulent buoyancy driven flow over a semi-infinite plate. By comparing the results with

experimental results and results from Chien'smodel, it is shown that the model gives resultswith reasonable accuracy. Better results

could be obtained by further refinement of themodel.

Since the model constants used in thismodel are exactly the same as those in thestandard k- model, away from the wall itreduces to the standard k- model. Thereforethis model could be used for both, the nearwall turbulence and high Reynolds numberturbulence.

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NOMENCLATURE

a1, a2, a3 Constants in the equation for damping functionck Constant in the equation for Kolmogorov time scale, 1.0

c1, c2 Model constantsc Coefficient in the equation for tf Damping functiong Gravitational accelerationGrx Local Grashof number, g (tw-t)x3 / 2k Turbulent kinetic energyNu x Local Nusselt numberp PressurePr Prandtl numberPr t Turbulent Prandtl number for tT t Turbulent time scale

Tk Kolmogorov time scalet Temperaturetw Wall temperaturet Free stream temperatureU Characteristic velocityUb Normalizing velocity, (g (tw- t) )1/3u Vertical velocity componentv Velocity component perpendicular to the platex Vertical coordinatey Horizontal coordinate Coefficient of thermal expansion Dissipation of kinetic energy Density Shear stressk Turbulent Prandtl number for k Turbulent Prandtl number for Kinematic viscosity t Turbulent kinematic viscosity

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REFERENCES

1. Heindel, T. J., Ramadhyani, S. andIncropera, P., Assessment of turbulencemodels for natural convection in an

enclosure, Numerical heat transfer, PartB, 26, 1994, pp. 147-172.

2. Henkes, R. A. W. M. and Hoogendoorn,Comparison of turbulence models for thenatural convection boundary layer along aheated vertical plate, Int. J. Heat MassTransfer, Vol. 32, No. 1, 1989, pp. 157-169.

3. Kays, W. M. and Crawford, M. E.,

Convective Heat and Mass Transfer, 3rd

Edition, McGraw-Hill Inc., New York,1993.

4. Kuei-Yuan Chien, Predictions of channeland boundary layer flows with a lowReynolds number turbulence model,

AIAA Journal, Vol. 20, 1982, pp. 33-38.

5. Tsuji, T. and Nagano, Y., Characteristicsof a turbulent natural convection boundarylayer along a vertical flat plate, Int. J.Heat Mass Transfer, Vol. 31, No. 8, 1988,pp. 1723-1734.

6. Yang, Z. and Shih, T. H., A new timescale based k- model for near wallturbulence, NASA Technical

Memorandum 105768, Sept. 1992.

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Figure 1 : Velocity Profile at Gr x= 10 12 (Variation of u/U with )

Figure 2: Temperature Profile at Gr x= 10 12 (Variation of (t-t )/(t w- t) with )

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 0.2 0.4 0.6 0.8 1

present low k-e

Chien's low k-e

00.10.20.30.40.50.60.70.80.9

1

0 0.2 0.4 0.6 0.8 1

present low k-e

Chien's low k-e

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Figure 3 : Wall Shear Stress (Variation of w / Ub2 with Gr x)

Figure 4 : Wall Heat Transfer (Variation of Nu x with Gr x)

1

10

1.E+09 1.E+10 1.E+11 1.E+12 1.E+13

present low k-e

experimental

Chien's low k-e

1.E+02

1.E+03

1.E+04

1.E+09 1.E+10 1.E+11 1.E+12 1.E+13

present low k-e

experimentalChien's low k-e

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