NASA Technical Memorandum 102293 

ICOMP-89- 18 

c 

Calculation of Reattaching Shear Layers 

in Divergent Channel With a MultipleTime-Scale Turbulence Model 

S.-W. Kim 

Institute for Computational Mechanics in Propulsion 

Lewis Research Center 

Cleveland, Ohio 

August 1989 

-- 

(NASA-TM-162293) CALCULATIQN OF 6ATTACHIkG 

SHEAR LAYERS IN UIV€RG€rJT CHANIvEL kT1H A 

HULTIPLE-TIpk-SCALE TUHHUltYCE M3PEl (NASA, 

Lewis Research tenter) 27 p cscc 200 

N93-2 

ICOMP CASE WfSTfRN 

REYRVf WNERSlTY 

18749 

Urlcl as 

G3/34 0225964 
CALCULATION OF REATTACHING SHEAR LAYERS IN DIVERGENT CHANNEL WITH 

A MULTIPLE-TIME-SCALE TURBULENCE MODEL 

S.-W. Kim* 

Institute for Computational Mechanics in Propulsion 

Lewis Research Center 

Cleveland, Ohio 44135 

SUMMARY 

Numerical calculations of turbulent reattaching shear layers in a 

divergent channel are presented. The turbulence is described by a 

multiple-time-scale turbulence model. The turbulent flow equations are solved 

by a control-volume based finite difference method. The computational results 

are compared with those obtained using k-E curbulence models and algebraic 

Reynolds stress turbulence models. It is shown that the multiple-time-scale 

turbulence model yields significantly improved computational results than the 

other turbulence models in the region where the turbulence is in a strongly 

inequilibrium state. 

*Work funded under Space Act Agreement C99066G. 
A, 

A, 

Cf 

cP 

CPf 

CPJ 

CtR 

CPf 

f € 

H 

k 

kP 

kt 

P 

Pr 

Rt 

u, 

a 

NOMENCLATURE 

coefficient for tangential velocity correction 

coefficient for transverse velocity correction 

friction coefficient (-rw/(0.5pU, 2 )) 

pressure coefficient (-p/(0.5pU, 2 )) 

constant coefficient for eddy viscosity equation (-0.09) 

turbulence model constants for E equation (R-1,3) 

turbulence model constants for Et equation (R-1,3) 

constant coefficient (-0.09) 

wall damping function for eddy viscosity equation 

wall damping function for ew equation 

height of backward-facing step 

turbulent kinetic energy (k-$ + kt) 

P 

turbulent kinetic energy of eddies in production range 

turbulent kinetic energy of eddies in dissipation range 

pres sure 

production rate of turbulent kinetic energy 

turbulent Reynolds number (-k 2 /(vel)) 

inlet flow velocity 

time averaged velocity (={u,v)) 

friction velocity (==J(rw/p)) 

Reynolds stress (i=l,2,3 and j=l,2,3) 

velocity vector (=(u,v)) 

spatial coordinates (=(x,y,z)) 

reattachment location 

wall coordinate (=u,y/v) 

deflection angle of the top wall 

2 
‘P 

Et 

‘1 

n 

energy transfer rate from production range to 

dissipation range 

dissipation rate of turbulent kinetic energy 

dissipation rate in near-Val1 equilibrium region 

von Karman constant (-0.41) 

molecular viscosity 

effective viscosity (=p+pt) 

turbulent viscosity 

kinematic viscosity of fluid 

turbulent eddy viscosity 

density 

turbulent Prandtl number for $ equation 

turbulent Prandtl number for kt equations 

turbulent Prandtl number for eP equation 

turbulent Prandtl number for et equation 

wall shearing stress 

3 
INTRODUCTION 

The experimental study of a reattaching shear layer in a divergent 

channel [l] was designed to test the predictive capability of various 

turbulence models, to identify any deficiency in turbulence closure models, 

and thus to improve predictive capability of turbulence models. The flow 

geometry is shown in Figure 1. The height of the backward-facing step is 

smaller than the boundary layer thickness of the incoming flow. Abrupt 

breakdown of the boundary layer generated a strongly inequilibrium 

turbulent flow. Furthermore, a strong pressure gradient was generated by 

varying the divergence angle of the top wall to study its effect on the 

development of the turbulence field, especially the Reynolds stress, and 

the reattachment process. A number of turbulence models, such as k-e 

turbulence models and algebraic Reynolds stress turbulence models (ARSM), 

were shown to yield poor computational results for the flow [1,2]. It is 

also shown in References 1 and 2 that a modified ARSM, with modifications 

in the dissipation rate equation, yielded computational results which are 

in good agreement with measured data. However, generality of the improved 

predictive capability for other complex turbulent flows has not been shown 

yet. 

It has been shown previously that the high Reynolds number 

multiple-time-scale turbulence model yields accurate computational results 

for a number of complex turbulent flows such as a wall jet flow, a 

wake-boundary layer interaction flow, a confined coaxial jet without swirl 

and a confined coaxial swirling jet to name a few [3,4]. In the 

single-time-scale turbulence models such as k-c turbulence models, 

algebraic stress turbulence models, and Reynolds stress turbulence models, 

a single time scale is used to express both the turbulent transport and the 

4 
dissipation of the turbulent kinetic energy. However, this practice is 

inconsistent with physically observed turbulence in the sense that the 

turbulent transport is related to the time scale of energy containing large 

eddies and the dissipation of turbulent kinetic energy is related to the 

time scale of fine scale eddies in the dissipation range. The 

single-time-scale turbulence models yield reasonably accurate computational 

results for simple turbulent flows; however, the predictive capability 

degenerates rapidly as turbulent flows to be solved become more complex. In 

the multiple-time-scale turbulence models [3-71, the turbulent transport of 

mass and momentum is described using the time scale of the large eddies and 

the dissipation rate is described using the time scale of the fine-scale 

eddies. The improved computational results obtained using the 

multiple-time-scale turbulence model for complex turbulent flows can be 

attributed to the physically consistent nature of the turbulence models 

discussed above. 

In numerical calculations of turbulent flows, wall function methods 

are most frequently used to model the near-wall region. These methods have 

been derived from the logarithmic velocity profile based on experimental 

observations that the turbulence in the near-wall region can be described 

in terms of the wall shearing stress. Therefore, these methods are not 

valid if the logarithmic velocity profile no longer prevails in the 

near-wall region. For example, the logarithmic velocity profile no longer 

prevails in the near-wall or in the wake regions of unsteady turbulent 

flows [8], therefore wall function methods can not be applied. Many other 

cases for which the wall function methods are invalid can be found in 

References 9 and 10. Due to this limited applicability of the wall function 

methods, numerous alternative approaches have been proposed. In the 

5 
alternative approaches, the near-wall low turbulent Reynolds number region 

is included into numerical analyses to overcome the shortcomings of the 

wall functions methods. Various turbulence models which include the 

near-wall low turbulence region can be classified as two-layer (or 

multi-layer) turbulence models [ll] and low Reynolds number turbulence 

models [lo] based on the way the near-wall region is treated. More detailed 

discussion on the advantages and disadvantages of various near-wall 

turbulence models can be found in References 9 and 10. 

In the present study, the near-wall turbulence is described by a 

"partially low Reynolds number approach." In the model [9], only the 

turbulent kinetic energy equations are extended to include the near-wall 

low turbulence region and the energy transfer rate and the dissipation rate 

inside the near-wall layer are obtained from algebraic equations. The 

algebraic equations were obtained from a k-equation turbulence model [12]. 

It would be appropriate to classify the method as a "partially low Reynolds 

number approach" to distinguish it from other classes of methods. This 

approach was first used in Chen and Pate1 to solve turbulent flows over 

airfoils [13]. Advantages of the partially low Reynolds number approach 

over the other methods can be summarized as follows. The turbulence length 

scale of the external flows is related to the flow field characteristics 

[14]. On the other hand, the turbulence length scale of boundary layer 

flows is strongly related to the normal distance from the wall. This 

characteristic of the wall bounded turbulent flows can be described quite 

naturally by the present class of turbulence models. The low Reynolds 

number turbulence models can also be used to describe the wall bounded 

turbulent flows; however, more grid points have to be used to resolve the 

steep dissipation rate in the near-wall region. It is also interesting to 

6 
note that various similar k-equation turbulence models, which form the 

basis of the present near-wall turbulence model, yield accurate 

computational results for a class of simple turbulent boundary layer flows 

[15], turbulent flows with drag reduction [16], and fully developed 

unsteady turbulent pipe flows [8]. However, the k-equation turbulence model 

itself is less useful for separated and/or swirling turbulent flows with 

complex geometry due to lack of a systematic method to evaluate the 

turbulence length scale. Development of the near-wall turbulence model and 

its application to fully developed turbulent channel and pipe flows can be 

found in Reference 9. It has been shown in the reference that the present 

near-wall turbulence model can resolve the over-$hoot phenomena of the 

turbulent kinetic energy and the dissipation rate in the region very close 

to the wall and that significantly improved computational results for the 

turbulence structure in the near-wall region are obtained. Incorporation of 

the same near-wall turbulence model into a k-e turbulence model and its 

application to complex turbulent flows such as a supersonic turbulent flow 

over a compression ramp and a transonic flow over an axisymmetric curved 

hill can be found in References 17 and 18, respectively. 

The numerical method used herein is based on the pressure correction 

method [19] which has been used most extensively to solve incompressible 

flows the domain of which can be discretized by an orthogonal mesh. 

However, the present numerical method is applicable for both incompressible 

and compressible flows with arbitrary, complex geometries. The capability 

to solve compressible flows is achieved by including a convective 

incremental pressure term into the pressure correction equation [17,18]. In 

the method, the velocities are located at the same grid points and the 

pressure is located at the centroid of the cell formed by the four adjacent 

7 
velocity grid points. This grid layout was found to be quite suitable to 

solve flows with complex geometries [17]. The accuracy and the convergence 

nature of the numerical method have been demonstrated by solving a number 

of flow cases. The example problems considered in References 17 and 18 

include: a developing channel flow, a developing pipe flow, a 

two-dimensional laminar flow in a 90 degree bent channel, polar cavity 

flows, a turbulent supersonic flow over a compression ramp, and a shock 

wave - turbulent boundary layer interaction in transonic flow over a curved 

hill. It was found that the numerical method used herein yielded accurate 

computational results even when highly skewed, unequally spaced, curved 

grids were used. 

TURBULENT PLOW EQUATIONS 

The incompressible turbulent flow equations are given as; 

a a 

-((pu) + -((pv) = 0. 

ax aY 

where eqs. (1-3) follow from the conservation of mass, u-momentum, and 

v-momentum, respectively. In numerical calculation, the conservation of 

mass equation is replaced by a pressure correction equation given as: 

8 
'- 

where the last term represents the mass imbalance, and the first two 

convection terms are unnecessary for incompressible flows. The pressure 

correction equation can be derived following the standard SIMPLE procedure 

[191. In the present numerical method, all flow variables, except pressure, 

are located at the same grid points and the preooure node has been located 

at the centroid of the cell. The control volume for the pressure correction 

equation is defined as the cell enclosed by the four neighboring grid 

points. The velocity-pressure decoupling is eliminated by treating the 

pressure correction equation as a continuous form partial differential 

equation rather than treating it as a constraint condition. In the former 

case, the discrete pressure correction obtained from eq. (4) becomes a 

five-diagonal system of equations for rectangular grids. On the other hand, 

the discrete pressure correction equation obtained by directly substituting 

the incremental pressure - incremental velocity relations into the 

conservation of mass equation yields a nine-diagonal system of equations. 

The latter discrete pressure correction equation can yield a 

velocity-pressure decoupled solution, whereas the former equation does not 

[171. 

In control-volume based finite difference methods, the discrete system 

of equations is derived by integrating the governing differential equations 

over the control volume [19]. For curvilinear grids, the number of 

interpolations required to obtain flow variables at the cell boundaries is 

significantly reduced by using the present grid layout. Enhanced 

convergence rate is partly attributed to the grid layout which required 

9 
fewer interpolations [17]. In solving the discrete system of equations, the 

off-diagonal terms are moved to the load vector term and the resulting 

system of equations can be solved using a tri-diagonal matrix algorithm 

(TDMA) . 

TURBULENCE EQUATIONS 

For clarity, the multiple-time-scale turbulence model supplemented 

with the near-wall turbulence model is summarized below. The turbulent 

kinetic energy and the energy transfer rate equations for the energy 

containing large eddies are given as; 

ut 8% 

cP ( (u +-) - ) - Pr - 

a 

akp axj "j- 

-- 

axj axj 

(5) 

where the production rate is given as; 

The turbulent kinetic energy equation and the dissipation rate equations 

for the fine scale eddies are given as: 

akt a ut akt 

uj- - - ( (u +-) - ) - €p - €t 

axj axj akt axj 

(7) 

LO 
The turbulent kinetic energy equations, eqs. (5) and (7), are defined for 

the entire flow domain while the energy transfer rate and the dissipation 

rate equations are valid for the flow domain away from the near-wall 

region. The turbulence model constants are given as; Ukp-o.75, Ukt-0.75, 

~,~-1.15, uCt-1.15, cpl-0.21, cp2-1.24, cp3- 1.84, ctl-0.29, ct2- 1.28, and 

ct3-1.66. These turbulence model constants approximately satisfy the 

near-wall equilibrium turbulence condition, the decay rate of the grid 

turbulence [20], and the turbulence intensity growth rate in a constant 

shear flow [21]. Further discussion on the establishment of these 

turbulence model constants can be found in References 3. 

The energy transfer rate and the dissipation rate inside the near-wall 

layer are given as; 

where 

f, - 1- exp( -A,Rt) 

k2 

Rt IVe 1 

Note that €1 in eq.(lO) represents the standard dissipation rate for 

near-wall turbulent flows in equilibrium state. The dissipation rate given 

11 
as eq. (9) is formally identical to the one proposed by Wolfshtein [12]. 

For y=O, eq. (9) takes the limit value given as 2uk/y2, which is an 

analytical solution of the turbulent kinetic energy equation for a limiting 

case as y approaches the wall. Slightly away from the wall where the 

turbulence is in the equilibrium state, f, becomes unity. For near-wall 

equilibrium turbulent flows, the production rate (P,) is approximately 

equal to dissipation rate (et) and hence the energy transfer rate (E ) from 

the low wave number production range to the high wave number dissipation 

range has to be approximately equal to the production and dissipation 

P 

rates. Recall that the production rate vanishes on the wall and grows to a 

peak value at y+=15. Hence eq. (9) may not be a good approximation for 

O<y+<15. 

turbulent kinetic energy on the wall yields a growth rate of turbulent 

kinetic energy and a production rate that are in good agreement with 

experimental data as well as theoretical analysis (91. 

However, use of the vanishing boundary condition for the 

The eddy viscosity away from the near-wall layer is given as; 

k2 

"t - Cpf- 

'P 

and that for the near-wall layer is given as; 

k2 

'1 

ut - Cpf fp - (13) 

where fp=l-l./exp(A1./Rt + A2Rt2) is a linear function of the distance from 

the wall in the viscous sublayer and becomes unity in the fully turbulent 

region. AI-0.025 

Reference 9. The 

and A2=0.00001 

eddy viscosity 

have been used for the near wall layer, see 

given as eq. (13) grows in proportion to 

12 
. 

the cubic power of the distance from the wall. It can be found in Reference 

10 that the near-wall analysis yields the same growth rate of the eddy 

viscosity in the region very close to the wall. However, there also exist a 

few low Reynolds number turbulence models in which the eddy viscosity 

varies in proportion to the fourth power of the distance from the wall, see 

References 9 and 10 for more discussion. 

COMPUTATIONAL RESULTS 

The experimental data for the reattaching shear layer can be found in 

Reference 1. The inlet free stream velocity was 40 m/sec, the boundary 

layer thickness was 0.019 meters, and the height of the backward-facing 

step was 0.0127 meters. The top wall was deflected from -2 degrees to 10 

degrees to generate a strong adverse pressure gradient. 

In numerical calculations, the inlet boundary was located at four 

step-heights upstream of the expansion corner and the exit boundary was 

located at approximately 35 step-heights downstream of the expansion 

corner. The flow domain was discretized by a 105 by 85 mesh with 

concentration of grid points near the expansion corner and in the bottom 

wall region, see Figure 2-(a). The grid in the vicinity of the expansion 

corner is fine enough to resolve details of the large eddies subjected to 

strong shear and sudden expansion, see Figure 2-(b). The inlet boundary 

conditions for the tangential velocity, the turbulent kinetic energies, and 

the dissipation rates (ep and et) were obtained from experimental data for 

a fully developed boundary layer flow over a flat plate [3,22]. The 

non-dimensional velocity and the turbulent kinetic energy profiles were 

scaled to 

boundary. 

yield a boundary layer thickness of 0.019 meters at the inlet 

The no-slip boundary condition for velocities and vanishing 

13 
turbulent kinetic energies were prescribed at the solid wall boundary. At 

the exit boundary, a vanishing gradient boundary condition was used for all 

flow variables except the pressure. A uniform pressure was prescribed at 

the exit boundary. The partition between the near-wall layer and the 

external region was located at approximately y+-lOO, 12 grid points were 

allocated inside the near-wall layer. The mesh size of the first grid point 

on the bottom wall was Ay+-2 and the grid size in the normal to the wall 

direction was increased by a factor of approximately 1.15. Further details 

on the computational procedure can be found in Reference 17. 

The calculated streamline contours are shown in Figure 3. The flow 

field consisted of two recirculation zones. The primary recirculation zone 

extended from the separation corner toward the downstream direction; and 

the secondary recirculation zone was very small and confined in the corner 

region. The reattachment location versus the top wall deflection angle is 

shown in Figure 4. It can be seen in the figure that the k-c and ARSM 

turbulence models largely under-predict the reattachment location. The 

modified ARSM yielded a significantly improved computational result, 

however, the present computational result compared more favorably with the 

measured data than did the modified ARSM. 

The static pressure contour lines are shown in Figure 5, where the 

pressure has been normalized by the inlet total pressure and the 

incremental pressure between the contour lines is 0.005. It can be seen in 

the figure that a few contour lines pass through the expansion corner, and 

thus there exists a mild base pressure in the backward-facing step region. 

The calculated static pressure on the wall is compared with experimental 

data as well as the numerical results of Reference 1 in Figure 6. The mild 

pressure drop at x/H-0 represent the base pressure. For a=Oo, the present 

14 
computational result compared slightly more favorably with the measured 

wall pressure than the other computational results. For a-6', all the 

computational results compared decently with the measured data; however, 

the slope of the wall pressure in the continuously diverging downstream 

region obtained in the present study compared more favorably with the 

measured data than the other results. This difference may due to the 

different numerical methods used. 

The calculated wall shearing stresses are shown in Figure 7. It can be 

seen in the figure that the location of the peak wall shearing stress 

obtained using the k-e turbulence model is grossly in error. It is 

interesting to note that the modified ARSH under-predicts the peak value 

and the present turbulence model over-predicts the peak value even though 

the relative differences are almost the same for both deflection angles. 

The mean velocity, the turbulent kinetic energy, and the Reynolds 

stress profiles at four downstream locations are compared with experimental 

data and with the calculated results using the modified ARSM [1,2] in 

Figures 8-10, respectively. The experimental turbulent kinetic energy shown 

in Figure 9 was estimated using the measured value of uf2+vf2 and an 

assumption that w' 2=(u82+v'2)/2. As shown in Figures 8-10, both 

computational results exhibit fair comparison with the experimental data. 

It can be seen in Figure 9 that the peak value of the turbulent kinetic 

energy and the shape of the turbulent kinetic energy profile obtained using 

the multiple-time-scale turbulence model compare slightly better with the 

measured data than those obtained using the modified ARSM at x/H-1.0 where 

the turbulence is in a strongly inequilibrium state. It has been shown 

previously that the improved computational results for complex turbulent 

flows are attributed to the capability of the multiple-time-scale 

15 
turbulence model to resolve the inequilibrium turbulence [3]. The same 

argument can be applied for the present flow case. At further downstream 

locations, the present computational results compared slightly less 

favorably with the experimental data, a result perhaps due to the near-wall 

turbulence model which can not take into account of the inequilibrium 

turbulence. 

The ratio of turbulent viscosity to molecular viscosity at three 

downstream locations are shown in Figure 11. It can be seen in the figure 

that the Jones-Launder k-6 turbulence model over-estimates the ratio so 

that the reattachment location is largely under-predicted. On the other 

hand, the present computational results compare quite favorably with the 

measured data so that the reattachment location is correctly predicted. The 

calculated production and dissipation rates of the turbulent kinetic energy 

at the same downstream locations were qualitatively and quantitatively 

almost the same as those of reference 1. 

CONCLUSIONS 

Numerical calculations of reattaching shear layers in a diverging 

channel using a multiple-time-scale turbulence model supplemented with a 

near-wall turbulence model have been presented. The calculated reattachment 

location versus the top wall deflection angle obtained using the present 

turbulence model was in excellent agreement with measured data. The 

calculated wall pressure and the wall shearing stress were also in good 

agreement with the experimental data. The rest of the present computational 

results such as the normalized velocity profiles and the Reynolds stress 

profiles compared favorably with experimental data. The computational 

results obtained using the multiple-time-scale turbulence model compared 

16 
slightly more favorably with the experimental data than those obtained 

using the modified algebraic Reynolds stress turbulence model. It has been 

shown that prediction of the correct reattachment location depends on the 

prediction of the correct level of the turbulent viscosity, which depends, 

in turn, on the capability of a turbulence model to resolve the entire 

turbulence structure of the flow field correctly. Thus the improved 

computational results are attributed to the capability of the 

multiple-time-scale turbulence model to resolve the strong inequilibrium 

turbulence in the vicinity of the expansion corner and in the following 

shear-layer region. 

17 
REFERENCES 

1. D. M. Driver and H. L. Seegmiller, "Features of a Reattaching Turbulent 

Shear Layer in Divergent Channel Flow," J. AIAA, vol. 23, pp. 163-171, 

1985. 

2. M. Sindir, "Numerical Study of Separating and Reattaching Flows in a 

Backward-Facing Step Geometry," Ph.D. Thesis, University of California 

at Davis, CA, 1982. 

3. S.-W. Kim and C.-P. Chen, "A Multiple-Time-Scale Turbulence model 

Based on Variable Partitioning of the Turbulent Kinetic Energy 

Spectrum", To appear in the Numerical Heat Transfer, 1989, Also 

available as NASA CR-179222, 1987 and AIAA Paper 88-0221, 1988. 

4. S.-W. Kim and C.-P. Chen, "A Review on Various Flow-Solid 

Interaction Analysis Methods with Emphasis on Recent Advances in 

Turbulence Models and Flow Analysis Methods," AIAA Paper 88-3685, 1988. 

5. K. Hanjelic, B. E. Launder, and R. Schiestel, "Multiple-Time-Scale 

Concepts in Turbulent Shear Flows" in L. J. S. Bradbury, F. Durst, 

B. E. Launder, F. W. Schmidt, and J. H. Whitelaw, (eds.), Turbulent 

Shear Flows, vol. 2, pp. 36-49, Springer-Verlag, New York, 1980. 

6. R. Schiestel, "Multiple-Time-Scale Modeling of Turbulent Flows in One 

point Closure", Phvsics of Fluids, vol. 30, pp. 722-731, 1987. 

7. R. Schiestel, "Multiple-Scale Concept in Turbulence Modelling, 

11. Reynolds Stresses and Turbulent Fluxes of a Passive Scalar, 

Algebraic Modelling and Simplified Model using Boussinesq Hypothesis", 

Journal de Mechaniaue theoriaue et appliauee, vol. 2, pp. 601-628, 

1983. 

8. S. W. Tu and B. R. Ramaprian, "Fully Developed Periodic Turbulent Pipe 

Flow, Part I. Main Experimental Results and Comparison with 

18 
Predictions," J. F1 uid Mech, , VO~. 137, pp. 31-58, 1983. 

9. S.-W. Kim, "A Near-Wall Turbulence Model and Its Application to Ful 

Developed Turbulent Channel and Pipe Flows," To appear in Numerical 

Beat Transfer, 1989. Also available as NASA TM-101399, 1988. 

10. V. C. Patel, W. Rodi and G. Scheuarer, "Turbulence Models for Near 

Wall and Low Reynolds Number Flows: A Review", J. AIM, 23, 

pp. 1308-1319 (1985). 

11. R. S. Amano, "Development of a Turbulent Near-Wall Model and Its 

Application to Separated and Reattached Flows," Pumeri cal He at 

Transfer, vol. 7, pp. 59-75, 1984. 

12. M. Wolfshtein, The Velocity and Temperature Distribution in 

One-Dimensional Flow with Turbulence Augmentation and Pressure 

Y 

Gradient, 'I Jnt. J . Heat and Mass Transf er, vol. 12, pp. 301-318, 1969. 

13. H. C. Chen and V. C. Patel, "Practical Near-Wall Turbulence Models for 

Complex Flows Including Separation," AIM-87-1300, 1987. 

14. A. Roshko, "Structure of Turbulent Shear Flows: A New Look," J. AIAA, 

VO~. 14, NO. 10, pp. 1349-1357, 1976. 

15. Gibson, M. M., Spalding, D. B., and Zinser, W., "Boundary Layer 

Calculations using the Hassid-Poreh One-Equation Energy Model," 

Letters in Heat and Mass Transfer , VO~. 5, pp. 73-80, 1978. 

16. Hassid, S., and Poreh, M., "A Turbulent Energy Model for Flows with 

Drag Reduction," J. Fluid Enei - neer ing , Transactions of ASME, 

pp. 234-241, 1975. 

17. S.-W. Kim, "A Control-Volume Based Reynolds Averaged Navier-Stokes 

Equation Solver Valid at All Flow Velocities," NASA TM-101488, 1989. 

18. S.-W. Kim, "Numerical Computation of Shock Wave - Turbulent Boundary 

Layer Interaction in Transonic Flow over an Axisymmetric Curved Hill," 

19 
NASA TM-101473, 1989. 

19. S. V. Patankar, Numerical Heat Transfer and Flui d Flow, McGraw-Hill, 

New York, 1980. 

20. F. H. Harlow and P. I. Nakayama, "Transport of Turbulence Energy Decay 

Rate," Los Alamos Sci. Lab., LA-3854, 1968. 

21. V. G. Harris, J. A. H. Graham, and S. Corrsin, "Further 

Experiments in Nearly Homogeneous Turbulent Shear Flow," J. Fluid 

Nech., vol. 81, pp. 657-687, 1977. 

22. P. S. Klebanoff, "Characteristics of Turbulence in a Boundary Layer 

with Zero Pressure Gradient," NACA Report 1247, 1955. 

20 
FIGURE 1. - NWNCLATURE OF THE REATTACHING SHEAR LAYER, 

H: HEIGHT OF THE BACKWARD-FACING STEP. a: TOP WALL DEFLECTION ANGLE. 

FIGURE 3. - STRWCIE ColllouR. 

12 

10 

8 

I 

x 

z 

6 

4 

(b) VICINITY OF EXPANSION CORNER. 

FIGURE 2. - DISCRETIZATIM OF FLCU WMIH. 

-0 EXPER IPTNTAL DATA 1 I 

PRESENT COMPUTATIONAL RESULT 

MODIFIED ARSN 

0 STANDARD k-E AND ARSN 

- 

I I I I I 1 I J 2 

a 

FIGURE 4. - REATTACHKNT LOCATION VERSUS DEFLECTION 

ANGLE. 

21 
FIGURE 5. - PRESSURE CONTOUR. 

-.15 

- 

0 EXPERIMENTAL DATA [ 1 I 

PRESENT COMPUTATIONAL %$ ---- STANDARD k-E AND ARSM 

RESULT 

-- MODIFIED ARSM 

0- 

\ 

'a 

I I I I I I I 

a 

V 

I I I 1 

5 10 15 20 25 30 

X/H 

(b) a = 6'. 

FIGURE 6. - PRESSURE ON BOTTOM WALL. 

22 
I- 2- 

0 EXPERIENTAL DATA 111 

RODIFIED ARM 

STANDARD k-E AND ARM 

- PRESENT CWUTATIONAL RESULT -- 

Do-- 

0 

0 EXPERIENTAL DATA 111 

RODIFIED ARM 

-1 - PRESENT CWUTATIONAL RESULT -- 

Do-- STANDARD k-E AND ARM 

-2 

U 

- (a) a = o0. 'r 

-2 

0 5 10 15 20 25 30 

X/H 

(b) a = 6'. 

FIGURE 7. - SHEAR STRESS ON Bolloll UALL. 

4 

- 

0 EXPERIEIITAL DATA 

PRESENT CWUTATIONAL RESULT 

- - - RODIFIED ARM 

- 

WH = 1.0 4.0 7.0 12.0 

(a) a = oo. % 

WH = 1.0 4.0 7.1 12.0 

u/u 00 

(b) a = 6'. 

FIGURE 8. - VELOCITY PROFILES. 

23 
- 

0 EXPERIENTM DATA 

PRESENT CWUTATIONAL RESULT 

--- NODIFIED ARM 

(a) a = 0'. 

k/Um2 

(b) a = 6'. 

FIGURE 9. - TURBMNT KlWETlC ElERGY PROFILES. 

- 0 EXPERIENTAL DATA PRESENT COMPUTATIONAL RESULT 

- - MODIFIED ARM 

' c WH = 1.0 4.0 7.0 12.0 4 

(a) a = 0'. 

e 

r, 

9- X/H = 1.0 4.0 7.1 12.0 

iT/Um2 

(b) a = 6'. 

F1W 10. - REYNOLDS STRESS PRQILES. 

24 
. 

1 

0 

e 

- 

0 EXPERIENTAL DATA 

PRESENT CWUTATIONAL RESULT - - 

---- CEBECI ALGEBRAIC WEL [1I 

JONES AND LAUNDER k-E IK)DEL [I1 

(a) a = oo. 

0 0 0 lo00 2000 

(b) a = 6'. 

CIt4 

FIGURE 11. - RATIOS OF TURBULENT VISCOSITY OVER MOLECULAR 

VISCOSITY. 

25 
Natlonal Aeronautics and Space Administration 

9. Security Classif. (of this report) 

Unclassified 

Report Documentation Page 

20. Security Classif. (of this page) 21. No of pages 22. Price' 

Unclassified 26 A03 

2. Government Accession No. NASA TM-102293 

ICOMP-89- 18 

1. Report No. 

4. Title and Subtitle 

Calculation of Reattaching Shear Layers in Divergent Channel 

With a Multiple-Time-Scale Turbulence Model 

7. Author(s) 

S.-W. Kim 

9. Performing Organization Name and Address 

National Aeronautics and Space Administration 

Lewis Research Center 

Cleveland, Ohio 44135-3191 

2. Sponsoring Agency Name and Address 

National Aeronautics and Space Administration 

Washington, D.C. 20546-0001 

3. Recipient's Catalog No. 

5. Report Date . 

August 1989 

6. Performing Organization Code 

8. Performing Organization Report No. 

E-4965 

10. Work Unit No. 

505-62-2 1 

11. Contract or Grant No. 

13. Type of Report and Period Covered 

Technical Memorandum 

14. Sponsoring Agency Code 

5. Supplementary Notes 

S.-W. Kim, Institute for Computational Mechanics in Propulsion, Lewis Research Center, Cleveland, Ohio 44135 

(work funded by Space Act Agreement C99066G). Space Act Monitor: Louis A. Povinelli 

6. Abstract 

Numerical calculations of turbulent reattaching shear layers in a divergent channel are presented. The turbulence is 

described by a multiple-time-scale turbulence model. The turbulent flow equations are solved by a control-volume 

based finite difference method. The computational results are compared with those obtained using k-c turbulence 

models and algebraic Reynolds stress turbulence models. It is shown that the multiple-time-scale turbulence model 

yields significantly improved computational results than the other turbulence models in the region where the 

turbulence is in a strongly inequilibrium state. 

7. Key Words (Suggested by Author@)) 

Multiple-time-scale turbulence model 

Algebraic Reynolds stress model 

Shear layer 

Diverging channel flow 

18. Distribution Statement 

Unclassified - Unlimited 

Subject Category 34