[American Institute of Aeronautics and Astronautics 35th AIAA Fluid Dynamics Conference and Exhibit...

American Institute of Aeronautics and Astronautics

1

A l−k TURBULENCE CLOSURE FOR WALL-BOUNDED FLOWS

U. Goldberg* and S. Chakravarthy** Metacomp Technologies, Inc., Agoura, California.

ABSTRACT

Among the several merits of a two-equation turbulence closure of the k-ℓ variety, two advantages stand out: (a) homogeneous wall boundary conditions for both variables since both vanish at solid surfaces and (b) direct access to the turbulence length-scale, often needed in turbulent flow calculations. The k-ℓ turbulence model proposed here differs from previous versions in two main aspects: (1) the formulation is not based on the k-ε model since the transport equation for ℓ is not derived from the corresponding k and ε equations but rather from assumed basic elements; (2) the entire formulation is local, in particular no topography-related parameters (such as wall distance) are used. The paper details (a) the steps taken to form the transport equations, (b) the treatment of the ℓ-equation’s destruction term to avoid wrong entrainment velocity at boundary layer edges, (c) derivation of model constants from appropriately chosen unit problems, (d) a new proposal for the variable coefficient

µC~ and (e) the unique approach to the eddy viscosity damping function, based on time-scale realisability and a wall-proximity indicator (rather than using explicit wall-distance). Several wall-proximity indicators are discussed and their merits and usage are pointed out. The paper concludes with a variety of wall-bounded flow validation cases which demonstrate the model’s performance compared to experimental data as well as to predictions by a k-ε closure.

Keywords: turbulence model, two-equation, wall-distance-free.

NOMENCLATURE

Constants in ℓ- transport equation

Constants in ε- transport equation

Streamline curvature Coefficient of damping function

Damping function

Turbulence kinetic energy Turbulence length-scale

Mach number Turbulence production

Pressure Reynolds number

Turbulence Reynolds number

Mean strain invariant Temperature

Time Mean velocity component in j-direction

Reynolds stress tensor

Friction velocity ( )wρτ / Cartesian coordinate in j-direction

Distance to nearest wall

Wall coordinate ( µρτ /yu ) *President* Scientist *Principal

ratio heats Specific function dampingin Parameter

γα

ijδ Kronecker Delta

ε Turbulence dissipation rate κ von Karman Constant (0.41) µ Kinematic viscosity

ν Dynamic viscosity

ρ Density

kσ Diffusion coefficient in k-equation

lσ Diffusion coefficient in ℓ-equation τ Turbulence time-scale (also shear stress) Ω Mean vorticity invariant ∇ Gradient operator Subscripts 0 Stagnation conditions ∞ Evaluated at freestream ,j Derivative with respect to jx t Turbulent w Evaluated at wall

INTRODUCTION Numerous alternatives to the k-ε type turbulence model have been suggested over the years. The motivation behind this effort is two-fold: (1) to avoid the nonhomogeneous wall boundary condition for ε and (2) to overcome the weakness exhibited by some

+yyxu

uu

UtTSR

pPM

kfC

CCC

CCC

j

ji

j

t

k

s

τ

µ

µ

εε

Re

~

,,,

21

321

l

35th AIAA Fluid Dynamics Conference and Exhibit6 - 9 June 2005, Toronto, Ontario Canada

AIAA 2005-4638

Copyright © 2005 by Metacomp Technologies, Inc. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.


2

versions of the k-ε closure in flows involving adverse pressure gradient. One proposed alternative is a k-ℓ type model. The main advantages of such a closure over a k-ε model are: (a) it admits homogeneous wall boundary conditions for both turbulence variables, (b) it enables direct prediction of the turbulence length-scale which, together with the velocity scale ( k ) is a natural scalar in forming the eddy viscosity ( lkt ∝µ ) and (c) proper formulation of the forcing terms in the ℓ-transport equation can lead to improved predictive performance under adverse pressure gradient flow conditions. The paper begins with descriptions of previous k-ℓ modeling proposals, then introduces the current approach and points out the differences between it and the older formulations. This is followed by a detailed discussion of determining various model constants. Next several flow examples, covering a wide range of Mach numbers, are described and finally conclusions are given.

PREVIOUS APPROACHES The k-ℓ model may be derived directly from the k-ε equations using the transformation

−= l

lldkdkkd

32ε . This leads to the following

transport equation for ℓ: ( ) ( )

)1(.ln32

32

32

sink term Source

2

2/3

2/3

2

21

44444444 344444444 21l

l

l

l

l

ll

l

l

l

+

∂∂

∂∂

−

∂∂

∂∂

×

+−

−+

−

+

∂∂

+

∂∂

=∂∂

+∂

∂

kxx

kxx

kk

kCPk

C

xxU

xt

jjjj

tk

j

t

jj

j

σµ

µρ

σµ

µρρ

εε

The current proposal does not use the term involving

kP , replaces the (complicated) source/sink term with

( )jjCk ,,3ˆ llρ− and new constants are derived as

described below. A different approach was suggested by Smith,1 in which a k-ℓ model was derived from a previous k-kℓ closure by an exact transformation. The resulting equation for ℓ was of the form ( ) ( )

.2

122

2jj

t

jj

t

j

t

jj

j

xk

xkyxxykC

xxU

xt

∂∂

∂∂

+

∂∂

∂∂

−

−

+

∂∂

+

∂∂

=∂∂

+∂

∂

llll

l

l

ll

l

ll

l

σµ

κσµ

κρ

σµ

µρρ

For near-wall eddy viscosity damping, Smith1 proposed the following:

µρχ

κ

χχχχ

µχµ

3

2

1

4/1

4222

41

42221

41

,50exp

,,

Ak

yf

AAAfA

t

ll=

−=

++

++=ΦΦ=

where the iA s are constants. The current proposal

differs from the above in three ways: (1) the formulation does not involve wall-distance, (2) the cross-diffusion term is not used and (3) the destruction term involves k , not l/tµ , thus eliminating the near-wall damping function as a factor. In addition, the proposed damping function (see below) resorts to a wall proximity indicator rather than to explicit wall-distance as in Smith’s model. There is evidently no unique way to model a transport equation for any turbulence variable and the one proposed here for ℓ retains both functionality and simplicity.

THE POINTWISE APPROACH Fig. 1(a) demonstrates the basic weakness of a modeling approach based on wall distance. When multiple intersecting walls occur, there is no longer a unique way to define wall distance and the adopted choice influences the crucial near-wall distribution of turbulence quantities, hence also of heat transfer and skin friction. In contrast, using a pointwise approach (see Fig. 1(b)) preserves uniqueness and does not depend on the particular topology at hand. This method is, therefore, also called “topography-parameter-free” approach. There are several wall proximity indicators to replace direct wall distance, as indicated in Fig. 1(b). In the formulation described below, the proposed model uses two of these: the turbulence Reynolds number,

tR (indicated as 1 in the figure) and the dot product of the gradients of two turbulence quantities (indicated as 3 in the figure). These convey wall proximity without the need for actual wall distance and they are pointwise (local) in nature.

Fig. 1(a) Elements of the pointwise approach


3

Fig. 1(b) Elements of the pointwise approach

PROPOSED MODEL EQUATIONS

The following describes the current k-ℓ model’s transport equations and related components.

( ) ( )

( ) ( )

( )

( )

,,,12

ˆ

3,2,1,

,),,,(,

,1165.0,3/1,,

,,,max,)(1

)(~

,,2,1max11

,21

,32

21

,32

,ˆ

,

~

33

1

3212

32

22

12

25.2

2

2

2

2

2

32

2/3

kxxC

C

UUK

UUUKKKC

BASS

AeSBC

C

kRRe

ef

xU

xU

xU

xU

xU

S

xU

kSxU

uuP

xxCCk

xxU

xt

k

Pxk

xkU

xtk

kfC

jj

s

C

tt

R

R

i

j

j

i

ijk

k

i

j

j

i

k

kt

j

ijik

jj

j

t

jj

j

kjk

t

jj

j

t

s

t

t

ll

rrrr

rr

l

ll

ll

l

l

l

l

l

=∂∂

∂∂

=

+=

=∇⋅−∇⋅=

==++=

==Ω=Ω=

=Ω=+

+=

=

−

−=

∂

∂−

∂∂

=Ω

∂∂

−∂

∂+

∂∂

=

∂∂

−=∂∂

−=

∂∂

∂∂

−

+

∂∂

+

∂∂

=∂∂

+∂

∂

−

+

∂∂

+

∂∂

=∂∂

+∂

∂

=

−

−

−

−

−

ττϕϕϕ

ηςςςς

ςςςς

ττ

φψφψ

φψ

µρ

δ

ρµρ

ρ

σµ

µρρ

ρ

σµ

µρρ

ρµ

ηηη

µµ

α

µ

µµ

.0

,41.0,09.0,1

,42.0,018.0,67.0,0.1

223

2

==

==

+=

====

ww

k

k

CCC

CC

C

l

l

l

κκσ

ασσ

µµ

µ

The turbulence production,kP , is formulated based on

the Boussinesq model. The coefficient µC~ is

sensitized to mean strain and vorticity, in the absence of which

µµ CC →~ . A similar proposal, using a

different formulation, was given by Craft et al,2 however, the current approach is further sensitized to the ratio of local eddy length-scale to radius of streamline curvature. For large values of this ratio

µµ CC →~ (see fig. 2 where the curvature radius

sc CR /1= ). This additional refinement enhances the model’s generality. The damping function,

µf , incorporates time-scale realisability which prevents the turbulence time-scale from falling below the corresponding Kolmogorov level. Thus

tR Rk /2,1max/l=τ . The coefficient 3C is

designed to limit the sink term to near-wall portions of boundary layers in order to avoid nonphysical behavior of the model in boundary layer edge regions due to potentially creating entrainment velocity directed toward the wall. This is because the untreated sink term (

3C instead of 3C ) replaces the

entrainment velocity, V, by ykCV ∂∂+ /3 l which reduces V since 0/ <∂∂ yl toward the boundary layer edge (here y is the normal-to-wall direction). The flat plate results (see below) demonstrate the effect of the

3C coefficient treatment.

Fig. 2 Effect of cR/l on

µC~

DETERMINATION OF MODEL CONSTANTS Representative unit flow cases are used to extract relationships for some model constants, following Zeierman and Wolfshtein.3 The remaining constants are calibrated based on channel and flat plate flows.


4

Decay of isotropic turbulence In the absence of all gradients (mean flow and turbulence) the transport equations reduce to

.

,

2

2/3

kCdtd

kdtdk

=

−=

l

l

Introduce kq = , then

.///)/( 21 dtdqqdtdqdtqd −− −= lll . The above transport equations become

,

,2

2

2

qCdtd

qdtdq

=

−=

l

l

and

).(21

21

22 tfAtCq

Cqdt

d≡+

+=⇒+=

ll

Writing qtf )(=l , taking time derivatives of ℓ and q and using the previous relationships leads, after a few steps, to the following results:

( )

( )

( ) ./2/1

/

,/2/1

/

,/2/1

/

2

2

0

2

2

2

2121

002

00

00

122

002

00

0

122

002

00

0

CC

t

t

CC

C

qtCq

qq

qtCq

qtCq

kk

+−

+−

+

++

==

++

=

++

=

l

l

l

l

l

l

l

l

l

l

µµ

In the above, 000 0 == ttt qq µµ ll . In the k-ε model the corresponding expression is

( ) ./1

/ 1

2

002

00 2

2

0

−

−

+−

=ε

ε

εµµ C

C

t

t

qtCql

l

where 92.12 =εC based on experimental data. Retaining this proven decay level with the current model dictates

.42.02 =C

Logarithmic overlap layer Eddy viscosity dominates here over the molecular one and the ℓ-transport equation reduces to:

. ,)/( ,/ ,/

)2( ,01

24/3

2

32

yuyuSCukCy

dydCCk

dyd

dyd

t

t

ττµτµ κνκκ

νσ

====

=

−+

l

ll

l

Substituting these values into transport equation (2) results in

+= 223

1 CC

CCκσ

µµ

l

.

The remaining constants,

lσσ ,k and α, are

determined from matching skin friction and velocity profile data over a flat plate as well as velocity and kinetic energy profiles in a fully-developed channel flow. This is described in the next section.

MODEL EVALUATION Several flow examples are described in this section. Results are compared with both experimental data and predictions by a k-ε model.4 Except for the first test case, all computations were done using the commercial flow solver CFD++.5 CFD++ is a Navier-Stokes solver for either compressible or incompressible fluid flows. It features a second order Total Variation Diminishing (TVD) discretization scheme based on a multi-dimensional interpolation framework. For the results presented here, an HLLC (Harten, Lax, van Leer, with Contact wave) Riemann solver6 was used to define the (limited) upwind fluxes. This Riemann solver is particularly suitable for high speed flow applications since, unlike classical linear solvers such as Roe's scheme, it automatically enforces entropy and positivity conditions. Fully developed channel flow The primary purpose of this flow case is to serve as a preliminary test for the model. As seen in Fig. 3, the time-scale (τ) and the length-scale (ℓ) possess the same slope from the channel wall to the centerline. Therefore in this case

33ˆ CC = in the entire domain.

The secondary purpose of this test case is to obtain preliminary values for the as yet undetermined model constants

lσσ ,k and α. Fig. 4 shows a velocity

profile in wall coordinates. Both the k-ε and k-ℓ models agree well with the data. Fig. 5 shows the corresponding turbulence kinetic energy profiles. By selecting the value 0.1=kσ the k-ℓ model predicts this profile similar to the k-ε prediction, especially toward the centerline. This case was computed using Wilcox’s1D flow solver.7


5

Fig. 3 Channel flow length- and time-scale profiles

Fig. 4 Channel flow velocity profile

Fig. 5 Channel flow turbulence kinetic energy profile

Flat plate flow This basic flow serves to calibrate the remaining model constants,

lσ and α, based on matching skin friction and velocity profile correlations. Figs. 6 and 7 show skin friction and velocity profiles, respectively. Note in Fig. 6 that the k-ℓ model is calibrated to follow the Karman-Schoenherr and Bradshaw skin friction correlations8 whereas the k-ε closure follows the White-Christoph correlation.8 Fig. 8 shows ℓ and τ profiles at the same downstream station where the velocity profiles in Fig. 7 were taken. The dot product

τ∇⋅∇l is observed to change sign from positive to negative inside the boundary layer, as indicated. This is in contrast to the channel flow where the dot product remains positive across the channel. Hence in the current case (and in most external flows)

33ˆ CC ≠ .

Figure 9 shows the effect of applying33 vs.ˆ CC .

Without the 3C treatment the velocity profile exhibits

accentuated curvature toward the boundary layer edge (dash-dot line in Fig. 9) as a result of incorrect entrainment (see discussion under proposed model equations section).

Finally, the result of this test case was to adopt the values .018.0,67.0 == ασ l

Fig. 6 Flat plate skin friction


6

Fig. 7 Flat plate velocity profile

Fig. 8 Flat plate ℓ and τ profiles

Fig. 9 Flat plate. Effect of 3C on velocity profile

Hypersonic flow over a curved ramp This example concerns hypersonic flow over a curved compression surface, with experimental data by Holden.9 The compression creates an oblique shock which induces a large increase in heat transfer to the cooled wall. Some flow conditions are: M=8.03, Re=16.31 610× /ft, ∞T =90.6R,

∞TTw / =5.89. The

experimental data indicate laminar-to-turbulent flow transition approximately between x=7 and x=10 inches. The calculations were carried out on a 39,000- size grid, with at least five cells inside the viscous sublayer and ≤+y 1.0 at the first layer of grid centroids away from the wall. Figure 10 shows geometry and main flow features. Figure 11 compares results of heat transfer prediction with experimental data. The calculations were done with the following initial/free-stream conditions: free-stream turbulence level T'=0.1% and turbulence length-scale ∞l =0.2 mm, corresponding to an eddy-to-molecular viscosity ratio µµ /t =1.4. Under these conditions the k-ε model postpones laminar-to-turbulent flow transition till .31 ′′≈x The k-ℓ closure, on the other hand, predicts the transition in close agreement with the data albeit more abruptly. Fig. 12 is a convergence history plot.

Fig. 10 Curved ramp schematic view (not to scale)

Fig. 11 Curved ramp wall heat transfer

Fig. 12 Curved ramp convergence history


7

Fig. 13 Impinging jet schematic view Impinging jet This case is a low speed flow of a turbulent jet impinging on an isothermally heated plate. The jet emanates from a pipe which ends 2 pipe diameters ahead of the plate, allowing the jet to form at the pipe exit and hit the plate. The shear layer from the pipe becomes the outer boundary of this jet. As the jet impinges on the plate, the highly turbulent shear layer hits the newly developing plate boundary layer about 2 diameters away from the stagnation point, where a secondary peak in heat transfer occurs due to the local increase in turbulence intensity (see figure 13). Several sets of experimental data are available for this flow. Here we compute the case for air at

000,23Re =D and compare predictions with data by Baugh et al.10 and by Yan.11 Flow conditions are:

.0403.0 ,9.314 ,293 ,105 mDKTKTPap w ==== Figure 13 is a sketch showing topology and main flow features. To predict this flow a 50 diameter long adiabatic wall pipe was used to insure that the flow becomes fully developed before the pipe end is reached. Tests show that the flow is fully developed when it reaches about 30 diameters from the inlet. The far-field boundaries allowed inflow or outflow and imposed the specified ambient pressure. A 60,000-size grid was employed for the calculations. The near-wall mesh was fine enough ( +y < 1) and sufficiently dense to integrate the equations to walls. Fig. 14 shows wall heat transfer predictions in Nusselt Number form. All computations were performed with 1% turbulence intensity at the pipe entrance and a turbulence length-scale of 1.6 cm. Once the flow in the pipe becomes fully developed, varying the inflow length-scale affects only the transition location near the pipe entrance; there is no effect on the plate heat transfer characteristics (however, if the pipe is not long enough to insure fully developed flow, heat transfer results exhibit significant sensitivity to inflow turbulence levels). The k-ℓ model is seen to capture the heat transfer distribution very well, including the secondary peak which is turbulence-induced due to the turbulence energy from the pipe boundary layer carried by the

jet. The k-ε closure fails to predict this behavior and performs unsatisfactorily. The superior performance of the k-ℓ model is due to the

µC~ function. The figure includes prediction by the SST model12 which performs here better than the k-ε closure but not as well as the k-ℓ model. Figure 15 is a typical convergence history plot.

Fig. 14 Impinging jet Nusselt number profiles

Fig. 15 Impinging jet residual history

Hypersonic flow over a flat plate Mach 10 flow calculations over a flat plate, with both adiabatic and cooled walls ( 3/0TTw = ), were carried out on a 122×85 grid with 1≤+y at the first off-wall centroids. As in the low speed calculation, the plate's


8

leading-edge was preceded by a section of freestream flow parallel to it. A freestream turbulence level, corresponding to µµ /t =50, was imposed at the inflow. The White-Christoph skin friction correlation8 as well as the one by van Driest,8 both corrected for compressibility effects, were used to compare with the computational results. Fig. 16 shows that both the k-ℓ and k-ε closures predict leading edge laminar-to-turbulent flow transition but its extent is larger with the latter model. Both models are in close agreement with the correlations data. Fig. 17 is a typical residual history plot.

Fig. 16 Hypersonic flow over plate. Skin friction profiles.

Fig. 17 Hypersonic flow over plate. Residual plot Transonic flow over an axisymmetric bump

An M=0.875 flow over an axisymmetric bump at

m/106.13Re 6×=∞ was computed. Experimental data are by Bachalo and Johnson.13 A normal shock, impinging on the bump, causes flow detachment from the bump surface with subsequent reattachment onto the cylindrical surface. Fig. 18 provides a schematic view with main flow features and some nomenclature. The calculations were performed on a 151×81 mesh with 1≤+y at the first off-wall centroids. Streamwise clustering was imposed, centered at x/c=0.7 where the shock impinges on the wall

according to the experiment. A computation on a twice larger mesh was used to confirm grid-independence of the reported results. Fig. 19 shows wall pressure profiles and Fig. 20 velocity profiles within the separation bubble and downstream of it. The k-ℓ model predicts the shock location slightly further downstream than the k-ε closure does whereas the velocity profiles are better predicted by the k-ℓ model. A calculation with the SST closure12 reveals best pressure prediction by this model albeit the shock is shifted slightly too upstream. A sluggish post-reattachment boundary layer recovery is predicted by all closures, and here SST produces the worst performance due to separation bubble overprediction. Fig. 21 is a typical residual convergence plot.

Fig. 18 Transonic bump flow. Schematic view.

Fig. 19 Tranrsonic bump flow. Wall pressure profiles.


9

Fig. 20 Transonic bump flow. Velocity profiles.

Fig. 21 Transonic bump flow. Residual history.

Onera M6 wing As a three-dimensional example, flow over the ONERA M6 wing14 was computed under the following conditions: M ∞ = 0.8395, T ∞ = 255.6 K,

p ∞ = 45.829 psia, Re c = 11.72 610× , based on the mean aerodynamic cord, and angle-of-attack α=3.06 o . Free-stream turbulence level was set to 2% and the corresponding length-scale was assumed to be 1 mm. The relatively coarse mesh consisted of 96,000 hexahedral elements with y ≅+ 100. The wing surface was treated as adiabatic and a wall function was invoked on it. Symmetry conditions were imposed on the base plate. Figure 22 is a composite pressure coefficient contour plot on the wing and the

44% section, showing the λ-shock footprint on the wing's suction side. Figure 23 compares calculated

pC distribution around the wing, at the 44% section,

with experimental data.14 The k-ℓ and k-ε model predictions are almost identical and both predict the suction peak and shock location well. Fig. 24 is a residual history plot.

Fig. 22 Onera M6 pressure contours

Fig. 23 Onera M6 wing pressure profiles at 44% location

Fig. 24 Onera M6 residual plot

Asymmetric diffuser Obi et al15 obtained experimental data for an asymmetric plane diffuser at a Reynolds number of 21,200 based on inflow centerline velocity and


10

channel height, h. Figure 25 shows the overall geometry including typical velocity contours. The diffuser is 21h long and the expansion angle of 10 degrees gives an overall expansion ratio of 4.7. Corners are rounded at both ends of the diffuser with an arc radius of 4.3h.

Fig. 25 Asymmetric diffuser topology and velocity contours

Velocity and normal stress profiles at the inlet are provided, (

CLU =15.9 m/s) and the experimental data include velocity components and Reynolds stresses at several streamwise locations. The computations were done on a 45,700 cell mesh with y ≅+ 0.1 on both walls and adequate grid density to capture the shear layers. The computational domain extended from x/h=-11 to x/h=60. The inflow kinetic energy was based on the approximation 2uk ≈ for fully developed channel flow. The turbulence length-scale was chosen as ¼h. Atmospheric conditions were used, with a constant temperature of 293K. In addition to the k-ε and k-ℓ models, the SST closure12 was also employed. Figure 26 shows the location and extent of the separation bubble on the diffuser. Note that the separation and reattachment points do not coincide with any geometrical features.

Fig. 26 Asymmetric diffuser separation bubble

While all three models predicted about the same bubble, differences in the details are seen in figures 27 and 28, showing velocity profiles at the approximate locations of incipient separation and reattachment.

Fig. 27 Asymmetric diffuser velocity profile at incipient separation point

Fig. 28 Asymmetric diffuser velocity profile at approximate reattachment point The k-ε model predicts the separation profile accurately whereas the SST closure exhibits early separation and poor agreement with the data. The k-ℓ model’s prediction is almost as good as that of k-ε. On the other hand the latter model predicts reattachment too early while the SST and k-ℓ closures capture it accurately. However, the k-ℓ model yields


11

the best reattachment profile overall. Figure 29 shows a typical convergence history.

Fig. 29 Asymmetric diffuser convergence history

CONCLUSIONS This paper takes a fresh look at the k-ℓ turbulence model formulation. Rather than derive the model from the k-ε closure equations or from the approach previously taken by Smith,1 the current model is based on an assumed destruction term. Some model constants are derived from unit flow problems whereas the remaining ones are determined from further scrutiny of the closure using a variety of flow cases. One of the main attributes of this closure is its novel variable

µC formulation. The resulting k-ℓ model is simple and it uses homogeneous wall boundary conditions. It proved superior to the k-ε closure in predicting some flows involving heat transfer and otherwise behaved similar to the k-ε model in wall-bounded flow cases.

REFERENCES [1] B. R. Smith, “A near wall model for the k-ℓ two equation turbulence model,” AIAA Paper 94-2386, 25th AIAA Fluid Dynamics Conference, Colorado Springs, June 20-23, 1994. [2] T. J. Craft, B. E. Launder and K. Suga, “Development and application of a cubic eddy-viscosity model of turbulence,” Int. J. Heat & Fluid Flow, Vol. 17, pp. 108-115, 1996. [3] S. Zeierman and M. Wolfshtein, “Turbulent time scale for turbulent-flow calculations,” AIAA J., Vol. 24, pp. 1606-1610, October 1986. [4] U. Goldberg, O. Peroomian and S. Chakravarthy, “A wall-distance-free k-ε model with enhanced near-wall treatment,” ASME J. Fluids Eng., Vol. 120, pp. 457-462, Sept. 1998. [5] O. Peroomian, S. Chakravarthy, S. Palaniswamy and U. Goldberg, “Convergence acceleration for unified-grid formulation using preconditioned implicit relaxation”, AIAA Paper 98-0116, Reno, NV, Jan. 1998.

[6] P. Batten, M. A. Leschziner and U. C. Goldberg, “Average-state Jacobians and implicit methods for compressible viscous and turbulent flows,” J. Computational Physics, Vol. 137, pp. 38-78, 1997. [7] D. C. Wilcox, Turbulence Modeling for CFD, 2nd Ed., DCW Industries, 1998. [8] F. M. White, Viscous Fluid Flow, 1st Ed., McGraw-Hill Book Company, pp. 498-499, 1974. [9] M. S. Holden, “Turbulent boundary layer development on curved compression surfaces," Calspan Report No. 7724-1, 1992. [10] J. Baughn, A. Hechanova and X. Yan, “An experimental study of entrainment effects on the heat transfer from a flat plate surface to a heated circular impinging jet,” ASME J. Heat Transfer, Vol. 113, pp. 1023-1025, 1991. [11] X.Yan, J. W. Baughn and M. Mesbah, “The effect of Reynolds number on the heat transfer distribution from a flat plate to an impinging jet,” ASME Heat Transfer Division, Vol. 226, pp. 1-7, 1992. [12] F. R. Menter, “Two-equation eddy-viscosity turbulence models for engineering applications,” AIAA Journal, Vol. 32, No. 8, pp. 1598-1605, August 1994. [13] W. D. Bachalo and D. A. Johnson, “An investigation of transonic turbulent boundary layer separation generated on an axisymmetric flow model,” AIAA Paper No. 79-1479, 1979. [14] V. Schmitt and F. Charpin, “Pressure distribution in the ONERA M6 wing at transonic Mach numbers,” AGARD AR138: Experimental Database for Computer Program Assessment, 1970. [15] S. Obi, K. Aoki, and S. Masuda, “Experimental and computational study of turbulent separating flow in an asymmetric plane diffuser,” 9th Symposium on Turbulent Shear Flows, Kyoto, Japan, 1993.

[American Institute of Aeronautics and Astronautics 35th AIAA Fluid Dynamics Conference and Exhibit...

Documents

Transcript of [American Institute of Aeronautics and Astronautics 35th AIAA Fluid Dynamics Conference and Exhibit...