NASNTM-2003-2 1243 1 

Summary of EASM Turbulence Models in 

CFL3D With Validation Test Cases 

Christopher L. Rumsey and Thomas B. Gatski 

Langley Research Center, Hampton, Virginia 

June 2003 L 
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NASA/TM-2003-2 1243 1 

Summary of EASM Turbulence Models in 

CFL3D With Validation Test Cases 

Christopher L. Rumsey and Thomas B. Gatski 

Langley Research Center, Hampton, Virginia 

National Aeronautics and 

Space Administration 

Langley Research Center 

Hampton, Virginia 2368 1-2 199 

June 2003 
Contents 

ABSTRACT iv 

1 INTRODUCTION 1 

2 OVERVIEW OF CFL3D 2 

3 THE EASM EXPLICIT ALGEBRAIC STRESS MODELS 4 

4 VALIDATION TEST CASES 7 

4.2 BackStep ........................................ 10 

4.3 Transonic Diffuser ................................... 13 

4.4 NACA 4412 Airfoil ................................... 19 

4.5 EjectorNozzle ..................................... 21 

4.6 Axisymmetric Bump .................................. 22 

4.7 ONERA M6 Wing ................................... 27 

4.8 2-D Airfoil Wake .................................... 30 

4.9 Curved Duct in Zero Pressure Gradient ....................... 31 

4.1 Flat Plate ........................................ 7 

5 CONCLUSIONS 36 

iii 
ABSTRACT 

This paper summarizes the Explicit Algebraic Stress Model in K-w form (EASM-ko) and in 

K-E form (EASM-ke) in the Reynolds-averaged Navier-Stokes code CFLSD. These models have 

been actively used over the last several years in CFL3D, and have undergone some minor 

modifications during that time. Details of the equations and method for coding the latest 

versions of the models are given, and numerous validation cases are presented. This paper 

serves as a validation archive for these models. 

iv 
1 INTRODUCTION 

A turbulence model must be employed in conjunction with the Reynolds-averaged NavierStokes (RANS) equations in order to close the equations. Over the last 30 years, many different 

turbulence models have been developed for use with RANS. Because there are so many to choose 

from, it is sometimes difficult for a CFD user to decide which model to use for a given flow 

situation. 

It is therefore very important that new (as well as existing) models be thoroughly validated 

in all the CFD codes into which they are implemented. This validation process should consist 

of applications to a suite of as many test cases as possible, which, taken together, thoroughly 

exercise the capabilities of the turbulence model. 

Most turbulence models for use with RANS today are linear eddy viscosity models, which 

assume a Boussinesq relationship between the turbulent stresses and mean strain rate tensor 

through the use of an isotropic eddy viscosity. Recently, however, nonlinear eddy viscosity 

models have been gaining widespread attention. This class of models assumes a higher-order 

tensor representation involving either powers of the mean velocity gradient or combinations of 

the mean strain rate and rotation rate tensors. 

One of the advantages of nonlinear eddy viscosity models over linear eddy viscosity models 

is that nonlinear models can predict differences in the turbulent normal stresses. Although this 

deficiency in linear models is not generally considered important for most external aerodynamic 

problems of interest, it has been shown to be crucial for capturing secondary motion in a corner 

flow, which is driven by the gradient of the turbulent normal stresses (see Abdol-Hamid et 

al. [ 11). 

Explicit algebraic stress models (EASM) belong to the class of nonlinear eddy viscosity 

models. However, unlike some nonlinear models which determine their expansion coefficients 

through calibration with experimental or numerical data, EASMs obtain their expansion coefficients through their rigorous relationship with their “parent)’ full differential Reynolds stress 

equations. See Gatski and Rumsey [2] for details. 

The EASM models, when originally conceived [3], were solved by assuming a fixed (equilibrium) value of P/e when determining the parameter g = (C1/2 + P/e - 1)-’. This “fixed-g” 

EASM model was subsequently improved [4, 5, 61 by allowing P/e to vary. It is only this 

latter method, sometimes referred to as “variable-g” EASM, that is discussed in this report. 

The “fixed-g” EASM method has also been coded into CFL3D, but it is generally no longer 

recommended and will not be discussed. Other applications of EASM (using variable-g) can be 

found in Carlson [7] and Carlson et a1.[8]. 

The purpose of this paper is primarily as a validation archive for the EASM-ko and EASM-ke 

models as coded in CFL3D. Thus, engineers who attempt to code these turbulence models for 

themselves in their own CFD codes can use this paper as a reference for the results they should 

expect to get with these models. This paper is not intended to advocate EASM, or to point 

out cases for which EASM yields improved (or worse) results than more conventional models. 

Therefore, no comparisons are presented between EASM and other models. That exercise is 

left to the reader. Here, the two versions of EASM-ko and EASM-ke are compared only with 

each other (and with experiment or theory). 

The next two sections describe the formulas for the EASM-ko and EASM-ke models, including details regarding the numerical method employed to solve them in CFL3D. Following that, 

1 
the models are applied to several test cases. Then, conclusions are drawn in the last section. 

2 OVERVIEW OF CFLSD 

The computer code CFLSD [9] solves the three-dimensional, time-dependent, Reynolds averaged 

compressible Navier-Stokes equations with an upwind finite-volume formulation (it can also be 

exercised in two-dimensional mode of operation for 2-D cases). It can solve flows over multiplezone grids that are connected in a one-to-one, patched, or overset manner, and can employ 

grid sequencing, multigrid, and local time stepping when accelerating convergence to steady 

state. Upwind-biased spatial differencing is used for the inviscid terms, and flux limiting is used 

to obtain smooth solutions in the vicinity of shock waves, when present. Viscous terms are 

centrally differenced, and cross-diffusion terms are neglected. For very low Mach number flows, 

preconditioning [lo] is used to insure convergence and accuracy of the solutions. 

The CFL3D code is advanced in time with an implicit approximate factorization method. 

The implicit derivatives are written as spatially first-order accurate, which results in block 

tridiagonal inversions for each sweep. However, for solutions that utilize Roe flux-difference 

splitting [ll], the block tridiagonal inversions are further simplified using a diagonal algorithm 

with a spectral radius scaling of the viscous terms. All solutions in this paper use Roe. 

Following Wilcox [12], Reynolds averaging can be used with the Navier-Stokes equations in 

Favre variables [13] to account for turbulent fluctuations. The resulting equations of motion 

can be written using the summation convention as follows. The full Navier-Stokes equations are 

shown here, but in CFL3D they are solved using the thin-layer approximation in pre-selected 

coordinate direction(s). The equations are given here in terms of dimensional quantities. Although not shown, the equations are subsequently nondimensionalized and solved in generalized 

coordinates (see Krist et a1.[9]). 

where 

1 

p = (y - 1) [pE - y(u2 + w2 + w2) - PK] 

1 

2 

H=E+p/p 

1 

E = e + -(u2+ w2 + w2) + K 

2 YP a =-. 

P 

(4) 

2 
where UK is defined in the next section. Define: 

The stress term r;j is composed of a laminar and a turbulent component as 

For linear eddy-viscosity models using the Boussinesq eddy viscosity assumption: 

where pz is the turbulent eddy viscosity computed by a linear turbulence model. For 

nonlinear eddy-viscosity models EASM-ko and EASM-ke: 

14) 

the 

where the terms u2, u3, and a4 are defined in the next section. 

In CFLSD, the kinetic energy of the fluctuating turbulent field K is ignored in the definition 

of p and E in equations (4) and (5). Furthermore, for all linear eddy-viscosity models, the term 

involving K is ignored in the definition of rz in equation (14)) and the $j term in the energy 

equation (3) is assumed to be zero. These approximations and assumptions have relatively little 

effect for most flows at low or transonic Mach numbers, but could have significant impact for 

hypersonic flows. From a coding perspective, for linear eddy-viscosity models, the end result is 

that the laminar Navier-Stokes equations are identical to the turbulent Navier-Stokes equations 

with the exception that p is replaced by p + p; and p/Pr is replaced by p/Pr + p;/Prt (in these 

equations Pr is taken as 0.72 and Prt is taken as 0.9). 

For the nonlinear models, the term involving K is included in the definition of rz in equation 

(15), and the $j term in the energy equation (3) is modeled via equation (9). However, K is 

still ignored in the definition of p and E in equations (4) and (5). Therefore, like the linear 

turbulence models, strictly speaking the nonlinear turbulence models are not applicable for 

3 
hypersonic flows either, as coded in CFL3D. From a coding perspective, the nonlinear models 

are similar to the linear models except that additional terms beside & and &/Prt must be 

added to the right-hand sides of the momentum and energy equations (2) and (3) when going 

from laminar to turbulent Navier-Stokes. 

All the turbulence models in CFL3D, including EASM-ko and EASM-ke, are solved uncoupled from the mean flow equations using implicit approximate factorization. Their advective 

terms are solved using first-order upwind differencing. Details for obtaining the pt term for the 

EASM-ko and EASM-ke models are given in the next section. 

3 THE EASM EXPLICIT ALGEBRAIC STRESS MODELS 

For background and summary on the derivation of EASM, see Gatski and Speziale [3], Jongen 

and Gatski [14], Rumsey et a1.[6], and Rumsey and Gatski [15]. The following section bypasses 

the derivation and jumps directly to the final resulting equations for the EASM-ko and EASM-ke 

models, and also outlines the method of solution in CFL3D. 

The kinematic eddy viscosity & is given by 

U: = p:/p = CZKT = -Kai, (16) 

with r K/E for EASM-ke. Thus, cq/r is equivalent to -CG, 

which is variable (as opposed to constant as in most linear two-equation models). The nominal 

level for C; in a zero-pressure-gradient log layer is approximately 0.09. The value of al/r is 

obtained from the solution to the following cubic equation at each point in the flow field: 

1/0 for EASM-ko and r 

where 

( 7)3 + (s) + (s) + = 0, 

The correct root to choose from this equation is the root with the lowest real part [16]. 

Also, the degenerate case when q2 + 0 must be avoided. See Rumsey and Gatski [15] for 

further details. In the current implementation, the resulting C; = -(al/r) is limited by 

C; = max(-(al/r), 0.0005). Other parameters are given by 

4 
a1 = 1 (! - c2) 23 

1 

a2 = - (2 - C4) 

2. 

1 

a3 = - (2 - C3) 2 

a4 = [ 71 * - 270 * (:) q2r2] -l r. 

Also, 

y; = c;/2 

and C,l = 1.44, CE2 = 1.83, Cy = 3.4, C; = 1.8, C2 = 0.36, C3 = 1.25, and C4 = 0.4. 

The preceding implementation is exactly the same for EASM-ko or EASM-ke. The two 

methods differ in the two equations solved (K-w vs. K-E). For EASM-ko, the explicit tensor 

representation for qj is coupled with the following K-w two-equation model: 

-=P- DK fpKw+-[(u+z)z] a 

Dt axk 

-=r-p-pw Dw w 28 +- 

Dt K dxk 

where 

and UK = 1, u, = tc2/[G(/3 - y)], K = 0.41, y = 0.53, /3 = 0.83, and C, = 0.0895. Note that 

for 2-D incompressible flows, P = 2uTq2 is exact. In CFLSD, the user has the option of using 

the exact or the approximate P term. We have found there to be very little difference for a 

wide variety of subsonic and transonic 2-D aerodynamic-type flows, and use of the approximate 

term is more robust during transient stages of the computation in some cases. Except where 

otherwise noted, all the results in this paper were obtained using the approximate term. 

Also, it should be noted that the values of UK and y in this model are different than reported 

in Rumsey and Gatski [15]. They were changed recently to improve the model’s capability for 

jet-type flows (see Georgiadis et al.[17]). The change was found to have relatively small impact 

for wall-bounded flows in general. In the current implementation, P in the K-equation is limited 

to be less than 20 times the destruction term fp* Kw. The function fp*, taken from from Wilcox 

[12], is given by 

where the C; term in the formula for Xk is necessary because w in the current model does not 

“absorb” C, as in Wilcox’s model. 

5 
For the EASM-ke two-equation model: 

where P is given by equation (32) and is limited in the IC-equation to be less than 20 times 

the destruction term E. Also, f, = [l - exp(-Re~/10.8)], ReK = K'/2d/v, OK = 1.0, u, = 

K~/[G(C,~ - C,1)], C,l = 1.44, CE2 = 1.83, C, = 0.0885, and d is the distance to the nearest 

wall. Additional wall damping functions (such as f,, to achieve expected asymptotic behavior of 

the turbulence quantities very near the wall) are not employed in the current EASM-ke model. 

Note that in equations (36) and (37), the diffusion terms are modeled using an equilibrium eddy 

viscosity ut = C,K2/&, where the constant C, = 0.0885 for this model. This is different than in 

the EASM-ko model, which uses the actual eddy viscosity ut* (with variable C;1) in its modeled 

diffusion terms. The diffusion terms for both EASM-ko and EASM-ke are approximate models 

in any case; see, for example, Warsi [HI. 

In CFL3D, the turbulence equations are nondimensionalized by the same reference variables 

as the Navier-Stokes equations: aref, pref, pref, and Lref, where a is speed of sound, p is density, p 

is molecular viscosity, and L is length scale corresponding to unit one of the computational grid. 

Thus, denoting nondimensional variables with a tilde: I? = K/(a:ef), 6 = (prefw)/(Prefa:ef), 

and 2 = (PrefE) / (prefa:er) - 

The turbulent boundary conditions applied at solid walls are K, = 0, E, = 2u,,,(dfi/dn)i, 

and w, = 10(6~,)/[p(An)~], where An is the distance to the first cell center away from the 

wall. The boundary condition for w, is from Menter [19]. This boundary condition simulates 

the analytical behavior of w near solid walls without the need for specifying the solution at 

interior points. At farfield inflow boundaries, the following nondimensional values are assigned 

as freestream reference values in CFL3D: I? = 9 x lo-' and 6 = 9 x for EASM-ko; 

and I? = 1 x lo-' and E = 1 x for EASM-ke. The end result for both models is that 

nondimensional jit = /it//iref M 0.009 at farfield inflow boundaries (based on C; = 0.09, which is 

only approximate because C;1 is variable in these models). At outflow boundaries, the turbulent 

quantities are extrapolated from the interior of the computational domain. 

Some additional details concerning the method for solving the two-equation turbulence 

models are now given. First, the following is an example showing the transformation from 

Cartesian coordinates to generalized coordinates: 

where 

6 
and [, q, and < are the three generalized coordinate directions. When transforming into generalized coordinates, cross-derivative terms are neglected in the diffusion-type terms that appear 

in the turbulence equations. 

The two turbulence equations are solved decoupled using implicit approximate factorization. 

Each sweep requires the solution of a scalar tridiagonal matrix. The production terms are 

treated explicitly, lagged in time, while the destruction and diffusion terms are treated implicitly. 

It should be noted here that in CFLSD, the K, w, and E terms are not allowed to go negative 

during the solution procedure. If the update yields a negative result, it is instead limited to 

a very small positive number, and a counter keeps track of the number of times this occurs 

in the flowfield as an indicator of non-convergence in the solution. Furthermore, q1, 722, and 

r33 are limited to be positive (this is the realizability constraint). In EASM-ke, q2r2 and 

R2q2r2 are each limited to be less than 1200 in magnitude. In both EASM-ko and EASM-ke, 

where w or E appears in the denominator of equation (15) (i.e., in the nonlinear terms added 

to the Navier-Stokes equations), these values are limited to be greater than or equal to their 

freestream reference levels in order to avoid dividing by unrealistically small numbers during 

the convergence process. 

4 VALIDATION TEST CASES 

This section contains a detailed account of a series of validation test cases to which the EASMko and EASM-ke models have been applied. The first 7 cases are also “standard” CFL3D test 

cases, described and available on the CFL3D website [20]. (In some instances, the case on the 

website has been set up for a different turbulence model other than EASM-ko or EASM-ke, 

and some adjustments to CFL number are necessary in the provided input file in order to get 

the cases to run.) The last 2 cases are not currently available on the website. For the first case 

(flat plate), grid sensitivity studies are described in this report. All of the subsequent cases use 

only a single grid size, with no parametric variations performed. However, many of the grids 

used in these validations were found to be fine enough to capture the flow features of interest 

in previous grid sensitivity studies performed by the authors or by other researchers. In each 

case, the boundary conditions are described in detail; unless otherwise noted, at all solid walls 

a no-slip, adiabatic wall boundary condition is used. At these walls, the turbulence boundary 

conditions are those described in section 3. For the mean flow variables, velocity components 

at the wall are set to zero, &”,/dn = 0, and p, is determined using linear extrapolation from 

the interior of the domain. Note that the EASM-ke model sometimes has difficulty establishing 

turbulence when started from freestream or weak turbulence initial conditions. Therefore, for 

all the results in this paper, EASM-ke was initialized from the corresponding EASM-ko solution. 

4.1 Flat Plate 

The flat plate case (zero pressure gradient) was performed at A4 = 0.3, Re = 6 x lo6 per unit 

length of the grid. The default grid size for this case was 65 x 97. Fig. 1 shows a picture of 

the grid. The viscous, adiabatic wall was unit one in length. Symmetry boundary conditions 

were imposed on a region of length x = 0.333 in front of the plate. Grid domain height 

was approximately 1.0, minimum normal grid spacing at the wall was 1 x and the grid 

was stretched at a rate of 1.18 until the vertical spacing first exceeded the horizontal spacing. 

Horizontal grid spacing was constant at Ax = 0.0208333. 

7 
At the inflow boundary, the total pressure and total temperature were set via pT/pref = 

1.02828 and TT/T~~~ = 1.0080, and a 1-D characteristic boundary condition was used to determine the other flow variables [9]. At the outflow boundary, linear extrapolation was employed. 

At the top boundary, a farfield Riemann-invariant 1-D characteristic boundary condition was 

employed. 

1'- 

0.9 

8 O.* 

C 0.7 

10.6 

N 0.5 

0.4 

0.3 

0.2 

0.1 

? 

1.21.1 

I- 

: 

- 

I- 

+ 

r 

5 

F 

005 

plate length 

x-distance 

Figure 1: Flat plate grid, 65 x 97 

Surface skin friction coefficient results using EASM-ko are shown in Fig. 2 for a grid density 

study. A total of four grids were used: each had 65 points in the streamwise direction, and the 

number of points in the normal direction was successively halved from 193 to 97 to 49 to 25. 

Thus, the default grid size (65 x 97) was included as well as one level finer and two levels coarser. 

Also shown in the figure are symbols giving three different theoretical curves from White [21], 

indicative of the type of spread that might be expected from experimental data of a turbulent 

flat plate boundary layer. As the grid is refined, the results approach the range defined by the 

theoretical curves. Looking at one particular 2-location on the plate (2 = 0.8125), and using 

second-order Richardson extrapolation on the finest two grids, one finds that the 65 x 193 grid 

yields a skin friction value 0.5% in error from the result on an infinitely-refined grid at this 

location. The default 65 x 97 grid is 2.0% in error, the 65 x 49 grid is 4.7% in error, and the 

65 x 25 grid is 10.6% in error. 

The same grid study was also conducted for EASM-ke. Results are shown in Fig. 3. Unlike 

EASM-ko, with EASM-ke the skin friction decreases with increasing grid density. In this case 

the 65 x 193 grid yields a skin friction value 0.1% in error from the result on an infinitely-refined 

grid at z = 0.8125. The default 65 x 97 grid is 0.6% in error, the 65 x 49 grid is 2.9% in error, 

and the 65 x 25 grid is 9.0% in error. Notice that the infinitely-refined skin friction levels 

approached by EASM-ke are slightly lower than those using EASM-ko. 

8 
Figure 2: Effect of grid size normal to the wall on surface skin friction, EASM-ko. 

X 

Figure 3: Effect of grid size normal to the wall on surface skin friction, EASM-ke. 

9 
Figs. 4 and 5 show the effect of varying the minimum normal spacing at the wall using 

EASM-ko and EASM-ke, while keeping grid size constant (at 65x97). The “standard” minimum 

spacing of 1 x corresponds with an average y+ value for the first cell-center off the wall 

of approximately 0.2. The minimum normal spacing was decreased to 1 x (average y+ of 

0.02) as well as increased to 2.2 x (average y+ of 0.5), 5 x (average y+ of 1.1), and 

1 x (average y+ of 2.2). In each case the stretching rate was varied to keep the grid height 

approximately 1.0. As the average y+ level decreases, the skin friction increases for EASM-ko 

and decreases for EASM-ke. Using the result at 2 = 0.8125, the difference between the smallest 

y+ and the largest y+ tested is approximately 5% for EASM-ko and 2.6% for EASM-ke. The 

difference between using y+ of 0.02 and 0.2 is only 0.3% for EASM-ko and less than 0.1% for 

EASM-ke. 

A plot of u+ vs. log(y+) is shown for the two models in Fig. 6 using the 65 x 97 grid, along 

with Spalding theory from White [all. Results agree well with the law-of-the-wall. 

Figure 4: Effect of average minimum y+ on surface skin friction, EASM-ko, 65 x 97 grid. 

4.2 Back Step 

The back step case compares against experimental data from Driver and Seegmiller [22]. In the 

experiment, several different upper wall angles were tested, but in the present computation, the 

upper wall was straight (0 deg). The test conditions were M = 0.128, Re = 37573 per unit step 

height H. In the grid for this configuration, the lower wall starts out at a height of 1.0, then 

drops to a height of 0.0 at x/H = 0. The upper wall is at a height of 9.0. The grid extends 

from s/H = -4.0 upstream to x/H = 35 downstream. 

10 
Figure 5: Effect of average minimum y+ on surface skin friction, EASM-ke, 65 x 97 grid. 

Figure 6: Velocity profiles for both models in wall variables at z = 0.8, 65 x 97 grid, y+=0.2. 

11 
The grid itself is made up of two zones: 25 x 65 and 129 x 113. These two zones abut along 

2 = 0 in a patched-grid (non 1-tel) fashion. Interpolation is used in CFL3D at the patched 

interface to provide communication between the zones. The minimum normal spacing at the 

walls varies within the range of approximately 0.006-0.007, which yields an average y+ value at 

the first cell center off the walls of between 0.6 and 0.9. The only exception to this is the back 

face of the step itself, which has a normal spacing of 0.1. The streamwise grid spacing varies. 

It is clustered near the step, and stretched as it approaches the upstream and downstream 

boundaries. Two pictures of the grid are given in Figs. 7 and 8. 

The lower wall, upper wall, and back of the step used viscous, adiabatic boundary conditions. 

At the upstream boundary, the density, velocity, and turbulence quantities were set to closely 

match experiment (see Rumsey et a1.[6] for details concerning this procedure). The pressure 

at the inflow was extrapolated from the interior of the domain. At the downstream boundary, 

the pressure was set to p/p,,f = 1.00149 and all other quantities were extrapolated from the 

interior of the domain. 

20 

: zone1 

patched interface 15 

zone 2 

I 

Figure 7: Backstep grid, 25 x 65 and 129 x 113. 

Figure 8: Close-up of the backstep grid near the interface between zones. 

12 
Lower wall pressure coefficient and skin friction coefficient for both EASM-ko and EASM-ke 

are given in Figs. 9 and 10. A series of velocity profiles is given in Figs. 11, 12, and 13, and 

turbulent shear stress is given in Figs. 14, 15, and 16. 

Reattachment length in the experiment was approximately in the range of z = 6 - 6.3. 

Using EASM-ko, the computed reattachment length was z = 6.30, whereas for EASM-ke it was 

z = 5.86. Both of these are in very good agreement with the experiment, considering that many 

turbulence models have been shown to seriously underpredict reattachment length by 30% or 

more [22]. The EASM-ko skin friction results agree extremely well with experiment in general, 

but EASM-ke underpredicts the levels in the separated region and overpredicts the levels after 

reattachment. This poorer prediction of EASM-ke is consistent with skin friction results that 

others have found for this case using different IC-€ models (see, e.g., Menter [19]). Velocity 

profiles for both models are similar. Both do a reasonably good job in the separated region, 

but then do not predict as rapid a recovery after reattachment as experiment. This too-slow 

recovery from separation is a chronic problem with almost all RANS turbulence models in use 

today [23]. Turbulent shear stress is predicted well by both models up through z = 2.5, but 

then the peak shear is underpredicted (and is also farther from the wall) through approximately 

z = 5.5. Downstream of this, the levels and shapes again agree fairly well with experiment. 

0.2 

0.1 

on 

0 

-0.1 

EASM-ko 

EASM-ke 

I I 

20 30 

X 

Figure 9: Backstep surface pressure coefficients. 

4.3 Transonic Diffuser 

This case models the strong-shock diffuser experiment of Sajben and Kroutil [24]. Fig. 17 shows 

the 81 x 51 grid, which was obtained from the NPARC Alliance Validation Archive website [25]. 

13 
X 

Figure 10: Backstep surface skin friction coefficients. 

x=o x=l .o XZl.5 

3 

2.5 

2 

3.5 

1 

0.5 

0 5 

x=2.0 

3 

2.5 

2 

6.5 * 

I 

0.5 

0 5 

~~2.5 

3 

2.5 

2 

b .5 * 

1 

0.5 

0 0.05 0.1 0.15 

“‘am1 

3 

2.5 

2 

6.5 * 

1 

0.5 

0 0.05 0.1 0.15 

U‘amt 

Figure 11: Backstep velocity profiles, set 1. 

14 
~~4.0 

0 0.05 0.1 0.15 

ula, 

~=6.0 

3 

2.5 

2 

3.5 

1 

0.5 

0 0.05 0.1 0.15 

want 

~=5.0 

3 

2.5 

2 

5.5 

1 

0.5 

0 0.05 0.1 0.15 

waml 

x=5.5 

3 

2.5 

2 

6 5 h' 

1 

0.5 

0 0.05 0.1 0.15 

want 

~~7.0 

3 

2.5 

2 

b .5 * 

1 

0.5 

0 0.05 0.1 0.15 

Uaml 

Figure 12: Backstep velocity profiles, set 2. 

~~8.0 

0 0.05 0.1 0.15 

waml 

3 

2.5 

2 

F15 h' 

1 

0.5 

0 0.05 0.1 0.15 

want 

3 x=lO.O 

2.5 

2 

b 5 *' 

1 

0.5 

0 0.05 0.1 0.15 

waml 

~~16.0 

3 

2.5 

2 

6 5 h' 

1 

0.5 

0 0.05 0.1 0.15 

waml 

XZ12.0 3 

2.5 

2 

5.5 

1 

0.5 

0 0.05 0.1 0.15 wml 

~~32.0 

3 

2.5 

2 

%I .5 

1 

0.5 

0 0.05 0.1 0.15 

~aml 

Figure 13: Backstep velocity profiles, set 3. 

15 
x=o 

-0.01 -0.005 0 0.005 

u’v’lu,2 

x=l .o ~~1.5 

3 

2.5 

2 

3.5 

1 

0.5 

-0.01 -0.005, 0 0.005 

u’v’lu, 

~=2.5 ~~3.0 

3 3 

2.5 2.5 2.5 

2 2 2 

xz2.0 

3 

6 5 

1 1 1 

*. h .5 6.5 * * 

0.5 0.5 0.5 

05 05 0 -0.01 -0.005 0 0.005 

u’w’lu,2 

Figure 14: Backstep turbulent shear stress profiles, set 1. 

~~4.0 xS.0 XZ5.5 

5 

~~6.5 ~=7.0 

3 3 

2.5 2.5 

2 2 

6.5 3.5 * 

1 1 

0.5 0.5 

0 -0.01 -0.005 0 0.005 05 

u’v’/u,2 

Figure 15: Backstep turbulent shear stress profiles, set 2. 

16 
~=8.0 xz10.0 xzl2.0 

5 

~~14.0 

3 

2.5 

2 

3.5 

1 

0.5 

-0.01 -0.005, 0 0.005 

u'v'/u, 

~~16.0 3 

2.5 

2 

?1.5 % 

1 

0.5 

-0.01 -0.005 0 0.005 

U'V'IU,' 

~~32.0 

3 

2.5 

2 

?l 5 %. 

1 

0.5 

-0.01 -0,005 0 0.005 

U'V'IU,? 

Figure 16: Backstep turbulent shear stress profiles, set 3. 

This configuration had an entrance to throat area ratio of 1.4 and an exit to throat area ratio 

of 1.5. This flow is characterized by the ratio, R, of exit static to inflow total pressure. For 

the strong-shock case R is 0.72. In the strong shock case, there is separated flow on the upper 

wall downstream of the shock wave [25]. The grid had a minimum spacing at the walls of 

approximately 0.0001, yielding an average minimum y+ value at the first cell center off the wall 

of approximately 2.5. 

When computing this case in CFL3D, the lower and upper walls were modeled as viscous, 

adiabatic walls. At the inflow plane, the total pressure divided by reference pressure was 1.894 

and total temperature divided by reference temperature was 1.201. At the outflow plane, the 

static pressure divided by reference pressure was 1.364. Reference conditions (at the throat) 

were: M=1.0 and Re=937,000 per throat width. (In the grid, the throat width was unit 1.) 

Computed wall pressures are shown compared to experiment in Figs. 18 and 19 for the lower 

and upper walls, respectively. There is a slight difference in predicted shock position, with 

EASM-ko downstream of EASM-ke, but both models agree well with experiment in general. 

Velocity profiles at four x-locations are shown in Fig. 20. Both EASM-ko and EASM-ke yield 

similar results in good agreement with experiment. Both models also correctly predict separated 

flow on the upper wall at the two stations x/H=2.662 and 4.611, although in both cases the 

maximum reverse flow velocity magnitude is lower than in the experiment. 

17 
4- 

3- 

2- 

> 

Figure 17: Transonic diffuser grid, 81 x 51. 

5 x/H 

Figure 18: Transonic diffuser lower wall pressures. 

I 

0 

18 
0.9 

0.8 

0.7 

0 4 0.6 

0.5 

0.4 ___- 

t I 'bay 

Figure 19: Transonic diffuser upper wall pressures. 

4.4 NACA 4412 Airfoil 

This case models the separated flow over a NACA 4412 airfoil, and compares with the experiment of Coles and Wadcock [26]. This flow was modeled computationally using a freestream 

Mach number of M = 0.2, Reynolds number Re = 1.52 x lo6 based on chord, and angle of 

attack of o = 13.87'. This flow separates on the rear part of the airfoil upper surface at these 

conditions. Fig. 21 shows a picture of the 257 x 81 grid used. The grid had a minimum spacing 

at the wall of 2 x chords, a farfield extent of approximately 17 - 20 chords, and a total of 

177 points on the airfoil surface. The average minimum y+ value of the first cell center off the 

wall was less than 2 for this grid. 

Surface pressure coefficients are shown in Fig. 22 and upper surface velocity profiles are 

shown in Fig. 23. Both turbulence models predict similar surface pressures, which are in good 

agreement with experiment everywhere except near the upper surface trailing edge. For velocity 

profiles, EASM-ko yields results in excellent agreement with experiment, whereas EASM-ke 

predicts velocity levels that are too low (the profile has greater inflection). However, both 

models predict separation at the same location of z/c = 0.830. 

Part of the reason for the poorer prediction of EASM-ke may be due to the fact that the K-E 

formulation in general does not handle wall-bounded adverse pressure gradients well, because 

of shortcomings in the E equation. See, for example, Rumsey and Gatski [15], Wilcox [12], 

and Rodi and Scheuerer [27]. Rodi and Scheuerer concluded that the generation term of the 

E equation has to be increased above its usual level. Fig. 24 shows a plot of the predicted 

velocities in wall variables as compared to Spalding theory at z/c = 0.5 on the upper surface 

19 
xIH=2.882 xIH=4.611 

xIH=6.340 xIH=7.493 

Figure 20: Transonic diffuser velocity profiles. 

20 
of the NACA 4412 airfoil. EASM-ke did not obtain the correct slope of the log layer in this 

adverse pressure gradient part of the flow, whereas EASM-ko did. This same problem was also 

shown and discussed in Rumsey and Gatski [15] and Nagano and Tagawa [as]. 

Figure 21: NACA 4412 grid, 257 x 81. 

4.5 Ejector Nozzle 

The ejector nozzle case models a subsonic 2-D jet flow that entrains and mixes with a secondary 

outer flow. The experimental data are from Gilbert and Hill [29]. See also Georgiadis et a1.[30]. 

This case has nozzle total pressure divided by atmospheric static pressure of 2.44, with 

total temperature = 644 R. The secondary flow has a total pressure equal to atmospheric 

and a temperature of 550 R. At the outflow used by this grid, the static pressure divided 

by the atmospheric static pressure was taken to be 0.9131. However, in CFL3D, a reference 

condition was chosen that corresponded with a non-zero Mach number: in this case it was 

taken to be M=0.22, which is the approximate Mach number at the secondary flow inlet in 

this grid. (Corresponding Reynolds number was taken to be approximately 1.64 million per 

foot.) Using M=0.22 and isentropic relations, total pressure divided by reference pressure is 

1.0343 and total temperature divided by reference temperature is 1.0097. These numbers are 

used as inflow conditions for the secondary flow. Also as a result of choosing this reference 

condition, static pressure divided by reference pressure at the outflow is 0.9444. At the nozzle 

inlet, total pressure divided by reference pressure is 2.5237 and total temperature divided by 

reference temperature is 1.1822. 

The grid consists of three zones with the following dimensions: 31 x 41 for primary nozzle, 

21 
e 

0 0.25 0.5 0.75 1 

X/C 

-2b ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' 

Figure 22: Surface pressure coefficient for NACA 4412. 

31 x 71 for secondary inlet, and 101 x 121 for downstream. The grid is shown in Fig. 25. 

Minimum grid normal spacing at the walls yielded an average y+ level of approximately 0.7. 

Fig. 26 shows velocity profiles at four stations downstream using EASM-ko and EASMke. Both models overpredict the centerline velocity at the upstream stations and somewhat 

underpredict the jet width in general, but qualitatively the overall agreement is good (it is 

similar to results on the same grid using different turbulence models and a different code in 

Georgiadis et a1.[30]). 

4.6 Axisymmetric Bump 

The axisymmetric bump case has served as a standard test case for transonic separated flow for 

many years. The experimental data are from Bachalo and Johnson [31], and have been used, 

for example, to validate Menter's widely-used SST turbulence model [32]. 

In this case, a turbulent boundary layer develops in the axial direction over an axisymmetric 

circular bump (smoothed at its leading and trailing edges). Wall effects in the experiment are 

minimal because the body does not generate any lift. For the particular case studied here, the 

Mach number is 0.875 and Reynolds number based on the length of the bump is 2.66 x lo6. At 

these conditions, there is a shock wave and the flow separates on the rear part of the bump, 

then reattaches a short distance downstream. 

This case is run in "3-D mode" in CFL3D on a grid with two planes separated by 1.0 degrees, 

and with periodic boundary conditions used on each of the two circumferential planes. Fig. 27 

shows a picture of one plane of the 181 x 101 x 2 grid. The grid had an approximate minimum 

22 
~k0.842 

0.12 

0.1 

0.08 

$0.06 

0.04 

0.02 

0 

0.12 

0.1 

0.08 

$0.06 

0.04 

0.02 

0 

~k0.675 

0.12 

0.1 

0.08 

$0.06 

0.04 

0.02 

0 

~k0.786 

0.12 

0.1 

0.08 

$0.06 

0.04 

0.02 

0 

0.12 

0.1 

0.08 

$0.06 

0.04 

0.02 

0 

Figure 23: Velocity profiles for NACA 4412. 

23 
H Spalding theory 

I I I I I 

Figure 24: Velocity profiles in wall variables for upper surface of NACA 4412 at x/c = 0.5. 

wall 

-0.2 / 0 0.2 0.4 

primary nozzle x, ft 

axis of symmetry 

Figure 25: Ejector nozzle grid. 

24 
x=0.25 ft 1 

0.8 

0.6 

0.4 

0.2 

50 

-0.2 

-0.4 

-0.8 

-0.8 

200 400 600 800 IO00 1200 

axbl wlocny (WS) 

X=0.58333 ft 1 

0.8 

0.6 

0.4 

0.2 

I 

so 

-0.2 

0.4 

-0.6 

-0.8 

200 400 600 800 1000 1200 

axbl vdodty (W.) 

X=0.41667 ft 1 

0.8 

0.6 

0.4 

0.2 

so 

-0.2 

-0.4 

-0.6 

-0.8 

200 400 600 800 too0 1200 

axbl velocny (Ws) 

XzO.875 ft 1 

0.8 

0.6 

0.4 

0.2 

50 

-0.2 

-0.4 

-06 

-0.8 

200 400 600 800 loo0 1200 

axbl velocny (We) 

Figure 26: Jet velocity profiles for ejector nozzle. 

25 
spacing at the wall of 1.5 x chords. Its farfield extent was 3.2 chords in front of the 

bump leading edge, 3.4 chords downstream of the bump trailing edge, and 4.1 chords domain 

height. There were 85 axial points on the bump part of the wall surface. The average minimum 

y+ value of the first cell center off the wall was approximately 1.3 for this grid. Farfield-type 

boundary conditions were used on the upstream, downstream, and top faces of the grid. 

Fig. 28 shows surface pressure coefficients on the body. Both EASM-ko and EASM-ke 

capture the shock position very accurately, as well as the shape of the pressure variation downstream. EASM-ke predicts the shock location slightly further forward. Velocity profiles and 

turbulent shear stress profiles at seven stations are shown in Figs. 29 and 30, respectively. At 

the station z/c = 0.688 upstream of separation, velocity profiles using EASM-ke agree with 

the experimental data somewhat better than EASM-ko, but EASM-ko recovers better than 

EASM-ke at the last three stations, downstream of reattachment. Both models yield excellent 

velocity profile results in the separated region. Also, both models predict turbulent shear stress 

profiles in reasonable agreement with experiment, with EASM-ke tending to give slightly lower 

peak levels in magnitude in the separated region. 

In section 3, the use of an approximate production term in the EASM-ko and EASM-ke 

models was discussed, and it, was asserted that use of the approximate term makes very little 

difference for a wide variety of subsonic and transonic 2-D aerodynamic-type flows. In fig. 31, 

shear stress profiles at two stations are shown using the two models with both the approximate 

P term as well as its exact form. Results are nearly identical. Pressure coefficients and velocity 

profiles (not shown) are also nearly identical. 

Figure 27: Axisymmetric bump grid, 181 x 101 x 2. 

26 
____ 

0 experiment 

X/C 

Figure 28: Surface pressure coefficients for axisymmetric bump. 

4.7 ONERA M6 Wing 

The ONERA M6 wing is a 3-D case often used to validate codes and turbulence models. The 

experimental data are from Schmitt and Charpin [33]. 

The particular case studied here has a Mach number of 0.84 and Reynolds number based on 

mean aerodynamic chord of 11.7 x lo6. The angle of attack is 3.06'. At these conditions, there 

are two shock waves across the inboard part of the wing upper surface, which merge to form a 

single shock wave at the outboard part of the wing. There is no significant flow separation for 

this case. 

The grid used for this case is a C-0 type grid of size 289 x 65 x 49. Part of the grid (with 

every other grid point removed) is shown in Fig. 32. The grid is scaled so that the chord of the 

wing varies from 0.6737 at the root to 0.3789 at the tip. In these units, the Reynolds number 

per unit length is 21.66 x lo6. The grid extends to approximately -6.4 unit lengths upstream, 

7.4 unit lengths downstream, 6.4 unit lengths high, and 7.4 unit lengths to the side. Symmetry 

conditions are applied along the center plane, and farfield conditions are applied at the farfield 

boundaries. There are 225 streamwise points and 49 spanwise points on the wing surface itself, 

for a total of 11025 surface points. 

unit lengths, 

which yields an average minimum y+ value of the first cell center off the wall of approximately 

3.2. This minimum spacing is larger than one would usually want to use for a turbulent flow 

computation. Based on the flat plate results in section 4.1, such a large wall spacing can yield 

skin friction levels that are more than 5% in error. However, it was still considered important 

The average minimum normal spacing at the wall is approximately 6 x 

27 
~k0.688 

1 1.5 

WC=l .o 

0.16 

0.12 

2008 h' 

0.04 

0 

-0.5 0 

XlCA .375 

1 1.5 

-05 O Ji., 

0.16 

0.1 2 

?008 r' 

0.04 

0 

-0.5 0 

Wc=l xlc=l.250 ,125 

0.1 6 

0.12 

2008 *' 

0.04 

0 

-0.5 0 

u Y5 u, ' lS 

Figure 29: Velocity profiles for axisymmetric bump. 

28 
xltkO.688 

‘0 5 10 15 20 

-u’v’/u:looo 

xk0.813 

0.1 6 

0.12 

$0.08 

0.04 

‘0 5 10 15 20 

-u’v’/u2*1000 

~h0.938 

0.16 

0.12 

2008 *. 

0.04 

‘0 5 10 15 20 

-u’v’/uy 000 

xlc=l .o xkl.125 xkl.250 

‘0 5 10 15 20 

-u’v’/u:looo 

xkl.375 

‘0 5 10 15 20 

-u’v’/u:l 000 

0.16 

0.12 

2008 *. 

0.04 

‘0 5 10 15 20 

-u’v’/u~1000 

0.16 

0.12 

2008 *’ 

0.04 

‘0 5 10 15 20 

-u’v’/u:looo 

Figure 30: Turbulent shear stress profiles for axisymmetric bump. 

29 
xlc=l .125 

0.1 6 

0.12 

0 -,0.08 

0.04 

4 

xlc=O .938 

EASM-ko, appmx P term 

EASM-ke, appmx P term 

EASM-ko, e~ect P tern 

m 

-. . -. .- EASM-ke, exact P bnn 

3 

Figure 31: Effect of approximate production term on turbulent shear stress profiles for axisymmetric bump. 

to include this case as a part of this validation archive because it demonstrates the capabilities 

of these models on a less-than-optimal grid (which is not an uncommon occurrence in many 

situations involving 3-D configurations). 

Fig. 33 shows surface pressure coefficients at six span stations for the two models. Both do 

an excellent job capturing the pressure levels, including shock location. 

4.8 2-D Airfoil Wake 

This test case investigates the near-field development and decay of the 2-D wake of an airfoil 

at low Mach number. The experimental data are from Nakayama [34]. 

The case studied here uses the “Model A” airfoil, which is a 10% thick conventional airfoil. 

It is computed at a Mach number of 0.1 and Reynolds number based on chord of 1.2 x lo6 

(the chordlength is 24 inches). The angle of attack is 0”. At these conditions, there is no flow 

separation on the airfoil. 

The grid used for this case is a C-grid, but it is divided into two zones. The first zone is 

257 x 97 and consists of the part of the C-grid containing points on the airfoil surface (thus, 

there are 257 points on the airfoil surface). The second zone is 153 x 193 and consists of the part 

of the C-grid in the wake, with the top and bottom halves put together. The grid is shown in 

Fig. 34. The grid extent is approximately 20 chords, and the average minimum normal spacing 

at the wall is approximately 1 x chords, which yields an average minimum y+ value of 

the first cell center off the wall of approximately 0.5. In the wake, the minimum grid spacing 

occurs along the wake cut; it starts out with the same minimum spacing as at the airfoil, then 

spreads further downstream. For example, at z/c = 1.2, the minimum normal spacing on the 

centerline is approximately 1 x chords. At that station there are approximately 97 grid 

30 
Figure 32: Schematic of ONERA M6 grid, 289 x 65 x 49 (every other grid point shown). 

points between y = -1 and 1 inches (y/c = -0.0417 and 0.0417). 

Fig. 35 shows wake velocity profiles at seven streamwise stations for the two models. The 

EASM-ke has a slightly deeper maximum wake deficit than EASM-ko at stations downstream of 

x/c = 1.2, in better agreement with experiment, but EASM-ko yields a slightly better spreading 

rate in this case. Both models predict the wake position to be somewhat too low compared to 

experiment at the last two stations. 

Fig. 36 shows wake turbulent shear stress profiles. Overall, both models agree well with 

experiment. The EASM-ke predicts higher peak levels than EASM-ko at the first two stations, 

then predicts slightly lower levels downstream. Both models underpredict the peak levels seen 

in the experiment at most of the stations downstream of x/c = 1.2. 

4.9 Curved Duct in Zero Pressure Gradient 

The curved duct test case was the subject of an extensive study in Rumsey et a1.[35] related 

to curvature effects, and included the application of curvature corrections both to EASM-ke as 

well as to a linear one-equation model. However, the subject of the present paper is solely a 

validation for the “standard” forms of EASM-ko and EASM-ke, so curvature-corrected results 

are not presented here. (Nonetheless, it was found in Rumsey et al. that the “standard” form of 

EASM performs reasonably well for this case because EASM still retains some of the invariance 

properties of the full differential stress model. The more exact curvature-corrected version 

yielded only modest improvements.) The experimental data for this case are from So and 

Mellor [36]. 

31 
. 

2y/B=0.20 

-'O 0.2 0.4 0.6 0.8 1 

XlC 

2y/B=0.44 

1.5 

1 

0.5 

0" 

0 

-0.5 

-11, ' 0!2 ' ' of4 L Of6 ' ' Of8 ' a 

XlC 

1.5 

1 

0.5 

0" 

0 

-0.5 

-1 0.8 1 

2y/B=0.90 2y/B=0.95 

1.5 

1 

0.5 

P 

0 

-0.5 

0.8 1 

Figure 33: Surface pressure coefficients for ONERA M6. 

32 
division belween zones 

(at trailing edge) 

i 

Figure 34: Nakayama model A grid, 257 x 97 and 153 x 193. 

This case is an internal flow with inner-wall convex curvature. The outer wall is shaped 

to yield approximately a zero streamwise pressure gradient along the inner wall in the region 

of interest. The Mach number at the inlet is 0.063, and the Reynolds number is taken to be 

3.6417 x lo4 per inch. 

The grid used for this case is 129 x 81, shown in Fig. 37. In this figure (and in those to 

follow), the coordinate s refers to distance measured along the inner wall, with s = 24 in. as 

the location of the inlet to the computational grid. A grid study was performed in Rumsey et 

a1.[35], where it was concluded that this grid size is adequate. Slip-wall boundary conditions 

were applied at the outer wall in the CFD simulation. This boundary condition allowed the 

simulation to be run without the complication of having to contend with tangential air or bleed 

boundary conditions. The inner wall boundary condition was no-slip, adiabatic wall. The 

minimum normal spacing at the lower wall was 0.00015 in., which yields an average minimum 

y+ value of the first cell center off the wall of less than 0.5. The outflow boundary condition 

set pressure at p/p,,j = 1.0 (where “rei” refers to inlet conditions), and extrapolated all other 

quantities. At the upstream boundary, the density, velocity, and turbulence quantities were set 

to closely match experiment (see Rumsey et a1.[6] for details concerning this procedure). The 

pressure at the inflow was extrapolated from the interior of the domain. 

Fig. 38 shows surface skin friction coefficient along the inner wall. Both EASM-ko and 

EASM-ke give reasonable agreement with experiment. Velocity profiles (referenced to inlet 

conditions and local boundary layer thickness) are shown in Fig. 39. These are predicted in 

good agreement with experiment. Turbulence quantities are given in Figs. 40, 41, and 42. Note 

that all turbulent shear and normal stresses are in the local body/normal coordinate system. 

33 
x/C=l.Ol 

2 

1 

cn 

Q) 

c 

- 20 

SE 

-1 

-2 

0.4 0.6 0.8 u/u_ 

x/c=l.4 

x/c=l .OS x/c=l.2 

x/c=2.19 

Figure 35: Wake velocity profiles for Nakayama model A. 

34 
x/C=l.Ol xkl.05 xkl.2 

2 

1 

v) 

0 

c 

.- 80 

k 

-1 

-2 -0.002 0 0.002 0.004 

-u’v’/uf 

x/c=l.4 

2 

1 

v) 

Q) 

c 

.- 20 

s 

-1 

-0.002 0 20.002 0.004 -2 

-u’v’lu_ 

2 

1 

v) 

Q) 

c 

.- go 

k 

-1 

04 -2 

2 

1 

v) 

Q) 

c 

Y Eo 

k 

-1 

-2 -0.002 0 20.002 0.004 

-u’v’lu_ 

x/c=i .a 

2 

1 

Q) 

Q) 

c 

Y Eo 

k 

-1 

-2 0.002 0 20.002 0.004 

-u’v’/u_ 

2 

1 

v) 

Q) 

c 

.- EO 

s 

-1 

-0.002 0 0.002 0.004 -2 

-u’v’/u: 

x/c=2.19 

Figure 36: Wake turbulent shear stress profiles for Nakayama model A. 

35 
Stresses in this frame are related to those in the Cartesian frame by the following relations: 

-- 

~/ut = ~/~/,cos~~ + v",sin20 + u'o/,sin(2O) 

w/w/ = ~/~/,cos~~ + u",sin20 - .u",sin (2O), -- (43) 

(44) 

where the subscript c indicates Cartesian frame, and 0 is the angle that the body tangent 

vector makes with the x-axis. 

The u'u' turbulent normal stress levels are predicted somewhat too large in magnitude by 

the EASM models in the curved region of the duct, but the turbulent shear stresses and the 

v'v' normal stresses are predicted in good agreement with experiment. 

10 - 

0- 

8 10: 

f, 

.- C 

SI 

20 - 

30 - 

Figure 37: Curved duct grid, 129 x 81. 

5 CONCLUSIONS 

In conclusion, this paper summarized the Explicit Algebraic Stress Model in K-w form (EASMko) and in K-E form (EASM-ke) in the Reynolds-averaged Navier-Stokes code CFLSD. Details 

of the equations and method for coding the latest versions of the models were given, and 

numerous validation cases were presented. Except for the well-known problem of the standard 

form of the E equation being ill-suited for wall-bounded adverse pressure gradient flows, both 

of these models were shown to yield good results for a wide variety of different cases. These 

cases included flow fields with shock waves, curvature, and significant regions of separation. 

36 
0.008 

0.007 

0.006 

0.005 

6 0.004 

0.003 

0.002 

0.001 

0 

20 30 40 50 60 70 80 

s, inches 

Figure 38: Surface skin friction coefficient, referenced to inlet conditions. 

9=24 in. 1 

0.8 

0.6 

la 

a 

0.4 

0.2 

'0 0.2 0.4 0.6 0.8 1 1 wa' 

e67 in. 1 

0.8 

0.6 

la 

a 

0.4 

0.2 

'0 0.2 0.4 0.6 0.8 1 1.2 

e59 in. 1 

0.8 

0.6 

la 

a 

0.4 

0.2 

'0 02 0.4 0.6 0.8 1 1.2 

"Umt 

9=71 in. 1 

0.8 

0.6 

5 

0.4 

0.2 

'0 02 0.4 0.6 0.6 1 1.2 

""mt 

Figure 39: Velocity profiles, referenced to inlet conditions. 

37 
e24 in. s=59 in. 

0002 0004 0006 

-u'v'/u,: 0.002 0.004 0.006 

-u'v'lu,: 

s=67 in. 

0.002 0.004 0.006 

-u'v'Iu,; 

9=71 in. 

0002 0004 ow6 

-u'v'lu,; 

Figure 40: Turbulent shear stress profiles, referenced to inlet conditions. 

s=24 in. 

0002 0004 0006 

u'u'lu$ 

9=67 in. 

0002 0004 0006 

U'u'lu,; 

e59 in. 

0002 ow4 ow8 

u'u'lu,: 

-71 in. 

0002 0004 0006 

u'u'lu,: 

Figure 41: Turbulent u'u) normal stress profiles, referenced to inlet conditions. 

38 
s=24 in. e59 in. 

0.002 OW4 0.006 

vvlu,: 

s=67 in. e71 in. 

0002 ow4 0006 

v”lu,a VVlU,? 

Figure 42: Turbulent v’v’ normal stress profiles, referenced to inlet conditions. 

References 

Abdol-Hamid, K. S., Lakshmanan, B., and Carlson, J. R., “Application of Navier-Stokes 

Code PAB3D with k-E Turbulence Model to Attached and Separated Flows,” NASA TP2480, January 1995. 

Gatski, T. B. and Rumsey, C. L., “Linear and Nonlinear Eddy Viscosity Models,” in closure 

Strategies for Turbulent and Transitional Flows, ed. B. E. Launder and N. D. Sandham, 

Cambridge University Press, Cambridge U. K., 2002. 

Gatski, T. B. and Speziale, C. G., “On Explicit Algebraic Stress Models for Complex 

Turbulent Flows,” Journal of Fluid Mechanics, Vol. 254, 1993, pp. 59-78. 

Ying, R., and Canuto, V. M., “Turbulence Modelling over Two-Dimensional Hills Using 

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2,” Journal of Propulsion and Power, Vol. 13, No. 5, 1997, pp. 610-628. 

39 
Carlson, J. R., Duquesne, N., Rumsey, C. L., and Gatski, T. B., “Computation of Turbulent 

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40 
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41 
1. AGENCY USE ONLY (Leave blank) 

I I 

4. TITLE AND SUBTITLE 

Summary of EASM Turbulence Models in CFL3D With Validation Test 

Cases 

2. REPORT DATE 3. REPORT TYPE AND DATES COVERED 

June 2003 Technical Memorandum 

6. AUTHOR(S) 

Christopher L. Rumsey and Thomas B. Gatski 

14. SUBJECT TERMS 

Turbulence model; Validation; Explicit algebraic stress; Separated flow 

17. SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATION 19. SECURITY CLASSIFICATION 

OF REPORT OF THIS PAGE OF ABSTRACT 

Unclassified Unclassified Unclassified 

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15. NUMBER OF PAGES 

48 

16. PRICE CODE 

20. LIMITATION 

OF ABSTRACT 

UL 

NASA Langley Research Center 

Hampton, VA 2368 1-2 199 

~~ 

9. SPONSORlNGlMONITORlNG AGENCY NAME(S) AND ADDRESS(ES) 

National Aeronautics and Space Administration 

Washington, DC 20546-000 1 

 SUPPLEMENTARY NOTES 

128. DISTRIBUTION/AVAILABlLlTY STATEMENT 

Unclassified-Unlimited 

Subject Category 02 Distribution: Nonstandard 

Availability: NASA CAS1 (301) 621-0390 

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5. FUNDING NUMBERS 

762-20- 1 1-06 

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REPORT NUMBER 

L-18311 

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AGENCY REPORT NUMBER 

NASA/TM-2003-21243 1 

12b. DISTRIBUTION CODE 

This paper summarizes the Explicit Algebraic Stress Model in k-omega form (EASM-ko) and in k-epsilon form 

(EASM-ke) in the Reynolds-averaged Navier-Stokes code CFL3D. These models have been actively used over 

the last several years in CFL3D, and have undergone some minor modifications during that time. Details of the 

equations and method for coding the latest versions of the models are given, and numerous validation cases are 

presented. This paper serves as a validation archive for these models.