April 2009 

NASA/TM–2009-215705 

Compressibility Considerations for κ-ω

Turbulence Models in Hypersonic Boundary 

Layer Applications 

C. L. Rumsey 

Langley Research Center, Hampton, Virginia 
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April 2009 

NASA/TM–2009-215705 

Compressibility Considerations for κ-ω

Turbulence Models in Hypersonic Boundary 

Layer Applications 

C. L. Rumsey 

Langley Research Center, Hampton, Virginia 
Available from: 

NASA Center for AeroSpace Information 

7115 Standard Drive 

Hanover, MD 21076-1320 

443-757-5802 

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National Aeronautics and Space Administration. 
Abstract

The ability of k-ω models to predict compressible turbulent skin friction in hypersonic boundary

layers is investigated. Although uncorrected two-equation models can agree well with correlations

for hot-wall cases, they tend to perform progressively worse - particularly for cold walls - as the

Mach number is increased in the hypersonic regime. Simple algebraic models such as BaldwinLomax perform better compared to experiments and correlations in these circumstances. Many of

the compressibility corrections described in the literature are summarized here. These include corrections that have only a small influence for k-ω models, or that apply only in specific circumstances.

The most widely-used general corrections were designed for use with jet or mixing-layer free shear

flows. A less well-known dilatation-dissipation correction intended for boundary layer flows is also

tested, and is shown to agree reasonably well with the Baldwin-Lomax model at cold-wall conditions. It exhibits a less dramatic influence than the free shear type of correction. There is clearly a

need for improved understanding and better overall physical modeling for turbulence models applied

to hypersonic boundary layer flows.

1 Introduction

Compressibility is typically not considered to be important for wall-bounded turbulent flows over a

wide range of Mach numbers. As stated in Wilcox [1] (p. 239): “Generally speaking, compressibility has a relatively small effect on turbulent eddies in wall-bounded flows. This appears to be true

for Mach numbers up to about 5 (and perhaps as high as 8), provided the flow doesn’t experience

large pressure changes over a short distance such as we might have across a shock wave. At subsonic speeds, compressibility effects on eddies are usually unimportant for boundary layers provided

Tw/Te < 6.”

The hypothesis of Morkovin [2] states that the compressibility effects on turbulence can be accounted for by mean density variations alone. For many applications, this hypothesis has proved

correct in that good results can be obtained for mean velocity and temperature fields using incompressible turbulence models extended directly to compressible turbulent boundary layers. Furthermore, So et al. [3] have shown the Morkovin hypothesis to be equally applicable for prediction of

the turbulence field itself, for flat plate boundary layers up to a Mach number of at least 10. They

state: “there is indeed a dynamic similarity of the incompressible and compressible mean and turbulence field, and the Morkovin hypothesis is valid for both fields.” In other words, for many subsonic

through hypersonic boundary layer applications, the incompressible forms of turbulence models

(with mean density variations accounted for) are expected to be reasonable approximations.

The most common classes of compressibility correction for Reynolds-averaged Navier-Stokes

(RANS) turbulence models were developed for the purpose of improving correlations with experiment for free shear layer or jet spreading rates. See, e.g., Refs. 4–6. However, what we are concerned

with here is primarily (attached) hypersonic boundary layer flow. In this paper, compressibility corrections (particularly applicable to boundary layer flows) from the literature are described. The

focus here is solely on the k-ω form of two-equation models. The claim that compressibility corrections are not required for hypersonic boundary layer flows is investigated for a wide range of Mach

numbers and wall-temperature boundary conditions.

The paper is organized as follows. First, several standard forms of the k-ω model are given.

Then, compressibility corrections from the literature are described. Finally, results for hypersonic

boundary layer flows are shown, and conclusions are made.

1
2 Standard Forms of Two-Equation Turbulence Models

2.1 Wilcox 1988

The original incompressible form of the Wilcox k-ω model [7], referred to here as Wilcox88, is

written as:

Dk

Dt

=

P

ρ

− β

∗ωk +

∂

∂xj



(ν + σkνt)

∂k

∂xj



(1)

Dω

Dt

=

γ

μt

P − βω2 +

∂

∂xj



(ν + σωνt)

∂ω

∂xj



(2)

where ν = μ/ρ and νt = μt/ρ. The eddy viscosity is given by:

μt = ρ

k

ω

(3)

and the production term P is given by

P = −τij

∂ui

∂xj

(4)

where

τij = ρuiuj = −2μt



Sij −

1

3

∂uk

∂xk

δij+

2

3

ρkδij (5)

Sij =

1

2



∂ui

∂xj

+

∂uj

∂xi



(6)

Note that the definition for τij varies in the literature: sometimes it is defined with the opposite

sign, and sometimes it is defined without the density. The definition does not matter as long as the

production term is defined appropriately in Eq. (4), with +2μtSij (∂ui/∂xj ) as the leading term in

P. In Eq. (5), the −(1/3)(∂uk/∂xk)δij term and the (2/3)ρkδij term are often ignored for lowspeed flows (the former term makes the strain rate tensor traceless in 3-D flows), but these both may

be non-negligible for higher speed flows, or near stagnation regions. The constants are β

∗ = 0.09,

σk = 0.5, γ = 5/9, β = 3/40, and σω = 0.5.

For clarity, the production term is expanded out here:

P = 2μtSij

∂ui

∂xj

−

2

3

μt



∂uk

∂xk

2

−

2

3

ρk ∂uk

∂xk

(7)

= μt

"

∂ui

∂xj

+

∂uj

∂xi



∂ui

∂xj

−

2

3



∂uk

∂xk

2

#

−

2

3

ρk ∂uk

∂xk

(8)

Note that

S

2 ≡ 2SijSij =

1

2



∂ui

∂xj

+

∂uj

∂xi

  ∂ui

∂xj

+

∂uj

∂xi



=



∂ui

∂xj

+

∂uj

∂xi



∂ui

∂xj

(9)

So:

2
P = μtS

2 −

2

3

μt



∂uk

∂xk

2

−

2

3

ρk ∂uk

∂xk

(10)

= 2μtSijSij −

2

3

μt



∂uk

∂xk

2

−

2

3

ρk ∂uk

∂xk

(11)

= 2μtSijSij −

2

3

ρk ∂uk

∂xk

(12)

where Sij is the traceless form of the strain rate tensor (in 3-D):

Sij = Sij −

1

3

∂uk

∂xk

δij (13)

2.2 Wilcox 2006

In 1998 Wilcox presented a modified version of the k-ω model [8]. Because it has been superseded,

this 1998 version is not described here. A newer form of the Wilcox k-ω model [1, 9], referred to

here as Wilcox06, was developed to improve the predictive accuracy compared to the 1988 and 1998

versions for free shear flows and strongly separated flows (and hence be more competitive with the

Menter SST model, described in the next section). Wilcox06 is given by:

Dk

Dt

=

P

ρ

− β

∗ωk +

∂

∂xj



(ν + σkν

0

t

)

∂k

∂xj



(14)

Dω

Dt

=

γ

μt

P − βω2 +

∂

∂xj



(ν + σων

0

t

)

∂ω

∂xj



+

σd

ω

∂k

∂xj

∂ω

∂xj

(15)

where ν

0

t = k/ω. The eddy viscosity is given by:

μt = ρ

k

ω ̃

(16)

where

ω ̃ = max



ω; Clims

2SijSij

β

∗



 (17)

and Clim = 7/8. The coefficients are γ = 13/25, β = β0fβ, β

∗ = 0.09, σω = 0.5, σk = 3/5,

σd0 = 1/8, β0 = 0.0708, and

σd = 0

∂k

∂xj

∂ω

∂xj

≤ 0 (18)

σd = σd0

∂k

∂xj

∂ω

∂xj

> 0 (19)

Also:

fβ =

1 + 85χω

1 + 100χω

(20)

χω =











ΩijΩjkSˆ

ki

(β

∗ω)3











(21)

3
where Sˆ

ki = Ski −

1

2

(∂um/∂xm)δkl. This form forces χω = 0 for 2-D flow (both incompressible

and compressible). The fβ parameter was added by Wilcox to account for the round-jet/plane-jet

anomaly [1]. For boundary layer applications, the vortex stretching parameter χω is sometimes

ignored (set to zero), yielding fβ = 1.

2.3 Menter SST

The two-equation SST model of Menter [10] is written as:

Dk

Dt

=

P

ρ

− β

∗ωk +

∂

∂xj



(ν + σkνt)

∂k

∂xj



(22)

Dω

Dt

=

γ

μt

P − βω2 +

∂

∂xj



(ν + σωνt)

∂ω

∂xj



+ 2(1 − F1)

σω2

ω

∂k

∂xj

∂ω

∂xj

(23)

The eddy viscosity is given by:

μt = ρ

a1k

max(a1ω, ΩF2)

(24)

where a1 = 0.31, Ω is the magnitude of vorticity, and F1 and F2 are blending functions (given

below).

The “shear stress transport” (SST) part of the model is based on Bradshaw’s assumption that

the principal shear stress is proportional to k, via: τ12 = −ρa1k. From Eq. (5), the primary term

in eddy viscosity models is: τ12 = −2μtS12. In adverse pressure gradient boundary layer flows,

the standard method often leads to too much eddy viscosity (an overprediction of τ12), inhibiting or

delaying separation. In these situations, it is better for the model to choose τ12 based on Bradshaw’

assumption. Using:

ρa1k = 2μtS12 (25)

we find how to set the eddy viscosity in order to recover values corresponding with Bradshaw’s

assumption:

μt =

ρa1k

2S12

≈

ρa1k

Ω

(26)

Functionally, Eq. (24) chooses the minimum eddy viscosity between the standard one and that dictated by Bradshaw’s assumption limited to within the boundary layer region.

In the SST model, there are two sets of coefficients, which are combined using a blending

function. The constants for set 1 are β

∗

1 = 0.09, σk1 = 0.85, β1 = 0.075, σω1 = 0.5, and

γ1 = β1/β∗

1 − σω1κ

2/

p

β

∗

1 ≈ 0.55317. The constants for set 2 are β

∗

2 = 0.09, σk2 = 1.0,

β2 = 0.0828, σω2 = 0.856, and γ2 = β2/β∗

2 − σω2κ

2/

p

β

∗

2 ≈ 0.44035. The constant κ is defined

as κ = 0.41. Set 1 and set 2 are blended via:

φ = F1φ1 + (1 − F1)φ2 (27)

and

F1 = tanh(arg4

1

) (28)

arg1 = min "

max √

k

0.09ωd;

500ν

d

2ω

!

;

4ρσω2k

CDkωd

2

#

(29)

CDkω = max 2ρσω2

1

ω

∂k

∂xj

∂ω

∂xj

; 10−20

(30)

4
where d is the distance to the nearest wall. The F2 term is given by:

F2 = tanh(arg2

2

) (31)

arg2 = max 2

√

k

0.09ωd;

500ν

d

2ω

!

(32)

2.4 Other Considerations for k-ω Models

For situations in which compressibility is important (and shocks may be present), the turbulence

equations are sometimes solved in conservation form. For example, making use of the continuity

equation ∂ρ/∂t + ∂(ρuj )/∂xj = 0, Eqs. (1) and (2) can be written:

∂(ρk)

∂t +

∂(ρujk)

∂xj

= P − β

∗

ρωk +

∂

∂xj



(μ + σkμt)

∂k

∂xj



(33)

∂(ρω)

∂t +

∂(ρujω)

∂xj

=

γρ

μt

P − βρω2 +

∂

∂xj



(μ + σωμt)

∂ω

∂xj



(34)

It is unclear whether the turbulence model equation form (conservative vs. non-conservative) makes

much difference in the common situation where the turbulence models are solved separately (loosely

coupled) from the conservative mean flow equations. But certainly if the equations are fully coupled,

all should be solved consistently in conservation form.

For flows in which the turbulent kinetic energy is non-negligible compared to the square of the

mean velocity, the k contributes to the conservation of total energy via ρE = ρ(e +

1

2

uiui + k) (see

Wilcox [1]). Also, the molecular and turbulent diffusion of k, typically modeled as

∂

∂xj



(μ + σkμt)

∂k

∂xj



(35)

in the mean flow energy equation [11], should be included. Furthermore, the perfect gas equation of

state then becomes [12]: p = (γ − 1)ρ(E −

1

2

uiui − k). Because many CFD codes include other

(simpler) turbulence models beside two-equation models, for which k is not available, the turbulent

kinetic energy contribution to total energy (and its explicit appearance in the energy equation and

equation of state) is often ignored.

In an often-used variant of the k-ω model, the production term is simplified by an approximation

that makes use of the local magnitude of vorticity Ω:

P = μtΩ

2 −

2

3

ρkδij

∂ui

∂xj

= μtΩ

2 −

2

3

ρk ∂uk

∂xk

(36)

This vorticity source term is often a good approximation of the exact source term in boundary layer

flows [13], and its use can avoid some numerical difficulties sometimes associated with the use of

the exact source term. Again, it is often common to ignore the (2/3)ρk term in the production source

of Eq. (36) for many applications, but this may have a non-negligible influence for high-speed flows.

The recommended wall boundary conditions for k-ω models are [10]: kwall = 0, ωwall =

60ν/[β1(∆d1)

2

]. In his boundary layer code, Wilcox [1] overwrites the computed value of ω with

the theoretical value ω ≈ 6ν/(β1y

2

) at the first grid point off the surface (for smooth walls), but this

method of overwriting field variables rather than specifying boundary conditions is undesirable in

general-purpose Navier-Stokes codes. The farfield boundary conditions are more difficult to define

with confidence. Part of the problem is that the freestream levels are not preserved; they decay

rapidly (both due to the equations themselves as well as due to typically coarse grid spacing in the

farfield). This decay, which occurs for k-ε equations as well, makes the local “ambient” levels near

5
the body a function of the farfield grid extent. In the freestream, the k-ω governing equations dictate

that the decay of eddy viscosity occurs according to:

μt = μt,∞



1 + βω∞

x

u∞

1−(0.09/β)

(37)

where x is the distance from the location where the boundary conditions are set. As discussed

in Spalart and Rumsey [14], real flow over external aerodynamic configurations has no reason to

obey the decay equations used to calibrate two-equation models in isotropic turbulence. In reality,

the kinetic energy (and eddy viscosity) relevant to the aircraft flow varies very little over the size

of the typical CFD domain. Thus, the behavior represented by decaying freestream turbulence is

not representative of reality. Ref. 14 describes the use of sustaining terms which, when added to

the k-ω equations, preserve the freestream levels without decay. Although maintaining freestream

turbulence is probably important from the point of view of numerical-transition consistency with

grid refinement, at high Reynolds numbers the effect is generally small. Therefore, results in this

paper will not use the sustaining terms.

For typical subsonic/transonic/supersonic applications, most CFD application codes have developed their own methodology for setting farfield boundary conditions for k and ω, in order to yield

reasonable results across a broad range of problems. For example, in CFL3D (Krist et al. [15]), the

boundary conditions are: k/a2

∞ = 9 × 10−9

and ωμ∞/(ρ∞a

2

∞) = 1 × 10−6

, which always gives

μt,∞/μ∞ = 0.009. Because the freestream turbulence level, Tu (in percent), is given by:

Tu, % = 100s

2

3

k

u

2∞

(38)

this means that a fixed k/a2

∞ = 9 × 10−9 yields (for example): Tu = 0.0387% for M = 0.2,

0.0039% for M = 2.0, 0.0016% for M = 5.0, and 0.0008% for M = 10.0. Thus, perhaps, it

makes better sense when solving over a broad range of Mach numbers to use a fixed Tu (i.e., fixed

k/u2

∞) in the freestream instead of fixed k/a2∞). Otherwise higher Mach number cases will have an

even greater tendency to become laminar.

3 Compressibility Corrections

Wilcox [1] describes many of the compressible-flow closure approximations. A few of them are

already employed in most compressible flow CFD codes. For example, the Reynolds stress tensor of

Eq. (5) is already written appropriately for compressible flows. The most commonly used turbulent

heat-flux vector (qT = −(μt/P rt)∂h/∂xj , where h is enthalpy and P rt is typically around 0.9

for boundary layers), has been in common use in compressible flow CFD codes for many years.

However, the models for pressure-diffusion, pressure-dilatation, and pressure-work are all either

under development, very little is known, or proposed models are too complex or have not gained

wide acceptance (see, e.g., Zeman [16], Grasso and Falconi [17], and Yoshizawa et al. [18]). Many

of these compressibility effects are presumed to be small in boundary layers [1]. As a result, most

widely-used models do not include them. For example, Sarkar’s model for the pressure-dilatation

correction in compressible flows [19] is rarely employed for boundary-layer computations. See

also Wilcox [1] and Grasso and Falconi [17]. In the Sarkar model, the pressure-dilatation adds the

following term to the k-equation (in the k-ε model):

(−α2P + α3ρε)M2

T

(39)

where α2 = 0.15 (0.4 in Ref. 17), α3 = 0.2, MT = (√

2k)/a, and a is the local speed of sound.

On the other hand, the Sarkar/Zeman compressibility corrections for dilatation-dissipation are

often employed for jets and free shear mixing layers, in spite of the fact that the reasoning behind

6
them is fundamentally flawed [1]. See also Sarkar [20]. It turns out that dilatation-dissipation

is small or negligible, and mixing-layer compressibility effects likely manifest themselves in the

pressure-strain redistribution term. Properly formulated corrections are still being explored. In

the mean time, the existing dilatation-dissipation corrections provide the desired trends for mixing

layers, albeit for the wrong reasons, and so are still considered useful for those cases. It should also

be noted that dilatation-dissipation models typically account for the pressure-dilatation correction

[21], so when employing both corrections the coefficient in the dilatation-dissipation model must be

reduced by about a factor of two from its standard value [17].

Unfortunately, the Sarkar/Zeman compressibility corrections can have a detrimental effect on

many boundary-layer predictions (they tend to produce wall skin frictions that are too low, and can

also negatively impact the size of the predicted separation region [22]). Wilcox [1, 23] developed a

modification that significantly decreases this detrimental effect, and Brown [24] further attempted to

eliminate its potential impact in very high Mach number boundary layers by combining it with the

F1 function of Menter (Eq. (28)). In the Wilcox correction, the coefficients of the k-ω destruction

terms are modified as follows:

β

∗

c = β

∗

[1 + ξ

∗F(MT )] (40)

βc = β − β

∗

ξ

∗F(MT ) (41)

where

F(MT ) = M2

T − M2

T0



H (MT − MT0) (42)

and ξ

∗ = 2, MT0 = 0.25, MT = (√

2k)/a, H(·) is the Heaviside function, and a is the local speed

of sound.

However, it has been observed 1that for cold-wall cases the skin friction is typically overpredicted to such an extent that including a dilatation-dissipation correction can yield improved results,

although possibly for the wrong reasons. Zeman [16] noted that the apparent unimportance of the

pressure-dilatation and dilatation-dissipation in boundary layers is “only a question of degree.” He

found that as freestream Mach number increases and wall cooling increases, compressibility effects

become increasingly important. In the Zeman dilatation-dissipation correction for boundary layers,

the coefficients of the k-ω destruction terms are modified as in Eqs. (40) and (41), only now F(MT )

is given by:

F(MT ) = "1 − exp −



MT − MT0

Λ

2

!# H (MT − MT0

) (43)

with ξ

∗ = 0.75, MT0 = 0.2, and Λ = 0.66.

In addition to a pressure-dilatation correction and a dilatation-dissipation correction, Grasso and

Falconi [17] also included a correction to the k-equation in their k-ε model, due to the scalar product

of the Favre-velocity and the mean pressure gradient. They believed that this term may be influential

in regions of large pressure and density gradients.

Both Wilcox [1] and Huang et al. [25] mention that the k-ε form of the two-equation model

exhibits more deviation from the compressible law-of-the-wall than k-ω at high Mach numbers.

Wilcox also points out that the k-ε form is more problematic for adverse pressure-gradient wallbounded flows. Huang et al. proposed a possible iterative procedure to reproduce the expected

profile, and also mentioned that alternative forms such as k-(ε

5/6/k) may reduce the sensitivity, but

neither of these proposals were widely used.

Catris et al. [26] extended the analysis of Huang et al. [25], and showed that specific corrections

to the diffusion terms are necessary to make the models consistent with the logarithmic law for

1White, J. A., private communication 2008.

7
compressible boundary layers. The corrections were derived for a variety of models. Here, we are

only concerned with the k-ω form. For example, Eqs. (33) and (34) get altered as follows:

∂(ρk)

∂t +

∂(ρujk)

∂xj

= P − β

∗

ρωk +

∂

∂xj



1

ρ

(μ + σkμt)

∂(ρk)

∂xj



(44)

∂(ρω)

∂t +

∂(ρujω)

∂xj

=

γρ

μt

P − βρω2 +

∂

∂xj



1

√ρ

(μ + σωμt)

∂(

√ρω)

∂xj



(45)

where the only changes are in the diffusion terms. However, Catris et al. point out that for k-ω, the

difference in results due to modifying the diffusion terms is only very slight. This further confirms

the low sensitivity of k-ω to compressibility effects in boundary layer flow, as described in Refs. 1

and 25.

A length-scale modification was proposed by Vuong and Coakley [27] and Huang and Coakley [28], to reduce the magnitude of heat transfer for high-speed separated boundary layers near

reattachment. See also Brown [24] and Coratekin et al. [29]. In this correction, the length scale

going into the eddy viscosity is limited based on Bradshaw’s relation τt/ρ ∝ a1k and the mixing

length relation ` = κd. The turbulent length scale is limited by:

` = min κd√

a1;

√

k

ω

!

(46)

(Note that in Vuong and Coakley and Huang and Coakley, there is an additional factor of Cμ = 0.09

present due to the different way that ω is defined.) Because ω =

√

k/`, the end result is that the

eddy viscosity is limited according to:

μt = min μt,std; ρκdpa1k



(47)

where μt,std is the eddy viscosity computed the usual way, via Eq. (3), (16), or (24). It should be

noted here that the assumption of constant turbulent Prandtl number has been recently questioned,

relative to its effect on heat flux for shock/boundary layer cases [30]. A variable turbulent Prandtl

number model was shown to improve heat flux near reattachment.

A rapid compression fix was implemented by Coakley and Huang [31] (see also Vuong and

Coakley [27], Coratekin et al. [29], and Forsythe et al. [32]). In this fix, the production term in the

ω-equation

γρ

μt

P = −

γρ

μt

τij

∂ui

∂xj

=

γρ

μt



μtS

2

−

2

3

ρkδij

∂ui

∂xj



(48)

= γρS

2

−

2

3

γρω

∂uk

∂xk

(49)

is altered to read:

γρ

μt

P = γρS

2

−

4

3

ρω

∂uk

∂xk

(50)

where

S

2

≡



∂ui

∂xj

+

∂uj

∂xi

−

2

3

∂uk

∂xk

δij

∂ui

∂xj

=



∂ui

∂xj

+

∂uj

∂xi



∂ui

∂xj

−

2

3



∂uk

∂xk

2

= S

2 −

2

3



∂uk

∂xk

2

(51)

or equivalently:

8
S

2

= 2SijSij (52)

where Sij was defined earlier in Eq. (13). Thus, in this rapid compression fix, the original (2/3)γ

(which is close to 1/3) gets increased to a fixed value of 4/3 for linear deformations.

This modification was made in order to increase the size of computed separation bubble regions,

by insuring that the turbulent length scale does not change too quickly when undergoing rapid compression. However, as discussed in Forsythe et al. [32], the shear-stress transport part of the Menter

SST model already improves correlations with experimental bubble size, so the ad hoc rapid compression fix was not used for that model. The Wilcox06 model is designed with a similar stress

limiter modification, so this model, too, probably would not benefit from the rapid compression fix.

In summary, when considering high Mach number compressible boundary layer flows using

k-ω models, the conservation of total energy should be configured to include the contribution of

the turbulent kinetic energy k, and the mean flow energy equation should include the molecular

and turbulent diffusion of k. It is sometimes common practice to ignore these effects, which is

certainly justified when k is significantly smaller than the square of the mean velocity. Furthermore,

turbulent production should officially include the (2/3)ρk term, which multiplies ∂uk/∂xk and

hence is identically zero for incompressible flows. Because this term typically has little effect over a

broad range of conditions, it is sometimes ignored, particularly when other approximations, limiting,

etc. are employed for turbulence production. Other than these considerations (which may or may

not be important, depending on the case), little extra appears to be called for – based on the currently

available literature – in terms of specific corrections for compressible Navier-Stokes codes applied

to boundary layer flows. Adopting the modified diffusion-term form of Catris et al. [26] has been

shown to make very little difference for k-ω models. The length-scale modification of Vuong and

Coakley [27] appears to only be important for the specific circumstance of predicting heat transfer

near reattachment after separation, but this flow feature may also be improved by adopting models

with variable turbulent Prandtl number. The rapid compression fix of Coakley and Huang [31] has

been negated by the more accepted stress limiters that appear in SST and Wilcox06.

However, Zeman [16] and practical experience indicates that the need for corrections in hypersonic boundary layers becomes increasingly evident as Mach number increases, particularly for cold

walls. But the “traditional” Sarkar / Zeman / Wilcox fixes for free shear flows tend to over-correct

in many cases when applied to boundary layer flows. In the Results section, the influence of the

less-widely-used Zeman correction (formulated for boundary layer flows) is explored.

4 Results

4.1 Experimental Correlations for Skin Friction and Heat Transfer on a Flat

Plate

Wall skin friction is given by the formula:

Cf =

τw

2ρeU2

e

(53)

where τw is the wall shear stress μw∂u/∂y|w and the subscript “e” represents “edge” or freestream

values. There have been a plethora of correlations for wall skin friction (and heat transfer) for the

flat plate over the years. See, for example, White [33], Peterson [34], and Hopkins and Inouye [35].

One of the reasons for the large number is the fact that there is a significant amount scatter in the

available experimental data, especially those with heat transfer, making certainty difficult. Many of

the skin friction correlations use a compressibility transformation idea:

Cf =

1

Fc

Cf,incomp(ReθFReθ) (54)

9
In other words, the formula for incompressible Cf,incomp, which is often expressed as a function of

Reθ, is instead computed using the altered variable ReθFReθ, and then divided by the function Fc.

A widely-used correlation for Cf,incomp is the Karman-Schoenherr relation (see Roy and Blottner

[36]):

Cf,incomp =

1

log10 (2Reθ) [17.075log10 (2Reθ) + 14.832] (55)

Here, we examine three correlations for Cf : van Driest [37], Spalding and Chi [38], and White and

Christoph [33]. For each of these, the Fc is defined the same way:

Fc =

Taw/Te − 1



sin−1A + sin−1B

2

(56)

where Te represents the “edge” or freestream temperature, and

Taw = Te



1 + r

γ − 1

2

M2



(57)

The recovery factor r is taken to be 0.9. This empirical factor is often introduced because in practice

energy recovery is not perfect. The numerator of Eq. (56) is thus simply 1

2

r(γ − 1)M2. The A and

B are given by:

A =

2a

2 − b

(b

2 + 4a2)1/2

(58)

B =

b

(b

2 + 4a2)1/2

(59)

where

a =



r

γ − 1

2

M2 Te

Tw

1/2

(60)

b =

Taw

Tw

− 1 =

Te

Tw



1 + r

γ − 1

2

M2



− 1 (61)

A contour plot of Fc as a function of Mach number and log10(Tw/Te) is shown in Fig. 1. A different

version of this same plot is also given in Spalding and Chi [38].

The three correlations differ in their definitions of FReθ:

FReθ =

μe

μw

van Driest (62)

FReθ =



Tw

Te

−0.702 

Taw

Tw

0.772

Spalding and Chi (63)

FReθ =

p

Fc



μe

μw

  Te

Tw

1/2

White and Christoph (64)

(Note the typographical error in White [33] in Table 7-3, where the term Q is inverted.) Both van

Driest and White/Christoph correlations are functions of μe/μw. This quantity can be obtained via

Sutherland’s law (see White [33]):

μ = μ0



T

T0

3/2

T0 + S

0

T + S0

(65)

10
where for air μ0 = 0.1716 mP, T0 = 491.6 R, and S

0 = 199 R. Thus:

μe

μw

=



Tw

Te

−3/2

(Tw/Te) + (S

0/Te)

1 + (S0/Te)

(66)

Therefore, when Sutherland’s law is used, both of these correlations are functions not only of the

ratio Tw/Te, but also of the freestream temperature Te itself. For all of the work herein, Te is chosen

to be 540 R. Contour plots of FReθfor the three correlations are shown in Figs. 2(a) - (c). Note that

a different version of Fig. 2(b) can also be found in Spalding and Chi [38].

Using Eq. (55) for the Cf,incomp value, results for the compressible Cf can be computed for each

of the correlations. It turns out that results using van Driest and White/Christoph are very similar,

so only the results of van Driest and Spalding/Chi are shown in Fig. 3 for clarity of presentation.

Plots of Cf /Cf,incomp are shown here for a variety of Tw/Te ratios, as well as for adiabatic wall

temperature. In all cases, an assumed Rex of 5 × 106 was used. In terms of Reθ, this translates

to approximately Reθ = 14200, using the formula Reθ ≈ 0.0142Re6/7

x from White [33]. For

the adiabatic case (for which the most experimental data exist), the correlations give nearly the

same result (black lines). But for fixed wall temperature ratios, the results can differ significantly.

Spalding and Chi [39] claim a smaller root mean square error compared to van Driest [37] using a

variety of experiments, but recall that relatively few experiments exist for walls with heat transfer.

Plots of compressible Cf vs. Rex are shown in Figs. 4(a) and (b) for several specific cases. In

Fig. 4(a), the results of the correlations agree well, whereas in Fig. 4(b) the correlations are seen to

differ by as much as 50% or more. The main point here is that there is some uncertainty regarding

Cf for walls with heat transfer, so it is more difficult to validate (or invalidate) turbulence models

for these cases with confidence. In the literature, most people tend to compare with the van Driest

correlation, but others have since developed correlations that may work better in specific cases. For

example, Huang et al. [40] developed a method for which skin friction for strongly cooled walls falls

below van Driest, in better agreement with data.

The wall heat flux is often expressed [33] in terms of the Stanton number:

St = Ch =

qw

ρeUecp(Taw − Tw)

(67)

where the heat flow at the wall qw = −k(∂T /∂y)|w. The so-called Reynolds analogy is usually

used to relate the local wall heat flux in terms of skin friction:

St ≈

1

2

CfRaf (68)

where Raf is the Reynolds analogy factor. This factor generally lies in the range 0.9 < Raf < 1.3,

and is believed to be close to unity for hypersonic flows [36] and for very cold walls [22]. Thus, St

is directly proportional to Cf , at approximately half its numerical value (with appropriate sign).

4.2 CFD Results on a Flat Plate

In order to test the ability of k-ω turbulence models to predict compressible boundary layer flow,

computations were performed for flow over a flat plate in zero pressure gradient for a variety of flow

conditions. Most of the computations were performed using the CFL3D code [15]. Note that in

CFL3D, the turbulence models are decoupled from the mean flow equations, k is not included in

the definition of total energy, and diffusion of k does not appear in the mean flow energy equation

for its models tested here. Furthermore, for the current applications, Eq. (36) is used for production,

with the (2/3)ρk term ignored. (Some computations were tried with this term included, and it was

found to make little difference even for M = 10 cases.) The (2/3)ρk term was also neglected in

τij , Eq. (5). For comparison, several results were also obtained using the VULCAN code [41], in

which the turbulence models are fully coupled to the mean flow equations and no approximations

are made for τij or the turbulence production terms.

11
Table 1. Flat plate cases computed

Mach Tw/T∞ Tw/Taw,ideal wall type

0.2 1.008 1.0 adiabatic

5.0 6.0 1.0 adiabatic

10.0 21.0 1.0 adiabatic

0.2 5.0 4.96 hot

2.0 1.0 0.556 cold

2.0 2.0 1.11 hot

5.0 1.0 0.167 cold

5.0 3.0 0.5 cold

5.0 20.0 3.33 hot

10.0 1.0 0.0476 cold

10.0 10.0 0.476 cold

10.0 40.0 1.905 hot

The majority of runs were performed using the Menter SST model. Full Navier-Stokes (as opposed to thin-layer) was employed. For subsonic Mach numbers, the inflow boundary condition set

total pressure and total temperature conditions (according to the particular Mach number). The pressure was extrapolated from the interior of the domain, and the remaining variables were determined

from the extrapolated pressure and the input data, using isentropic relations. The outflow boundary

condition set p/p∞ = 1 and extrapolated all other quantities from the interior of the domain. For

supersonic Mach numbers, the inflow boundary condition set all primitive variables, and the outflow

boundary condition extrapolated all variables from the interior of the domain. In all cases the top

boundary, located a nondimensional distance y = 1 from the wall, used a farfield Riemann invariant

boundary condition. The wall boundary condition enforced no slip, and set temperature either (1)

according to a desired Tw/T∞, or (2) according to Tw = Taw,ideal, where Taw,ideal is the ideal

adiabatic wall temperature computed from Taw,ideal = T∞(1 + 1

2

(γ − 1)M2). This latter method

yielded almost the same results as enforcing zero wall temperature gradient, which insured no heat

flux at the wall. The freestream T∞ was taken to be 540 R.

As mentioned earlier, the wall boundary conditions for turbulent quantities were those recommended by Menter [10]. Although not shown, the boundary condition on ωwall was varied by a

factor of 10 in both directions, but this change did not have an appreciable influence on the results.

The farfield boundary conditions for turbulent quantities were determined from Tu = 0.08165%

and μt,∞/μ∞ = (2 × 10−7)Re. For the results to be shown, Re = 107 over the length of a plate 2

nondimensional units long. Thus, Re per unit length was 5 × 106and μt,∞/μ∞ = 1.0.

The finest grid employed was 273 × 193, with 225 points on the plate and 49 points leading up

to the plate (where symmetry was imposed). Nondimensional minimum normal spacing at the wall

was approximately ∆y = 1 × 10−6, yielding average y

+ = 0.2 (or less for higher Mach numbers).

There was streamwise clustering near the plate leading edge, as shown in Fig. 5, for which only

every fourth grid point is shown for clarity. For supersonic Mach number cases, it was necessary to

employ a flux limiter in the computations.

Table 1 summarizes the cases computed. For the purposes of this study, the wall temperature

is defined as “hot” or “cold” depending on whether it is above or below the ideal adiabatic wall

temperature. Note that the M = 2, Tw/T∞ = 2 case is only slightly hot, with wall temperature

only slightly above adiabatic.

A grid study for the M = 5 adiabatic wall case was conducted using the SST model on the

273×193 (fine grid), along with two successively coarser grids for which every other grid point was

removed in each coordinate direction (medium: 137 × 97; coarse: 69 × 49). Results are shown in

Fig. 6. The biggest differences were near the plate leading edge, particularly for the coarse level. But

12
over most of the plate the medium and fine grids yielded very close results. For example, between

Rex = 5 × 106and the trailing edge of the plate, the Cf results on the medium and fine grids

differed by less than 0.2%.

In order to get an idea about the magnitude of the computed turbulent kinetic energy relative to

the square of the velocity, profiles of local k/U2are shown as a function of y in the boundary layer

at the x-location where Rex = 5 × 106for several different cases in Fig. 7(a). The maximum level

was only about 3%. Fig. 7(b) shows the value of another quantity sometimes used to ascertain the

compressibility effects of turbulence (albeit most commonly for free shear layer applications) [21],

the turbulence Mach number MT = (√

2k)/a, where a is the local speed of sound. The highest

levels in the boundary layer at a given freestream Mach number occur for the cold-wall cases. For

example, for M = 10, Tw/T∞ = 1, the peak MT reaches approximately 0.5.

Results for the adiabatic wall cases using the finest grid are shown in comparison with van Driest

and Spalding/Chi correlations in Fig. 8. The CFD results captured the correct trends compared to

theory, although the tendency of numerically-induced transition in the CFD to occur further aft

with increasing Mach number should be noted [42]. Compared to the correlations, the SST model

slightly underpredicted turbulent skin friction for the subsonic Mach number case and overpredicted

the correlations for the hypersonic Mach numbers. The Wilcox06 model produced similar Cf as

SST for all three cases, although it had a greater tendency to remain laminar than SST as the Mach

number increased. (Although not shown, increasing freestream Tu could shift the transition location

for Wilcox06 forward.)

Effects of a different code (VULCAN with SST model), as well as effects of including two

different compressibility corrections, are shown in Fig. 9. The VULCAN results are seen to be

relatively close to the CFL3D results on the same grid, yielding slightly lower Cf levels. Regarding

the compressibility corrections, generally speaking, for adiabatic wall with freestream M ≤ 10 there

is only a fairly small influence on the results. Both the Wilcox and the Zeman corrections reduce the

skin friction, with the Wilcox correction having the larger effect. Note that Wilcox [1] reported a

larger decrease in Cf with the Zeman correction because he used different coefficients (his version

of the Zeman model was designed for free shear flows).

Results for two cold-wall cases are shown in Fig. 10. Again, both CFL3D and VULCAN (using

SST) yield skin friction levels that are very close. These SST results are significantly high in comparison with the correlations. It is generally well-known that simple algebraic turbulence models

such as Baldwin-Lomax [43] can perform reasonably well for attached hypersonic boundary layer

flows, provided that the definition of y

+ in the van Driest damping function uses local values for ρ

and μ (rather than wall values) [44], as follows: y

+ =

p

(ρτw)y/μ. As shown here, using this version of Baldwin-Lomax for the two cold-wall cases yields better predictions than SST, in reasonably

good agreement with the van Driest correlation. Employing SST with the Wilcox compressibility

correction (SST + Wilcox cc) lowers skin friction significantly. Both CFL3D and VULCAN produce

nearly identical results. Although results still lie within the band defined by the two correlations, the

SST + Wilcox cc results are quite a bit lower than those of Baldwin-Lomax. Results with the Zeman

correction agree better with Baldwin-Lomax results and the van Driest correlation. As shown in

Fig. 11, the Baldwin-Lomax model and SST + Zeman cc produce lower eddy viscosity values than

uncorrected SST very near the wall (corresponding to y

+ < O(100)). The lower levels produce a

“less turbulent” profile, and consequently lower wall skin friction.

Results for two hot-wall cases are shown in Fig. 12. In these cases, the results using SST generally fell within the shaded band defined by the two correlations; for M = 0.2, Tw/T∞ = 5 results

were closer to the van Driest correlation, and for M = 5, Tw/T∞ = 20 results were closer to the

Spalding/Chi correlation. For these cases, the compressibility corrections made almost no difference

in the results.

Using the same plot format shown earlier in Fig. 3, CFD results using the SST model (without

and with the Zeman compressibility correction) for various cases in terms of the ratio Cf /Cf,incomp

are over-plotted alongside the correlations for Rex = 5 × 106in Figs. 13(a) and (b). In all cases the

Cf,incomp used was the CFD result for M = 0.2 with adiabatic wall temperature. The SST results

13
are given by the large filled-in symbols. The trends discussed above can be clearly discerned in this

plot. Although overall the general effects of Mach number and wall temperature on skin friction

can be qualitatively predicted by SST with no explicit compressibility correction, Fig. 13(a) shows

that adiabatic and cold-wall cases are consistently overpredicted quantitatively as Mach number

is increased (up to about 30% at M = 10 for adiabatic walls, and up to as much as 40 − 100% at

M = 10 for cold walls). On the other hand, hot-wall cases yielded results that lay in or near the band

defined by the two correlations. Fig. 13(b) shows that SST results using the Zeman compressibility

correction are significantly closer to the correlations.

Results can also be plotted as a function of Tw/Taw,ideal. Fig. 14 shows van Driest and Spalding/Chi correlations for both M = 5 and M = 10, along with current SST results. The SST results

with no compressibility correction mostly lie above the shaded regions defined by the two correlations, except for the hot wall cases where results start to fall below the Spalding/Chi correlation.

When the Zeman compressibility correction is employed, cold wall results are improved relative to

the correlations, while hot wall results are essentially not changed at all.

5 Conclusions

The most widely-used compressibility corrections for k-ω models for high Mach number boundary

layer flows are based on improvements intended for free shear applications. As such, these corrections are often unacceptable for boundary layer flows, and many researchers prefer to employ no

corrections at all. Although it should be borne in mind that there is some uncertainty in the theoretical correlations (especially at Mach numbers well above 5), it appears that the uncorrected k-ω

models perform progressively worse – particularly for cold walls – as the Mach number is increased

in the hypersonic regime. As is well-known, simple algebraic models such as Baldwin-Lomax perform better compared to experiment and correlations in these circumstances.

There is still no clarity about whether dilatation-dissipation and pressure-dilatation effects are

important in boundary layers or not, particularly at the higher Mach numbers and for cold-wall

hypersonic cases. Anything that reduces near-wall eddy viscosity in these situations can help obtain

better agreement with correlations, but there is currently no strong physical argument for choosing

one “fix” over another. Corrections designed for improving free shear flows tend to over-correct in

the boundary layer and yield wall skin friction (and heat transfer) values that are too low. In this

paper, it was shown that a dilatation-dissipation correction designed by Zeman specifically for use

in boundary layer flows works reasonably well for cold wall cases. Its influence is smaller in the

boundary layer than the popular Wilcox correction.

The physical modeling needed to improve wall skin friction predictions in highly compressible

boundary layer flows has yet to be formulated and accepted for k-ω turbulence models. Currently,

omitting explicit compressibility corrections works reasonably well only for lower Mach numbers

(e.g., M < 5) or for hot-wall cases. Using the Zeman compressibility correction (formulated for

boundary layers) improves high-Mach-number cold wall results, but it would be an insufficient

correction for free shear flows. Better overall physics-based compressible turbulence modeling is

clearly needed.

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Compressible Turbulent Boundary Layers,” AIAA Journal, Vol. 31, No. 9, 1993, pp. 1600–1604.

41. White, J. A., and Morrison, J. H., “Pseudo-Temporal Multi-Grid Relaxation Scheme for Solving

the Parabolized Navier-Stokes Equations,” AIAA Paper 99-3360, June 1999.

16
42. Rumsey, C. L. and Spalart, P. R., “Turbulence Model Behavior in Low Reynolds Number Regions of Aerodynamic Flowfields,” AIAA Paper 2008-4403, June 2008.

43. Baldwin, B. S. and Lomax, H., “Thin-Layer Approximation and Algebraic Model for Separated

Turbulent Flows,” AIAA Paper 78-257, June 1978.

44. Dilley, A. D., “Evaluation of CFD Turbulent Heating Prediction Techniques and Comparison

with Hypersonic Experimental Data,” NASA/CR-2001-210837, March 2001.

17
Figure 1. Contours of Fc using Eq. (56).

18
Figure 2. Contours of FReθ

using (a) Eq. (62), (b) Eq. (63), and (c) Eq. (64).

19
Figure 3. Theoretical compressible wall skin friction compared to incompressible level for Rex =

5 × 106, Te = 540 R, using two different correlations.

Figure 4. Theoretical values of compressible wall skin friction as a function of Rex; (a) for selected

cases where the van Driest (solid lines) and Spalding/Chi (dashed lines) correlations agree well, (b)

for selected cases where the correlations differ.

20
Figure 5. Flat plate grid with every fourth grid point removed in each coordinate direction for clarity.

21
Figure 6. Wall skin friction coefficient grid study for M = 5, adiabatic wall, using SST on 3

successive grid sizes.

Figure 7. Profiles of local turbulent quantities in the boundary layer at Rex = 5 × 106, SST model:

(a) k/U2, (b) MT = (√

2k)/a.

22
Figure 8. Wall skin friction coefficients for adiabatic wall cases.

Figure 9. Effect of code and Wilcox/Zeman compressibility corrections on wall skin friction coefficients for adiabatic wall cases.

23
Figure 10. Wall skin friction coefficients for two cold-wall wall cases.

Figure 11. Profiles of nondimensional μt for M = 5 and M = 10 cases, Tw/T∞ = 1, at Rex =

5 × 106.

24
Figure 12. Wall skin friction coefficients for two hot-wall wall cases.

Figure 13. Theoretical compressible wall skin friction compared to incompressible level as a function of Mach number for Rex = 5 × 106

, Te = 540 R, including (a) SST (no compressibility

correction), and (b) SST (Zeman compressibility correction).

25
Figure 14. Compressible wall skin friction as a function of Tw/Taw,ideal for Rex = 5 × 106

,

Te = 540 R, with SST results included.

26

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Technical Memorandum

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4. TITLE AND SUBTITLE

Compressibility Considerations for k-ω Turbulence Models in Hypersonic Boundary Layer

Applications

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644423.02.39.04.10.03

6. AUTHOR(S)

C. L. Rumsey

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NASA Langley Research Center

Hampton, Virginia 23681-2199

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L–19589

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14. ABSTRACT

The ability of k-ω models to predict compressible turbulent skin friction in hypersonic boundary layers is investigated. Although uncorrected two-equation

models can agree well with correlations for hot-wall cases, they tend to perform progressively worse - particularly for cold walls - as the Mach number is

increased in the hypersonic regime. Simple algebraic models such as Baldwin-Lomax perform better compared to experiments and correlations in these

circumstances. Many of the compressibility corrections described in the literature are summarized here. These include corrections that have only a small

influence for k-ω models, or that apply only in specific circumstances. The most widely-used general corrections were designed for use with jet or

mixing-layer free shear flows. A less well-known dilatation-dissipation correction intended for boundary layer flows is also tested, and is shown to agree

reasonably well with the Baldwin-Lomax model at cold-wall conditions. It exhibits a less dramatic influence than the free shear type of correction. There is

clearly a need for improved understanding and better overall physical modeling for turbulence models applied to hypersonic boundary layer flows.

15. SUBJECT TERMS

CFD, hypersonic, turbulence model, compressibility, skin friction

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