714 D. Guézengar et al.
1. Introduction
One of the leading effect of compressibility in supersonic boundary layers is the occurrence of large density gradients across the layer. As a result, mean velocity scaling differs from its subsonic equivalent and the correct
inner similarity law is the van Driest compressible law of the wall van Driest [1]: V
u
τ
= 1
κ ln
yu
τ
ν
w
+ C
∗
, 1
where V is the van Driest transformed velocity: V
=
Z
U
ρ ρ
w 12
dU. 2
Although the constants in 1 can in principle be functions of the friction Mach number Bradshaw [2], the extensive review by Fernholz and Finley [3] shows that most of the available experimental data follow 1 with
κ = 0.41 and C
∗
∼ 5.2. Expression 1 can therefore be considered as the present consensus that turbulence models should reproduce.
Another effect of supersonic flow is the presence of additional terms in the turbulent kinetic energy budget. These terms are either explicitly present in the exact averaged equations pressure dilatation correlation
p
′
∂u
i
′′
∂x
i
or implicitly included in the definition of the turbulent kinetic energy rate of dissipation dilatation dissipation. In principle, an appropriate compressible modelling should not ignore these terms.
Despite these well-known facts, compressible shear flows are usually computed with little or no modification of the models originally developed for incompressible flows. A standard attitude might be just to move to
the Favre mass-weighted averages and hope for the best. However, several recent works have questioned the validity of such an approach for supersonic boundary layers: Huang et al. [4] shows that the standard k–ε
model is inconsistent with the van Driest law 1 and that the coefficients in turbulence models must be a function of density gradients. They proposed models to reduce this density gradient effect to an insignificant
level. Following the analysis of Huang et al. [4], Aupoix and Viala [5] have experimented with the inclusion of density gradient terms in the dissipation equation.
Due to their success for compressible mixing layer predictions, the use of the so-called compressibility models to evaluate the pressure–dilatation correlations or the dilatation dissipation have also received a lot
of attention Aupoix and Viala [5], Huang et al. [4], Mohammadi and Pironneau [6] for boundary layer computations.
Our goal in this paper is to review the use of different variants of the k–ε model for supersonic boundary layers up to Mach 5. Aside from the basic k–ε model with wall functions, we have studied the use of three
different low-Reynolds models and a two-layer model inspired by the work of Chen and Patel [7]. We will also report on the use of compressibility models and test an original model allowing the Prandtl number σ
ε
introduced to model the diffusion of dissipation rate, to be dependent on density variations in a way originally suggested by Huang et al. [4]. The test-cases on which these studies have been conducted are the Mach 1.76
experiment of Dussauge et al. Fernholz and Finley [8] and the Mach 4.52 experiment of Mabey et al. Fernholz and Finley [9].
Computations of turbulent flows are done with a numerical method that adds to turbulence modelling an extra numerical modelling. Many reports on turbulence model comparisons give little or no indications on
the numerical conditions boundary and initial conditions, mesh size, numerical dissipation, etc. used for the computations. As a result, it is sometimes difficult to clearly identify what the respective influences
EUROPEAN JOURNAL OF MECHANICS – BFLUIDS, VOL.
18
, N
◦
4, 1999
Supersonic boundary layer computations 715
of models and of numerics are. We believe that such effects are more important in compressible flows than in incompressible ones. Therefore to make fair comparisons, for each physical modelling near-wall,
compressibility and density variation effects, test-cases were performed with the same numerical method, in order to solve the averaged Navier–Stokes equations coupled to the k–ε model. In this paper, we also give
some examples of the influence of the numerical conditions on the results and emphasize the role that proper boundary conditions can have.
2. Description of the two-equation models
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