International Review of Mechanical Engineering (I.RE.M.E), Vol. 2, N. 4
July 2008
Numerical Study of Transonic Flows Using Various Turbulence
Models
N.Bekka1*, R.Bessaïh2, M. Sellam1
Abstract - In this paper, validation and assessment of various turbulence models are performed
in transonic flows, including an algebraic Baldwin Lomax model, Spalart Allmaras one equation,
and two equation (k-ε, k-ω and SST k-ω) turbulence models. The NACA 0012 airfoil has been
chosen for the turbulence model validation studies. The three test cases selected here included
both attached and separated transonic flows. This study shows that the five turbulence models
provide satisfactory results for transonic attached flows. However, all models, fail to predict the
location of the shock correctly when a strong shock and a shock induced flows separation are
present. Compared to other models, it has been shown that the SST k-ω model is the most robust
model in the prediction of lift coefficient for all cases. Computed results are performed with the
CFD-FASTRAN code by using the fully implicit scheme for time integration, and the upwind Roe
flux difference splitting scheme for space discretization augmented by a high order Min-Mod
limiter to damp spurious oscillations and to allow for sharp shock resolution.
Keywords: Numerical study ,Transonic flows, Turbulence models, Airfoil
Nomenclature
I.
CD drag coefficient
CL lift coefficient
E total energy
k turbulent kinetic energy
M∞ Mach number
P pressure
Pr Prandtl number
Prt turbulent Prandtl number
q j total heat flux
Modern transport and passenger aircraft operate at
cruise flight transonic speeds, corresponding to a Mach
number interval M∞ =0.75-0.85 and a very high
Reynolds numbers.
Transonic flow occurs when there is a mixed sub and
supersonic local flow in the same flow field [1]-[2]. It is
characterized by the development of a closed, wallbounded supersonic flow region at the upper wing
surface. Depending on wing curvature and the angle of
incidence, the appearance of a closed wall-bounded
supersonic flow region at the lower wing surface is also
possible. Transonic airfoil flow is to a large degree
affected by a strong adverse pressure gradient,
shock/boundary-layer interaction, and separation
downstream of the shock. These features as well as the
compressible properties of the flow pose a special
challenge not only for the numerical treatment, but also
for turbulence models.
The aerodynamics of transonic turbulent flows over
airfoils and wings has been studied for decades in
experimental and numerical works. A total of 23 sets of
numerical results from 15 different author groups are
summarized and compared by Holst [3]. These methods
vary from momentum integral boundary-layer methods
coupled with transonic potential inviscid codes to a full
Navier-Stokes methods [4]-[7]. Lai et al. [8] simulated
numerically the turbulent transonic flow over the
NACA0012 and RAE2822 airfoils with and without
shocks, using a pressure-based method, including the
Baldwin-Lomax and k-ε models. Four popular
Re∞ Reynolds number
T temperature
t time
uj component of the velocity
Greek symbols
αc corrected angle of attack
αg geometric angle of attack
γ specific heat ratio
ε dissipation rate of the turbulent kinetic energy
µ dynamic viscosity
µt
eddy viscosity
density of the fluid
τij Reynolds stresses
τˆij
Introduction
total stress tensor
599
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