The Scale-Adaptive Simulation (SAS) is an improved URANS formulation, which allows the resolution of the turbulent spectrum in unstable flow conditions. The SAS concept is based on the introduction of the von Karman length-scale into the turbulence scale equation. The information provided by the von Karman length-scale allows SAS models to dynamically adjust to resolved structures in a URANS simulation, which results in a LES-like behavior in unsteady regions of the flowfield. At the same time, the model provides standard RANS capabilities in stable flow regions.
In recent years the turbulent length-scale equation has been revisited by Menter and Egorov [130 (p. 289)]. It was shown that the exact transport equation for the turbulent length-scale, as derived by Rotta [134 (p. 290)], does actually introduce the second derivative of the velocity field and thereby LvK into the turbulent scale equation. In Menter and Egorov [130 (p. 289)], a two-equation turbulence model was presented using a k −νt formulation, which can be operated in RANS and SAS mode. While the further development of this model is still ongoing, it was considered desirable to investigate if the SAS term in the k−νt model could be transformed to existing two-equation models. The target two-equation model is the SST model and this leads to the formulation of the SST-SAS model.
The original version of the SST-SAS model (Menter and Egorov [131 (p. 289)]) has undergone certain evolution and the latest model version has been presented in Egorov and Menter [197 (p. 297)]. One model change is the use of the quadratic length scale ratio
(
L L/ vK)
2in the Equation 2.209 (p. 85) below, rather than the linear form of the original model version. The use of the quadratic length scale ratio is more consistent with the derivation of the model and no major differences to the original model version are expected. Another new model aspect is the explicitly calibrated high wave number damping to satisfy the requirement for an SAS model that a proper damping of the resolved turbulence at the high wave number end of the spectrum (resolution limit of the grid) must be provided.
In the following the latest model version of the SST-SAS model (Egorov and Menter [197 (p. 297)]) will be discussed, which is also the default version in ANSYS CFX.
The governing equations of the SST-SAS model differ from those of the SST RANS model [129 (p. 289)] by the additional SAS source term QSAS in the transport equation for the turbulence eddy frequency ω:
(Eq. 2.207)
+ = − + ⎡
⎣⎢⎛
⎝⎜ + ⎞
⎠⎟
⎤
⎦⎥
∂
∂
∂
∂
∂
∂
∂
ρU k P ρc kω μ ∂
( )
ρk
t x j k μ x
μ σ
k
j j x
t
k j
(Eq. 2.208)
+ = − + + ⎡
⎣⎢ ⎛
⎝⎜ + ⎞
⎠⎟
⎤
⎦⎥ + −
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
ρU ω α P ρβω Q μ F ∂
( ) (
1)
ρω
t x j ω
k k SAS x
μ σ
ω x
ρ σ ω
k x
ω x
2 1 2 1
j j
t
ω j ω2 j j
where σω2 is the σω value for the k −ε regime of the SST model.
The additional source term QSAS reads for the latest model version Egorov and Menter [197 (p. 297)]:
Boundary Conditions
(Eq. 2.209)
= ⎡
⎣
⎢⎢
⎛
⎝⎜
⎞
⎠⎟ − ⋅ ⎛
⎝⎜ ⎞
⎠⎟
⎤
⎦
⎥⎥
∂
∂
∂
∂
∂
∂
∂
QSAS max ρζ κS LL C σρkmax , ∂ , 0
ω ω x
ω x k
k x
k
2 2 2 x
2 1 1
vK Φ 2 j j 2 j j
This SAS source term originates from a term
″
∫
⋅ ′ ′ +−∞
∞
U r uy ( )y v
(
y ry)
dryin Rotta’s transport equation for the correlation-based length scale, see Menter and Egorov [130 (p. 289)]. Since the integral is zero in homogeneous turbulence, it should in general be proportional to a measure related to inhomogeneity. The second velocity derivative U ″ was selected as this measure to ensure that the integral alone is modeled to zero at a constant shear rate, thus leading to U ″2 and ultimately to
(
L L/ vk)
2in the SAS source term (Equation 2.209 (p. 85)).
This model version (Egorov and Menter [197 (p. 297)], Model Version=2007 ) is used as default. In order to recover the original model formulation (Menter and Egorov [131 (p. 289)]), the parameter Model Version must be set to 2005 directly in the CCL:
FLUID MODELS:
...
TURBULENCE MODEL:
Model Version = 2005 Option = SAS SST END
...
END
The model parameters in the SAS source term Equation 2.209 (p. 85) are
= = =
ζ2 3.51, σΦ 2/ 3, C 2
Here L is the length scale of the modeled turbulence
(Eq. 2.210)
= ⋅
L k /
(
cμ1/4 ω)
and the von Karman length scale LvK given by
(Eq. 2.211)
= ′ ′ LvK κS
U
is a three-dimensional generalization of the classic boundary layer definition
= ′ ′ ′
LvKBL κU ( ) /y U ( )y
The first velocity derivative U ′( )y is represented in LvK by S, which is a scalar invariant of the strain rate tensor Sij:
(Eq. 2.212)
= = ⎡
⎣⎢ + ⎤
⎦⎥
∂
∂
∂
S 2S Sij ij, Sij 12 Uxi ∂Ux j
j i
Note, that the same S also directly participates in QSAS (Equation 2.209 (p. 85)) and in the turbulence production term Pk=μtS2.
The second velocity derivative U ′ ′( )y is generalized to 3-D using the magnitude of the velocity Laplacian:
Scale-Adaptive Simulation Theory
(Eq. 2.213)
′ ′ =
∑
⎛⎝
⎜⎜
∂
∂ ∂
⎞
⎠
⎟⎟
U U
x x
i
i
j j
( )
2 2
As a result, L and LvK are both equal to (κy) in the logarithmic part of the boundary layer, where κ=0.41 is the von Karman constant.
Beside the use of the quadratic length scale ratio
(
L L/ vK)
2, the latest model version provides for the direct control of the high wave number damping. Two formulations are available:
1. The first formulation is the default and is realized by a lower constraint on the LvK value in the following way:
(Eq. 2.214)
= ⎛
⎝
⎜⎜ ⋅ ⎞
⎠
⎟⎟ =
′ ′ −
LvK max κS ,C Δ , Δ Ω
U S κζ
β c α CV
(
/ μ)
2 1/3This limiter is proportional to the grid cell size Δ, which is calculated as the cubic root of the control volume size ΩCV. The purpose of this limiter is to control damping of the finest resolved turbulent fluctuations. The structure of the limiter is derived from analyzing the equilibrium eddy viscosity of the SST-SAS model.
Assuming the source term equilibrium (balance between production and destruction of the kinetic energy of turbulence) in both transport equations, one can derive the following relation between the equilibrium eddy viscosity μteq, LvK and S:
(Eq. 2.215)
= ⋅ ⎛
⎝⎜ − ⋅ ⎞
⎠⎟ ⋅ μteq ρ
( ( )
β/c α)
/( )
κζ L Sμ 2 vK
2
This formula has a similar structure as the subgrid scale eddy viscosity in the LES model by Smagorinsky:
= ⋅ ⋅ ⋅
μtLES ρ
(
CS Δ)
2 STherefore it is natural to adopt the Smagorinsky LES model as a reference, when formulating the high wave number damping limiter for the SST-SAS model. The limiter, imposed on the LvK value, must prevent the SAS eddy viscosity from decreasing below the LES subgrid-scale eddy viscosity:
(Eq. 2.216)
≥ μteq μtLES
Substitution of μteq and μtLES in condition above (Equation 2.216 (p. 86)) results in the LvK limiter value used in Equation 2.214 (p. 86). Similar to LES, the high wave number damping is a cumulative effect of the numerical dissipation and the SGS eddy viscosity. The model parameter CS has been calibrated using decaying isotropic turbulence. The default value of CS is 0.11 which provides nearly the same energy spectrum as the Smagorinsky LES model.
2. The second formulation limits the eddy viscosity directly:
(Eq. 2.217)
=
μt max
(
μtSAS,μ)
tLES
The LES-WALE model is used for the calculation of the LES eddy viscosity μtLES, since this model is suitable for transitional flows and does not need wall damping functions. This limiter can be turned on by setting the expert parameter limit sas eddy viscosity = t. The limiter (Equation 2.214 (p. 86) is then automatically turned off. The default value for the WALE model coefficient is 0.5 and can be specified by the expert parameter limit sas eddy viscosity coef=0.5.
Similar to the DES formulation, the SAS model also benefits from a switch in the numerical treatment between the steady and the unsteady regions. In DES, this is achieved by a blending function as proposed by Strelets [58 (p.
281)], which allows the use of a second order upwind scheme with the CFX-Solver in RANS regions and a second order central scheme in unsteady regions. The blending functions are based on several parameters, including the grid spacing and the ratio of vorticity and strain rate, as explained below.
Scale-Adaptive Simulation Theory
Discretization of the Advection Terms
The discretization of the advection is the same as that for the SST-DES model, beside the fact that no RANS-shielding is performed for the SAS-SST model. For details, see Discretization of the Advection Terms (p. 83).