The k-ω Turbulence Model
The k-ω model solves for the turbulent kinetic energy, k, and for the dissipation per unit turbulent kinetic energy, ω. ω is also commonly know as the specific dissipation rate. The CFD Module has the Wilcox revised k-ω model (Ref. 1)
(3-92)
where
(3-93)
(3-94)
where in turn Ωij is the mean rotation-rate tensor
and Sij is the mean strain-rate tensor
Pk is given by Equation 3-76. The following auxiliary relations for the dissipation, ε, and the turbulent mixing length, l, are also used:
(3-95)
Mixing Length Limit and Realizability Constraints
The implementation of the k-ω model relies on the same concepts as the k-ε model (Ref. 10). This means that the following approximations have been used:
where lr is the limit given by the realizability constraints (Equation 3-81 and Equation 3-82).
Wall Boundary Conditions
Automatic Wall Treatment
The automatic wall treatment is a way to obtain an accurate low-Reynolds-number formulation when the mesh allows it, and to fall back on a wall function formulation when the mesh is coarse. It is a blending between the solutions in the linear sublayer and the logarithmic layer respectively. For the specific dissipation, these solutions read
(3-96)
The corresponding expressions for the velocity is
(3-97)
where δw is the distance to the nearest wall, κv, is the von Kárman constant (default value 0.41), U|| is the velocity parallel to the wall and B is a constant that by default is set to 5.2. Menter et al. (Ref. 9) suggested the following smooth blending expressions for ω and uτ:
(3-98)
and
(3-99)
These expression can be combined with the lift-off concept shown in Figure 3-7 which gives δw = hw/2. The wall condition for ω is given by Equation 3-98 and the conditions for the momentum equations are a no-penetration condition u n = 0 and a shear stress condition
(3-100)
The k-equation formally fulfills both at the wall and in the log-layer, so this condition is applied for all δw+.
The system given byEquation 3-83 through Equation 3-100 are, however, nonlinear in uτ and not very stable. To circumvent this, a variable u∗, log is introduced (see Ref. 10 and Ref. 11) such that
(3-101)
This is then used in an alternative expression for the friction velocity:
(3-102)
Equation 3-102 is in turn is used to calculate an alternative dimensionless wall distance
(3-103)
Equation 3-102 is used instead of uτ in the expression for ωlog and Equation 3-103 is used instead of δw+ in the expression for uτlog. The traction condition in Equation 3-100 is replaced by
(3-104)
Observe that the variable u+ is calculated using equation Equation 3-99.
The resulting wall resolution, δw+, is available as the postprocessing variable Delta_wPlus.
Wall Functions
When Wall Treatment is set to Wall functions, wall boundaries are treated with the same type of boundary conditions as for the k-ε model (see Wall Functions) with Cμ replaced by and the boundary condition for ω given by
(3-105)
Low Reynolds Number
The k-ω turbulence model can be integrated all the way down to the wall and is consistent with the no-slip condition u = 0. Since all velocities must disappear on the wall, so must k. Hence, k=0 on the wall.
The corresponding boundary condition for ω is
(3-106)
To avoid the singularity at the wall, ω is not solved for in the cells adjacent to a solid wall. Instead, its value is prescribed by Equation 3-106 (using the variable ωw, which only exists in those cells). Accurate solutions in the near-wall region require that,
(3-107)
where uτ is the friction velocity which is calculated from the wall shear-stress τw,
(3-108)
The boundary variable Distance to cell center in viscous units, lplus_cc, is available to ensure that the mesh is fine enough. According to Equation 3-107, should be about 0.5. Observe that very small values of can reduce the convergence rate.
Since the ωw requires the wall distance, a wall distance equation must be solved prior to solving a k-ω model with low-Reynolds-number wall treatment.
Initial Values
The default initial values are the same as for the k-ε model (see Initial Values) but with the initial value of ω given by
Scaling for Time-Dependent Simulations
The k-ω model applies absolute scales of the same type as the k-ε model (see Scaling for Time-Dependent Simulations) except that the scale for ω is given by
Model Properties
The k-ω model can in many cases give results that are superior to those obtained with the k-ε model (Ref. 1). It behaves, for example, much better for flat plate flows with adverse or favorable pressure gradients. However, there are two main drawbacks. The first is that the k-ω model can display a relatively strong sensitivity to free stream inlet values of ω. The other is that the k-ω model is numerically less robust than the k-ε model.