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)

where in turn Ωij is the mean rotation-rate tensor

and Sij is the mean strain-rate tensor

Pk is given by Equation 3-78. The following auxiliary relations for the dissipation, ε, and the turbulent mixing length, l∗, are also used:

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 3-83 and Equation 3-84).

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:

These expression can be combined with the lift-off concept shown in Figure 3-9 which gives δw = hw/2. The wall condition for ω is given by Equation 3-93 and the conditions for the momentum equations are a no-penetration condition u ⋅ n = 0 and a shear stress condition

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 3-85 through Equation 3-95 are however non-linear in uτ and not very stable. To circumvent this, a variable u∗, log is introduced (see Ref. 10 and Ref. 11) such that

Equation 3-97 is in turn is used to calculate an alternative dimensionless wall distance

Equation 3-97 is used instead of uτ in the expression for ωlog and Equation 3-98 is used instead of δw+ in the expression for uτlog. The traction condition in Equation 3-95 is replaced by

Observe that the variable u+ is calculated using equation Equation 3-94.

The resulting wall resolution, δw+, is available as the postprocessing variable Delta_wPlus.

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

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

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 3-101 (using the variable ωw, which only exists in those cells). Accurate solutions in the near-wall region require that,

where uτ is the friction velocity which is calculated from the wall shear-stress τw,

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 3-102, 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.

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

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

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.