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 3-91. 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 3-96 and Equation 3-97).

where δw is the distance to the nearest wall, κv, is the von Kármán 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 and others (Ref. 9) suggested the following smooth blending expressions for ω and uτ:

These expressions 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 3-113 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 by Equation 3-98 through Equation 3-115 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

Equation 3-117 is in turn is used to calculate an alternative dimensionless wall distance

Equation 3-117 is used instead of uτ in the expression for ωlog and Equation 3-118 is used instead of δw+ in the expression for uτlog. The traction condition in Equation 3-115 is replaced by

Observe that the variable is based on uτ from Equation 3-114.

The postprocessing variable for friction velocity uτw (u_tauWall) is based on the effective traction at the wall and is defined as

while the postprocessing variable for the resulting wall resolution (Delta_wPlus) is based on uτw and is defined as

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 3-121 (using the variable ωw, which only exists in those cells). Accurate solutions in the near-wall region require that the wall resolution in viscous units

where the resolution height is based on the distance between the first computational node inside the domain and the wall. uτ is the friction velocity which is calculated from the wall shear-stress τw,

and is available as the postprocessing variable (u_tauWall). Also, the boundary variable Distance to cell center in viscous units, (lplus_cc), is available to ensure that the mesh is fine enough. is the distance, measured in viscous units, from the wall to the center of the wall adjacent cell, thus and according to Equation 3-122 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 postprocessing variable Wall resolution in viscous units, (Delta_wPlus), is available.

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.