12.15 Providing an Initial Guess for $k$ and $\epsilon$ (or $k$ and $\omega$)

For flows using one of the $k$- $\epsilon$ models, one of the $k$- $\omega$ models, or the RSM, the converged solutions or (for unsteady calculations) the solutions after a sufficiently long time has elapsed should be independent of the initial values for $k$ and $\epsilon$ (or $k$ and $\omega$). For better convergence, however, it is beneficial to use a reasonable initial guess for $k$ and $\epsilon$ (or $k$ and $\omega$).

In general, it is recommended that you start from a fully-developed state of turbulence. When you use the enhanced wall treatment for the $k$- $\epsilon$ models or the RSM, it is critically important to specify fully-developed turbulence fields. Guidelines are provided below.

• If you were able to specify reasonable boundary conditions at the inlet, it may be a good idea to compute the initial values for $k$ and $\epsilon$ (or $k$ and $\omega$) in the whole domain from these boundary values. (See Section  26.9 for details.)

• For more complex flows (e.g., flows with multiple inlets with different conditions) it may be better to specify the initial values in terms of turbulence intensity. 5-10% is enough to represent fully-developed turbulence. The values of $k$ can then be computed from the turbulence intensity and the characteristic mean velocity magnitude of your problem ( $k = 1.5(I u_{\rm avg})^2$).
  
  

  You should specify an initial guess for $\epsilon$ so that the resulting eddy viscosity ( $C_{\mu} \frac{k^2}{\epsilon}$) is sufficiently large in comparison to the molecular viscosity. In fully-developed turbulence, the turbulent viscosity is roughly two orders of magnitude larger than the molecular viscosity. From this, you can compute $\epsilon$.

where $\frac{\mu_T}{\mu}$ is the turbulent viscosity ratio which can be prescribed at the inlet and then used for the domain initialization.

Note that, for the RSM, Reynolds stresses are initialized automatically using Equations  12.14-1 and 12.14-2.