ε, ω, k
An estimation of the turbulent mainstream quantities should take place to have a stable solver.
Turbulent external flow approximations
External flows can be a bit tricky to approximate because it is hard to evaluate the flow downstream considering anything that could have affected the turbulence. This could be other objects, convection over land or the position of the domain in a boundary layer. A good technique for approximation turbulence is by using the Turbulent Viscosity Ratio—the ratio between molecular viscosity and turbulent viscosity. This ratio can be used along with the Turbulence Intensity, freestream velocity and molecular viscosity to determine k, ε and ω using the following technique. To start the calculation, the Turbulent Intensity I, and Viscosity Ratio β need to be approximated by using the table below.
| Turbulence | Reynolds number | Turbulent Intensity |
|---|---|---|
| Low turbulence | 3000 < Re < 5000 | 1% |
| Med turbulence | 5000 < Re < 15000 | 1-5% |
| High turbulence | 15000 < Re < 20000 | 5-20% |
| High turbulence | Re > 100000 | 5-20% |
To calculate k, the following equation can be used:
\[ k = \frac{3}{2}\left( \text{UI} \right)^{2} \]
\[ ε = C_{\mu}\frac{\text{ρk}^{2}}{\mu}β ^{- 1}\ \]
\[ ω = \frac{ρk}{\mu}\ β^{-1}\ \]
Turbulent internal flow approximations
For a fully developed inlet flow, approximating turbulent boundary conditions can be extrapolated using the Reynolds number Re to determine the turbulent intensity I to define the intensity length scale l, the required turbulent boundary conditions can be calculated.
\[ I = 0.16 \ Re_L ^{-1/8} \]
This can be used to calculate a mean approximation for the turbulent boundary conditions
\[ k = \frac{3}{2} U I^{2} \]
\[ ε = C_{\mu}\frac{k^{\frac{3}{2}}}{l} = C_{\mu}\frac{k^{\frac{3}{2}}}{0.07L} \]
\[ ω = \frac{\varepsilon}{kC_{\mu}}\ \]
Turbulent Wall functions
Keeping the focus only in the treatment of the wall, the corresponding wall functions exist:
- εWallFuncion for ε( (fixed value e=0 or better e=1e-8(?) for lowRe calculations):
calculate (for each timestep) the first grid point value by using an algebraic expression derived from the classical logarithmic law-of the wall approach
- kqRWallFunction for k, q, R
in code: Boundary condition for turbulence k, Q, and R when using wall functions. Simply acts as a zero-gradient condition. It appears to be applicable down to yPlus ~ 1, but one should use a fixed value with k=0 or a very small value for y+ <1)
\ omegaWallFunction for omega; Not really a wall function but the b.c. defined by Menter for Omega, i.e. should be used always for kOmega model, independent of y+)
omegawall=60*nu/(β *y+2), with nu=kinematic viscosity at the wall, β =0.075 and y+normal distance between the first fluid node and the nearest wall-> very large value for omega)
The “value” which is specified for the wall functions is only an initial condition