Turbulent Flow

y_p= \Large \frac {y^+  \mu} {u_{\tau}  \rho}

y_H= 2  y^+

The friction velocity ( u_{\tau} ) is calculated as:

u_{\tau} = \sqrt{ \Large \frac {\tau_{\omega}} {\rho}}

Where wall shear stress ( \tau_{\omega} ) depends on the skin friction coefficient ( C_f):

\tau_{\omega} = \Large \frac {1}{2}\normalsize \rho  U^2  C_f

Where U is the free stream velocity of the fluid and the skin friction coefficient ( C_f ) must be estimated since we do not know it, so we will take the flat plate one in turbulent regime.

C_f = (2 \log_{10} (Re) - 0.65)^{-2.3}

For internal flows: C_f= 0.079 Re^{-0.25}

For external flows: C_f= 0.058 Re^{-0.2}

Re = \Large \frac {\rho U L} {\mu}

k = \Large \frac {3} {2} \normalsize U^2 I^2

If the maximum free stream velocity is defined as U_{max} , then the turbulence intensity (I) is calculated as:

I (\%) = \Large \frac {U_{max}-U} {U} \normalsize \cdot 100 = (G-1)\cdot 100

U_{max} = G \cdot U

  \epsilon = C_{\mu} \Large \frac {\rho k^2}{\mu} \normalsize \beta^{-1}

Where C_{\mu} = 0.09 in the methods k-ϵ and k-ω, and β is the turbulent viscosity ratio.

\beta = \Large \frac{\mu_t}{\mu}

  \omega = \Large \frac {\rho k}{\mu} \normalsize \beta^{-1}

1< \beta < 10