# A roughness-dependent model

## Two-equation $k$-$\epsilon$ eddy viscosity model

 $\nu _t = C_{\mu} {{k^2 } \over \epsilon }$ (1)

where: $C_{\mu} = 0.09$

## One-equation eddy viscosity model

 $\nu _t = k^{{1 \over 2}} l$ (2)

## Algebraic eddy viscosity model

 $\nu _t(y) = {C_{\mu}}^{{1 \over 4}} l_m(y) k^{{1 \over 2}}(y)$ (3)

$l_m$ is the mixing length.

### Algebraic model for the turbulent kinetic energy

 $k^{{1 \over 2}}(y) = {1 \over {C_{\mu}}^{{1 \over 4}}} u_\tau e^{\frac{-y}{A}}$ (4)

$u_\tau$ is the shear velocity and $A$ a model parameter.

### Algebraic model for the mixing length, based on (4) [Absi (2006)]

 $l_m(y) = \kappa \left( A - \left(A - y_0\right) e^{\frac{-(y-y_0)}{A}} \right)$ (5)

$\kappa = 0.4$, $y_0$ is the hydrodynamic roughness

### the algebraic eddy viscosity model is therefore

 $\nu _t(y) = \kappa \left( A - \left(A - y_0\right) e^{\frac{-(y-y_0)}{A}} \right) u_\tau e^{\frac{-y}{A}}$ (6)

for a smooth wall ($y_0 = 0$):

 $\nu _t(y) = \kappa A \left( 1 - e^{\frac{-y}{A}} \right) u_\tau e^{\frac{-y}{A}}$ (7)