# A roughness-dependent model

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 Revision as of 15:07, 19 June 2007 (view source)R.absi (Talk | contribs) (→Algebraic eddy viscosity model)← Older edit Revision as of 15:10, 19 June 2007 (view source)R.absi (Talk | contribs) (→Algebraic eddy viscosity model)Newer edit → Line 20: Line 20: $l_m$ is the mixing length. $l_m$ is the mixing length. - Algebraic model for the turbulent kinetic Energy: + ===Algebraic model for the turbulent kinetic Energy===
$[itex] k^{{1 \over 2}}(y) = {1 \over {C_{\mu}}^{{1 \over 4}}} u_\tau e^{\frac{-y}{A}} k^{{1 \over 2}}(y) = {1 \over {C_{\mu}}^{{1 \over 4}}} u_\tau e^{\frac{-y}{A}}$(4)
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(4)
- $u_\tau$ is the shear velocity + $u_\tau$ is the shear velocity and $A$ a model parameter. - Algebraic model for the mixing length, based on (4) : + ===Algebraic model for the mixing length, based on (4)===
$[itex] Line 34: Line 34: [itex]\kappa = 0.4$, $y_0$ is the hydrodynamic roughness $\kappa = 0.4$, $y_0$ is the hydrodynamic roughness - the algebraic eddy viscosity model is therefore: + ===the algebraic eddy viscosity model is therefore===
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## Two-equation eddy viscosity model

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

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

## References

• Absi, R. (2006), "A roughness and time dependent mixing length equation", Journal of Hydraulic, Coastal and Environmental Engineering, Japan Society of Civil Engineers, (Doboku Gakkai Ronbunshuu B), Vol. 62, No. 4, pp.437-446.