# A roughness-dependent model

(Difference between revisions)
 Revision as of 14:37, 19 June 2007 (view source)R.absi (Talk | contribs) (→Kinematic Eddy Viscosity)← Older edit Revision as of 14:57, 19 June 2007 (view source)R.absi (Talk | contribs) (→Kinematic Eddy Viscosity)Newer edit → Line 1: Line 1: - ==Kinematic Eddy Viscosity== + ==Two-equation eddy viscosity model== - Two-equation model: +
$[itex] \nu _t = C_{\mu} {{k^2 } \over \varepsilon } \nu _t = C_{\mu} {{k^2 } \over \varepsilon } -$ + (1)
- where: $C_{\mu} = 0.09$ + where: + $C_{\mu} = 0.09$ - One-equation model: + ==One-equation eddy viscosity model== +
$[itex] - \nu _t = l k^{{1 \over 2}} = {C_{\mu}}^{1/4} l_m k^{{1 \over 2}} + \nu _t = k^{{1 \over 2}} l - +$(2)
- Algebraic model: + ==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. + + where: +
$[itex] - k^{{1 \over 2}} = {1 \over {C_{\mu}}^{1/4}} + k^{{1 \over 2}}(y) = {1 \over {C_{\mu}}^{{1 \over 4}}} u_\tau e^{\frac{-y}{A}} -$ + [/itex](4)
+ $u_\tau$ is the shear velocity + and: +
+ $+ l_m(y) = \kappa \left( A - \left(A - y_0\right) e^{\frac{-(y-y_0)}{A}} \right) +$(5)
+ $\kappa = 0.4$, $y_0$ is the hydrodynamic roughness + + 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 == == References ==

## 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.

where:

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

$u_\tau$ is the shear velocity

and:

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

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

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.