# A roughness-dependent model

(Difference between revisions)
 Revision as of 20:31, 19 June 2007 (view source)R.absi (Talk | contribs) (→Two-equation eddy viscosity model)← Older edit Revision as of 12:12, 21 June 2007 (view source)R.absi (Talk | contribs) (→Algebraic eddy viscosity model)Newer edit → Line 27: Line 27: $u_\tau$ is the shear velocity and $A$ a model parameter. $u_\tau$ is the shear velocity and $A$ a model parameter. - ===Algebraic model for the mixing length, based on (4) [[#References|[Absi (2006)]]]=== + For steady open channel flows in local equilibrium, where the energy production is balanced by the dissipation, from the modeled k-equation [[#References|[Nezu and Nakagawa (1993)]]] obtained a similar semi-theoretical equation. + + ===Algebraic model for the mixing length=== + + For local equilibrium, an extension of von Kármán’s similarity hypothesis allows to write, with equation (4) [[#References|[Absi (2006)]]]: +
$[itex] l_m(y) = \kappa \left( A - \left(A - y_0\right) e^{\frac{-(y-y_0)}{A}} \right) l_m(y) = \kappa \left( A - \left(A - y_0\right) e^{\frac{-(y-y_0)}{A}} \right)$(5)
[/itex]
(5)
- $\kappa = 0.4$, $y_0$ is the hydrodynamic roughness + $\kappa = 0.4$, $y_0$ is the hydrodynamic roughness. + For a smooth wall ($y_0 = 0$): +
+ $+ l_m(y) = \kappa A \left( 1 - e^{\frac{-y}{A}} \right) +$(6)
===the algebraic eddy viscosity model is therefore=== ===the algebraic eddy viscosity model is therefore=== Line 39: Line 49: \nu _t(y)  = \kappa \left( A - \left(A - y_0\right) e^{\frac{-(y-y_0)}{A}} \right) \nu _t(y)  = \kappa \left( A - \left(A - y_0\right) e^{\frac{-(y-y_0)}{A}} \right) u_\tau  e^{\frac{-y}{A}} u_\tau  e^{\frac{-y}{A}} - [/itex]
(6)
+ [/itex]
(7)
+ + + ==The mean velocity profile== + + In local equilibrium region, we are able to find the mean velocity profile from the mixing length $l_m$ and the turbulent kinetic energy $k$ by: - for a smooth wall ($y_0 = 0$):
$[itex] - \nu _t(y) = \kappa A \left( 1 - e^{\frac{-y}{A}} \right) + {{d U} \over {d y}} = C_{\mu}^{1 \over 4} {{k^{1 \over 2}} \over {l_m}} - u_\tau e^{\frac{-y}{A}} +$(8)
- [/itex]
(7)
+ With equations (4) and (5), we obtain: + + [[Image:fig7a.jpg]] + [[Image:fig7b.jpg]] + + Fig. Vertical distribution of mean flow velocity. + $A = {{h} \over {c_1}}$; $c_1 = 1$; + Dash-dotted line: logarithmic profile; solid line: obtained from equation (8); symbols: experimental data (Sukhodolov et al). a) profile 2: $y_0 = 0.062 cm$; $h = 145 cm$; $U_f = 3.82 cm/s$. b) profile 4: $y_0 = 0.113 cm$; $h = 164.5 cm$; $U_f = 3.97 cm/s$ ; values of $y_0 , h, U_f$  are from  [[#References|[Sukhodolov et al. (1998)]]]. == References == == References ==

## 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.

For steady open channel flows in local equilibrium, where the energy production is balanced by the dissipation, from the modeled k-equation [Nezu and Nakagawa (1993)] obtained a similar semi-theoretical equation.

### Algebraic model for the mixing length

For local equilibrium, an extension of von Kármán’s similarity hypothesis allows to write, with equation (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. For a smooth wall ($y_0 = 0$):

 $l_m(y) = \kappa A \left( 1 - e^{\frac{-y}{A}} \right)$ (6)

### the algebraic eddy viscosity model is therefore

 $\nu _t(y) = \kappa \left( A - \left(A - y_0\right) e^{\frac{-(y-y_0)}{A}} \right) u_\tau e^{\frac{-y}{A}}$ (7)

## The mean velocity profile

In local equilibrium region, we are able to find the mean velocity profile from the mixing length $l_m$ and the turbulent kinetic energy $k$ by:

 ${{d U} \over {d y}} = C_{\mu}^{1 \over 4} {{k^{1 \over 2}} \over {l_m}}$ (8)

With equations (4) and (5), we obtain:

Fig. Vertical distribution of mean flow velocity. $A = {{h} \over {c_1}}$; $c_1 = 1$; Dash-dotted line: logarithmic profile; solid line: obtained from equation (8); symbols: experimental data (Sukhodolov et al). a) profile 2: $y_0 = 0.062 cm$; $h = 145 cm$; $U_f = 3.82 cm/s$. b) profile 4: $y_0 = 0.113 cm$; $h = 164.5 cm$; $U_f = 3.97 cm/s$ ; values of $y_0 , h, U_f$ are from [Sukhodolov et al. (1998)].