# A roughness-dependent model

(Difference between revisions)
 Revision as of 15:09, 21 June 2007 (view source)R.absi (Talk | contribs) (→Algebraic model for the turbulent kinetic energy)← Older edit Revision as of 15:12, 21 June 2007 (view source)R.absi (Talk | contribs) (→One-equation eddy viscosity model)Newer edit → Line 12: Line 12: \nu _t  = k^{{1 \over 2}}  l \nu _t  = k^{{1 \over 2}}  l [/itex]
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+ + [http://www.cfd-online.com/Wiki/Prandtl%27s_one-equation_model One-equation model] ==Algebraic eddy viscosity model== ==Algebraic eddy viscosity 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.

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 $u$ 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 [Absi (2006)]:

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: $y_0 = 0.062 cm$; $h = 145 cm$; $u_\tau = 3.82 cm/s$. b) profile 4: $y_0 = 0.113 cm$; $h = 164.5 cm$; $u_\tau = 3.97 cm/s$ ; values of $y_0 , h, u_\tau$ are from [Sukhodolov et al. (1998)].

## References

• Nezu, I. and Nakagawa, H. (1993), "Turbulence in open-channel flows", A.A. Balkema, Ed. Rotterdam, The Netherlands.
• Sukhodolov A., Thiele M. and Bungartz H. (1998), "Turbulence structure in a river reach with sand bed", Water Resour. Res., 34, pp. 1317-1334.