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.

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

For local equilibrium, we are able to find the mean velocity profile $u$ from the turbulent kinetic energy $k$ (equation 4) and the mixing length $l_m$ (equation 5), by: ${{d u} \over {d y}} = C_{\mu}^{1 \over 4} {{k^{1 \over 2}} \over {l_m}}$ (8)

Figure (1) shows that the velocity profile obtained from equations (8), (4) and (5) (solid line) is more accurate than the logarithmic velocity profile (dash-dotted line).

Figure 1, 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. (1998)]). 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)]); Figure from [Absi (2006)].