## 2.2.1 Riverbed

The slip boundary condition is assumed to solve the velocity near the riverbed, and the production term of kinetic energy and the turbulence kinetic energy dissipation rate are calculated by wall-function approach [9]:

\({G_k} \approx {\tau _w}\frac{{{\tau _w}}}{{\kappa \rho c_{\mu }^{{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0pt} 4}}}k_{p}^{{1/2}}\Delta {z_p}}}\) , \(\varepsilon =\frac{{c_{\mu }^{{{3 \mathord{\left/ {\vphantom {3 4}} \right. \kern-0pt} 4}}}k_{p}^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0pt} 2}}}}}{{\kappa \Delta {z_p}}}\) (7)

The resultant wall shear stress \({\tau _w}\) is related to \({u_p}\) is the velocity parallels the riverbed by log law:

\(\frac{{{u_p}c_{\mu }^{{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0pt} 4}}}k_{p}^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}}}}{{{{{\tau _w}} \mathord{\left/ {\vphantom {{{\tau _w}} \rho }} \right. \kern-0pt} \rho }}}=\left\{ {\begin{array}{*{20}{c}} {\frac{1}{\kappa }\ln \left( {E{z^*}} \right){\kern 1pt} {\kern 1pt} ,\quad {z^*}>11.225} \\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} \quad \;{\kern 1pt} {\kern 1pt} {z^*}{\kern 1pt} {\kern 1pt} {\kern 1pt} \quad {\kern 1pt} {\kern 1pt} ,\quad \,{\kern 1pt} {z^*}<11.225} \end{array}{\kern 1pt} } \right.\,\) with \({z^*}={{c_{\mu }^{{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0pt} 4}}}k_{p}^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}}\Delta {z_p}} \mathord{\left/ {\vphantom {{c_{\mu }^{{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0pt} 4}}}k_{p}^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}}\Delta {z_p}} \upsilon }} \right. \kern-0pt} \upsilon }\) (8)

Where \(\kappa\) is van Karman constant (\(\kappa\)= 0.42); *E* is roughness coefficient, and determined by:

$$E=\exp \left[ {\kappa \left( {B - \Delta B} \right)} \right]$$

9

where *B* is additive constant (*B* = 5.2);\(\Delta B\)is roughness function, and calculated as flows [10]:

\(\Delta B=\left\{ {\begin{array}{*{20}{c}} {\quad \quad \quad \quad \;\;\;\;\quad \quad 0\quad \quad \quad \quad \quad \quad \quad \quad \,\,\;\;,\,\;k_{s}^{+}<2.25} \\ {\left( {B - 8.5+\frac{1}{\kappa }\ln k_{s}^{+}} \right)\sin \left[ {0.4258\left( {\ln k_{s}^{+} - 0.811} \right)} \right]\;,\;2.25 \leqslant k_{s}^{+}<90\;} \\ {\,\quad \quad \;\quad \,B - 8.5+\frac{1}{\kappa }\ln k_{s}^{+}\quad \,\,\quad \quad \quad \quad \,,\;\,k_{s}^{+} \geqslant 90} \end{array}} \right.\) with \(k_{s}^{+}={u_*}{k_s}/\upsilon\) (10)

where \({u_*}\) is bed shear velocity (\({u_*}\)\(=\sqrt {{{{\tau _w}} \mathord{\left/ {\vphantom {{{\tau _w}} \rho }} \right. \kern-0pt} \rho }}\));\({k_s}\) is equivalent roughness height.