CMS-Flow:Subgrid Turbulence Model

Subgrid Turbulence Model

The eddy viscosity is calculated as $\nu _{t}=(1-\theta _{m})\nu _{tc}+\theta _{m}\nu _{m}$ where $\theta _{m}$ is weighting factor equal to $\theta _{m}=(H_{s}/h)^{3}$ in which $H_{s}$ is the significant wave height and and are the current- and wave-related eddy viscosity components respectively. The wave contribution is included using the equation of KRAUS and LARSON (1991) , where is an empirical coefficient (set to 0.5 here), and is the wave bottom orbital velocity. The current-related eddy viscosity is calculated using a subgrid turbulence in which the turbulent eddy viscosity is a function of the flow gradients and the grid size model of the form (A1) where is a base value approximately equal to the dynamic viscosity, and is an empirical coefficient and is the subgrid mixing length. The mixing length is calculated here as where is an empirical coefficient (Smagorinsky coefficient). Note that if and are set to zero and is set to 0.578 , where is the bed friction coefficient, (A1) reduces to the FALCONER (1980) equation originally used in CMS-Flow.

        ${\frac {\partial }{\partial t}}{\biggl (}{\frac {hC_{t}}{\beta _{t}}}{\biggr )}+{\frac {\partial (U_{j}hC_{t})}{\partial x_{j}}}={\frac {\partial }{\partial x_{j}}}{\biggl [}\nu _{s}h{\frac {\partial (r_{s}C_{t})}{\partial x_{j}}}{\biggr ]}+\alpha _{t}\omega _{s}(C_{t*}-C_{t})$

where $h$ is the total water depth ($h=\zeta +\eta$), $C_{t}$ is the total load concentration, $C_{t*}$ is the sediment transport capacity, $\beta _{t}$ is the total load correction factor, $\nu _{s}$ is the diffusion coefficient, $r_{s}$ is the fraction of suspended sediments, $\alpha _{t}$ is the total load adaptation coefficient, and $\omega _{s}$ is the sediment fall velocity. The concentration capacity may be calculated with either the Lund-CIRP (Carmenen and Larson 2007), the van Rijn (1985), or the Watanabe (1987) transport equations. The advantage of a bed-material approach is that the suspended- and bed-load transport equations are combined into a single equation and there is one less empirical parameter to estimate.