THEORY OF FULLY DISPERSIVE MODEL¶

Equations¶

The 3D equations, describing fully dispersive non-hydrodynamic motion, can be written in a generalized conservative form as

(142)¶\[\frac{\partial {\bf \Psi}}{\partial t} + \nabla \cdot {\bf \Theta}({\bf \Psi}) = {\bf S}\]

where $\nabla = \left(\frac{\partial }{\partial x}, \frac{\partial }{\partial y}, \frac{\partial }{\partial \sigma} \right)$. ${\bf \Psi}$ and ${\bf \Theta}({\bf \Psi})$ are the vector of conserved variables and the flux vector function respectively.: (143)¶\[ \begin{align}\begin{aligned}{\bf \Psi} = \begin{bmatrix}\\\begin{split} D \\\end{split}\\\begin{split} Du \\\end{split}\\\begin{split} Dv \\\end{split}\\ D\omega\\ \end{bmatrix}\end{aligned}\end{align} \]

(144)¶\[ \begin{align}\begin{aligned}{\bf \Theta} = \begin{bmatrix}\\\begin{split} Du {\bf i} + Dv {\bf j} + \omega {\bf k} \\ (Duu+(\frac{1}{2}g \eta^2+gh\eta)) {\bf i} + Duv{\bf j} + u\omega {\bf k} \\ Duv {\bf i} + (Dvv +(\frac{1}{2}g \eta^2+gh\eta) ){\bf j} + v\omega {\bf k} \\ Duw {\bf i} + Dvw{\bf j} + w \omega {\bf k}\end{split}\\ \end{bmatrix}\end{aligned}\end{align} \]

where $D = h + \eta$, $h$ is water depth and $\eta$ is surface elevation, ($u,v,w$) represent velocity components in the Cartesian coordinate system ($x^*, y^*, z^*$) and $\omega$ is the vertical velocity defined in the $\sigma$ coordinate image domain. The reason for using the $\sigma$ coordinate system will be described in the Flow Surface Technique in the next section.

The source term on the right hand side includes three source components:

(145)¶\[{\bf S} = {\bf S}_h +{\bf S}_p+{\bf S}_\tau\]

where ${\bf S}_h, {\bf S}_p$, and ${\bf S}_\tau$ represent the bottom slope term, dynamic pressure gradient and turbulent mixing respectively. Associated with the splitting method in the study, the dynamic pressure term is expressed below.

(146)¶\[ \begin{align}\begin{aligned}{\bf S}_h = \begin{bmatrix}\\\begin{split} g\eta\frac{\partial{h}}{\partial{x}} \\ g\eta\frac{\partial{h}}{\partial{y}} \\ 0\end{split}\\ \end{bmatrix}\end{aligned}\end{align} \]

(147)¶\[ \begin{align}\begin{aligned}{\bf S}_p = \begin{bmatrix}\\\begin{split} -\frac{D}{\rho}(\frac{\partial p}{\partial x}+\frac{\partial p}{\partial \sigma} \frac{\partial \sigma}{\partial x^*}) \\ -\frac{D}{\rho}(\frac{\partial p}{\partial y}+\frac{\partial p}{\partial \sigma} \frac{\partial \sigma}{\partial y^*}) \\ -\frac{1}{\rho}\frac{\partial p}{\partial \sigma}\end{split}\\ \end{bmatrix}\end{aligned}\end{align} \]

(148)¶\[ \begin{align}\begin{aligned}{\bf S}_{\tau} = \begin{bmatrix}\\\begin{split} DS_{\tau x} \\ DS_{\tau y} \\ DS_{\tau z}\end{split}\\ \end{bmatrix}\end{aligned}\end{align} \]

where $p$ represents the dynamic pressure.

Flow Surface Technique¶

Different from a traditional CFD model where the water surface is governed by a separate surface equation such as in the Volume of Fluid method (VOF) or the Marker and Cell (MAC) method, a so-called``surface flow” technique was used by transforming the Navier-Stokes equations in Cartesian coordinates into the $sigma$ coordinates. The governing equation for free surface can be written as

(149)¶\[\frac{\partial D}{\partial t}+\frac{\partial }{\partial x} (D \int_0^1 u d \sigma)+\frac{\partial }{\partial y} (D \int_0^1 v d \sigma)=0\]

The use of $\sigma$ coordinate is consistent with FUNWAVE-TVD, in which the reference elevation $z_\alpha$ is in the $\sigma$ coordinate as in Kennedy et al. (2001).