B.2 Governing Equations
TUFLOW FV solves the Non-Linear Shallow Water Equations (NLSWE), including viscous flux terms and various source terms on a flexible mesh comprised of triangular and quadrilateral elements.
The NLSWE describe the conservation of mass and momentum in an incompressible fluid under hydrostatic pressure and the Boussinesq approximation. The equations relate the time-derivative of the conserved variables to flux-gradient and source terms:
\[\begin{equation} \frac{\partial \mathbf{U}}{\partial t} + \nabla \cdot \mathbf{F}(\mathbf{U}) = \mathbf{S}(\mathbf{U}) \tag{B.1} \end{equation}\]
The finite-volume schemes are derived from the conservative integral form of the NLSWE, obtained by integrating the conservation equations over a control volume \(\Omega\):
\[\begin{equation} \int_{\Omega} \frac{\partial \mathbf{U}}{\partial t} d\Omega + \int_{\Omega} \nabla \cdot \mathbf{F}(\mathbf{U}) d\Omega = \int_{\Omega} \mathbf{S}(\mathbf{U}) d\Omega \tag{B.2} \end{equation}\]
Applying Gauss’s theorem, the flux-gradient volume integral is rewritten as a boundary integral:
\[\begin{equation} \frac{\partial}{\partial t} \int_{\Omega} \mathbf{U} d\Omega + \oint_{\partial \Omega} (\mathbf{F} \cdot \mathbf{n}) ds = \int_{\Omega} \mathbf{S}(\mathbf{U}) d\Omega \tag{B.3} \end{equation}\]
where \(\int_{\Omega} d\) represents volume integrals, \(\oint_{\partial \Omega} ds\) represents a boundary integral, and \(\mathbf{n}\) is the unit-normal vector.
The NLSWE conserved variables are volume (depth), x-momentum, and y-momentum:
\[\begin{equation} \mathbf{U} = \begin{bmatrix} h \\ hu \\ hv \end{bmatrix} \tag{B.4} \end{equation}\]
where: - \(h\) is depth - \(u\) is x-velocity - \(v\) is y-velocity
The x, y, and z components of the inviscid flux \(\mathbf{F}^I\) and viscous flux \(\mathbf{F}^V\) terms in the NLSWE are given via Equation (B.5)
\[\begin{equation} \mathbf{F}_x^I = \begin{bmatrix} hu \\ hu^2 + \frac{1}{2} gh^2 \\ huv \end{bmatrix}, \quad \mathbf{F}_x^V \approx \begin{bmatrix} 0 \\ -h \nu_{t,h} \frac{\partial u}{\partial x} \\ -h \nu_{t,h} \frac{\partial v}{\partial x} \end{bmatrix} \end{equation}\]
\[\begin{equation} \mathbf{F}_y^I = \begin{bmatrix} hv \\ huv \\ hv^2 + \frac{1}{2} gh^2 \end{bmatrix}, \quad \mathbf{F}_y^V \approx \begin{bmatrix} 0 \\ -h \nu_{t,h} \frac{\partial u}{\partial y} \\ -h \nu_{t,h} \frac{\partial v}{\partial y} \end{bmatrix} \tag{B.5} \end{equation}\]
\[\begin{equation} \mathbf{F}_z^I = \begin{bmatrix} hw \\ hwu \\ hwv \end{bmatrix}, \quad \mathbf{F}_z^V \approx \begin{bmatrix} 0 \\ -\nu_{t,v} \frac{\partial u}{\partial z} \\ -\nu_{t,v} \frac{\partial v}{\partial z} \end{bmatrix} \end{equation}\]
The source terms \(\mathbf{S}(\mathbf{U})\) include bed slope effects, Coriolis forces, atmospheric pressure gradients, and other external forces.
\[\begin{equation} \mathbf{S} = \begin{bmatrix} 0 \\ gh \frac{\partial z_b}{\partial x} + fv h - \frac{h}{\rho_0} \frac{\partial p_a}{\partial x} - \frac{hg}{\rho_0} \int_{z}^{\eta} \frac{\partial \rho}{\partial x} dz - \frac{1}{\rho_0} \left( \frac{\partial s_{xx}}{\partial x} + \frac{\partial s_{xy}}{\partial y} \right) + \frac{\tau_{sx}}{\rho_0} - \frac{\tau_{bx}}{\rho_0} \\ gh \frac{\partial z_b}{\partial y} - fu h - \frac{h}{\rho_0} \frac{\partial p_a}{\partial y} - \frac{hg}{\rho_0} \int_{z}^{\eta} \frac{\partial \rho}{\partial y} dz - \frac{1}{\rho_0} \left( \frac{\partial s_{yx}}{\partial x} + \frac{\partial s_{yy}}{\partial y} \right) + \frac{\tau_{sy}}{\rho_0} - \frac{\tau_{by}}{\rho_0} \end{bmatrix} \tag{B.6} \end{equation}\]
where:
- \(\frac{\partial z_b}{\partial x}, \frac{\partial z_b}{\partial y}\) are the x- and y-components of bed slope,
- \(f\) is the Coriolis coefficient,
- \(\rho\) is the local fluid density, \(\rho_0\) is the reference density, and \(p_a\) is the mean sea level pressure,
- \(s_{ij}\) is the short-wave radiation stress tensor, and
- \(\tau_s\) and \(\tau_b\) are, respectively, the surface and bottom shear stress terms (where applicable).
Other source terms not included above include inflow and outflow to and from the water column.