Governing Equations

Cabernet2D uses the following 2D shallow-water equations.

\[\frac{\partial h}{\partial t}+\frac{\partial (uh)}{\partial x}+ \frac{\partial (vh)}{\partial y}=0\]
\[\frac{\partial u}{\partial t}+u \frac{\partial u}{\partial x}+ v \frac{\partial u}{\partial y}= -g \frac{\partial H}{\partial x} -\frac{\tau_x}{\rho h}+ \frac{\partial}{\partial x} (\nu_t \frac{\partial u}{\partial x})+ \frac{\partial}{\partial y} (\nu_t \frac{\partial u}{\partial y})\]
\[\frac{\partial v}{\partial t}+u \frac{\partial v}{\partial x}+ v \frac{\partial v}{\partial y}= -g \frac{\partial H}{\partial y} -\frac{\tau_y}{\rho h}+ \frac{\partial}{\partial x} (\nu_t \frac{\partial v}{\partial x})+ \frac{\partial}{\partial y} (\nu_t \frac{\partial v}{\partial y})\]

Here \(x, y\) are the mutually orthogonal planar coordinate axes, \(t\) is time, \(u, v\) are the depth-averaged velocity components in the \(x, y\) directions, \(h\) is the water depth, \(H\) is the water surface elevation, \(g\) is gravitational acceleration, \(\tau_x, \tau_y\) are the bed shear stresses in the \(x, y\) directions, \(\rho\) is the density of water, and \(\nu_t\) is the eddy viscosity coefficient.

Using Manning’s formula as the resistance law, the bed shear stress is given by:

\[\frac{\tau_x}{\rho h}= \frac{gn^2}{h^{4/3}} u\sqrt{u^2+v^2}, \ \ \ \frac{\tau_y}{\rho h}= \frac{gn^2}{h^{4/3}} v\sqrt{u^2+v^2}\]

where \(n\) is Manning’s roughness coefficient. The eddy viscosity coefficient is evaluated with a zero-equation model:

\[\nu_t = \frac{\kappa}{6} u_\ast h\]

where \(u_\ast\) is the friction velocity

\[u_\ast=\sqrt{ghI_f}, \ \ \ I_f=\frac{n^2 (u^2+v^2)}{h^{4/3}}\]

and \(I_f\) is the friction slope.

The governing equations for sediment transport and bed variation are the same as in Nays2DH and are omitted here.

Numerical Method

As in Nays2DH, the governing equations above are transformed into boundary-fitted 2D general coordinates and solved on a structured grid with a staggered arrangement, in which water depth is computed at cell centers and velocity at cell edges. At each time step, the momentum equations are split into advection and non-advection parts: the advection part is solved with the CIP method, and the non-advection part is solved iteratively together with the momentum equations so that \(u, v, h\) are simultaneously satisfied.

Cabernet2D computes confluent tributaries and bifurcating branches together with the main channel as a single integrated computational domain. For each river network, the location and condition of inflow and outflow boundaries are specified through the GUI.

Computation Procedure

The computation proceeds through the following steps.

  1. Create the computational grid

  2. Set boundary conditions such as inflow and outflow

  3. Set the calculation conditions

  4. Run the computation

  5. Display the computation results