CMS-Wave:TR3-Chap2

Overview

The analytical and idealized cases described in this chapter were selected for verification of CMS-Flow to confirm that the intended numerical algorithms have been correctly implemented. These cases have an ID, the first two characters identifies Category number, followed by a dash and the Example number under the Category. For example, test case C1-Ex1 refers to Category 1 - Example 1. This notation is used henceforth in this report. Four goodness-of-fit statistics are used to assess the model performance and are defined in Appendix A. The Category 1 V&V test cases completed are listed below. Additional cases are under investigation and will be included in future reports. Category 1 tests cases completed are:

1. Wind setup in a flat basin
2. Wind-driven flow in a circular basin
3. Tidal propagation in a quarter annulus
4. Transcritical flow over a bump
5. Long-wave runup over a frictionless slope

Test C1-Ex1: Wind Setup in a Flat Basin

Purpose

This verification test is designed to test the most basic model capabilities by solving the most reduced or simplified form of the governing equations in which only the water level gradient balances the wind surface drag. The specific model features/aspects to be tested are (1) spatially constant wind fields, (2) water surface gradient implementation, and (3) land-water boundary condition.

Problem and Analytical Solution

Assuming a closed basin with a spatially constant, steady state wind in one direction, no advection, diffusion, bottom friction, waves or Coriolis force, the momentum equations reduce to

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \rho g h \frac{\partial \eta}{\partial y} = \rho_a C_d \left|W\right| W}

(1)

where $h=\zeta +\eta$ is the total water depth, $\zeta$ is the still water depth, $\eta$ is the water surface elevation (water level) with respect to the still water level, $C_{d}$ is the wind drag coefficient, $y$ is the coordinate in the direction of the wind, $g$ is the gravitational acceleration, $\rho$ is the water density, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \rho_a} is the air density, and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle W} is the wind speed. Assuming a constant wind drag coefficient, the following analytical expression for the water level may be obtained by integrating the above equation (Dean and Dalrymple 1984)

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \eta = \sqrt{\frac{2\rho_aC_d\left|W\right|W}{\rho g}\left(y+C\right)+\zeta^2 }-\zeta}

(2)