Analytical solution of a dam-break problem on a wet bed without friction (Stocker)

This example presents the one-dimensional analytical solution proposed by Stocker for the dam-break problem on a wet, horizontal bed without friction.

Model Assumptions

• Instantaneous dam break at position \(x = x_0\) and at time \(t = 0\).

• The bed is flat and horizontal (no slope).

• No bed friction is considered.

• A finite volume of still water of height \(h_0\) is located to the left of the dam at \(0 < x \leq x_0\).

• A shallow layer of water of height \(h_r\) (with \(h_r < h_0\)) exists to the right of the dam (\(x_0 < x\)).

• The fluid is incompressible and inviscid, subject only to gravity.

Initial Conditions

\[\begin{split}h(x, 0) = \begin{cases} h_0 > 0 & \text{for } 0 < x \leq x_0, \\\\ 0 < h_r < h_0 & \text{for } x_0 < x, \end{cases}\end{split}\]

\[u(x, 0) = 0\]

where \(x_0\) is the initial dam location and \(h_0\) is the height of the water column at the left of the dam and \(h_r\) is the height of the water column at the right of the dam.

Analytical Solution

The water height and velocity profiles for \(t > 0\) are given by:

\[\begin{split}h(x, t) = \begin{cases} h_0 & \text{if } x \leq x_A(t), \\\\ \frac{\left( 2 \sqrt{g h_0} - \frac{x}{t} \right)^2}{9 g} & \text{if } x_A(t) < x \leq x_B(t), \\\\ h_m = \frac{1}{2} h_r \left( \sqrt{1 + \frac{8 c_m^2}{g h_r}} - 1 \right) & \text{if } x_B(t) < x \leq x_C(t), \\\\ h_r & \text{if } x_C(t) < x, \end{cases}\end{split}\]

\[\begin{split}u(x,t) = \begin{cases} 0 & \text{if } x \leq x_A(t), \\\\ \frac{2}{3} \left( \frac{x}{t} + \sqrt{g h_0} \right) & \text{if } x_A(t) < x \leq x_B(t), \\\\ 2 \sqrt{g h_0} - 2 \sqrt{g h_m} & \text{if } x_B(t) < x \leq x_C(t), \\\\ 0 & \text{if } x_C(t) < x, \end{cases}\end{split}\]

where the locations separating the flow regions evolve in time according to:

\[\begin{split}\begin{cases} x_A(t) = - t \sqrt{g h_0}, \\\\ x_B(t) = t \left( 2 \sqrt{g h_0} - 3 \sqrt{g h_m} \right), \\\\ x_C(t) = c_m t \end{cases}\end{split}\]

with \(c_m\) is the front shock wave speed, obtained as the solution of the nonlinear equation:

\[c_m h_r - h_r \left( \sqrt{1 + \frac{8 c_m^2}{g h_r}} - 1 \right) \left( \frac{c_m}{2} - \sqrt{g h_0} + \sqrt{\frac{g h_r}{2} \left( \sqrt{1 + \frac{8 c_m^2}{g h_r}} - 1 \right)} \right) = 0\]

Implementation

First import required packages and define the spatial domain for visualization. For following examples we will use a 1D space from -5.5 to 6 m.

import numpy as np
from tilupy.analytic_sol import Stoker_SARKHOSH_wet

x = np.linspace(-5.5, 6, 1000)

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Case: Stocker’s solution with dam at \(x_0 = 0 m\), initial fluid height \(h_0 = 0.5 m\) and initial domain height \(h_r = 0.05 m\)

case = Stoker_SARKHOSH_wet(h_0=0.5, h_r=0.025)

Compute and plot fluid height at times \(t = {0, 0.5, 1, 1.5, 2} s\).

case.compute_h(x, [0, 0.5, 1, 1.5, 2])
ax = case.plot(show_h=True, linestyles=["", ":", "-.", "--", "-"])

Compute and plot fluid velocity at times \(t = {0, 0.5, 1, 1.5, 2} s\).

case.compute_u(x, [0, 0.5, 1, 1.5, 2])
ax = case.plot(show_u=True, linestyles=["", ":", "-.", "--", "-"])

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Original reference:

Stoker, J.J., 1957, Water Waves: The Mathematical Theory with Applications, Pure and Applied Mathematics, vol. 4, Interscience Publishers, New York, USA.

Papier original (pas d’accès) https://www.researchgate.net/publication/230513364_Water_Waves_The_Mathematical_Theory_with_Applications

Code matlab: https://github.com/psarkhosh/Stoker_solution

Article utilisant les travaux https://www.mdpi.com/2073-4441/15/21/3841#

Article domaine humide sur pente inclinée (pas d’accès) https://ascelibrary.org/doi/10.1061/%28ASCE%29HY.1943-7900.0001683