Analytical solution of a dam-break problem on a dry slope with friction (Chanson)

This example presents the classical one-dimensional analytical solution proposed by Chanson (2005) for the dam-break problem on a dry, horizontal bed with friction.

Model Assumptions

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

• The domain is initially dry for \(x > x_0\), with a finite volume of still water (with height \(h_0\)) on the left of the dam (\(0 < x \leq x_0\)).

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

• Basal friction is modeled via Darcy friction factor \(f\).

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

Basic equations

As a reminder, the general formula of the Saint-Venant equation system is:

\[\begin{split}\begin{cases} \partial_t h + \partial_x (hu) = 0 \\\\ \partial_t (hu) + \partial_x (hu^2) + \frac{1}{2}g\cos{\theta} \partial_x (h^2) = gh\sin{\theta} - S \end{cases}\end{split}\]

with:

• \(h\) the fluid depth

• \(u\) the fluid velocity

• \(g\) the gravitational acceleration

• \(\theta\) the surface slope

• \(S\) source term

Here is equation 1 from Chanson (2005) with the same notation and in 1D:

\[\begin{split}\begin{cases} \partial_t h + h \partial_x (u) + u \partial_x (h) = 0 \\\\ \partial_t u + u \partial_x u + g \partial_x h + g S_f = 0 \end{cases}\end{split}\]

with \(S_f = \frac{f}{2} \frac{u^2}{g D_H}\), \(f\) being Darcy friction factor and \(D_H\) the hydraulic diameter.

By transforming these equations, we find the Saint-Venant equations:

\[\begin{split}\begin{cases} \partial_t h + \partial_x (hu) = 0 \\\\ h \partial_t u + hu \partial_x u + hg \partial_x h = - S \end{cases}\end{split}\]

with \(S = \frac{h f u^2}{2 D_H}\) the source term integrating the dissipative effects due to friction. In those conditions (approximate to a wide rectangular channel), the hydraulic diameter can be expressed \(D_H = 4 h\) (see footnote number 5 of Chanson (2005)).

With this we have finally the source term \(S = \frac{f u^2}{8}\).

In fluid simulation, hydraulic models can be used to express the source term \(S\). For example, we can cite an equation combining the Darcy-Weisbach and Manning laws:

\[S_g = g n^2 \frac{u^2}{h^{1/3}}\]

where \(n\) is Manning coefficient (in \(s.m^{-1/3}\)).

These forms diverge in that the fluid height \(h\) is not present in the expression found for Chanson. However, it can be compared to Voellmy’s expression by taking a friction coefficient \(\mu = 0\) and with an empirical coefficient \(\xi = \frac{8}{f}\):

\[S = h \mu \left( g \cos{\theta} + \gamma u^2 \right) + \frac{u^2}{\xi}\]

Initial Conditions

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

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

Analytical Solution

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

\[\begin{split}h(x, t) = \begin{cases} h_0 & \text{if } x \leq x_A(t), \\\\ \frac{4}{9g} \left( \sqrt{g h_0} - \frac{x - x_0}{2t} \right)^2 & \text{if } x_A(t) < x \leq x_B(t), \\\\ \sqrt{\frac{f}{4} \frac{U(t)^2}{g h_0} \frac{x_C(t)-x}{h_0}} & \text{if } x_B(t) < x \leq x_C(t), \\\\ 0 & \text{if } x_C(t) < x \end{cases}\end{split}\]

where the positions of the rarefaction wave front, the tip position and the dry front are:

\[\begin{split}\begin{cases} x_A(t) = x_0 - t \sqrt{g h_0}, \\\\ x_B(t) = x_0 + \left( \frac{3}{2} \frac{U(t)}{\sqrt{g h_0}} - 1 \right) t \sqrt{g h_0}, \\\\ x_C(t) = x_0 + \left( \frac{3}{2} \frac{U(t)}{\sqrt{g h_0}} - 1 \right) t \sqrt{\frac{g}{h_0}} + \frac{4}{f\frac{U(t)^2}{g h_0}} \left( 1 - \frac{U(t)}{2 \sqrt{g h_0}} \right)^4 \end{cases}\end{split}\]

with the celerity of the wave front \(U(t)\) solution of:

\[\left( \frac{U}{\sqrt{g h_0}} \right)^3 - 8 \left( 0.75 - \frac{3 f t \sqrt{g}}{8 \sqrt{h_0}} \right) \left( \frac{U}{\sqrt{g h_0}} \right)^2 + 12 \left( \frac{U}{\sqrt{g h_0}} \right) - 8 = 0\]

Implementation

First import required packages and define the spatial domain for visualization. For following examples we will use a 1D space from -600 to 600 m.

import numpy as np
from tilupy.analytic_sol import Chanson_dry

x = np.linspace(-600, 600, 1000)

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Case: Chanson’s solution with dam at \(x_0 = 0 m\), initial height \(h_0 = 10 m\) and friction coefficient \(f = 0.05\)

case = Chanson_dry(h_0=10, x_0=0, f=0.05)

Compute and plot fluid height at times \(t = {0, 1, 20, 30, 50} s\).

case.compute_h(x, [0, 1, 20, 30, 50])
ax = case.plot(show_h=True, linestyles=["", ":", "-.", "--", "-"])

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Original reference:

Chanson, H., 2005, Applications of the Saint‑Venant Equations and Method of Characteristics to the Dam Break Wave Problem. https://espace.library.uq.edu.au/view/UQ:9438