Analytical solution of a dam-break problem on a dry slope with friction (Mangeney et al.)

This script presents a one-dimensional analytical solution of a dam-break problem, as proposed by Mangeney et al. (2000), for gravity-driven flow on a dry, inclined plane, including basal friction.

Model Assumptions

• Instantaneous dam break at time \(t = 0\).

• The downstream domain (\(x > 0\)) is initially dry, while the upstream side (\(x < 0\)) contains a fluid column of height \(h_0\).

• The flow occurs on a constant slope inclined at an angle \(\theta\).

• Basal friction is modeled via a constant friction angle \(\delta\), leading to a constant horizontal acceleration.

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

• The initial fluid volume is considered infinite.

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 and 2 from Mangeney (2000) with the same notation and in 1D:

\[\begin{split}\begin{cases} \partial_t h + \partial_x (hu) = 0 \\\\ h \partial_t u + hu \partial_x u + hg\cos{\theta} \partial_x h = gh\sin{\theta} - F \end{cases}\end{split}\]

with \(F = g \cos{\theta} \tan{\delta}\), the source term integrating the dissipative effects due to friction for a Coulomb-type friction law.

The general form for an hydrostatic model with basal friction (Coulomb-type friction law) can be expressed:

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

with \(\gamma = \frac{1}{R}\), \(R\) being the radius of curvature.

Since we are on a flat surface and with a flow restricted only by internal friction, we have \(\gamma = 0\) and \(\mu = \tan{\delta}\), allowing to obtain Mangeney’s relation.

Initial Conditions

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

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

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

Analytical Solution

The analytical expressions for fluid height and velocity at time t are given by:

\[\begin{split}h(x, t) = \begin{cases} h_0 & \text{if } x \leq x_A(t), \\\\ \frac{1}{9g cos(\theta)} \left(2 c_0 - \frac{x-x_0}{t} + \frac{1}{2} m t \right)^2 & \text{if } x_A(t) < x \leq x_B(t), \\\\ 0 & \text{if } x_B(t) < x, \end{cases}\end{split}\]

\[\begin{split}u(x,t) = \begin{cases} 0 & \text{if } x \leq x_A(t), \\\\ \frac{2}{3} \left( \frac{x-x_0}{t} + c_0 + mt \right) & \text{if } x_A(t) < x \leq x_B(t), \\\\ 0 & \text{if } x_B(t) < x, \end{cases}\end{split}\]

where

\[\begin{split}\begin{cases} x_A(t) = x_0 + \frac{1}{2}mt^2 - c_0 t, \\\\ x_B(t) = x_0 + \frac{1}{2}mt^2 + 2 c_0 t \end{cases}\end{split}\]

with \(c_0\) the initial wave propagation speed defined by:

\[c_0 = \sqrt{g h_0 \cos{\theta}}\]

and \(m\) the constant horizontal acceleration of the front defined by:

\[m = g \sin{\theta} - g \cos{\theta} \tan{\delta}\]

Implementation

The following example replicates the case shown in Figure 3 of Mangeney et al. (2000).

---

First import required packages and define the spatial domain. For following examples we will use a 1D space from -100 to 1000 m.

import numpy as np
from tilupy.analytic_sol import Mangeney_dry

x = np.linspace(-100, 1000, 1000)

---

Case 1: No friction (\(\delta = 0°\)), slope \(\theta = 30°\), initial height \(h_0 = 20 m\).

case_1 = Mangeney_dry(h_0=20, x_0=0, theta=30, delta=0)

Compute and plot the fluid height at times \(t = {0, 5, 10, 15} s\).

case_1.compute_h(x, [0, 5, 10, 15])
ax = case_1.plot(show_h=True, linestyles=["", ":", "--", "-"])

Compute and plot the fluid velocity at times \(t = {0, 5, 10, 15} s\).

case_1.compute_u(x, [0, 5, 10, 15])
ax = case_1.plot(show_u=True, linestyles=["", ":", "--", "-"])

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Case 2: Add basal friction (\(\delta = 20°\)), same slope and initial height.

case_2 = Mangeney_dry(h_0=20, x_0=0, theta=30, delta=20)

Compute and plot the fluid height at times \(t = {0, 5, 10, 15} s\).

case_2.compute_h(x, [0, 5, 10, 15])
ax = case_2.plot(show_h=True, linestyles=["", ":", "--", "-"])

Original reference:

Mangeney, A., Heinrich, P., & Roche, R., 2000, Analytical solution for testing debris avalanche numerical models, Pure and Applied Geophysics, vol. 157, p. 1081-1096.