Analytical solution of a dam-break problem on a dry bed without friction (Ritter)

This example presents the classical one-dimensional analytical solution proposed by Ritter (1892) for the dam-break problem on a dry, horizontal bed without friction.

Model Assumptions

• Instantaneous dam break at position \(x = x_0\) 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).

• No bed friction is considered.

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

• The solution is valid until the rarefaction wave reaches the left boundary of the water column.

Initial Conditions

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

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

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

Analytical Solution

The water height and velocity profiles at any time t are given by:

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

\[\begin{split}u(x,t) = \begin{cases} 0 & \text{if } x \leq x_A(t), \\\\ \frac{2}{3} \left( \frac{x - x_0}{t} + \sqrt{g h_0} \right) & \text{if } x_A(t) < x \leq x_B(t), \\\\ 0 & \text{if } x_B(t) < x, \end{cases}\end{split}\]

where the positions of the rarefaction wave front and the dry front are:

\[\begin{split}\begin{cases} x_A(t) = x_0 - t \sqrt{g h_0}, \\\\ x_B(t) = x_0 + 2 t \sqrt{g h_0} \end{cases}\end{split}\]

Implementation

First import required packages and define the spatial domain for visualization. For following examples we will use a 1D space from -25 to 45 m.

import numpy as np
from tilupy.analytic_sol import Ritter_dry

x = np.linspace(-25, 45, 1000)

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Case 1: Ritter’s solution with dam at \(x_0 = 0 m\) and initial height \(h_0 = 0.5 m\)

case_1 = Ritter_dry(x_0=0, h_0=0.5)

Compute and plot fluid height at times \(t = {0, 2, 4, 6, 8, 10} s\).

case_1.compute_h(x, [0, 2, 4, 6, 8])
ax = case_1.plot(show_h=True, linestyles=["", ":", "-.", "--", "-"])

Compute and plot fluid velocity at times \(t = {0, 2, 4, 6, 8, 10} s\).

case_1.compute_u(x, [0, 2, 4, 6, 8])
ax = case_1.plot(show_u=True, linestyles=["", ":", "-.", "--", "-"])

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Case 2: Specific example from SWASHES benchmark database Dam at \(x_0 = 5 m\), initial height \(h_0 = 0.005 m\), domain length \(L = 10 m\), solution at \(t = 6 s\).

x = np.linspace(0, 10, 1000)

case_2 = Ritter_dry(x_0=5, h_0=0.005)
case_2.compute_h(x, 6.0)
ax = case_2.plot(show_h=True, linestyles=["-"])

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Original reference:

ID Poisson. Swashes. ID Poisson, [online]. Available at: https://www.idpoisson.fr/swashes/ ; accessed June 2025.

Ritter, A., 1892, Die Fortpflanzung der Wasserwellen, Zeitschrift des Vereines Deutscher Ingenieure, vol. 36(33), p. 947–954.