M100 · Fluid mechanics

Shallow-Water Waves & Dam Break

Does a dam-break flood front move at the speed of small water waves? No. Small waves travel at a speed set by the local depth: the deeper the water, the faster they go. A dam-break front is a large, nonlinear wave, and its speed comes from the full solution. When the flood runs onto dry ground, the front moves at twice the small-wave speed of the original reservoir.

Open a gate, launch a pulse, hold a still lake over a ridge or look at a hydraulic jump. Scrub through the finite-volume solution, compare it with the exact dam-break reference and refine the mesh. The water in the flume plus what has flowed out through the ends should stay constant.

Solution time $\text{Ready}$

Gauge depth $\text{Ready}$

Depth-mean velocity $\text{Ready}$

Discharge per width $\text{Ready}$

Local long-wave speed $\text{Ready}$

Gauge Froude number $\text{Ready}$

Stored water volume $\text{Ready}$

Net exported volume $\text{Ready}$

Maximum Courant number $\text{Ready}$

Cell width $\text{Ready}$

Solver steps $\text{Ready}$

Mean depth error against exact solution $\text{Ready}$

Volume integrated from raw scan $\text{Ready}$

Recorded stations $\text{Ready}$

Force or volume balance residual $\text{Ready}$

Model state $\text{Ready}$

Floating stability and shallow-water flow

Can the numerical flood be trusted?

Initial left surface height

H_L

2.00\,m

Initial right surface height

H_R

0.50\,m

Full-width bed ridge height

b_0

0.00\,m

Gravity

g

9.81\,m/s^2

Finite volume cells

N

160.00\,

Solution time

t

2.00\,s

Gauge position fraction

\xi_p

0.60\,

Scenario: dam, lake, pulse, jump

s

0.00\,

Ends: transmissive or reflecting

b

0.00\,

Sensor perturbation gain

\epsilon

0.00\,

The channel is eighty metres long and six metres wide, and every length in the scene is drawn to the same scale. The ridge spans the full width, so the bed profile is one-dimensional. Each water column is a finite-volume average, and all velocity arrows share one display scale. The gate disappears instantly at release.

The solver steps depth and discharge forward with a first-order local Lax-Friedrichs flux and hydrostatic reconstruction, using an adaptive Courant number of at most 0.4. A cell counts as dry below a tiny numerical threshold, and any significant negative depth is reported. Each end either reflects waves or copies the neighbouring cell so water can flow out; that outflow boundary still reflects a little. Friction, cross-channel flow, dispersion and the roller of a breaking jump are not resolved.

The dam-break presets start at rest. The exact flat-bed solution is shown for reference only until its waves reach an end of the channel. The still-lake and pulse cases use the left surface height. For a hydraulic jump the depth must increase to the right, and the two conjugate depths set a constant discharge. A bed ridge changes these starting states. An acquisition reads every cell and integrates the raw depths into a volume estimate. Choosing a gauge keeps the records; changing the setup or the time clears them. Sensor perturbations are bounded and repeatable, and no statistical confidence is claimed.

The flow is one-dimensional, depth-averaged and inviscid. The scheme uses hydrostatic reconstruction with a diffusive first-order flux. The 3D flume shows cell averages and does not resolve vertical turbulence or the rollers of breaking waves.

Start with the essentials

Focus question: Does the flood on screen obey the conservation laws?
One-sentence intuition: Compare the full physical model, its approximation and the separately recorded data.

Core mathematical model

Conservation of depth and momentum

\begin{aligned}h_t+(hu)_x&=0\\(hu)_t+\left(hu^2+\frac{gh^2}{2}\right)_x&=-ghb_x\end{aligned}

The flow is hydrostatic and depth-averaged in a channel of constant width. The bed enters as a source term in the momentum equation.

Local speeds of small waves

\lambda_\pm=u\pm\sqrt{gh}

A flood shock does not in general travel like a small long wave. The front over a dry bed moves at yet another speed.

Froude number

\mathrm{Fr}=\frac{|u|}{\sqrt{gh}}

In subcritical flow, signals can travel both upstream and downstream. Supercritical flow carries both families of signals downstream. A dry cell has no Froude number.

Volume conservation with flow through the ends

\begin{aligned}V&=W\sum_i h_i\Delta x\\V(t)+\int_0^t W(q_R-q_L)\,ds&=V(0)\end{aligned}

Water leaving through the ends can change the volume in the flume without breaking conservation. Scanning the raw cell depths gives a separate estimate of the integral.

Adaptive step and grid error

\begin{aligned}\mathrm{CFL}&=\frac{\Delta t}{\Delta x}\max_i\left(|u_i|+\sqrt{gh_i}\right)\leq0.4\\E_h&=\frac1N\sum_i|h_i-h_{\mathrm{exact}}(x_i,t)|\end{aligned}

The first-order scheme smears shocks out. The mean depth error is available only for a flat-bed dam break, before the waves reach the ends.

Stationary hydraulic jump

\begin{aligned}q_L&=q_R=q\\\frac{q^2}{h_L}+\frac{gh_L^2}{2}&=\frac{q^2}{h_R}+\frac{gh_R^2}{2}\end{aligned}

The preset balances the mass and momentum fluxes across a depth increase that satisfies the entropy condition. The numerical solution smears the jump over several cells.

Common difficulties

A reference curve is not a measurement

Typical misconceptionThe reference curve on screen is a measured result.

Better mental modelOnly the sensor channels you record count as observations, and changing the setup clears them. The perturbations are deterministic, so they give no statistical certification.

Run the experiment

01
Release the gate
Compare the wet-bed and dry-bed dam presets, and move the time slider.
What to observe: The rarefaction, the shock and the dry front all come from the same balance laws.
02
Refine and compare
Double the cell count, keeping the same initial state and time.
What to observe: The reference curve stays the same, and the numerical smearing shrinks.
03
Test a still lake
Select Lake over a ridge, then change the ridge height.
What to observe: The well-balanced hydrostatic reconstruction keeps this lake at rest, with no false currents.
04
Check the volume balance
Compare reflecting and transmissive ends, and acquire every cell.
What to observe: The volume in the flume plus the net outflow stays constant. The raw heights integrate into a separate volume estimate that carries its own sensor perturbations.