M100 · Fluid mechanics

Shallow-Water Waves & Dam Break

Release a gate, launch a pulse, hold a still lake over a ridge or inspect a hydraulic jump. Scrub the finite volume solution, compare an exact dam reference and refine the mesh while closing the boundary-flux volume ledger.

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 closure residual $\text{Ready}$

Model state $\text{Ready}$

GEOMETRY / CONSERVATION / ACQUISITION

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; all physical scene lengths share one scale. The ridge spans the full width and represents a one-dimensional bed profile. Water columns are finite volume averages; arrows use one common velocity display gain. The gate vanishes instantaneously at release.

The solver advances depth and discharge with a first-order local Lax-Friedrichs flux, hydrostatic reconstruction and adaptive Courant number at most 0.4. Dry cells use a tiny numerical threshold; significant negative depth is reported, not hidden. Boundaries reflect or copy the adjacent state for transmissive outflow; the latter is not perfectly nonreflecting. No friction, transverse flow, dispersion or roller structure is resolved.

Dam presets start at rest. The exact flat-bed reference appears only before its waves reach an end. Lake and pulse scenarios use the left surface height; jump depths must increase to the right and set a conjugate constant discharge. A bed ridge modifies these candidates. Acquire every cell to integrate raw depth into a volume estimate; selecting a gauge preserves records, while configuration or time changes clear them. Sensor perturbations are bounded and deterministic, without statistical confidence claims.

One-dimensional depth-averaged inviscid flow; hydrostatic reconstruction with diffusive first-order flux. The 3D flume displays cell averages, without resolving vertical turbulence or breaking-wave rollers.

Start with the essentials

Focus question: Is the visible flood also a conservative solution?
One-sentence intuition: Compare the full physical model, its approximation and the separately acquired record.

Core mathematical model

Depth and momentum balances

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

Hydrostatic depth-averaged flow along a constant-width channel; the bed enters the momentum source.

Local characteristic speeds

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

A flood shock is not generally a small long wave. The dry-bed front has a different speed.

Froude number

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

Subcritical flow permits signals both ways; supercritical flow carries both families downstream. Dry cells have no Froude value.

Conservative volume ledger

\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}

Boundary export can change stored volume without violating conservation. Raw cell scans provide a separate 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 diffuses shocks. Mean depth error is available only for a flat dam before boundary interactions.

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 mass and momentum flux across an entropy-admissible depth increase; the numerics spread its interface.

Common difficulties

Reference and acquisition

Typical misconceptionA displayed reference is an observed measurement.

Better mental modelRecord sensor channels explicitly. Changing the configuration clears observations; no statistical certification follows from deterministic perturbations.

Run the experiment

01
Release the gate
Compare wet-bed and dry-bed dam presets; move the time slider.
What to observe: Rarefaction, shock or dry front arise from the same balances.
02
Refine and compare
Double the cell count with the same initial state and time.
What to observe: The reference overlay stays fixed while numerical diffusion decreases.
03
Test a still lake
Select Lake over a ridge, then alter the ridge height.
What to observe: A well-balanced hydrostatic reconstruction avoids spurious currents in this equilibrium.
04
Close the ledger
Compare reflecting and transmissive ends; acquire every cell.
What to observe: Stored volume plus net export closes; raw heights integrate into a separately perturbed volume estimate.