Standing gravity wave

The potential free-surface solver had no reference behind it at all. This case gives it one, and as with the dam break and the shock tube the reference is arithmetic rather than a transcribed table, so there is nothing to mis-copy: a standing wave of wavelength L in still water of depth h oscillates at ω² = g k tanh(k h), with k = 2π/L.

It is not an idealisation of what this solver solves — it is the closed-form solution of it. The solver does not track a moving interface. The mesh top stays flat and the free surface enters as a boundary condition there, integrating dζ/dt = U·n over the patch and setting the pressure to −g·ζ. Those two are the linearised kinematic and dynamic free-surface conditions the dispersion relation comes from.

Three wavelengths are run in the same depth of water, so that what is tested is the dispersion relation and not one frequency three times. The phase speed spans 1.7638 m/s to 3.1193 m/s across them, from deep water to shallow.

The harness builds the relation instead of asserting it. Take the velocity potential of a standing wave over a flat bed,

    phi = cosh(k(y + h)) cos(k x) cos(omega t)

and check numerically, by finite differences, that it does the three things it must:

    harmonic       d2phi/dx2 + d2phi/dy2 = 0                  residual 1.10e-07
    bed            dphi/dy = 0 at y = -h                       residual 0
    surface        d2phi/dt2 + g dphi/dy = 0 at y = 0          residual 8.82e-09

    which closes only if    omega^2 = g k tanh(k h)

Both limits are then confirmed against their own series rather than against a round number. The deep-water limit is exactly √(g/k). The shallow-water limit approaches √(g h) at 1.6666661 × 10⁻⁷ against the 1.6666667 × 10⁻⁷ its own (k h)²/6 expansion demands — agreement to 0.0000317 %.

The surface is released from rest as ζ = a cos(k x) and then left entirely alone. At every write, the patch values are projected onto cos(k x) using each face’s own x coordinate, read from the mesh, so that no assumption about face ordering can put the comparison against a shifted mode. That gives a modal amplitude, which is fitted to a damped cosine by least squares over the whole record.

The fit is the headline. The zero crossings of the same signal are carried alongside as a cross-check that shares none of the fitting machinery — only the samples. The two periods agree to between 0.00003 % and 0.00094 % on every run, and every healthy run has the eight crossings four periods should produce.

How much of the surface is still the mode that was released is measured too, summed over the record. The worst figure anywhere in the nine runs is 0.9999956.

Nine runs landing within a per cent of a closed form would be a plausible coincidence of mesh and scheme. An error that falls at close to the second order the schemes claim, over a factor of four in both the mesh and the time step, is not.

The wave loses amplitude as it runs, and it is worth being precise about where that goes. The molecular viscosity the product writes damps this wave at 2νk², which accounts for 0.034 % of the amplitude per period at L = 2 m and 0.0010 % at L = 40 m. The measured loss is far larger than that, and it halves with each refinement while the viscous figure does not move at all.

Neither fixture below was manufactured. Each is the same nine runs with one thing left exactly as the application writes it, rather than as the benchmark needs it.

• The tank is two-dimensional and one cell thick. There is no third direction, no side-wall boundary layer and no cross-mode.
• The free surface never leaves the mesh top. This solver does not track an interface, which is exactly why the linear relation is its exact solution — and equally why nothing here establishes anything about a solver that does.
• Four periods is the whole record. The damping is measured over that window, and no claim is made about long-time behaviour beyond it.
• The case that ships is not the case that was run. The geometry, the boundary types, the time-step control, the turbulence setting and the class of one field all had to be changed for the closed form to apply. The discretisation was not touched — the schemes and the linear solvers are the product’s own — but a user staging this solver from the application does not get the case measured here.