Kolmogorov flow 2D (`kolmogorov_flow_2d`)

Forced two-dimensional turbulence. A steady sinusoidal band force injects energy at a single wavenumber, the nonlinearity cascades it across scales and viscosity dissipates it, and at low viscosity the result is a statistically steady turbulent state rather than a decaying one. That steadiness is what makes this the right model for long-horizon work: the statistics of the target do not drift with \(T\), so a horizon can be lengthened without also changing what is being asked.

Equation

The Navier-Stokes vorticity dynamics of ns_vorticity_2d driven by \(\mathbf{f} = (f_0 \sin(n y),\, 0)\), which in vorticity form contributes

\[\nabla \times \mathbf{f} = -f_0\,n\,\cos(n y)\]

on the \(2\pi\)-periodic box.

Operator learning task

\[w(x, y, 0) \mapsto w(x, y, T)\]

Parameters

Parameter	Default	Range	Description
`viscosity`	0.025	(1e-4, 1.0)	Kinematic viscosity, \(1/\mathrm{Re}\); lower is more turbulent
`forcing_wavenumber`	4	(1, 16)	Band-forcing wavenumber \(n\)
`forcing_amplitude`	1.0	(0.0, 10.0)	Forcing amplitude \(f_0\)
`time_horizon`	10.0	(0.1, 200.0)	Final time \(T\)

forcing_wavenumber sets how many bands the forcing lays across the box, and so the scale at which energy enters. It is visible in the output: the figure above is \(n = 4\).

Usage

from pdeforge import generate_dataset

dataset = generate_dataset(
    model="kolmogorov_flow_2d",
    n_samples=500,
    resolution={"x": 128, "y": 128},
    params={"viscosity": 1/70, "forcing_wavenumber": 4, "time_horizon": 30.0},
    backend="jax",
    seed=7,
)

The JAX backend is worth using here: the horizons that reach a developed turbulent state are long, and the solver is a good fit for accelerator execution.

Solver

Inherited from ns_vorticity_2d: ETDRK4 with exact viscous diffusion and dealiased explicit advection, plus the steady forcing term added to the right-hand side.

Behaviour

At \(\nu = 0.025\) the flow is smooth and the band structure of the forcing remains visible in the solution. Around \(\nu = 1/70\) the cascade takes over and the field develops the filamentary vorticity texture of two-dimensional turbulence. Below roughly \(\nu = 10^{-3}\) at \(128^2\) the finest structures reach the grid scale, and the dataset begins to record the discretisation.

Data shapes

dataset.inputs.shape   # (n_samples, ny, nx)
dataset.outputs.shape  # (n_samples, ny, nx)

ns_vorticity_2d: the unforced parent model and the published benchmark family.
ks_1d: chaos in one dimension, where the attractor is small enough to characterise.
cylinder_flow_2d_turbulent: wall-bounded turbulence under a large-eddy closure.

Kolmogorov flow 2D (kolmogorov_flow_2d)