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Pinball

Advanced workflow examples

The scripts on this page access the Firedrake solver directly — they do not use the standard env.reset() / env.step() interface. For RL training, see Getting Started.

The fluidic pinball is a multi-body benchmark consisting of three circular cylinders arranged in an equilateral triangle. The configuration is an influential test case for multi-input flow control: each cylinder can be actuated independently via rotation, giving the controller three degrees of freedom to shape a wake that exhibits rich dynamics including mode switching, asymmetry, and chaos at moderate Reynolds numbers.

All scripts live in examples/firedrake/advanced/pinball/.

Physical setup​

  • Geometry: three cylinders of radius 0.5 in an equilateral triangle, separated by one diameter
  • Inflow: uniform velocity U∞ = 1.0 from the left boundary
  • Reynolds number: Re = 30 – 150

The flow behaviour changes qualitatively across this range:

  • Re ≈ 30: near-steady, weakly asymmetric wake
  • Re ≈ 80: onset of periodic vortex shedding
  • Re ≈ 150: complex unsteady wake with three-body interactions; mode switching between symmetric and asymmetric states, with chaotic dynamics at the upper end

Workflow overview​

Step 1 — Compute the steady base flow​

Solve for the steady equilibrium at Re = 80 using Newton iteration. Direct convergence from Re = 80 is difficult, so the solver ramps through intermediate Reynolds numbers:

python solve-steady.py

Source: solve-steady.py

Reynolds ramping schedule: 40 → 60 → 80. Outputs:

  • output/pinball_Re80_steady.h5 — restart checkpoint
  • Paraview files for visualisation
  • Force coefficients for all three cylinders at the steady state

Step 2 — Run a transient simulation​

Simulate the unsteady three-cylinder wake and observe the inter-body interactions:

python run-transient.py

Source: run-transient.py

The console prints the lift coefficient time series for each cylinder. The output file coeffs.dat records the full CL history for post-processing.

Step 3 — Observe wake transition​

unsteady.py chains an internal steady solve with a long transient integration, demonstrating the full sequence from equilibrium to complex unsteady dynamics in one run:

python unsteady.py

Source: unsteady.py

  • Stage 1: Newton solver with Reynolds ramping (40 → 60 → 80 → 100)
  • Stage 2: Perturbed transient integration for T_f = 200 time units
  • Outputs: time series, Paraview animations, force data for all three cylinders

This script has no prerequisites and provides a self-contained end-to-end demonstration.

MPI parallelisation​

All scripts support MPI-parallel execution:

mpirun -np 4 python solve-steady.py
mpirun -np 4 python run-transient.py
mpirun -np 4 python unsteady.py