Cylinder

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.

Flow past a circular cylinder is the canonical benchmark for flow instability and active flow control. At Re = 100 the steady symmetric flow bifurcates via a supercritical Hopf bifurcation, and the wake develops into a periodic von Kármán vortex street. This makes the cylinder an ideal test case for studying the transition mechanism, validating control strategies, and benchmarking RL agents.

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

Physical setup

• Geometry: circular cylinder of radius 0.5 in a 2-D domain
• Inflow: uniform velocity U∞ = 1.0 from the left boundary
• Reynolds number: Re = 100 (default)

Two independent actuation mechanisms are available:

• Jet blowing/suction (Cylinder): two 10° tangential jets at ±90° from the stagnation point — used by solve-steady.py, unsteady.py, step_input.py, and pressure-probes.py.
• Rotary control (RotaryCylinder): a prescribed tangential surface velocity — used by run-transient.py, pd-control.py, pd-phase-sweep.py, and lti_system.py.

Both mechanisms are capable of suppressing vortex shedding, but they operate through different physical mechanisms and are suited to different control design approaches.

Workflow overview

Step 1 — Observe natural vortex shedding

Run the uncontrolled cylinder flow to generate a time-periodic reference trajectory and a restart checkpoint for subsequent steps:

python run-transient.py

Source: run-transient.py

The console prints the CL/CD time series; checkpoints are written for use in pd-control.py and related scripts.

Step 2 — Compute the unstable steady state

At Re = 100 the symmetric, non-shedding equilibrium exists but is unstable. It can be computed with a Newton solver and used as the base flow for stability analysis and LTI modelling:

python solve-steady.py

Source: solve-steady.py

Outputs: output/cylinder_Re100_steady.h5 and associated Paraview files.

Step 3 — Observe the transition to vortex shedding

unsteady.py chains the Newton solver and a transient simulation into a single run, demonstrating how a small perturbation to the steady state grows into the limit cycle:

python unsteady.py

Source: unsteady.py

• Stage 1: Newton solve → unstable equilibrium
• Stage 2: Perturbed transient integration → limit cycle

Step 4 — Linear stability analysis

Extract the leading eigenvalue (growth rate and shedding frequency) from the linearised Navier-Stokes operator around the steady base flow:

python stability.py

Source: stability.py

The script computes both direct and adjoint global modes via Arnoldi iteration. The optional checkpoint from solve-steady.py can be supplied to skip the internal steady solve.

Step 5 — Apply PD feedback control

With a vortex-shedding trajectory available, demonstrate open-loop-to-closed-loop switching with a proportional-derivative controller:

python pd-control.py

Source: pd-control.py

Prerequisite: a checkpoint produced by run-transient.py.

The script runs the flow uncontrolled for an initial period, then activates the PD controller and shows the oscillations decaying over time.

Complete script reference

Script	Purpose	Key features	Prerequisites
solve-steady.py	Compute steady-state base flow	Newton solver, Reynolds ramping	None
unsteady.py	Steady → unsteady transition	Two-stage: Newton + transient	None
run-transient.py	Basic time integration	Simple vortex-shedding demo	None
stability.py	Linear stability analysis	Eigenvalues, eigenmodes (direct/adjoint)	Optional steady checkpoint
step_input.py	System identification	Step-response for control design	None
pressure-probes.py	Point measurements	Sparse sensing at pressure probes	None
pd-control.py	Feedback control	PD controller with on/off phases	Checkpoint from run-transient.py
pd-phase-sweep.py	Controller tuning	Sweeps phase angles for optimal gain	Checkpoint from run-transient.py
lti_system.py	LTI model extraction	Base flow + actuation influence matrix	None

MPI parallelisation

All scripts support MPI-parallel execution:

mpirun -np 4 python run-transient.py
mpirun -np 4 python solve-steady.py
mpirun -np 4 python stability.py
mpirun -np 4 python pd-control.py