Flow Past Cylinder

The von Kármán vortex street is a canonical setup in which flow separation behind a cylinder induces periodic vortex shedding and fluctuating forces on the cylinder. The objective is to reduce the drag coefficient \(C_D\) while keeping the lift \(C_L\) small.

The system is parametrized via the Reynolds number \(\mathrm{Re} = \frac{\overline{U} D}{\nu}\), where \(\overline{U}\) is the mean incoming velocity, \(D\) the cylinder diameter, and \(\nu\) the kinematic viscosity.

Environment List

Environment ID	Actuation	Reynolds	Resolution
CylinderJet2D-easy-v0	Jet	100	24
CylinderJet2D-medium-v0	Jet	250	32
CylinderJet2D-hard-v0	Jet	500	32
CylinderRot2D-easy-v0	Rotation	100	24
CylinderRot2D-medium-v0	Rotation	250	32
CylinderRot2D-hard-v0	Rotation	500	32
CylinderJet3D-easy-v0	Jet	100	24
CylinderJet3D-medium-v0	Jet	250	32
CylinderJet3D-hard-v0	Jet	500	48

Reward

The reward at step \(t\) is:

\[r_t = C_{D,\mathrm{ref}} - \langle C_D \rangle_{T_{\mathrm{act}}} - \omega \, |\langle C_L \rangle_{T_{\mathrm{act}}}|\]

where \(\langle \cdot \rangle_{T_{\mathrm{act}}}\) is the temporal average over the actuation interval, \(C_{D,\mathrm{ref}}\) is the reference drag coefficient of the uncontrolled flow, and the lift regularization weight \(\omega = 1.0\).

The drag and lift coefficients are computed as:

\[C_D = \frac{F_D}{\frac{1}{2} \rho \overline{U}^2 D}, \qquad C_L = \frac{F_L}{\frac{1}{2} \rho \overline{U}^2 D}\]

with forces acting on the cylinder surface \(S\):

\[F_D = \int_S (\boldsymbol{\sigma} \cdot \mathbf{n}) \cdot \mathbf{e}_x \, \mathrm{d}S, \qquad F_L = \int_S (\boldsymbol{\sigma} \cdot \mathbf{n}) \cdot \mathbf{e}_y \, \mathrm{d}S\]

where \(\boldsymbol{\sigma}\) is the Cauchy stress tensor and \(\mathbf{n}\) the outward unit normal at the cylinder surface.

MARL Reward

In MARL mode, the reward for agent \(i\) is a weighted combination of a local and global term:

\[r_t^i = \beta \, r_t^{i,\mathrm{local}} + (1 - \beta) \, r_t^{\mathrm{global}}\]

The local reward is computed over the cylinder segment controlled by agent \(i\), while the global reward covers the full cylinder. The local weight \(\beta = 0.8\).

Action Space

Two actuation modes are available:

• Jet actuation (CylinderJet): opposing synthetic jets on the top and bottom surfaces with a parabolic velocity profile and a total deflection angle of 10°. In 3D, the domain is extended spanwise, yielding eight individual jet pairs. The maximum jet velocity is \(\overline{U}\).

• Rotation actuation (CylinderRot): rigid cylinder rotation with a maximum absolute angular velocity of \(\overline{U}\).

The raw control signal is temporally smoothed via exponential filtering:

\[c_s = c_{s-1} + \alpha \, (a_t - c_{s-1}), \qquad \alpha = 0.1\]

where \(c_s\) is the applied value at simulation sub-step \(s\) and \(a_t\) the action at episode step \(t\).

Observation Space

Observations consist of the horizontal and vertical velocity components at sensor locations surrounding the cylinder (indicated by pink dots in 2D / pink planes in 3D). In 3D, the spanwise velocity component is additionally included. To enable policy transfer from 2D to 3D, the number of sensor planes and included components can be configured to match the 2D observation layout.

Difficulty Levels

Difficulty is controlled by the Reynolds number:

Level	Reynolds number
Easy	Re = 100
Medium	Re = 250
Hard	Re = 500

Higher Reynolds numbers increase turbulence intensity and flow unsteadiness. The medium and hard settings introduce three-dimensional flow interactions.

API Reference

fluidgym.envs.cylinder.CylinderJetEnv2D(...)	Environment for flow around a cylinder with jet actuation.
fluidgym.envs.cylinder.CylinderRotEnv2D(...)	Environment for flow around a cylinder with rotating cylinder actuation.
fluidgym.envs.cylinder.CylinderJetEnv3D(...)	3D Environment for flow around a cylinder with jet actuation.