Cylinder Flow 2D Turbulent (`cylinder_flow_2d_turbulent`)

Time-dependent turbulent flow around a circular cylinder with parameterized cylinder position and Smagorinsky LES turbulence modeling for high Reynolds number flows.

Re 2000 wake vorticity with the base shear removed (cylinder_flow_2d_turbulent): the cylinder position is a per-sample input.

Equations

LES-filtered incompressible Navier-Stokes:

\[\frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} - \nabla \cdot ((\nu + \nu_t) \nabla \mathbf{u}) + \nabla p = 0\]

\[\nabla \cdot \mathbf{u} = 0\]

Smagorinsky Turbulence Model

The turbulent eddy viscosity is computed as:

\[\nu_t = (C_s \Delta)^2 |S|\]

where: - \(C_s \approx 0.1\) is the Smagorinsky constant - \(\Delta\) is the filter width (mesh cell size) - \(|S| = \sqrt{2 S_{ij} S_{ij}}\) is the strain rate magnitude - \(S_{ij} = \frac{1}{2}\left(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i}\right)\)

Reynolds Number

\[Re = \frac{U \cdot D}{\nu}\]

where: - \(U\) = inlet velocity - \(D = 2r\) = cylinder diameter - \(\nu\) = kinematic viscosity

Flow Regimes

Reynolds Number	Regime	Characteristics
Re < 5	Creeping flow	No separation
5 < Re < 40	Steady separated	Twin vortices behind cylinder
40 < Re < 200	Laminar vortex shedding	Von Kármán street
200 < Re < 300,000	Turbulent wake	Irregular vortex shedding
Re > 300,000	Fully turbulent	Turbulent boundary layer

Boundary Conditions

Inlet: Parabolic velocity profile \(u(y) = \frac{4 U_{max} y (H-y)}{H^2}\)
Outlet: Zero-stress (do-nothing)
Walls: No-slip
Cylinder surface: No-slip

Operator Learning Task

Map inlet velocity scale and cylinder position to flow field trajectory:

\[(s, c_x, c_y) \mapsto \{(u, v, p)_t\}_{t=0}^{T}\]

This enables learning: 1. Flow patterns for different cylinder positions 2. Turbulent dynamics and vortex shedding 3. Effect of Reynolds number on wake structure

Parameters

Parameter	Default	Range	Description
`viscosity`	0.0001	(1e-6, 0.01)	Kinematic viscosity ν [m²/s]
`inlet_velocity`	1.0	(0.1, 5.0)	Mean inlet velocity [m/s]
`cylinder_radius`	0.05	(0.02, 0.1)	Cylinder radius [m]
`cx_range`	(0.15, 0.5)	-	Range for cylinder x-position
`cy_range`	(0.15, 0.26)	-	Range for cylinder y-position
`use_les`	True	-	Enable Smagorinsky LES
`smagorinsky_constant`	0.1	(0.05, 0.2)	C_s constant
`time_end`	10.0	(1.0, 50.0)	Simulation time [s]
`n_time_steps`	101	(21, 501)	Output time steps

Usage

Requires FEniCSx. See FEniCSx Setup.

Basic Usage

from pdeforge import generate_dataset

dataset = generate_dataset(
    model="cylinder_flow_2d_turbulent",
    n_samples=5,
    resolution={"x": 220, "y": 82},
    params={
        "inlet_velocity": 1.0,
        "viscosity": 0.0001,  # Re ~ 1000
        "use_les": True,
        "time_end": 10.0,
        "n_time_steps": 51,
    },
    seed=42,
)

Direct Model Access

from pdeforge import get_model

Model = get_model("cylinder_flow_2d_turbulent")
model = Model(
    resolution={"x": 220, "y": 82},
    inlet_velocity=1.0,
    viscosity=0.0001,
    cx_range=(0.2, 0.4),
    cy_range=(0.15, 0.25),
)

print(model.describe())
# Reynolds number: 1000

# Generate trajectory with specific cylinder position
trajectory = model.solve(inlet_scale=1.0, cx=0.25, cy=0.2)
# trajectory.shape: (n_time_steps, nx, ny, 3)

Solver

Time-dependent finite element method using FEniCSx:

Spatial discretization: Taylor-Hood elements (P2-P1)
Time integration: Backward Euler (implicit)
Nonlinear solver: Newton with line search
Turbulence: Smagorinsky LES subgrid-scale model
Mesh: gmsh with cylinder hole, finer resolution for turbulence

Time Stepping

Adaptive time step based on CFL condition:

\[\Delta t = C_{safety} \cdot \frac{h_{min}}{U_{max}}\]

where \(C_{safety} \approx 0.5\) ensures stability.

Domain

Default channel: 2.2 × 0.41 m with cylinder of radius 0.05 m.

Cylinder position can vary within: - x ∈ [0.15, 0.5] (default) - y ∈ [0.15, 0.26] (default)

Data Shapes

# For time-dependent output
dataset.inputs.shape   # (n_samples, 3)  # [inlet_scale, cx, cy]
dataset.outputs.shape  # (n_samples, n_time, nx, ny, 3)  # u, v, p over time

Physical Behavior

Vortex Shedding

At moderate Reynolds numbers (Re > 40), alternating vortices shed from the cylinder, forming the von Kármán vortex street. The shedding frequency is characterized by the Strouhal number:

\[St = \frac{f \cdot D}{U} \approx 0.2\]

Turbulent Wake

At higher Reynolds numbers (Re > 200), the wake becomes turbulent with: - Irregular vortex shedding - Increased mixing - Higher drag coefficient - Broadband frequency spectrum

Comparison with Other Cylinder Models

Model	Steady/Unsteady	Cylinder Position	Turbulence	Re Range
`cylinder_flow_2d`	Steady	Fixed	No	< 100
`cylinder_flow_2d_unsteady`	Unsteady	Fixed	No	< 500
`cylinder_flow_2d_parameterized`	Steady	Variable	No	< 100
`cylinder_flow_2d_turbulent`	Unsteady	Variable	LES	Up to 10,000

Notes

Computational cost: Higher Re requires finer mesh and smaller time steps
Initialization: Flow is initialized with steady Stokes solution
Validation: Solutions checked for NaN/Inf, max velocity reported
Grid interpolation: FEM solutions interpolated to regular grid for ML
Cylinder interior: Filled with zeros, use domain_mask in metadata

References

Smagorinsky, J. (1963). "General circulation experiments with the primitive equations"
Williamson, C.H.K. (1996). "Vortex dynamics in the cylinder wake"
FEniCSx documentation: https://docs.fenicsproject.org/

Cylinder Flow 2D Turbulent (cylinder_flow_2d_turbulent)