Backward-facing Step

Steady channel flow over a backward-facing step at $\mathrm{Re}=800$ — a boundary-value problem with separation and reattachment:

$$(\mathbf{u}\cdot \mathrm{\nabla })\mathbf{u}+\mathrm{\nabla }p-\nu {\mathrm{\nabla }}^2\mathbf{u}=0,\mathrm{\nabla }\cdot \mathbf{u}=0.$$

Problem setup

The channel spans $x\in [0,15]$, $y\in [-\frac{1}{2},\frac{1}{2}]$. Flow enters only through the upper half of the inlet plane: the parabolic profile $u(y)=24y(\frac{1}{2}-y)$, $v=0$ is positive for $y\in (0,\frac{1}{2})$ (peaking at $u=1.5$ at $y=\frac{1}{4}$) and zero below, so the lower half of the inlet acts as the vertical face of a backward-facing step of height $\frac{1}{2}$ — an expansion ratio of 2. The kinematic viscosity is $\nu =1/800$, giving $\mathrm{Re}=800$. Downstream of the step the flow separates, forms a recirculation bubble along the lower wall, and reattaches several step heights downstream — the quantity PINN training must get right.

Geometry and boundary conditions (streamwise axis compressed; the channel is 15 units long).

Boundary conditions, exactly as they enter the loss:

• Inflow ($x=0$): $u=24y(\frac{1}{2}-y)$ for $y>0$ (zero on the step face), $v=0$ — terms u_in, v_in.
• Walls ($y=\pm \frac{1}{2}$, and the step face): no-slip $u=v=0$ — terms u_noslip, v_noslip.
• Outflow ($x=15$): $p=0$ — term p_out.

Together with the three momentum/continuity residuals, that is eight loss terms balanced by grad-norm weighting; collocation points are drawn from the unstructured reference mesh with MeshSampler since the geometry is non-rectangular.

Reference velocity magnitude: recirculation bubble behind the step.

Results

relative L2 error 0.016recipe fixed pseudo-timeRe 800, 100k steps, single GPU

The baseline never finds the recirculation bubble: it settles at a relative L2 error of 0.28 in $u$ (0.83 in $v$), an attached-flow spurious solution. Pseudo-time stepping recovers the separation, and here the constant-weight variant wins: 0.029 in $u$ and 0.106 in $v$ — 0.016 in the velocity magnitude, a ~17× improvement over the baseline. The adaptive variant helps but lands mid-way (0.15 in $u$) and is seed-sensitive on this problem; the recirculating flow develops slowly enough that the adaptive step-size estimate keeps shrinking the damping too early.

Reference and predicted velocity magnitude, and the absolute error — concentrated along the shear layer and reattachment region.

Training losses and per-variable relative L2 errors of the showcase (fixed pseudo-time) run.

Run

cd examples/bfs_flow
python3 main.py --config=configs/fixed_pseudo_time.py

Config	Description
configs/baseline.py	PirateNet baseline
configs/pseudo_time.py	Adaptive pseudo-time stepping
configs/fixed_pseudo_time.py	Constant pseudo-time weights (showcase)

Notes

• Reading the .vtu reference data requires pip install pyvista.