This is a genuinely different computation from the 3D companion sim, not the same scene viewed flat. Here a real, mass-conserving 2D velocity field u(x,y), v(x,y) is solved in a longitudinal slice of the vessel — a recirculation vortex is built from a closed-form velocity perturbation behind each strut, and wall shear stress is read off the actual velocity gradient at the wall, ∂u/∂y, rather than an empirical curve fit.
Baseline (2D channel Poiseuille) profile & wall shear:
u_base(y) = 6V·(y/D)·(1 − y/D)
τ₀ = μ·du_base/dy|_(y=0) = 6μV/D
Wake vortex (0 ≤ xr ≤ Lr behind each strut, 0 ≤ y ≤ h):
u_eddy = −U_max·sin(πxr/Lr)·sin(πy/h)
v_eddy = +U_max·(h/Lr)·cos(πxr/Lr)·(1−cos(πy/h))
→ satisfies continuity: ∂u_eddy/∂x + ∂v_eddy/∂y = 0 everywhere
Local wall shear behind a strut:
τ(xr) = τ₀ − μ·U_max·(π/h)·sin(πxr/Lr)
τ(0) = τ(Lr) = τ₀ (bubble closes smoothly, no discontinuity)
Vortex depth scales with strut height via a shear-layer growth law:
ratio(h) = τ_min/τ₀ = 1 − 6·√(h/D) (Blasius-type √-scaling, not a fixed constant)
- Mass conservation — whatever forward flux the vortex blocks near the wall is added back as a small uniform speed-up above the vortex, so total flow through every cross-section stays exactly V·D. That correction is verified numerically to 1e-6 relative error.
- Strut protrusion h — sets both the reattachment length Lr = 5h (same literature constant used by the 3D sim) and, independently, the vortex's own depth via ratio(h): a 20 μm strut barely disturbs the wall (ratio ≈ +0.5, shear stays positive throughout), while a 320 μm strut drives the minimum shear to ≈ −0.96·τ₀ — genuinely deeper reversal, not just a longer wake, because a taller step grows a thicker, more energetic shear layer.
- Strut spacing — when spacing is shorter than Lr, neighboring vortices overlap and the wall never fully recovers to τ₀ between struts.
- Velocity & viscosity — both set τ₀ = 6μV/D directly; this 2D longitudinal slice uses the channel-flow coefficient 6 (parallel walls), not the pipe coefficient 8 the 3D sim's circular cross-section uses — a real, intentional difference between the two geometries, not a mismatch to fix.
- The wall color strip shows τ(x) directly from this field; the moving particles are advected through the actual (u,v) vortex — watch them loop backward near the wall inside each wake and forward again past reattachment, then compare to the schematic bulk-flow band above, drawn at true scale relative to the near-wall zoom.