This is a 2D-native view of the same Shields / Sumer-Fredsøe scour physics as the 3D lab, built from a top-down bed-shear grid instead of a rendered seabed mesh. An analytic 2D potential-flow solution around the pile gives a local flow speed-up factor at every grid cell — doubled at the flanks, zero at the up/downstream stagnation points, exactly the pattern that seeds a real horseshoe vortex — and each cell erodes independently toward its own local Sumer-Fredsøe equilibrium depth. Eroded volume is tracked and redeposited downstream as a mound, so sediment mass is conserved cell-by-cell instead of using a fixed prescribed profile.
Potential flow around a cylinder (radius a, free stream U along +x):
u(x,y) = U·[1 − a²(x²−y²)/r⁴], v(x,y) = −U·2a²xy/r⁴, r² = x²+y²
surface speed |u| = 2U|sinθ| → max at flanks (θ=90°), 0 at stagnation (θ=0°,180°)
Local Shields threshold (same as 3D lab):
τ꜀ᵣ = θ꜀ᵣ(ρₛ−ρw)g·D₅₀, U꜀ᵣ = (√(τ꜀ᵣ/ρw)/κ)·ln(12h/ks)
Per-cell equilibrium (Sumer & Fredsøe), driven by LOCAL speed |u(x,y)|:
Sₑ(x,y)/D ramps 0→1.3 as |u(x,y)|/U꜀ᵣ goes 0.5→1, plateau above
Cell relaxation: z(x,y,t) → zₑ(x,y)·(1 − e^(−t/T))
Downstream mound height scaled so ∫mound dA ≈ 0.55·∫pit dA (mass-conserving)
- Grid heatmap (top) — plan view of the seabed: dark teal = deepest scour, tan = undisturbed bed, pale = the redeposited downstream mound. The pit is widest at the flanks and narrowest fore/aft of the pile because the potential-flow speed-up is largest at the flanks — a genuinely different emergent shape than a hand-authored axisymmetric pit.
- Cross-section (bottom) — the actual grid data sliced along the flow centerline (y=0), the same bed-elevation numbers driving the heatmap above, so the two views are always consistent with each other.
- Flow lines & drifting grains — traced directly from the same analytic velocity field (not a canned swirl), so grains genuinely accelerate at the flanks and slow near the stagnation points.
- Current speed / grain size / rock-armor — identical physics to the 3D lab: faster current or finer sand raises U/U꜀ᵣ and deepens the pit toward the ≈1.3D live-bed plateau; armor scales every cell's equilibrium down to a shallow residual.