Debris Flow Cross-Section: 2D Bingham Solve & Levee Accretion

Rheology Controls

Slope angle θ 18°

Yield stress τ_y 1200 Pa

Plastic viscosity μ 120 Pa·s

Flow depth h 1.4 m

Channel width W 6.0 m

Flow State (from the 2D solve)

Solving the cross-section…

Surface centerline velocity

— m/s

Rigid plug area (of cross-section)

— %

Stalled margin width (each bank)

— m

Bingham number Bi = τ_y/(ρg sinθ·h)

—

How this 2D model differs from the 3D version

This is not a flattened copy of the 3D scene. The 3D sim solves the Bingham profile only in the vertical direction and fakes the lateral (bank-to-bank) variation with a hand-picked parabolic multiplier. This 2D sim instead numerically solves the real governing PDE over the whole channel cross-section (x = lateral position, z = height above bed):

0 = ρg·sinθ + ∂/∂x(μ_eff ∂u/∂x) + ∂/∂z(μ_eff ∂u/∂z)

μ_eff(γ̇) = μ + (τ_y/γ̇)·(1 − e^(−m·γ̇))     [Papanastasiou regularization]
γ̇ = |∇u|                                    (local shear-rate magnitude)

u = 0 on bed (z=0) and both banks (x=±W/2)
∂u/∂z = 0 at the free surface (z=h)

The regularization replaces the true rigid/yielded switch with a smooth apparent viscosity that grows huge (≈ μ + τ_y·m) wherever the local shear rate is tiny, and relaxes to μ where shear is strong — so a cell "locks up" into the plug purely because the PDE finds it energetically favorable, not because a formula told it to. The solver runs a Picard iteration: freeze μ_eff from the current velocity field, solve the resulting linear finite-volume system with over-relaxed line sweeps (an exact tridiagonal solve down each vertical column, swept bank to bank), recompute μ_eff, repeat to convergence.

Far from the banks (x ≈ 0, wide W) this 2D solve converges to the same closed-form vertical profile the 3D sim uses — that's the physics check, not a coincidence.
Near the banks, wall drag genuinely drops the local velocity to zero rather than being scaled by an ad-hoc factor — the stalled margin width you see is a real PDE output.
The lower strip accretes a physical levee: wherever the converged surface velocity stays below 5% of the centerline peak, material is deposited there frame by frame, building the bank the way a real debris flow leaves natural-hazard deposits after repeated pulses.

Rigid plug core Fast shear layer Stalled margin / levee