Paper boat stream — 2D river flow lab

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Controls
Live readouts
Numerical checks
What to try
  1. Watch the white arrows: the water is fastest in the middle of the channel and slows towards the banks.
  2. Press "Fast mountain brook": the Manning velocity and the Froude number jump when slope rises and depth falls.
  3. Press "Straight canal": the boat settles on one streamline; the meanders disappear and the sinuosity goes to 1.
  4. Raise the boat response time: a heavy, slow-responding boat overshoots the bends and grounds against the bank.
The science

In a wide open channel the depth-averaged momentum balance is gravity along the slope against bed friction. At steady state it gives the Manning equation:

dV/dt = g S − g n² V|V| / d4/3   ⇒   V = d2/3 S1/2 / n Fr = V / √(g d),   Q = V W d,   u(n) = 1.5 V (1 − (2n/W)²)

The cross-channel profile is a parabola that averages to V. The centreline is a sine meander y = A sin(2πx/λ); sinuosity is arc length over wavelength. The boat is a particle that relaxes towards the local water velocity, dv/dt = (u − v)/τ, with small time-dependent eddies added, and bounces off the banks.

Checks: relaxed momentum balance versus Manning, profile normalisation, boat relaxation after one time constant, and the meander arc length computed two independent ways.

Simplifications: depth-averaged flow with a prescribed (not computed) cross-channel profile, no secondary helical flow in bends, no wind or waves, drag in the lag-time form only; eddies are a smooth analytic pattern, not real turbulence.