Incompressible Wake and the Projection Method

Chorin's projection method made visible: the incompressible Navier-Stokes equations $\partial_t\vec u+(\vec u\cdot\nabla)\vec u=-\nabla p+\nu\nabla^2\vec u$ with the constraint $\nabla\cdot\vec u=0$ , solved on a MAC staggered grid (semi-Lagrangian advection, implicit diffusion, an SOR pressure-Poisson projection, the gate-tested shared engine). Flow past a bluff body, coloured by speed $|\vec u|$ over the dark field: the uniform stream, the bright acceleration over the shoulders of the body, the dark wake deficit, and the discrete shed cores. The headline is the live $\max|\nabla\cdot\vec u|$ readout: the pressure solve drives it small every step, so the flow stays incompressible. Sweep the Reynolds number $Re=UD/\nu$ from a glassy creep, through a steady recirculating bubble, to a genuine periodic von Karman vortex street. (The live path switches on the engine's BFECC low-dissipation advection and Steinhoff vorticity confinement, both default-off so the offline invariants are unaffected: these cut the semi-Lagrangian numerical viscosity so the effective $Re$ tracks the nominal one and the wake genuinely sheds. The shed period gives an approximate Strouhal number; the precise $St(Re)$ stays a documented finer-grid quantity.)

Figure 1. Speed field of incompressible 2D flow past a bluff body across the Reynolds-number regimes (creeping, steady recirculating bubble, shedding von Karman street, broadband wake); the live readout is the post-projection discrete divergence, the incompressibility constraint the Chorin projection enforces each step. Method: BFECC low-dissipation semi-Lagrangian advection, implicit diffusion, Steinhoff vorticity confinement, red-black SOR pressure-Poisson on a MAC staggered grid (the gate-tested shared engine; BFECC and confinement default-off so the offline invariants run the unmodified scheme), speed colormap over the dark theme.

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Reynolds300

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WHAT TO TRY

Raise the Reynolds number through the regimes: creeping Stokes flow is fore-aft symmetric, a steady wake forms two attached eddies, and past Re about 100 the wake sheds a von Karman vortex street. The regime label tracks the transition.
Switch the obstacle between a disk and a square: the sharp corners fix the separation points and change the shedding, while the round cylinder lets the separation point wander.
Toggle the field to vorticity: the alternating red and blue patches are the signed vortices peeling off each side, and the max-divergence readout near zero confirms the projection step is keeping the flow incompressible.