Lid-Driven Cavity: Reynolds Number, Vortex and Grid Convergence

The length and velocity scales are $L$ and $U$, giving $\mathrm{Re}=UL/\nu$. The coordinates start at the bottom left; the lid moves right. Define $u=\psi_y$, $v=-\psi_x$ and $\omega=-\nabla^2\psi$. The steady equations are

These are the equations discussed in WiredWhite post 39865. Post 45218 describes a different, pressure-projection solution of the same cavity problem.

The implementation here is derived from these equations. Uniform grids use fourth-order centred first and second derivatives, with fourth-order one-sided derivatives at the first interior line. A nine-point compact Poisson stencil satisfies $L_9\psi=-\omega-h^2 L_5\omega/12$. At a stationary wall the one-sided vorticity closure is $\omega_w=(-6\psi_1+1.5\psi_2-2\psi_3/9)/h^2$; the moving top adds $-11/(3h)$. These coefficients follow from Taylor expansion, not benchmark fitting. The lid corners are assigned the stationary-wall velocity.

Keep streamfunction and scaled vorticity $\theta=h^2\omega$ as coupled unknowns. Group them by grid row and factor the nonzero bands of the analytic Jacobian. The largest dense pivot is 62×62, rather than a full-grid dense system. Mesh-only derivative matrices and the coarse-grid Stokes estimate are cached separately. The coarse solve uses Reynolds-number continuation with increments no larger than 600. Bilinear prolongation of its converged streamfunction and vorticity initializes the fine solve at the requested Re.

A damped modified Newton iteration reuses a factorization while the residual contracts by at least a factor of 0.4; otherwise it rebuilds the Jacobian. A halving line search reduces the transport imbalance. Each stage stops below a maximum dimensionless transport residual of $10^{-6}$ or after 40 updates. The reported residual is evaluated from the final fields, including when the iteration limit is reached. The compact Poisson residual and discrete divergence are evaluated separately. This is a steady solver, not an RK2/SOR transient reproduction.

The primary vortex is the most negative streamfunction node, refined by a local two-dimensional quadratic. Both velocity minima are sampled on the complete centreline, including the walls. Compare both grid profiles at common nodes: $\Delta_g=\max(|u_f-u_c|,|v_f-v_c|)/U$. The screening targets are $R<10^{-6}$ on both grids and $\Delta_g<0.03$. A small residual does not measure discretization error. Tracers use midpoint integration and bilinear sampling of a cached velocity field; their positions alone change during play.

Steady, two-dimensional, incompressible Newtonian flow; constant kinematic viscosity; a square cavity; a uniform rightward lid speed; three stationary no-slip walls; no body forces. All solver variables are dimensionless. The finite-difference equations and wall coefficients are derived here. No empirical vortex-position correlation or benchmark table is used in the solver. The two-grid comparison uses identical discretization methods. The tracer animation is a visualization of the frozen field and restarts after 30 units of tU/L.

Use over 1 ≤ Re ≤ 1000. Fast mode uses 13×13 and 25×25 nodes; refinement mode uses 17×17 and 33×33. Browser cost rises with mesh size and Reynolds number. Both mesh options are screened across the supported range; the tracer animation reuses cached fields. The Re = 100 and 400 checks accept finite-grid differences explicitly: up to 0.01–0.02 L in vortex coordinates and 0.007–0.025 U in the centreline minima. These tolerances are test criteria, not confidence intervals. Above Re = 400 the small grids can significantly under-resolve wall shear and corner vortices. The lid/wall velocity discontinuity makes global formal order uncertain. Two grids cannot establish an observed order, GCI or rigorous error bound; the grid-change target concerns centreline velocities only. This tool does not calculate pressure, forces, turbulence, three-dimensional instability, heat transfer or actual startup time. No symmetry of the lid-driven velocity profiles is assumed. Smooth contours alone do not validate a solution.