Incompressible 2D Lattice (Decaying Vortex Array)

Overview

This case verifies the unsteady incompressible solver against an exact time-dependent solution of the Navier–Stokes equations: a doubly-periodic lattice of counter-rotating vortices that decays in place under viscous diffusion. Because the spatial pattern is fixed and only its amplitude relaxes, every velocity component scaling as $e^{-2\nu t}$, the exact field is known at any instant, so the solver's error can be measured directly rather than inferred from a reference code. Integrating the flow forward in time and refining the mesh uniformly lets the discretization error in each conservative equation be tracked as the grid is resolved, with the design order of the spatial discretization being two. The second-order implicit time integration keeps the temporal error below the spatial error being measured.

Problem Setup

The domain is the unit square $[0,1{]}^2$, discretised as a single spanwise layer of hexahedra and refined uniformly through ${16}^2,{32}^2,{64}^2,{128}^2$ cells. Both the streamwise and transverse boundaries are translational periodic pairs, making the domain an infinite lattice; the two spanwise faces are symmetry planes, enforcing the quasi-two-dimensional flow. The fluid is treated as constant-density and laminar, and the simulation is advanced from the exact initial field at $t=0$ to $t=1$ with a fixed time step using the second-order implicit scheme. The solver is initialised and scored against the built-in analytic lattice solution, so no external mesh or reference data is required.

Quantities of Interest

The verification metric is the discretization error of each conservative equation, continuity and the momentum components, measured in the $L_2$ norm at the final time and reported by the solver on each grid. The error is plotted against the representative cell size $h=(1/N{)}^{1/2}$, and the observed order of accuracy $p$ is computed between successive grids. Across the refinement family the observed orders for continuity range from 2.4 to 2.7, and those for the two momentum equations range from 2.8 to 3.8; both exceed the design order of two, which is drawn for reference.