Taylor’s Vortex Array: A New Test Problem for Navier-Stokes Solution Procedures

A direct simulation of the exact, nonparallel flow solution of the Navier-Stokes equations published by G.I. Taylor in 1923 was performed. The solution is a two-dimensional double array of vortices which decay exponentially with time. The unsteady Navier-Stokes equations were solved numerically for a subdomain of four vortices along the boundaries of which the exact solution was specified. The simulation was compared with Taylor’s solution to verify a new computational method. A moderate size (1200 lines) program was written to solve a system of coupled, time dependent, nonlinear partial differential equations. The algorithm developed implements a variable time step Crank-Nicolson method with Richardson’s extrapolation to discretize the timewise integration. It implements a 4th-order collocation method to solve the linear elliptic problems at the core of the computations. An iterative method is applied to solve for the nonlineari ties. The coded algorithm was designed to solve large scale fluid dynamics problems. It was applied to solve the Navier-Stokes equations and, in particular, to simulate the flow in a subdomain of Taylor’s vortex array. This simulation was a tests to verify the new 4th-order in time, 4th-order in space Navier-Stokes code. This code was executed on an Alliant FX/80 superminicomputer and on the IBM 3090-600E supercomputer. The simulation reproduced the exact solution ‘exactly,’ i.e., to within the precision of the machine.