Turbulence Simulation with Spectral Method in Ideal and Density-Stratified Fluids

Numerical simulation of the evolution of two- and three-dimensional vortex arrays in incompressible and weakly density-stratified fluids is performed. The Navier -Stokes equations for the free 3-D vortex arrays are solved with the Galerkin-Fourier spectral method and periodic boundary conditions. The initial conditions include the elliptical variations of Taylor-Green vortex, which has closed plane streamlines, and the full 3-D Deissler vortex. The results, including the lifetime of main initial vortices, the generation of offspring structures and the evolutions of property spectra, shows that the interaction between neighboring vortex sectors with different orientations is an important mechanism in the turbulence generation. The numerical simulation of the 3-D vortex flow in a density-stratified fluid is based on a generalization of the Boussinesq approximation. The generalization makes the continuity relation become nonhomogeneous, but still solvable with the Navier-Stokes equations. The analysis and calculation of the evolutions of vortex arrays in stratified fluid show that (1) the potential energy change may contribute to the flow development; (2) a negative real term in the governing equation explains the trend towards two-dimensional flow; and (3) an imaginary term in the governing equation describes a weak inhibition of stratification on turbulence generation.