Three-dimensional vortex simulation of rollup and entrainment in a shear layer

The transport element method is extended to three dimensions to study the evolution of scalar fields in a flow with high vorticity concentration. The numerical scheme is based on following Lagrangian computational elements employed in the transport of vorticity and local scalar gradients. The formulation of the numerical scheme is first presented as a direct generalization of the three-dimensional vortex element method. The numerical algorithms required to implement this scheme are then developed. Problems associated with severe distortion of the flow map due to the growth of perturbations are shown to cause difficulties including loss of numerical accuracy and resolution. Means to overcome this problem are discussed and are shown to yield accurate solutions. Two grid-based on two grid-free methods for the computation of vorticity stretching are implemented. The accuracy of the methods is discussed in the light of numerical results which reveal the need for a careful treatment of the discrete form of the vorticity transport equation. The methods are applied to study the evolution of an initially two-dimensional shear layer, perturbed in the streamwise and spanwise directions. Attention is focused on the role of the spanwise instability in enhancing the rate of scalar entrainment into the large-scale structures which form as the streamwise instability develops. Two mechanisms, associated with the onset of three-dimensional instability, are responsible for this enhancement: vorticity intensification within the large eddy core due to spanwise stretching which delays its collapse; and generation of transverse entrainment currents towards the eddy core due to the formation of streamwise vortex structures within the core and along the braids between neighboring cores. Preferential entrainment is detected along the spanwise direction due to the streamwise vorticity.