Computation of High Reynolds Number Flows Using Vorticity Confinement: I. Formulation

A computational method is described that has been d esigned to capture thin vortical regions in high Re ynolds number incompressible flows. The principal objecti ve of the method—Vorticity Confinement (VC)—is to capture the essential features of these small-scale vortical structures and model them with a very efficient difference met hod directly on an Eulerian computational grid. Essentially, th e small scales are modeled as nonlinear solitary waves that “live” on the lattice indefinitely. The method allo ws convecting structures to be modeled over as few as 2 grid cells with no numerical spreading as they convect indefin itely over long distances, with no special logic re quired for merging or reconnection. It also serves as a very efficient substitute for RANS models of attached a nd separating boundary layers and vortex sheets and filaments. Fu rther, the method easily allows boundaries with no- slip conditions to be treated as “immersed” surfaces in uniform, non-conforming grids, with no requirements for complex logic involving “cut” cells. In this paper a description of the basic VC method is given. This is more comprehensive than has been previously available. There are close analogies between VC and well-known shock and contact discontinuity capturi ng methodologies. These are discussed to explain the b asic ideas behind VC, since it is somewhat differen t than conventional CFD methods. Some of the possibilitie s that VC offers towards very efficient computation of turbulent flows in the LES approximations are explored. Thes e stem from the ability of VC to act as a negative dissipation at scales just above a grid cell, but that saturates a nd does not lead to divergence. This feature allows 1. approximate cancellation of numerical diffusion, so that more complex, high order-low dissipation s chemes can be avoided. Small-scale vortical structures at the grid cell level can then be captured, resulting in very efficient use of the available degrees of freedom o n the grid. 2. approximate treatment of backscatter. This invol ves the addition of (modeled) subgrid kinetic energ y to the flow in a natural way, without requiring stochastic forcing, and which restores some of the instabilit ies that are removed by the (implicit) filtering. Although used for a number of years for complex, at tached and separating flows, and trailing vortices, its use as an LES method is relatively recent. In Ref. [0], some initial LES results are presented.