On the behavior of velocity gradient tensor invariants in direct numerical simulations of turbulence

Abstract

Studies of direct numerical simulations of incompressible, homogeneous, and inhomogeneous turbulence indicate that, in regions of high kinetic energy dissipation rate, the geometry of the local velocity gradient field has a universal character. The velocity gradient tensor satisfies the nonlinear evolution equation (dAij/dt)+AikAkj−1/3 (AmnAnm)δij=Hij where Aij=∂ui/∂xj and the tensor Hij contains terms involving the action of cross derivatives of the pressure field and viscous diffusion of Aij. The restricted Euler equation corresponding to Hij=0 can be solved in closed form [Cantwell, Phys. Fluids A 4, 782 (1992)] and the solution has the property that, for any initial condition, Aij(t) evolves to an asymptotic state of the form Aij(t)≂Kij[R(t)]1/3 where R(t) is a function which becomes singular in a finite time and Kij is a constant matrix. A number of the universal features of fine-scale motions observed in direct numerical simulations are reproduced by Kij. In the simulation studies the first invariant of Aij is zero by incompressibility. The second and third invariants, Q and R, are determined at every grid point in the flow and the entire data set is cross-plotted to search for significant features in the space of tensor invariants. Such features can then be associated with corresponding local flow structures in physical space. When a variety of incompressible simulations are studied, scatter plots of Q vs R reveal that a fairly significant fraction of the data lies in the lower right quadrant. This is consistent with behavior predicted by the restricted Euler solution. However, the bulk of the data lies more or less uniformly distributed in a slightly elliptical region about the origin. In a direct numerical simulation of a plane, time-developing, mixing layer a small fraction of the data collects along a very pronounced, nearly straight, ridgeline extending into the upper left quadrant. This data can be traced to regions with local vorticity much larger than the local strain lying within streamwise rib vortices which connect adjacent spanwise rollers in the mixing layer simulation. Neither the predominant tendency for Q and R to lie near the origin nor the possibility for Q and R to lie far from the origin in the upper left quadrant are predicted by the restricted Euler solution. The purpose of this paper is to show that, by relaxing the assumption Hij=0 while retaining a model of dAij/dt motivated by the asymptotic form of the restricted Euler solution, one can begin to account for these features of the (Q,R) scatter plots. The results suggest that the velocity gradient tensor in three-dimensional flow tends to evolve toward an attractor in the space of tensor invariants. A significant feature of the model is that, although Hij≠0, singular behavior of Aij(t) can still occur along specific paths in the (Q,R) plane corresponding to zero values of the discriminant of Hij.