EXACT SOLUTION OF A RESTRICTED EULER EQUATION FOR THE VELOCITY-GRADIENT TENSOR

The velocity gradient tensor satisfies a nonlinear evolution equation of the form (dA(ij)/dt) + A(ik)A(kj) - (1/3) (A(mn)A(nm)) delta(ij) = H(ij), where A(ij) = partial derivative u(i)/partial derivative x(j) and the tensor H(ij) contains terms involving the action of cross derivatives of the pressure field and viscous diffusion of the velocity gradient. The homogeneous case (H(ij) = 0) considered previously by Vielliefosse [J. Phys. (Paris) 43, 837 (1982); Physica A 125, 150 (1984)] is revisited here and examined in the context of an exact solution. First the equations are simplified to a linear, second-order system (d2A(ij)/dt2) + (2/3)Q(t)A(ij) = 0, where Q(t) is expressed in terms of Jacobian elliptic functions. The exact solution in analytical form is then presented providing a detailed description of the relationship between initial conditions and the evolution of the velocity gradient tensor and associated strain and rotation tensors. The fact that the solution satisfies both a linear second-order system and a nonlinear first-order system places certain restrictions on the solution path and leads to an asymptotic velocity gradient field with a geometry that is largely but not wholly independent of initial conditions and an asymptotic vorticity which is proportional to the asymptotic rate of strain. A number of the geometrical features of fine-scale motions observed in direct numerical simulations of homogeneous and inhomogeneous turbulence are reproduced by the solution of the H(ij) = 0 case.