Exact solution of a restricted Euler equation for the velocity gradient tensor

Abstract

The velocity gradient tensor satisfies a nonlinear evolution equation of the form (dA_ij/dt)+A_ikA_kj− (1/3)(A_mnA_nm)δ_ij=H_ij, where A_ij=∂u_i/∂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) 4 3, 837 (1982); Physica A 1 2 5, 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.