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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 (*d* *A* _{ i j }/*d* *t*)+*A* _{ i k } *A* _{ k j }− (1/3)(*A* _{ m n } *A* _{ n m })δ_{ i j }=*H* _{ i j }, where *A* _{ i j }=∂*u* _{ i }/∂*x* _{ j } and the tensor *H* _{ i j } contains terms involving the action of cross derivatives of the pressure field and viscous diffusion of the velocity gradient. The homogeneous case (*H* _{ i j }=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 (*d* ^{2} *A* _{ i j }/*d* *t* ^{2})+(2/3)*Q*(*t*)*A* _{ i j }=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* _{ i j }=0 case.

© 1992 American Institute of Physics

Received 30 July 1991
Accepted 14 November 1991

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http://aip.metastore.ingenta.com/content/aip/journal/pofa/4/4/10.1063/1.858295

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/content/aip/journal/pofa/4/4/10.1063/1.858295

1992-04-01

2016-09-24

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