Dynamically stretched vortices as solutions of the 3D Navier-Stokes equations

A well known limitation with stretched vortex solutions of the 3D Navier-Stokes (and Euler) equations, such as those of Burgers type, is that they possess uni-directional vorticity which is stretched by a strain field that is decoupled from them. It is shown here that these drawbacks can be partially circumvented by considering a class of velocity fields of the type u = (u(1)(x, y, t), u(2)(x, y, t), gamma(x, y, t)z + W(x, y, t)) where u(1), u(2), y and W are functions of x, y and t but not z. It turns out that the equations for the third component of vorticity omega(3) and W decouple. More specifically, solutions of Burgers type can be constructed by introducing a strain field into u such that u = (-(gamma/2)x - (gamma/2)y, yz) + (-psi(y), psi(x), W). The strain rate, y (t), is solely a function of time and is related to the pressure via a Riccati equation (gamma) over dot + gamma(2) + p(zz)(t) = 0. A constraint on p(zz)(t) is that it must be spatially uniform. The decoupling of omega(3) and W allows the equation for omega(3) to be mapped to the usual general 2D problem through the use of Lundgren's transformation, while that for W can be mapped to the equation of a 2D passive scalar. When omega(3) stretches then W compresses and vice versa. Various solutions for W are discussed and some 2 pi-periodic theta-dependent solutions for W are presented which take the form of a convergent power series in a similarity variable. Hence the vorticity omega = (r(-1) W-theta, -W-r, omega(3)) has nonzero components in the azimuthal and radial as well as the axial directions. For the Euler problem, the equation for W can sustain a vortex sheet type of solution where jumps in W occur when a passes through multiples of 2 pi. (C)1999 Elsevier Science B.V. All rights reserved.