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

Abstract 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=(u1(x,y,t),u2(x,y,t),γ(x,y,t)z+W(x,y,t)) where u1,u2,γ and W are functions of x,y and t but not z. It turns out that the equations for the third component of vorticity ω3 and W decouple. More specifically, solutions of Burgers type can be constructed by introducing a strain field into u such that u = −(γ/2)x−(γ/2)y,γz + −ψ y ,ψ x ,W . The strain rate, γ(t), is solely a function of time and is related to the pressure via a Riccati equation γ +γ 2 +p zz (t)=0 . A constraint on pzz(t) is that it must be spatially uniform. The decoupling of ω3 and W allows the equation for ω3 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 ω3 stretches then W compresses and vice versa. Various solutions for W are discussed and some 2π-periodic θ-dependent solutions for W are presented which take the form of a convergent power series in a similarity variable. Hence the vorticity ω = r −1 W θ ,−W r ,ω 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 θ passes through multiples of 2π.