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 u u = (u 1 (x, y, t), u 2 (x, y, t), γ (x, y, t)z + W (x, y, t)) where u 1 , u 2 , 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 ω 3 and W decouple. More specifically, solutions of Burgers type can be constructed by introducing a strain field into u u u such that u u u = (−(γ /2)x − (γ /2)y, γ z) + −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 equationγ + γ 2 + p zz (t) = 0. A constraint on p zz (t) is that it must be spatially uniform. The decoupling of ω 3 and W allows the equation for ω 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 ω 3 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 ω 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 .

Local and nonlocal strain rate fields and vorticity alignment in turbulent flows

“…We also showed that even with limited R and n values, the local contribution to the total strain rate tensor field can be sufficiently removed to eliminate the anomalous alignment switching throughout the flow field. This conclusion is expected to also apply to the more realistic case of a non-uniformly stretched vortex where S zz = f (z) [16,19,20,21].…”

On the Euler equations of incompressible fluids

“…The three-dimensional vorticity is carried as well, but its magnitude is amplified or diminished by the gradient of the flow map. If one allows for infinite kinetic energy solutions, then one can find blowup ( [32], [39], [85], [86], [113], [122]). If one considers complex solutions, then again one can find blowup ( [22]).…”

A geometric interpretation of coherent structures in Navier–Stokes flows

“…Known as Burgers' vortices (Burgers 1948), these correspond to either straight tubes or flat sheets depending on whether stretching is chosen in one or two directions (see Moffatt et al 1994;Gibbon et al 1999). These exact solutions are highly idealized, whereas computations and experiments show that the reality is closer to a tangle of spaghetti.…”