Extreme vortex states and the growth of enstrophy in three-dimensional incompressible flows

Abstract

In this investigation we study extreme vortex states defined as incompressible velocity fields with prescribed enstrophy E 0 which maximize the instantaneous rate of growth of enstrophy dE/dt. We provide an analytic characterization of these extreme vortex states in the limit of vanishing enstrophy E 0 and, in particular, show that the Taylor-Green vortex is in fact a local maximizer of dE/dt in this limit. For finite values of enstrophy, the extreme vortex states are computed numerically by solving a constrained variational optimization problem using a suitable gradient method. In combination with a continuation approach, this allows us to construct an entire family of maximizing vortex states parameterized by their enstrophy. We also confirm the findings of the seminal study by Lu & Doering (2008) that these extreme vortex states saturate (up to a numerical prefactor) the fundamental bound dE/dt < C E 3 , for some constant C > 0. The time evolution corresponding to these extreme vortex states leads to a larger growth of enstrophy than the growth achieved by any of the commonly used initial conditions with the same enstrophy E 0 . However, based on several different diagnostics, there is no evidence of any tendency towards singularity formation in finite time. Finally, we discuss possible physical reasons why the initially large growth of enstrophy is not sustained for longer times.

Maximum amplification of enstrophy in three-dimensional Navier–Stokes flows

“…A typical time evolution of enstrophy E(u(t)) is presented in figure 2 where we show the results produced by solving the Navier-Stokes system (2) with the initial data u E 0 and u 0;E 0 ,T obtained as the solutions of the instantaneous and finite-time optimization problems 2.1 and 3.1 for E 0 = 50 and E 0 = 200, and different T . We note that when the instantaneously optimal initial data u E 0 is used, then the enstrophy grows very rapidly for short times which is followed by an immediate depletion of its growth, as already analyzed by Ayala & Protas (2017). On the other hand, when the optimal initial data u 0;E 0 ,T obtained as solution of the finite-time optimization problem 3.1 with "long" time windows T is used, then the enstrophy grows very slowly at first (or even decreases when T is sufficiently large), but eventually a much larger growth is achieved at the end of the time window, i.e., at t = T .…”

“…Such an approach allows us to simultaneously assess both the intensity and the structure of the vorticity field. In figure 8 we see that as T increases the structure of the optimal initial condition gradually changes from two colliding vortex rings characterizing the instantaneous maximizers u E 0 (Lu & Doering, 2008;Ayala & Protas, 2017) to a more complex vorticity distribution filling the entire flow domain. There are also evident differences between the optimal initial conditions belonging to the symmetric and asymmetric branches, cf.…”

“…We now move on to characterize the structure of the optimal initial conditions which give rise to the maximum enstrophy growth reported in figures 2, 5 and 6. First, in figure 8 we analyze the structure of both the symmetric and asymmetric initial conditions u 0;E 0 ,T obtained by solving the finite-time optimization problem 3.1 for a fixed E 0 = 100 and different T , and also show the instantaneously optimal initial condition u E 0 obtained as a solution of Problem 2.1 for the same value of E 0 (Ayala & Protas, 2017). These initial conditions are presented in figure 8 in terms of their vorticity ∇ × u 0;E 0 ,T and ∇ × u E 0 , whose magnitude is visualized via volume rendering and, in addition, we also show a number of vortex lines (i.e., lines everywhere tangent to the vorticity field) chosen to pass through regions with strong vorticity.…”

“…where the vector u = [u 1 , u 2 , u 3 ] T is the velocity field, p is the pressure and ν > 0 is the coefficient of kinematic viscosity (hereafter we will set ν = 0.01 which is the same value as used in earlier studies of closely-related problems (Lu & Doering, 2008;Ayala & Protas, 2017)). The velocity gradient ∇u is the tensor with components [∇u] ij = ∂ j u i , i, j = 1, 2, 3.…”

“…Since the flow evolution is governed by the Navier-Stokes system of partial differential equations (PDEs), this leads to a PDE-constrained optimization problem for the initial data u 0 which is amenable to solution using a gradient approach with gradient information obtained from the solutions of an adjoint system. The motivation for this investigation comes from our earlier study (Ayala & Protas, 2017), see also Lu & Doering (2008), where families of vortex states maximizing the instantaneous rate of growth of enstrophy were found. Although these vector fields did saturate the rigorous upper bounds on the instantaneous rate of growth of enstrophy, this maximal growth was in fact very rapidly depleted during the subsequent flow evolution, resulting in a very small only increase of enstrophy over finite times.…”