Controlling the Dual Cascade of Two-dimensional Turbulence

Abstract

<p>p.p1 {margin: 0.0px 0.0px 0.0px 0.0px; font: 11.5px Times} p.p2 {margin: 0.0px 0.0px 0.0px 0.0px; font: 8.0px Times} span.s1 {font: 12.0px Times} span.s2 {font: 8.0px Times} span.s3 {font: 6.0px Times}</p> <p>The Kraichnan-Leith-Batchelor (KLB) theory of statistically stationary forced homogeneous isotropic 2-D turbulence predicts the existence of two inertial ranges: an energy inertial range with an energy spectrum scaling of k⁻³ , and an enstrophy inertial range with an energy spectrum scaling of k⁻³. However, unlike the analogous Kolmogorov theory for 3-D turbulence, the scaling of the enstrophy range in 2-D turbulence seems to be Reynolds number dependent: numerical simulations have shown that as Reynolds number tends to infinity the enstrophy range of the energy spectrum converges to the KLB prediction, i.e. E ~ k⁻³.</p> <p>We develop an adjoint-equation based optimal control approach for controlling the energy spectrum of incompressible fluid flow. The equations are solved numerically by a highly accurate method. The computations are carried out on parallel computers in order to achieve a reasonable computational time.</p> <p>The results show that the time-space structure of the forcing can significantly alter the scaling of the energy spectrum over inertial ranges. This effect has been neglected in most previous numerical simulations by using a randomphase forcing. A careful analysis of the resulting forcing suggests that it is unlikely to be realized in nature, or by a simple numerical model. Therefore, we conjecture that the dual cascade is unlikely to be realizable at moderate</p> <p>Reynolds numbers without resorting to forcings that depend on the instantaneous flow structure or are not band-limited.</p>