Date of Award


Degree Type


Degree Name

Master of Applied Science (MASc)


Computational Engineering and Science


Nicholas Kevlahan




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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.

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.

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

Reynolds numbers without resorting to forcings that depend on the instantaneous flow structure or are not band-limited.

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