Discussion Room, Newton Institute
This abstract is being written after the talk. The talk was planned as a report on recent results. Instead, it became a rather informal discussion of the approach to predicting streaks advocated in (Chernyshenko & Baig, The mechanism of streak formation in near-wall turbulence, J. Fluid Mech. 2005, V.544, pp.99-131). It is appropriate, therefore, simply to state the main ideas put forward and discussed.
The approach is based on linearised equations but applies to fully-developed turbulent flow, which is non-linear. The reason why linearised equations apply to nonlinear turbulence can be explained in the following way. The full (non-linear) Navier-Stokes equations can be rewritten in terms of perturbations, uí, with all the terms occurring in the linearised equations placed in the left-hand side and all other terms in the right-hand side: Luí=N[uí]. The non-linear operator N acts as a mixer, conserving energy but destroying structure. The linear operator L acts as a filter-amplifier, amplifying particular structures (streaks) and filtering out the rest (so that the total energy remains constant). Hence, streaks can be predicted by studying L. One way of studying its properties is to solve an optimization problem: maximize ||uí|| for fixed ||Luí||. From the mathematical viewpoint this leads to a version of the optimal perturbation problem and is related to non-normality and transient growth. However, the interpretation given above requires that the norm of uí should measure the property of the solution which is being predicted. For predicting streak spacing at a certain distance from the wall this should be the energy of uí measured within a plane at that distance from the wall. Such an approach proved to have significant predictive ability. Note also that the optimal perturbation solution itself is not a model of the developed turbulent flow but rather a means to investigate the properties of L. More details can be found in the paper cited.