Mitor Projects 2009-2010 & 2011-2012

MISTI Global Seed Funds

Title: Long-term interaction in fluid systems

Massachusetts Institute of Technology (Mathematics Department, Prof. G. Staffilani) and Politecnico di Torino (Department of Aeronautics and Space Engineering, Prof. D. Tordella)

In order to understand whether, and to what extent, the turbulent power spectrum reflects the effect of the nonlinear interaction among different wave-lengths, we propose to study the state that precedes the onset of turbulence in a flow. In this condition, a system is still technically stable (i.e.: linear perturbations eventually decay). However, the transient behavior of the linearized problem is very complicated, and poorly understood: there seems to be a multitude of multiple and or degenerate eigenvalues, with associated eigenfunctions whose behavior is not purely exponential.

This causes dramatic initial growth of the perturbations, with complicated patterns that eventually may trigger 3-D nonlinear instabilities that prevent their eventual decay. The set of all the possible 3D small perturbations that arise in this fashion constitutes a system of multiple spatial and temporal scales which are subject to all the processes included in the perturbative Navier-Stokes equations: linearized convective transport, linearized vortical stretching and tilting, and molecular diffusion. With the exception of the nonlinear interactions among the different scales, these are the features present in a turbulent state. An interesting question is then: to what extent can one think as the role of the nonlinear interactions as "merely" being the engine that keeps the transient behavior of the linearized system ever active, postponing and preventing the eventual exponential decay that the linear system alone has?

In order to answer the question at the end of a last paragraph, the idea is to observe the behavior of the linearized system, while keeping it artificially within the transient window. Then one can observe what spectra such a situation eventually produces, when starting from featureless data (say, white noise: all wavelengths have the same probability). If a power spectrum law arises, then is the exponent the same (or different) from the expected (-5/3) for a developed turbulence state? The exponent difference would then be a quantitative measure of how important the nonlinear interactions are. A small difference would be very interesting, as it would indicate a higher level of universality for the value of the exponent of the inertial range, not necessarily associated with the nonlinear interactions in any detailed fashion. Our work of the last few months strongly suggests that the second situation is the actual one.

We propose to do a careful verification of the result described above, by building a temporal observation window for the transient evolution of a large number (order of 103) of arbitrary small 3D perturbations acting on a typical shear flow. In particular we can take advantage of a recently numerically obtained set of solutions given by the initial value problem for 3D perturbations of a plane bluff-body wake. These solutions have revealed the existence of a very complicated and varied set of transient behaviors, many highly non-trivial. These transients, combined with a selection of white-noise initial amplitudes, can then be used to produce the situation where the eventual linear decay is suppressed. The simplest way to do this is to compute the evolution of each input wave-length till right before the time where the exponential decay kicks in. With a large set of data, it would then be possible to verify if the situation described at the end of the prior paragraph actually happens. Assuming that the answer from the study above is a yes, and then the question is why? What is the relationship of this linear process with turbulence? Is there some self-similarity of the intermediate wave numbers solutions of the linear problem that produces this, analogous to the Kolmogorov’s argument for fully developed homogeneous turbulence?