## Lagrange Project PhD Fellowship 2011-2013

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- Last Updated on Thursday, 12 February 2015 17:27

**To what extent spectra of turbulent flows are linked to the nonlinear interaction among their different modes?**

**Why the project belongs to the complex system research**

Turbulence is historically known as one of the archetype of the complex system research.

In particular, turbulent flows contain a wide range of scales, each range being characterized by its own physics. For instance the energy dissipation takes place at small-scales. Yet, the process is linked to the large scales of the system. One central problem of turbulence is to compute the large scale phenomena by modeling or parametrizing the small scales; this is the goal of subgrid scale models. Another example deals with modeling micro-mixing (relevant to chemical industry and combustion), in which small scales are the important feature.

Laboratory and numerical results are continuously being generated on the small scale features of turbulent dynamics. One fundamental question is: are the small scales universal? If so, under which conditions? If not, when? In particular, what is their connection to the large scale motion? The basis of the near-universal behavior of small scales is provided by Kolmogorovâ€™s theory (1941 and 1962). The gaps in this theory are becoming increasingly certain. New ideas, post-Kolmogorov, dealing with the non-universality of small scales dynamics are needed.

**Project description**

In order to understand whether, and to what extent, spectral representation can effectively highlight the nonlinear interaction among different scales, it is necessary to consider the state that precedes the onset of instabilities and turbulence in flows. In this condition, a system is still stable, but is however subject to swarming of arbitrary three-dimensional small perturbations. These can arrive any instant, and then undergo a transient evolution which is ruled out by the initial value problem associated to the Navier-Stokes linearized formulation. The set of all possible 3D small perturbations constitutes a system of multiple spatial and temporal scales which are subject to all processes included in the perturbative Navier-Stokes equations: linearized convective transport linearized vertical stretching and tilting, and the molecular diffusion. Leaving aside nonlinear interaction among different scales, these features are tantamount to the features of the turbulent state.

If it were possible to observe such a system in a temporal window and obtain the instantaneous 3D wave number spectra, it would be possible, among others, to determine the exponent of the inertial range of the arbitrary perturbation evolution, and to compare it with the exponent of the corresponding developed turbulent state (notoriously equal to -5/3). Two possible situations can therefore appear:

- The exponent difference is large, and as such, is a quantitative measure of the nonlinear interaction in spectral terms.

- The difference is small. This would be even more interesting, because it would indicate a higher level of universality on the value of the exponent of the inertial range, not necessarily associated to the non linear interaction.

We propose building temporal observation window for the transient evolution of a large number (order 10^^{3}) of arbitrary small 3D perturbations acting on a typical shear flow. In particular we take advantage of a recently -numerically obtained- set of solutions yielded by the initial value problem applied to 3D perturbations of a plane bluff-body wake. These solutions have revealed the existence of many different kind of transient behavior, not all of which is trivial. If these transients, obtained in association with arbitrary, statistically selected, initial conditions, are injected in a statistical way into the temporal observation window, we can obtain a close representation of the perturbation state that precedes the onset of instability-turbulence.

**PhD Fellow 2011-2013:** Francesca De Santi