1. A new feature of nonlinear processes in smooth shear flows: Angular redistribution of Nonlinear perturbations G. D. Chagelishvili M. Nodia institute of Geophysics at the Javakhishvili State university, Tbilisi, Georgia Georgian National Astrophysical Observatory at the Ilia State University, Tbilisi, Georgia Co-workers: Horton W., Kim J-H., Bowman J., Lominadze J.G. Bangalore- 2011

2. The aim of my talk is to describe the general behavior of nonlinear processes of the Navier- Stokes equations in smooth shear flow by direct numerical simulation, following the dynamics of localised coherent and stochastic vortical perturbations in a prototypical 2D constant shear flow. A Model of Turbulence in Unbounded Smooth Shear Flows:- Weak Turbulence Approach, Chagelishvili G.D., Chanishvili R.G. and Lominadze J.G., A Model of Turbulence in Unbounded Smooth Shear Flows:- Weak Turbulence Approach, JETP, 2002, 94, 434 Angular redistribution of nonlinear perturbations: a universal feature of nonuniform flows, Horton W., Kim J-H., Chagelishvili G.D., Bowman J. and Lominadze J., Angular redistribution of nonlinear perturbations: a universal feature of nonuniform flows, Phys. Rev. E., 2010, 81, 066304 This study highlights a novel feature of nonlinear processes in shear flows.

3. The non-normal nature of shear flow and its consequences became well understood by the hydrodynamic community in the 1990s. Classical modal analysis misses finite time behaviors (transient phenomena) that can be important. The transient phenomena may be grasped refusing spectral/classical analysis and using so-called nonmodal approach.

4. The first formulation of the nonmodal approach consists in the studying of the evolution of spatial Fourier harmonics of perturbations (SFHs), or otherwise, Kelvin modes/waves (without any spectral expansion in time). The second formulation of the nonmodal approach is a generalized stability theory (GST) – pseudo spectral method – to account comprehensively transient growth processes in nonuniform flows. SFHs (Kelvin modes) represent the “basic elements” of dynamical processes at constant shear rate and greatly help to understand transient phenomena in shear flows (Yoshida, Phys. Plasmas 2005, Chagelishvili et al, JETP Lett., 1996).

7. Usually, Usually, linear processes in different systems depend on some combination of wavenumber values, say (in the 2D case) kx 2 and ky 2. Therefore nonlinear processes result in changes of kx 2 and ky 2 increasing or decreasing them, leading to direct or inverse cascades. In our case, In our case, flow velocity shear introduces dependency of the linear processes on the combination kxky/k 2. Consequently, nonlinear processes can lead to changes in the value and sign of kxky. We call the nonlinear redistribution of perturbation Fourier harmonics over wavenumber angle a nonlinear transverse redistribution (NTR).

8. It is important to stress: It is important to stress: the nonlinear processes redistribute perturbations between different quadrants of the wavenumber plane (from kxky>0 to kxky<0, or the reverse) while leaving the total energy unchanged. Consequence of NTR: Consequence of NTR: the nonlinearity produces positive or negative feedback, and the interplay of linear and nonlinear phenomena becomes intricate. The nonlinearity acquires a vital role: it enables the self-sustenance or self-suppression of regular or stochastic perturbations.

13. Re=1000 and |B|=3

19. - The nonlinear transverse redistribution of perturbations in wavenumber plane (NTR) is a general feature of nonlinear processes in shear flows and should be inherent to 3D flows; -The conventional characterization of turbulence in terms of the direct and inverse cascades, which neglects NTR, is misleading for shear flow turbulence; -- NTR ensures positive feedback and the selfsustenance of different perturbations (including turbulent spots) during the simulation time. Consequently, NTR is naturally entered in the scheme of the bypass concept.

21. Formation of turbulent patterns near the onset of transition in plane Couette flow, Duguet Y., Schlatter P. and Henningson D. S., Formation of turbulent patterns near the onset of transition in plane Couette flow, J. Fluid Mech., 2010, 650, 119 Localised disturbances Given the role of growing turbulent spots in the formation of banded patterns and the finite correlation distance mentioned above, it is investigated the dynamics of a single localized disturbance free from any mutual interaction. The numerical domain: (L x,L z )=(800 h, 356 h) The numerical resolution: 2048x33x1024 The spots is triggered using a localized initial condition. The very large size of the domain ensures that the growing spot (at least in the early stage of its spatial development) is not affected by the ‘neighbours’ resulting from the periodic boundary conditions. The flow outside a circle of radius 10 h is initially strictly laminar. The growth of a spot at Re =350 is visualized using the streamwise velocity in the midplane y =0. The corresponding Supplementary movie is below. The flow evolves quickly into a localized structure with a rhomb/ellipse shape, dominated by streaks. The fronts of the spot initially travel with a constant velocity, resulting in a spatial expansion of the turbulent phase. The spot starts to distort and takes an S-shape, symmetric around the origin (despite the lack of initial symmetry). The structure later continues to grow by increasing in length, but keeping a constant width.

23. The end