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Jet noise constitutes an important component of the sound radiated by modern civil transport aircrafts.
The stringent regulations imposed by regulatory agencies have led manufacturers and research establishments to investigate novel aerodynamic noise modelling concepts. The availability of powerful computers has resulted in the emergence of the new field of Computational Aeroacoustics, whereby one seeks to solve the full compressible Navier-Stokes equations numerically both in the turbulent part of the flow and in the acoustic far field.
Figure 1 - Sound emitted by a jet in a self-sustained oscillation régime.
Idealised theoretical models have recently led to the definition of so-called superdirective noise radiation due to the presence of instability wavepackets evolving in the jet. This phenomenon is distinct from the classical quadrupole description implicit in Lightill's theory. A research project carried out in collaboration with ONERA - Châtillon and the French Defense Agency (DGA) is currently underway to identify superdirective noise sources in direct numerical simulations of hot jets. Such flows are known to give rise to self-excited global oscillations generated by a pocket of absolute instability immediately downstream of the norzle exit. An example of simulation is illustrated in Figure 1.
An important part of the energy in wall bounded turbulent flows is associated with large scale coherent streaks. Since for these flows the turbulent mean velocity profile is stable, these coherent structures are not issued from a linear instability as is the case for turbulent shear layers or wakes but from processes including transient energy growth. We therefore compute the optimal amplification of coherent perturbations sustained by turbulent channel and boundary layer mean flows. The turbulent stresses are modelled using the eddy viscosity associated with the mean flow. The most amplified perturbations correspond to coherent streamwise streaks generated by coherent streamwise vortices. For sufficiently large Reynolds numbers two distinct peaks of the optimal growth exist, respectively scaling in inner and outer units. The optimal structures associated with the peak scaling in inner units correspond well with the most probable streaks and vortices observed in the buffer layer. The energy growth associated with the peak scaling in outer units increases with the Reynolds number. The most amplified perturbations are very large scale coherent streaks that scale in outer variables in the outer region and in wall units in the inner region where they are proportional to the mean flow velocity. These outer streaks protrude far into the near wall region, having still 50% of their maximum amplitude at y+=20. These findings could be relevant for the control of turbulent flows with optimal streaks, as shown in figure 5 above.
Figure 6 - (a - b) spatio-temporal evolution of the axial velocity measured on a radius at z = 0.42 in the boundary layer of the lower disc, a) globally stable case b) globally unstable case; c) local amplitude of the mode showing the exponential trump of the elephant mode with the predicted spatial growth rate -k0i.
Elephant Mode riding the boundary layer of a 3D rotating disc: This study is the subject of the thesis of Bertrand Viaud and is part of a collaboration with Eric Serre from the MSNM-GP Laboratory in Marseilles. The objective is to study instabilities and transition in confined flows subjected to rotation. Rotating wall flows exhibit fundamental physical phenomena relevant for a variety of configurations: crossflow instability of flow over a wing, vortex breakdown, turbulent Ekman layers in the oceans and atmosphere.
We have conducted direct numerical simulations of flow between two co-rotating annuli, which in the limit of large rotation, involves two Ekman boundary layers separated by a core in solid body rotation, similar to those observed in geophysics (Hide, Journal of Fluid Mechanics, vol. 32, pp. 737-764, 1968). When the global Reynolds number increases, self-sustained cross-flow vortices are observed and their spatial structure (Figure 6) is characteristic of an elephant mode (Pier & Huerre, Journal of Fluids and Structures, vol. 15, pp. 471-480, 2001). This numerical observation of an elephant mode agrees with Pier's conjecture (Journal of Fluid Mechanics, vol. 487, pp, 315-343 2003) on the existence of a nonlinear global mode due to the presence of an absolutely unstable region, even when the flow is linearly globally stable (Davies & Carpenter, Journal of Fluid Mechanics, vol. 486, pp. 287-329, 2003).
Non-modal instabilities; Schmid (2007) has presented a summary of the theory of hydrodynamic stability with a non-modal perspective. A general approach was proposed to formulate the response to initial perturbations, external disturbances or uncertainties. The approach allows to treat instabilities at short times and long times and it avoids the drawbacks of the classical theory of stability based on the calculation of eigenvalues for a highly non-normal system. Moreover this formulation allows to address a wider range of stability problems such as unsteady flows with, as a special case, the periodic flows, stochastically forced flows, flows with fluctuating mean profiles, spatially inhomogeneous flows, etc. The nonlinear stability properties can also be treated by introducing minor changes in the approach. The article by Schmid (2007) introduces all the conceptual tools necessary to analyze the stability and response of shear flows: energy growth, frequency response, impulse response (Figure 7) standard H2 transfer function norm, etc. The article concludes by presenting the characteristics of global modes in flows of complex geometry and by a brief introduction of interactive methods for calculating eigenvalues for large-scale problems.
Figure 7 - Impulse response of plane Poiseuille flow for Re = 1000 at t = 120. The colors represent the intensity of the longitudinal velocity component. This result shows the dominance of streaks in the wave packet (Schmid 2007).
In systems where boundary effects are too weak to freeze the spatial structure, it is impossible to approach the problem of transition to turbulence in terms of dynamical systems with a few degrees of freedom. Progress in this field has mainly relied on the study of models such as the Swift-Hohenberg model in convection or the complex Ginzburg-Landau equation. Our investigation of plane Couette flow falls within the class of semi-realistic models whereby one seeks to approach a given class of physical phenomena by avoiding the difficulties inherent to the original governing equations.
Plane Couette flow refers to a configuration whereby fluid is entrained by two parallel plates which are moving in opposite directions. According to experiments, the turbulent state, which coexists with the laminar state of uniform shear consists in large unstable turbulent streaks aligned along the stream, which disintegrate and regenerate. A simplified system of partial differential equations has been obtained by truncating a projection of the original equations on an appropriate basis of functions. A satisfactory agreement between the laboratory experiments of the "Instabilities and Turbulence" group at Centre d'Etudes de Saclay and numerical simulations of our model has been qualitatively obtained. The turbulent state relaxes toward the laminar state through long chaotic transients that are typical of a discontinuous transition scenario (Figure 8). The inverse process of nucleation of a turbulent state within the laminar flow is currently under investigation. Figure 9 represents a snapshot of a typical turbulent spot observed at a Reynolds number of 50, by integrating the simplified system of partial differential equations. The turbulent spot consists of sharp fronts which delineate its boundary. Further developments are necessary in order to arrive at a satisfactory understanding of such a widespread phenomenon in hydrodynamics.
Figure 8 - Numerical simulation of Couette flow model. Variations of the kinetic energy K contained in the perturbed flow as a function of time t, at different values of the Reynolds number in a quenching experiment. The flow is initialised in a turbulent state at R = 180. R = 100, 60, 50 : statistically stationary turbulence ; R= 45, 40 : relaxation towards a laminar state after a turbulent transient of variable time interval ; R = 30 : direct and sudden relaxation towards the laminar state.
Figure 9 - Numerical simulation of Couette flow model. Turbulent spot within flow at Reynolds number R = 50. Colors represent kinetic energy levels K of the perturbed flow.