Pre-unstable set of multiple transient three-dimensional perturbation waves and the associated turbulent state in a shear flow

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 a swarming of arbitrary 3D 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 3D small perturbations 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 the molecular diffusion. Leaving aside nonlinear interaction among the different scales, these features are tantamount to the features of the turbulent state. We determine the exponent of the inertial range of arbitrary longitudinal and transversal perturbations acting on a typical shear flow, i.e. the bluff-body wake. Then, we compare the present results with the exponent of the corresponding developed turbulent state (notoriously equal to -5/3). For longitudinal perturbations, we observe a decay rate of -3 in the inertial range, typically met in two-dimensional turbulence. For purely 3D perturbations, instead, the energy decreases with a factor of -5/3. If we consider a combination of longitudinal and transversal perturbative waves, the energy spectrum seems to have a decay of -3 for larger wavenumbers ([50, 100]), while for smaller wavenumbers ([3,50]) the decay is of the order -5/3. We can conclude that the value of the exponent of the inertial range has a much higher level of universality, which is not necessarily associated to the nonlinear interaction.Comment: Proceedings of the 17th Australasian Fluid Mechanics Conference, 5-9 December 2010, Auckland, New Zealan

Complex Networks Unveiling Spatial Patterns in Turbulence

Numerical and experimental turbulence simulations are nowadays reaching the size of the so-called big data, thus requiring refined investigative tools for appropriate statistical analyses and data mining. We present a new approach based on the complex network theory, offering a powerful framework to explore complex systems with a huge number of interacting elements. Although interest on complex networks has been increasing in the last years, few recent studies have been applied to turbulence. We propose an investigation starting from a two-point correlation for the kinetic energy of a forced isotropic field numerically solved. Among all the metrics analyzed, the degree centrality is the most significant, suggesting the formation of spatial patterns which coherently move with similar vorticity over the large eddy turnover time scale. Pattern size can be quantified through a newly-introduced parameter (i.e., average physical distance) and varies from small to intermediate scales. The network analysis allows a systematic identification of different spatial regions, providing new insights into the spatial characterization of turbulent flows. Based on present findings, the application to highly inhomogeneous flows seems promising and deserves additional future investigation.Comment: 12 pages, 7 figures, 3 table

A synthetic perturbative hypothesis for multiscale analysis of bluff-body wake instability

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Collective behaviour of linear perturbation waves observed through the energy density spectrum

Abstract: We consider the collective behaviour of small three-dimensional transient perturbations in sheared flows. In particular, we observe their varied life history through the temporal evolution of the amplification factor. The spectrum of wave vectors considered fills the range from the size of the external flow scale to the size of the very short dissipative waves. We observe that the amplification factor distribution is scale-invariant. In the condition we analyze, the system is subject to all the physical processes included in the linearized Navier-Stokes equations. With the exception of the nonlinear interaction, these features are the same as those characterizing the turbulent state. The linearized perturbative system offers a great variety of different transient behaviours associated to the parameter combination present in the initial conditions. For the energy spectrum computed by freezing each wave at the instant where its asymptotic condition is met, we ask whether this system is able to show a power-law scaling analogous to the Kolmogorov argument. At the moment, for at least two typical shear flows, the bluff-body wake and the plane Poiseuille flow, the answer is ye

Travelling perturbations in sheared flows: sudden transition infrequency and phase speed asymptotics

We present recent findings concerning angular frequency discontinuities in the transient evolution of three-dimensional perturbations in two sheared ows, the plane channel and the wake ows. By carrying out a large number of initial-value problem simulations1;2 we observe a discontinuity which appears toward the end of the perturbation transient life. Both the frequency, ω, and the phase speed, C, decrease to zero when ϕ, the angle of obliquity between the perturbation and the base flow, approaches π/2. A few examples of transient of the frequency are reported in Fig. 1(a-b) for the channel and wake flows, respectively. When the transient is close to the end, the angular frequency suddenly jumps to the asymptotic value, which is in general higher than the transient one. The relative variation between the transient and asymptotic values can change from a few percentages to values up to 30-40%. Whenever it occurs, the emergence of a frequency discontinuity can be considered as a particular range of the temporal evolution which separates the transient (algebraic) dynamics from the asymptotic (exponential) regime. Within this temporal range, the perturbation suddenly changes its behavior by increasing its phase velocity. Independently to what observed for the amplification factor, one can assume that beyond this temporal instant the asymptotic state sets in. The investigation of the dispersion relation, C(k) (see an example in Fig. 1c for the channel ow case), reveals that longitudinal short waves are non-dispersive (C const as k is large enough), while longitudinal long waves and all the perturbations not aligned with the base ow present a dispersive behavior (C varies either with the angle of obliquity, ϕ, or the polar wavenumber, k). Moreover, orthogonal waves (ϕ = π/2), which can experience a quick initial growth of energy, are standing waves (C = 0). This result can be explained in terms of the system symmetry. A possible interpretation for the morphology of turbulent spots 3;4 can be drawn in the case of wall flow