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

Dispersive to non-dispersive transition and phase velocity transient for linear waves in plane wake and channel flows

In this study we analyze the phase and group velocity of three-dimensional linear traveling waves in two sheared flows, the plane channel and the wake flows. This was carried out by varying the wave number over a large interval of values at a given Reynolds number inside the ranges 20-100, 1000-8000, for the wake and channel flow, respectively. Evidence is given about the possible presence of both dispersive and non-dispersive effects which are associated with the long and short ranges of wavelength. We solved the Orr-Sommerfeld and Squire eigenvalue problem and observed the least stable mode. It is evident that, at low wave numbers, the least stable eigenmodes in the left branch of the spectrum beave in a dispersive manner. By contrast, if the wavenumber is above a specific threshold, a sharp dispersive to non-dispersive transition can be observed. Beyond this transition, the dominant mode belongs to the right branch of the spectrum. The transient behavior of the phase velocity of small three-dimensional traveling waves was also considered. Having chosen the initial conditions, we then show that the shape of the transient highly depends on the transition wavelength threshold value. We show that the phase velocty can oscillate with a frequency which is equal to the frequency width of the eigenvalue spectrum. Furthermore, evidence of intermediate self-similarity is given for the perturbation field.Comment: 19 pages, 11 figures. Text and discussion improved with respect to the first version. Accepted for publication on Physical Review

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

Perturbation dynamics in laminar and turbulent flows. Initial value problem analysis.

Stability and turbulence are often studied as separate branches of fluid dynamics, but they are actually the two faces of the same coin: the existence of equilibrium, laminar in one case and steady in the mean in the other. The link between these two faces is transition. Initial value problems are considered to analyse the dynamics of disturbances in the three phases. In the context of stability, linearised equations of motion can be used. Although this is a substantial simplification, the results that are obtained with this analysis are far from being trivial. The transition to turbulence through the dynamics of disturbances is discussed in the context of the zig-zag instability: a particular kind of instability that occurs in geophysical flows. Eventually, the perturbations dynamics in turbulent flows is used to analyse the mixing process between water-vapour in clouds and clear air in the surroundings, in the presence of a meteorological inversion

Intermittency layers associated to turbulent interfaces

In this study we focus on the transport across an interface which separates two regions with homogeneous and isotropic turbulence in absence of a mean shear. The turbulent transport resulting presents an internal structure. Indeed, in the case of turbulent self-diffusion, both experiments and simulations show that the fluid velocity field is marked by a high intermittency front located aside the interface, which is the source of turbulent bursts penetrating the low turbulence region. The presence of an inner structure inside a layer of turbulence self-transport highlights the different nature of the turbulent transport with respect to the Gaussian diffusion. By including other effects, for instance a passive scalar transport or a mass transport in presence of a density stratification, the phenomenology is much enriched. For instance, our preliminary numerical experiments on the passive scalar transport reveals the presence of two intermittency fronts, one on each side of the interface. As can be seen if the figure below, the intermittency level in the fronts is high. This is true both for the scalar and the scalr derivative statistics. A gradual decay in time is observed while they propagate toward the lateral isotropic regions of the flow. In the presence of a kinetic energy gradient across the interface, the locations and intensity of the intermittency fronts are no more symmetric to respect to the interface. The front on the high energy side of the mixing region penetrates deeper and exhibits stronger intermittency. Analogous features are observed also in two dimension