Parallel Wake Research Articles

Linear instability of tilted parallel shear flow in a strongly stratified and viscous medium

A linear stability analysis is performed on a tilted parallel wake in a strongly stratified fluid at low Reynolds numbers. A particular emphasis of the present study is given to the understanding of the low-Froude-number mode observed by the recent experiment (Meunier in J Fluid Mech 699:174–197, 2012). In the limit of low Froude number, the linearised equations of motion can be reduced to the Orr–Sommerfeld equation on the horizontal plane, except the viscous term that contains vertical dissipation. Based on this equation, it is proposed that the low-Froude-number mode would be a horizontal inflectional instability and should remain two-dimensional at small tilting angles as long as the Reynolds number is sufficiently low. To support this claim, the asymptotic regime where this analysis is strictly valid is subsequently discussed in relation to previous work on the proper vertical length scale. The absolute and convective instability analysis of parallel wake is further performed, showing qualitatively good agreement with the experimental result. The low-Froude-number mode is found to be stabilised on increasing Froude number, as in the experiment. It is shown that the emergence of small vertical velocity at finite Froude number, the size of which is proportional to the square of Froude number, plays the key role in the stabilisation by modifying the inflectional instability and paradoxically creating stabilising buoyancy effect with the increase of Froude number.

Probing plasma wakefields using electron bunches generated from a laser wakefield accelerator

We show experimental results of probing the electric field structure of plasma wakes by using femtosecond relativistic electron bunches generated from a laser wakefield accelerator. Snapshots of laser-driven linear wakes in plasmas with different densities and density gradients are captured. The spatiotemporal evolution of the wake in a plasma density up-ramp is recorded. Two parallel wakes driven by a laser with a main spot and sidelobes are identified in the experiment and reproduced in simulations. The capability of this new method for capturing the electron- and positron-driven wakes is also shown via 3D particle-in-cell simulations.

Three-dimensional effects on flag flapping dynamics

We examine three-dimensional (3D) effects on the flapping dynamics of a flag, modelled as a thin membrane, in uniform fluid inflow. We consider periodic spanwise variations of length $S$ (ignoring edge effects), so that the 3D effects are characterized by the dimensionless spanwise wavelength ${\it\gamma}=S/L$, where $L$ is the chord length. We perform linear stability analysis (LSA) to show increase in stability with ${\it\gamma}$, with the purely 2D mode being the most unstable. To confirm the LSA and to study nonlinear responses of 3D flapping, we obtain direct numerical simulations, up to Reynolds number 1000 based on $L$, coupling solvers for the Navier–Stokes equations and that for a thin membrane structure undergoing arbitrarily large displacement. For nonlinear flapping evolution, we identify and characterize the effect of ${\it\gamma}$ on the distinct flag motions and wake vortex structures, corresponding to spanwise standing wave (SW) and travelling wave (TW) modes, in the absence and presence of cross-flow respectively. For both SW and TW, the response is characterized by an initial instability growth phase (I), followed by a nonlinear development phase (II) consisting of multiple unstable 3D modes, and tending, in long time, towards a quasi-steady limit-cycle response (III) dominated by a single (most unstable) mode. Phase I follows closely the predictions of LSA for initial instability and growth rates, with the latter increased for TW due to suppression of restoring forces by the cross-flow. Phase II is characterized by multiple competing flapping modes with energy cascading towards the more unstable mode(s). The wake is characterized by interwoven (SW) and oblique continuous (TW) shed vortices. For phase III, the persistent single dominant mode for SW is the (most unstable) 2D flag displacement with a continuous parallel wake structure; and for TW, the fundamental oblique travelling-wave flag displacement corresponding to the given ${\it\gamma}$ with continuous oblique shedding. The transition to phase III occurs slower for greater ${\it\gamma}$. For the total forces, drag decreases for both SW and TW with decreasing ${\it\gamma}$, while lift is negligible in phase I and II and comparable in magnitude to drag in phase III for any ${\it\gamma}$.

Optimal perturbations of non-parallel wakes and their stabilizing effect on the global instability

We compute the spatial optimal energy amplification of steady inflow perturbations in a non-parallel wake and analyse their stabilizing action on the global mode instability. The optimal inflow perturbations, which are assumed spanwise periodic and varicose, consist in streamwise vortices that induce the downstream spatial transient growth of streamwise streaks. The maximum energy amplification of the streaks increases with the spanwise wavelength of the perturbations, in accordance with previous results obtained for the temporal energy growth supported by parallel wakes. A family of increasingly streaky wakes is obtained by forcing optimal inflow perturbations of increasing amplitude and then solving the nonlinear Navier-Stokes equations. We show that the linear global instability of the wake can be completely suppressed by forcing optimal perturbations of sufficiently large amplitude. The attenuation and suppression of self-sustained oscillations in the wake by optimal 3D perturbations is confirmed by fully nonlinear numerical simulations. We also show that the amplitude of optimal spanwise periodic (3D) perturbations of the basic flow required to stabilize the global instability is much smaller than the one required by spanwise uniform (2D) perturbations despite the fact that the first order sensitivity of the global eigenvalue to basic flow modifications is zero for 3D spanwise periodic modifications and non-zero for 2D modifications. We therefore conclude that first-order sensitivity analyses can be misleading if used far from the instability threshold, where higher order terms are the most relevant.

Stabilizing effect of optimally amplified streaks in parallel wakes

AbstractWe show that optimal perturbations artificially forced in parallel wakes can be used to completely suppress the absolute instability and to reduce the maximum temporal growth rate of the inflectional instability. To this end we compute optimal transient energy growths of stable streamwise uniform perturbations supported by a parallel wake for a set of Reynolds numbers and spanwise wavenumbers. The maximum growth rates are shown to be proportional to the square of the Reynolds number and to increase with spanwise wavelengths with sinuous perturbations slightly more amplified than varicose ones. Optimal initial conditions consist of streamwise vortices and the optimally amplified perturbations are streamwise streaks. Families of nonlinear streaky wakes are then computed by direct numerical simulation using optimal initial vortices of increasing amplitude as initial conditions. The stabilizing effect of nonlinear streaks on temporal and spatiotemporal growth rates is then determined by analysing the linear impulse response supported by the maximum amplitude streaky wakes profiles. This analysis reveals that at $\mathit{Re}= 50$, streaks of spanwise amplitude ${A}_{s} \approx 8\hspace{0.167em} \% {U}_{\infty } $ can completely suppress the absolute instability, converting it into a convective instability. The sensitivity of the absolute and maximum temporal growth rates to streak amplitudes is found to be quadratic, as has been recently predicted. As the sensitivity to two-dimensional (2D, spanwise uniform) perturbations is linear, three-dimensional (3D) perturbations become more effective than the 2D ones only at finite amplitudes. Concerning the investigated cases, 3D perturbations become more effective than the 2D ones for streak amplitudes ${A}_{s} \gtrsim 3\hspace{0.167em} \% {U}_{\infty } $ in reducing the maximum temporal amplification and ${A}_{s} \gtrsim 12\hspace{0.167em} \% {U}_{\infty } $ in reducing the absolute growth rate. However, due to the large optimal energy growths they experience, 3D optimal perturbations are found to be much more efficient than 2D perturbations in terms of initial perturbation amplitudes. Despite their lower maximum transient amplification, varicose streaks are found to be always more effective than sinuous ones in stabilizing the wakes, in accordance with previous findings.

Laser-driven collimated tens-GeV monoenergetic protons from mass-limited target plus preformed channel

Proton acceleration by ultra-intense laser pulse irradiating a target with cross-section smaller than the laser spot size and connected to a parabolic density channel is investigated. The target splits the laser into two parallel propagating parts, which snowplow the back-side plasma electrons along their paths, creating two adjacent parallel wakes and an intense return current in the gap between them. The radiation-pressure pre-accelerated target protons trapped in the wake fields now undergo acceleration as well as collimation by the quasistatic wake electrostatic and magnetic fields. Particle-in-cell simulations show that stable long-distance acceleration can be realized, and a 30 fs monoenergetic ion beam of >10 GeV peak energy and <2° divergence can be produced by a circularly polarized laser pulse at an intensity of about 1022 W/cm2.

Transient growth of secondary instabilities in parallel wakes: Anti lift-up mechanism and hyperbolic instability

This paper investigates the three-dimensional temporal instabilities and the transient growth of perturbations on a Von Kármán vortex street, issuing from the development of the primary instability of a parallel Bickley velocity profile typical of a wake forming behind a thin flat plate. By solving iteratively the linearized direct Navier Stokes equations and its adjoint equations, we compute the optimal perturbations that exhibit the largest transient growth of energy between the initial instant and different time horizons. At short time horizons, optimal initial perturbations are concentrated on the points of maximal strain of the base flow. The optimal gain of energy and the mechanism of instability are well predicted by local theories that describe the lagrangian evolution of a perturbation wave packet. At time of order unity, hyperbolic region leads the dynamics. Only at large time (t ≥ 20), the growth is led by the most amplified eigenmode. This eigenmode evolves, when the wavenumber increases, from perturbation centred in the core of the vortices, to perturbations localised on the stretching manifold of the hyperbolic points. At finite and large time, the gain in energy is initially associated with a mechanism reminiscent to the anti lift-up mechanism described by Antkowiak and Brancher [J. Fluid Mech. 578, 295 (2007)] in the context of an axisymmetric vortex. Presently, the optimal initial condition (the adjoint modes at large time) corresponding to streamwise streaks localised on the contracting manifold of the hyperbolic point induces streamwise vortices aligned with the stretching manifold of the hyperbolic point (the direct modes). The localisation on distinct manifolds of direct and adjoint eigenmodes is more pronounced when the Reynolds number is increased. An interpretation is proposed based on a balance between diffusion and stretching effects that predicts the thickness of the energy containing region for the adjoint and the direct mode decreasing as 1/Re. The extra gain of energy due to non normal effects grows, since direct and adjoint modes are localised in different regions of space, i.e., the stretching and contracting manifold, a novel effect of the so called convective non normality associated with the transport of the perturbation by the base flow.

Global linear and nonlinear stability of viscous confined plane wakes with co-flow

The global stability of confined uniform density wakes is studied numerically, using two-dimensional linear global modes and nonlinear direct numerical simulations. The wake inflow velocity is varied between different amounts of co-flow (base bleed). In accordance with previous studies, we find that the frequencies of both the most unstable linear and the saturated nonlinear global mode increase with confinement. For wake Reynolds number Re = 100 we find the confinement to be stabilising, decreasing the growth rate of the linear and the saturation amplitude of the nonlinear modes. The dampening effect is connected to the streamwise development of the base flow, and decreases for more parallel flows at higher Re. The linear analysis reveals that the critical wake velocities are almost identical for unconfined and confined wakes at Re ≈ 400. Further, the results are compared with literature data for an inviscid parallel wake. The confined wake is found to be more stable than its inviscid counterpart, whereas the unconfined wake is more unstable than the inviscid wake. The main reason for both is the base flow development. A detailed comparison of the linear and nonlinear results reveals that the most unstable linear global mode gives in all cases an excellent prediction of the initial nonlinear behaviour and therefore the stability boundary. However, the nonlinear saturated state is different, mainly for higher Re. For Re = 100, the saturated frequency differs less than 5% from the linear frequency, and trends regarding confinement observed in the linear analysis are confirmed.

Global linear and nonlinear stability of viscous confined plane wakes with co-flow

The global stability of confined uniform density wakes is studied numerically, using two-dimensional linear global modes and nonlinear direct numerical simulations. The wake inflow velocity is varied between different amounts of co-flow (base bleed). In accordance with previous studies, we find that the frequencies of both the most unstable linear and the saturated nonlinear global mode increase with confinement. For wake Reynolds numberRe= 100 we find the confinement to be stabilising, decreasing the growth rate of the linear and the saturation amplitude of the nonlinear modes. The dampening effect is connected to the streamwise development of the base flow, and decreases for more parallel flows at higherRe. The linear analysis reveals that the critical wake velocities are almost identical for unconfined and confined wakes atRe≈ 400. Further, the results are compared with literature data for an inviscid parallel wake. The confined wake is found to be more stable than its inviscid counterpart, whereas the unconfined wake is more unstable than the inviscid wake. The main reason for both is the base flow development. A detailed comparison of the linear and nonlinear results reveals that the most unstable linear global mode gives in all cases an excellent prediction of the initial nonlinear behaviour and therefore the stability boundary. However, the nonlinear saturated state is different, mainly for higherRe. ForRe= 100, the saturated frequency differs less than 5% from the linear frequency, and trends regarding confinement observed in the linear analysis are confirmed.

Stability of parallel wake flows in quasigeostrophic and frontal regimes

Recent laboratory experiments [G. Perret, A. Stegner, M. Farge, and T. Pichon, Phys. Fluids 18, 036603 (2006)] have shown that the vortex-street formed in the wake of a towed cylinder in a rotating shallow-water layer could present a strong cyclone-anticyclone asymmetry. In extreme cases, only large-scale anticyclones were observed in the far wake. This asymmetry occurs in the so-called frontal regime when the Rossby number is small and the surface deviation is large. This asymmetry may have various origins and in particular may be attributed to the asymmetry of the flow around the cylinder, to the linear stability property of the wake, or to its nonlinear evolution. To discriminate between these mechanisms, we study the stability of two idealized parallel flows in the quasigeostrophic and in the frontal regimes. These parallel flows correspond to two velocity profiles measured just behind the cylinder in a region where the perturbations are negligible. According to our linear stability analysis, the most unstable mode, in the frontal regime, is localized in the anticyclonic shear region whether the base flow profile is symmetric or not. On a linear basis, it is thus more the instability that imposes the asymmetry than the base flow. Direct numerical simulations of the synthetic parallel wake flows show that nonlinearity exacerbates the dominance of the anticyclonic mode linearly selected. By numerically studying the spatio-temporal evolution of a small perturbation localized in space, we show that, unlike incompressible two–dimensional wake flows and the symmetric wake in the quasigeostrophic regime, the parallel asymmetric wake is strongly convectively unstable in the frontal regime, and not absolutely unstable. When the surface deformation becomes large, the wake instability changes from the absolute instability in the quasi-geostrophic regime to the strongly convective instability of the frontal regime. This explains well the changes.

Control of absolute instability by basic-flow modification in a parallel wake at low Reynolds number

In this paper, we investigate the effect of basic-flow modification on the absolute instability in a two-dimensional wake at low Reynolds number with the parallel-flow approximation. Using the method of calculus of variation, we investigate how to modify the basic flow to suppress or enhance the absolute instability and suggest an optimal modification of the basic flow for stabilizing a bluff-body wake. In order to validate the present approach, we also measure the sensitivity of all the eigenvalues including the absolute-instability frequency, using the � -pseudo-spectrum, showing that small modifications in the basic flow do not destabilize other eigenvalues by more than the original absolute-instability frequency, at least for the Reynolds number considered here. For a two-dimensional parallel model wake and a circularcylinder wake, the present approach shows that the positive and negative velocity perturbations to the basic flows, respectively, at the wake centreline and separating shear layer suppress the absolute instability.

Secondary instability mechanisms in the wake of a flat plate

We extend the work of Sutherland and Peltier (Phys. Fluids 6 (1994) 1267) and investigate numerically the three-dimensional (3D) secondary stability of a wake forming behind a thin flat plate. The primary flow is a Kármán vortex street numerically computed from the two-dimensional (2D) even instability of a parallel wake based on the Bickley velocity profile. Considering the symmetries of the von Kármán Street, the 3D modes are classified into two families, whether symmetric or antisymmetric. For each family, we determine the leading eigenmodes using a Krylov method. The growth rate curves show that both the most unstable symmetric and antisymmetric modes are stationary and present a maximum of amplification for a wavelength of the order of the primary vortex spacing. The maximum growth rate, corresponding wavelength and cutoff wavelength are well predicted by the elliptic instability of the vortex core. The eigenmode structure of the most unstable wavenumber is centered in the core and is typical of the elliptic instability. The hyperbolic instability of the braid region gives a growth rate five times larger and a cutoff two times higher than the ones computed. As recently discussed for mixing layers by Caulfield and Kerswell (Phys. Fluids 12 (5) (2000) 1032), this is not surprising since the hyperbolic instability applies for an unbounded hyperbolic flow. When the region of hyperbolic flow is bounded, intense transient growth is generated, but when time goes to infinity, the instability becomes small or even dies out. Finally, good qualitative and quantitative agreement is found with the experiments previously done by Julien, Lasheras and Chomaz (J. Fluid Mech. 479 (2003) 155) on the secondary instability in the wake of a flat plate for the symmetry selection, the most amplified wavenumber and growth rate.

Fully nonlinear dynamics of parallel wakes

The fully nonlinear theory of global modes in open flows, proposed in recent analyses of amplitude equations, is extended to the case of Navier–Stokes equations using direct numerical simulations. The basic flow under consideration is a parallel wake in a finite domain generated by imposing the wake profile at the inlet boundary and by adding a body force to compensate the basic flow diffusion. The link between the global bifurcation, the absolute or convective nature of the local linear instability, and the theory of speed selection for the front separating an unperturbed domain of the flow from a fully saturated solution is elucidated. In particular, thanks to the parallelism of the flow, the bifurcation scenario and the associated scaling laws for the frequency, the healing length, and the slope at the origin predicted by a previous analysis of amplitude equations are recovered with great precision.

A study on 2-D wake flow control by acoustic excitation

In a low speed open-type wind tunnel, a group of parallel wakes downstream of two dimensional grid model consisting of several circular cylinders were experimentally investigated to study the response of the wake flows to the acoustic excitation, in hoping to promote the understanding of the underlying mechanism behind the gross flow change due to artificial excitation. In the unexcited wake flows, the development of the individual wakes behind cylinders was almost uniform for the ratio of the spacing to the cylinder diameter of s/d.geq.1.5. For smaller s/d, however, the jet streams issued through the gaps between the cylinders became biased in one side and the cylinders had wakes of different sizes. At s/d=1.25, the gap flow directions change in time, leading to unstable wake patterns. Further reduction in s/d made this unstable flip-flopping of the jets stable. The most effective excitation frequency was found to be in the Strouhal number range of St=0.5-0.6. This frequency was related to the vortex shedding. At s/d=1.75, the excitation frequency was 2 or 4 times the vortex shedding frequency. When the flow was excited at this frequency, the vortex sheddings were energized, and pairings between neighboring vortices were generated. Also, the merging process between individual wakes was accelerated. The unstable and unbalanced wake patterns at s/d=2.15 were made stable and balanced. The unstable and unbalanced wake patterns at s/d=2.15 were made stable and balanced. For smaller spacing of s/d .leq,1.0, the acoustic excitation became less effective in controlling the flow.