Three-dimensional Secondary Instability Research Articles

The effects of boundary proximity on Kelvin–Helmholtz instability and turbulence

Studies of Kelvin–Helmholtz (KH) instability have typically modelled the initial flow as an isolated shear layer. In geophysical cases, however, the instability often occurs near boundaries and may therefore be influenced by boundary proximity effects. Ensembles of direct numerical simulations are conducted to understand the effect of boundary proximity on the evolution of the instability and the resulting turbulence. Ensemble averages are used to reduce sensitivity to small variations in initial conditions. Both the transition to turbulence and the resulting turbulent mixing are modified when the shear layer is near a boundary: the time scales for the onset of instability and turbulence are longer, and the height of the KH billow is reduced. Subharmonic instability is suppressed by the boundary because phase lock is prevented due to the diverging phase speeds of the KH and subharmonic modes. In addition, the disruptive influence of three-dimensional secondary instabilities on pairing is more profound as the two events coincide more closely. When the shear layer is far from the boundary, the shear-aligned convective instability is dominant; however, secondary central-core instability takes over when the shear layer is close to the boundary, providing an alternate route for the transition to turbulence. Both the efficiency of the resulting mixing and the turbulent diffusivity are dramatically reduced by boundary proximity effects.

The butterfly effect and the transition to turbulence in a stratified shear layer

In a stably stratified shear layer, multiple competing instabilities produce sensitivity to small changes in initial conditions, popularly called the butterfly effect (as a flapping wing may alter the weather). Three ensembles of 15 simulated mixing events, identical but for small perturbations to the initial state, are used to explore differences in the route to turbulence, the maximum turbulence level and the total amount and efficiency of mixing accomplished by each event. Comparisons show that a small change in the initial state alters the strength and timing of the primary Kelvin–Helmholtz instability, the subharmonic pairing instability and the various three-dimensional secondary instabilities that lead to turbulence. The effect is greatest in, but not limited to, the parameter regime where pairing and the three-dimensional secondary instabilities are in strong competition. Pairing may be accelerated or prevented; maximum turbulence kinetic energy may vary by up to a factor of 4.6, flux Richardson number by 12 %–15 % and net mixing by a factor of 2.

Optimal growth over a time-evolving variable-density jet at Atwood number

Secondary instabilities growing over a time-evolving variable-density round jet subject to the primary Kelvin–Helmholtz (KH) instability at Atwood number $\vert \textit {At} \vert = 0.25$ are investigated with a non-modal linear stability analysis. Despite local modifications of the base flow vorticity induced by the baroclinic torque, these disturbances experience a short-term universal growth due to a combination of the Orr and lift-up mechanisms, whatever the azimuthal wavenumber $m$ . At $\textit {Re}=1000$ , the secondary energy growth stems from the development of elliptical and hyperbolic instabilities, with an E-type-to-H-type transition as $m$ and $\textit {Re}$ increase, as in the homogeneous case (Nastro et al., J. Fluid Mech., vol. 900, 2020, A13). In the light jet at $\textit {Re} = 1000$ , after the KH mode saturation, the high- $m$ H-type instability is replaced by a perturbation organised as counter-rotating streamwise vortices located in the base flow region of promoted strain rate. Increasing the Reynolds number up to $\textit {Re} = 10\,000$ yields larger energy growths and a strong anisotropy among energy and enstrophy components with a preferential increase of axial velocity and azimuthal vorticity. Both come from the linearised baroclinic source that drives the optimal response towards folded sheets of axial velocity that differ from those observed in the variable-density plane shear layers. When the perturbation is injected around the KH saturation time for $\textit {Re}=10\,000$ , the response to optimal perturbation takes the form of fast growing secondary KH instabilities whatever $m$ . We find these three-dimensional secondary KH instabilities to be good candidates for the transition to turbulence in variable-density jet flows.

On secondary instability of a transitional separation bubble

It is well established in the natural transition of an attached boundary layer that the transition process starts with a two–dimensional primary instability (Tollmien–Schlichting wave, denoted as TS wave), followed by usually a three-dimensional secondary instability (fundamental mode or subharmonic mode) leading to the breakdown to turbulence. However, the transition process of a separation bubble (laminar flow or laminar boundary layer at separation and transition occurs downstream of the separation, leading to turbulence at reattachment) is less well understood, especially on the nature of secondary instability.The focus of this paper is on trying to advance our understanding of secondary instability of a transitional separation bubble on a flat plate with a blunt leading edge (separation is induced geometrically at the leading edge) under a very low free-stream turbulence level (< 0.1%). Large-Eddy Simulation (LES) is employed in the current study with a dynamic sub-grid-scale model. The numerical flow visualisation together with the spectral analysis has indicated that a three dimensional secondary instability, the elliptical instability, which occurs for fundamental frequency is the main mechanism at work whereas the subharmonic mode in the form of vortex-pairing is hardly active. There is no evidence for the existence of hyperbolic instability in the braid region either.

Optimal perturbations in time-dependent variable-density Kelvin–Helmholtz billows

We analyse the influence of the specific features of time-dependent variable-density Kelvin–Helmholtz (VDKH) roll-ups on the development of three-dimensional secondary instabilities. Due to inertial (high Froude number) baroclinic sources of spanwise vorticity at high Atwood number (up to 0.5 here), temporally evolving mixing layers exhibit a layered structure associated with a strain field radically different from their homogeneous counterpart. We use a direct-adjoint non-modal linear approach to determine the fastest growing perturbations over a single period of the time-evolving two-dimensional base flow during a given time interval $[t_{0},T]$. When perturbations are seeded at the initial time of the primary KH mode growth, i.e. $t_{0}=0$, it is found that additional mechanisms of energy growth are onset around a bifurcation time $t_{b}$, a little before the saturation of the primary two-dimensional instability. The evolution of optimal perturbations is thus observed to develop in two distinct stages. Whatever the Atwood number, the first period $[t_{0},t_{b}]$ is characterised by a unique route for optimal energy growth resulting from a combination of the Orr and lift-up transient mechanisms. In the second period $[t_{b},T]$, the growing influence of mass inhomogeneities raises the energy gain over the whole range of spanwise wavenumbers. As the Atwood number increases, the short spanwise wavelength perturbations tend to benefit more from the onset of variable-density effects than large wavelength ones. The extra energy gain due to increasing Atwood numbers relies on contributions from spanwise baroclinic sources. The resulting vorticity field is structured into two elongated dipoles localised along the braid on either side of the saddle point. In return they yield two longitudinal velocity streaks of opposite sign which account for most of the energy growth. This transition towards three-dimensional motions is in marked contrast with the classic streamwise rib vortices, so far accepted as the paradigm for the transition of free shear flows, either homogeneous or not. It is argued that the emergence of these longitudinal velocity streaks is generic of the transition in variable-density shear flows. Among them, the light round jet is known to display striking side ejections as a result of the loss of axisymmetry. The present analysis helps to renew the question of the underlying flow structure behind side jets, otherwise based on radial induction between pairs of counter-rotating longitudinal vortices (Monkewitz &amp; Pfizenmaier, Phys. Fluids A, vol. 3 (5), 1991, pp. 1356–1361). Instead, it is more likely that side ejections would result from the convergence of the longitudinal velocity streaks near the braid saddle point. When the injection time is delayed so as to suppress the initial stage of energy growth, a new class of perturbations arises at low wavenumber with energy gains far larger than those observed so far. They correspond to the two-dimensional Kelvin–Helmholtz secondary instability of the baroclinically enhanced vorticity braid discovered by Reinaud et al. (Phys. Fluids, vol. 12 (10), pp. 2489–2505), leading potentially to another route to turbulence through a two-dimensional fractal cascade.

Three-dimensional instabilities of quasi-solitary waves in a falling liquid film

AbstractThe stability of $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\gamma _2$ travelling waves at the surface of a film flow down an inclined plane is considered experimentally and numerically. These waves are fast, one-humped and quasi-solitary. They undergo a three-dimensional secondary instability if the flow rate (or Reynolds number) is sufficiently high. Rugged or scallop wave patterns are generated by the interplay between a short-wave and a long-wave instability mode. The short-wave mode arises in the capillary region of the wave, with a mechanism of capillary origin which is similar to the Rayleigh–Plateau instability, whereas the long-wave mode deforms the entire wave and is triggered by a Rayleigh–Taylor instability. Rugged waves are observed at relatively small inclination angles. At larger angles, the long-wave mode predominates and scallop waves are observed. For a water film the transition between rugged and scallop waves occurs for an inclination angle around 12°.

The ‘zoo’ of secondary instabilities precursory to stratified shear flow transition. Part 2 The influence of stratification

AbstractThe linear stability analyses described in Mashayek &amp; Peltier (J. Fluid Mech., vol. 708, 2012, 5–44, hereafter MP1) are extended herein in an investigation of the influence of stratification on the evolution of secondary instabilities to which an evolving Kelvin–Helmholtz (KH) wave is susceptible in an initially unstable parallel stratified shear layer. We show that over a wide range of background stratification levels, the braid shear instability has a higher probability of emerging at early stages of the flow evolution while the secondary convective instability (SCI), which occurs in the eyelids of the individual Kelvin ‘cats eyes’, will remain a relevant and dominant instability at high Reynolds numbers. The evolution of both modes is greatly influenced by the background stratification. Various other three-dimensional secondary instabilities are found to exist over a wide range of stratification levels. In particular, the stagnation point instability (SPI), which was discussed in detail in MP1, may be of great potential importance providing alternate routes for transition of an initially two-dimensional KH wave into fully developed turbulence. The energetics of the secondary instabilities revealed by our simulations are analysed in detail and the preturbulent mixing properties are studied.

Three-dimensionalization of the stratified mixing layer at high Reynolds number

Numerical analyses of the stability of a stratified two-dimensional Kelvin-Helmholtz billow against three dimensional disturbances provide a theoretical means of identifying the primary mechanism(s) that induce transition to turbulence. We identify, through fully resolved three-dimensional numerical simulations, secondary modes of instability which have been suggested by recent observations to be responsible for this transition. Our analyses lead us to two primary conclusions. First, as the Reynolds number is increased at a fixed stratification level, the vortex pairing process may be entirely suppressed by the rapid growth of three-dimensional secondary instabilities. Second, the new transition mechanisms identified herein have significant implications for the efficiency of mixing in the turbulent flow that develops subsequent to saturation of the secondary instabilities.

Three-dimensional secondary instability of a spatially developing von Kármán vortex street in a far wake

This paper presents studies of a three-dimensional secondary instability of a spatially developing von Kármán vortex street. It develops owing to the nonlinear interaction between a two-dimensional mean far-wake flow and its most unstable disturbances. This forms a nonlinear primary wake flow. Sections of this flow are selected to perform a temporal secondary stability study under the assumption of parallel flow. The eigenvalue characteristics of the secondary instability are compared with the results from the use of a linear primary flow comprising unmodified mean wake flow coexisting with a linear primary fundamental disturbance with an empirical amplitude as a parameter, resulting in a simpler Floquet analysis. The maximum amplification rates occur at about the same spanwise wavenumber for both the nonlinear and linear primary flows, in qualitative agreement. But the amplification rate versus the spanwise wavenumber spectrum are both qualitatively and quantitatively different, the nonlinear primary flow results in a lower magnitude of the amplification rates. Some interpretations of controlled experiments are made, and it is concluded that the two- and three-dimensional disturbances so obtained appeared to be from the primary instability, where the amplification mechanisms come from the unmodified mean flow. A general discussion of the nonlinear interaction between the primary two-dimensional flow and the three-dimensional secondary instability is given, which may well form the basis for further nonlinear studies.

Possible states in the flow around two circular cylinders in tandem with separations in the vicinity of the drag inversion spacing

The possible states in the flow around two identical circular cylinders in tandem arrangements are investigated for configurations in the vicinity of the drag inversion separation. By means of numerical simulations, the hysteresis in the transition between the shedding regimes is studied and the relationship between (three-dimensional) secondary instabilities and shedding regime determination is addressed. The differences observed in the behavior of two- and three-dimensional flows are analyzed, and the regions of bistable flow are delimited. Very good agreement is found between the proposed scenario and results available in the literature.

Structural sensitivity of the secondary instability in the wake of a circular cylinder

The sensitivity of the three-dimensional secondary instability of a circular-cylinder wake to a structural perturbation of the associated linear equations is investigated. In particular, for a given flow condition, the region of maximum coupling between the velocity components is localized by using the most unstable Floquet mode and its adjoint mode. The variation of this region in time is also found by considering a structural perturbation which is impulsively applied in time at a given phase of the vortex-shedding process. The analysis is carried out for both mode A and mode B types of transition in the wake of a circular cylinder using a finite-difference code. The resulting regions identified as the core of the instability are in full agreement with the results reported in the literature and with the a posteriori checks documented here.

Numerical investigation of the three-dimensional secondary instabilities in the time-developing compressible mixing layer

Mixing layers are present in very different types of physical situations such as atmospheric flows, aerodynamics and combustion. It is, therefore, a well researched subject, but there are aspects that require further studies. Here the instability of two-and three-dimensional perturbations in the compressible mixing layer was investigated by numerical simulations. In the numerical code, the derivatives were discretized using high-order compact finite-difference schemes. A stretching in the normal direction was implemented with both the objective of reducing the sound waves generated by the shear region and improving the resolution near the center. The compact schemes were modified to work with non-uniform grids. Numerical tests started with an analysis of the growth rate in the linear regime to verify the code implementation. Tests were also performed in the non-linear regime and it was possible to reproduce the vortex roll-up and pairing, both in two-and three-dimensional situations. Amplification rate analysis was also performed for the secondary instability of this flow. It was found that, for essentially incompressible flow, maximum growth rates occurred for a spanwise wavelength of approximately 2/3 of the streamwise spacing of the vortices. The result demonstrated the applicability of the theory developed by Pierrehumbet and Widnall. Compressibility effects were then considered and the maximum growth rates obtained for relatively high Mach numbers (typically under 0.8) were also presented.

The primary and secondary instabilities of flow generated by an oscillating circular cylinder

Two- and three-dimensional instabilities of the two-dimensional symmetric flow generated by a circular cylinder oscillating with simple harmonic motion in quiescent fluid, or, alternatively, by oscillatory flow past a stationary cylinder at low Keulegan-Carpenter and Stokes numbers are investigated via Floquet analysis and direct numerical simulation. Previous experimental visualization has found that the flows produced at low amplitudes and frequencies of motion can be grouped by their visual characteristics into a number of distinct regimes. At low values of Keulegan-Carpenter and Stokes numbers, the flow is two-dimensional and has a reflection symmetry about the axis of oscillation, in addition to a pair of spatio-temporal symmetries. This study isolates and classifies the symmetry-breaking instabilities from these two-dimensional basic states as functions of these control parameters. It is found that while the initial bifurcations produced by increasing the parameters can be to three-dimensional flows, much of the behaviour can be explained in terms of two-dimensional symmetry-breaking instabilities. These have two primary manifestations: at low Stokes numbers, the instability is synchronous with the imposed oscillation, and gives rise to a boomerang-shaped mode, while at higher Stokes numbers, the instability is quasi-periodic, with a well-defined second period, which becomes infinite as Stokes numbers are reduced along the marginal stability boundary, ‘freezing’ the quasi-periodic mode into a synchronous one. These two-dimensional modes are, with further small increase in control parameter, unstable to three-dimensional secondary instabilities, and these are the flows which have been reported in previous experimental studies. In contrast, the mode first reported by Honji (J. Fluid Mech. vol. 107, 1981, p. 509), which arises at high Stokes numbers, and lower Keulegan-Carpenter numbers than the two-dimensional quasi-periodic mode, has a three-dimensional primary instability arising directly from the symmetrical two-dimensional basic state.

Global destabilization of flow over a backward-facing step

Global destabilization of two-dimensional flow over a backward-facing step embedded in a channel, i.e., flow in a plane channel with a sudden expansion, is investigated by numerical simulations using a spectral element method. In the low Reynolds number regime where the two-dimensional flow is steady without manipulation, self-excited oscillations of the entire flow are induced by appropriate simultaneous suction at the step face and blowing at the wall adjacent to the step. The boundary between steady and time-dependent (destabilized) flow is determined as a function of the streamwise extent of the blowing region and its position relative to the step, for an expansion ratio of approximately two, a Reynolds number fixed at Re=1000, and equal suction and blowing mass flow rates. The computed periodic, globally synchronized flow regimes are characterized using instantaneous streamline patterns, time traces of physical quantities, and proper orthogonal decomposition of the velocity fields. The global flow behavior is also related to the absolute instability properties of the local streamwise velocity profiles. Preliminary three-dimensional simulations finally suggest that the 2-D unsteady flows obtained in the present study are, in general, susceptible to three-dimensional secondary instabilities. Nevertheless, the first experiments suggest that the 2-D analysis provides a qualitatively correct description of the flow transitions and that practical applications of this wall-blowing and -suction scheme, such as mixing enhancement, may be feasible.

Three-dimensional instability and vorticity patterns in the wake of a flat plate

We investigated experimentally the dynamics of the three-dimensional secondary instability developing in the wake of a thin flat plate at moderate Reynolds numbers. The wake is formed as the two laminar boundary layers developing on each side merge at the trailing edge of the flat plate. Both the spatial and temporal evolution of the two- and three-dimensional instabilities are analysed by means of laser-induced visualizations of the deformation of the interface separating the two streams. It was found that although the wake may exhibit two distinct three-dimensional modes with different symmetry characteristics, Modes 1 and 2 (Lasheras & Meiburg 1990), the latter appears to be amplified first, thereafter dominating the evolution of the near wake. By varying the forcing frequency of the primary two-dimensional instability, we found that the wavelength of the three-dimensional mode is selected by the wavelength of the two-dimensional Karman vortices, with a ratio () of order one. In the far-wake region, both modes appear to grow and co-exist. Furthermore, by analysing the response of the wake to spanwise-periodic and impulsive perturbations applied at the trailing edge of the plate, we demonstrate that the nature of the secondary instability of the wake behind a thin flat plate is convective. In addition, both modes are shown to have comparable wavelengths and to be the result of the same instability mechanism.

Effect of nonuniform wall heating on the three-dimensional secondary instability of falling films

Secondary three-dimensional instabilities of nearly sinusoidal waves on vertically falling and nonuniformly heated films are studied by using a long-wave evolution equation. Two-dimensional waves are unstable with respect to transverse modulations with sufficiently long spanwise wavelength. Two distinct three-dimensional modes of instability are examined: a synchronous mode which does not alter the wave number of the basic two-dimensional waves and a subharmonic mode with one-half of the streamwise wave number. According to a Floquet analysis, the subharmonic instability is most likely to be dominant for streamwise wavenumbers close to the neutral curve. The three-dimensional instability mechanism depends on film heating. The secondary growth rate increases (decreases) with increasing (decreasing) film heating downstream, but the contribution of thermocapillarity to synchronous and subharmonic instabilities is different.

A Nonlinear Dynamics Analysis of Vortex-Structure Interaction Models

Vortex-structure interaction models are studied in the work presented here. The third- order model by Hartlen and Currie (HC model) can reproduce the correct response amplitude, while a fifth-order model by Landl predicts the observed hysterisis effect. Using concepts from nonlinear dynamics and bifurcation theory, the range of possible dynamics of the models is investigated in parameter space; essentially, a class of nonlinear oscillators deriving “naturally” from the HC model is studied. It is found that perturbations of the HC model in parameter space lead to qualitatively physically meaningful dynamics. Forced excitation of the HC model is the highlight of the work. In this case, it is shown that a subharmonic lock-in predicted by the model may be related to a three-dimensional secondary subharmonic instability of a periodic flow. Experimental results are presented for comparison.

Three-dimensional secondary instability in plane current-vortex sheets

It is shown that three-dimensional ideal instabilities can occur for the plane current-vortex sheet in the magnetically dominated regime. For the parameters under consideration, increasing the mean flow strength leads to decreased secondary instability growth rates. A calculation is presented which tracks the evolution of the secondary instability into the fully nonlinear regime, in which it is found that spanwise modes are dominant.

Transition to turbulent electric current sheet reconnection

Electric current sheets develop in the solar corona when different flux systems come into contact. At these sheets magnetic energy is transformed into heat and kinetic energy by means of reconnection. We have previously demonstrated how to accelerate neutral sheet energy conversion by means of a transition to turbulent reconnection via ideal, three-dimensional secondary instabilities, as conjectured by Montgomery. In this paper we describe how our previous results are modified by the presence of a finite mean sheetwise magnetic field. We find a stabilization from this field, due to a decrease in energy transfer from the basic magnetic field to the three-dimensional perturbed fields. An increase in perturbed dissipative energy losses is also observed.

Transition to turbulence in solar surges

view Abstract Citations (24) References (34) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Transition to Turbulence in Solar Surges Dahlburg, Russell B. ; Karpen, Judith T. Abstract Transition to turbulence in magnetohydrodynamic (MHD) tearing jets has been invoked as a mechanism underlying some of the complex behavior observed in solar surges, including deceleration of the upflowing plasma and temporal correlations with types I and III radio bursts. In this paper we investigate a possible mechanism for this transition: three-dimensional secondary instabilities on two-dimensional saturated states. We find through linear analysis that these MHD configurations -- in particular, the tearing jet -- are secondarily unstable, with the dominant energy transfer from the one-dimensional field into the 3-dimensional fields. Using nonlinear simulations, we also investigate the system evolution after the secondary modes attain finite amplitude. When the tearing jet transitions to turbulence, the total kinetic energy drops rapidly corresponding to the deceleration of the jet. The electric field grows rapidly as the primary mode saturates and the three-dimensional secondary mode develops, and then decays quickly as the tearing jet becomes turbulent, providing a possible explanation for the finite duration of the associated meter-wave bursts. The electric field decays as the magnetic and velocity fields both decay. The system is dominated at late times by spanwise modes, which strongly resemble the magnetic field-aligned filamentary flows characteristic of many surges. Publication: The Astrophysical Journal Pub Date: October 1994 DOI: 10.1086/174779 Bibcode: 1994ApJ...434..766D Keywords: Magnetohydrodynamic Flow; Magnetohydrodynamic Turbulence; Plasma Jets; Solar Activity; Stellar Models; Surges; Tearing Modes (Plasmas); Type 3 Bursts; Computerized Simulation; Electric Fields; Fourier Analysis; Numerical Analysis; Solar Magnetic Field; Velocity Distribution; Solar Physics; MAGNETOHYDRODYNAMICS: MHD; SUN: RADIO RADIATION; TURBULENCE full text sources ADS |