Hyperbolic Instability Research Articles

Stability of Stuart vortices in rotating stratified fluids

The linear stability of the Stuart vortices, which is a model of arrays of vortices often observed in the atmosphere and the oceans, in rotating stratified fluids is investigated by local and modal stability analysis. As in the case of the two-dimensional (2-D) Taylor–Green vortices, five types of instability appear in general: the pure-hyperbolic instability, the strato-hyperbolic instability, the rotational-hyperbolic instability, the centrifugal instability and the elliptic instability. The condition for each instability and the estimate of the growth rate derived by Hattori & Hirota (J. Fluid Mech., vol. 967, 2023, A32) are shown to also be useful for the Stuart vortices, which supports their applicability to general flows. The properties of each instability depend on stratification and rotation in a way similar to the case of the 2-D Taylor–Green vortices. For the Stuart vortices, however, the centrifugal instability and the elliptic instability become more dominant than the three hyperbolic instabilities in comparison to the 2-D Taylor–Green vortices; this is explained by the larger ratios of the maximum vorticity and the strain rate at the elliptic stagnation points to the strain rate at the hyperbolic stagnation points. Direct correspondence between the modal and local stability results is further established by comparing unstable modes to solutions to the local stability equations; this is useful for identifying the types of modes since the mechanism of instability is readily known in the local stability analysis. This helps us to discover the modes of the ring-type elliptic instability, which have been predicted only theoretically.

Stability of two-dimensional Taylor–Green vortices in rotating stratified fluids

The linear stability of the two-dimensional Taylor–Green vortices, which is a spatially periodic array of vortices, in rotating stratified fluids is investigated by local and modal stability analysis. Five types of instability appear in general: the pure hyperbolic instability, the strato-hyperbolic instability, the rotational-hyperbolic instability, the centrifugal instability and the elliptic instability. The condition for each instability and the estimate of the growth rate, which are useful in interpreting numerical results, are obtained in the framework of local stability analysis. Realizability of an instability is introduced to predict whether an unstable mode corresponding to an unstable region found in the local stability analysis exists at finite Reynolds numbers. In the absence of stratification, the pure hyperbolic instability is dominant for weak rotation; it is stabilized for strong rotation. For strong anti-cyclonic rotation, the elliptic instability or the centrifugal instability becomes dominant depending on the parameter values; further stronger rotation stabilizes both instabilities. For strong cyclonic rotation, the rotational-hyperbolic instability or the elliptic instability becomes dominant, although the growth rate is smaller than the anti-cyclonic cases. Strong stratification changes the stability properties. The strato-hyperbolic instability occurs for weak rotation. The rotational-hyperbolic instability and the elliptic instability are weakened under cyclonic rotation, while the latter survives and extends the unstable range under anti-cyclonic rotation. The pure hyperbolic instability and the centrifugal instability are less affected by stratification. The mode structures of each instability are in good agreement with the corresponding solution to local stability equations, confirming the physical mechanism of the instability.

Local stability analysis of homogeneous and stratified Kelvin–Helmholtz vortices

We perform a three-dimensional short-wavelength linear stability analysis of numerically simulated two-dimensional Kelvin–Helmholtz vortices in homogeneous and stratified environments at a fixed Reynolds number of $Re = 300$ . For the homogeneous case, the elliptic instability at the vortex core dominates at early times, before being taken over by the hyperbolic instability at the vortex edge. For the stratified case of Richardson number $ Ri = 0.08$ , the early-time instabilities comprise a dominant elliptic instability at the core and a hyperbolic instability influenced strongly by stratification at the vortex edge. At intermediate times, the local approach shows a new branch of (convective) instability that emerges at the vortex core and subsequently moves towards the vortex edge. A few more convective instability bands appear at the vortex core and move away, before coalescing to form the most unstable region inside the vortex periphery at large times. In addition, the stagnation point instability is also recovered outside the periphery of the vortex at intermediate times. The dominant instability characteristics from the local approach are shown to be in good qualitative agreement with the results based on global instability studies for both homogeneous and stratified cases. A systematic study of the dependence of the dominant instability characteristics on $ Ri $ is then presented. While $ Ri = 0.1$ is identified as most unstable (with convective instability being dominant), another growth rate maximum at $ Ri = 0.025$ is not far behind (with the hyperbolic instability influenced by stratification being dominant). Finally, the local stability approach is shown to predict the potential orientation of the flow structures that would result from hyperbolic and convective instabilities, which is found to be consistent with three-dimensional numerical simulations reported previously.

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.

Transition control of cylinder wake via Lorentz force

The flow around a cylinder is a typical flow acting as the oscillator, and the perturbations can grow with time at a fixed position. This phenomenon can induce the laminar-turbulent transition in the wake, thus increasing the vibrations of the cylinder as well as the noise in the flow system. There exist three control strategies, i.e. the streamwise Lorentz force from the electromagnetic actuator set on the windward surface named windward control, the leeward surface named leeward control, and the whole surface of cylinder named global control, which are adopted to modify the two-dimensional base flow, thereby reducing the growth rates of perturbations in the wake and changing the transition mode. According to the Floquet stability analysis, it is found that the growth rates of the transition modes A and B present small changes in the windward control, while the growth rates of the two modes decrease with the increase of the control number in the other two control cases. Comparing the inviscid growth rates induced by the elliptic instability and the hyperbolic instability with each other, it is observed that the high inviscid growth rate in the windward control can be similar to those without control, while the inviscid growth rates can decrease with the increase of control number in the other two control cases. Three-dimensional direct numerical simulations are performed to validate the control effects. The results shows that the three-dimensional shape of the wake is changed from mode B to mode A when the actuator is set on the leeward surface or the whole surface of the cylinder. This is consistent with the result from the Floquet stability analysis. In addition, the drag of the cylinder reduces 15.2% for the leeward control and 14.4% for the the global control.

Instability margin analysis for parametrized LTI systems with application to repressilator

This paper is concerned with a robust instability analysis for the single-input-single-output unstable linear time-invariant (LTI) system under dynamic perturbations. The nominal system itself is possibly perturbed by the static gain of the uncertainty, which would be the case when a nonlinear uncertain system is linearized around an equilibrium point. We define the robust instability radius as the smallest H∞ norm of the stable linear perturbation that stabilizes the nominal system. There are two main theoretical results: one is on a partial characterization of unperturbed nominal systems for which the robust instability radius can be calculated exactly, and the other is a numerically tractable procedure for calculating the exact robust instability radius for nominal systems parametrized by a perturbation parameter. The results are applied to the repressilator in synthetic biology, where hyperbolic instability of a unique equilibrium guarantees the persistence of oscillation phenomena in the global sense, and the effectiveness of our linear robust instability analysis is confirmed by numerical simulations.

Dynamics and stability of the wake behind a circular cylinder in the vicinity of a plane moving wall

This paper identifies and characterizes modes of the three-dimensional instability of the flow over a circular cylinder of diameter D near a plane moving wall with a gap height of G=0.1D beyond the onset of vortex shedding through the Floquet stability analysis. The associated effects of the transitions on the hydrodynamic forces of the cylinder are examined by performing direct numerical simulations. Six distinct three-dimensional modes are identified and their characteristics are discussed in detail. It is found that the physical mechanism responsible for the synchronous mode 1 appears to be associated to an elliptic instability. The quasi-periodic mode 4 is observed to transform to a synchronous mode (mode 6) on increasing the Reynolds number, indicating that they are likely to be resulted from the same physical mechanism. For mode 6, its physical nature seems to be a hyperbolic instability. Fully three-dimensional simulations show that the wake undergoes a rapid transition to complex chaotic flow state after the most dominant mode starts to saturate, due to the strongly nonlinear interactions of many modes with different spanwise wavelengths.

Modal stability analysis of arrays of stably stratified vortices

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Optimal perturbations in viscous round jets subject to Kelvin–Helmholtz instability

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Numerical investigation of separation-induced transition in a low-pressure turbine cascade in a low-disturbance environment

In this paper, the separation-induced transition in an LPT (low-pressure turbine) cascade is investigated at low Reynolds number with DNS (direct numerical simulation). The transition process is accurately predicted giving good agreements between the DNS and experimental results. To illustrate the secondary instability of separation-induced transition in a low-disturbance environment, the results are comprehensively analyzed in both Fourier space and physical space. It is illustrated that the effect of hyperbolic instability dominates around the saddle point of hyperbolic streamlines. This instability mechanism is responsible for the emergence of the streamwise vortices in the braid region. Elongated and intensified because of the “stretching” effect of the background flow, these vortices become the most noticeable characteristic of the flow field. Fundamental modes of small spanwise wavelength are excited in the braid region, so as some low-frequency modes. The elliptical instability plays a minor role than hyperbolic instability. It is also observed that the fundamental mode with a larger spanwise wavelength is unstable in the vortex core which is associated with the deformation of the vortex core via elliptical instability. There is no convincing evidence for the existence of subharmonic instability.

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.

Integrated orbit determination and maintenance in Earth-Moon unstable dynamics environment

The hyperbolic instability of collinear libration point orbits dominates the distribution of orbit uncertainty, which is closely related with the unstable manifold of the local orbit. Exploiting the distribution of orbit uncertainty in an unstable orbital environment, an integrated orbit determination and maintenance approach is proposed. X-ray pulsar navigation embedded in the Kalman filtering architecture is adopted to determine the orbit of a spacecraft flying along the Earth-Moon L2 halo orbit. To keep the spacecraft in the vicinity of the nominal trajectory, the newly proposed orbit maintenance approach based on the covariance matrix is proposed to design the orbit maintenance maneuver. The numerical simulations show that the proposed approach has a good performance in orbit determination and maintenance.

Strato-hyperbolic instability: a new mechanism of instability in stably stratified vortices

The stability of stably stratified vortices is studied by local stability analysis. Three base flows that possess hyperbolic stagnation points are considered: the two-dimensional (2-D) Taylor–Green vortices, the Stuart vortices and the Lamb–Chaplygin dipole. It is shown that the elliptic instability is stabilized by stratification; it is completely stabilized for the 2-D Taylor–Green vortices, while it remains and merges into hyperbolic instability near the boundary or the heteroclinic streamlines connecting the hyperbolic stagnation points for the Stuart vortices and the Lamb–Chaplygin dipole. More importantly, a new instability caused by hyperbolic instability near the hyperbolic stagnation points and phase shift by the internal gravity waves is found; it is named the strato-hyperbolic instability; the underlying mechanism is parametric resonance as unstable band structures appear in contours of the growth rate. A simplified model explains the mechanism and the resonance curves. The growth rate of the strato-hyperbolic instability is comparable to that of the elliptic instability for the 2-D Taylor–Green vortices, while it is smaller for the Stuart vortices and the Lamb–Chaplygin dipole. For the Lamb–Chaplygin dipole, the tripolar instability is found to merge with the strato-hyperbolic instability as stratification becomes strong. The modal stability analysis is also performed for the 2-D Taylor–Green vortices. It is shown that global modes of the strato-hyperbolic instability exist; the structure of an unstable eigenmode is in good agreement with the results obtained by local stability analysis. The strato-hyperbolic mode becomes dominant depending on the parameter values.

Instability of cylinder wake under open-loop active control

Instability of a wake controlled by a streamwise Lorentz force is investigated through a Floquet stability analysis. The streamwise Lorentz force, which is a two-dimens- ional control input created by an electromagnetic actuator located on the cylinder surface, adjusts the base flow to affect the three-dimensional wake instability and achieve wake stabilization and transition delay. The instability mode at a Reynolds number Re = 300 can be transformed from B to A with N = 1.0, where N is an interaction number repre- senting the strength of the Lorentz force relative to the inertial force in the fluid. The wake flow is Floquet stable when N increases to 1.3. The spanwise perturbation wavelengths are 3.926D and 0.822D in the modes A and B, respectively, where D is the cylinder diameter. In addition, the oscillating amplitudes of drag and lift are reduced with the increase in the interaction number. Particle tracing is used to explore the essential physical mechanism for mode transformation. The path lines show that suppression of flow separation hinders the fluid deformation and rotation, leading to the decrease in elliptic and hyperbolic instability regions, which is the material cause of mode transformation. All of the results indicate that wake stabilization and transition delay can be achieved under open-loop active control via the streamwise Lorentz force.

A universal three-dimensional instability of the wakes of two-dimensional bluff bodies

Linear stability analysis of a wide range of two-dimensional and axisymmetric bluff-body wakes shows that the first three-dimensional mode to became unstable is always mode E. From the studies presented in this paper, it is speculated to be the universal primary 3D instability, irrespective of the flow configuration. However, since it is a transition from a steady two-dimensional flow, whether this mode can be observed in practice does depend on the nature of the flow set-up. For example, the mode E transition of a circular cylinder wake occurs at a Reynolds number of $\mathit{Re}\simeq 96$, which is considerably higher than the steady to unsteady Hopf bifurcation at $\mathit{Re}\simeq 46$ leading to Bénard–von-Kármán shedding. On the other hand, if the absolute instability responsible for the latter transition is suppressed, by rotating the cylinder or moving it towards a wall, then mode E may become the first transition of the steady flow. A well-known example is flow over a backward-facing step, where this instability is the first global instability to be manifested on the otherwise two-dimensional steady flow. Many other examples are considered in this paper. Exploring this further, a structural stability analysis (Pralits et al.J. Fluid Mech., vol. 730, 2013, pp. 5–18) was conducted for the subset of flows past a rotating cylinder as the rotation rate was varied. For the non-rotating or slowly rotating case, this indicated that the growth rate of the instability mode was sensitive to forcing between the recirculation lobes, while for the rapidly rotating case, it confirmed sensitivity near the cylinder and towards the hyperbolic point. For the non-rotating case, the perturbation, adjoint and structural stability fields, together with the wavelength selection, show some similarities with those of a Crow instability of a counter-rotating vortex pair, at least within the recirculation zones. On the other hand, at much higher rotation rates, Pralits et al. (J. Fluid Mech., vol. 730, 2013, pp. 5–18) have suggested that hyperbolic instability may play a role. However, both instabilities lie on the same continuous solution branch in Reynolds number/rotation-rate parameter space. The results suggest that this particular flow transition at least, and probably others, may have a number of different physical mechanisms supporting their development.

Effects of an axial flow on the centrifugal, elliptic and hyperbolic instabilities in Stuart vortices

AbstractLinear stability of the Stuart vortices in the presence of an axial flow is studied. The local stability equations derived by Lifschitz &amp; Hameiri (Phys. Fluids A, vol. 3 (11), 1991, pp. 2644–2651) are rewritten for a three-component (3C) two-dimensional (2D) base flow represented by a 2D streamfunction and an axial velocity that is a function of the streamfunction. We show that the local perturbations that describe an eigenmode of the flow should have wavevectors that are periodic upon their evolution around helical flow trajectories that are themselves periodic once projected on a plane perpendicular to the axial direction. Integrating the amplitude equations around periodic trajectories for wavevectors that are also periodic, it is found that the elliptic and hyperbolic instabilities, which are present without the axial velocity, disappear beyond a threshold value for the axial velocity strength. Furthermore, a threshold axial velocity strength, above which a new centrifugal instability branch is present, is identified. A heuristic criterion, which reduces to the Leibovich &amp; Stewartson criterion in the limit of an axisymmetric vortex, for centrifugal instability in a non-axisymmetric vortex with an axial flow is then proposed. The new criterion, upon comparison with the numerical solutions of the local stability equations, is shown to describe the onset of centrifugal instability (and the corresponding growth rate) very accurately.

Instabilities in laminar separation bubbles

AbstractWall-bounded flows, in their transition from a laminar state to turbulence, pass through a set of particular stages characterized by different physical processes. Among wall-bounded flows, separated flows have a special place because their dynamics can either be noise amplifiers or oscillators. For several years Marxen and co-workers have been studying the evolution of two- and three-dimensional perturbations in the laminar part of a laminar separation bubble. In Marxen et al. (J. Fluid Mech., vol. 728, 2013, p. 58) they study vortex formation and its evolution in laminar–turbulent transition in a forced separation bubble. By the combined use of numerical and experimental methods, different mechanisms of secondary instabilities have been highlighted: elliptic instability of vortex cores and hyperbolic instability responsible for three-dimensionality in the braid region. This work shows, for the first time in laminar separation bubbles, the first nonlinear stages of transition to turbulence of such a flow. However, since this type of flow is very sensitive to various environmental stresses, several scenarios for transition to turbulence remain to be explored.

Three-dimensionality in the wake of a rapidly rotating cylinder in uniform flow

AbstractThe flow around an isolated cylinder spinning at high rotation rates in free stream is investigated. The existence of two steady two-dimensional states is confirmed, as is the existence of a secondary mode of vortex shedding. The stability of the two steady states to three-dimensional perturbations is established using linear stability analysis. At lower rotation rates on the first steady state, two three-dimensional modes are confirmed, and their structure and curves of marginal stability as a function of rotation rate and Reynolds number are determined. One mode (named mode $E$) appears consistent with a hyperbolic instability in the wake, while the second (named mode $F$) appears to be a centrifugal instability of the flow very close to the cylinder surface. At higher rotation rates on the second steady state, a single three-dimensional mode due to centrifugal instability (named mode ${F}^{\prime } $) is found. This mode becomes increasingly difficult to excite as the rotation rate is increased.

Suppression of collision-induced dephasing by periodic, erratic, or noisy driving

We compare different driving scenarios for controlling the loss of single particle coherence of an initially coherent preparation in the vicinity of the hyperbolic instability of the two-mode bose-Hubbard model. In particular we contrast the quantum Zeno suppression of decoherence by broad-band erratic or noisy driving, with the Kapitza effect obtained for high frequency periodic monochromatic driving.

Transient perturbation growth in time-dependent mixing layers

AbstractWe investigate numerically the transient linear growth of three-dimensional (3D) perturbations in a homogeneous time-evolving mixing layer in order to identify which perturbations are optimal in terms of their kinetic energy gain over a finite, predetermined time interval. We model the mixing layer with an initial parallel velocity distribution $\mathbi{U}(y)= {U}_{0} \tanh (y/ d)\mathbi{e}_{x}$ with Reynolds number $Re= {U}_{0} d/ \nu = 1000$, where $\nu $ is the kinematic viscosity of the fluid. We consider a range of time intervals on both a constant ‘frozen’ base flow and a time-dependent two-dimensional (2D) flow associated with the growth and nonlinear saturation of two wavelengths of the most-unstable eigenmode of linear theory of the initial parallel velocity distribution, which rolls up into two classical Rayleigh instabilities commonly referred to as Kelvin–Helmholtz (KH) billows, which eventually pair to form a larger vortex. For short times, the most-amplified perturbations on the frozen $\tanh $ profile are inherently 3D, and are most appropriately described as oblique wave ‘OL’ perturbations which grow through a combination of the Orr and lift-up mechanisms, while for longer times, the optimal perturbations are 2D and similar to the KH normal mode, with a slight enhancement of gain. For the time-evolving KH base flow, OL perturbations continue to dominate over sufficiently short time intervals. However, for longer time intervals which involve substantial evolution of the primary KH billows, two broad classes of inherently 3D linear optimal perturbation arise, associated at low wavenumbers with the well-known core-centred elliptical translative instability, and at higher wavenumbers with the braid-centred hyperbolic instability. The hyperbolic perturbation is relatively inefficient in exploiting the gain of the OL perturbations, and so only dominates the smaller wavenumber (ultimately) core-centred perturbations when the time evolution of the base flow or the start time of the optimization interval does not allow the OL perturbations much opportunity to grow. When the OL perturbations can grow, they initially grow in the braid, and then trigger an elliptical core-centred perturbation by a strong coupling with the primary KH billow. If the optimization time interval includes pairing of the primary billows, the secondary elliptical perturbations are strongly suppressed during the pairing event, due to the significant disruption of the primary billow cores during pairing.