Elliptical Instability Research Articles

Precessing non-axisymmetric ellipsoids: bi-stability and fluid instabilities

This study explores precession-driven flows in a non-axisymmetric ellipsoid spinning around its medium axis. Using an experimental approach, we focus on two aspects of the flow: the base flow of uniform vorticity and the development of fluid instabilities. In contrast to a preceding paper (J. Fluid. Mech., vol. 932, 2022, A24), where the ellipsoid rotated around its shortest axis, we do not observe bi-stability or hysteresis of the base flow, but a continuous transition from small to large differential rotation and tilt of the fluid rotation axis. We then use the model developed by Noir & Cébron (J. Fluid. Mech., vol. 737, 2013, pp. 412–439) to numerically determine regions in the parameter space of axial and equatorial deformations for which bi-stability may exist. Concerning fluid instabilities, we use three independent observations to track the onset of both boundary layer and parametric instabilities. Our results clearly show the presence of a parametric instability, yet the exact nature of the underlying mechanism (conical shear layer instability, shear instability and elliptical instability) is not unambiguously identified. A coexisting boundary layer instability, although unlikely, cannot be ruled out based on our experimental data. To make further progress on this topic, a new generation of experiments at significantly lower Ekman numbers (ratio of rotation and viscous time scales) is clearly needed.

Experimental evidence of the correlation between the flow rolling structures and momentum transfer in Rayleigh–Bénard convection

The formation of large-scale circulating structures (LSC) is one of the critical characteristics of turbulent Rayleigh–Bénard convection (RBC). Although the effect of LSC in turbulent RBC was disputed due to conflicting results, recently, the results of three-dimensional direct numerical simulation [Zwirner et al., “Elliptical instability and multiple-roll flow modes of the large-scale circulation in confined turbulent Rayleigh-Bénard convection,” Phys. Rev. Lett. 125, 54502 (2020)] confirmed that in RBC flows with a low Prandtl number, Pr=0.1 within Oberbeck–Boussinesq assumption, the formation of higher number of LSC leads to a decrease in heat and momentum transfer. However, it was shown that for higher Prandtl numbers, heat/momentum transfer is not correlated with the number of LSC. Experimental evidence is investigated of an inverse correlation between the momentum transfer and small-scale rolling structures for high Prandtl non-Oberbeck–Boussinesq condition. Experiments were undertaken at a Prandtl number of Pr=7 and Rayleigh number of Ra=5.3×107 in a cubical convection cell with an unit aspect ratio. Particle image velocimetry along with a robust combinatorial vortex detection algorithm was used to capture the flow field, detect the rolling structures, and estimate their size. It was found that although the flow structures were dominated by the LSC, the number of smaller rolling structures was significant. The results also showed that after the initiation of convection while the flow was still undeveloped, the majority of rolling structures were small scale. For this state, an inverse correlation between the number of rolling structures and momentum transfer was observed highlighting the influence of flow rolling structures regardless of the formation of the LSC.

Modelling the Elliptical Instability of Magnetic Skyrmions

Two recently developed methods of modelling chiral magnetic soliton elliptical instability are applied in two novel scenarios: the tilted ferromagnetic phase of chiral magnets dominated by easy-plane anisotropy and the general case of the chiral magnet with tilted applied field and arbitrary uniaxial anisotropy. In the former case, the analytical predictions are found to exactly match previous numerical results. In the latter case, the instability of isolated chiral skyrmions has not yet been studied, although interestingly, the predictions correspond to previous numerical investigation into the phase diagram.

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.

Velocity gradient analysis of a head-on vortex ring collision

We simulate the head-on collision between vortex rings with circulation Reynolds numbers of 4000 using an adaptive, multiresolution solver based on the lattice Green's function. The simulation fidelity is established with integral metrics representing symmetries and discretization errors. Using the velocity gradient tensor and structural features of local streamlines, we characterize the evolution of the flow with a particular focus on its transition and turbulent decay. Transition is excited by the development of the elliptic instability, which grows during the mutual interaction of the rings as they expand radially at the collision plane. The development of antiparallel secondary vortex filaments along the circumference mediates the proliferation of small-scale turbulence. During turbulent decay, the partitioning of the velocity gradients approaches an equilibrium that is dominated by shearing and agrees well with previous results for forced isotropic turbulence. We also introduce new phase spaces for the velocity gradients that reflect the interplay between shearing and rigid rotation and highlight geometric features of local streamlines. In conjunction with our other analyses, these phase spaces suggest that, while the elliptic instability is the predominant mechanism driving the initial transition, its interplay with other mechanisms, e.g. the Crow instability, becomes more important during turbulent decay. Our analysis also suggests that the geometry-based phase space may be promising for identifying the effects of the elliptic instability and other mechanisms using the structure of local streamlines. Moving forward, characterizing the organization of these mechanisms within vortices and universal features of velocity gradients may aid in modelling turbulent flows.

Non-periodic shear layer instabilities driving local extinction in premixed ramjet cavity flames

The local extinction characteristics of turbulent premixed ethylene-air flames are experimentally investigated in a high-speed cavity combustor to provide a full characterization of the temporally evolving non-periodic hydrodynamic instabilities that result in local extinction. The local flame extinction phenomenon is captured temporally using high-speed particle image velocimetry (PIV) and broadband chemiluminescence. The evolution of the flame topology, turbulent structures, and flame strain rate are analyzed through the extinction process to distinguish the turbulent shear layer mechanisms contributing to flame instabilities and extinction. A frequency analysis of the extinction event was performed and found the flame extinction to occur at non-periodic temporal intervals. A further wavelet decomposition revealed the presence of a non-periodic hydrodynamic instability in the shear layer causing extinction. The instability mechanism is related to the stability of vortex pairing in the shear layer, which were found to correspond to large-amplitude wave generation on the flame surface. In instances of flame extinction, the vortex pairs exhibit a core deformation from an elliptic instability mechanism, leading eventually to the shredding of the vortical cores, which is the primary mechanism for excessive turbulent flame strain and resulting extinction. Although the local flame extinction mechanism is driven from flame and shear layer interactions, the condition for local flame extinction to occur requires the flame to interact with an elliptical instability in the shear layer. The results can guide strategies to enhance combustion stability, performance, and efficiency.

On the propeller wake evolution using large eddy simulations and physics-informed space-time decomposition

A novel modal analysis methodology, denoted as the physics informed sparsity-promoting dynamic mode decomposition (pi-SPDMD) model, was introduced for the reduction and reconstruction analysis of intricate propeller wake flows, aiming to provide insight into the inherent flow structures spanning diverse temporal and spatial scales. Large-Eddy Simulation (LES) was employed to numerically model the wake dynamics of a four-bladed propeller, providing a comprehensive resolution from the proximate to the distant wake regions. The findings indicate that the pi-SPDMD model enhances the efficiency of the sparse-promoting algorithm, producing modes that gravitate towards stability, and the resulting decomposition maintains commendable physical fidelity. Integrating the results from the LES solution and the modal decomposition of pi-SPDMD, the tip vortex exhibits a uniform topological configuration with notable coherence in the proximate domain. In this region, the large-scale vortex is the dominant feature of the propeller wake, and there is a marked intermittency in the turbulence. In the mid-field, the tip vortex system transitions into fine-scale vortices, rapidly diminishing in coherence due to the onset of elliptic instability and subsequent secondary vortex generation. As the tip vortex structures related to physical quantities become fully discretized, the small-scale turbulent patterns quickly intermingle, leading to a more homogeneous distribution in the distant wake.

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.

Tidal dissipation due to the elliptical instability and turbulent viscosity in convection zones in rotating giant planets and stars

ABSTRACT Tidal dissipation in star–planet systems can occur through various mechanisms, among which is the elliptical instability. This acts on elliptically deformed equilibrium tidal flows in rotating fluid planets and stars, and excites inertial waves in convective regions if the dimensionless tidal amplitude (ϵ) is sufficiently large. We study its interaction with turbulent convection, and attempt to constrain the contributions of both elliptical instability and convection to tidal dissipation. For this, we perform an extensive suite of Cartesian hydrodynamical simulations of rotating Rayleigh–Bénard convection in a small patch of a planet. We find that tidal dissipation resulting from the elliptical instability, when it operates, is consistent with ϵ3, as in prior simulations without convection. Convective motions also act as an effective viscosity on large-scale tidal flows, resulting in continuous tidal dissipation (scaling as ϵ2). We derive scaling laws for the effective viscosity using (rotating) mixing-length theory, and find that they predict the turbulent quantities found in our simulations very well. In addition, we examine the reduction of the effective viscosity for fast tides, which we observe to scale with tidal frequency (ω) as ω−2. We evaluate our scaling laws using interior models of Hot Jupiters computed with mesa. We conclude that rotation reduces convective length-scales, velocities, and effective viscosities (though not in the fast tides regime). We estimate that elliptical instability is efficient for the shortest period Hot Jupiters, and that effective viscosity of turbulent convection is negligible in giant planets compared with inertial waves.

Modeling of wake features of a propeller using the vorticity confinement method

The instability and evolution mechanisms of propeller wakes are of vital significance to the development of next-generation propulsion devices with better hydrodynamic and noise performances. The temporal–spatial scales and the vortex details are important for the understanding of the vortex features and their dynamic responses to the propeller. In the present study, the vorticity confinement (VC) method was employed on the numerical simulations achieved by the improved delay detached eddy simulation with various advance coefficients to characterize the underlying features of wake flows. Comparisons were made between the results computed with and without the VC model from different perspectives. The analyses showed that the VC method captures more high-frequency power spectral density results as well as more small-scale vortical topology on the far downstream field based on the same spatial resolution and indicates the multi-scale interference on the tip vortex evolutionary trajectories. The VC method also elucidates rich small vortical structures with low advance coefficient and elliptical instability with high advance coefficient. This paper further widens our knowledge on the propeller wake evolution mechanisms and highlights the value of the VC method in the investigation of propeller wakes.

Influence of kinematics on the growth of secondary wake structures behind oscillating foils

The wake of an oscillating hydrofoil in combined heaving and pitching motion is numerically evaluated at a range of phase offset (90∘≤ϕ≤270∘) and chord based Strouhal number (0.32 ≤Stc≤ 0.64), while the main analysis is currently reported at Stc=0.32. The Reynolds number is fixed at Re=8000. At ϕ=90∘, a dominant spanwise elliptic instability feature is observed due to the presence of a paired primary and secondary leading edge vortex (LEV) roller over the foil boundary. Subsequent outflux of vorticity from the secondary LEV leads to the formation of streamwise hairpin structures that evolve downstream on account of straining in the braid vorticity region. The spanwise wavelength (λz) of the growing streamwise secondary hairpin legs is approximately 0.86c before the LEV structures shed from the foil. This is followed by a reduction in λz, which coincided with continuous ejection of vorticity from the secondary leading edge structures, and the rearrangement of growing hairpin legs. However, the dominant association between spanwise instability and growth of secondary streamwise structures at ϕ=90∘ reduces with increasing ϕ towards 270∘. At ϕ=180∘, there is a delay in the emergence of dominant secondary hairpin structures that evolve through deformed trailing edge vortex rather than the secondary leading edge structure. Further increase in ϕ to 225∘ and 270∘ depicts the absence of secondary hairpin structures, despite the apparent local instability features on the leading edge rollers.

The onset of turbulence in decelerating diverging channel flows

This work investigates the stability and transition to turbulence in a diverging channel subjected to a time-varying trapezoidal-shaped inflow boundary condition. Numerical simulations are performed for different deceleration rates and Reynolds numbers while maintaining a constant acceleration rate. The flow transition begins with two-dimensional primary instability with the formation of inflectional velocity profiles, followed by local separation and the emergence of an array of shear layer vortices. We divide simulation cases systematically into three categories based on the onset of secondary instability and the generation of streamwise vorticity. At low and medium Reynolds numbers (type I), the spanwise vortex rolls formed by inflectional instability remain two-dimensional and diffuse at the channel centre without exhibiting further instabilities. At high Reynolds numbers and deceleration rates (type II), the rolled shear layer exhibits secondary instability during the zero mean inflow phase, followed by local incipient turbulent structure formation. The streamwise vorticity that develops over the shear layer structures causes oscillations with a spanwise wavelength similar to those associated with the elliptic instability in a counter-rotating vortex pair. Using the Lamb–Oseen approximation of vortices in conjunction with the dynamic mode decomposition algorithm of the three-dimensional flow field, we captured successfully the characteristics of the secondary instability. The third category (type III) is characterized by periodic unsteady separation, secondary instability, and merging of shear layer vortices, which occurs when Reynolds numbers are high and deceleration rates are low.

Numerical Analysis of Propeller Wake Evolution under Different Advance Coefficients

Propeller wake fields in an open-water configuration were compared between two loading circumstances using large-eddy simulation (LES) with a computational domain of 48 million grids and an overset mesh technique. To validate the results of the numerical simulation, available experimental data are compared, which indicates that the grid systems are suitable for the present study. The results indicate that the present LES simulations describe the inertial frequency range well for both high and low-loading conditions. Under high-loading conditions, the interlaced spirals and secondary vortices that connect adjacent tip vortices amplify the effects of mutual inductance, ultimately triggering the breakdown of the propeller wake systems. At a great distance from the propeller, the vortex system loses all coherence and turns into a collection of smaller vortices that are equally scattered across the wake. In contrast, under light-loading conditions, the wake vortex system exhibits strong coherence and has a relatively simple topology. The elliptic instability and pairing processes are only observed at a far distance from the propeller. The convection velocity transferring tip vortices downstream is larger under the light-loading condition, which leads to the larger pitch of the helicoidal vortices. The larger pitch weakens the mutual inductance or interaction effects among tip vortices, which delays the instability behaviors of the whole vortex system. The results and implications of this study serve as a guide for the development and improvement of next-generation propellers that function optimally when operating behind aquaculture vessels.

The interactions of the elliptical instability and convection

Elliptical instability is an instability of elliptical streamlines, which can be excited by large-scale tidal flows in rotating fluid bodies and excites inertial waves if the dimensionless tidal amplitude (ε) is sufficiently large. It operates in convection zones, but its interactions with turbulent convection have not been studied in this context. We perform an extensive suite of Cartesian hydrodynamical simulations in wide boxes to explore the interactions of elliptical instability and Rayleigh–Bénard convection. We find that geostrophic vortices generated by the elliptical instability dominate the flow, with energies far exceeding those of the inertial waves. Furthermore, we find that the elliptical instability can operate with convection, but it is suppressed for sufficiently strong convection, primarily by convectively driven large-scale vortices. We examine the flow in Fourier space, allowing us to determine the energetically dominant frequencies and wavenumbers. We find that power primarily concentrates in geostrophic vortices, in convectively unstable wavenumbers, and along the inertial wave dispersion relation, even in non-elliptically deformed convective flows. Examining linear growth rates on a convective background, we find that convective large-scale vortices suppress the elliptical instability in the same way as the geostrophic vortices created by the elliptical instability itself. Finally, convective motions act as an effective viscosity on large-scale tidal flows, providing a sustained energy transfer (scaling as ε2). Furthermore, we find that the energy transfer resulting from bursts of elliptical instability, when it operates, is consistent with the ε3 scaling found in prior work.

Dynamic mode decomposition and reconstruction of the transient propeller wake under a light loading condition

This research aims to extend our understanding of propeller wake dynamics under a light loading condition, thereby laying a foundation for design optimization and flow control of the propeller. Dynamic mode decomposition (DMD) and reconstruction are used to analyze the transient vortical wake structures obtained by large eddy simulation. The propeller wake includes stable tip and hub vortices without interacting evolution at the light loading condition, and elliptical instabilities are observed downstream of the tip vortices. DMD describes the most energetic modes and the corresponding dominant frequencies are the blade passing frequency and its multiples. The coherent structures identified via DMD are primarily associated with the ordered convection of the tip vortices and have little correlation with the hub vortices. Additionally, the propeller wake flow is reconstructed using the first four DMD modes, and the primary wake features are well restored with a maximum reconstructed error of 7.98%. This demonstrates that the flow-field reconstruction based on the DMD reduced-order model is promising for predicting the propeller wake and controlling the propeller operation.

Vortex dynamics impact on the wake flow of a marine rudder with leading-edge tubercles

The impact of two tubercle leading-edge (TLE) modifications on the turbulent wake of a reference marine rudder at Reynolds number 2.26 × 106 was analysed numerically using Detached Eddy Simulations (DES). This paper studies the counter-rotating vortex pair formation around the TLE and their impact on the wake structures behind the rudder to find out if the vortex interaction can accelerate the tip vortex dissipation. According to the results, the tubercles enhanced lift for angles of attack (AOA) 10º and above, but at the cost of a drag penalty which reduced the rudders’ lift-to-drag ratio. The formation of the distinctive stream-wise counter-rotating vortex pairs occurred behind the tubercles, which then interacted with the dominant tip vortex. Due to the inherent spanwise flow component of finite-span lifting surfaces the counter-rotating vortex pairs were generated at unequal strength and soon merged into singular vortices co-rotating with the tip vortex. The vortices facilitated flow compartmentalisation over the rudder suction side which broke up the trailing-edge vortex sheet and confined the spanwise flow separation over the rudder surface as AOA increased. The tubercles confined flow separation closer to the rudder tip which reduced the lift generation in the tip area and minimised the initial tip vortex strength. Large elements of stream-wise counter-rotating vorticity formed around the localised stall cells of the TLE rudders that interacted with the tip vortex downstream, introducing elliptical instabilities further weakening the tip vortex and changing its trajectory.

Instability mechanisms in meandering streamwise vortex pairs of upswept afterbody wakes

Wakes of upswept afterbodies are often characterized by counter-rotating streamwise vortex pairs which meander in space. One application concerns aft regions of cargo aircraft, which are characterized by a relatively flat upswept base. Here we consider a canonical configuration comprised of a cylinder with upswept basal surface. The resulting longitudinal vortices, which are much closer to each other than wing-tip vortices, can adversely influence paratrooper and cargo drop operations as well as trailing aircraft. The unsteady dynamics of these vortices are examined using spatio-temporally resolved Large-Eddy Simulations (LES) and stability considerations. Emphasis is placed on understanding the potential instability dynamics responsible for meandering, which was observed, characterized and quantified at a representative location downstream of the body. The dynamics is then successfully mapped to a matched Batchelor vortex pair, and spatial and temporal stability analyses are performed with both counter-rotating vortices in the computational domain. Both spatial and temporal analyses reveal dipole structures associated with |m|=1 elliptic modes as dominant modes in afterbody vortices. A short-wave elliptic instability mode is found to dominate the meandering motion in the vortex pair; this mode was stable in the case of an isolated vortex. Further, the strain due to axial velocity plays a key role in the instability and therefore breakdown. The low frequency of the unstable mode (Strouhal number StD≃0.3 based on cylinder diameter) is consistent with the spectral analysis of meandering in the LES. Stability analyses at very low-wavenumber do not exhibit any unstable mode suggesting an absence of the Crow instability.

Transition mechanisms in a boundary layer controlled by rotating wall-normal cylindrical roughness elements

The transition to turbulence induced by counter-rotating wall-normal rotating cylindrical roughness pairs immersed within a laminar boundary layer on a flat plate is investigated with direct numerical simulations, dynamic mode decomposition (DMD) and perturbation kinetic energy (PKE) analysis. As long as the cylinder stub is rotating, the wake contains a steady dominating inner vortex (DIV) surrounded by a secondary inner vortex. Its circumferential velocity accelerates the fluid on one side of the cylinder and decelerates it on the other side. With low rotation speed, the perturbation is initiated by a combination of elliptical and centrifugal instabilities in the near wake. At medium rotation speeds, Taylor–Couette-like streamwise vortices are generated on the decelerated side, resulting in a protruding reverse-flow zone. Results from DMD analysis and corresponding PKE analysis reveal the unstable nature of the deceleration region and the wake. At the largest rotation speed investigated, the onset of perturbations is directly located on the decelerated side of the cylinder stubs, where a deceleration mechanism feeds the instability. In the near wake, the mechanism gradually changes to a pure centrifugal instability when the rotation speed increases. In the far wake, both elliptical and centrifugal instabilities fade away, and the streaky flow featuring a vigorous DIV is then only subject to inviscid inflectional instability.

Transition to three-dimensional flow in thermal convection with spanwise rotation

Convection in a frame of reference rotating rapidly about an axis perpendicular to the direction of gravity is a model system for convection in the equatorial region of gaseous planets and is realized in recent laboratory experiments. It is found that the transition from rotationally constrained two-dimensional to three-dimensional flows in this system does not occur at some fixed Rossby number. Instead, the Rossby number at the transition depends on the Reynolds number of the flow. The transition occurs when the Taylor columns which constitute the two-dimensional flow become unstable to elliptical instability.

Non-modal growth of finite-amplitude disturbances in oscillatory boundary layer

The essence of sub-critical transition of oscillatory boundary-layer flows is the non-modal growth of finite-amplitude disturbances. The current understanding of the mechanisms of the orderly and bypass transitions of oscillatory boundary-layer flows is limited. The present study adopts optimisation approaches to predict the maximum energy amplification of two- and three-dimensional perturbations in response to the optimal initial disturbance with or without external forcing. A series of direct numerical simulations are also performed to compare with the results obtained from the stability analyses. In particular, the optimal initial perturbation similar to a Tollmien–Schlichting (T–S) wave yields the largest transient growth under the combined effects of the Orr mechanism and inflectional point instability. With a considerable level of two-dimensional disturbance, the vortex tube nonlinearly develops from the T–S-like wave, and then either deforms into a $\varLambda$ -vortex in the near-wall region or rolls up to the free shear region. The further burst of turbulence can follow the first pathway as K-type transition or the second one as vortex tube breakdown due to the elliptical instability. Additionally, non-modal growth can initiate the inception of streaky structures by favourable three-dimensional initial perturbations and/or forcing. The secondary instabilities responsible for the streak breakdown are classified as the varicose (symmetric) and sinuous (anti-symmetric) modes. Under a sufficiently high level of three-dimensional disturbance, the bypass transition is predominantly characterised by the formation of the sinuous mode and turbulent spots, which leads to the suppression of inflection point instability.