Three-dimensional Instability Research Articles

Local gravitational instability of two-component thick discs in three dimensions

Aims. The local gravitational instability of rotating discs is believed to be an important mechanism in different astrophysical processes, including the formation of gas and stellar clumps in galaxies. We aim to study the local gravitational instability of two-component thick discs in three dimensions. Methods. We use as a starting point a recently proposed analytic three-dimensional (3D) instability criterion for discs with non-negligible thickness that takes the form Q3D < 1, where Q3D is a 3D version of the classical 2D Toomre Q parameter for razor-thin discs. Here, we extend the 3D stability analysis to two-component discs, considering first the influence on Q3D of a second unresponsive component, and then the case in which both components are responsive. We present the application to two-component discs with isothermal vertical distributions, which can represent, for instance, galactic discs with both stellar and gaseous components. Finally, we relax the assumption of vertical isothermal distribution, by studying one-component self-gravitating discs with polytropic vertical distributions for a range of values of the polytropic index corresponding to convectively stable configurations. Results. We find that Q3D < 1, where Q3D can be computed from observationally inferred quantities, is a robust indicator of local gravitational instability, depending only weakly on the presence of a second component and on the vertical gradient of temperature or velocity dispersion. We derive a sufficient condition for local gravitational instability in the midplane of two-component discs, which can be employed when both components have Q3D > 1.

A novel finite element limit equilibrium method and its application in instability analysis for 3-D high core rockfill dams

The safe operation of high rockfill dams is an important issue related to human life and property safety. However, the stability analysis of high core rockfill dams with complex spatial morphology is mostly confined to two-dimensional, and appropriate stability analysis methods are scarce. Therefore, it is necessary to develop the three-dimensional stability analysis method for high core rockfill dams and study the spatial stability mechanism after impoundment. Firstly, a finite element limit equilibrium framework suitable for stability analysis of high rockfill dams is introduced, and the horn-shaped three-dimensional failure mechanism is presented for optimization. Then, according to the statistical results of the actual core rockfill dam above 100 m, 15 high dam models with actual Duncan-Chang parameters between 100 m and 300 m are established to study the three-dimensional instability mechanism. Meanwhile, grey relation analysis (GRA) is used for quantitative analysis of stability sensitive factors. Finally, the stability analysis of the Quxue core rockfill dam (173.2 m) was completed within the established stability framework. The results show that the combined failure mechanism with the horn-shaped is suitable for dam stability analysis, and with the increase of height and slope ratio of dam, the most dangerous sliding surface evolves to shallow layer. In addition, GRA results show that slope ratio is the most sensitive parameter compared to dam height and material parameters. The practical engineering stability results of Quxue show that the research results can provide a reliable reference for the stability of the high core rockfill dam.

Nonmodal stability analysis of Poiseuille flow through a porous medium

We unravel the nonmodal stability of a three-dimensional nonstratified Poiseuille flow in a saturated hyperporous medium constrained by impermeable rigid parallel plates. The primary objective is to broaden the scope of previous studies that conducted modal stability analysis for two-dimensional disturbances. Here, we explore both temporal and spatial transient disturbance energy growths for three-dimensional disturbances when the Reynolds number and porosity of the material are high, based on evolution equations with respect to time and space, respectively. Modal stability analysis reveals that the critical Reynolds number for the onset of shear mode instability increases as porosity increases. Moreover, the Darcy viscous drag term stabilizes shear mode instability, resulting in a delay in the transition from laminar flow to turbulence. In addition, it demonstrates the suppression of three-dimensional shear mode instability as the spanwise wavenumber increases, thereby confirming the statement of Squire’s theorem. By contrast, nonmodal stability analysis discloses that both temporal and spatial transient disturbance energy growths curtail as the effect of the Darcy viscous drag force intensifies. But their maximum values behave like O(Re2) for a fixed porous material, where Re is the Reynolds number. However, for different porous materials, the scalings for both temporal and spatial transient disturbance energy growths are different. Furthermore, increasing porosity also suppresses both temporal and spatial disturbance energy growths. Finally, we observe that temporal transient disturbance energy growth becomes larger for a spanwise perturbation, while spatial transient disturbance energy growth becomes larger for a steady perturbation when angular frequency vanishes. The initial disturbance that excites the largest temporal energy amplification generates two sets of alternating high-speed and low-speed elongated streaks in the streamwise direction.

Direct Numerical Simulation of Tandem-Wing Aerodynamic Interactions at Low Reynolds Number

A three-dimensional direct numerical simulation was conducted to investigate the vortex–wing interactions through two NACA 0012 stationary wings placed in tandem at a low Reynolds number Re=10,000. The aerodynamic characteristics and three-dimensional flow structures were analyzed for the tandem wings. The back wing disturbed by the upstream vortices gained an evident increase in aerodynamic performance, where the advantage is related to the suppression of the large-scale vortex formation near the trailing edge. The Liutex method was applied to visualize the vortical structures for investigating the three-dimensional evolution and instability when interacting with the back wing. The upstream wake triggered dual-secondary vortices and intensified the secondary instability on the back wing. The induced vortices contributed to the lift enhancement because they provided an extra low-pressure region when propagating downstream on the suction side of the back wing. Because of the three-dimensional destabilization, the vortex interaction in the evolution process accelerated the transition and injection of the high-momentum flow into the boundary layer attached to the back wing, energizing the turbulent boundary layer and eliminating the large-scale separation near the trailing edge. This study provided a new perspective on the enhanced aerodynamic performance of tandem layout.

Linear stability of stratified, rotating, viscous plane Couette–Poiseuille flow

The linear stability of plane Couette–Poiseuille flow (CPF) is studied with the physical effects of stratification, rotation and viscosity all included for the first time together. With no stratification, two instability mechanisms are present due to the shear and rotation which, for the most part, do not interact as they favour different forms of two-dimensionality. However, there are some small parts of parameter space where new three-dimensional instability appears indicating that Rayleigh's criterion is also violated in parameter space beyond where shear instability is expected. No fully localised centrifugal instabilities can be found for CPF, but they are shown to exist if the base flow shear has a maximum in the domain (the base flow needs to be at least cubic in the cross-stream variable rather than just quadratic as in CPF). With stable stratification present, new instabilities are found due to the combined effects of stratification and rotation, but only some appear to be of the resonance-type associated with the strato-rotational instability. The other unstable branches are more accurately interpreted as a stratification-modified centrifugal instability. Three-dimensional versions of this violate Rayleigh's criterion even when this is extended to include stratification. When stratification is stronger than rotation, the resonance-type instabilities are only dominant for cyclonic flows.

Suppression of lift-up effect in the 3D Boussinesq equations around a stably stratified Couette flow

In this paper, we establish linear enhanced dissipation results for the three-dimensional Boussinesq equations around a stably stratified Couette flow, in the viscous and thermally diffusive setting. The dissipation rates are faster compared to those observed in the homogeneous Navier-Stokes equations, in light of the interplay between velocity and temperature, driven by buoyant forces. Our approach involves introducing a change of variables grounded in a Fourier space symmetrization framework. This change elucidates the energy structure inherent in the system. Specifically, we handle non-streaks modes through an augmented energy functional, while streaks modes are amenable to explicit solutions. This explicit treatment reveals the oscillatory nature of shear modes, providing the elimination of the well-known three-dimensional instability mechanism known as the “lift-up effect”.

Positivity-preserving discontinuous spectral element methods for compressible multi-species flows

We introduce a novel positivity-preserving numerical stabilisation approach for high-order discontinuous spectral element approximations of compressible multi-species flows. The underlying stabilisation method is the adaptive entropy filtering approach (Dzanic and Witherden, J. Comput. Phys., 468, 2022), which is extended to the conservative formulation of the multi-species flow equations. We show that the straightforward enforcement of entropy constraints in the filter yields poor results around species interfaces and propose an adaptive switch for the entropy bounds based on the convergence properties of the pressure field which drastically improves its performance for multi-species flows. The proposed approach is shown in a variety of numerical experiments applied to the multi-species Euler and Navier–Stokes equations computed on unstructured grids, ranging from shock-fluid interaction problems to three-dimensional viscous flow instabilities. We demonstrate that the approach can retain the high-order accuracy of the underlying numerical scheme even at smooth extrema, ensure the positivity of the species density and pressure in the vicinity of shocks and contact discontinuities, and accurately predict small-scale flow features with minimal numerical dissipation.

Elasto-inertial instability in torsional flows of shear-thinning viscoelastic fluids

It is well known that inertia-free shearing flows of a viscoelastic fluid with curved streamlines, such as the torsional flow between a rotating cone and plate or the flow in a Taylor–Couette geometry, can become unstable to a three-dimensional time-dependent instability at conditions exceeding a critical Weissenberg ( $Wi$ ) number. However, the combined effects of fluid elasticity, shear thinning and finite inertia (as quantified by the Reynolds number $Re$ ) on the onset of elasto-inertial instabilities are not fully understood. Using a set of cone–plate geometries, we experimentally explore the entire $Wi$ – $Re$ phase space for a series of nonlinear viscoelastic fluids (with the dependence on shear rate $\dot{\gamma}$ quantified using a shear-thinning parameter $\beta _P(\dot {\gamma })$ ). We tune $\beta _P(\dot {\gamma })$ by varying the dissolved polymer concentration in solution. This progressively reduces shear thinning but leads to finite inertial effects before the onset of elastic instability, and thus naturally results in elasto-inertial coupling. Time-resolved rheometric measurements and flow visualization experiments allow us to investigate the effects of flow geometry, and document the combined effects of varying $Wi, Re$ and $\beta _P(\dot {\gamma })$ on the emergence of secondary motions at the onset of instability. The resulting critical state diagram quantitatively depicts the competition between the stabilizing effects of shear thinning and the destabilizing effects of inertia. We extend the curved streamline instability criterion of Pakdel & McKinley (Phys. Rev. Lett., vol. 77, no. 12, 1996, p. 2459) for the onset of purely elastic instability in curvilinear geometries by using scaling arguments to incorporate shear thinning and finite inertial effects. The augmented condition facilitates predictions of the onset of instability over a broader range of flow conditions, and thus bridges the gap between purely elastic and elasto-inertial curved streamline instabilities.

Analysis of single-mode Richtmyer–Meshkov instability using high-order incompressible vorticity—streamfunction and shock-capturing simulations

Two- and three-dimensional simulation results obtained using a new high-order incompressible, variable-density vorticity–streamfunction (VS) method and data from previous ninth-order weighted essentially nonoscillatory (WENO) shock-capturing simulations [M. Latini and O. Schilling, “A comparison of two- and three-dimensional single-mode reshocked Richtmyer-Meshkov instability growth,” Physica D 401, 132201 (2020)] are used to investigate the nonlinear dynamics of single-mode Richtmyer–Meshkov instability using a model of a Mach 1.3 air(acetone)/SF6 shock tube experiment [J. W. Jacobs and V. V. Krivets, “Experiments on the late-time development of single-mode Richtmyer–Meshkov instability,” Phys. Fluids 17, 034105 (2005)]. A comparison of the density fields from both simulations with the experimental images demonstrates very good agreement in the large-scale structure with both methods but differences in the small-scale structure. The WENO method captures the small-scale disordered structure observed in the experiment, while the VS method partially captures such structure and yields a strong rotating core. The perturbation amplitude growth from the simulations generally agrees well with the experiment. The simulation bubble and spike amplitudes agree well at early times. At later times, the WENO bubble amplitude is smaller than the VS amplitude and vice versa for the spike amplitude. The predictions of nonlinear single-mode instability growth models are shown to agree with the simulation amplitudes at early-to-intermediate times but underpredict the amplitudes at later times in the nonlinear regime. Visualizations of the mass fraction and enstrophy isosurfaces, velocity and vorticity fields, and baroclinic vorticity production and vortex stretching terms from the three-dimensional simulations indicate that, with the exception of the small-scale structure within the rollups, the VS and WENO results are in good agreement.

Three-dimensional diffusive-thermal instability of flames propagating in a plane Poiseuille flow

The three-dimensional diffusive-thermal stability of a two-dimensional flame propagating in a Poiseuille flow is examined. The study explores the effect of three non-dimensional parameters, namely the Lewis number Le, the Damköhler number Da, and the flow Peclet number Pe. Wide ranges of the Lewis number and the flow amplitude are covered, as well as conditions corresponding to small-scale narrow (Da≪1) to large-scale wide (Da≫1) channels. The instability experienced by the flame appears as a combination of the traditional diffusive-thermal instability of planar flames and the recently identified instability corresponding to a transition from symmetric to asymmetric flame. The instability regions are identified in the Le-Pe plane for selected values of Da by computing the eigenvalues of a linear stability problem. These are complemented by two- and three-dimensional time-dependent simulations describing the full evolution of unstable flames into the non-linear regime. In narrow channels, flames are found to be always symmetric about the mid-plane of the channel. Additionally, in these situations, shear flow-induced Taylor dispersion enhances the cellular instability in Le<1 mixtures and suppresses the oscillatory instability in Le>1 mixtures. In large-scale channels, however, both the cellular and the oscillatory instabilities are expected to persist. Here, the flame has a stronger propensity to become asymmetric when the mean flow opposes its propagation and when Le<1; if the mean flow facilitates the flame propagation, then the flame is likely to remain symmetric about the channel mid-plane. For Le>1, both symmetric and asymmetric flames are encountered and are accompanied by temporal oscillations.

Floquet-based analysis on three-dimensional non-axisymmetric instabilities in oscillatory-driven Taylor–Couette flows and their low-frequency asymptotic behavior using Wentzel–Kramers–Brillouin method

This paper revisits the linear stability analysis of oscillatory-driven flows between two oscillating cylinders against non-axisymmetric disturbances. This study is motivated by the lack of a sufficiently reliable theoretical analysis giving insight into the experimentally observed spiral-like non-axisymmetric patterns when the cylinders are counter-oscillating. A new generalized time-dependent algebraic eigenvalue problem is constructed from the linearized set of the three-dimensional Navier–Stokes equations around the purely azimuthal basic state. Numerical evaluation of the critical eigenvalues combining both Floquet theory and spectral method reveals the existence of frequency ranges where this basic state becomes unstable against three-dimensional non-axisymmetric disturbances before it does so for two-dimensional axisymmetric ones. Indeed, as the oscillation frequency of the cylinders increases, the azimuthal wave number of the critical eigensolution is found to change from 0 to 2 to 1 and then back to 0. The primary bifurcation exchange between two instability modes with different azimuthal wave numbers occurs via different types of codimension-2 bifurcation points giving rise to discontinuities in the critical axial wave number where reversing and non-reversing non-axisymmetric Taylor vortex flows are identified. In addition, by extending our numerical calculations to the co-oscillating case, we show that the axisymmetric disturbances are the most unstable confirming thus existing experimental findings. Furthermore, a Wentzel–Kramers–Brillouin (WKB) analysis is performed to shed light on the asymptotic behavior of these time-dependent flows in the low-frequency limit when the cylinders are slowly oscillating.

Investigation of three-dimensional instability behind a circular cylinder via low-dimensional space spanned by optimal proper orthogonal decomposition modes

Investigation of three-dimensional instability behind a circular cylinder via low-dimensional space spanned by optimal proper orthogonal decomposition modes

Three-dimensional bond-order instability in infinite-layer nickelates due to nonlocal quantum interference

Three-dimensional bond-order instability in infinite-layer nickelates due to nonlocal quantum interference

A self-sustaining mechanism for plane Couette flow with a flexible lower boundary

Abstract The stability of plane Couette flow to travelling-wave disturbances is studied asymptotically at high Reynolds numbers Re when the lower boundary possesses a degree of flexibility modelled by a spring-backed plate. First, it is shown that a three-dimensional (3D) linear instability exists, with streamwise and spanwise wavelengths comparable with the channel width. Building on this, nonlinear effects from the self-interaction of the wave are introduced, leading to a self-sustaining interaction between a roll/streak flow and the 3D wave. Governing nonlinear vortex-wave interaction (VWI) equations are derived and a perturbation analysis is carried out to guide a numerical investigation of the equations. The co-existence of two families of finite-amplitude solutions, each with different flow structures, is found. Numerical solutions of the VWI equations in each case show that a small wave amplitude of O(Re−1(log Re)−1/2) is all that is necessary to provoke an O(1) change to the basic Couette flow.

Three-dimensional global instability analysis for high-speed boundary layer flow

Herein, a numerical approach for the three-dimensional global instability analysis of high-speed boundary layer is developed. The approach is based on the Krylov–Schur method for large-scale eigenproblems. A shift-and-invert spectral transformation is employed to accelerate the convergence for eigenvalues of interest. Iterative linear solver is adopted in the transformation to reduce memory usage by avoiding the fill-in problem of direct solver and the explicit construction of matrices. A Robin condition based on BiGlobal instability analysis is imposed at the inflow boundary, and a non-reflecting condition derived from inviscid flow is used for the far-field boundary. The approach is validated in a Blasius boundary layer. According to a comparison between the performances of several algorithms in this case, the conjugate gradient squared method with no preconditioning is chosen as the linear solver. The results provided by three-dimensional global instability analysis consist with those from linear stability theory and BiGlobal instability analysis. A further demonstration of our approach is provided for a three-dimensional roughness-disturbed boundary layer. The fluctuations extracted from direct numerical simulation results illustrate that the symmetric component is the dominant unstable mode in roughness wake, as it grows twice as fast as the antisymmetric component. Two eigenvalues are obtained from the three-dimensional global instability analysis, and these represent the symmetric and antisymmetric unstable modes. The wavelengths and growth rates of both modes delivered by instability analysis agree well with the direct numerical simulation results, and the profiles of disturbance amplitude match the features of the varicose and sinuous modes.

A shock-stable numerical scheme accurate for contact discontinuities: Applications to 3D compressible flows

The low-dissipation Roe scheme is a popular shock capturing scheme but the shock instability, such as the infamous carbuncle phenomenon, seriously affects its application in high Mach number flows. Although considerable research has been undertaken in two dimensions, mechanism analysis of the three-dimensional shock instability is relatively scarce. Numerical simulations of the Sedov blast wave indicate that the shock instability is more prominent in three dimensions, so it has theoretical and practical significance to analyze and heal the numerical instability. The mechanism for three-dimensional shock instability has been thoroughly studied by means of the linearized stability analysis and the dissipation-controlling approach which introduces the required transverse dissipation in the numerical shock layer and is adopted to enhance the Roe scheme’s shock stability. Furthermore, the unphysical expansion shock resulting from the violation of entropy condition is eliminated by a simple modification of numerical signal speed. For the improvement of the resolution for contact discontinuities and shear waves, an algebraic method which combines the THINC reconstruction scheme and the BVD algorithm is employed to minimize the density difference in the numerical diffusion term. A series of benchmark numerical experiments fully demonstrate that the proposed scheme is endowed with excellent robustness against the shock instability and high accuracy for contact discontinuities and shear waves.

Three-dimensional shear-flow instability saturation via stable modes

Turbulence in three dimensions (3D) supports vortex stretching that has long been known to accomplish energy transfer to small scales. Moreover, net energy transfer from large-scale, forced, unstable flow-gradients to smaller scales is achieved by gradient-flattening instability. Despite such enforcement of energy transfer to small scales, it is shown here that the shear-flow-instability-supplied 3D-fluctuation energy is largely inverse-transferred from the fluctuation to the mean-flow gradient, and such inverse transfer is more efficient for turbulent fluctuations in 3D than in two dimensions (2D). The transfer is due to linearly stable eigenmodes that are excited nonlinearly. The stable modes, thus, reduce both the nonlinear energy cascade to small scales and the viscous dissipation rate. The vortex-tube stretching is also suppressed. Up-gradient momentum transport by the stable modes counters the instability-driven down-gradient transport, which also is more effective in 3D than in 2D (≈70% vs ≈50%). From unstable modes, these stable modes nonlinearly receive energy via zero-frequency fluctuations that vary only in the direction orthogonal to the plane of 2D shear flow. The more widely occurring 3D turbulence is thus inherently different from the commonly studied 2D turbulence, despite both saturating via stable modes.

Positivity-preserving entropy filtering for the ideal magnetohydrodynamics equations

In this work, we present a positivity-preserving adaptive filtering approach for discontinuous spectral element approximations of the ideal magnetohydrodynamics equations. This approach combines the entropy filtering method (Dzanic and Witherden, 2022) for shock capturing in gas dynamics along with the eight-wave method for enforcing a divergence-free magnetic field. Due to the inclusion of non-conservative source terms, an operator-splitting approach is introduced to ensure that the positivity and entropy constraints remain satisfied by the discrete solution. Furthermore, a computationally efficient algorithm for solving the optimization process for this nonlinear filtering approach is presented. The resulting scheme can robustly resolve strong discontinuities on general unstructured grids without tunable parameters while recovering high-order accuracy for smooth solutions. The efficacy of the scheme is shown in numerical experiments on various problems including extremely magnetized blast waves and three-dimensional magnetohydrodynamic instabilities.

On the onset of long-wavelength three-dimensional instability in the cylinder wake

We study the onset of the three-dimensional mode A instability in the near wake behind a circular cylinder. We show that long-wavelength perturbations organise in a time-shifting pattern such that the in-plane velocity in each streamwise slice corresponds to the base flow solution at shifted times. This observation introduces an additional unifying characteristic for certain mode A type instabilities. We then analyse the mechanisms which control the growth or decay of these perturbations and highlight the crucial role played by the tilting mechanism which operates via non-local interactions in a manner similar to Biot–Savart induction. We characterise its domain of influence using a Green's function-based approach which allows us to rationalise the non-trivial dependence of the growth rate on the spanwise wavenumber. We connect this behaviour to the subtle balance between the local growth of the perturbations as they are swept along by the flow and the feedback on the perturbations that are generated during the next period of the time-periodic base flow. Finally, we discuss generalisations of our findings to other types of flows.

Asymmetric cellular bi-stability in the gap between tandem cylinders

In the gap region of tandem cylinders, within the reattachment regime, bi-stability is seen to be cellular. Direct numerical simulations at a Reynolds number of 500, and a gap ratio of 3.0, show that shedding of gap vortices occurs in spanwise cells, with lengths between 0.3 and 2.7 cylinder diameters. These unstable vortex cells tend to be asymmetric with respect to the gap centreline, so that vortices are repeatedly shed from just one gap shear layer for several periods. The unstable cells appear within the basic spanwise cell structure dictated by the three-dimensional instability, and their cell lengths do not exceed those of the basic cells. Unstable cells intermittently become spanwise coherent, and this leads to a significant increase in drag amplitude. The mode change in the gap is associated with low-frequency variation of the reattachment and separation points on the downstream cylinder, causing low-frequency modulation of the vortex formation length.