Linear Stability Results Research Articles

Temporal evolution of vorticity staircases in randomly strained two-dimensional vortices

The evolution of a Gaussian vortex subject to a weak-external-random n-fold multipolar strain field is examined using fully nonlinear simulations. The simulations show that at large Reynolds numbers, fine scale steps form at the periphery of the vortex, before merging, generally leaving one large step, which acts as a barrier between the vorticity within the coherent core and the surrounding, well mixed, “surf zone.” It is shown for n = 2 that the width and the number of fine scale steps which initially form at the periphery of the vortex is dependent on the strain parameters, but that the range of radial values for which steps initially occur is only dependent on n and the amplitude of the strain field. A criteria is developed which can predict this range of radial values using the linear stability results of Le Dizès [“Non-axisymmetric vortices in two-dimensional flows,” J. Fluid Mech. 406, 175 (2000)]. This criteria is based upon the perturbation vorticity needing to be larger than some fraction of the vorticity gradient to flatten the vortex profile. For n = 3 and 4, the radial step range is again predicted, and it is observed that for these higher wavenumbers the long lasting steps are narrower than the n = 2 case. For n = 4 the steps which form are so narrow that they do not persist very long before they are destroyed by the strain field and viscosity.

Verification of gyrokinetic microstability codes with an LHD configuration

We extend previous benchmarks of the GS2 and GKV-X codes to verify their algorithms for solving the gyrokinetic Vlasov-Poisson equations for plasma microturbulence. Code benchmarks are the most complete way of verifying the correctness of implementations for the solution of mathematical models for complex physical processes such as those studied here. The linear stability calculations reported here are based on the plasma conditions of an ion-ITB plasma in the LHD configuration. The plasma parameters and the magnetic geometry differ from previous benchmarks involving these codes. We find excellent agreement between the independently written pre-processors that calculate the geometrical coefficients used in the gyrokinetic equations. Grid convergence tests are used to establish the resolution and domain size needed to obtain converged linear stability results. The agreement of the frequencies, growth rates, and eigenfunctions in the benchmarks reported here provides additional verification that the algorithms used by the GS2 and GKV-X codes are correctly finding the linear eigenvalues and eigenfunctions of the gyrokinetic Vlasov-Poisson equations.

Three-dimensional stability, receptivity and sensitivity of non-Newtonian flows inside open cavities

We investigate the stability properties of flows over an open square cavity for fluids with shear-dependent viscosity. Analysis is carried out in context of the linear theory using a normal-mode decomposition. The incompressible Cauchy equations, with a Carreau viscosity model, are discretized with a finite-element method. The characteristics of direct and adjoint eigenmodes are analyzed and discussed in order to understand the receptivity features of the flow. Furthermore, we identify the regions of the flow that are more sensitive to spatially localized feedback by building a spatial map obtained from the product between the direct and adjoint eigenfunctions. Analysis shows that the first global linear instability of the steady flow is a steady or unsteady three-dimensionl bifurcation depending on the value of the power-law index n. The instability mechanism is always located inside the cavity and the linear stability results suggest a strong connection with the classical lid-driven cavity problem.

Linear stability analysis for periodic travelling waves of the Boussinesq equation and the Klein–Gordon–Zakharov system

The question of the linear stability of spatially periodic waves for the Boussinesq equation (in the cases p = 2, 3) and the Klein–Gordon–Zakharov system is considered. For a wide class of solutions, we completely and explicitly characterize their linear stability (instability) when the perturbations are taken with the same period T. In particular, our results allow us to completely recover the linear stability results, in the limit T → ∞, for the whole-line case.

Dynamical stability of algebraic Ricci solitons

Abstract We consider dynamical stability for a modified Ricci flow equation whose stationary solutions include Einstein and Ricci soliton metrics. Our focus is on homogeneous metrics on non-compact manifolds. Following the program of Guenther, Isenberg, and Knopf, we define a class of weighted little Hölder spaces with certain interpolation properties that allow the use of maximal regularity theory and the application of a stability theorem of Simonett. With this, we derive two stability theorems, one for a class of Einstein metrics and one for a class of non-Einstein Ricci solitons. Using linear stability results of Jablonski, Petersen, and the first author, we obtain dynamical stability for many specific Einstein and Ricci soliton metrics on simply connected solvable Lie groups.

Global stability analysis of the axisymmetric wake past a spinning bullet-shaped body

AbstractWe analyze the global linear stability of the axisymmetric flow around a spinning bullet-shaped body of length-to-diameter ratio $L/D=2$, as a function of the Reynolds number, $Re=\rho w_{\infty } D /\mu $, and of the rotation parameter $\varOmega =\omega D/(2 w_{\infty })$, in the ranges $Re<450$ and $0\leq \varOmega \leq 1$. Here, $w_{\infty }$ and $\omega $ are the free-stream and the body rotation velocities respectively, and $\rho $ and $\mu $ are the fluid density and viscosity. The two-dimensional eigenvalue problem (EVP) is solved numerically to find the spectrum of complex eigenvalues and their associated eigenfunctions, allowing us to explain the different bifurcations from the axisymmetric state observed in previous numerical studies. Our results reveal that, for the parameter ranges investigated herein, three global eigenmodes, denoted low-frequency (LF), medium-frequency (MF) and high-frequency (HF) modes, become unstable in different regions of the $(Re,\varOmega )$-parameter plane. We provide precise computations of the corresponding neutral curves, that divide the $(Re,\varOmega )$-plane into four different regions: the stable axisymmetric flow prevails for small enough values of $Re$ and $\varOmega $, while three different frozen states, where the wake structures co-rotate with the body at different angular velocities, take place as a consequence of the destabilization of the LF, MF and HF modes. Several direct numerical simulations (DNS) of the nonlinear state associated with the MF mode, identified here for the first time, are also reported to complement the linear stability results. Finally, we point out the important fact that, since the axisymmetric base flow is $SO(2)$-symmetric, the theory of equivariant bifurcations implies that the weakly nonlinear regimes that emerge close to criticality must necessarily take the form of rotating-wave states. These states, previously referred to as frozen wakes in the literature, are thus shown to result from the base-flow symmetry.

The SBP-SAT technique for initial value problems

A detailed account of the stability and accuracy properties of the SBP-SAT technique for numerical time integration is presented. We show how the technique can be used to formulate both global and multi-stage methods with high order of accuracy for both stiff and non-stiff problems. Linear and non-linear stability results, including A-stability, L-stability and B-stability are proven using the energy method for general initial value problems. Numerical experiments corroborate the theoretical properties.

An Efficient, Nonlinear Stability Analysis for Detecting Pattern Formation in Reaction Diffusion Systems

Reaction diffusion systems are often used to study pattern formation in biological systems. However, most methods for understanding their behavior are challenging and can rarely be applied to complex systems common in biological applications. I present a relatively simple and efficient, nonlinear stability technique that greatly aids such analysis when rates of diffusion are substantially different. This technique reduces a system of reaction diffusion equations to a system of ordinary differential equations tracking the evolution of a large amplitude, spatially localized perturbation of a homogeneous steady state. Stability properties of this system, determined using standard bifurcation techniques and software, describe both linear and nonlinear patterning regimes of the reaction diffusion system. I describe the class of systems this method can be applied to and demonstrate its application. Analysis of Schnakenberg and substrate inhibition models is performed to demonstrate the methods capabilities in simplified settings and show that even these simple models have nonlinear patterning regimes not previously detected. The real power of this technique, however, is its simplicity and applicability to larger complex systems where other nonlinear methods become intractable. This is demonstrated through analysis of a chemotaxis regulatory network comprised of interacting proteins and phospholipids. In each case, predictions of this method are verified against results of numerical simulation, linear stability, asymptotic, and/or full PDE bifurcation analyses.

Kinetic effects on transversal instability of planar fronts in packed-bed reactors

We derive a new criterion for transversal instability of planar fronts in packed bed reactors (PBR’s) in which nth order or Langmuir–Hinshelwood kinetics reaction occurs using a pseudo-homogeneous two-variables (C,T)-model. The derivation follows the analysis of combustion of reaction–diffusion systems by Sivashinsky, Comb. Sci. Tech. 15 (1977). The new criterion is expressed as a complicated relation of the ratio of the heat to mass dispersivities on the kinetic parameters. A necessary (but not sufficient) condition for emerging patterns in the reactor cross-section for the kinetic models studied is that (ΔTad/ΔTm)/(PeC/PeT)>1, where ΔTad and ΔTm are the adiabatic and the maximal temperature rise, respectively, PeC and PeT are the mass and the heat Peclet numbers, respectively. This condition agrees with our previous results that were limited to first order kinetics and is unlikely to be satisfied in PBR’s. Also, unlike the previous condition, the new criterion allows to determine the critical wave number (minimal reactor radius) that can exhibit transversal patterns. The new criterion is verified by comparison with the linear stability results and with 3-D simulations.

Linear Stability Analysis for Periodic Standing Waves of the Klein–Gordon Equation

We present linear stability result for periodic standing wave solutions for the Klein–Gordon equation. We find conditions on parameters of the waves which imply linear stability and instabilty of periodic standing waves.

Viscous liquid films on a porous vertical cylinder: Dynamics and stability

In this paper, liquid films flowing down a porous vertical cylinder were investigated by an integral boundary layer model. Linear stability and nonlinear evolution were studied. Linear stability results of the integral boundary layer model were in good agreement with the linearized Navier-Stokes equations which indicated that the permeability of the porous medium enhanced the instability of the flow system. The growth rate and cut-off wave number increased with increasing the permeability and the Reynolds number. Linear stability analysis showed that the system was more unstable for a larger Reynolds number Re. Nonlinear studies showed that, for a very small Re, the film evolved with time while a saturated state was not observed. In addition, it was observed that the film ruptured when the permeability parameter β > 0, and the rupture time decreased with increasing β. However, for a moderate Reynolds number, a small finite harmonic disturbance evolved to a saturated traveling wave. Further investigation was conducted on the droplet-like wave solution. Results showed that the wave speed increased as the permeability parameter increased.

The Dean instability for shear-thinning fluids

We investigate the Dean instability for a generalised Newtonian fluid which satisfies an approximately power-law viscosity model, albeit modified to incorporate a low-shear Newtonian plateau. Infinite aspect ratio linear stability results are presented for both a narrow-gap width and a finite radius of curvature. These results reveal a surprising sensitivity to the details of the low-shear Newtonian region. Finite element solutions of the axisymmetric Navier–Stokes equations for flow through a finite aspect ratio duct confirm this sensitivity and, in addition, demonstrate the potential for hysteresis on the primary branch of vortices. A detailed bifurcation analysis over a range of the aspect ratio reveals that the nonlinear structure of the problem is qualitatively similar to that for a Newtonian fluid despite the apparently quite distinctive behaviour when a comparison is made at a fixed aspect ratio.

On the Nonlinear Stability of Inviscid Homogeneous Shear Flows in Sea Straits of Arbitrary Cross Sections

In this paper we study the nonlinear stability of steady flows of inviscid homogeneous fluids in sea straits of arbitrary cross sections. We use the method of Arnol'd [1] to obtain two general stability theorems for steady basic flows with respect to finite amplitude disturbances. For the special case of plane parallel shear flows we find a finite amplitude extension of the linear stability result of Deng et al [2]. We also present some examples of basic flows which are stable to finite amplitude disturbances.

Long waves between a subsonic gas flow-film interface flowing down an inclined plate

The present work deals with temporal stability properties of a falling liquid film down an inclined plane in the presence of a parallel subsonic gas flow. The waves are described by evolution equation previously derived as a generalization of the model for the Newtonian liquid. We confirm linear stability results of the basic flow using the Orr–Sommerfeld analysis to that obtained by long wave approximation analysis. The non-linear stability criteria of the model are discussed analytically and stability branches are obtained. Finally, the solitary wave solutions at the liquid–gas interface are discussed, using specially envelope transform and direct ansatz approach to Ginzburg–Landau equation. The influence of different parameters governing the flow on the stability behavior of the system is discussed in detail.

First-order energy-integral model for thin Newtonian liquids falling along sinusoidal furrows

An average modeling methodology under the lubrication approach is used to formulate a set of three coupled nonlinear partial differential equations based on the Nusselt scales. This system, known as the energy-integral method in literature, simplifies the Navier-Stokes equation at the first order and analyzes the dynamics of a thin sheet of fluid flowing over a topography with sinusoidally varying longitudinal furrows. Limiting cases of the linear stability results are mathematically discussed and the complete linear system is numerically handled by means of finite differences to approximate the eigenfunctions and their derivatives in a periodic domain. In a geometry which resembles a vertical shift of a topography, with the amplitude being equal to the shift length, it is found that such a geometry stabilizes the flow compared to its counterpart with no shift, such that the wave characteristics get affected. To confirm the stability results, a numerical investigation is performed.

Linear stability analysis of the onset of sublithospheric convection

We consider the onset of small scale convection associated with the cooling of the oceanic lithosphere. This process is investigated by examining the linear stability of a 2-D shear flow that is cooled from the top and obeys an Arrhenius rheology. The perturbation equations are derived assuming a parallel flow, local 1-D analysis. The linear stability results are able to reproduce the structure and growth rate produced by solving the full non-linear equations. In addition, linear stability analysis predicts a similar dependence of onset times on activation energy and Rayleigh number as previous studies when small plate velocities are specified. Our linear stability analysis predicts a significant increase in the onset time with increasing plate velocity. Suggestions are given for the dynamical influence of plate velocity in reducing the convective forcing and delaying onset times.

Three-dimensional numerical simulation of the wake flow of an afterbody at subsonic speeds

We numerically investigate the wake flow of an afterbody at low Reynolds number in the incompressible and compressible regimes. We found that, with increasing Reynolds number, the initially stable and axisymmetric base flow undergoes a first stationary bifurcation which breaks the axisymmetry and develops two parallel steady counter-rotating vortices. The critical Reynolds number (Recs) for the loss of the flow axisymmetry reported here is in excellent agreement with previous axisymmetric BiGlobal linear stability (BiGLS) results. As the Reynolds number increases above a second threshold, Reco, we report a second instability defined as a three-dimensional peristaltic oscillation which modulates the vortices, similar to the sphere wake, sharing many points in common with long-wavelength symmetric Crow instability. Both the critical Reynolds number for the onset of oscillation, Reco, and the Strouhal number of the time-periodic limit cycle, Stsat, are substantially shifted with respect to previous axisymmetric BiGLS predictions neglecting the first bifurcation. For slightly larger Reynolds numbers, the wake oscillations are stronger and vortices are shed close to the afterbody base. In the compressible regime, no fundamental changes are observed in the bifurcation process. It is shown that the steady state planar-symmetric solution is almost equal to the incompressible case and that the break of planar symmetry in the vortex shedding regime is retarded due to compressibility effects. Finally, we report the developments of a low frequency which depends on the afterbody aspect ratio, as well as on the Reynolds and on the Mach number, prior to the loss of the planar symmetry of the wake.

Sediment-laden fresh water above salt water: linear stability analysis

AbstractWhen a layer of particle-laden fresh water is placed above clear, saline water, both Rayleigh–Taylor and double diffusive fingering instabilities may arise. For quasi-steady base profiles, we obtain linear stability results for such situations by means of a rational spectral approximation method with adaptively chosen grid points, which is able to resolve multiple steep gradients in the base state density profile. In the absence of salinity and for a step-like concentration profile, the dominant parameter is the ratio of the particle settling velocity to the viscous velocity scale. As long as this ratio is small, particle settling has a negligible influence on the instability growth. However, when the particles settle more rapidly than the instability grows, the growth rate decreases inversely proportional to the settling velocity. This damping effect is a result of the smearing of the vorticity field, which in turn is caused by the deposition of vorticity onto the fluid elements passing through the interface between clear and particle-laden fluid. In the presence of a stably stratified salinity field, this picture changes dramatically. An important new parameter is the ratio of the particle settling velocity to the diffusive spreading velocity of the salinity, or alternatively the ratio of the unstable layer thickness to the diffusive interface thickness of the salinity profile. As long as this quantity does not exceed unity, the instability of the system and the most amplified wavenumber are primarily determined by double diffusive effects. In contrast to situations without salinity, particle settling can have a destabilizing effect and significantly increase the growth rate. Scaling laws obtained from the linear stability results are seen to be largely consistent with earlier experimental observations and theoretical arguments put forward by other authors. For unstable layer thicknesses much larger than the salinity interface thickness, the particle and salinity interfaces become increasingly decoupled, and the dominant instability mode becomes Rayleigh–Taylor-like, centred at the lower boundary of the particle-laden flow region.

Energy transient growth in curved channel flow

Transient growth of perturbations due to non-normality for the curved channel Poiseuille flow (CCPF) is presented. The study covers a wide range of the radii ratios η as well as axial and azimuthal modes, with the purpose of complementing the results of linear stability for this flow with a study of the optimal linear growth possible in the linearly stable parameter regions. For the wide-gap case, the transient growth of streamwise-azimuthal modes that grow most is of low level and suppressed by curvature. It is also found that as curved channel flow approaches the flow in a straight channel enough, both the normal and non-nor- mal stability characteristics are almost identical to that of plane Poiseuille flow. The modulation of the basic circular Poiseuille flow by the presence of azimuthal streaks resulting from the significant growth of initial perturbations can be clearly visualized. For the transition region between the wide-gap case and the narrow-gap case, the sensitivity of eigenvalues is shown to be closely related to the magnitude of transient growth, which is tuned by curvature in a smooth way.

Inertial effects on rotating Hele-Shaw flows

We examine the finite surface tension radial viscous fingering problem in a rotating Hele-Shaw cell, and focus on the action of inertial effects on the stability and morphology of the emerging patterns. To study such a flow we use an alternative version of the usual Darcy's law derived by gap-averaging the Navier-Stokes equation, but retaining its inertial terms. The importance of inertial forces is pertinently characterized by a rotational Reynolds number. Linear and weakly nonlinear stages of the dynamics are described analytically through a mode coupling approach. The linear stability results indicate that inertia has a stabilizing role. However, the characteristic number of fingers and the width of the band of linearly unstable modes are not altered by inertia. In the early nonlinear regime we find that inertia acts to favor the development of fingers with narrower tips than those obtained in its absence. We have also verified that finger competition events are affected by inertial effects which tend to restrain finger length variability among inward-moving fingers.