Linear Instability Modes Research Articles

Vertical convection regimes in a two-dimensional rectangular cavity: Prandtl and aspect ratio dependence

Vertical convection is the fluid motion that is induced by the heating and cooling of two opposed vertical boundaries of a rectangular cavity (see e.g. Wang et al., J. Fluid Mech., vol. 917, 2021, A6). We consider the linear stability of the steady two-dimensional flow reached at Rayleigh numbers of O( $10^8$ ). As a function of the Prandtl number, $Pr$ , and the height-to-width aspect ratio of the domain, $A$ , the base flow of each case is computed numerically and linear simulations are used to obtain the properties of the leading linear instability mode. Flow regimes depend on the presence of a circulation in the entire cavity, detachment of the thermal layer from the boundary or the corner regions and on the oscillation frequency relative to the natural frequency of oscillation in the stably temperature-stratified interior, allowing for the presence of internal waves or not. Accordingly, the regime is called slow or fast, respectively. Either the global circulation or internal waves in the interior may couple the top and bottom buoyancy currents, while their absence implies asymmetry in their perturbation amplitude. Six flow regimes are found in the range of $0.1 \leq Pr \leq 4$ and $0.5 \leq A \leq 2$ . For $Pr \lessapprox 0.4$ and $A>1$ , the base flow is driven by a large circulation in the entire cavity. For $Pr \gtrapprox 0.7$ , the thermal boundary layers are thin and the instability is driven by the motion along the wall and the detached boundary layer. A transition between these regimes is marked by a dramatic change in oscillation frequency at $Pr = 0.55 \pm 0.15$ and $A <2$ .

Hydrodynamic α-effect in a rotating stratified moist atmosphere driven by small-scale non-helical force

In this paper we have studied the large-scale instability in a rotating stratified moist atmosphere with a small-scale turbulence. The axis of rotation of the medium is deviated from the vertical direction. The turbulence is excited by an external small-scale force with zero helicity and a low Reynolds number. The nonlinear equations of a vortex dynamo were obtained on the basis of the method of multiscale asymptotic expansions. The linear instability and stationary nonlinear modes were studied. The solutions in a form of localized vortex structures were obtained.

Nonlinear Vortex Structures Driven by Small-Scale Non-helical Forces in Obliquely Rotating Stratified Fluids

We study a new type of large-scale instability in obliquely rotating stratifi ed fl uids with small-scale nonhelical turbulence. The small-scale turbulence is generated by the external force with zero helicity and low Reynolds number. The theory uses the method of multiscale asymptotic expansions. The nonlinear equations for large-scale motions are obtained in the third order of perturbation theory. We consider a linear instability and stationary nonlinear modes. Solutions in the form of nonlinear Beltrami waves and localized vortex structures such as kinks of a new type are obtained.

Sideband thermocapillary instability of a thin film flowing down the outside of a thick walled cylinder with finite thermal conductivity

The sideband thermocapillary instability of a thin liquid film flowing down the outside of a hot vertical cylinder is investigated. The wall of the cylinder is thick and has finite thermal conductivity. In contrast to previous work, it is shown that the presence of a main flow represented by the Reynolds number promotes the linear axial instability mode as the most unstable one. Thermocapillarity excites the appearance of linear azimuthal modes but it is not able to make them the most unstable when the wavenumber is small but finite. Only when the wavenumber tends to zero the growth rate of the azimuthal modes is larger than that of the axial mode. It is found that the thickness and thermal conductivity of the wall have an important influence on the stability. Even though the linear axial mode is the most unstable one, azimuthal modes are observed in the experiment. Therefore, it is of interest here to investigate the nonlinear sideband instability of the flow to find out the parameters region where the first azimuthal mode is able to prevail. It was possible to group for the first time the Reynolds, surface tension and Marangoni numbers with wall effect in only one parameter.

Hunting Stability of High-Speed Railway Vehicles Under Steady Aerodynamic Loads

With the rising speed of high-speed trains, the aerodynamic loads become more significant and their influences on the hunting stability of railway vehicles deserve to be considered. Such an effect cannot be properly considered by the conventional model of hunting stability analysis. To this end, the linear hunting stability of high-speed railway vehicles running on tangent tracks is studied. A model considering the steady aerodynamic loads due to the joint action of the airflow facing the moving train and the crosswind, is proposed for the hunting stability analysis of a railway vehicle with 17 degrees of freedom (DOF). The key factors considered include: variations of the wheel–rail normal forces, creep coefficients, gravitational stiffness and angular stiffness due to the actions of the aerodynamic load, which affects the characteristics of hunting stability. Using the computer program developed, numerical calculations were carried out for studying the behavior of the linear hunting stability of vehicles under steady aerodynamic loads. The results show that the aerodynamic loads have an obvious effect on the linear critical speeds and instability modes. The linear critical speed decreases monotonously as the crosswind velocity increases, and the influences of pitch moment and lift force on the linear critical speed are larger than the other components of the aerodynamic loads.

Convective instabilities of Maxwell–Cattaneo fluids

Motivated by the need to understand better the dynamics of non-Fourier fluids, we examine the linear and weakly nonlinear stabilities of a horizontal layer of fluid obeying the Maxwell–Cattaneo relationship of heat flux and temperature using three different forms of the time derivative of the heat flux. Linear stability mode regime diagrams in the parameter plane have been established and used to summarize the linear instabilities. The energy balance of the system is used to identify the mechanism by which the Maxwell–Cattaneo effect (i) introduces overstability, (ii) leads to preferred stationary modes with the critical Rayleigh and wavelengths either both increasing or both decreasing, (iii) gives rise to instabilities in a layer heated from above, and (iv) enhances heat transfer. A formal weakly nonlinear analysis leads to evolution equations for the amplitudes of linear instability modes. It is shown that the amplitude of the stationary mode obeys an equation of the Landau–Stuart type. The two equally excitable overstable modes obey two equations of the same type coupled by an interaction term. The evolution of the different amplitudes leads to supercritical stability, supercritical instability or subcritical instability depending on the model and parameter values. The results are presented in regime diagrams.

Comparing kinetic Monte Carlo and thin-film modeling of transversal instabilities of ridges on patterned substrates

We employ kinetic Monte Carlo (KMC) simulations and a thin-film continuum model to comparatively study the transversal (i.e., Plateau-Rayleigh) instability of ridges formed by molecules on pre-patterned substrates. It is demonstrated that the evolution of the occurring instability qualitatively agrees between the two models for a single ridge as well as for two weakly interacting ridges. In particular, it is shown for both models that the instability occurs on well defined length and time scales which are, for the KMC model, significantly larger than the intrinsic scales of thermodynamic fluctuations. This is further evidenced by the similarity of dispersion relations characterizing the linear instability modes.

Parabolized stability analysis of jets from serrated nozzles

We study the viscous spatial linear stability characteristics of the time-averaged flow in turbulent subsonic jets issuing from serrated (chevroned) nozzles, and compare them to analogous round jet results. Linear parabolized stability equations (PSE) are used in the calculations to account for the non-parallel base flow. By exploiting the symmetries of the mean flow due to the regular arrangement of serrations, we obtain a series of coupled two-dimensional PSE problems from the original three-dimensional problem. This reduces the solution cost and manifests the symmetries of the stability modes. In the parallel-flow linear stability theory (LST) calculations that are performed near the nozzle to initiate the PSE, we find that the serrated nozzle reduces the growth rates of the most unstable eigenmodes of the jet, but their phase speeds are approximately similar. We obtain encouraging validation of our linear PSE instability wave results vis-à-vis near-field hydrodynamic pressure data acquired on a phased microphone array in experiments, after filtering the latter with proper orthogonal decomposition (POD) to extract the energetically dominant coherent part. Additionally, a large-eddy simulation database of the same serrated jet is investigated, and its POD-filtered pressure field is found to compare favourably with the corresponding PSE solution within the jet plume. We conclude that the coherent hydrodynamic pressure fluctuations of jets from both round and serrated nozzles are reasonably consistent with the linear instability modes of the turbulent mean flow.

Efficient jet noise models using the one-way Euler equations

Experimental and numerical investigations have correlated large-scale coherent structures in turbulent jets with acoustic radiation to downstream angles, where sound is most intense. These structures can be modeled as linear instability modes of the turbulent mean flow. The parabolized stability equations have been successfully used to estimate the near-field evolution of these modes, but are unable to properly capture the acoustic field. We have recently developed an efficient method for calculating these linear modes that properly captures the acoustic field. The linearized Euler equations are modified such that all upstream propagating acoustic modes are removed from the operator. The resulting equations, called one-way Euler equations, can be stably and efficiently solved in the frequency domain as a spatial initial value problem in which initial perturbations are specified at the flow inlet and propagated downstream by integrating the equations. We demonstrate the accuracy and efficiency of the metho...

Rapid Annihilation of the Kármán Vortex Street behind a Rectangular Cylinder

The annihilation and reappearance of the Karman vortex street behind a rectangular cylinder are investigated by wind-tunnel experiments, numerical simulations and linear stability analyses. Both experiments and numerical simulations show that the Karman vortex street (primary vortex street) is annihilated at a much shorter distance from the cylinder in the wake of a rectangular cylinder with a small aspect ratio of cross section than in the wake of a circular cylinder, and that another vortex street (secondary vortex street) appears further downstream. The locations where the primary vortex street is annihilated and the secondary one appears are determined numerically, and the mechanism underlying the rapid annihilation of the vortex street is examined on the basis of the linear stability theory. We find that the rapid annihilation of the primary vortex street results from the inherent characteristics of the linear instability mode for the steady symmetric flow.

Nonlinear stability, bifurcation and vortical patterns in three-dimensional granular plane Couette flow

AbstractThe effects of three-dimensional (3D) perturbations, having wave-like modulations along both the streamwise and spanwise/vorticity directions, on the nonlinear states of five types of linear instability modes, the nature of their bifurcations and the resulting nonlinear patterns are analysed for granular plane Couette flow using an order-parameter theory which is an extension of our previous work on two-dimensional (2D) perturbations (Shukla & Alam, J. Fluid Mech., vol. 672, 2011b, pp. 147–195). The differential equations for modal amplitudes (the fundamental mode, the mean-flow distortion, the second harmonic and the distortion of the fundamental mode), up to cubic-order in perturbation amplitude, are solved using a spectral-based numerical technique, yielding an estimate of the first Landau coefficient that accounts for the leading-order nonlinear effect on finite-amplitude perturbations. In the near-critical regime of flows, we found evidence of mean-flow resonance, characterized by the divergence of the first Landau coefficient, that occurs due to the interaction/resonance between a linear instability mode and a mean-flow mode. The nonlinear solutions are found to appear via both pitchfork and Hopf bifurcations from the underlying linear instability modes, leading to supercritical nonlinear states of stationary and travelling wave solutions. The subcritical travelling wave solutions have also been uncovered in the linearly stable regimes of flow. It is shown that multiple nonlinear states of both stationary and travelling waves can coexist for a given parameter combination of mean density and Couette gap. The 3D nonlinear solutions persist for a range of spanwise wavenumbers up to ${k}_{z} = O(1)$ that originate from 2D instabilities which occur beyond a moderate value of the mean density. For purely 3D instabilities in dilute flows (having no analogue in 2D flows), the supercritical finite-amplitude solutions persist for a much larger range of spanwise wavenumber up to ${k}_{z} = O(10)$. For all instabilities, the vortical motion on the cross-stream plane has been characterized in terms of the fixed/critical points of the underlying flow field: saddles, nodes (sources and sinks) and vortices have been identified. While the cross-stream velocity field for supercritical solutions in dilute flows contains nodes and saddles, the subcritical solutions are dominated by large-scale vortices in the background of saddle-node-type motions. The latter type of flow pattern also persists at moderate densities in the form of supercritical nonlinear solutions that originate from the dominant 2D instability modes for which the vortex appears to be driven by two nearby saddles. The location of this vortex is found to be correlated with the local maxima of the streamwise vorticity.

Geometrically Nonlinear Stability Study of Self-Anchored Arch-Suspension Bridges

The self-anchored arch-suspension bridge is characterized by innovative structure and beautiful shape. In order to research the mechanical characteristics and stability of this structure that has never been built, the proposed preliminary design of the combined system bridge with a main span of 406m is studied. On the basis of the transfer mechanism of live load based on the deflection theories of suspension bridges and arch bridges, the stability of the combined system bridge is analyzed in this paper. The universal finite element software ANSYS is applied to study the linear elastic stability and the geometrically non-linearity stability under six load cases. The study indicates that the transfer mechanism of live load is determined by the distribution of the stiffness. There is no possibility of in-plane instability under live load. The first-order linear elastic instability mode shows the shape of lateral flexure of the junction at mid-span, and the geometrically nonlinear stability is controlled by the lateral flexure. These conclusions can be of some value for the design, calculation and operation management of this structure.

Convective instabilities in films of binary mixtures

We present a model for the evolution of films of isothermal binary liquid mixtures with a free evolving surface. The model is based on model-H supplemented by appropriate boundary conditions at the free surface and the solid substrate. The equations account for the coupled transport of the concentration of a component (convective Cahn-Hilliard equation) and the momentum (Korteweg-Navier-Stokes equation). The inclusion of convective motion makes surface deflections possible, i.e., the model allows to study couplings between the decomposition of the mixture and the evolving surface corrugations. We present selected steady layered film states for representative polymer mixtures, and show that convective motion favors their destabilization and qualitatively changes the linear instability modes in experimentally accessible ranges of parameters.

Three-dimensional evolution of magnetic and velocity shear driven instabilities in a compressible magnetized jet

The problem of three-dimensional combined magnetic and velocity shear driven instabilities of a compressible magnetized jet modeled as a plane neutral/current double vortex sheet in the framework of the resistive magnetohydrodynamics is addressed. The resulting dynamics given by the stream+current sheet interaction is analyzed and the effects of a variable geometry of the basic fields are considered. Depending on the basic asymptotic magnetic field configuration, a selection rule of the linear instability modes can be obtained. Hence, the system follows a two-stage path developing either through a fully three-dimensional dynamics with a rapid evolution of kink modes leading to a final turbulent state, or rather through a driving two-dimensional instability pattern that develops on parallel planes on which a reconnection+coalescence process takes place.

Eddy diffusivity in convective hydromagnetic systems

An eigenvalue equation, for linear instability modes involving large scales in a convective hydromagnetic system, is derived in the framework of multiscale analysis. We consider a horizontal layer with electrically conducting boundaries, kept at fixed temperatures and with free surface boundary conditions for the velocity field; periodicity in horizontal directions is assumed. The steady states must be stable to short (fast) scale perturbations and possess symmetry about the vertical axis, allowing instabilities involving large (slow) scales to develop. We expand the modes and their growth rates in power series in the scale separation parameter and obtain a hierarchy of equations, which are solved numerically. Second order solvability condition yields a closed equation for the leading terms of the asymptotic expansions and respective growth rate, whose origin is in the (combined) eddy diffusivity phenomenon. For about 10% of randomly generated steady convective hydromagnetic regimes, negative eddy diffusivity is found.

Three-dimensional instabilities and transition of steady and pulsatile axisymmetric stenotic flows

A straight tube with a smooth axisymmetric constriction is an idealized representation of a stenosed artery. We examine the three-dimensional instabilities and transition to turbulence of steady flow, steady flow plus an oscillatory component, and an idealized vascular pulsatile flow in a tube with a smooth 75 % stenosis using both linear stability analysis and direct numerical simulation. Steady flow undergoes a weak Coanda-type wall attachment and turbulent transition through a subcritical bifurcation, leading to hysteretic behaviour with respect to changes in Reynolds number. The pulsatile flows become unstable through a subcritical period-doubling bifurcation involving alternating tilting of the vortex rings that are ejected from the throat with each pulse. These tilted vortex rings rapidly break down through a self-induction mechanism within the confines of the tube. While the linear instability modes for pulsatile flow have maximum energy well downstream of the stenosis, we have established using direct numerical simulation that breakdown can gradually propagate upstream until it occurs within a few tube diameters of the constriction, in agreement with previous experimental observations. At the Reynolds numbers employed in the present study, transition is localized, with relaminarization occurring further downstream. A non-exhaustive investigation has also been undertaken into the receptivity of the axisymmetric shear layer in the idealized physiological pulsatile flow, with the results suggesting it has localized convective instability over part of the pulse cycle.

The extended GrtlerHmmerlin model for linear instability of three-dimensional incompressible swept attachment-line boundary layer flow

A simple extension of the classic Gortler–Hammerlin (1955) (GH) model, essential for three-dimensional linear instability analysis, is presented. The extended Gortler–Hammerlin model classifies all three-dimensional disturbances in this flow by means of symmetric and antisymmetric polynomials of the chordwise coordinate. It results in one-dimensional linear eigenvalue problems, a temporal or spatial solution of which, presented herein, is demonstrated to recover results otherwise only accessible to the temporal or spatial partial-derivative eigenvalue problem (the former also solved here) or to spatial direct numerical simulation (DNS). From a numerical point of view, the significance of the extended GH model is that it delivers the three-dimensional linear instability characteristics of this flow, discovered by solution of the partial-derivative eigenvalue problem by Lin & Malik (1996a), at a negligible fraction of the computing effort required by either of the aforementioned alternative numerical methodologies. More significant, however, is the physical insight which the model offers into the stability of this technologically interesting flow. On the one hand, the dependence of three-dimensional linear disturbances on the chordwise spatial direction is unravelled analytically. On the other hand, numerical results obtained demonstrate that all linear three-dimensional instability modes possess the same (scaled) dependence on the wall-normal coordinate, that of the well-known GH mode. The latter result may explain why the three-dimensional linear modes have not been detected in past experiments; criteria for experimental identification of three-dimensional disturbances are discussed. Asymptotic analysis based on a multiple-scales method confirms the results of the extended GH model and provides an alternative algorithm for the recovery of three-dimensional linear instability characteristics, also based on solution of one-dimensional eigenvalue problems. Finally, the polynomial structure of individual three-dimensional extended GH eigenmodes is demonstrated using three-dimensional DNS, performed here under linear conditions.

On the behaviour of organised disturbances in a turbulent boundary-layer

This paper is in continuation of our earlier work on the role of hydrodynamic stability theory in understanding wall-bounded turbulent flows. Work in this area was pioneered by Malkus, followed by Reynolds, Tiederman and Hussain. Numerical results showed that the linear instability modes are damped, a result also confirmed by our earlier work for the boundary layer flow case. This led to waning of interest in this approach.In the present work the problem is reformulated using an improved nonisotropic model for the stress tensor based on the model of Pope. This improved model does yield unstable wall modes. A wide range of unstable wavenumbers is observed and these unstable modes mimic some of the key features of wall-bounded turbulent flows. Comparisons with experimental data are also presented. The present work keeps alive the question of relevance of stability theory to wall-bounded turbulent flows.

Large-Eddy Simulation of Transition to Turbulence in Boundary Layers

Large-eddy simulation (LES) results for laminar-to-turbulent transition in a spatially developing boundary layer are presented. The disturbances are ingested into a laminar flow through an unsteady suction-and- blowing strip. The filtered, three-dimensional time-dependent Navier-Stokes equations are integrated numerically using spectral, high-order finite-differences, and a three-stage low-storage Runge- Kutta/Crank-Nicolson time-advancement method. The buffer-domain technique is used for the outflow boundary condition. The localized dynamic model used to parametrize the subgrid-scale (SGS) stresses begins to have a significant impact at the beginning of the nonlinear transition (or intermittency) region. The flow structures commonly found in experiments are also observed in the present simulation; the computed linear instability modes and secondary instability lambda- vortex structures are in agreement with the experiments, and the streak-like structures and turbulent statistics compare with both the experiments and theory. The physics captured in the present LES are consistent with the experiments and the full Navier-Stokes simulation (DNS), at a signficant fraction of the DNS cost. A comparison of the results obtained with several SGS models shows that the localized model gives accurate results both in a statistical sense and in terms of predicting the dynamics of the energy- carrying eddies, while requiring fewer ad hoc adjustments than the other models.

Nonlinear interaction of shear flow with a free surface

In this paper the nonlinear evolution of two-dimensional shear-flow instabilities near the ocean surface is studied. The approach is numerical, through direct simulation of the incompressible Euler equations subject to the dynamic and kinematic boundary conditions at the free surface. The problem is formulated using boundary-fitted coordinates, and for the numerical simulation a spectral spatial discretization method is used involving Fourier modes in the streamwise direction and Chebyshev polynomials along the depth. An explicit integration is performed in time using a splitting scheme. The initial state of the flow is assumed to be a known parallel shear flow with a flat free surface. A perturbation having the form of the fastest growing linear instability mode of the shear flow is then introduced, and its subsequent evolution is followed numerically. According to linear theory, a shear flow with a free surface has two linear instability modes, corresponding to different branches of the dispersion relation: Branch I, at low wavenumbers; and Branch II, at high wavenumbers for low Froude numbers, and low wavenumbers for high Froude numbers. Our simulations show that the two branches have a distinctly different nonlinear evolution.Branch I: At low Froude numbers, Branch I instability waves develop strong oval-shaped vortices immediately below the ocean surface. The induced velocity field presents a very sharp shear near the crest of the free-surface elevation in the horizontal direction. As a result, the free-surface wave acquires steep slopes, while its amplitude remains very small, and eventually the computer code crashes suggesting that the wave will break.Branch II: At low Froude numbers, Branch II instability waves develop weak vortices with dimensions considerably smaller than their distance from the ocean surface. The induced velocity field at the ocean surface varies smoothly in space, and the free-surface elevation takes the form of a propagating wave. At high Froude numbers, however, the growing rates of the Branch II instability waves increase, resulting in the formation of strong vortices. The free surface reaches a large amplitude, and strong vertical velocity shear develops at the free surface. The computer code eventually crashes suggesting that the wave will break. This behaviour of the ocean surface persists even in the infinite-Froude-number limit.It is concluded that the free-surface manifestation of shear-flow instabilities acquires the form of a propagating water wave only if the induced velocity field at the ocean surface varies smoothly along the direction of propagation.