Distinct Modes Of Instability Research Articles

Instability-induced patterns and their post-buckling development in soft particulate composites

In this paper, we investigated elastic instabilities in soft particulate composites under finite strains. Through our numerical analysis, we find distinct instability modes developing in the composites upon the critical deformation level. In particular, fully predetermined by its initial geometry, the microstructures transform into (i) strictly doubled periodicity, (ii) seemingly non-periodic state, or (iii) longwave pattern. The latter may give rise to highly ordered domain formations. We analyze the various mechanisms leading to the development of the different instability modes and specifically focus on the seemingly non-periodic one. We illustrate the distinct features of their corresponding eigenmodes obtained from the Bloch-Floquet analysis. We further employ the quasi-convexification analysis and examine the energy landscapes of the soft composites developing the different instability modes. Finally, we examine the development of the predicted instability modes in the post-buckling regime. We find that, depending on the characteristic critical wavenumber, the post-buckling patterns can significantly diverge from those predicted by the Bloch-Floquet analysis. In the post-buckling regime, the instabilities may develop into various scenarios: from a combination of different repeating inclusion sets to disordered, seemingly chaotic patterns.

Sorting and bed waves in unidirectional shallow-water flows

The stability of a uniform flow above an erodible bed composed of a bimodal mixture of sediments is investigated by means of linear analysis. Results show that, for any given set of the flow and sediment parameters, two distinct modes of instability arise, each one characterized by its own wave speed, growth rate and longitudinal wavelength, each one involving spatial variations of both grain size density and bed elevation. Although at a linear level no information on the amplitude of the perturbations is gathered, the analysis of the eigenvectors associated with the two modes of instability allows for an easy classification in terms of the relative amplitudes of the perturbations of bed elevation and size density. One eigenvalue is shown to be associated with the modifications of bed forms induced by the presence of the heterogeneous mixture, such as the local accumulation of finer and coarser material along the unit wavelength, the other with the formation of the low-amplitude sorting waves known as bedload sheets. In the present unidirectional shallow-water framework, only the sorting wave is found to be unstable, since dunes and antidunes, the relevant bed forms for this case, require a more refined rotational flow model in order to become unstable. On the other hand, the simple flow model adopted allows for the formulation of an algebraic eigenvalue problem that can be solved analytically, allowing for a deep insight into the mechanisms that drive both instabilities.

Viscous dissipation effects on the linear stability of Rayleigh-Bénard-Poiseuille/Couette convection

We investigate how viscous dissipation changes the linear stability characteristics of Rayleigh-Bénard-Poiseuille/Couette mixed convection flows. In addition to the contribution of the vertical temperature gradient imposed in the external boundaries, thermal stratification is also a consequence of the volumetric heating induced by viscous dissipation. The governing equations for small perturbations after normal-mode expansion are solved numerically. Results showed that the most unstable perturbations are three dimensional longitudinal rolls. Viscous dissipation induced changes of the critical Rayleigh number and critical wave number at the onset of convection are determined as a function of the governing parameters Λ=GePe2 and Pr, where Ge,Pe and Pr are respectively the Gebhart number, the Péclet number and the Prandtl number. Two distinct modes of instability, with quite different characteristics, can occur. For moderate Λ, the linear instability properties are almost similar to Rayleigh-Bénard convection. When Λ is large enough, the viscous dissipation mechanism induced a strong destabilization and triggered instability even if heating is from above. A thermal energy budget analysis is proposed to understand the physical mechanism underlying the smooth transition between the two regimes of convection. Finally, the obtained results are discussed in connection with existing experiments and with hydrodynamic instability.

Effect of soluble surfactants on the stability of stratified flows through soft-gel-coated walls.

The effect of a soluble surfactant on the linear stability of layered two-phase Poiseuille flows through soft-gel-coated parallel walls is studied in this paper. The focus is on determining the effect of the elastohydrodynamic coupling between the fluids and the soft-gel layers on the various flow instabilities. The fluids are assumed Newtonian and incompressible, while the soft gels are modeled as linear viscoelastic solids. The effect of a soluble surfactant on the different instabilities is specifically investigated. The soft-gel-coated plates are maintained at two different solute concentrations. The dynamics of the soluble surfactant in the fluids is captured using a species transport equation. A linear stability analysis is carried out to identify different instabilities in the system. The linearized governing equations are solved numerically using a Chebyshev spectral Collocation technique. The effect of deformability of the soft gels on three distinct instability modes, (a) a liquid-liquid long-wave mode, (b) a liquid-liquid short-wave mode, and (c) a liquid-liquid Marangoni short-wave mode, is analyzed. An analytical expression for the growth rate is obtained in the long-wave length limit using an asymptotic analysis. From the long-wave analysis a stability map is obtained, in which dominant effects in different regions are identified. The Marangoni stresses can either stabilize or destabilize the interfacial instability depending on the direction of mass transfer. They have a predominantly stabilizing effect on the interfacial instability when the mass transfer is from the more viscous broader fluid to the less viscous thinner fluid. Placing a gel closer to the more viscous fluid has a stabilizing effect on this instability. The Marangoni stresses and soft-gel layers can have opposing effects on the stability of the long-wave mode. The dominant of these two opposing effects is determined by the prevailing parameters. Insights into the dominant physical causes of different instabilities are presented.

Detached-eddy simulation of supersonic flow past a spike-tipped blunt nose

Space vehicle in atmosphere travels mostly at supersonic speed and generates a very strong bow shockwave around its blunt nose. Oblique shock and conical separated flow zone generated by a forward disk-tip spike significantly reduce the drag by reducing the high pressure area on the blunt nose. This study employs improved delayed detached eddy simulation to investigate the characteristic flow structures around a spike-tipped blunt nose at Mach number of 3 and Reynolds number (based on the blunt-body diameter) of 2.72 × 108. The calculated time-averaged quantities agree well with experimental data. Characteristic frequencies in different flow regions are extracted using fast Fourier transform. It is found that two distinct instability modes exist: oscillation mode and pulsation mode. The former is related to the foreshock/turbulence interaction with non-dimensional frequency at around 0.004. The latter corresponds to the interaction between turbulence and shock structures around the blunt nose, with a typical coherent structure shedding frequency at 0.092.

Linear stability of layered two-phase flows through parallel soft-gel-coated walls.

The linear stability of layered two-phase Poiseuille flows through soft-gel-coated parallel walls is studied in this work. The focus is on determining the effect of the elastohydrodynamic coupling between the fluids and the soft-gel layers on the different instabilities observed in flows between parallel plates. The fluids are assumed Newtonian and incompressible, while the soft gels are modeled as linear viscoelastic solids. A long-wave asymptotic analysis is used to obtain an analytical expression for the growth rate of the disturbances. A Chebyshev collocation method is used to numerically solve the general linearized equations. Three distinct instability modes are identified in the flow: (a) a liquid-liquid long-wave mode; (b) a liquid-liquid short-wave mode; (c)a gel-liquid short-wave mode. The effect of deformability of the soft gels on these three modes is analyzed. From the long-wave analysis of the liquid-liquid mode a stability map is obtained, in which four different regions are clearly demarcated. It is shown that introducing a gel layer near the more viscous fluid has a predominantly stabilizing effect on this mode seen in flows between rigid plates. For parameters where this mode is stable for flow between rigid plates, introducing a gel layer near the less viscous and thinner fluid has a predominantly destabilizing effect. The liquid-liquid short-wave mode is destabilized by the introduction of soft-gel layers. Additional instability modes at the gel-liquid interfaces induced by the deformability of the soft-gel layers are identified. We show that these can be controlled by varying the thickness of the gel layers. Insights into the physical mechanism driving different instabilities are obtained using an energy budget analysis.

Linear stability analysis for monami in a submerged seagrass bed

The onset of monami – the synchronous waving of seagrass beds driven by a steady flow – is modelled as a linear instability of the flow. Unlike previous works, our model considers the drag exerted by the grass in establishing the steady flow profile, and in damping out perturbations to it. We find two distinct modes of instability, which we label modes 1 and 2. Mode 1 is closely related to Kelvin–Helmholtz instability modified by vegetation drag, whereas mode 2 is unrelated to Kelvin–Helmholtz instability and arises from an interaction between the flow in the vegetated and unvegetated layers. The vegetation damping, according to our model, leads to a finite threshold flow for both of these modes. Experimental observations for the onset and frequency of waving compare well with model predictions for the instability onset criteria and the imaginary part of the complex growth rate respectively, but experiments lie in a parameter regime where the two modes can not be distinguished.

Transitional hysteresis loop and coexistence of synchronized shedding in coupled wakes

Hysteretic in-phase ↔ anti-phase exchange of vortex shedding and co-existence of reverse-synchronized bistable wake structures past two side-by-side elliptic/circular cylinders are examined through extensive numerical simulations and bifurcation analysis. Wake characteristics and synchronizations past two side-by-side cylinders have often been demarcated in terms of the gap-ratio “G” and the Reynolds number “Re.” The focus here is the “in-phase ↔ anti-phase” two-way transition of oppositely synchronized bistable shedding states. In a remarkable parallel to discontinuous shifts of Strouhal frequency (prompting growth of two distinct instability modes past a single cylinder), the present work reveals interesting in-phase ↔anti-phase transitional switching of vortex shedding past two side-by-side symmetric cylinders, as facilitated by “discontinuous jumps of combined lift-force CL,1+2,” and preceding bistable wake evolution via both of these two reverse-synchronized phases. The hysteresis loops are demarcated (for cylinders of different aspect-ratios A) through extended computations of two anti-synchronized solution branches by slowly increasing/decreasing the Re at fixed gap-ratio (G) and increasing/decreasing G minutely at a constant Re, thereby facilitating transitions and using the computed discontinuous jumps of CL,1+2. Simulations conducted with various A (0.5 ≤ A ≤ 2.0) exhibit, both in-phase and anti-phase shedding co-exist over significantly wide ranges of G-space/Re-space, and the exchange of vortex synchronization at the ends of hysteresis loop occurs through discontinuous variation of the CL,1+2. The “gap-biased” anti-phase → in-phase transition gets gradually delayed, as the cylinder aspect-ratio A is decreased. However, the “Re-biased” in-phase → anti-phase transition is advanced with the decrease of A. The tolerance width “HW” of gap-biased hysteresis loop increases fairly linearly, as A decreased over the range 1.0 ≤ A ≤ 2.0.

Two-dimensional instability of the bottom boundary layer under a solitary wave

The objective of this paper is to establish a detailed map for the temporal instability of the bottom boundary layer (BBL) flow driven by a soliton-like wave induced pressure gradient in a U-tube-shaped tunnel, which serves as an approximation to the BBL under surface solitary waves. Both linear stability analysis and fully nonlinear two-dimensional simulations using high-order numerical methods have been carried out. The process of delineation of the stability regions as a function of boundary layer thickness-based Reynolds number of the temporally evolving base flow, Reδ, consists of two parts. In the first part, we assess the lower limit of the Reδ range within which the standard, quasi-steady, linear stability analysis is applicable when considering individual base flow profiles sampled during its transient evolution. Below this limit, transient linear stability analysis serves as a more accurate predictor of the stability properties of the base flow. In the second step, above the Reδ limit where the BBL is determined to be linearly unstable, the base flow is further classified as unconditionally stable, conditionally unstable, or unconditionally unstable in terms of its sensitivity to the amplitude and the insertion time of perturbations. Two distinct modes of instability exist in this case: post- and pre-flow-reversal modes. At a moderate value of Reδ, both modes are first observed in the wave deceleration phase. The post flow reversal mode dominates for relatively low Re and it is the one observed in Sumer et al. [“Coherent structures in wave boundary layers. Part 2. Solitary motion,” J. Fluid Mech. 646, 207 (2010)]. For Re above a threshold value of the base flow in the unconditionally unstable regime, the pre-flow reversal mode, which is longer in wavelength than its post-reversal counterpart, becomes dominant. In the same regime, the threshold Reδ value above which instability is observed in the acceleration phase of the wave is also identified. In this case, the base flow velocity profiles lack any inflection point, suggesting that the origin of such an instability is viscous. Finally, the lower Reδ limit above which quasi-steady linear stability analysis is valid may be independently obtained by adapting to the surface solitary wave BBL, an instability criterion which links the average growth rate and wave event timescale, as previously proposed in studies of the instability of the interior of progressive and solitary internal waves.

Physical and numerical aspects of the high-speed unsteady flow around concave axisymmetric bodies

The axisymmetric concave body is a typical configuration about which shock/shock interactions appear. Various shapes of axisymmetric concave bodies are used in a variety of applications in aeronautics, for example, axisymmetric jet inlets with conical centerbody, ballistic missiles drag reduction by spike, plasma or hot gas injection, parachutes for pilot-ejection capsules. However, it is well known that two distinct modes of instability appear around a concave body in the high-speed flow regime for a certain range of geometric parameters. These instabilities can cause undesirable effects such as severe vibration of the structure, heating and pressure loads. According to the experimental evidence, the unsteady flow is characterised by periodic radial inflation and collapse of the conical separation bubble formed around the forebody (pulsation). Various explanations have been given for the driving mechanism of the instabilities. In the present, merging of the leading explanations is done, and basic rules for the passive suppression of the instabilities are applied, in order to enforce their proposed driving. In addition, the effect of the flow initialisation method on the flow structure predicted by numerical simulations is examined. For certain configurations, bifurcation of the time-dependent flow has been found. This behaviour is explained with recourse to the phenomenon of hysteresis, which is an inherent feature of the examined flows.

Galerkin analysis of kinematic dynamos in the von Kármán geometry

We investigate dynamo action by solving the kinematic dynamo problem for velocity fields of the von Kármán type between two coaxial counter-rotating propellers in a cylinder. A Galerkin method is implemented that takes advantage of the symmetries of the flow and their subsequent influence on the nature of the magnetic field at the dynamo threshold. Distinct modes of instability have been identified that differ by their spatial and temporal behaviors. Our calculations give the result that a stationary and antisymmetric mode prevails at the dynamo threshold. We then present a quantitative analysis of the results based on the parametric study of four interaction coefficients obtained by reduction of our initially large eigenvalue problem. We propose these coefficients to measure the relative importance of the different mechanisms at play in the von Kármán kinematic dynamo.

Instability and Dynamics of Thin Liquid Bilayers

Interfacial instability of two layers of thin (<100-nm) immiscible liquid films on a solid substrate is studied using the long-wave equations. Instability is derived from the van der Waals interactions among the substrate and the films. The stability characteristics are classified on the basis of the different combinations of surface tensions of the liquid layers and the solid. Depending on the surface tensions and the film thicknesses, two distinct modes of instability, namely, squeezing and bending, are manifested. Nonlinear simulations are presented to describe the various modes of initial evolution and film rupture in such systems.

The temporal stability of a developing jet: a model problem

AbstractThe temporal instability of a developing swirling incompressible jet is considered. The jet development (in the streamwise direction) is modelled by combining a near-field and far-field approximation to the jet velocity profile into a one parameter family of basic velocity fields. The single parameter in the jet velocity field then allows us to model the radial spreading of the jet and the decay of swirl observed experimentally. Two distinct modes of instability of this model profile are found. The first is that found from a stability analysis of a fully developed swirling jet in the far field whilst the second is relevant to a “top-hat” jet with an imposed rigid body rotation. We demonstrate that the effect of azimuthal swirl is to destabilise both modes of instability. Additionally our results suggest that the near-nozzle modes of instability will dominate; indeed the growth rates of these modes are significantly larger than those found from previous studies of a fully developed jet in the far-field region.

Primary and secondary instabilities of the asymptotic suction boundary layer on a curved plate

The temporal growth of Görtler vortices and the associated secondary instability mechanisms are investigated numerically in the case of an asymptotic suction boundary layer on a curved plate. Highly inflectional velocity profiles are generated in both the spanwise and vertical directions. The inflectional velocity profile develops earlier in the spanwise direction. There exist two distinct modes of instability that eventually lead to the breakdown of Görtler vortices: the sinuous mode and the varicose mode. The temporal secondary instability analysis of the three-dimensional inflectional velocity profile reveals that the sinuous mode becomes unstable earlier than the varicose mode. The sinuous mode is shown to be primarily related to shear in the spanwise direction, ∂U/∂z, and the varicose mode to shear in the vertical direction, ∂U/∂y.

The observation of the simultaneous development of a long- and a short-wave instability mode on a vortex pair

Experiments on the stability of vortex pairs are described. The vortices (ratio of length to core diameter L/c of up to 300) were generated at the edge of a flat plate rotating about a horizontal axis in water. The vortex pairs were found to be unstable, displaying two distinct modes of instability. For the first time, as far as it is known to the authors, a long-wave as well as a short-wave mode of instability were observed to develop simultaneously on such a vortex pair. Experiments involving single vortices show that these do not develop any instability whatsoever. The wavelengths of the developing instability modes on the investigated vortex pairs are compared to theoretical predictions. Observed long wavelengths are in good agreement with the classic symmetric long-wave bending mode identified by Crow (1970). The developing short waves, on the other hand, appear to be less accurately described by the theoretical results predicted, for example, by Windnall, Bliss &amp; Tsai (1974).

The flow over a backward-facing step under controlled perturbation: laminar separation

The flow over a backward-facing step with laminar separation was investigated experimentally under controlled perturbation for a Reynolds number of 11000, based on a step height h and a free-stream velocity UO. The reattaching shear layer was found to have two distinct modes of instability: the ‘shear layer mode’ of instability at Stθ ≈ 0.012 (Stθ ≡ fθ/UO, θ being the momentum thickness at separation and f the natural roll-up frequency of the shear layer); and the ‘step mode’ of instability at Sth ≈ 0.185 (Sth ≡ fh/U0). The shear layer instability frequency reduced to the step mode one via one or more stages of a vortex merging process. The perturbation increased the shear layer growth rate and the turbulence intensity and decreased the reattachment length compared to the unperturbed flow. Cross-stream measurements of the amplitudes of the perturbed frequency and its harmonics suggested the splitting of the shear layer. Flow visualization confirmed the shear layer splitting and showed the existence of a low-frequency flapping of the shear layer.

The linear inviscid secondary instability of longitudinal vortex structures in boundary layers

The inviscid instability of a longitudinal vortex structure within a steady boundary layer is investigated. The instability has wavelength comparable with the boundary-layer thickness so that a quasi-parallel approach to the instability problem can be justified. The generalization of the Rayleigh equation to such a flow is obtained and solved for the case when the vortex structure is induced by curvature. Two distinct modes of instability are found; these modes correspond with experimental observations on the breakdown process for Görtler vortices.

Magnetic buoyancy instabilities incorporating rotation

Plane layer stability analysis is extended to incorporate the effects of uniform rotation. Detailed studies are made of the interchange, or 'axisymmetric modes' and of the undular or wavelike motions, considering both high and low frequency modes. The force due to rotation on the fluid motions are determined. The types of instability relevant to astrophysical problems, are also discussed with attention given to the occurrence of two distinct modes of instability in a bottom heavy field gradient. Some peculiar stability boundaries, which are due to the absence of a dominating force, are described.

Low wavenumber instability on the equatorial beta‐plane

Unstable waves on the equatorial beta‐plane are studied when the mean flow U(y) is a function of latitude only, which permits separation of variables. The additional restrictions of (i) a linear shear U(y) = Sy where S is a constant and y is non‐dimensional latitude and (ii) small zonal wavenumber k permit simple approximations that allow identification of two distinct modes of instability. The "mixed Kelvin‐inertial" mode is an unstable Kelvin wave for |S| &lt; 2 but an inertial instability for larger shear. The "mixed gravity‐Kelvin" mode is the neutral, n=0 positive frequency gravity wave for small shear but an unstable Kelvin wave for |S| &gt; 2. In contrast to normal hydrodynamics problems, there is no minimum shear for instability but instead Im(c) is proportional to exp[−5.34/|S|] for small S. Although the inertial instability was known previously, the two branches of unstable Kelvin waves and the connection between them and the neutral and inertially unstable gravity waves are original discoveries. For weak shear, only the Kelvin wave is unstable.

The Double-Diffusive Modes of Symmetric Instability on an Equatorial Beta-Plane

Abstract The symmetric instability due to horizontal shear on an equatorial beta-plane exhibits two distinct modes of instability. The classical monotonic non-oscillatory instability exists for all Prandtl numbers but is favored when the Prandtl number is approximately less than 3/2. For values of Prandtl number approximately larger than this we find that an oscillating “overstability” is the preferred mode of instability. This result contrasts with the baroclinic centrifugally stable case in which overstabilities exist but are never preferred. Similar results can be demonstrated analytically on an artificially bounded f-plane which mimics the finite latitudinal scale imposed by the equatorial beta-plane geometry. Radiative relaxation would favor the monotonic mode, but the effect might be insignificant if breaking internal gravity waves are present.