Three-dimensional Linear Instability Research Articles

Electrohydrodynamic instability in an annular liquid layer with radial conductivity gradients.

In this paper, the electrohydrodynamic stability in an annular liquid layer with a radial electrical conductivity gradient is investigated. A weak shear flow arises from a constant pressure gradient in the axial direction. In the radial direction, an electric field is applied. The three-dimensional linear instability analysis is implemented to study the influence of the inner radius, electrical conductivity gradient, shear flow, and ionic diffusion on the dynamics of the fluid layer. It is found that the critical unstable mode may either be oscillatory or stationary. The system becomes more unstable as the dimensionless inner radius a increases. When the inner radius a is small, the critical unstable mode is stationary, while it is given by three-dimensional oblique waves when a is large. When the conductivity gradient is small, the critical unstable mode is the three-dimensional oblique wave, while when the conductivity gradient is large, it would switch to the stationary mode rather than the oscillatory mode. The system becomes more unstable when the Reynolds number is slightly increased from zero. Additionally, it is found that the electrical Schmidt number has dual effects. The liquid layer becomes either more unstable or stable as the electric Schmidt number increases.

砕波の三次元渦構造, フィンガージェットの形成から飛沫の放出に至る渦力学的不安定

This paper presents vorticity instabilities occurring during wave breaking process, which interprets formations of the initial three-dimensional vortex structure, scarifying and fingering surface via surface-vorticity interactions, and the sea-spray fragmenting from the finger jets. Three-dimensional linear instability analyses are performed for identifying these instability modes to describe a series of local surface deformations featuring the splashing process in this study.

Linear Instability Analysis of an Annular Swirling Viscous Liquid Jet

A three-dimensional linear instability analysis was carried out for an annular swirling viscous liquid jet with solid vortex swirl velocity profile. An analytical form of dispersion relation was derived and then solved by a direct numerical procedure. A parametric study was performed to explore the instability mechanisms that affect the maximum spatial growth rate. It is observed that the liquid swirl enhances the breakup of liquid sheet. The surface tension stabilizes the jet in the low velocity regime. The aerodynamic force intensifies the developing of disturbance and makes the jet unstable. Liquid viscous force holds back the growing of disturbance and the makes the jet stable, especially in high liquid velocity regime.

Three-dimensional linear instability in pressure-driven two-layer channel flow of a Newtonian and a Herschel–Bulkley fluid

The three-dimensional linear stability characteristics of pressure-driven two-layer channel flow are considered, wherein a Newtonian fluid layer overlies a layer of a Herschel–Bulkley fluid. We focus on the parameter ranges for which Squire’s theorem for the two-layer Newtonian problem does not exist. The modified Orr–Sommerfeld and Squire equations in each layer are derived and solved using an efficient spectral collocation method. Our results demonstrate the presence of three-dimensional instabilities for situations where the square root of the viscosity ratio is larger than the thickness ratio of the two layers; these “interfacial” mode instabilities are also present when density stratification is destabilizing. These results may be of particular interest to researchers studying the transient growth and nonlinear stability of two-fluid non-Newtonian flows. We also show that the “shear” modes, which are present at sufficiently large Reynolds numbers, are most unstable to two-dimensional disturbances.

Stability analysis in spanwise-periodic double-sided lid-driven cavity flows with complex cross-sectional profiles

Three-dimensional linear instability analyses are presented of steady two-dimensional laminar flows in the lid-driven cavity defined by [15] and further analyzed in the present volume [1], as well as in a derivative of the same geometry. It is shown that in both of the geometries considered three-dimensional BiGlobal instability leads to deviation of the flow from the two-dimensional solution; the analysis results are used to define low- and high-Reynolds number solutions by reference to the flow physics. Critical conditions for linear global instability and neutral loops are presented in both geometries.

Three-Dimensional Linear Instability Analysis of Thermocapillary Return Flow on a Porous Plane

A three-dimensional linear instability analysis of thermocapillary convection in a fluid-porous double layer system, imposed by a horizontal temperature gradient, is performed. The basic motion of fluid is the surface-tension-driven return flow, and the movement of fluid in the porous layer is governed by Darcy's law. The slippery effect of velocity at the fluid-porous interface has been taken into account, and the influence of this velocity slippage on the instability characteristic of the system is emphasized. The new behavior of the thermocapillary convection instability has been found and discussed through the figures of the spectrum.

Linear instability of compound jets with nonaxisymmetric disturbances

A three-dimensional linear instability analysis is carried out for a double infinitely, incompressible, and viscous compound jet moving in an inviscid surrounding gas. An analytical form of dispersion relation is derived and then solved by a direct numerical procedure. A detailed parametric study is performed to explore the instability mechanisms that cause the growth of nonaxisymmetric disturbances in compound jets. The results show that the Weber number defined on the outer interface between the ambient gas and the shell-liquid layer and the gas-to-shell density ratio both have significant influences on the growth of nonaxisymmetric instability modes, whereas the other parameters including the radius ratio, the core-to-shell density ratio, the Reynolds numbers, and the Weber number defined on the inner interface are relatively less important to the onset of the nonaxisymmetric modes. Particularly, it is found that the nonaxisymmetric sinuous mode may prevail over the axisymmetric mode for a compound jet operating at high Weber number but with low Reynolds number. The instability characteristics exhibit that the aerodynamic drag is the dominant mechanism resulting in the onset of nonaxisymmetric instability in compound jets. The present results complement the existing axisymmetric investigations and give a more complete theoretical understanding for the instability behaviors of compound jets.

Survey of instability thresholds of flow between exactly counter-rotating disks

The three-dimensional linear instability of axisymmetric flow between exactly counter-rotating disks is studied numerically. The dynamics are governed by two parameters, the Reynolds number . Axisymmetric instabilities are also calculated; these may be stationary or Hopf bifurcations. Their thresholds are always higher than those of non-axisymmetric modes.

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.

Three-dimensional instability of Kirchhoff’s elliptic vortex

The three-dimensional linear instability of Kirchhoff’s elliptic vortex in an inviscid incompressible fluid is investigated numerically. Any elliptic vortex is shown to be unstable to an infinite number of short-wave bending modes, with azimuthal wave number m=1. In the limit of small ellipticity, the axial wave number of each unstable mode approaches the value obtained by the asymptotic theory of Vladimirov and Il’in, indicating that the instability is caused by a resonance phenomena. As the ellipticity increases, the bandwidth broadens and neighboring bands overlap each other. The maximum growth rate of each mode, except for that of the longest one, agrees fairly with that of the elliptical instability modified by the influence of a Coriolis force. The growth rate of these three-dimensional modes are larger than those of the two-dimensional modes when ellipticity is smaller than a certain value.

Three-Dimensional Linear Instability Modeling of the Cloud Level Venus Atmosphere

Based on the success of several two-dimensional (latitude, longitude) linear barotropic instability models at matching some of the observed characteristics of the cloud level, polar region of the Venus atmosphere, a more realistic, linear, three-dimensional (height, latitude and longitude) model has been developed to further test the hypothesis that the observed features can be described by linear instability theory. The approach taken is to vary the model input parameters to see whether it is possible to produce modes that resemble the observations of wave activity and to compare those input parameters with other observations of the mean state. Sensitivity studies show that in addition to a well-documented dependence on the mean zonal wind, the growth and propagation of unstable modes depends on the latitude variation of the mean temperature (and hence static stability). These studies have led to the specification of a model basic state wind and temperature field that produces modes which are matched to observations of spatial structure, preferred wavenumber and phase speed of the polar disturbances. Wavenumber 2 is found to have the shortest growth time and unlike the two-dimensional results, wavenumbers 1-3 share a nearly common period of about 3 days. The derived basic state has a temperature field that is quite similar to Pioneer Venus observations; however, in some regions the model basic state wind field departs from cyclostrophic values based on temperature observations.

Stability bounds on turbulent Poiseuille flow

For steady-state turbulent flows with unique mean properties, we determine a sense in which the mean velocity is linearly supercritical. The shear-turbulence literature on this point is ambiguous. As an example, we reassess the stability of mean profiles in turbulent Poiseuille flow. The Reynolds & Tiederman (1967) numerical study is used as a starting point. They had constructed a class of one-dimensional flows which included, within experimental error, the observed profile. Their numerical solutions of the resulting Orr-Sommerfeld problems led them to conclude that the Reynolds number for neutral infinitesimal disturbances was twenty-five times the Reynolds number characterizing the observed mean flow. They found also that the first nonlinear corrections were stabilizing. In the realized flow, this latter conclusion appears incompatible with the former. Hence, we have sought a more complete set of velocity profiles which could exhibit linear instability, retaining the requirement that the observed velocity profile is included in the set. We have added two dynamically generated modifications of the mean. The first addition is a fluctuation in the curvature of the mean flow generated by a Reynolds stress whose form is determined by the neutrally stable Orr-Sommerfeld solution. We find that this can reduce the stability of the observed flow by as much as a factor of two. The second addition is the zero-average downstream wave associated with the above Reynolds stress. The three-dimensional linear instability of this modification can even render the observed flow unstable. Those wave amplitudes that just barely will ensure instability of the observed flow are determined. The relation of these particular amplitudes to the limiting conditions admitted by an absolute stability criterion for disturbances on the mean flow is found. These quantitative results from stability theory lie in the observationally determined Reynolds-Tiederman similarity scheme, and hence are insensitive to changes in Reynolds number.

Three-Dimensional Linear Instability on a Sphere: Resolution Experiments with a Model Using Vertical Orthogonal Basis Functions

A new form of the linear, quasi-geostrophic model is derived on a sphere. The new feature is the use of empirically defined orthogonal basis functions (OBFs) to represent the vertical structure of the perturbation solutions. The prescribed basic state is expressed using a third vertical structure function. Spherical harmonics are used for the horizontal structure. The nonseparablc eigenvalue problem is derived. Solutions are presented using various vertical OBFs, basic flows (both zonally varying and zonally uniform) and horizontal truncations(both rhomboidal and triangular). One OBF (labeled "MOBF') is patterned after the structure found in the most unusable solution of a simpler problem. Another OBF (labeled "2-L") is intended to simulate a two-layer model. Increasing the horizontal resolution from RI5 (rhomboidal truncation at zonal wavenumber 15) to R30 is found to decrease the growth rates in nearly all cas. (One exception is solid body rotation.) Surprisingly high resolution is needed to properly represent the instability of most of the basic flow jets studied herein. For some of these flows, R30 may not be high-enough resolution. In none of the flows examined did we conclude that RI 5 was adequate. The phase speeds in the "MOBF" cases are frequently much faster.than the most unstable modes in the "2-L" cases. In a few instances, the "2-U" version of the model obtains nearly stationary, rapidlygrowing modes whose counterpart is not found in the "MOBF" model. Initially, our results suggested that much higher resolution may be needed than suggested by a previous researcher. This contradiction was seemingly resolved by our obse~vatlon that the convergence to the correct solution was faster when the basic jet was centered at a lower latitude. Some implications for low-resolution general circulaton models are made.

Three-dimensional linear instability of parallel shear flows

The three-dimensional linear instability of parallel shear flows is investigated through simple geometrical constructions based on the classical Squire transformation. It is shown that if a profile is unstable for a finite Reynolds number and some value of (total) horizontal wavenumber, this profile is also unstable for all larger Reynolds numbers and the same wavenumber. In particular, the direction of the unstable mode tends toward the perpendicular as the Reynolds number increases, thus providing the viscous/shear balance required for resistive instability. The example of plane Poiseuille flow is discussed in detail, and the results compared with classical viscous and inviscid stability criteria.