Stability Of Shear Flows Research Articles

Linear stability analysis of parallel shear flows for an inviscid generalized two-dimensional fluid system

The linear stability of parallel shear flows for an inviscid generalized two-dimensional (2D) fluid system, the so-called α turbulence system, is studied. This system is characterized by the relation q = −( − Δ)α/2ψ between the advected scalar q and the stream function ψ. Here, α is a real number not exceeding 3 and q is referred to as the generalized vorticity. In this study, a sufficient condition for linear stability of parallel shear flows is derived using the conservation of wave activity. A stability analysis is then performed for a sheet vortex that violates the stability condition. The instability of a sheet vortex in the 2D Euler system (α = 2) is referred to as a Kelvin–Helmholtz (KH) instability; such an instability for the generalized 2D fluid system is investigated for 0 < α < 3. The sheet vortex is unstable in the sense that a sinusoidal perturbation applied to it grows exponentially with time. The growth rate is finite and depends on the wavenumber of the perturbation as k3 − α for 1 < α < 3, where k is the wavenumber of the perturbation. In contrast, for 0 < α ⩽ 1, the growth rate is infinite. In other words, a transition of the growth rate of the perturbation occurs at α = 1. A physical model for KH instability in the generalized 2D fluid system, which can explain the transition of the growth rate of the perturbation at α = 1, is proposed.

Hydromagnetic Stability of Homogeneous Shear Flows in Sea Straits Type Region of Arbitrary Cross Sections

In this paper, the hydromagnetic stability of homogeneous shear flows in sea straits type region of arbitrary cross section has been discussed. A weak m agnetic field is applied parallel to the flow of incompressible fluid. Different bounds of unstable mode s have been obtained.

Eigenvalue problems modelling the stability of a plane-parallel shear in a two-layer viscous composite

We present the derivation of a precise statement of a linearized eigenvalue problem that models the stability of a plane-parallel shear flow in a composite consisting of two different layers of heavy Newtonian media. The peculiarities of the influence on the stability of five dimensionless criteria participating in the equations and boundary conditions are discussed. The specific feature of the power conditions of contact is that a spectral parameter α enters these conditions nonlinearly.

Flow shear stabilization of rotating plasmas due to the Coriolis effect

A radially decreasing toroidal rotation frequency can have a stabilizing effect on nonaxisymmetric magnetohydrodynamic (MHD) instabilities. We show that this is a consequence of the Coriolis effect that induces a restoring pressure gradient force when plasma is perturbed radially. In a rotating cylindrical plasma, this Coriolis-pressure effect is canceled by the centrifugal effect responsible for the magnetorotational instability. In a magnetically confined toroidal plasma, a large aspect ratio expansion shows that only half of the effect is canceled. This analytical result is confirmed by numerical computations. When the plasma rotates faster toroidally in the core than near the edge, the effect can contribute to the formation of transport barriers by stabilizing MHD instabilities.

Global gyrokinetic turbulence simulations of MAST plasmas

Electrostatic gyrokinetic analyses are presented for an L-mode discharge with an internal transport barrier, from the spherical tokamak, MAST. Local and global microstability analysis finds similar linear growth rates for ion temperature gradient (ITG) driven modes. When the electron response is assumed to be adiabatic, growth rates are found to be lower than the experimental E × B flow shearing rate. Including kinetic electrons, without collisions, increases the ITG growth rates above the flow shearing rate, and these modes are found to be linearly unstable in the outer part of the plasma only. In global simulations the flow shear stabilization is found to be asymmetric with respect to the direction of the flow: there is a small destabilizing effect at low flow shear when the flow is in the co-direction.Global non-linear simulations with kinetic electrons and including the flow shear effects predict turbulent ion heat transport that is well above the neoclassical level in the region outside the internal transport barrier in this MAST plasma. In non-linear simulations we also find turbulence extending from the outer part of the plasma into the linearly stable core region.

Stability of quasi-Keplerian shear flow in a laboratory experiment

Context: Subcritical transition to turbulence has been proposed as a source of turbulent viscosity required for the associated angular momentum transport for fast accretion in Keplerian disks. Previously cited laboratory experiments in supporting this hypothesis were performed either in a different type of flow than Keplerian or without quantitative measurements of angular momentum transport and mean flow profile, and all of them appear to suffer from Ekman effects, secondary flows induced by nonoptimal axial boundary conditions. Such Ekman effects are expected to be absent from astronomical disks, which probably have stress-free vertical boundaries unless strongly magnetized. Aims: To quantify angular momentum transport due to subcritical hydrodynamic turbulence, if exists, in a quasi-Keplerian flow with minimized Ekman effects. Methods: We perform a local measurement of the azimuthal--radial component of the Reynolds stress tensor in a novel laboratory apparatus where Ekman effects are minimized by flexible control of axial boundary conditions. Results: We find significant Ekman effects on angular momentum transport due to nonoptimal axial boundary conditions in quasi-Keplerian flows. With the optimal control of Ekman effects, no statistically meaningful angular momentum transport is detected in such flows at Reynolds number up to two millions. Conclusions: Either a subcritical transition does not occur, or, if a subcritical transition does occur, the associated radial transport of angular momentum in optimized quasi-Keplerian laboratory flows is too small to directly support the hypothesis that subcritical hydrodynamic turbulence is responsible for accretion in astrophysical disks. Possible limitations in applying laboratory results to astrophysical disks due to experimental geometry are discussed.

On the Stability of Inviscid Parallel Shear Flows with a Free Surface

We consider the linear stability of inviscid parallel shear flow with a free surface. Gravity is included in the analysis, but surface tension is not. We focus on several specific profiles including Poiseuille flow and various profiles with inflection points. Instabilities are found in all the profiles we studied. At neutral limiting modes, the wave speed of the disturbance is equal to one of three choices: the fluid speed at the bottom, an extremum of the fluid speed, or an inflectional value. The results help to clarify conflicting claims in prior literature.

Instability and hydraulics of turbulent stratified shear flows

AbstractThe Taylor–Goldstein (T–G) equation is extended to include the effects of small-scale turbulence represented by non-uniform vertical and horizontal eddy viscosity and diffusion coefficients. The vertical coefficients of viscosity and diffusion, ${A}_{V} $ and ${K}_{V} $, respectively, are assumed to be equal and are expressed in terms of the buoyancy frequency of the flow, $N$, and the dissipation rate of turbulent kinetic energy per unit mass, $\varepsilon $, quantities that can be measured in the sea. The horizontal eddy coefficients, ${A}_{H} $ and ${K}_{H} $, are taken to be proportional to the dimensionally correct form, ${\varepsilon }^{1/ 3} {l}^{4/ 3} $, found appropriate in the description of horizontal dispersion of a field of passive markers of scale $l$. The extended T–G equation is applied to examine the stability and greatest growth rates in a turbulent shear flow in stratified waters near a sill, that at the entrance to the Clyde Sea in the west of Scotland. Here the main effect of turbulence is a tendency towards stabilizing the flow; the greatest growth rates of small unstable disturbances decrease, and in some cases flows that are unstable in the absence of turbulence are stabilized when its effects are included. It is conjectured that stabilization of a flow by turbulence may lead to a repeating cycle in which a flow with low levels of turbulence becomes unstable, increasing the turbulent dissipation rate and so stabilizing the flow. The collapse of turbulence then leads to a condition in which the flow may again become unstable, the cycle repeating. Two parameters are used to describe the ‘marginality’ of the observed flows. One is based on the proximity of the minimum flow Richardson number to the critical Richardson number, the other on the change in dissipation rate required to stabilize or destabilize an observed flow. The latter is related to the change needed in the flow Reynolds number to achieve zero growth rate. The unstable flows, typical of the Clyde Sea site, are relatively further from neutral stability in Reynolds number than in Richardson number. The effects of turbulence on the hydraulic state of the flow are assessed by examining the speed and propagation direction of long waves in the Clyde Sea. Results are compared to those obtained using the T–G equation without turbulent viscosity or diffusivity. Turbulence may change the state of a flow from subcritical to supercritical.

Dynamics of vorticity defects in stratified shear flow

AbstractWe consider the linear stability and nonlinear evolution of two-dimensional shear flows that take the form of an unstratified plane Couette flow that is seeded with a localized ‘defect’ containing sharp density and vorticity variations. For such flows, matched asymptotic expansions furnish a reduced model that allows a straightforward and computationally efficient exploration of flows at sufficiently high Reynolds and Péclet numbers that sharp density and vorticity gradients persist throughout the onset, growth and saturation of instability. We are thereby able to study the linear and nonlinear dynamics of three canonical variants of stratified shear instability: Kelvin–Helmholtz instability, the Holmboe instability, and the lesser-considered Taylor instability, all of which are often interpreted in terms of the interactions of waves riding on sharp interfaces of density and vorticity. The dynamics near onset is catalogued; if the interfaces are sufficiently sharp, the onset of instability is subcritical, with a nonlinear state existing below the linear instability threshold. Beyond onset, both Holmboe and Taylor instabilities are susceptible to inherently two-dimensional secondary instabilities that lead to wave mergers and wavelength coarsening. Additional two-dimensional secondary instabilities are also found to appear for higher Prandtl numbers that take the form of parasitic Holmboe-like waves.

Non-Modal Stability and Optimal Perturbations in Unbounded Granular Shear Flow: Three-Dimensionality and Particle Spin

The effects of three-dimensional perturbations and particle rotations are analyzed on the non-modal stability characteristics of an unbounded granular shear flow for which the stability problem is solved as an initial value problem. A kinetic-theory constitutive model is used that incorporates the spin degrees of freedom along with certain micro-polar effects of granular materials. The singular values of the underlying non-normal linear operator play a central role in the non-modal analysis in contrast to the standard modal analysis where the eigenvalues determine the asymptotic (in)stability of the flow. For linearly stable flows, it is shown that the perturbation energy can be amplified by a few orders of magnitude at short times before decaying in the asymptotic time limit. Optimal perturbations, that correspond to maximum energy growth over all possible initial conditions, are found to be two-dimensional in a smooth granular fluid. The effect of particle rotation has been assessed by varying the tangential restitution coefficient (β) for smooth particles (β = −1) to perfectly rough particles (β = 1), with significant enhancement of maximum energy for rough particles. Since the non-modal mechanism can significantly amplify perturbation energy, this provides a viable alternate route for pattern formation in a sheared granular fluid.

Instability in Stratified Shear Flow: Review of a Physical Interpretation Based on Interacting Waves

Instability in homogeneous and density stratified shear flows may be interpreted in terms of the interaction of two (or more) otherwise free waves in the velocity and density profiles. These waves exist on gradients of vorticity and density, and instability results when two fundamental conditions are satisfied: (I) the phase speeds of the waves are stationary with respect to each other (“phase-locking“), and (II) the relative phase of the waves is such that a mutual growth occurs. The advantage of the wave interaction approach is that it provides a physical interpretation to shear flow instability. This paper is largely intended to purvey the basics of this physical interpretation to the reader, while both reviewing and consolidating previous work on the topic. The interpretation is shown to provide a framework for understanding many classical and nonintuitive results from the stability of stratified shear flows, such as the Rayleigh and Fjørtoft theorems, and the destabilizing effect of an otherwise stable density stratification. Finally, we describe an application of the theory to a geophysical-scale flow in the Fraser River estuary.

On the stability of shear flows in nematics

A theoretical study of the effect of an applied magnetic field on the stability of the flow of nematic slabs subjected to an arbitrary shear is presented. Homeotropic boundary conditions with strong anchoring and a constant magnetic field applied perpendicular to the plates are considered. We discuss the general conditions on the control parameters under which the flow is stable, for a low molecular weight liquid crystal and for a polymer liquid crystal, and obtain estimations of the critical values.

Pellet injection into H-mode ITER plasma with the presence of internal transport barriers

The impacts of pellet injection into ITER type-1 ELMy H-mode plasma with the presence of internal transport barriers (ITBs) are investigated using self-consistent core-edge simulations of 1.5D BALDUR integrated predictive modeling code. In these simulations, the plasma core transport is predicted using a combination of a semi-empirical Mixed B/gB anomalous transport model, which can self-consistently predict the formation of ITBs, and the NCLASS neoclassical model. For simplicity, it is assumed that toroidal velocity for ωE×B calculation is proportional to local ion temperature. In addition, the boundary conditions are predicted using the pedestal temperature model based on magnetic and flow shear stabilization width scaling; while the density of each plasma species, including both hydrogenic and impurity species, at the boundary are assumed to be a large fraction of its line averaged density. For the pellet’s behaviors in the hot plasma, the Neutral Gas Shielding (NGS) model by Milora-Foster is used. It was found that the injection of pellet could result in further improvement of fusion performance from that of the formation of ITB. However, the impact of pellet injection is quite complicated. It is also found that the pellets cannot penetrate into a deep core of the plasma. The injection of the pellet results in a formation of density peak in the region close to the plasma edge. The injection of pellet can result in an improved nuclear fusion performance depending on the properties of pellet (i.e., increase up to 5% with a speed of 1 km/s and radius of 2 mm). A sensitivity analysis is carried out to determine the impact of pellet parameters, which are: the pellet radius, the pellet velocity, and the frequency of injection. The increase in the pellet radius and frequency were found to greatly improve the performance and effectiveness of fuelling. However, changing the velocity is observed to exert small impact.

The linear stability of inviscid shear flow of a vibrationally excited diatomic gas

The problem of the linear stability of plane-parallel shear flows of a vibrationally excited compressible diatomic gas is investigated using a two-temperature gas dynamics model. The necessary and sufficient conditions for stability of the flows considered are obtained using the energy integrals of the corresponding linearized system for the perturbations. It is proved that thermal relaxation produces an additional dissipation factor, which enhances the flow stability. A region of eigenvalues of unstable perturbations is distinguished in the upper complex half-plane. Numerical calculations of the eigenvalues and eigenfunctions of the unstable inviscid modes are carried out. The dependence on the Mach number of the carrier stream, the vibrational relaxation time τ and the degree of non-equilibrium of the vibrational mode is analysed. The most unstable modes with maximum growth rate are obtained. It is shown that in the limit there is a continuous transition to well-known results for an ideal fluid as the Mach number and τ approach zero and for an ideal gas when τ → 0.

Hydromagnetic stability of stratified shear flows in the presence of cross flow

Hydromagnetic stability of stratified shear flows in the presence of cross flow

Generalized Joseph estimates of stability of plane shear flows with scalar nonlinearity

Possible statements of eigenvalue problems generalizing the classical Orr-Sommerfeld problem are given for incompressible stationary flows of non-Newtonian fluids; these problems are interpreted within the mechanics of a continuum as problems of the shear stability of such flows. A schematic diagram of the integral relation method as applied to these flows is described. In the case of an unperturbed Couette flow, Joseph estimates are generalized for any type of rheological curve and domains of guaranteed stability are constructed on a plane with the axes showing Reynolds numbers corresponding to the tangent and secant dynamic viscosity at one point.

Impacts of pellets injected from the low-field side on plasma in ITER

Impacts of pellets injected from the low-field side (LFS) on plasma in ITER are investigated using the 1.5D BALDUR integrated predictive modeling code. In these simulations, the pellet ablation is described using the neutral gas shielding (NGS) model. The pellet ablation model is coupled with the plasma core transport model, which is a combination of the MMM95 anomalous transport model and NCLASS neoclassical transport model. The boundary conditions are assumed to be at the top of the pedestal, in which the pedestal parameters are predicted using a pedestal model based on the theoretical-based pedestal width scaling (either magnetic and flow shear stabilization width scaling, or flow shear stabilization width scaling, or normalized poloidal pressure width scaling) and the infinite-n ballooning mode pressure gradient limit. These pedestal models depend sensitively on the density at the top of the pedestal, which can be strongly influenced by the injection of pellets. The combination of the MMM95 and NCLASS models, together with the pedestal and NGS models, is used to simulate the time evolution of the plasma current, ion and electron temperatures, and density profiles for ITER standard type-I ELMy H-mode discharges during the injection of LFS pellets. It is found that the injection of pellets results in a complicated plasma scenario, especially in the outer region of the plasma and the plasma conditions at the boundary in which the pellet has an impact on increasing the plasma edge density, but reducing the plasma edge temperature. The LFS pellet has a stronger impact on the edge as compared to the center. For fusion performance, the pellet can result in either enhancement or degradation, depending sensitively on the pellet parameters; such as the pellet size, pellet velocity, and pellet frequency. For example, when a series of deuterium pellets with a size of 0.5 cm, velocity of 1 km/s, and frequency of 2 Hz are injected into the ITER plasma from the LFS, the plasma performance, evaluated in terms of Q fusion, can increase to 72% of that before the use of pellets. It is also found that the injection of pellets results in an increase in the ion and electron densities, but does not enhance the central plasma density. On the other hand, it results in the formation of another peak of the plasma density in the outer region near the plasma edge. The formation of the density peak results in the reduction of plasma transports near the edge by decreasing the contributions of ion-temperature-gradient and trapped electron modes, as well as kinetic ballooning modes.

Generation of Ultrafast Jets Through the Implosion of a Nested Conical Wire Array

It is shown that in contrast to the electric pulse power driven implosion of a single conical wire array, the implosion of a nested conical wire array with opposite alternate opening angles can lead to the generation of fast jets, with velocities of the order 108 cm/s. This technique can be applied for the supersonic shear flow stabilization of a dense z-pinch, but possibly also for the fast ignition of a pre-compressed dense deuterium–tritium target.

On the stability of plane parallel viscoelastic shear flows in the limit of infinite Weissenberg and Reynolds numbers

Abstract Elastic effects on the hydrodynamic instability of inviscid parallel shear flows are investigated through a linear stability analysis. We focus on the upper convected Maxwell model in the limit of infinite Weissenberg and Reynolds numbers. We study the effects of elasticity on the instability of a few classes of simple parallel flows, specifically plane Poiseuille and Couette flows, the hyperbolic-tangent shear layer and the Bickley jet. The equation for stability is derived and solved numerically using the spectral Chebyshev collocation method. This algorithm is computationally efficient and accurate in reproducing the eigenvalues. We consider flows bounded by walls as well as flows bounded by free surfaces. In the inviscid, nonelastic case all the flows we study are unstable for free surfaces. In the case of wall bounded flow, there are instabilities in the shear layer and Bickley jet flows. In all cases, the effect of elasticity is to reduce and ultimately suppress the inviscid instability.

On short wave stability and sufficient conditions for stability in the extended Rayleigh problem of hydrodynamic stability

We consider the extended Rayleigh problem of hydrodynamic stability dealing with the stability of inviscid homogeneous shear flows in sea straits of arbitrary cross section. We prove a short wave stability result, namely, if k > 0 is the wave number of a normal mode then k > kc (for some critical wave number kc) implies the stability of the mode for a class of basic flows. Furthermore, if \( K(z) = \frac{{ - (U''_0 - T_0 U'_0 )}} {{U_0 - U_{0s} }} \), where U0 is the basic velocity, T0 (a constant) the topography and prime denotes differentiation with respect to vertical coordinate z then we prove that a sufficient condition for the stability of basic flow is \( 0 < K(z) \leqslant \left( {\frac{{\pi ^2 }} {{D^2 }} + \frac{{T_0^2 }} {4}} \right) \), where the flow domain is 0 ≤ z ≤ D.