Stability Of Shear Flows Research Articles

MHD Kelvin-Helmholtz instability in non-hydrostatic equilibrium

The present work deals with the linear stability of a magnetohydrodynamic shear flow so that a stratified inviscid fluid rotating about a vertical axis when a uniform magnetic field is applied in the direction of the streaming or zonal flow. In geophysical flow, the stability of the flow is determined by taking into account the nonhydrostatic condition depending on Richardson number Ri and the deviation δ from hydrostatic equilibrium.According to Stone (Stone P H 1971 J. Fluid. Mech. 45 659), it is shown that such deviation δ decreases the growth rates of three kinds of instability which can appear as geostrophic (G), symmetric (S) and Kelvin-Helmholtz (K-H) instabilities.To be specific, the evolution of the flow is therefore considered in the light of the influence of magnetic field, particularly, on K-H instability. The results of this study are presented by the linear stability of a magnetohydrodynamic, with horizontal free-shear flow of stratified fluid, subject to rotation about the vertical axis and uniform magnetic field in the zonal direction. Results are discussed and compared to previous works as Chandrasekhar (Chandrasekhar S 1961 Hydrodynamic and hydromagnetic stability(Oxford: Clarendon Press) chapter 11 pp 481–513) and Stone [1].

Experimental and numerical investigation of wave ferrofluid convection

The stability of buoyancy-driven shear flow in an inclined layer of a ferrocolloid is investigated for different values of inclinations and homogeneous longitudinal magnetic fields. Near the onset of Rayleigh convection of ferrofluid layer inclined with respect to gravity, the wave oscillatory regimes were observed in experiments and numerical simulations. Visualization of convection patterns is provided by a temperature-sensitive liquid crystal film. As experiments testify, the origin of traveling wave regimes in ferrofluid is due to concentration gradients caused by gravity sedimentation of the magnetic particles. To study the effects of initial concentration gradient of particles, on convective instabilities, finite volume numerical simulations using a two-phase mixture model were carried out for the same setup. The most fascinating effect in ferrofluid convection is spontaneous formation of localized states, those where the convection chaotically focuses in confined regions and is absent in the remainder of cavity.

Equilibrium Evolution in the ZaP Flow Z-Pinch

Investigations of sheared flow stabilization in a Z-pinch geometry have generated Z-pinch plasmas that exhibit long-lived stability during a quiescent period. Holographic interferometry measurements show a discrete pinch. Heat conduction analysis reveals a high temperature plasma. Internal magnetic fields are measured using Zeeman splitting of impurity carbon line emission. The measurements are consistent with a well-confined pinch plasma.

Effects of a fluctuating sheared flow on cross phase in passive-scalar turbulent diffusion

Transport barriers are key elements concerning energy and particle confinement in fusion devices. They play a fundamental role in the L→H transition observed in most tokamaks' edges. It has been shown that a shear in the E×B velocity could trigger and sustain such a barrier. The E×B velocity shear model has proven to be of great interest in the study of the formation and characteristics of transport barriers. Here we address a particular case of flow shear stabilization, namely the effect of a shear flow on the diffusion of a passive scalar. A shear flow reduces the radial flux (radial transport) Γ of a passive scalar field (we consider the pressure field) via the reduction of the turbulence energy ⟨p2⟩ and/or via the reduction of the cross phase cosδ between the fluctuations of the pressure and velocity fields. We compare our results with those of different analytical models for passive-scalar advection or diffusion [Terry et al., Phys. Rev. Lett. 87, 185001 (2001); Kim and Diamond, Phys. Rev. Lett. 91, 075001 (2003)]. However, these studies yielded contradictory results. The purpose of this study is to shed light on this particular issue using numerical simulations to clarify the role of the reduction of the amplitude of turbulence and cross phase in regulating the radial transport.

The effect of confinement on the stability of two-dimensional shear flows

It has been shown recently that the instability of a two-dimensional wake increases when it is confined in the transverse direction by two flat plates. Confinement causes the transition from convective to absolute instability to occur at lower values of shear. This paper examines this effect comprehensively and concludes that it is due to the constructive interaction of modes with zero group velocity in the wake (or jet) and in the surrounding flow. Maximum instability occurs when the wavenumber of the fundamental mode in the wake (or jet) matches that of the fundamental mode in the surrounding flow. Other regions of high instability occur when the harmonics of one mode interact with the fundamental of the other. This effect is examined at density ratios from 0.001 to 1000. At each density ratio, the confinement which causes maximum absolute instability can be predicted. This study also shows that it is vital to examine the wavenumber of absolutely unstable modes in order to avoid over-predicting the absolute instability. In some situations this wavenumber is vanishingly small and the mode must be discounted on physical grounds.

Edge transport barriers in magnetic fusion plasmas

The present level of understanding of the physics of the formation and sustainment of edge transport barriers in magnetically confined fusion plasmas is presented. The formation of edge transport barriers is studied through evolution of mechanisms which can suppress plasma turbulence and so reduce turbulent driven transport, such as E × B flow shear stabilization of turbulence. Comparisons of theoretical studies with experimental results are described including investigations of zonal flows, which are considered important for saturation and self-regulation of turbulence and turbulence-driven transport. Processes that affect the dynamics and spatial structure of the edge barrier are described with emphasis on the width of the transport barrier. To cite this article: P. Gohil, C. R. Physique 7 (2006).

Experiments on the linear instability of flow in a wavy channel

The effects of wall corrugation on the stability of wall-bounded shear flows have been examined experimentally in plane channel flows. One of the channel walls has been modified by introduction of the wavy wall model with the amplitude of 4% of the channel half height and the wave number of 1.02. The experiment is focused on the two-dimensional travelling wave instability and the results are compared with the theory [J.M. Floryan, Two-dimensional instability of flow in a rough channel, Phys. Fluids 17 (2005) 044101 (also: Rept. ESFD-1/2003, Dept. of Mechanical and Materials Engineering, The University of Western Ontario, London, Ontario, Canada, 2003)]. It is shown that the flow is destabilized by the wall corrugation at subcritical Reynolds numbers below 5772, as predicted by the theory. For the present corrugation geometry, the critical Reynolds number is decreased down to about 4000. The spatial growth rates, the disturbance wave numbers and the distribution of disturbance amplitude measured over such wavy wall also agree well with the theoretical results.

Stability of an MHD shear flow with a piecewise linear velocity profile

In this paper we present the results of the stability analysis of a simple shear flow of an incompressible fluid with a piecewise linear velocity profile in the presence of a magnetic field. In the flow, a finite transitional magnetic-free layer with a linear velocity profile is sandwiched by two semi-infinite regions. One of these regions is magnetic-free and the flow velocity in the region is constant. The other region is magnetic and the fluid in it is quiescent. The magnetic field is constant and parallel to the flow in the transitional layer. The fluid density is constant both in the magnetic as well as the magnetic-free regions, while it has a jump-type discontinuity at the boundary between the transitional layer and the magnetic region. The effect of gravity is included in the model, and it is assumed that the lighter fluid is overlaying the heavier one, thus no Rayleigh-Taylor instability is present. The dispersion equation governing the normal-mode stability of the flow is derived and its properties are analysed. We study stability of two cases: (i) magnetic-free flow in the presence of gravity, and (ii) magnetic flow without gravity. In the first case, the flow stability is controlled by the Rayleigh number, In the second case, the control parameter is the inverse squared Alfvénic Mach number, H . Stability of a particular monochromatic perturbation also depends on its dimensionless wavenumber α . We combine the analytical and numerical approaches to obtain the neutral stability curves in the -plane in the case of the magnetic-free flow, and in the -plane in the case of the magnetic flow. The dependence of the instability increment on R in the first case, and on H in the second case is treated. We apply the results of the analysis to the stability of a strongly subsonic portion of the heliopause. Our main conclusion is as follows: The inclusion of a transitional layer near the heliopause into the model increases by an order of magnitude the strength of the interstellar magnetic field required to stabilize this portion of the heliopause in comparison with the corresponding stabilizing strength of the magnetic field required when modelling the heliopause as a tangential discontinuity.

On Magnetic Field Control Experiments of Ferrofluid Convection Motion

Summary The experiments have been performed on the stability of buoyancy-driven flows of a ferrofluid for the cases of an inclined and vertical orientation of thin cylindrical encloser. Visualization of flow patterns is provided by a temperature-sensitive liquid crystal sheet which serves as the lower surface of a transparent heat exchanger. The local temperature sensor is used for measurement of heat transport across the fluid layer. The results indicate that with the help of an external homogeneous transversal magnetic field it is possible to control the stability of thermogravitational shear flow under vertical orientation of the layer, i.e. induce the thermomagnetic mode of instability with vertical orientation of longitudinal convection rolls. As numerous experimental investigations show the origin of concentration density gradients due to gravity sedimentation of magnetic particles and their aggregates results in traveling wave regimes of Rayleigh and thermomagnetic convection flows.

Effects of hydrophobic surface on stability and transition

The effects of a hydrophobic surface on stability and transition in wall-bounded shear flows are investigated. The hydrophobic surface is represented by a slip-boundary condition on the surface. The linear stability analysis with slip-boundary conditions shows that the critical Reynolds number increases with streamwise slip. The effects of slip-boundary conditions on the transient growth of initial disturbances are investigated through the singular value decomposition (SVD) analysis of the linearized Navier–Stokes equations. The maximum transient growth (i.e., the amplification factor for the optimal disturbance) is reduced with streamwise slip, indicating that non-normality of the linearized Navier–Stokes equations is reduced with streamwise slip. Finally, it is shown that the transition to turbulence is delayed significantly with streamwise slip, whereas spanwise slip induces an earlier transition. The present results suggest that it is desirable to develop a hydrophobic surface with specified directional sensitivity in order to meet a particular need for specific applications.

GLF23 Modeling of Turbulent Transport in DIII-D

During the past decade, there has been significant progress made in our predictive understanding of turbulent transport in tokamaks. Theoretical advances have led to the development of comprehensive theoretical transport models based on drift wave physics. This paper summarizes the development of the GLF23 drift wave transport model, its application to modeling of DIII-D experiments, and burning plasma projections. The model predicts the transport due to ion temperature gradient, trapped electron, and electron temperature gradient modes and includes the effects of E × B shear flow and Shafranov shift stabilization. GLF23 has been successful in predicting the core profiles in a wide variety of discharges. Examples of published results are given along with a discussion of some outstanding physics issues.

Transport Studies in DIII-D with Modulated Heat and Particle Sources

DIII-D has studied thermal and particle transport in International Thermonuclear Experimental Reactor (ITER)-relevant regimes. In order to better distinguish between thermal transport models, it is important to test both the steady-state and time-dependent predictions of models against experimental results. Based on experiments in DIII-D, models containing the full spectral range of drift wave physics from ion temperature gradient to electron temperature gradient modes were in closest agreement with experimental observations. Inclusion of E × B flow shear stabilization effects was found to be important. Although some aspects of the experimental observations were well matched by various models, no individual model did well matching both the equilibrium and time-dependent electron and ion behavior, which clearly indicates that further improvement in transport models is required. Helium transport studies in DIII-D are encouraging for ITER in that they indicate that the measured particle diffusivity is sufficient to remove helium ash fast enough to avoid deleterious fuel dilution, but other factors for ITER such as divertor geometry and pumping speed must also be assessed.

Stability of the inviscid plane couette flow in bubbly fluids

The aim of this article is to study the stability of shear flows in bubbly fluids. A mathematical model of bubbly fluids is presented. The stability of shear flows is studied by two methods: by using a spectral approach and by solving the initial-value problem. It is proved that the linear velocity profile is stable in the long wave approximation.

Stability of magnetohydrodynamic shear flows with and without bounding walls

We have solved the shear flow stability problem of compressible electrically conducting fluid (plasma) in a magnetic field. A comparative analysis is made of the influence of different boundary conditions on shear flow stability. We consider the problems for flows in an unbounded medium and a flow between fixed walls, as well as in the presence of a fixed wall on one side of the shear flow only. Shear flow velocity profiles are treated both in the form of a tangential discontinuity and as being described by a hyperbolic tangent function. For the case of an unbounded flow with a velocity vortex sheet, analytical solutions are found; for all other cases, the solutions are found numerically. Problems are solved for two different positions of the tangential wave vector of oscillations and magnetic field . For shear flows bounded by a fixed wall we found an unstable mode of oscillations produced by the wave, reflected from the wall and transmitted through the shear layer.

On Holmboe’s instability for smooth shear and density profiles

The linear stability of a stratified shear flow for smooth density profiles is studied. This work focuses on the nature of the stability boundaries of flows in which both Kelvin–Helmholtz and Holmboe instabilities are present. For a fixed Richardson number the unstable modes are confined to finite bands between a smallest and a largest marginally unstable wavenumber. The results in this paper indicate that the stability boundary for small wavenumbers is comprised of neutral modes with phase velocity equal to the maximum/minimum wind velocity whereas the other stability boundary, for large wavenumbers, is comprised of singular neutral modes with phase velocity in the range of the velocity shear. We show how these stability boundaries can be evaluated without solving for the growth rate over the entire parameter space as was previously done. The results indicate further that there is a new instability domain that has not been previously noted in the literature. The unstable modes, in this new instability domain, appear for larger values of the Richardson number and are related to the higher harmonics of the internal gravity wave spectrum.

Stability of an oscillatory shear flow in a differentially heated vertical channel

The linear stability of an oscillatory shear flow in a differentially heated vertical channel was investigated numerically for Re = 1000 and Pr = 0.7. The Galerkin method is used to solve the disturbance momentum and energy equations. The results show that the least stable disturbance could be three-dimensional for higher flow oscillation frequency and larger flow oscillation amplitude, while it is two-dimensional in the isothermal oscillatory and heated steady channel flows. The flow oscillation acts to stabilize the flow at moderate and high oscillation frequencies, where the degree of stabilization increases with the oscillation amplitude; but, it acts to destabilize the flow and the amount of destabilization increases with the oscillation amplitude at low oscillation frequency. It is shown from the balance of disturbance kinetic energy budget that shear production is responsible for the flow instability. For the 2-D wave initiated instability, almost all the shear production is generated during a very short time interval at low oscillation frequency, while it is generated during most of the time of a cycle for the 3-D disturbance.

Linear analysis of magnetic and flow shear stabilization of Z-pinch instabilities

A global normal mode stability analysis on the effect of radially sheared azimuthal flow and radially sheared magnetic fields on magnetohydrodynamic (MHD) instabilities in Z-pinch plasmas is presented. A linearized set of ideal MHD equations allows the investigation of plasmas with both magnetic shear and flow shear included in the plasma equilibrium. The stabilizing effects of sheared azimuthal flow and sheared axial magnetic field are presented. Here we show that radial shear in the functional form of a vortex is particularly efficient at reducing instability growth rates.

A note on nonlinear stability of plane parallel shear flows

We present a generalized energy functional E for plane parallel shear flows which provides conditional nonlinear stability for Reynolds numbers Re below some value Re E depending on the shear profile. In the case of the experimentally important profiles, viz. combinations of laminar Couette and Poiseuille flow, Re E is shown to be at least 174.

The effect of stable thermal stratification on shear flow stability

We study the effect of buoyancy on shear flow stability. The vertical body force is induced by a vertical, constant, and positive thermal gradient. A linear stability analysis is carried out, in the spatial framework, focusing on both exponential and transient growth. In both cases, positive thermal stratification is found to stabilize the disturbances.

Hydrodynamic stability of unidirectional shear flow of linear and branched polymeric melts

In this study, the linear stability of homogeneous shear flow of polymer melts has been investigated by using a prototypical reptation-based model, namely, the Pompom model [1]. The hydrodynamic stability characteristics of the flow have been determined by using an eigenvalue analysis. Particular attention has been paid to accurate determination of the discrete and continuous eigenvalues as well as identification and isolation of the spurious eigenvalues. Specifically, it has been shown that the eigenspectrum for the Pompom model has four continuous spectra, three of which are associated with the orientation tensor, S, i.e., one regular and two branch-cuts. One spectrum arises as a result of the stretch, λ evolution equation, which always occurs as the leftmost spectrum. The discrete modes are classified as centered and non-centered eigenvalues, depending on their imaginary parts. It is shown that there is only one pair of non-centered eigenvalues (a non-centered eigenvalue possesses σ i ≠ − α/2, where σ i denotes the imaginary part of the eigenvalue and α denotes the wavenumber in the streamwise direction). For large deformations rates, the real parts of these eigenvalues decay in a similar fashion as the Gorodstov–Leonov eigenvalue pair [2], observed in planar flows of the Upper Convected Maxwell model (UCM). The number of centered eigenvalues however, depends strongly on the choice of parameters. We observe that the leading eigenvalues always belong to the right-most continuous mode (branch-cut), which possess highly singular eigenfunctions. Although, the flow is linearly stable when a maximum in the shear stress versus shear rate curve is not observed, the ballooning of the right most eigenspectrum that arises due to the singular nature of its eigenfunctions could lead to erroneous determination of onset conditions for the instability.