Stability Of Parallel Flows Research Articles

Inviscid stability of parallel flows of a compressible fluid as an initial-value problem

An initial-value problem is posed for the linearised equations governing small disturbances superposed on a parallel flow of an inviscid compressible fluid to obtain a prescription for the proper path of integration across the critical pointy=yc, (whereu=c), in the complexy-plane, associated with the inviscid solutions.

Stability of parallel flow of a multi-component cylindrical plasma to electrostatic perturbations

Sufficient conditions for the stability of parallel flow of a warm N-component cylindrical plasma to electrostatic perturbations are obtained. In the unperturbed state the jth plasma component is assumed to have axial velocity Vj0(r), r being the radial co-ordinate, and the equilibrium quantities are permitted to be arbitrary functions of r consistent with the zeroth-order equations. The L2-norms of certain system variables are shown to be bounded uniformly in time. Circle theorems are obtained for the complex eigenfrequencies of any normal mode.

Stability of shear flow of stratified fluids with fine dust

The linear stability of plane parallel shear flows of a stably stratified incompressible fluid laden with uniformly distributed fine dust particles is studied. It is shown that the Miles criterion for stability, the Rayleigh–Synge criterion for instability, and Howard’s semicircle theorem can be extended under the assumptions that the mass concentration is very small and that the relaxation time of the dust particles is very much less than the time scale characterizing the basic flow. The effect of fine dust is found to increase the region of instability. In addition, a semi-ellipse criterion for instability has been given as an improvement over Howard’s semicircle criterion.

Stability of parallel flow of a multicomponent plasma in a transverse magnetic field

A Liapunov functional is constructed for the plane parallel flow of a warm multicomponent plasma immersed in a transverse magnetic field. Sufficient conditions for stability to electrostatic perturbations are obtained. The L2-norms of certain system variables are shown to be bounded uniformly in time.

Spatial stability of the non-parallel Bickley jet

The spatial stability of the plane, two-dimensional jet flow to infinitesimal disturbances is investigated by taking into account the effects of transverse velocity component and the streamwise variations of the basic flow and of the disturbance amplitude, wave-number and spatial growth rate. This renders the growth rate dependent on the flow variable as well as on the transverse and streamwise co-ordinates. Growth rates for the energy density of the disturbance and the associated neutral curves are provided as a function of the streamwise co-ordinate. Variation of growth rate of the disturbance stream function and streamwise component of velocity with the transverse co-ordinate is also given for different disturbance frequencies and streamwise locations. Results are compared with those for the parallel-flow stability analysis, and also with those for an analysis that accounts for only some of the non-parallel effects. It is found that the critical Reynolds number based on the growth of energy density of the disturbance depends on the streamwise co-ordinate and lies within the range (around 20) found experimentally, while the parallel-flow theory yields a rather low value of 4·0.

The stability of parallel flow of a multi-component plasma

A Liapunov functional is constructed for the plane parallel flow of a multicomponent plasma in an external gravitational field. Sufficient conditions for stability to electrostatic perturbations are obtained. The L2-norms of certain system variables are shown to be bounded uniformly in time.

Nonparallel stability analysis of axisymmetric stagnation-point flow

The method of multiple scales has been used to extend linear-stability theory to nonparallel axisymmetric boundary-layer flow in the neighborhood of the stagnation point on a blunt body of revolution. The extended theory includes nonparallel planar effects as well as the effects of such possibly destabilizing three-dimensional mechanisms as the stretching of vortex filaments in the rapidly diverging boundary-layer flow. These mechanisms have not been previously incorporated into stability theory. Stability predictions based on the extended theory are found to be in close agreement with predictions of parallel-flow stability theory based on the Orr–Sommerfeld equation; axisymmetric and planar nonparallel effects are concluded to be of negligible importance in the axisymmetric stagnation-point boundary layer.

Inviscid stability of magnetogasdynamic parallel flows

Inviscid stability of magnetogasdynamic parallel flows

Inviscid parallel flow stability with mean profile distortion

Inviscid parallel flow stability with mean profile distortion

Nonlinear Stability of Parallel Flows by High-Ordered Amplitude Expansions

* * The Landau-Stuart theory and its subsequent modifications suffer from some restrictions and from the nonuniqueness in determining higher-order terms of the amplitude expansions, which limit the range of applicability as well as the validity of the results. In the present paper, a well-defined amplitude is introduced a priori. In this way, uniqueness for terms of any order is achieved. Moreover, Watson's method is no longer restricted to almost neutral disturbances. This offers not only more accurate approximations and numerical studies on convergence but, as a consequence, a whole series of new applications. As a first example, the nonlinear equilibrium states of the plane Poiseuille flow are investigated. The numerical results are discussed in context with the author's solutions of the nonlinear equations and with special emphasis on the convergence of Landau's series. f

A mechanical analog to parallel flow stability as modeled by the Orr–Sommerfeld equation

A mechanical analog to the Orr–Sommerfeld equation is presented in the form of a vibrating elastic plate resting on a locally reacting foundation of specific form. In the analog, Reynolds number becomes the cube of the slenderness ratio of the plate. Some aspects of the solutions to the Orr–Sommerfeld equation, as it is applied to boundary layer transition, are interpreted in terms of the analogous system.

Stability of plane parallel flows in a magnetized plasma

We present an investigation of the problem of stability of plane parallel flows in a magnetized plasma, based on magnetohydrodynamics equations. The corresponding eigenvalue problem, for incompressible velocity perturbations, is solved numerically, by using continuous profiles for both plasma density and velocity at equilibrium.

Stability of parallel flows by the finite element method

AbstractThis paper presents and results obtained from the stabilitystudies of plane Poiseuille flow and magnetohydrodynamic flow by the finite element method. Applying Galerkin's weighted residual method and introducing the interpolation function in the exponetial form with respect to time, the governing flow equations are reduced to an eigenvalue problem. This formulation is much simpler than that of the asymptotic expansion method. Solutions are obtained directly in terms of velocity and pressure. Results of the critical Reynolds number obtained by this method compare well with those of other methods for plane Poiseuille flow and magnetohydrodynamic flow.

Nonparallel Effects on the Stability of Jet Flows

A theoretical analysis of the linear, spatial stability of Bickley’s jet is presented. The analysis takes into account the effects of transverse velocity component and the axial variations of the basic flow and of the disturbance amplitude, wavenumber and spatial growth rate. The integration of stability equations is started from the outer region of the jet toward the jet axis using the solution of the asymptotic forms of the governing equations. Results are compared with those for the parallel-flow stability analysis. It is found that the nonparallel effects decrease the wave number at low frequencies but increase it at high frequencies. Thus the nonparallel effects make Bickley’s jet unstable over a wider frequency range.

Numerical calculation of the stability of parallel flows

Abstract : A numerical approach to solving the two-dimensional, incompressible Navier-Stokes equations is presented and is used to study the stability and evolution of disturbance in a boundary layer. Previous numerical approaches have used streamfunction-vorticity formulations that take advantage of the special properties of two-dimensional flow. The present method solves a finite-difference form of the momentum equations and may therefore be extended to three-dimensions more readily. Computations using this technique are compared with the result of linear stability theory for small amplitude disturbances and are found to give satisfactory agreement, while calculations at large amplitude support the conjecture that non-linear effects can be destabilizing. (Author)

Nonlinear stability of parallel flows with subcritical Reynolds numbers. Part 2. Stability of pipe Poiseuille flow to finite axisymmetric disturbances

A theoretical study is presented of the spatial stability of flow in a circular pipe to small but finite axisymmetric disturbances. The disturbance is represented by a Fourier series with respect to time, and the truncated system of equations for the components up to the second-harmonic wave is derived under a rational assumption concerning the magnitudes of the Fourier components. The solution provides a relation between the damping rate and the amplitude of disturbance. Numerical calculations are carried out for Reynolds numbers R between 500 and 4000 and βR [les ] 5000, β being the non-dimensional frequency. The results indicate that the flow is stable to finite disturbances as well as to infinitesimal disturbances for all values of R and βR concerned.

Nonlinear stability of parallel flows with subcritical Reynolds numbers. Part 1. An asymptotic theory valid for small amplitude disturbances

A strict distinction is made between the two fundamental assumptions in the Stuart-Watson theory of nonlinear stability, one of which is that the amplitude of disturbance is sufficiently small, while the other is that the damping or amplification rate for an infinitesimal disturbance is small. This distinction leads to classification of the nonlinear stability theory into two asymptotic theories: the theory based on the first assumption can be applied to subcritical flows with Reynolds numbers away from the neutral curve, even to flows with no neutral curve, such as plane Couette flow or pipe Poiseuille flow, while the theory based on the second assumption is available only for Reynolds numbers and wavenumbers in the neighbourhood of the neutral curve. In the theory based on the first assumption the concept of trajectories in phase space, together with the method of eigenfunction expansion, is introduced in order to display nonlinear behaviour of the disturbance amplitude and to provide the most rational definition of the Landau constant available for classification of the behaviour patterns.

Linear Stability of Jet Flows

A theoretical investigation into the linear, spatial stability of plane laminar jets is presented. The three cases studied are: 1. Inviscid stability of Sato’s velocity profile. 2. Viscous stability of the Bickley’s jet using parallel-flow stability theory. 3. Viscous stability of the Bickley’s jet using a theory modified to account for the inflow terms. The integration of stability equations is started from the outer region of the jet toward the jet axis using the solution of the asymptotic forms of the governing equations. An eigenvalue search technique is employed to find the number of eigenvalues and their approximate location in a closed region of the complex eigenvalue plane. The accurate eigenvalues are obtained using secant method. The inviscid spatial stability theory is found to give results that are in better agreement with Sato’s experimental results than those obtained by him after transformation of the temporal theory results. For the viscous case the critical Reynolds number found by using the theory accounting for inflow is in better agreement with the experimental value than that given by the parallel-flow theory, implying thereby that the parallel-flow approximation for a jet is erroneous for the stability analysis.

Instability due to three-dimensional disturbances

In the linear theory of the stability of parallel flows of a viscous fluid, most attention is usually given to plane-wave disturbances. The reason is the validity in many cases of the Squire theorem, which states that the critical Reynolds number R is determined by two-dimensional disturbances [1]. It is shown in the present paper that for large R the region generating the turbulence in the initial stage of its development is formed by three-dimensional disturbances. This feature applies both to the generating range of wave numbers and the dimension of the wall layer, where the fluctuating energy is produced. The consequences of the Squire transformations for parallel flows are analyzed. The contribution of resonant nonlinear triad coupling to the rapid growth of fluctuating energy is studied for the case of an explosive instability in an extended laminar mode. It is shown that the rate of turbulent energy production is not governed by the small derivatives of linear theory, but by nonlinear triad coupling of neutral and growing disturbances, with their three-dimensional nature playing an important role.

Influence of heat transfer on the stability of parallel flow between concentric cylinders

The stability of the parallel flow of water between concentric cylinders at different temperatures is investigated for infinitesimal velocity and pressure disturbances. Primary interest is in the effect of heat transfer and the radius ratio a/b on the critical point of the neutral stability curve. The results indicate a strong dependence of the critical eigenvalues on both the heat transfer and the radius ratio. The critical Reynolds number of the nonisothermal flow appears to approach a finite value as the inner radius approaches zero (pipe flow) by showing an inflection point on the curve Rc vs a/b.