Spatial Stability Problem Research Articles

Asymptotic boundary conditions for the analysis of hydrodynamic stability of flows in regions with open boundaries

Abstract A new approach to formulation of asymptotic boundary conditions for eigenvalue problems arising in numerical analysis of hydrodynamic stability of such shear flows as boundary layers, separations, jets, wakes, characterized by almost constant velocity of the main flow outside the shear layer or layers is proposed and justified. This approach makes it possible to formulate and solve completely the temporal and spatial stability problems in the locally-parallel approximation, reducing them to ordinary algebraic eigenvalue problems.

A Newton-type method for non-linear eigenproblems

Abstract This work is devoted to computations of invariant pairs associated with separated groups of finite eigenvalues of large regular non-linear matrix pencils. It is proposed to combine the method of successive linear problems with a Newton-type method designed for partial linear eigenproblems and a deflation procedure. This combination is illustrated with a typical hydrodynamic spatial stability problem.

Stability of stratified downslope flows with an overlying stagnant isolating layer

We investigate the dynamic stability of stratified flow configurations characteristic of hydraulically controlled downslope flow over topography. Extraction of the correct ‘base state’ for stability analysis from spatially and temporally evolving flows that exhibit instability is not easy since the observed flow in most cases has already been modified by nonlinear interactions between the instability modes and the mean flow. Analytical studies, however, can yield steady solutions under idealized conditions which can then be analysed for stability. Following the latter approach, we study flow profiles whose essential character is determined by recently obtained solutions of Winters & Armi (J. Fluid Mech., vol. 753, 2014, pp. 80–103) for topographically controlled stratified flows. Their condition of optimal control necessitates a streamline bifurcation which then naturally produces a stagnant isolating layer overlying an accelerating stratified jet in the lee of the topography. We show that the inclusion of the isolating layer is an essential component of the stability analysis and further clarify the nature and mechanism of the instability in light of the wave-interaction theory. The spatial stability problem is also briefly examined in order to estimate the downstream location where finite-amplitude features might be manifested in streamwise slowly varying flows over topography.

Necessary conditions for spatial inviscid instability

Rayleigh’s inflection point theorem and Fjortoft’s theorem provide necessary conditions for inviscid temporal instability of a plane parallel flow. Although these theorems have been assumed to hold in the spatial framework also, a rigorous theoretical basis for such an application is not available in the literature. In this paper, we provide such a basis by carrying out a formal analysis of the Rayleigh equation. We prove that, under certain conditions satisfied by a wide class of flows, the Rayleigh and Fjortoft theorems are applicable to the spatial stability problem also. This work thus fills the lacuna in the spatial stability theory with regard to these classical theorems.

Spatial stability of braced thin-walled members of steel structures

Abstract The solutions of the spatial stability problems for: (1) monosymmetrical columns and beams with a prescribed axis of rotation, compressed and bent under transverse loading and (2) monosymmetrical beams with a horizontal (lateral) brace, loaded with an arbitrary number of concentrated forces are provided. The considered structural members were additionally point braced along their length to prevent them from twisting and warping. The spatial stability problems were formulated as distributional differential equations, which were solved approximately using the Bubnov–Galerkin orthogonalization method and then the solutions of the problems were derived for the structural members for different support conditions at their ends. The general solutions were reduced to eigenvalue problems. The numerical calculations can be performed using in-house computer programs. Each of the considered problems is illustrated by numerical examples and ready-made particular solutions. In the case of the beams with a lateral brace, also the effect of stiffeners on the lateral buckling load was examined. The theoretical solutions have been validated by model tests carried out on laterally and longitudinally braced steel I-section columns subjected to compression and bending and on longitudinally braced steel I-beams.

NUMERICAL SOLUTION OF THREE-DIMENSIONAL STABILITY PROBLEMS FOR ELASTIC BODIES

A solution in Cartesian coordinates to plane and spatial stability problems for composites is obtained within the framework of the second variant of the three-dimensional linearized theory of stability of deformable bodies. Two mechanical models are used: a homogeneous anisotropic medium with averaged mechanical characteristics and a piecewise-homogeneous medium with orthotropic linearly elastic components. To solve the problems, a mesh approach is applied. Discrete models are constructed using the concept of a base scheme. The calculated results are analyzed

Linear instability in the near wake of channels cascade

The growth of linear disturbances in the high-Reynolds number laminar wake of a channels cascade is investigated. Disturbances in the near wake respond according to the Rayleigh equation which is solved numerically for a spatial stability problem. The unperturbed flow is studied using a double deck analysis. The results of This analysis allow for the basic flow profiles at different streamwise positions to be numerically obtained as solutions of the wake boundary-layer equations.

Linear instability in the near wake of a symmetrical junction of two channels

The growth of linear disturbances in the high-Reynolds number laminar wake of a symmetrical junction of two channels is investigated. Disturbances in the near wake respond according to the Rayleigh equation which is solved numerically for a spatial stability problem. The unperturbed flow is studied in the same way as Goldstein's analysis. The results of this analysis allow for the basic profiles at different streamwise positions to be numerically obtained as solutions of the wake boundary-layer equations.

On the Stability of Parallel Bubbly Cavitating Flows

This paper illustrates the effects of the dynamics of bubbles with arbitrary vapor-gas contents on the inviscid and viscous stability of two-dimensional parallel bubbly flows of low void fraction. The linear perturbation equations derived for the stability analysis include the effects of bubble compressibility, inertia, and energy dissipation due to the viscosity of the liquid and the transfer of heat and mass as a consequence of compression/expansion of the noncondensable gas and evaporation/condensation of the vapor contained in the bubbles. Numerical solution of the spatial stability problem for two-dimensional inviscid shear layers and Blasius boundary layers confirms that the presence of the dispersed phase is generally in favor of stability. Significant deviations from the classical results for compressible and incompressible single phase fluids are observed, especially when the occurrence of large compliant and/or resonant oscillations of the bubbles greatly enhances their dynamic coupling with the perturbation field. More importantly, the present analysis points out some major differences in the stability of parallel flows with noncondensable gas bubbles with respect to cavitating flows containing bubbles with a dominant content of vapor. Unconditional stability is predicted in the travelling bubble cavitation limit for low pressures and high vapor mass fraction of the bubble contents. Results are shown to illustrate these effects for some representative flow configurations and conditions. [S0098-2202(00)00603-9]

Spatial buckling of thin-walled steel-construction beam-columns with discrete bracings

General closed-form solutions of distributional differential equations of the elastic flexural-torsional buckling of monosymmetric, constant-section, thin-walled members fitted with discrete elastic bracings over their length and subjected to compression and bending in their symmetry plane have been worked out. The bracings confine the warping and displacements of the intermediate sections of the members from their symmetry plane. Besides the general case, members with a prescribed axis of rotation are considered. Particular solutions of spatial stability problems for members optionally supported at their ends have been worked out on the basis of the general solutions. The practical application of the presented solutions, incorporated into appropriate computer programs, is illustrated with numerical examples in which the effect of the particular bracings on the critical load of the flexural-torsional buckling of columns or that of the lateral buckling of steel structural beams is investigated. The theoretical solutions have been validated experimentally by the author's own tests on laterally and longitudinally braced steel columns and by Milner's tests on laterally braced beams [24].

The continuous spectrum of the Orr-Sommerfeld equation: note on a paper of Grosch & Salwen

Grosch & Salwen (1978) discuss the continuous-spectrum contribution in both temporal and spatial stability problems that are governed by the linear Orr–Sommerfeld equation. Their computed temporal continuum eigenfunctions for the Blasius boundary layer and for a laminar jet profile show surprising differences. This note provides an improved physical understanding of the results through a simple model, and shows that these differences are more apparent than real.

The continuous spectrum of the Orr-Sommerfeld equation. Part 2. Eigenfunction expansions

The expansion of an arbitrary two-dimensional solution of the linearized stream-function equation in terms of the discrete and continuum eigenfunctions of the Orr-Sommerfeld equation is discussed for flows in the half-space, y ε [0, ∞). A recent result of Salwen is used to derive a biorthogonality relation between the solution of the linearized equation for the stream function and the solutions of the adjoint problem.For the case of temporal stability, the orthogonality relation obtained is equivalent to that of Schensted for bounded flows. This relationship is used to carry out the formal solution of the initial-value problem for temporal stability. It is found that the vorticity of the disturbance at t = 0 is the proper initial condition for the temporal stability problem. Finally, it is shown that the set consisting of the discrete eigen-modes and continuum eigenfunctions is complete.For the spatial stability problem, it is shown that the continuous spectrum of the Orr-Sommerfeld equation contains four branches. The biorthogonality relation is used to derive the formal solution to the boundary-value problem of spatial stability. It is shown that the boundary-value problem of spatial stability requires the stream function and its first three partial derivatives with respect to x to be specified at x = 0 for all t. To be applicable to practical problems, this solution will require modification to eliminate disturbances originating at x = ∞ and travelling upstream to x = 0.For the special case of a constant base flow, the method is used to calculate the evolution in time of a particular initial disturbance.

The continuous spectrum of the Orr-Sommerfeld equation. Part 1. The spectrum and the eigenfunctions

It is shown that the Orr-Sommerfeld equation, which governs the stability of any mean shear flow in an unbounded domain which approaches a constant velocity in the far field, has a continuous spectrum. This result applies to both the temporal and the spatial stability problem. Formulae for the location of this continuum in the complex wave-speed plane are given. The temporal continuum eigenfunctions are calculated for two sample problems: the Blasius boundary layer and the two-dimensional laminar jet. The nature of the eigenfunctions, which are very different from the Tollmien-Schlichting waves, is discussed. Three mechanisms are proposed by which these continuum modes could cause transition in a shear flow while bypassing the usual linear Tollmien-Schlichting stage.

Finite-Amplitude Baroclinic Disturbances in Downstream Varying Currents

Abstract The finite-amplitude dynamics of baroclinic disturbances in currents whose cross-stream structure varies in the downstream direction is investigated. It is first shown under what circumstances downstream variations of the current properties influence the local stability of the current. For flows near the neutral curve only the potential vorticity in one of the fluid layers is significant in determining the local stability. For currents which are locally stable at some downstream locations and unstable at others, it is shown that the disturbance amplitude depends on the entire upstream structure of the current. In particular, simple examples illustrate the lack of a local relationship between “local” stability characteristics and the disturbance intensity. The linear initial value problem for uniform (in the downstream direction) currents is also discussed to elucidate the relation between the temporal and spatial stability problems.