The role of boundary conditions in the instability of one-dimensional systems

We investigate the instability properties of one-dimensional systems of finite length that can be described by a local wave equation and a set of boundary conditions. A method to quantify the respective contributions of the local instability and of wave reflections in the global instability is proposed. This allows to differentiate instabilities that emanate from wave propagation from instabilities due to wave reflections. This is illustrated on three different systems, that exhibit three different behaviors. The first one is a model system in fluid mechanics (Ginzburg–Landau equation), the second one is the fluid-conveying pipe (Bourrières equation), the third one is the fluid-conveying pipe resting on an elastic foundation (Roth equation).     

Local and global instabilities of flow in a flexible-walled channel

We consider laminar high-Reynolds-number flow through a long finite-length planar channel, where a segment of one wall is replaced by a massless membrane held under longitudinal tension. The flow is driven by a fixed pressure difference across the channel and is described using an integral form of the unsteady boundary-layer equations. The basic flow state, for which the channel has uniform width, exhibits static and oscillatory global instabilities, having distinct modal forms. In contrast, the corresponding local problem (neglecting boundary conditions associated with the rigid parts of the system) is found to be convectively, but not absolutely, unstable to small-amplitude disturbances in the absence of wall damping. We show how amplification of the primary global oscillatory instability can arise entirely from wave reflections with the rigid parts of the system, involving interacting travelling-wave flutter and static-divergence modes that are convectively stable; alteration of the mean flow by oscillations makes the onset of this primary instability subcritical. We also show how distinct mechanisms of energy transfer differentiate the primary global mode from other modes of oscillatory instability.     

The stability of a circumferential crack in a pipe I. The influence of the boundary conditions imposed at the ends

The paper examines the case where a pipe of length L is built-in at one end and the other end is subject to an imposed displacement or rotation. The criterion for instability of growth of a circumferential through-wall crack is shown to depend on the pipe-end boundary conditions as well as the pipe geometry, crack size and crack location. The worst possible case is that where there is only a force, but no moment, at the pipe-end. However, this is probably an artificial situation which is unlikely to arise in practice. A pipe is more likely to be built-in at both ends, and for this situation, it is concluded that the instability criterion is the same irrespective of whether a displacement or rotation is imposed at a built-in end.     

Finite element analysis of post-buckling dynamics in plates—Part I: An asymptotic approach

Various static and dynamic aspects of post-buckled thin plates, including the transition of buckled patterns, post-buckling dynamics, secondary bifurcation, and dynamic snapping (mode jumping phenomenon), are investigated systematically using asymptotical and non-stationary finite element methods. In part I, the secondary dynamic instability and the local post-secondary buckling behavior of thin rectangular plates under generalized (mechanical and thermal) loading is investigated using an asymptotic numerical method which combines Koiter’s nonlinear instability theory with the finite element technique. A dynamic multi-mode reduction method—similar to its static single-mode counterpart: Liapunov–Schmidt reduction—is developed in this perturbation approach. Post-secondary buckling equilibrium branches are obtained by solving the reduced low-dimensional parametric equations and their stability properties are determined directly by checking the eigenvalues of the resulting Jacobian matrix. Typical post-secondary buckling forms—transcritical, supercritical and subcritical bifurcations are observed according to different combinations of boundary conditions and load types. Geometric imperfection analysis shows that not only the secondary bifurcation load but also changes in the fundamental post-secondary buckling behavior are affected. The post-buckling dynamics and the global analysis of mode jumping of the plates are addressed in part II.     

Absolute, convective, and global instability of a magnetic liquid jet

An infinite or semi-infinite jet of non-conductive magnetic liquid in a uniform longitudinal magnetic field can be absolutely or convectively unstable for different values of the flow parameters. Though the higher field inhibits the absolute instability, this inhibition is maximum at some field intensity. A critical value of the surface tension exists, above which the instability is absolute for any intensity of the field. If the jet has a large but finite length and proper boundary conditions are held at its beginning and end, it is always globally unstable. The unstable global mode is based on a pair of waves that propagate in opposite directions and reflect from one into the other at the flow boundaries.     

On the absolute instability in a boundary-layer flow with compliant coatings

The question of absolute instabilities occuring in a boundary-layer flow with compliant coatings is reassessed. Compliant coatings of the Kramer's type are considered. Performing a local, linear absolute/convective stability analysis, a family of spring-backed elastic plates with damping is shown to be absolutely unstable for sufficiently thin plates. The absolute instability arises from the coalescence between an upstream propagating evanescent mode and the Tollmien–Schlichting wave. To reinforce the local, linear stability results the global stability behaviour of the system is investigated, integrating numerically the full nonparallel and nonlinear two-dimensional Navier–Stokes system coupled to the dynamical model. Injecting Gaussian-type, spatially localized flow disturbances as initial conditions, the spatio-temporal evolution of wave packets is computed. The absolute stability behaviour is retrieved in the global system, for a compliant panel of finite length. It is demonstrated numerically that the global stability behaviour of the wall, triggered by finite-end-effects, may be independent of the disturbance propagation in the flow.     

Writhing instabilities of twisted rods: from infinite to finite length

We use three different approaches to describe the static spatial configurations of a twisted rod as well as its stability during rigid loading experiments. The first approach considers the rod as infinite in length and predicts an instability causing a jump to self-contact at a certain point of the experiment. Semi-finite corrections, taken into account as a second approach, reveal some possible experiments in which the configuration of a very long rod will be stable through out. Finally, in a third approach, we consider a rod of real finite length and we show that another type of instability may occur, leading to possible hysteresis behavior. As we go from infinite to finite length, we compare the different information given by the three approaches on the possible equilibrium configurations of the rod and their stability. These finite size effects studied here in a 1D elasticity problem could help us guess what are the stability features of other more complicated (2D elastic shells for example) problems for which only the infinite length approach is understood.     

Numerical analysis of secondary instabilities of the incompressible boundary layer flow with suction

Based on the Euler–Maclaurin formula, a compact finite difference scheme is employed to solve a two-point boundary value problem for studying the secondary instabilities of the boundary layer flow. The parametric resonance of unstable waves is explored using the Floquet method. For both subharmonic and fundamental modes, two additional Fourier terms are added in the analysis, and the spatial growth rates are determined. The effect of suction mechanism on the secondary instability waves is also investigated. From numerical experiments, it is shown that the proposed numerical scheme is very promising. © 1998 John Wiley & Sons, Ltd.     

Thermally induced dynamic instability of laminated composite conical shells

Thermally induced dynamic instability of laminated composite conical shells is investigated by means of a perturbation method. The laminated composite conical shells are subjected to static and periodic thermal loads. The linear instability approach is adopted in the present study. A set of initial membrane stresses due to the elevated temperature field is assumed to exist just before the instability occurs. The formulation begins with three-dimensional equations of motion in terms of incremental stresses perturbed from the state of neutral equilibrium. After proper nondimensionalization, asymptotic expansion and successive integration, we obtain recursive sets of differential equations at various levels. The method of multiple scales is used to eliminate the secular terms and make an asymptotic expansion feasible. Using the method of differential quadrature and Bolotin's method, and imposing the orthonormality and solvability conditions on the present asymptotic formulation, we determine the boundary frequencies of dynamic instability regions for various orders in a consistent and hierarchical manner. The principal instability regions of cross-ply conical shells with simply supported–simply supported boundary conditions are studied to demonstrate the performance of the present asymptotic theory.     

Application of cellular automata modeling to seismic elastodynamics

This article details application of a physics-based cellular automata (CA) computational approach to model seismic events in an idealized linear-elastic medium. Application of rectangular-celled CA to the seismic problem is shown to yield discrete equations equivalent to the centered-difference finite difference (FD) approach. However, it is emphasized that the discrete equations are arrived at from the ‘bottom up’ using local rules vice ‘top-down’ discretization of global partial differential equations. A further distinction between the two methods concerns the location of stresses and its impact on boundary conditions: the CA approach assigns stresses to the cell faces while the FD approach assigns stress collocated with displacement components at a single node. These differences may provide important perspective on modeling arbitrary geometry with a finite difference-like approach based on cell assembly, similar to finite element analysis. Implementation of the CA paradigm using autonomous, local cells fits naturally with object-oriented programming practices and lends itself readily to distributed computing. Results are provided for an example ground-shock simulation in which a differentiated Gaussian pulse acts on the surface of a linear-elastic half-space. The CA perspective suggests a simple treatment for the free-surface boundary condition. Comparison of the computed pressure, shear, and surface waves to those computed using a staggered-grid finite difference approach demonstrates very good agreement. In addition, the simulation results suggest that the CA approach may exhibit less ‘ringing’ as waves pass, and more symmetry in left-ward and right-ward moving waves. Future directions exploiting attractive attributes of the CA approach are suggested, to include large-scale simulation, multi-resolution analysis, and coupled-field modeling.     

Small Reynolds number instabilities in two-layer Couette flow

Instabilities in two-layer Couette flow are investigated from a small Reynolds number expansion of the eigenvalue problem governing linear stability. The wave velocity and growth rate are given explicitly, and previous results for long waves and short waves are retrieved as special cases. In addition to the inertial instability due to viscous stratification, the flow may be subject to the Rayleigh–Taylor instability. As a result of the competition of these two instabilities, inertia may completely stabilise a gravity-unstable flow above some finite critical Froude number, or conversely, for a gravity-stable flow, inertia may give rise to finite wavenumber instability above some finite critical Weber number. General conditions for these phenomena are given, as well as exact or approximate values of the critical numbers. The validity domain of the many asymptotic expansions is then investigated from comparison with the numerical solution. It appears that the small-Re expansion gives good results beyond Re = 1, with an error less that 1%. For Reynolds numbers of a few hundred, we show from the eigenfunctions and the energy equation that the nature of the instability changes: instability still arises from the interfacial mode (there is no mode crossing), but this mode takes all the features of a shear mode. The other modes correspond to the stable eigenmodes of the single-layer Couette flow, which are recovered when one fluid is rigidified by increasing its viscosity or surface tension.     

Instabilities of wrinkled membranes with pressure loadings

Wrinkling can affect the functionality of thin membranes subjected to various loadings or boundary conditions. The concept of relaxed strain energy was studied for isotropic, hyperelastic, axisymmetric membranes pressurized by gas or fluid. Non-intuitive instabilities were observed when axisymmetric wrinkled membranes were perturbed with angle dependent displacement fields. A linearized theory showed that static equilibrium states of pressurized membranes, modelled by a relaxed strain energy formulation, are unstable, when the wrinkled surface is subjected to pressure loadings. The theory is extended to the non-axisymmetric membranes and it is shown that these instabilities are local phenomena. Simulations for the pressurized cylindrical membranes with non-uniform thickness and hemispherical membranes support the claims in both theoretical and numerical contexts including finite element simulations.     

Running and Standing Waves of Timoshenko Beam

Standing waves of a Timoshenko beamof finite length and their connectionwith running waves for an infinite beam are considered. It is shown that the principle of a “closed cycle” of a running wave is completely identical to the usual procedure of direct satisfaction from the side of the general solution for an infinite Timoshenko beam, to the boundary conditions of a beam of finite length. The question of the existence of a second frequency spectrum under arbitrary boundary conditions of a beam is discussed. A “relaxed approach” to the concept of the second frequency spectrum is proposed. The results of the theoretical analysis are confirmed by numerical calculations for the Timoshenko beam with elastic supports and elastic sealing of its end sections.     

Transient mixed convection flow instabilities in a vertical pipe

The present paper investigates a numerical study of flow instabilities in transient mixed convection in a vertical pipe. At the entrance of the pipe, the flow is suddenly submitted to a temperature step. The convection heat transfer on the outer surface of the pipe is taken into account. The governing equations are solved using a finite difference explicit scheme. The numerical results show that the time development of streamlines and isotherms is strongly dependent on the inlet temperature steps. For positive temperature steps, the unsteady vortex is significant in the vicinity of the wall and the reversal flow appears below the wave instability. In the case of negative temperature steps and especially for the low Reynolds number, the reversal flow appears on top of the wave instability. During the transient, the apparition of the vortical structures along the wall leads to the wall boundary layer instability. This phenomenon is due to the transient mixed convection flows. The temperature step effects on the heat transfer of the flow are presented in our paper.     

The viability of a simple procedure for determining the criterion for the instability of circumferential growth of a through-wall crack in a stainless steel piping system,Part 1: The applied displacements produce only bending deformation

This paper investigates the stability of a circumferential through-wall crack in a straight segment of stainless steel pipe which is built-in at both ends. One built-in end is fixed, while the other is subject to both a rotation and a transverse displacement. In analysing this situation, it is assumed that the material is non-work-hardening, with plasticity being confined to the cracked cross-section which is fully yielded; the remainder of the pipe deforms elastically. The plasticity is in the form of a rotation about a neutral axis, but allowance is made for the axial displacement produced by this rotation, and in this respect the analysis is an improvement upon some earlier analyses. The instability criterion, obtained with the aid of the tearing modulus methodology, is shown to be independent of the details of the imposed boundary conditions, but it is different to the criterion obtained by ignoring the axial displacement. The paper discusses the implications of the results to the problem of crack instability in stainless steel nuclear reactor piping systems, particularly with regards to the viability of a simple procedure that is currently used to assess the integrity of piping systems.     

Stabilization of a laminar boundary layer by an active surface-localized action

A method for rapidly damping instability waves is proposed as a means of actively controlling a perturbed gas boundary layer flow. The method is based on the use of an active body surface segment which reacts to an instantaneous local pressure variation by producing a proportional local wall displacement normal to the surface with a constant time lag calculated to result in the optimal suppression of unstable disturbances. It is shown that in the one-frequency case the wave number spectrum of the optimal control law contains multiple eigenvalues. The effectiveness of the method is demonstrated over a wide range of variation of the instability wave frequencies and directions. The propagation of an instability wave over an active segment of finite length is calculated using an integral-equation method based on solving the problem of boundary layer flow receptivity to surface vibration. Explicit formulas describing the process of scattering of the instability wave into stable modes at the junction point of the rigid and active surfaces are obtained using the Fourier method and the integral Cauchy theorem.     

Propagation of near-limit gaseous detonations in small diameter tubes

In this study, detonation limits in very small diameter tubes are investigated to further the understanding of the near-limit detonation phenomenon. Three small diameter circular tubes of 1.8, 6.3, and 9.5 mm inner diameters, of 3 m length, were used to permit the near-limit detonations to be observed over long distances of 300 to 1500 tube diameters. Mixtures with high argon dilution (stable) and without dilution (unstable) are used for the experiments. For stable mixtures highly diluted with argon for which instabilities are not important and where failure is due to losses only, the limit obtained experimentally appears well to be in good agreement in comparison to that computed by the quasi-steady ZND theory with flow divergence or curvature term modeling the boundary layer effects. For unstable detonations it is suggested that suppression of the instabilities of the cellular detonation due to boundary conditions is responsible for the failure of the detonation wave. Different near-limit propagation regimes are also observed, including the spinning and galloping mode. Based on the present experimental results, an attempt is made to study an operational criterion for the propagation limits of stable and unstable detonations.     

A theoretical mode for the instability of turbulent boundary layer over compliant surface

A theoretical model for the instability of turbulent boundary layer over compliant surfaces is described. The investigation of instability is carried out from a time-asymptotic space-time perspective that classifies instabilities as either convective or absolute. Results are compared against experimental observations of surface waves on elastic and viscoelastic compliant layers.     

Dynamic stability analysis of composite plates including delaminations using a higher order theory and transformation matrix approach

A refined higher order shear deformation theory is used to investigate the dynamic instability associated with composite plates with delamination that are subject to dynamic compressive loads. Both transverse shear and rotary inertia effects are taken into account. The theory is capable of modeling the independent displacement field above and below the delamination. All stress free boundary conditions at free surfaces as well as delamination interfaces are satisfied by this theory. The procedure is implemented using the finite element method. Delamination is modeled through the multi-point constraint approach using the transformation matrix technique. For validation purposes, the natural frequencies and the critical buckling loads are computed and compared with three-dimensional NASTRAN results and available experimental data. The effect of delamination on the critical buckling load and the first two instability regions is investigated for various loading conditions and plate thickness. As expected, the natural frequencies and the critical buckling load of the plates with delaminations decrease with increase in delamination length. Increase in delamination length also causes instability regions to be shifted to lower parametric resonance frequencies. The effect of edge delamination on the static and dynamic stability as well as of delamination growth is investigated.     

On two-dimensional linear waves in Blasius boundary layer over viscoelastic layers

This work concerns the direct numerical simulation of small-amplitude two-dimensional ribbon-excited waves in Blasius boundary layer over viscoelastic compliant layers of finite length. A vorticity-streamfunction formulation is used, which assures divergence-free solutions for the evolving flow fields. Waves in the compliant panels are governed by the viscoelastic Navier's equations. The study shows that Tollmien–Schlichting (TS) waves and compliance-induced flow instability (CIFI) waves that are predicted by linear stability theory frequently coexist on viscoelastic layers of finite length. In general, the behaviour of the waves is consistent with the predictions of linear stability theory. The edges of the compliant panels, where abrupt changes in wall property occur, are an important source of waves when they are subjected to periodic excitation by the flow. The numerical results indicate that the non-parallel effect of boundary-layer growth is destabilizing on the TS instability. It is further demonstrated that viscoelastic layers with suitable properties are able to reduce the amplification of the TS waves, and that high levels of material damping are effective in controlling the propagating CIFI.