Study Of Linear Stability Research Articles

A Linear Stability Study of the Gradient Zone of a Solar Pond

The linear stability of a plane layer with horizontal temperature and concentration stratification corresponding to gradient zone of a solar pond is investigated. The problem is described by Navier-Stokes equations with Boussinesq-Oberbeck approximation. Two source terms are introduced in the energy equations: the absorption of solar energy characterized by the extinction radiative coefficient μe and by the parameter f defined as the ratio of extracted heat flux to absorbed heat flux in the lower convective zone. The influence of the parameters μe and f on the onset of thermosolutal convection in the case of confined and infinite layers is analyzed. It is found that convection starts in an oscillatory state, independently of the RaS value. Different convection solutions were found for marginal stability and steady state.

Stability of Runge–Kutta–Nyström methods

In this paper, a general and detailed study of linear stability of Runge–Kutta–Nyström (RKN) methods is given. In the case that arbitrarily stiff problems are integrated, we establish a condition that RKN methods must satisfy so that a uniform bound for stability can be achieved. This condition is not satisfied by any method in the literature. Therefore, a stable method is constructed and some numerical comparisons are made.

Solutocapillary convection in spherical shells

A linear stability study of solutocapillary driven Marangoni instabilities in small spherical shells is presented. The shells contain a binary fluid with an evaporating solvent. The viscosity is a strong function of the solvent concentration, the inner surface of the shell is assumed impermeable and stress free, while nonlinear boundary conditions are modeled and prescribed at the receding outer boundary. A time-dependent diffusive state is possible and may lose stability through the Marangoni mechanism due to surface tension dependence on solvent concentration (buoyant forces are negligible in this microscale problem). A frozen-time or quasisteady state linear stability analysis is performed to compute the critical Reynolds number and degree of surface harmonics, as well as the maximum growth rate of perturbations at specified parameters. The development of maximum growth rates in time was also computed by solving the initial value problem with random initial conditions. Results from both approaches are in good agreement except at short times where there is dependence on initial conditions. The physical problem models the manufacturing of spherical shells used as targets in inertial confinement fusion experiments where perfect sphericity is demanded for efficient fusion ignition. It is proposed that the Marangoni instability might be the source of observed surface roughness. Comparisons with the available experiments are made with reasonable qualitative and quantitative agreement.

Existence and Stability of Multidimensional Shock Fronts in the Vanishing Viscosity Limit

In this paper we present a new approach to the study of linear and nonlinear stability of inviscid multidimensional shock waves under small viscosity perturbation, yielding optimal estimates and eventually an extension to the viscous case of the celebrated theorem of Majda on existence and stability of multidimensional shock waves. More precisely, given a curved Lax shock solution u0 to a hyperbolic system of conservation laws, we construct nearby viscous shock solutions u ε to a parabolic viscous perturbation of the hyperbolic system which converge to u0 as viscosity ε→0 and satisfy an appropriate (conormal) version of Majda’s stability estimate. The main new feature of the paper is the derivation of maximal and optimal estimates for the linearization of the parabolic problem about a highly singular approximate solution. These estimates are more robust than the singular estimates obtained in our previous work, and permit us to remove an earlier assumption limiting how much the inviscid shock we start with can deviate from flatness. The key to the new approach is to work with the full linearization of the parabolic problem, that is, the linearization with respect to both u ε and the unknown viscous front, and to allow variation of the front at all stages – not only in the construction of the approximate solution as done in previous work, but also in the final error equation. After reformulating the problem as a transmission problem, we show that the linearized problem can be desingularized and optimal estimates obtained by adding an appropriate extra boundary condition involving the front. The extra condition determines a local evolution rule for the viscous front.

A Godunov-type method in Lagrangian coordinates for computing linearly-perturbed planar-symmetric flows of gas dynamics

Linear stability studies of complex flows require that efficient numerical methods be devised for predicting growth rates of multi-dimensional perturbations. For one-dimensional (1D) basic flows – i.e. of planar, cylindrical or spherical symmetry – a general numerical approach is viable which consists in solving simultaneously the one-dimensional equations of gas dynamics and their linearized forms for three-dimensional perturbations. Extensions of artificial viscosity methods have thus been used in the past. More recently [Equations aux dérivées partielles et applications, articles dédiés à J.-L. Lions, 1998], Godunov-type schemes for single-fluid flows of gas dynamics and magnetohydrodynamics have been proposed. Pursuing this effort, we introduce, within the Lagrangian perturbation approach, a class of Godunov-type schemes which is well suited for solving multi-material problems of gas dynamics. These schemes are developed here for the planar-symmetric case and comprise two second-order extensions. The numerical capabilities of these methods are illustrated by computations of Richtmyer–Meshkov instabilities occurring at a single material interface. A systematic comparison of numerically computed growth rates with results of the linear theory for the Richtmyer–Meshkov instability is provided.

Stability of multidimensional undercompressive shock waves

This paper is devoted to the study of linear and nonlinear stability of undercompressive shock waves for first order systems of hyperbolic conservation laws in several space dimensions. We first recall the framework proposed by Freistühler to extend Majda's work on classical shock waves to undercompressive shock waves. Then we show how the so-called uniform stability condition yields a linear stability result in terms of a maximal $L^2$ estimate. We follow Majda's strategy on shock waves with several improvements and modifications inspired from Métivier's work. The linearized problems are solved by duality and the nonlinear equations by mean of a Newton type iteration scheme. Finally, we show how this work applies to phase transitions in an isothermal van der Waals fluid.

Linear stability analysis of time-dependent algorithms with spectral element methods for the simulation of viscoelastic flows

In order to better understand the sources of numerical instabilities occurring in the simulation of time-dependent flows of viscoelastic fluids at high Weissenberg numbers, we have used various types of linear stability analyses. In complement to the limited number of investigations related to the well-posedness of problems involving the flows of viscoelatic fluids, linear stability studies of steady viscoelastic flows have been carried out by various authors in both theoretical and numerical fields. In contrast to most published work where the equations have only been discretized in the cross-stream direction using a formulation with the stream function, we have used the full spatial discretization with spectral elements since we are more interested in computing the eigenvalue spectra generated by the spatial and temporal discretizations than the ones inherent in the partial differential equations. Our analysis enables us to investigate the influence of the spatial discretization, the time schemes, the various operators present in the conservation and constitutive equations and boundary conditions on the linear stability of the constitutive models.

Linear stability analysis of time-dependent algorithms with spectral element methods for the simulation of viscoelastic flows

In order to better understand the sources of numerical instabilities occurring in the simulation of time-dependent flows of viscoelastic fluids at high Weissenberg numbers, we have used various types of linear stability analyses. In complement to the limited number of investigations related to the well-posedness of problems involving the flows of viscoelatic fluids, linear stability studies of steady viscoelastic flows have been carried out by various authors in both theoretical and numerical fields. In contrast to most published work where the equations have only been discretized in the cross-stream direction using a formulation with the stream function, we have used the full spatial discretization with spectral elements since we are more interested in computing the eigenvalue spectra generated by the spatial and temporal discretizations than the ones inherent in the partial differential equations. Our analysis enables us to investigate the influence of the spatial discretization, the time schemes, the various operators present in the conservation and constitutive equations and boundary conditions on the linear stability of the constitutive models.

Computational analysis of the two-dimensionalthree-dimensional transition in forward-facing step flow

Results are presented from a computational study of the flow over a forward-facing step in a plane channel. The aim of the study is to gain better insight into the three-dimensionality that is typically observed in the separation region of flows over steps and ribs, and in similar configurations. We consider laminar flow at a Reynolds number of 330, based on step height and bulk velocity of the oncoming flow, and the step is assumed to be infinitely extended in the spanwise direction. High-resolution simulations are undertaken using a mixed spectral/spectral-element code. Moreover, a linear stability study of the flow at the step is performed. The results show that, in the case considered, the three-dimensionality is not related to some absolute instability of the separation bubble in front of the step; rather, it is a sensitive reaction of the flow to three-dimensional perturbations present in the oncoming stream. It is demonstrated that disturbance amplitudes of less than 1% of the mean flow (at, say, 10 step heights ahead of the step) already suffice to produce a visibly three-dimensional structure of the separation zone. If the disturbance level is systematically decreased, the three-dimensional state evolves to an almost two-dimensional recirculation. Here, the key finding is that the intensity of the flow response is proportionate to the amplitude of the inflow disturbance, meaning that the breakup of the flow in the step region is a linear (i.e. small) perturbation of the two-dimensional base flow. A comparison of the present simulation results with experimental data shows close agreement concerning, for example, the flow topology in the step region, and the spanwise spacing of the characteristic streaks that form further downstream.

The onset of thermal convection in EkmanCouette shear flow with oblique rotation

The onset of convection of a Boussinesq fluid in a horizontal plane layer is studied. The system rotates with constant angular velocity Ω, which is inclined at an angle V to the vertical. The layer is sheared by keeping the upper boundary fixed, while the lower boundary moves parallel to itself with constant velocity U 0 normal to the plane containing the rotation vector and gravity g (i.e. U 0 ∥ g × Ω). The system is characterized by five dimensionless parameters: the Rayleigh number Ra, the Taylor number τ 2 , the Reynolds number Re (based on U 0 ), the Prandtl number Pr and the angle V. The basic equilibrium state consists of a linear temperature profile and an Ekman-Couette flow, both dependent only on the vertical coordinate z. Our linear stability study involves determining the critical Rayleigh number Ra c as a function of r and Re for representative values of V and Pr. Our main results relate to the case of large Reynolds number, for which there is the possibility of hydrodynamic instability

Channel flow induced by wall injection of fluid and particles

The Taylor flow is the laminar single-phase flow induced by gas injection through porous walls, and is assumed to represent the flow inside solid propellant motors. Such a flow is intrinsically unstable, and the generated instabilities are probably responsible for the thrust oscillations observed in the aforesaid motors. However particles are embedded in the propellants usually used, and are released in the fluid by the lateral walls during the combustion, so that there are two heterogeneous phases in the flow. The purpose of this paper is to study the influence of these particles on stability by comparison with stability results from the single-phase studies, in a plane two-dimensional configuration. The particles are supposed to be chemically inert and of a uniform size. In order to carry out a linear stability study for this flow modified by the presence of particles, the mean particle velocity field is first determined, assuming that only the gas exerts forces on the particles. This field is sought in a self-similar form, which imposes a limit on the size of the particles. However, the particle mass concentration cannot be obtained in a self-similar form, but can only be described by a partial differential equation. The mean flow characteristics being determined, the spectrum of the discretized linear stability operator shows first that particle addition does not trigger any new “dangerous” modes compared with the single-phase flow case. It also shows that the most amplified mode in the case of the single-phase flow remains the most amplified mode in the case of the two-phase flow. Moreover, the addition of particles acts continuously upon stability results, behaving linearly with respect to the particle mass concentration when the latter is small. The linear correction to the monophasic mode, as well as the evolution of the modes with weak values of the particle mass concentration at the wall, are shown to be proportional to the ejection velocity of the particles. Then, the evolution of the eigenmodes from a given injection speed of the particles to another one is deduced by affinity, all other parameters being fixed. With a fixed Stokes number, stability results for a finite Reynolds number and results for the inviscid flow bring together when augmenting the particle mass concentration at the wall. Therefore, by knowing single-phase flow results and the evolution of stability characteristics of the two-phase flow in the inviscid case, it is easy to determine whether particle-laden Taylor flow is more or less stable than the monophasic Taylor flow for large particle mass concentration.

Revue bibliographique sur les écoulements de Poiseuille–Rayleigh–Bénard : écoulements de convection mixte en conduites rectangulaires horizontales chauffées par le bas

An exhaustive bibliographical review on the Poiseuille–Rayleigh–Bénard (PRB) flows, the mixed convection flows in horizontal rectangular ducts uniformly heated from below, is presented. This review covers the period 1920–2001 and counts 154 references. The concerned parameter range is 0< Pr⩽1000, 0⩽ Re⩽1000 and 0⩽ Ra⩽10 6. The interest of these flows in the industrial world is presented for the chemical vapor deposition studies (CVD) and for the cooling of the electronic equipments. After having done a synthesis of the temporal or spatial linear stability studies in the thermal entrance zone and in the fully-developed thermoconvective zone, we consider the linear or weakly non-linear spatio-temporal stability analyses, which take into account the concepts of convective and absolute instabilities and which are more appropriate to represent the dynamics of the PRB flows. All the stability results are systematically compared with the experimental and numerical results. The theoretical or experimental correlation formulae giving the variations of the Nusselt number, of the wave length, of the frequency and of the velocity of the transversal ( R ⊥) and/or longitudinal rolls ( R //), as a function of Re, Ra, Pr and B= l/ h, are collected. The influence of the inlet and of the outlet boundary conditions on the space and time development of the R ⊥ in the experiments and in the numerical simulations is also analysed. A synthesis of the very recently discovered complex thermoconvective patterns (superposition of R ⊥ and R //, periodic and aperiodic patterns, wavy or oblique rolls) is presented. To end, the badly known or badly understood aspects of the PRB flows are summed up and subjects for future investigations are proposed.

On the spectral problem in the linear stability study of flows on a sphere

A normal mode instability study of a steady nondivergent flow on a rotating sphere is considered. A real-order derivative and family of the Hilbert spaces of smooth functions on the unit sphere are introduced, and some embedding theorems are given. It is shown that in a viscous fluid on a sphere, the operator linearized about a steady flow has a compact resolvent, that is, a discrete spectrum with the only possible accumulation point at infinity, and hence, the dimension of the unstable manifold of a steady flow is finite. Peculiarities of the operator spectrum in the case of an ideal flow on a rotating sphere are also considered. Finally, as examples, we consider the normal mode stability of polynomial (zonal) basic flows and discuss the role of the linear drag, turbulent diffusion and sphere rotation in the normal mode stability study.

The dynamics of a laminar flow in a symmetric channel with a sudden expansion

Bifurcation analysis, linear stability study, and direct numerical simulations of the dynamics of a two-dimensional, incompressible, and laminar flow in a symmetric long channel with a sudden expansion with right angles and with an expansion ratio D/d (d is the width of the channel inlet section and D is the width of the outlet section) are presented. The bifurcation analysis of the steady flow equations concentrates on the flow states around a critical Reynolds number Rec(D/d) where asymmetric states appear in addition to the basic symmetric states when Re [ges ] Rec(D/d). The bifurcation of asymmetric states at Rec has a pitchfork nature and the asymmetric perturbation grows like √Re − Rec(D/d). The stability analysis is based on the linearized equations of motion for the evolution of infinitesimal two-dimensional disturbances imposed on the steady symmetric as well as asymmetric states. A neutrally stable asymmetric mode of disturbance exists at Rec(D/d) for both the symmetric and the asymmetric equilibrium states. Using asymptotic methods, it is demonstrated that when Re &lt; Rec(D/d) the symmetric states have an asymptotically stable mode of disturbance. However, when Re &gt; Rec(D/d), the symmetric states are unstable to this mode of asymmetric disturbance. It is also shown that when Re &gt; Rec(D/d) the asymmetric states have an asymptotically stable mode of disturbance. The direct numerical simulations are guided by the theoretical approach. In order to improve the numerical simulations, a matching with the asymptotic solution of Moffatt (1964) in the regions around the expansion corners is also included. The dynamics of both small- and large-amplitude disturbances in the flow is described and the transition from symmetric to asymmetric states is demonstrated. The simulations clarify the relationship between the linear stability results and the time-asymptotic behaviour of the flow. The current analyses provide a theoretical foundation for previous experimental and numerical results and shed more light on the transition from symmetric to asymmetric states of a viscous flow in an expanding channel. It is an evolution from a symmetric state, which loses its stability when the Reynolds number of the incoming flow is above Rec(D/d), to a stable asymmetric equilibrium state. The loss of stability is a result of the interaction between the effects of viscous dissipation, the downstream convection of perturbations by the base symmetric flow, and the upstream convection induced by two-dimensional asymmetric disturbances.

Linear stability of shock profiles for systems of conservation laws with semi-linear relaxation

The Evans function theory, which has recently been applied to the study of linear stability of viscous shock profiles, is developed below for semi-linear relaxation. We study the linear stability of shock profiles in the Lax, undercompressive and overcompressive cases. The results we obtain are similar to those found for viscous approximations by Gardner and Zumbrun [Commun. Pure Appl. Math. 51 (7) (1998) 797].

On the normal mode instability of modons and Wu-Verkley waves

The normal mode instability of steady Wu-Verkley (1993) wave and modons by Verkley (1984, 1987, 1990) and Neven (1992) is considered. All these flows are solutions to the vorticity equation governing the motion of an ideal incompressible fluid on a rotating sphere. A conservation law for infinitesimal perturbations to each solution is derived and used to obtain a necessary condition for its exponential instability. By these conditions, Fjörtoft's (1953) average spectral number of the amplitude of an unstable mode must be equal to a specific number that depends on the degree of the solution in its inner and outer regions as well as on spectral distribution of the mode energy in these regions. Some properties of the conditions for different types of modons are discussed. The maximum growth (and decay) rate of the modes is estimated, and the orthogonality of the amplitude of each unstable, decaying, or non-stationary mode to the basic solution is shown in the energy inner product. The new instability conditions confine the unstable disturbances of the WV wave and modon to a hypersurface in the perturbation space and allow interpretation of their energy structure. They are also useful both in estimating the maximum growth rate of unstable modes and in testing the numerical algorithms designed for the linear stability study.

Experimental instability study in a thermoacoustic prime-mover

An experimental linear stability study of a thermoacoustic prime-mover is performed for different values of the mean pressure between 0.5 and 10 bars. The damping rate is carefully obtained as a function of the temperature gradient |∇T| enforced along the stack, up to the instability onset at |∇T|c. These results are then confronted with the predictions of a numerical model based on Rotts’ theory used with complex frequencies, in which the prime-mover is seen as a feedback loop in the electrical analogy. The experimental results are found to reasonably comply with Rott’s theory as soon as the mean pressure exceeds 2 bars. Below this value substantial discrepancies are found, as confirmed by the work of Yazaki. Above the instability onset, the saturated wave amplitude is measured as a function of |∇T|−|∇T|c for fixed pressure. The bifurcation to the nonlinear saturated wave has been tentatively determined as subcritical, although thermal inertia effects make it look supercritical.

Asymptotic Analysis and Stability of Inviscid Liquid Sheets

Asymptotic methods are employed to determine the leading-order equations that govern the fluid dynamics of slender, and thin and slender, inviscid, irrotational, planar liquid sheets subject to pressure differences and gravity. Two flow regimes have been identified depending on the Weber number, and analytical solutions to the steady state equations are provided. Linear stability studies indicate that the sinuous mode corresponds to Weber numbers on the order of unity, while the varicose mode is associated with small Weber numbers. For small Weber numbers, the nonlinear stability of liquid sheets is determined analytically in terms of elliptic integrals of the first and second kinds. It is also shown that the sinuous mode of thin and slender liquid sheets is identical to the same mode for slender sheets.

A new formulation of the resistive magnetohydrodynamics stability problem for finite β toroidal plasmas

A new approach to the study of linear resistive magnetohydrodynamics stability is described. The approach is based on the traditional toroidal plasma model where the plasma resistivity and mass effects are essential only in thin layers around resonance surfaces, whereas the outer plasma is ideal and inertia free. This leads to differential equations with singular points. A new technique to solve these equations is proposed and it is shown that it has superior numerical convergence and accuracy properties to previous methods. The new technique is generally applicable to other problems in which differential equations with singular points arise.

On the linear stability study of zonal incompressible flows on a sphere

On the linear stability study of zonal incompressible flows on a sphere