Large Initial Perturbation Research Articles

Numerical modeling of perturbation propagation in a supersonic boundary layer

The growth of two-dimensional disturbances generated in a supersonic (M∞ = 6) boundary layer on a flat plate by a periodic perturbation of the injection/suction type is investigated on the basis of a numerical solution of the Navier-Stokes equations. For small initial perturbation amplitudes, the secondmode growth rate obtained from the numerical modeling coincides with the growth rate calculated using linear theory with account for the non-parallelism of the main flow. Calculations performed for large initial perturbation amplitudes reveal the nonlinear dynamics of the perturbation growth downstream, with rapid growth of the higher multiple harmonics.

Direct numerical simulations of two-layer viscosity-stratified flow

Two-dimensional simulations of flow instability at the interface of a two-layer, density-matched, viscosity-stratified Poiseuille flow are performed using a front-tracking/finite difference method. We present results for the small-amplitude (linear) growth rate of the instability at small to medium Reynolds number for varying thickness ratio n, viscosity ratio m, and wavenumber. We also present results for large-amplitude non-linear evolution of the interface for varying viscosity ratio and interfacial tension. For the linear case, the interfacial mode is neutrally stable for n = m as predicted by analysis. The growth rate is proportional to Reynolds number for small Re, and increases with viscosity ratio. The growth rate also increases when the thickness of the more viscous layer is reduced. Strong non-linear behavior is observed for relatively large initial perturbation amplitude. The higher viscosity fluid is drawn out as a finger that penetrates into the lower viscosity layer. The simulated interface shape compares well with previously reported experiments. Increasing interfacial tension retards the growth rate of the interface as expected, whereas increasing the viscosity ratio enhances it. Drop formation at the small Reynolds number considered in this study is precluded by the two-dimensional nature of the calculations.

Low mode surface perturbation tolerance of ignition capsule implosions for the National Ignition Facility

The stability of a capsule intended to produce ignition on the National Ignition Facility (NIF) [J. A. Paisner, J. D. Boyes, S. A. Kumpan, W. H. Lowdermilk, and M. S. Sorem, Laser Focus World 30, 75 (1994)] is examined. Sensitivity to target fabrication defects in spherical harmonics modes L⩽12 is quantified in terms of a mode dependent tolerance to surface perturbations. Simulations of NIF capsule implosions with single mode perturbations on a single surface allow the determination of modal growth factors. Simulations with large initial perturbations were also done to determine how large a final perturbation could be tolerated. Combining the growth factors and tolerances determines specifications on initial perturbations. This allows an estimate of modal tolerance for each capsule surface and thickness variation for a polyimide ablator capsule design with central gas fills of 0.3 or 0.6 mg/cm3.

Study of the membrane effect on turbulent mixing measurements in shock tubes

The effect of the thin membrane on the time evolution of the shock wave induced turbulent mixing between the two gases initially separated by it is investigated using two different sets of experiments. In the first set, in which a single-mode large-amplitude initial perturbation was studied, two gas combinations (air/SF \(_6\) and air/air) and two membrane thicknesses were used. The main conclusion of these experiments was that the tested membrane has a negligible effect on the evolution of the mixing zone, which evolves as predicted theoretically. In the second set, in which similar gas combinations and membrane thicknesses were used, small amplitude random-mode initial perturbation, caused by the membrane rupture, rather than the large amplitude single-mode initial perturbation used in the first set, was studied. The conclusions of these experiments were: (1) The membrane has a significant effect on the mixing zone during the initial stages of its growth. This has also been observed in the air/air experiment where theoretically no growth should exist. (2) The membrane effect on the late time evolution, where the mixing zone width has reached a relatively large-amplitude, was relatively small and in good agreement with full numerical simulations. The main conclusion from the present experiments is that the effect of the membrane is important only during the initial stages of the evolution (before the re-shock), when the perturbations have very small amplitudes, and is negligible when the perturbations reach relatively large amplitudes.

Nonlinear behavior of localized space-charge waves in space-charge dominated electron beams

We present experimental observations of the abnormal growth of localized nonlinear space-charge waves in space-charge dominated electron beams passing through a resistive channel. The energy width of the space-charge waves is measured on both ends of the channel. Previous experiments had shown that, for small initial perturbations, the energy width of the slow waves increases, while the energy width of the fast waves decreases, in agreement with linear theory. We report that in the nonlinear regime (large initial perturbations), the energy width of the fast wave increases, which is unexpected, and, to the best of our knowledge, no theory exists that would predict this phenomenon.

Adsorption Kinetics of Ionic Surfactants after a Large Initial Perturbation. Effect of Surface Elasticity

This theoretical study is devoted to the relaxation of surface tension of an ionic surfactant solution for submicellar concentrations. The effects of added nonamphiphilic electrolyte and counterion binding are taken into account. We consider a large initial deviation from equilibrium, which is defined as the formation of a new interface: there is no adsorbed surfactant and electric double layer at the initial moment. Next, the surfactant solution and its interface are allowed to relax without any subsequent perturbation. The electrodiffusion equations, which describe the adsorption kinetics, are nonlinear, and it is impossible to find a general analytical solution, especially in the case of large initial deviations. Nevertheless, the problem can be linearized in the asymptotic case of long times. The derived theoretical expressions show that the relaxation times in the cases of large and small initial perturbations are numerically close to each other. For that reason the relaxation time can be considered as a general kinetic property of the adsorption monolayer. The theory predicts also the slope of the experimental plot of dynamic surface tension vs inverse square root of time; this makes the theory useful for interpretation of experimental data. The theoretical expressions involve the surface (Gibbs) elasticity, whose definition for adsorption monolayers of soluble ionic surfactants is discussed in detail. The Gibbs elasticity of such monolayers is found to increase strongly with the rise of salt concentration. The derived asymptotic expressions are verified against an exact computer solution of the electrodiffusion problem, and excellent agreement is found.

Wave breaking in cold plasma

Strongly nonlinear plasma oscillations in a cold plasma are investigated in the wave breaking regime using the two-fluid model. The electrons are described in Lagrangian coordinates; the ions are described by the usual Eulerian coordinates. The model is extended to describe the motion after wave breaking has occurred. Numerical solutions are followed up to and beyond wave breaking for both driven and undriven cases. In a driven cold plasma, the plasma waves are excited from fluctuations by the modulational instability, growing to very large amplitudes, which leads to wave breaking. For the undriven cold plasma, large initial perturbations are assumed, which also lead to wave breaking. Multistreams and the generation of fast electrons are observed for both cases after wave breaking.

Mixing in ejecta of supernovae. I - General properties of two-dimensional Rayleigh-Taylor instabilities and mixing width in ejecta of supernovae

view Abstract Citations (53) References (20) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Mixing in Ejecta of Supernovae. I. General Properties of Two-dimensional Rayleigh-Taylor Instabilities and Mixing Width in Ejecta of Supernovae Hachisu, Izumi ; Matsuda, Takuya ; Nomoto, Ken'ichi ; Shigeyama, Toshikazu Abstract Nonlinear growth of two-dimensional Rayleigh-Taylor (R-T) instabilities are numerically studied to apply to the mixing in the supernova ejecta. We first present much refined calculations of mixing in the realistic model of SN 1987A with a better code and various mesh resolutions. The results show that the mixing width (or the extent of mixing) due to Rayleigh-Taylor instabilities is still too small to account for the observations even with relatively large initial perturbation. The mixing width is found to depend only slightly on the mesh resolution when the initial amplitude is larger than ~5% of the expansion speed. To clarify the basic properties of the R-T instabilities and the dependence of the mixing width on the initial density structure, initial perturbation, and numerical resolution, we consider much simplified ideal models of R-T instabilities of compressible gas with an adiabatic constant γ = 4/3. The ideal R-T instabilities are calculated for various mesh resolutions, the numerical accuracy (second-order and third- order), the density ratio, the initial amplitude of the perturbations, and the mode of the initial perturbation (random from mesh to mesh and sinusoidal waves). It is found that when the initial amplitude of the velocity perturbation is larger than 1% of the sound speed, the mixing width in time depends hardly on the mesh resolution, numerical accuracy, or the mode of the initial perturbation. The mixing width depends mainly on the initial amplitudes and the density ratio. This suggests that the mixing width in the supernova ejecta depends mainly on the initial amplitude of the perturbation and on the density structure of the presupernova models (density ratio), which confirm the results on the real mixing in ejecta of SN 1987A. Publication: The Astrophysical Journal Pub Date: May 1992 DOI: 10.1086/171274 Bibcode: 1992ApJ...390..230H Keywords: Ejecta; Hydrodynamics; Stellar Interiors; Supernova 1987a; Taylor Instability; Computational Astrophysics; Mixing; Scaling Laws; Astrophysics full text sources ADS | data products SIMBAD (1) NED (1) Related Materials (1) Part 2: 1994A&AS..104..341H

Large growth, planar Rayleigh–Taylor experiments on Nova

A set of indirect-drive experiments to study large growth Rayleigh–Taylor instability using shaped laser pulses at the Nova laser facility has been conducted. Planar foils of fluorosilicone were accelerated by x-ray ablation. The foil trajectory was measured using edge-on radiography. In separate experiments using face-on radiography, contrast in optical depth was measured as a function of time, from which the evolution of 50 μm wavelength initially sinusoidal surface perturbations was deduced. Holding other parameters fixed, the amplitude of the initial perturbation was varied by up to a factor of 30 in separate shots. The foils with the smallest initial perturbation exhibited growth factors of 75 in contrast. Foils with large initial amplitude perturbation gave growth factors of 6 or less, and displayed the ‘‘bubble-and-spike’’ shape characteristic of the nonlinear Rayleigh–Taylor instability. Comparisons of two-dimensional computer simulations with both the measured foil trajectory and the perturbation growth show good agreement, provided that a suitable opacity model is chosen. In the linear regime the observed growth rates are 60%–75% of classical, the reduction attributed primarily to ablative stabilization. The observed onset of harmonic generation, signaling the transition into the nonlinear regime, is well predicted by third-order theory.

Magnetohydrodynamic stability and nonlinear evolution of the m=1 mode in toroidal geometry for safety factor profiles with an inflection point

Magnetohydrodynamic (MHD) stability and nonlinear evolution of the m=1 mode are studied in toroidal geometry for safety factor profiles having an inflection point at the q=1 surface. For ideally stable cases, linear growth rates of the resistive m=1 mode are decreased by any combination of the following variations: decreasing the aspect ratio, decreasing shear at the q=1 surface, increasing S, decreasing q0, and/or decreasing beta. Transitions from resistive kink to tearing mode to complete stability are accompanied by increasing localization as the mode is stabilized. The ideal stability of the m=1 mode for low-shear inflection-point profiles is broken at very low, but finite, beta by the appearance of a radially localized ideal interchange mode. As beta is increased, the m=1 mode becomes a global ideal internal kink. These ideal modes are stabilized by decreasing the aspect ratio and, to a lesser extent, by increasing shear at the q=1 surface. Nonlinear evolution of the m=1 mode is found to follow the Kadomtsev complete reconnection process or to be stable, depending on the linear stability of the mode. However, all nonlinear calculations were carried out at Lundquist number S=105 and hence contain substantial resistive kink contributions. Consequently, nonlinear evolution in the m=1 tearing mode regime is not examined. Even for cases in which ideal effects dominate the linear mode, Kadomtsev reconnection is observed. The nonlinear stability of linearly stable modes was also uniformly observed, even for several cases having large initial perturbations. Hence complete linear stabilization resulting from a combination of toroidal effects and low shear at the q=1 surface can suppress sawteeth, as described by the Kadomtsev reconnection process, in tokamaks having safety factor profiles with q0<1 at the magnetic axis; but, for the S values considered, nonlinear calculations of unstable modes result in complete Kadomtsev reconnection.

An analytical nonlinear theory of thin film rupture and its application to wetting films

A formalism is developed to investigate the instability and the time of rupture of thin films which are subjected to finite amplitude perturbations. The hydrodynamic and the interaction force nonlinearities cannot be ignored in such an event. The key idea of the approach is that, for a finite amplitude perturbation, one needs to study the instability of the corresponding finite amplitude, spatially nonhomogeneous stationary solution of the governing equations. It is in the event of infinitesimal perturbations that the nonhomogeneous stationary state reduces to the planar interface, the stability issue of which is addressed by the linear stability analysis. Thus the essential idea is to account for a part of the finite disturbance of the interface in the base state itself, thereby facilitating the solution for a finite amplitude disturbance. This procedure brings the base state (or the zeroth order solution) closer to the initial perturbation, in contrast to the choice of the base state as an undisturbed, planar interface. The formalism gives the growth rate of the initial perturbations as a function of various system parameters, the wave number, and the amplitude of the initial perturbation. As opposed to the linear theory then, both the neutrally stable state of the thin film and the dominant growth rate of the perturbations are functions of the amplitude of the perturbations. The time of rupture is derived for a thin film in contact with a solid and bounded by a thick bulk phase, with soluble surfactants present in both the film and the bounding fluid. The analytical expressions for a thin film devoid of surfactants are in good agreement with the numerical calculations of Williams and Davis for the same case. The nonlinear effects of the surfactant concentration driven Marangoni-motion (surface elasticity) and the surface shear viscosity are then considered. It is shown that for large initial perturbations, both the Marangoni-flow and the surface viscosity may prolong the time of rupture of a thin film by as much as a factor of 10, whereas the linear theory predicts this factor to be 4. Finally, the time of rupture of the mucus layer covering the corneal epithelium is derived and the effects of alterations in various characteristics of the external eye on the time of tear film breakup are studied. It is shown that large quantitative discrepancies exist between the time of tear film breakup computed by the linear and the nonlinear theories, if the amplitude of perturbations at the mucousaqueous interface is relatively large.

Stability with respect to a specified number of variables

The stability with respect to a specified number of variables (specified quantitatively) is considered, the set of which changes depending on the initial conditions. The stability conditions of the form considered are derived using the Lyapunov-function method. The problem of the quenching of rotational motions with respect to the centre of mass of an asymptotic solid is examined; it is shown that the “twist” of a body with respect to one of the principal axes of its ellipsoid of inertia is possible using only one “fixed jet engines” for any large initial perturbations.

Motion of an electron bunch through a plasma

The motion of a tenuous bunch of fast electrons through an isotropic plasma has been studied using a one-dimensional Vlasov code [J. Comp. Phys. 27, 315 (1978)]. Given a fairly large initial perturbation from equilibrium, the moving bunch sheds a train of nonlinear plasma oscillations, which appear to be sustained by a mild instability.

4391027 Method of making a heat exchanger assembly : James Fleury, assigned to Ex-Cell-O Corporation

We study the large-time behavior of solutions to the compressible Navier–Stokes equations for a viscous and heat-conducting ideal polytropic gas in the one-dimensional half-space. A rarefaction wave and its superposition with a non-degenerate stationary solution are shown to be asymptotically stable for the outflow problem with large initial perturbation and general adiabatic exponent.

Oscillatory phenomena in a model of infectious diseases

Oscillations of the number of cases around an average endemic level are common in several infectious diseases. In this paper we study simple deterministic models, where the oscillations arise either solely from periodically varying contact rates or from the combined effect of large initial perturbation, small periodic variation of the contact rate, and the destabilizing nature of infectious and latent periods when described as time delays. The main results are: (a) For a model with a periodically varying contact rate and a recovery rate, a threshold amplitude of variation is found by numerical and analytic methods at which 2-year subharmonic resonance appears. (b) Approximate analytic relationships are derived for the amplitude and phase of the forced 1-year oscillations below this threshold and for the 2-year oscillations above it—in terms of the reproduction rate of the infection. (c) Similar calculations are performed when the recovery rate is replaced by a fixed infectious period represented by a pure time delay. The threshold amplitude of variation in the contact rate is found here to be smaller than in the recovery rate model. (d) A model with a fixed infectious period and a constant contact rate is considered. The nontrivial steady state is shown to be locally stable for the parameter range of interest. However, the ratio of the imaginary to real parts of the eigenvalues in the characteristic equation is increased as compared to the corresponding model with a recovery rate. (e) For the model with a fixed infectious period and a constant contact rate an approximation method indicates consistency in a certain range of contact rates with the existence of an unstable periodic solution about the locally stable steady state. The actual existence of such a solution is not verified. The interpretation is that the destabilizing effect of the introduction of a pure delay into the model becomes more significant as the distance in the variables space from the endemic steady state is increased. (f) For a fixed infectious period and very small subthreshold variation in the contact rate, two different types of solutions are found numerically: yearly small-amplitude oscillations about an endemic average and large-amplitude oscillations of a subharmonic period. The pattern seen depends on the initial conditions. For a sufficiently large initial deviation from the endemic level even very small seasonal variations lead to regular recurrent outbreaks of the disease. The effect of latent periods and of changing the form of the interaction are also considered.

Nonlinear equatorial spread F: Spatially large bubbles resulting from large horizontal scale initial perturbations

Motivated by the observations of large horizontal scale length equatorial spread F ‘bubbles,’ we have performed numerical simulations of the nonlinear evolution of the collisional Rayleigh‐Taylor instability in the nighttime equatorial ionosphere, using large horizontal scale length initial perturbations. The calculations were performed using a new, improved numerical code which utilizes the recently developed, fully multidimensional flux‐corrected transport (FCT) techniques. We find that large horizontal scale initial perturbations evolve nonlinearly into equally large horizontal scale spread F bubbles, on a time scale as fast as that of the corresponding small horizontal scale length perturbations previously used. Further, we find the level of plasma depletion inside the large‐scale bubbles to be appreciably higher than that of the smaller‐scale bubbles, approaching 100%, in substantial agreement with the observations. This level of depletion is due to the fact that the plasma comprising the large‐scale bubbles has its origin at much lower altitudes than that comprising the smaller scale bubbles. Analysis of the polarization electric fields produced by the vertically aligned ionospheric irregularities show this effect to be due to fringe fields similar in structure to those produced at the edge of a parallel plate capacitor.

Dissipative Effect on Formation of Ion Acoustic Solitons

Effect of dissipation on the formation of ion acoustic solitons is investigated by experiments and by numerical calculations. In the experiments, a large amplitude initial perturbation does not evolve into solitons, but into a triangular wave with ripples on a leg when a neutral gas pressure is increased or when a random low frequency noise is superposed. Numerical analyses of the K-dV equation with dissipation terms show that an initial perturbation develops into a triangular wave when the dissipation depends on the wavenumber. Therefore the formation of the triangular wave in the experiment is not due to an increase of collisional damping, but to an increase of collusionless dissipation, i.e. Landau damping resulting from a decrease of the temperature ratio between the electrons and ions.