Rayleigh–Taylor and Richtmyer–Meshkov instabilities of flat and curved interfaces

2009 ◽  
Vol 625 ◽  
pp. 387-410 ◽  
Author(s):  
R. KRECHETNIKOV
Keyword(s):  
Growth Rate ◽  
Stability Analysis ◽  
Velocity Field ◽  
Systematic Approach ◽  
Ad Hoc ◽  
Frame Of Reference ◽  
Boundary Perturbation ◽  
The Stability ◽  
Selection Of ◽  
Wavenumber Selection

In this work we discuss a non-trivial effect of the interfacial curvature on the stability of uniformly and suddenly accelerated interfaces, such as liquid rims. The new stability analysis is based on operator and boundary perturbation theories and allows us to treat the Rayleigh–Taylor and Richtmyer–Meshkov instabilities as a single phenomenon and thus to understand the interrelation between these two fundamental instabilities. This leads, in particular, to clarification of the validity of the original Richtmyer growth rate equation and its crucial dependence on the frame of reference. The main finding of this study is the revealed and quantified influence of the interfacial curvature on the growth rates and the wavenumber selection of both types of instabilities. Finally, the systematic approach taken here also provides a generalization of the widely accepted ad hoc idea, due to Layzer (Astrophys. J., vol. 122, 1955, pp. 1–12), of approximating the potential velocity field near the interface.

Stability of bounded rapid shear flows of a granular material

1996 ◽  
Vol 308 ◽  
pp. 31-62 ◽  
Author(s):  
Chi-Hwa Wang ◽  
R. Jackson ◽  
S. Sundaresan
Keyword(s):  
Growth Rate ◽  
Stability Analysis ◽  
Granular Material ◽  
Bulk Density ◽  
Particulate Material ◽  
Dominant Mode ◽  
Marginal Stability ◽  
Sheared Layer ◽  
The Stability ◽  
Unusual Type

This paper presents a linear stability analysis of a rapidly sheared layer of granular material confined between two parallel solid plates. The form of the steady base-state solution depends on the nature of the interaction between the material and the bounding plates and three cases are considered, in which the boundaries act as sources or sinks of pseudo-thermal energy, or merely confine the material while leaving the velocity profile linear, as in unbounded shear. The stability analysis is conventional, though complicated, and the results are similar in all cases. For given physical properties of the particles and the bounding plates it is found that the condition of marginal stability depends only on the separation between the plates and the mean bulk density of the particulate material contained between them. The system is stable when the thickness of the layer is sufficiently small, but if the thickness is increased it becomes unstable, and initially the fastest growing mode is analogous to modes of the corresponding unbounded problem. However, with a further increase in thickness a new mode becomes dominant and this is of an unusual type, with no analogue in the case of unbounded shear. The growth rate of this mode passes through a maximum at a certain value of the thickness of the sheared layer, at which point it grows much faster than any mode that could be shared with the unbounded problem. The growth rate of the dominant mode also depends on the bulk density of the material, and is greatest when this is neither very large nor very small.

Stability of time-dependent rotational Couette flow Part 2. Stability analysis

1971 ◽  
Vol 48 (2) ◽  
pp. 365-384 ◽  
Author(s):  
C. F. Chen ◽  
R. P. Kirchner
Keyword(s):  
Growth Rate ◽  
Stability Analysis ◽  
Kinetic Energy ◽  
Initial Value Problem ◽  
Basic Flow ◽  
The Other ◽  
Time Dependent ◽  
Stability Criteria ◽  
Initial Value ◽  
The Stability

The stability of the flow induced by an impulsively started inner cylinder in a Couette flow apparatus is investigated by using a linear stability analysis. Two approaches are taken; one is the treatment as an initial-value problem in which the time evolution of the initially distributed small random perturbations of given wavelength is monitored by numerically integrating the unsteady perturbation equations. The other is the quasi-steady approach, in which the stability of the instantaneous velocity profile of the basic flow is analyzed. With the quasi-steady approach, two stability criteria are investigated; one is the standard zero perturbation growth rate definition of stability, and the other is the momentary stability criterion in which the evolution of the basic flow velocity field is partially taken into account. In the initial-value problem approach, the predicted critical wavelengths agree remarkably well with those found experimentally. The kinetic energy of the perturbations decreases initially, reaches a minimum, then grows exponentially. By comparing with the experimental results, it may be concluded that when the perturbation kinetic energy has grown a thousand-fold, the secondary flow pattern is clearly visible. The time of intrinsic instability (the time at which perturbations first tend to grow) is about ¼ of the time required for a thousandfold increase, when the instability disks are clearly observable. With the quasi-steady approach, the critical times for marginal stability are comparable to those found using the initial-value problem approach. The predicted critical wavelengths, however, are about 1½ to 2 times larger than those observed. Both of these points are in agreement with the findings of Mahler, Schechter & Wissler (1968) treating the stability of a fluid layer with time-dependent density gradients. The zero growth rate and the momentary stability criteria give approximately the same results.

Linear Stability Analysis in a Compressible Axisymmetric Swirling Jet

2014 ◽  
Vol 574 ◽  
pp. 15-20
Author(s):  
Zhi Wei Guo ◽  
Si Min Shen ◽  
Wei Min Feng ◽  
Bo Fu Wang
Keyword(s):  
Growth Rate ◽  
Stability Analysis ◽  
Linear Stability ◽  
Previous Analysis ◽  
Optimal Growth ◽  
Flow Dynamics ◽  
Swirl Number ◽  
Swirling Jet ◽  
Optimal Growth Rate ◽  
The Stability

Temporal linear stability of a compressible axisymmetric swirling jet is investigated. The present work extends a previous analysis to include the effects of swirl number on the stability of flow dynamics. Results obtained show that the optimal growth rate of disturbance for azimuthal wavenumber n = -1 is larger than that for n = -2 while the corresponding frequencies for both n increases as axial wavenumber increases. As swirl number q increases, the optimal growth rate of disturbance also increases. What is more, there is an optimal swirl number for small axial wavenumbers, which is different from the situation for medium and large axial wavenumbers.

Feedback stabilization of tokamaks against slow axisymmetric MHD instabilities

1981 ◽  
Vol 26 (2) ◽  
pp. 351-357 ◽  
Author(s):  
Torkil H. Jensen
Keyword(s):  
Growth Rate ◽  
Control System ◽  
Stability Analysis ◽  
Vacuum Chamber ◽  
Feedback Stabilization ◽  
Wall Material ◽  
Growth Time ◽  
Conducting Wall ◽  
Mhd Instabilities ◽  
The Stability

Single axis tokamaks as well as doublets may be unstable toward axisymmetric MHD instabilities. Such instabilities may, for the case of a single-axis tokamak, be slow when the plasma is surrounded by a relatively close fitting conducting wall, such as a vacuum chamber; the growth rate may be proportional to the resistivity of the wall material. For the case of doublets, slowly growing instabilities with growth rates proportional to the plasma resistivity exist. Such slow instabilities can be stabilized by feedback control of the currents through coils surrounding the plasma; since it is only required that the amplifiers used in the circuits respond fast compared with the growth time of the slow instabilities, this feedback stabilization is not technologically demanding. This paper describes a formalism for the stability analysis of such a system consisting of the plasma, surrounded by a conducting wall or vacuum chamber and coils with their control system.

scholarly journals Mullins–Sekerka stability analysis for melting-freezing waves in helium

1994 ◽  
Vol 5 (1) ◽  
pp. 21-37
Author(s):  
Joseph D. Fehribach
Keyword(s):  
Surface Tension ◽  
Growth Rate ◽  
Stability Analysis ◽  
Original Problem ◽  
The Other ◽  
Field Conditions ◽  
System Of Equations ◽  
Far Field ◽  
Other Hand ◽  
The Stability

This paper considers the stability of melt-solid interfaces to eigenfunction perturbations for a system of equations which describe the melting and freezing of helium. The analysis is carried out in both planar and spherical geometries. The principal results are that when the melt is freezing, under certain far-field conditions, the interface is stable in the sense of Mullins and Sekerka. On the other hand, when the solid is melting (at least when the melting is sufficiently fast), the interface is unstable. In some circumstances these instabilities are oscillatory, with amplitude and growth rate increasing with surface tension and frequency. The last section considers the original problem of Mullins and Sekerka in the present notation.

The effect of surfactant on the stability of a fluid filament embedded in a viscous fluid

1999 ◽  
Vol 382 ◽  
pp. 331-349 ◽  
Author(s):  
S. HANSEN ◽  
G. W. M. PETERS ◽  
H. E. H. MEIJER
Keyword(s):  
Surface Tension ◽  
Growth Rate ◽  
Reynolds Number ◽  
Stability Analysis ◽  
Viscous Fluid ◽  
High Elasticity ◽  
Fluid Filament ◽  
The Stability ◽  
Elasticity Number

The effect of surfactant on the breakup of a viscous filament, initially at rest, surrounded by another viscous fluid is studied using linear stability analysis. The role of the surfactant is characterized by the elasticity number – a high elasticity number implies that surfactant is important. As expected, the surfactant slows the growth rate of disturbances. The influence of surfactant on the dominant wavenumber is less trivial. In the Stokes regime, the dominant wavenumber for most viscosity ratios increases with the elasticity number; for filament to matrix viscosity ratios ranging from about 0.03 to 0.4, the dominant wavenumber decreases when the elasticity number increases. Interestingly, a surfactant does not affect the stability of a filament when the surface tension (or Reynolds) number is very large.

scholarly journals Application of the Multiple Exp-Function, Cross-Kink, Periodic-Kink, Solitary Wave Methods, and Stability Analysis for the CDG Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Haifa Bin Jebreen ◽  
Yurilev Chalco-Cano
Keyword(s):  
Stability Analysis ◽  
Exponential Function ◽  
Periodic Wave ◽  
Hyperbolic Function ◽  
Kink Solution ◽  
Engineering Sciences ◽  
Hirota Bilinear Form ◽  
The Stability ◽  
Wave Methods ◽  
Selection Of

In this article, the exact wave structures are discussed to the Caudrey-Dodd-Gibbon equation with the assistance of Maple based on the Hirota bilinear form. It is investigated that the equation exhibits the trigonometric, hyperbolic, and exponential function solutions. We first construct a combination of the general exponential function, periodic function, and hyperbolic function in order to derive the general periodic-kink solution for this equation. Then, the more periodic wave solutions are presented with more arbitrary autocephalous parameters, in which the periodic-kink solution localized in all directions in space. Furthermore, the modulation instability is employed to discuss the stability of the available solutions, and the special theorem is also introduced. Moreover, the constraint conditions are also reported which validate the existence of solutions. Furthermore, 2-dimensional graphs are presented for the physical movement of the earned solutions under the appropriate selection of the parameters for stability analysis. The concluded results are helpful for the understanding of the investigation of nonlinear waves and are also vital for numerical and experimental verification in engineering sciences and nonlinear physics.

scholarly journals Stability analysis of the Kosciuszko Mound using terrestrial laser scanner and numerical modelling

2018 ◽  
Vol 66 ◽  
pp. 01018
Author(s):  
Elżbieta Pilecka ◽  
Karolina Tomaszkiewicz
Keyword(s):  
Stability Analysis ◽  
Laser Scanner ◽  
Technical Condition ◽  
Point Of View ◽  
Terrestrial Laser Scanner ◽  
The Finite Element Method ◽  
Anthropogenic Soils ◽  
Differential Model ◽  
The Stability ◽  
Selection Of

Landslides which form in anthropogenic soils are complicated from a geological engineering and geotechnical point of view. Each case requires a detailed investigation and the selection of effective reinforcements is a difficult project issue. The study presents the problem of the stability analysis of landslides occurring in the anthropogenic soils of the Kosciuszko Mound in Cracow. The previously performed protections are discussed to highlight their ineffectiveness and the current technical condition of the mound is also presented. By overlapping the results of displacement measurements made with a terrestrial laser scanner, a differential model of the terrain was created which made it possible to determine the size and direction of the deformation of the slopes of the mound and the tendencies for the development of landslide movements in this area. A cross-section, selected on the basis of the model, was numerically analysed using the finite element method (FEM) in the Midas GTS NX program. As a result of the analysis, the values of the displacements and strains occurring in the Mound were calculated. On the basis of the value of the safety factor obtained, it was also possible to assess the risk of landslide movements.

Thermoacoustic Instability Model With Porous Media: Linear Stability Analysis and the Impact of Porous Media

2018 ◽  
Author(s):  
Cody S. Dowd ◽  
Joseph W. Meadows
Keyword(s):  
Growth Rate ◽  
Porous Media ◽  
Stability Analysis ◽  
Frequency Shift ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Growth Rates ◽  
Thermoacoustic Instability ◽  
The Stability ◽  
The Impact

Lean premixed (LPM) combustion systems are susceptible to thermoacoustic instability, which occurs when acoustic pressure oscillations are in phase with the unsteady heat release rates. Porous media has inherent acoustic damping properties, and has been shown to mitigate thermoacoustic instability; however, theoretical models for predicting thermoacoustic instability with porous media do not exist. In the present study, a 1-D model has been developed for the linear stability analysis of the longitudinal modes for a series of constant cross-sectional area ducts with porous media using a n-Tau flame transfer function. By studying the linear regime, the prediction of acoustic growth rates and subsequently the stability of the system is possible. A transfer matrix approach is used to solve for acoustic perturbations of pressure and velocity, stability growth rate, and frequency shift without and with porous media. The Galerkin approximation is used to approximate the stability growth rate and frequency shift, and it is compared to the numerical solution of the governing equations. Porous media is modeled using the following properties: porosity, flow resistivity, effective bulk modulus, and structure factor. The properties of porous media are systematically varied to determine the impact on the eigenfrequencies and stability growth rates. Porous media is shown to increase the stability domain for a range of time delays (Tau) compared to similar cases without porous media.

Thermoacoustic Instability Model With Porous Media: Linear Stability Analysis and the Impact of Porous Media

2018 ◽  
Vol 141 (4) ◽  
Author(s):  
Cody S. Dowd ◽  
Joseph W. Meadows
Keyword(s):  
Growth Rate ◽  
Porous Media ◽  
Stability Analysis ◽  
Frequency Shift ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Growth Rates ◽  
Thermoacoustic Instability ◽  
The Stability ◽  
The Impact

Lean premixed (LPM) combustion systems are susceptible to thermoacoustic instability, which occurs when acoustic pressure oscillations are in phase with the unsteady heat release rates. Porous media has inherent acoustic damping properties and has been shown to mitigate thermoacoustic instability; however, theoretical models for predicting thermoacoustic instability with porous media do not exist. In the present study, a one-dimensional (1D) model has been developed for the linear stability analysis of the longitudinal modes for a series of constant cross-sectional area ducts with porous media using a n-Tau flame transfer function (FTF). By studying the linear regime, the prediction of acoustic growth rates and subsequently the stability of the system is possible. A transfer matrix approach is used to solve for acoustic perturbations of pressure and velocity, stability growth rate, and frequency shift without and with porous media. The Galerkin approximation is used to approximate the stability growth rate and frequency shift, and it is compared to the numerical solution of the governing equations. Porous media is modeled using the following properties: porosity, flow resistivity, effective bulk modulus, and structure factor. The properties of porous media are systematically varied to determine the impact on the eigenfrequencies and stability growth rates. Porous media is shown to increase the stability domain for a range of time delays (Tau) compared to similar cases without porous media.

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