Comparison of linear stability results with flight transition data

AIAA Journal ◽  
1995 ◽  
Vol 33 (1) ◽  
pp. 161-163 ◽  
Author(s):  
J. A. Masad ◽  
M. R. Malik
Keyword(s):  
Linear Stability ◽  
Stability Results

Density-driven instabilities of miscible fluids in a Hele-Shaw cell: linear stability analysis of the three-dimensional Stokes equations

2002 ◽  
Vol 451 ◽  
pp. 261-282 ◽  
Author(s):  
F. GRAF ◽  
E. MEIBURG ◽  
C. HÄRTEL
Keyword(s):  
Experimental Data ◽  
Growth Rate ◽  
Linear Stability ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Interface Thickness ◽  
Gap Width ◽  
Stability Results ◽  
Hele Shaw Cell

We consider the situation of a heavier fluid placed above a lighter one in a vertically arranged Hele-Shaw cell. The two fluids are miscible in all proportions. For this configuration, experiments and nonlinear simulations recently reported by Fernandez et al. (2002) indicate the existence of a low-Rayleigh-number (Ra) ‘Hele-Shaw’ instability mode, along with a high-Ra ‘gap’ mode whose dominant wavelength is on the order of five times the gap width. These findings are in disagreement with linear stability results based on the gap-averaged Hele-Shaw approach, which predict much smaller wavelengths. Similar observations have been made for immiscible flows as well (Maxworthy 1989).In order to resolve the above discrepancy, we perform a linear stability analysis based on the full three-dimensional Stokes equations. A generalized eigenvalue problem is formulated, whose numerical solution yields both the growth rate and the two-dimensional eigenfunctions in the cross-gap plane as functions of the spanwise wavenumber, an ‘interface’ thickness parameter, and Ra. For large Ra, the dispersion relations confirm that the optimally amplified wavelength is about five times the gap width, with the exact value depending on the interface thickness. The corresponding growth rate is in very good agreement with the experimental data as well. The eigenfunctions indicate that the predominant fluid motion occurs within the plane of the Hele-Shaw cell. However, for large Ra purely two-dimensional modes are also amplified, for which there is no motion in the spanwise direction. Scaling laws are provided for the dependence of the maximum growth rate, the corresponding wavenumber, and the cutoff wavenumber on Ra and the interface thickness. Furthermore, the present results are compared both with experimental data, as well as with linear stability results obtained from the Hele-Shaw equations and a modified Brinkman equation.

Sediment-laden fresh water above salt water: linear stability analysis

2011 ◽  
Vol 691 ◽  
pp. 279-314 ◽  
Author(s):  
P. Burns ◽  
E. Meiburg
Keyword(s):  
Growth Rate ◽  
Linear Stability ◽  
Fresh Water ◽  
Settling Velocity ◽  
Interface Thickness ◽  
Particle Settling ◽  
Salinity Field ◽  
Unstable Layer ◽  
Stability Results ◽  
Double Diffusive

AbstractWhen a layer of particle-laden fresh water is placed above clear, saline water, both Rayleigh–Taylor and double diffusive fingering instabilities may arise. For quasi-steady base profiles, we obtain linear stability results for such situations by means of a rational spectral approximation method with adaptively chosen grid points, which is able to resolve multiple steep gradients in the base state density profile. In the absence of salinity and for a step-like concentration profile, the dominant parameter is the ratio of the particle settling velocity to the viscous velocity scale. As long as this ratio is small, particle settling has a negligible influence on the instability growth. However, when the particles settle more rapidly than the instability grows, the growth rate decreases inversely proportional to the settling velocity. This damping effect is a result of the smearing of the vorticity field, which in turn is caused by the deposition of vorticity onto the fluid elements passing through the interface between clear and particle-laden fluid. In the presence of a stably stratified salinity field, this picture changes dramatically. An important new parameter is the ratio of the particle settling velocity to the diffusive spreading velocity of the salinity, or alternatively the ratio of the unstable layer thickness to the diffusive interface thickness of the salinity profile. As long as this quantity does not exceed unity, the instability of the system and the most amplified wavenumber are primarily determined by double diffusive effects. In contrast to situations without salinity, particle settling can have a destabilizing effect and significantly increase the growth rate. Scaling laws obtained from the linear stability results are seen to be largely consistent with earlier experimental observations and theoretical arguments put forward by other authors. For unstable layer thicknesses much larger than the salinity interface thickness, the particle and salinity interfaces become increasingly decoupled, and the dominant instability mode becomes Rayleigh–Taylor-like, centred at the lower boundary of the particle-laden flow region.

Linear stability analysis for periodic travelling waves of the Boussinesq equation and the Klein–Gordon–Zakharov system

2014 ◽  
Vol 144 (3) ◽  
pp. 455-489 ◽  
Author(s):  
Sevdzhan Hakkaev ◽  
Milena Stanislavova ◽  
Atanas Stefanov
Keyword(s):  
Stability Analysis ◽  
Linear Stability ◽  
Wide Class ◽  
Boussinesq Equation ◽  
Travelling Waves ◽  
Periodic Waves ◽  
Zakharov System ◽  
Klein Gordon ◽  
Spatially Periodic ◽  
Stability Results

The question of the linear stability of spatially periodic waves for the Boussinesq equation (in the cases p = 2, 3) and the Klein–Gordon–Zakharov system is considered. For a wide class of solutions, we completely and explicitly characterize their linear stability (instability) when the perturbations are taken with the same period T. In particular, our results allow us to completely recover the linear stability results, in the limit T → ∞, for the whole-line case.

scholarly journals On the instabilities of a potential vortex with a free surface

2017 ◽  
Vol 824 ◽  
pp. 230-264 ◽  
Author(s):  
J. Mougel ◽  
D. Fabre ◽  
L. Lacaze ◽  
T. Bohr
Keyword(s):  
Free Surface ◽  
Linear Stability ◽  
Swirling Flow ◽  
Simple Wave ◽  
Unstable Wave ◽  
Bottom Plate ◽  
Coupling Mechanism ◽  
Wave Coupling ◽  
Potential Vortex ◽  
Stability Results

In this paper, we address the linear stability analysis of a confined potential vortex with a free surface. This particular flow has been recently used by Tophøj et al. (Phys. Rev. Lett., vol. 110(19), 2013, article 194502) as a model for the swirling flow of fluid in an open cylindrical container, driven by rotating the bottom plate (the rotating bottom experiment) to explain the so-called rotating polygons instability (Vatistas J. Fluid Mech., vol. 217, 1990, pp. 241–248; Jansson et al., Phys. Rev. Lett., vol. 96, 2006, article 174502) in terms of surface wave interactions leading to resonance. Global linear stability results are complemented by a Wentzel–Kramers–Brillouin–Jeffreys (WKBJ) analysis in the shallow-water limit as well as new experimental observations. It is found that global stability results predict additional resonances that cannot be captured by the simple wave coupling model presented in Tophøj et al. (2013). Both the main resonances (thought to be at the root of the rotating polygons) and these secondary resonances are interpreted in terms of over-reflection phenomena by the WKBJ analysis. Finally, we provide experimental evidence for a secondary resonance supporting the numerical and theoretical analysis presented. These different methods and observations allow to support the unstable wave coupling mechanism as the physical process at the origin of the polygonal patterns observed in free-surface rotating flows.

scholarly journals Resonant instability in two-dimensional vortex arrays

2010 ◽  
Vol 467 (2128) ◽  
pp. 1164-1185 ◽  
Author(s):  
Paolo Luzzatto-Fegiz ◽  
Charles H. K. Williamson
Keyword(s):  
Linear Stability ◽  
Stability Boundary ◽  
Oscillatory Instability ◽  
Two Dimensional ◽  
Complex Flows ◽  
Stability Calculation ◽  
Resonant Instability ◽  
The Stability ◽  
Stability Results ◽  
Good Agreement

We examine conditions for the development of an oscillatory instability in two-dimensional vortex arrays. By building on the theory of Krein signatures for Hamiltonian systems, and considering constraints owing to impulse conservation, we show that a resonant instability (developing through coalescence of two eigenvalues) cannot occur for one or two vortices. We illustrate this deduction by examining available linear stability results for one or two vortices. Our work indicates that a resonant instability may, however, occur for three or more vortices. For these more complex flows, we propose a simple model, based on an elliptical vortex representation, to detect the onset of an oscillatory instability. We provide an example in support of our theory by examining three co-rotating vortices, for which we also perform a linear stability analysis. The stability boundary in our model is in good agreement with the full stability calculation. In addition, we show that eigenmodes associated with an overall rotation or an overall displacement of the vortices always have eigenvalues equal to zero and ±i Ω , respectively, where Ω is the angular velocity of the array. These results, for overall rotation and displacement modes, can also be used to immediately check the accuracy of a detailed stability calculation.

scholarly journals Stability of Solutal Convection in a Gravity Modulated Mushy Layer During the Solidification of Binary Alloys

2005 ◽  
Author(s):  
S. Govender
Keyword(s):  
Linear Stability ◽  
Transition Point ◽  
Binary Alloys ◽  
Linear Stability Theory ◽  
Mushy Layer ◽  
Gravity Modulation ◽  
Solutal Convection ◽  
Mushy Layers ◽  
Stefan Number ◽  
Stability Results

The linear stability theory is used to investigate the effects of gravity modulation on solutal convection in mushy layers found in solidifying binary alloys. The gravitational field is modeled to consist of constant part and a sinusoidally varying part. The linear stability results are presented for both the synchronous and subharmonic solutions, and it is demonstrated that up to the transition point between the synchronous and subharmonic regions, increasing the frequency of vibration rapidly stabilizes the solutal convection. However, beyond the transition point, further increases in the frequency tends to destabilize convection. It is also demonstrated that the effect of increasing the ratio of the Stefan number and the solid composition (ηo) is to destabilize the solutal convection.

scholarly journals On the stability of a falling liquid curtain

2002 ◽  
Vol 463 ◽  
pp. 163-171 ◽  
Author(s):  
PETER J. SCHMID ◽  
DAN S. HENNINGSON
Keyword(s):  
Linear Stability ◽  
Low Frequency ◽  
Frequency Range ◽  
Linear Superposition ◽  
Air Cushion ◽  
Global Modes ◽  
The Stability ◽  
Linear Stability Problem ◽  
Stability Results ◽  
Good Agreement

The stability of a falling liquid curtain is investigated. The sheet of liquid is assumed two-dimensional, driven by gravity and influenced by a compressible cushion of air enclosed on one side of the curtain. The linear stability problem is formulated in the form of an integro-differential eigenvalue problem. Although experimental efforts have consistently reported a peak in the low-frequency range of the spectrum, the linear stability results do not show instabilities at these frequencies. However, a multi-modal approach combined with a projection onto low-frequency modes reveals a dominant and robust instability feature that is in good agreement with experimental measurements. This instability manifests itself as a wave packet, consisting of a linear superposition of linear global modes, that travels down the curtain and causes a strong pressure signal in the enclosed air cushion.

scholarly journals Influence of viscosity contrast on buoyantly unstable miscible fluids in porous media

2015 ◽  
Vol 780 ◽  
pp. 388-406 ◽  
Author(s):  
Satyajit Pramanik ◽  
Tapan Kumar Hota ◽  
Manoranjan Mishra
Keyword(s):  
Linear Stability ◽  
Injection Velocity ◽  
External Excitation ◽  
Initial Value ◽  
Miscible Fluids ◽  
Viscosity Contrast ◽  
Stability Method ◽  
Linearized Operator ◽  
Stability Results ◽  
Dimensionless Formulation

The influence of viscosity contrast on buoyantly unstable miscible fluids in a porous medium is investigated through a linear stability analysis (LSA) as well as direct numerical simulations (DNS). The linear stability method implemented in this paper is based on an initial value approach, which helps to capture the onset of instability more accurately than the quasi-steady-state analysis. In the absence of displacement, we show that viscosity contrast delays the onset of instability in buoyantly unstable miscible fluids. Further, it is observed that by suitably choosing the viscosity contrast and injection velocity a gravitationally unstable miscible interface can be stabilized completely. Through LSA we draw a phase diagram, which shows three distinct stability regions in a parameter space spanned by the displacement velocity and the viscosity contrast. DNS are performed corresponding to parameters from each regime and the results obtained are in accordance with the linear stability results. Moreover, the conversion from one dimensionless formulation to another and its importance when comparing the different type of flow problem associated with each dimensionless formulation are discussed. We also calculate${\it\epsilon}$-pseudo-spectra of the time dependent linearized operator to investigate the response to external excitation.

Fluid Dynamical and Modeling Issues of Chemical Flooding for Enhanced Oil Recovery

2013 ◽  
Author(s):  
Prabir Daripa
Keyword(s):  
Porous Media ◽  
Linear Stability ◽  
Enhanced Oil Recovery ◽  
Oil Recovery ◽  
Chemical Flooding ◽  
Model Based ◽  
Optimal Injection ◽  
Exhaustive Study ◽  
Stability Results

This paper evaluates the relevance of Hele-Shaw (HS) model based linear stability results to fully developed flows in enhanced oil recovery (EOR). In a recent exhaustive study [Transport in Porous Media, 93, 675–703 (2012)] of the linear stability characteristics of unstable immiscible three-layer “Hele-Shaw” flows involving regions of varying viscosity, an optimal injection policy corresponding to the smallest value of the highest rate of growth of instabilities was identified among several injection policies. Relevance of this HS model based result to EOR is established by performing direct numerical simulations of fully developed tertiary displacement in porous media. Results of direct numerical simulation are succinctly summarized including characterization of the optimal flooding scheme that leads to maximum oil recovery. These results have been compared with the HS model based linear stability results. The scope for potential application of the HS model based results to the development of fast methods for optimization of various chemical flooding schemes is discussed. Numerical experiments with more complex flooding schemes in both homogeneous and heterogeneous reservoir are also performed and results analyzed to test the universality of the generic optimal viscous families in a broader setting.

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