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An interpretation of absolutely and convectively unstable waves using series solutions
Abstract
Integral solutions are found for linear wave equations, which, depending on the parameters, exhibit either absolute or convective instability. Series solutions are constructed to examine the instability behavior on a bounded domain. Solutions with non-real wave-numbers can be interpreted as a superposition of eigenmodes with jump-periodic boundary conditions. We show that an initial disturbance can be represented by not only periodic modes, but also spatially growing (or decaying) modes, and arbitrary modes using a Galerkin approach. For the examples presented here, growth in any such series solution matches the growth predictions obtained from the long-time asymptotic behavior of the integral solutions.
... This characterization assumes that the examination of individual modes is sufficient to characterize system stability. However, there are several neutrally stable flows in the literature [2][3][4] whose corresponding PDEs admit solutions that decay algebraically (typically like t −1/2 ), shown through the method of stationary phase or steepest descent in a long-time asymptotic analysis [5,6]. For our proposed work, we are interested in the less-understood case of algebraic growth (i.e., perturbations that grow as t s , s > 0). ...
... The delta function may be thought of as an idealized Gaussian [33]. If one were to use an actual Gaussian (or another localized continuous function) to initiate the disturbances examined in this paper, the long-time asymptotic stability result would be identical [4,23]. In thin liquid sheets, for example, the delta function may be motivated as an idealized Gaussian pressure disturbance in the dynamic boundary condition across the surface of the sheet [34]. ...
... (2) can be solved completely using a Fourier series expansion [4,23,34], such an approach does not provide precise growth rates, and thus conclusions regarding system stability, as well as the structure of the response, may be obscured. Spatiotemporal stability theory enables one to explicitly extract the details of growth through asymptotic analysis [35,36]. ...
A largely unexplored type of hydrodynamic instability is examined: long-time algebraic growth. Such growth is possible when the dispersion relation extracted from classical stability analysis indicates neutral stability. A physically motivated class of partial differential equations that describes the response of a system to disturbances is examined. Specifically, the propagation characteristics of the response are examined in the context of spatiotemporal stability theory. Morphological differences are identified between system responses that exhibit algebraic growth and the more typical case of exponential growth. One key attribute of predicted algebraically growing solutions is the prevalence of transient growth in almost all of the response, with the long-time growth occurring asymptotically at precisely one wave speed.
... Following the procedure given in [Barlow et al., 2010], the integral solution of (2) is obtained by first taking the successive Fourier and Laplace transforms such that the partial differential equation in h(x, t) becomes an algebraic equation in H(k, s). It is implicit in this notation that x transforms to k, t transforms to s, and h transforms in aggregate to H. ...
... The first integration (in ω or k) is carried out by applying the residue theorem, which leads to the substitution of either k(ω) roots or ω(k) roots of D(k, ω)=0 in to the argument of the exponential in (3). Although both approaches have been taken in the literature to analyze the response to an initial disturbance [Ashpis & Reshotko, 1990, Huerre, 2000, Schmid & Henningson, 2001, performing the Laplace integral first is generally easier [Barlow et al., 2010, Barlow et al., 2011 and this simplicity is even more apparent in the signaling problem [Gordillo & Pérez-Saborid, 2002, Barlow et al., 2012. ...
... The transfer function is compared with the Fourier series solution to (2) in Figure (10), where it can be seen that V E clearly separates regions of spatial and temporal growth. The Fourier series is constructed on a periodic domain, following the approach in [Barlow et al., 2010]. Observe that V E falls between the velocities V − and V + , which bound the time-growing packet; this makes sense for the case of forced spatial growth (ω f < ω f,c ). ...
The response of convectively unstable equations to a localized oscillatory forcing (i.e. the ‘signalling problem’) is studied. The full mathematical structure of this class of problems is elucidated by examining partial differential equations of second-order (the linear Ginzburg–Landau equation) and fourth-order (the linear Kuramoto–Sivashinsky equation) in space. The long-time asymptotic behaviours of the Fourier–Laplace integral solutions are obtained via contour integration and the method of steepest
descent. In the process, a general algorithm is developed to extract the important physical characteristics of such problems. The algorithm allows one to determine the velocities that bound the transient and spatially growing portions of the response, as well as a closed-form transfer function that relates the oscillatory disturbance amplitude to that of the spatially growing solution. A new velocity is identified that provides the most meaningful demarcation of the two regions. The algorithm also provides a straightforward criterion for identifying ‘contributing’ saddles that determine the long-time asymptotic behaviour and ‘non-contributing’ saddles that give errant solutions. Lastly, a discontinuity that arises in the long-time asymptotic solution, identified in prior studies, is resolved.
... The response of convectively unstable flows to oscillatory forcing 119 have also constructed the same solution using complex wavenumbers, as outlined in Barlow et al. (2010). Solutions are valid provided that the leading edge of the downstream response does not reach the boundaries of the domain at x = ±L. ...
... To obtain the double integral solution of a constant coefficient linear partial differential equation, the Fourier and Laplace transforms are taken, the transformed dependent variable is then solved for and successive inverse transforms are taken to return to the original variable; detailed examples are given in Barlow et al. (2010) for similar wave evolution equations. Following this procedure, the integral solution ...
... Evaluation of the steepest descent path through the saddle of maximum aggregate growth usually provides the asymptotic description of waves in the unforced problem as t → ∞ (Barlow et al. 2010). However, this saddle does not provide any information about the envelope of steady spatial growth induced by forcing; for that, we will need to consider the spectrum of all possibly growing saddles and identify x/t rays that are critical to determining the ultimate evolution of the forced solution. ...
The complete integral solution is found for the convectively unstable and oscillatory-forced linear Klein–Gordon equation as a function of spatial variable, , and time, . A comparison of the integral solution with series solutions of the Klein–Gordon equation elucidates salient features of both the transient and long-time spatially growing solutions. A rigorous method is developed for identifying the key rays associated with saddle points that can be used to characterize the transition between transient temporally growing and long-term spatially growing waves. This method effectively combines the procedure given by Gordillo & Pérez-Saborid (Phys. Fluids, vol. 14, 2002, pp. 4329–4343) for determining the ray at which the forced spatial growth response affects the observed waveform and competes with the transient response, with an established methodology for identifying the leading and trailing edge rays of an impulse response. The method is applied to a linearized system describing an oscillatory-forced liquid sheet and asymptotic predictions are obtained. Series solutions are used to validate these predictions. We establish that the portion of the solution responsible for spatial growth in the signalling problem is correctly identified by Gordillo & Pérez-Saborid (Phys. Fluids, vol. 14, 2002, pp. 4329–4343), and that this interpretation is in contrast with the classical literature. The approach provided here can be applied in multiple ways to study a convectively unstable oscillatory-forced medium. In cases where numerical or series solutions are readily available, the proposed method is used to extract key features of the solution. In cases where only the forced long time behaviour is needed, the dispersion relation is used to extract: (i) the time required to see the forced solution; (ii) the amplitude, phase and spatial growth of the forced solution; and (iii) the breadth of the transient.
... Algebraic growth is defined as a system response that obeys h ∼ Ct s , where C is a constant and the exponent s is a positive rational number. It should be noted that algebraic decay of disturbances (s < 0) has also been identified in previous work when exponential modes are neutrally stable for all real k, having both integer (Case 1960) and non-integer (Whitham 2011;Lighthill 2001;Barlow et al. 2010) character. It is apparent that the classification of neutral stability via classical means is deficient and warrants further study. ...
... a periodic domain, following the approach given in Barlow et al. (2010). In figure 3 the peak is shown to be growing in accordance with the exact solution given by (2.12) along x/t = c, as indicated by a black line in the figure. ...
This paper reports a breakdown in linear stability theory under conditions of neutral stability that is deduced by an examination of exponential modes of the form $h\approx {{e}^{i(kx-\omega t)}}$, where $h$ is a response to a disturbance, $k$ is a real wavenumber, and $\omega(k)$ is a wavelength-dependent complex frequency. In a previous paper, King et al (Stability of algebraically unstable dispersive flows, \textit{Phys. Rev. Fluids}, 1(073604), 2016) demonstrates that when Im$[\omega(k)]$=0 for all $k$, it is possible for a system response to grow or damp algebraically as $h\approx {{t}^{s}}$ where $s$ is a fractional power. The growth is deduced through an asymptotic analysis of the Fourier integral that inherently invokes the superposition of an infinite number of modes. In this paper, the more typical case associated with the transition from stability to instability is examined in which Im$[\omega(k)]$=0 for a single mode (i.e., for one value of $k$) at neutral stability. Two partial differential equation systems are examined, one that has been constructed to elucidate key features of the stability threshold, and a second that models the well-studied problem of rectilinear Newtonian flow down an inclined plane. In both cases, algebraic growth/decay is deduced at the neutral stability boundary, and the propagation features of the responses are examined.
... The Fourier series solution to (2.1) is shown in Fig. 3 for u 0 + f 0 = 1. All Fourier series solutions displayed in this paper are constructed on a periodic domain, following the approach given in Barlow et al. (2010). In Fig. 3 the peak is shown to be growing in accordance with the exact solution given by (2.12) along x/t = c, as indicated by a black line in the figure. ...
This paper reports a breakdown in linear stability theory under conditions of neutral stability that is deduced by an examination of exponential modes of the form h≈ei(kx−ωt), where h is a response to a disturbance, k is a real wavenumber and ω(k) is a wavelength-dependent complex frequency. In a previous paper, King et al. (2016, Stability of algebraically unstable dispersive flows. Phys. Rev. Fluids, 1, 073604) demonstrates that when Im[ω(k)]=0 for all k, it is possible for a system response to grow or damp algebraically as h≈ts where s is a fractional power. The growth is deduced through an asymptotic analysis of the Fourier integral that inherently invokes the superposition of an infinite number of modes. In this paper, the more typical case associated with the transition from stability to instability is examined in which Im[ω(k)]=0 for a single mode (i.e. for one value of k) at neutral stability. Two partial differential equation systems are examined, one that has been constructed to elucidate key features of the stability threshold, and a second that models the well-studied problem of rectilinear Newtonian flow down an inclined plane. In both cases, algebraic growth/decay is deduced at the neutral stability boundary, and the propagation features of the responses are examined.
We consider the genesis and dynamics of interfacial instability in gas-liquid flows, using as a model the two-dimensional channel flow of a thin falling film sheared by counter-current gas. The methodology is linear stability theory (Orr-Sommerfeld analysis) together with direct numerical simulation of the two-phase flow in the case of nonlinear disturbances. We investigate the influence of three main flow parameters (density contrast between liquid and gas, film thickness, pressure drop applied to drive the gas stream) on the interfacial dynamics. Energy budget analyses based on the Orr-Sommerfeld theory reveal various coexisting unstable modes (interfacial, shear, internal) in the case of high density contrasts, which results in mode coalescence and mode competition, but only one dynamically relevant unstable internal mode for low density contrast. The same linear stability approach provides a quantitative prediction for the onset of (partial) liquid flow reversal in terms of the gas and liquid flow rates. A study of absolute and convective instability for low density contrast shows that the system is absolutely unstable for all but two narrow regions of the investigated parameter space. Direct numerical simulations of the same system (low density contrast) show that linear theory holds up remarkably well upon the onset of large-amplitude waves as well as the existence of weakly nonlinear waves. In comparison, for high density contrasts corresponding more closely to an air-water-type system, although the linear stability theory is successful at determining the most-dominant features in the interfacial wave dynamics at early-to-intermediate times, the short waves selected by the linear theory undergo secondary instability and the wave train is no longer regular but rather exhibits chaotic dynamics and eventually, wave overturning.
There are three types of linear stability theories which are currently being used to predict the onset of breakup of liquid jets or sheets. Temporal theory which is most commonly used, because of its simplicity, assumes that the disturbance responsible for the breakup grows temporally at the same rate everywhere in space. A less commonly used spatial theory assumes the disturbance grows in space, because the breakup appears to take place in the region downstream of the location where the liquid is introduced. The most complete theory is that of spatio-temporal instability. This theory has not yet been applied widely because of its mathematical and numerical complexity. It is demonstrated here with an example that a flow may be predicted to be neutral according to pure spatial or pure temporal theory, while it is actually stable according to the spatio-temporal theory. The prediction of the latter theory is shown to agree with the numerical solution of the initial value problem.
The linear stability of an inviscid two-dimensional liquid sheet falling under gravity in a still gas is studied by analysing the asymptotic behaviour of a localized perturbation (wave-packet solution to the initial value problem). Unlike previous papers the effect of gravity is fully taken into account by introducing a slow length scale which allows the flow to be considered slightly non-parallel. A multiple-scale approach is developed and the dispersion relations for both the sinuous and varicose disturbances are obtained to the zeroth-order approximation. These exhibit a local character as they involve a local Weber number We[eta]. For sinuous disturbances a critical We[eta] equal to unity is found below which the sheet is locally absolutely unstable (with algebraic growth of disturbances) and above which it is locally convectively unstable. The transition from absolute to convective instability occurs at a critical location along the vertical direction where the flow Weber number equals the dimensionless sheet thickness. This critical distance, as measured from the nozzle exit section, increases with decreasing the flow Weber number, and hence, for instance, the liquid flow rate per unit length. If the region of absolute instability is relatively small it may be argued that the system behaves as a globally stable one. Beyond a critical size the flow receptivity is enhanced and self-sustained unstable global modes should arise. This agrees with the experimental evidence that the sheet breaks up as the flow rate is reduced. It is conjectured that liquid viscosity may act to remove the algebraic growth, but the time after which this occurs could be not sufficient to avoid possible nonlinear phenomena appearing and breaking up the sheet.
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A review is given of the general theory describing the linear evolution of spatio-temporal instability waves in fluid media. According to the character of the impulse response, one can distinguish between absolutely unstable (closed) flows and convectively unstable (open) flows. These notions are then applied to several evolution models of interest in weakly nonlinear stability theory. It is argued that absolutely unstable flows, convectively unstable flows and mixed flows exhibit a very different sensitivity to external perturbations. Implications of these concepts to frequency selection mechanisms in mixed flows and the onset of chaos in convectively unstable flows are also discussed.
Preface; Overview; 1. An introduction to hydrodynamics Bernard Castaing;
2. Hydrodynamical instabilities in open flows P. Huerre and M. Rossi; 3.
Asymptotic techniques in nonlinear problems: some illustrative examples
Vincent Halim; 4. Pattern forming instabilities Stephan Fauve; 5. An
introduction to the instability of flames, shocks and detonations G.
Joulin and P. Vidal.
With applications ranging from modelling the environment to automotive
design and physiology to astrophysics, conventional textbooks cannot
hope to give students much information on what topics in fluid dynamics
are currently being researched, or how to choose between them. This book
rectifies matters. It consists of eleven chapters that introduce and
review different branches of the subject for graduate-level courses, or
for specialists seeking introductions to other areas. Hb ISBN (2001):
0-521-78061-6
Betchov and Criminale found certain singularities in the eigenvalue relations for the combined stability problem in space and time. It is shown why these singularities must occur, and how they influence the perturbation generated by a pulse.
Necessary and sufficient criteria are derived for classifying waves in complex media into evanescent, propagating, and growing waves (or instabilities). An absolute or nonconvective instability is shown to exist only if the complex group velocity dω∕dk vanishes at some point in the upper half of the frequency plane. A physical interpretation of this result is given. The analysis is extended to include slowly varying inhomogeneous media.
This paper is concerned with the problem of distinguishing between amplifying and evanescent waves. These have, in the past, been distinguished by considerations of energy transfer or of the initial and boundary conditions with which a wave must be associated. Both procedures are open to criticism. The problem is here interpreted kinematically: we investigate the classes of wave functions which a given propagating system may support, without inquiring into the way these disturbances may be set up, and postponing inquiry into the boundary conditions necessary. In this way, we may distinguish between amplifying and evanescent waves by determining whether a wave function which may be analyzed into "real-frequency" waves may also be analyzed into "real-wave-number" waves. This question may be answered by means of a certain diagram, which may be constructed from knowledge of the dispersion relation. Interchange of the roles of time and space leads to the statement and solution of a further problem. If a propagating system is unstable, the instability may be such that a disturbance grows, but is propagated away from the point of origin: this is termed "convective instability." On the other hand, the instability may be such that the disturbance grows in amplitude and in extent, but always embraces the original point of origin: this is termed "nonconvective instability." The statement that the system supports amplifying waves is synonymous with the statement that the system exhibits convective instability. A system which exhibits nonconvective instability may not be used as an amplifier, but may be used as an oscillator. It is possible to distinguish between convective and nonconvective instability by a further diagram which also may be constructed from knowledge of the dispersion relation. Our theory enables us to make the following assertions. If omega is real for all real k, then any complex k, for real omega, denotes an evanescent wave. Conversely, if k is real for all real omega, then any complex omega, for real k, denotes nonconvective instability. The theory is illustrated by certain simple examples and by discussion of the result of weak coupling between certain types of waves.
This book gives a self-contained presentation of the methods of asymptotics and perturbation theory, methods useful for obtaining approximate analytical solutions to differential and difference equations. Parts and chapter titles are as follows: fundamentals - ordinary differential equations, difference equations; local analysis - approximate solution of linear differential equations, approximate solution of nonlinear differential equations, approximate solution of difference equations, asymptotic expansion of integrals; perturbation methods - perturbation series, summation series; and global analysis - boundary layer theory, WKB theory, multiple-scale analysis. An appendix of useful formulas is included. 147 figures, 43 tables. (RWR)
This book is a detailed look at some of the more modern issues of hydrodynamic stability, including transient growth, eigenvalue spectra, secondary instability. Analytical results and numerical simulations, linear and (selected) nonlinear stability methods will be presented. By including classical results as well as recent developments in the field of hydrodynamic stability and transition, the book can be used as a textbook for an introductory, graduate-level course in stability theory or for a special-topics fluids course. It will also be of value as a reference for researchers in the field of hydrodynamic stability theory or with an interest in recent developments in fluid dynamics. Since the appearance of "Drazin & Reid", no book on hydrodynamic stability theory has been published. However, stability theory has seen a rapid development over the past decade. Direct numerical simulations of transition to turbulence and linear analysis based on the initial-value problem are two of many such developments that will be included in the book.
1 Introduction 2 Mathematical Preliminaries, Integral Formulations, and Variational Methods 3 Second-order Differential Equations in One Dimension: Finite Element Models 4 Second-order Differential Equations in One Dimension: Applications 5 Beams and Frames 6 Eigenvalue and Time-Dependent Problems 7 Computer Implementation 8 Single-Variable Problems in Two Dimensions 9 Interpolation Functions, Numerical Integration, and Modeling Considerations 10 Flows of Viscous Incompressible Fluids 11 Plane Elasticity 12 Bending of Elastic Plates 13 Computer Implementation of Two-Dimensional Problems 14 Prelude to Advanced Topics
The theory of Bohm and Gross and the experiments of Looney and Brown
upon the excitation of plasma oscillations by the two-stream mechanism,
which appear superficially to be in disagreement, are here shown to be
compatible with each other and with related experiments.
The paper reviews recent developments in the hydrodynamic stability theory of spatially developing flows pertaining to absolute/convective and local/global instability concepts. It is demonstrated how these notions can be used effectively to obtain a qualitative and quantitative description of the spatial-temporal dynamics of open shear flow, such as mixing layers, jets, wakes, boundary layers, and plane Poiseuille flow. Only 'open flows' are considered, where fluid particles do not remain within the physical domain of interest, but are advected through downstream flow boundaries. In addition, the implications of local/global and absolute/convective instability concepts for geophysical flows are alluded to briefly.
The frequency and amplification rates for a disturbance growing with respect to time are compared with those of a spatially-growing wave having the same wave-number. For small rates of amplification it is shown that the frequencies are equal to a high order of approximation, and that the spatial growth is related to the time growth by the group velocity.
The evolution of wave packets generated by impulsive disturbances of the rotating-disk boundary-layer flow is studied. It is shown, by comparison with Briggs' method, that there are certain difficulties associated with the steepest-descent time-asymptotic method of evaluating the impulse response. The rotating-disk boundary layer provides examples of saddle points through which the steepest-descent path can and cannot be made to pass. It is shown that it is critically important to establish, from the topography of the phase function, whether the steepest-descent path passes through particular saddle points. If this procedure is not carried out, calculations of the wave-packet evolution can be majorly flawed and incorrect conclusions can be drawn about the stability of the flow, i.e., whether it is convectively or absolutely unstable.
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Criteria for instability and gain
- R V Polovin
R. V. Polovin, Criteria for instability and gain, Soviet Phys. -Tech. Phys. 6 (10) (1962, translated from 1961 paper) 889-895.
Perspectives in Fluid Dynamics, Ch. 4: Open Shear Flow Instabilities
- P Huerre
P. Huerre, Perspectives in Fluid Dynamics, Ch. 4: Open Shear Flow Instabilities, Cambridge University Press, Cambridge, United Kingdom, 2000.
Criteria for determining absolute instabilities and distinguishing between amplifying and evanescent waves
- Bers
A. Bers, R.J. Briggs, Criteria for determining absolute instabilities and distinguishing between amplifying and evanescent waves, Bull. Am. Phys. Soc. 9 (1964) 304.
Instabilities and Nonequilibrium Structures
- Huerre
- P Huerre
P. Huerre, Instabilities and Nonequilibrium Structures, Ch. 6: Spatio-temporal Instabilities in Closed and Open Flows, D. Reidel Publishing Company,
Dordrecht, Holland, 1987.
Hydrodynamics and Nonlinear Instabilities
- Huerre
Handbook of Plasma Physics
- Bers