Leonid Brevdo

Linear pulse structure and signalling in a film flow on an inclined plane

The film flow down an inclined plane has several features that make it an interesting prototype for studying transition in a shear flow: the basic parallel state is an exact explicit solution of the Navier–Stokes equations; the experimentally observed transition of this flow shows many properties in common with boundary-layer transition; and it has a free surface, leading to more than one class of modes. In this paper, unstable wavepackets – associated with the full Navier–Stokes equations with viscous free-surface boundary conditions – are analysed by using the formalism of absolute and convective instabilities based on the exact Briggs collision criterion for multiple k -roots of D ( k , omega ) = 0, where k is a wavenumber, omega is a frequency and D ( k , omega ) is the dispersion relation function. The main results of this paper are threefold. First, we work with the full Navier–Stokes equations with viscous free-surface boundary conditions, rather than a model partial differential equation, and, guided by experiments, explore a large region of the parameter space to see if absolute instability—as predicted by some model equations—is possible. Secondly, our numerical results find only convective instability, in complete agreement with experiments. Thirdly, we find a curious saddle-point bifurcation which affects dramatically the interpretation of the convective instability. This is the first finding of this type of bifurcation in a fluids problem and it may have implications for the analysis of wavepackets in other flows, in particular for three-dimensional instabilities. The numerical results of the wavepacket analysis compare well with the available experimental data, confirming the importance of convective instability for this problem. The numerical results on the position of a dominant saddle point obtained by using the exact collision criterion are also compared to the results based on a steepest-descent method coupled with a continuation procedure for tracking convective instability that until now was considered as reliable. While for two-dimensional instabilities a numerical implementation of the collision criterion is readily available, the only existing numerical procedure for studying three-dimensional wavepackets is based on the tracking technique. For the present flow, the comparison shows a failure of the tracking treatment to recover a subinterval of the interval of unstable ray velocities V whose length constitutes 29% of the length of the entire unstable interval of V . The failure occurs due to a bifurcation of the saddle point, where V is a bifurcation parameter. We argue that this bifurcation of unstable ray velocities should be observable in experiments because of the abrupt increase by a factor of about 5.3 of the wavelength across the wavepacket associated with the appearance of the bifurcating branch. Further implications for experiments including the effect on spatial amplification rate are also discussed.

Three-dimensional absolute and convective instabilities, and spatially amplifying the waves in parallel shear flows

SummaryA formalism for absolute and convective instabilities in parallel shear flows is extended to the three-dimensional case. Assuming that the dispersion relation function is given byD(k, l, ω), wherek andl are wave numbers, andω is a frequency, the analytic criterion is formulated by which a point (k0,l0,ω0) with Imω0>0 contributes to the absolute instability if and only if one of the two equivalent conditions is satisfied:(i)At least two roots inl of the systemD(k, l, ω)=0,Dk(k, l, ω)=0, originating on opposite sides of the reall-axis, collide on thel-plane for the parameter valuesk0,l0,ω0, asω is brought down toω0. Every point on thek-plane, that corresponds to a point on the collision paths on thel-plane, is itself a coalescence point ofk-roots for a fixedl ofD(k, l, ω)=0, that originate on opposite sides of the realk-axis.(ii)At least two roots ink of the systemD(k, l, ω)=0,Dl,(k, l, ω)=0, originating on opposite sides of the realk-axis, collide on thek-plane for the parameter valuesk0,l0,ω0, asω is brought down toω0. Every point on thel-plane, that corresponds to a point on the collision paths on thek-plane, is itself a coalescence point ofl-roots for a fixedk ofD(k, l, ω)=0, that originate on opposite sides of the reall-axis. Consequently, the causality condition for spatially amplifying 3-D waves in absolutely stable, but convectively unstable flow is derived as follows. We denote by (α, β) a unit vector on the (x, y) plane. The contributions to amplification in the direction of this vector come from the end points of the trajectories that consist of the coalescence roots on thel1-plane, given byl1,=−βk+αl, of the systemD=0,−βDk+gaD1=0. Thek1-components of these trajectories have to pass from above to below the real axis on ak1-plane, given byk1=αk+βl, asω moves down toω0. Hereω0 is the real frequency of excitation. At each point of such trajectories the group velocity vector (Dk,Dl) is collinear with the direction vector (α, β). There exists a direction for which the spatial amplification rate reaches its maximum.The formalism is illustrated with a simple model example. A procedure for computing theN-factor in theeN-method, which is based on the wave packet approach is developed.

Local and global instabilities of spatially developing flows: cautionary examples

In the analysis of the linear stability of basic states in fluid mechanics that are slowly varying in space, the quasi–homogeneous hypothesis is often invoked, where the stability exponents are defined locally and treated as slowly varying functions of a spatial coordinate. The set of local stability exponents is then used to predict the global perturbation dynamics and an implicit hypothesis is that the local analysis provides at least a conservative estimate of the global stability properties of the flow. In this paper cautionary examples are presented that demonstrate a contradiction between the results of the local and global analyses. For example, a local analysis may predict stability everywhere even when the exact PDE with non–constant coefficients is ill–posed, demonstrating that global stability exponents are not, in general, bounded by the maximal local stability exponents. A key observation in this paper is the importance of distinguishing between the discrete spectrum and the continuous spectrum when comparing global and local stability exponents. This distinction is particularly significant for spatially periodic flows where, for the global flow, only the continuous spectrum is present and, hence, instability arises always in the absence of discrete spectra. New exact definitions for global absolute and convective instabilities are also given for a class of spatially periodic basic states and applied to an example based on the complex Ginzburg–Landau equation. The consequences of this example, and of the argument involved for basic states that are slowly varying in space but non–periodic, and for some problems in fluid mechanics are also presented.

Wave Packets, Signaling and Resonances in a Homogeneous Waveguide

We solve the initial-boundary-value linear stability problem for small localised disturbances in a homogeneous elastic waveguide formally by applying a combined Laplace – Fourier transform. An asymptotic evaluation of the solution, expressed as an inverse Laplace – Fourier integral, is carried out by means of the mathematical formalism of absolute and convective instabilities. Wave packets, triggered by perturbations localised in space and finite in time, as well as responses to sources localised in space, with the time dependence satisfying e−iωt + O(e−ɛt), for t → ∞, where Im ω0 = 0 and ω > 0 , that is, the signaling problem, are treated. For this purpose, we analyse the dispersion relation of the problem analytically, and by solving numerically the eigenvalue stability problem. It is shown that due to double roots in a wavenumber k of the dispersion relation function D(k, ω), for real frequencies ω, that satisfy a collision criterion, wave packets with an algebraic temporal decay and signaling with an algebraic temporal growth, that is, temporal resonances, are present in a neutrally stable homogeneous waveguide. Moreover, for any admissible combination of the physical parameters, a homogeneous waveguide possesses a countable set of temporally resonant frequencies. Consequences of these results for modelling in seismology are discussed.

A Phenomenological Model of the Increase in Solute Concentration in Ground Water Due to Evaporation

We present a new phenomenological model of the evaporation of ground water containing a polluting material in the dissolved form. Only the one-dimensional case is treated. It is assumed that there exists a sharp evaporation front separating between the dry and the water-saturated soil. The water-saturated soil is assumed to occupy a semi-infinite domain x > X(t), where x is a vertical coordinate directed downward and X(t) is a position of the evaporation front. The mathematical description is based on four linear diffusion equations coupled through four boundary conditions, one of which is nonlinear, on the free moving evaporation front. We use a similarity solution of the governing equations and analyze it qualitatively showing that the solute concentration increases in the upward vertical direction and reaches its maximum on the evaporation front. The dependence of the solute concentration at the evaporation front and of the velocity of the front on the initial solute concentration and the temperature of the ground surface are computed. It is shown that for not high values of the initial solute concentration that are below the concentration value cd at which a deposition of the pollutant sets in, the solute concentration on the evaporation front can reach values that are above the deposition value cd. These results point to a possible mechanism of pollutant deposition in ground water caused by the evaporation.

Axisymmetric resonant disturbances in a homogeneous wave guide

Abstract The initial boundary-value linear stability problem for small localised axisymmetric disturbances in a homogeneous elastic wave guide, with the free upper surface and the lower surface being rigidly attached to a half-space, is formally solved by applying the Laplace transform in time and the Hankel transforms of zero and first orders in space. An asymptotic evaluation of the solution, expressed as a sum of inverse Laplace-Hankel integrals, is carried out by using the approach of the mathematical formalism of absolute and convective instabilities. It is shown that the dispersion-relation function of the problem D 0 ( κ , ω ), where the Hankel parameter κ is substituted by a wave number (and the Fourier parameter) κ, coincides with the dispersion-relation function D 0 ( k , ω ) for two-dimensional (2-D) disturbances in a homogeneous wave guide, where ω is the frequency (and the Laplace parameter) in both cases. An analysis for localised 2-D disturbances in a homogeneous wave guide is then applied. We obtain asymptotic expressions for wave packets, triggered by axisymmetric perturbations localised in space and finite in time, as well as for responses to axisymmetric sources localised in space, with the time dependence satisfying e − iω 0 t + O ( e − ϵt ) for t → ∞, where Im ω 0 = 0, ϵ > 0, and t denotes time, i.e. for signalling with frequency ω 0 . We demonstrate that, for certain combinations of physical parameters, axisymmetric wave packets with an algebraic temporal decay and axisymmetric signalling with an algebraic temporal growth, as √ t , i.e., axisymmetric temporal resonances, are present in a neutrally stable homogeneous wave guide. The set of physically relevant wave guides having axisymmetric resonances is shown to be fairly wide. Furthermore, since an axisymmetric part of any source is L 2 -orthogonal to its non-axisymmetric part, a 3-D signalling with a non-vanishing axisymmetric component at an axisymmetric resonant frequency will generally grow algebraically in time. These results support our hypothesis concerning a possible resonant triggering mechanism of certain earthquakes, see Brevdo, 1998 , J. Elasticity, 49, 201–237.