Absolute Instability Research Articles

Multibeam absolute stimulated Raman scattering and two-plasmon decay.

Multibeam absolute instability thresholds for stimulated Raman scattering (SRS) and two-plasmon decay (TPD) are calculated in three dimensions for conditions relevant to direct-drive inertial confinement fusion experiments on the OMEGA laser and at the National Ignition Facility (NIF). Although multibeam effects are found to be significant for both instabilities, SRS is found to have less efficient multibeam coupling than TPD. The results are consistent with the observation of a TPD-dominated regime on the OMEGA laser and a SRS-dominated regime on the NIF despite the single-beam SRS threshold being lower than the single-beam TPD threshold on both facilities. The minimum instability threshold for NIF plasma parameters occurs for SRS near quarter-critical densities with a shared electromagnetic wave propagating along the beam axis.

Linear instability of tilted parallel shear flow in a strongly stratified and viscous medium

A linear stability analysis is performed on a tilted parallel wake in a strongly stratified fluid at low Reynolds numbers. A particular emphasis of the present study is given to the understanding of the low-Froude-number mode observed by the recent experiment (Meunier in J Fluid Mech 699:174–197, 2012). In the limit of low Froude number, the linearised equations of motion can be reduced to the Orr–Sommerfeld equation on the horizontal plane, except the viscous term that contains vertical dissipation. Based on this equation, it is proposed that the low-Froude-number mode would be a horizontal inflectional instability and should remain two-dimensional at small tilting angles as long as the Reynolds number is sufficiently low. To support this claim, the asymptotic regime where this analysis is strictly valid is subsequently discussed in relation to previous work on the proper vertical length scale. The absolute and convective instability analysis of parallel wake is further performed, showing qualitatively good agreement with the experimental result. The low-Froude-number mode is found to be stabilised on increasing Froude number, as in the experiment. It is shown that the emergence of small vertical velocity at finite Froude number, the size of which is proportional to the square of Froude number, plays the key role in the stabilisation by modifying the inflectional instability and paradoxically creating stabilising buoyancy effect with the increase of Froude number.

Conditions for the Absolute Instability of the Solutions of Difference Equations with Self-Adjoint Operator Coefficients

We establish conditions for the absolute instability of the solutions of linear and nonlinear difference equations with self-adjoint operator coefficients.

Dissipative Instability of Shock Waves

A new condition for the linear dissipative instability of the strong plane shock wave in an arbitrary medium is obtained. The instability of the shock is realized due to the flow instability behind its front, which is similar to the known dissipative instability in the boundary layer for the Tollmien-Schlichting waves. It is found that within the limit of low viscosity the one-dimensional longitudinal disturbances grow much faster than the two-dimensional corrugation ones. It points to a better correspondence to experiment of the new condition for the absolute instability of the shock in comparison with theory, which does not take viscosity into account.

Linear Functional-Differential Equations with Absolutely Unstable Solutions

For linear functional-differential equations of retarded and neutral types with infinitely many deviations and self-adjoint operator coefficients, we establish necessary and sufficient conditions for the absolute instability of the trivial solutions.

Kinetics of Nonequilibrium Melting of a Macrosystem Under the Action of a Volume Heat Source

A kinetic model of nonequilibrium melting of a macrosystem under the action of a volume heat source formed in its bulk has been constructed. In this model, the melting of such a system is defined with account of the thermodynamic fluctuations of its order parameter, playing a great role in the superheating of the system and in the attainment by it of the absolute instability point at which the initially metastable solid phase of the system becomes absolutely unstable relative to the infinitely small fluctuations of its liquid phase. Analytical solutions of the kinetic equations defining the melting of a macrosystem have been obtained for different rates of its heating by an internal volume heat source.

Stimulated Brillouin scattering of backward stimulated Raman scattering

The rescattering of backward stimulated Raman scattering (BSRS) by stimulated Brillouin scattering (SBS) is found in the high electron density region by relativistic Vlasov-Maxwell simulation and particle-in-cell (PIC) simulation, where the BSRS is in the regime of absolute instability and dominates in all the scatterings. Both one dimension (1D) Vlasov simulation and two dimension (2D) PIC simulation have been given to verify that there exists SBS of BSRS in the regime of absolute instability for BSRS. The SBS of BSRS will be even stronger than forward stimulated Raman scattering (FSRS) and SBS in regime of absolute instability for BSRS. Thus, besides Langmuir decay instability and laser energy absorption, the SBS of BSRS is also an important saturation mechanism of BSRS in high electron density region.

Effect of Thermocapillary on Absolute and Convective Instability of Film Flow of Self-Rewetting Fluid

The dynamics of a self-rewetting film falling along a vertical fiber under the influence of gravity are considered. The evolution equation of the interface of the self-rewetting film is established in the framework of a long-wave approximation theory. The effect of thermocapillarity (Marangoni effect) on the absolute/convective instability (AI/CI) is investigated for self-rewetting fluids of which the surface tension is a quadratic function of the temperature at the surface. The effect of thermocapillarity on the Rayleigh-Plateau instability is investigated by examining the dispersion relation. The characteristics of self-rewetting fluids are considered for different Marangoni numbers (Ma) in different regions of absolute/convective instability using a spatio-temporal stability analysis. Numerical simulations of the nonlinear evolution in various regions of absolute/convective instability are also performed. The results of numerical simulations are in excellent agreement with the spatio-temporal stability analysis. The effect of thermocapillarity on absolute and convective instability depends on the difference between the temperature at the interface $\bar {\varTheta }_{i}$ , and the temperature corresponding the minimum surface tension Θ0. The results indicate that the thermocapillarity suppresses the absolute instability and enhances the convective instability as Ma increases when $\bar {\varTheta }_{i}-{\varTheta }_{0}> 0$ , and enhances the absolute instability as Ma increases when $\bar {\varTheta }_{i}-{\varTheta }_{0}< 0$ .

Two- and Three-Dimensional Absolute Instabilities in a Porous Medium with Inclined Temperature Gradient and Vertical Throughflow

A linear stability analysis for the onset of mixed convection in a saturated porous medium through an absolute instability of both two- and three-dimensional disturbances is performed. Relevant control parameters associated with the inclined temperature gradient and the vertical throughflow are the vertical and horizontal Rayleigh numbers, $$R_{\mathrm{v}}$$ and $$R_{\mathrm{h}}$$, and the vertical Peclet number, $$Q_{\mathrm{v}}$$, respectively. This work extends previous studies on the very same problem in two fronts. For two-dimensional disturbances, the present results do not agree with the literature for a few of the parametric conditions reported. This is caused by the collision of the convectively unstable downstream propagating branch with multiple upstream propagating branches, which generates several saddle points and, hence, makes the identification of the correct pinching point more difficult. In other words, literature results are all saddle points but not always pinching points. For three-dimensional disturbances, this issue is not present and the current results agree with the literature. On the other hand, due to the inherent difficulties associated with a three-dimensional absolute instability analysis, literature results have only been able to report the group velocities at the onset of convective instability. When their real parts are zero, transition occurs directly from stable to absolutely unstable. Otherwise, transition occurs from stable to convectively unstable first and nothing can be said about the onset of absolute instability. In this work, a novel technique recently developed by the authors allowed the identification of the onset of absolute instability under all parametric conditions investigated in the literature, extending earlier results. Doing so confirmed the dichotomy already observed in these earlier studies, i.e., the onset of absolute instability for two- and three-dimensional longitudinal modes indeed differs.

Absolute instability of free-falling viscoelastic liquid jets with surfactants

The effect of surfactants on the absolute instability of a viscoelastic liquid jet falling under gravity is examined for axisymmetrical disturbances. In general, the inclusion of surfactants to the interface of a viscoelastic liquid jet allows for the possibility of further processing droplet sizes and breakup lengths. We use the upper-convected Maxwell model to provide a mathematical description of the dynamics of the jet. An asymptotic approach, based on the slenderness of the jet, is used to render the problem more tractable and obtain steady-state solutions and then perform a linear analysis of the convective and absolute instability on these base solutions. By considering travelling wave modes, we derive a dispersion relationship, which is then solved numerically using the Newton-Raphson method. We show the effect of changing a number of dimensionless parameters, including the initial surfactant concentration, on convective and absolute instability. In this work, we use a mapping technique known as the cusp map method to explore absolute instability. The convective/absolute instability boundary is identified for various parameter regimes.

The Stability of a Radial Convergence of a Cylindrical Shell Consisting of Viscous Incompressible Liquid

The problem of the nonlinear stability of the radial collapse of a cylindrical shell, which is filled with a viscous incompressible fluid of uniform density, is studied. A number of assumptions are made: (1) vacuum is contained inside the shell; (2) it is surrounded by a layer of compressed polytropic gas, which serves as a product of instant detonation and exerts constant pressure on the outer surface of the shell; (3) vacuum is also behind the gas layer. The absolute instability of the radial collapse of the considered viscous cylindrical shell with respect to finite perturbations of the same symmetry type is established by the direct Lyapunov method. A Lyapunov function that satisfies all of the conditions of the first Lyapunov instability theorem, regardless of the specific mode of radial convergence, is constructed. This result fully confirms Trishin’s corresponding hypothesis and is a rigorous mathematical proof that the cumulation of kinetic energy of a viscous incompressible fluid of uniform density in the process of radial collapse of the studied cylindrical shell to its axis occurs exclusively at its impulse stage.

Mechanisms for Absolute Instabilities in Uniform and Nonuniform Density Swirling Flows

In this paper, transitions from convective to absolute instability in post-vortex-breakdown swirling flow with nonuniform density are investigated using linear stability analysis. Addition of swirl to flow in devices of practical interest achieves rapid mixing, but also renders such flows absolutely unstable, leading to self-excited flow oscillations. Through the spatiotemporal analysis, the onset of absolute instability is systematically analyzed in parametric space for uniform and nonuniform density cases. Five unstable azimuthal wavenumbers (, , ) studied show that, although mode first becomes absolutely unstable, it is the mode associated with the precessing vortex core that turns out to be more unstable for high swirl numbers and reduced shear layer thickness. The imposition of density field, representative of reacting flows, suppresses the absolute instability mode by shifting its onset to higher backflow velocities. Further, to investigate the cause of instabilities, a total disturbance energy equation is obtained with various energy feeding/depleting terms. Although the axial shear is the dominant production term, the alignment of pressure redistribution and density variation terms provides the necessary stabilization to weaken the absolute instabilities.

Generalized nonlinear Schrödinger equations describing the Second Harmonic Generation of femtosecond pulse, containing a few cycles, and their integrals of motion.

An interaction of laser pulse, containing a few cycles, with substance is a modern problem, attracting attention of many researches. The frequency conversion is a key problem for a generation of such pulses in various ranges of frequencies. Adequate description of such pulse interaction with a medium is based on a slowly evolving wave approximation (SEWA), which has been proposed earlier for a description of propagation of the laser pulse, containing a few cycles, in a medium with cubic nonlinear response. Despite widely applicability of the frequency conversion for various nonlinear optics problems solutions, SEWA has not been applied and developed for a theoretical investigation of the frequency doubling process until present time. In this study the set of generalized nonlinear Schrödinger equations describing a second harmonic generation of the super-short femtosecond pulse is derived. The equations set contains terms, describing the pulses self-steepening, and the second order dispersion (SOD) of the pulse, a diffraction of the beam as well as mixed derivatives. We propose the transform of the equations set to a type, which does not contain both the mixed derivatives and time derivatives of the nonlinear terms. This transform allows us to derive the integrals of motion of the problem: energy, spectral invariants and Hamiltonian. We show the existence of two specific frequencies (singularities in the Fourier space) inherent to the problem. They may cause an appearance of non-physical absolute instability of the problem solution if the spectral invariants are not taken into account. Moreover, we claim that the energy preservation at the laser pulses propagation may not occur if these invariants do not preserve. Developed conservation laws, in particular, have to be used for developing of the conservative finite-difference schemes, preserving the conservation laws difference analogues, and for developing of adequate theory of the modulation instability of the laser pulses, containing a few cycles.

Absolute instability of impinging leading edge vortices in a submodel of a bileaflet mechanical heart valve

Mechanical heart valves have been linked to the production of unphysiological turbulent blood flow. As a first step toward a model for laminar-turbulent transition in this flow, an absolute instability in the impinging leading-edge vortices that develop between the valve leaflets is identified.

On Stationary Solutions Of Delay Differential Equations: A Model Of Local Translation In Synapses

The results of analytical analysis of stationary solutions of a differential equation with two delayed arguments τ1 and τ2 are presented. Such equations are used in modeling of molecular-genetic systems where the delay of arguments appear naturally. Conditions of existence of non-negative solutions are described, and dependence of stability of these solutions on the values of delayed arguments is studied. This stability theory allows to give complete characterization of these solutions for all values of the parameters of the model, and ensures instability of a positive equilibrium point for any values of the delays τ2 ≥ τ1 ≥ 0 in the case when it is unstable for τ2 = τ1 = 0 (absolute instability). If this positive equilibrium point is stable only for τ2 = τ1 = 0, then this domain τ2 ≥ τ1 ≥ 0 is the domain of absolute instability as well. For positive equilibrium points which are stable at τ2 = τ1 = 0, we find domains of absolute stability were the equilibrium points remain stable for all values of the parameters τ1 and τ2. The domains of relative stability, where these points become unstable for some values of these parameters are also described. We show that when the efficiency of translation, and non-linearity and complexity of its regulation mechanisms grow, the domains of the absolute and relative stability of the positive equilibrium point shrink, while the domains of its instability expand. So, enhanced activity of the local translation system can be a factor of its instability and that of the risk of neuro-psychical diseases related to distortions of plasticity of the synapse and memory, where importance of stability of the proteome in the synapse is postulated.

Wavepacket instability in a rectangular porous channel uniformly heated from below

This paper is aimed to investigate the transition to absolute instability in a porous layer with horizontal throughflow. The importance of this analysis is due to the possible experimental failure to detect growing perturbations which are localised in space and which may be convected away by the throughflow. The instability of the uniform flow in a horizontal rectangular channel subject to uniform heating from below and cooled from above is studied. While the lower wall is modelled as an impermeable isoflux plane, the upper wall is assumed to be impermeable and imperfectly conducting, so that a Robin temperature condition with a given Biot number is prescribed. The sidewalls are assumed to be adiabatic and impermeable. The basic state considered here is a stationary parallel flow with a vertical uniform temperature gradient, namely the typical configuration describing the Darcy–Bénard instability with throughflow. The linear instability of localised wavepackets is analysed, thus detecting the parametric conditions for the transition to absolute instability. The absolute instability is formulated through an eigenvalue problem based on an eighth–order system of ordinary differential equations. The solution is sought numerically by utilising the shooting method. The threshold to absolute instability is detected versus the Péclet number associated with the basic flow rate along the channel.

Accurate numerical approximation of the absolute stability of unbounded flows

The initial stage of the laminar–turbulent transition of semi-infinite flows can be characterized as either an absolute or convective instability, naturally associated with localized wave packets. A convective instability is directly linked to an absolute instability in a different reference frame. Therefore, our aim is to determine the absolute stability of a flow in a given but arbitrary reference frame, which can only be directly inferred from the absolute eigenvalue spectrum. If advective processes are present, the associated absolute eigenfunctions grow exponentially in space in the advective direction. The eigenvalue spectrum is usually computed numerically, which requires truncating the domain and prescribing artificial boundary conditions at these truncation boundaries. For separated boundary conditions, the resulting spectrum approaches the absolute spectrum as the domain length tends to infinity. Since advective processes result in spatially exponentially growing eigenfunctions, it becomes increasingly difficult to represent these functions numerically as the domain length increases. Hence, a naive numerical implementation of the eigenvalue problem may result in a computed spectrum that strongly deviates from the (mathematically correct) absolute spectrum due to numerical errors. To overcome these numerical inaccuracies, we employ a weighted method ensuring the convergence to the absolute spectrum. From a physical point of view, this method removes the advection-induced spatial exponential growth from the eigenfunctions. The resulting (absolute) spectrum allows for a direct interpretation of the character of the pertinent perturbations and the eigensolutions can be used to construct and analyse the evolution of localized wave packets in an efficient way.

A quantification of penetration length of steam jet condensation in turbulent water flow in a vertical pipe

An experimental research is performed to investigate the penetration length of steam jet condensation in turbulent water flow in a vertical pipe. The penetration length of the condensing jet, with steam mass velocities ranging from subsonic to supersonic, was measured via utilizing a visualization window and a high-speed camera. At low steam flow rate a steam bubble plume forms, while at relatively high flow rate the bubbles in the plume coalesce and a steam jet is produced. Quantitative analyses of the interface behavior of the condensing jet are performed by exploring the transient and statistical characteristics of the jet penetration length. The waveforms of the transient jet penetration length exhibit that in the bubbling regime the period is approximately constant, while in the jetting regime the period varies a lot. The statistical analysis of the transient jet penetration length confirms that significant variations of the power distribution function (PDF) and power spectrum density (PSD) between the bubbling and jetting regimes occur at the transonic condition. The ratio of steam mass flux to critical steam mass flux at ambient pressure is found to have good linearity with the mean of the jet penetration length. And a linear correlation is fitted with a maximum absolute deviation of 5.5% between the measured and predicted values of mean of the jet penetration length. The relative unsteadiness of jet penetration length, defined as the ratio of the standard deviation to mean of transient penetration length, is approximately invariable in the bubbling and jetting regime. The bubbling regime contains more relative unsteadiness, while the jetting regime yield less relative unsteadiness. Because absolute instability exists in the subsonic region while no absolute instability occurs in the supersonic region, the transition from the bubbling to jetting regime occurs in the transonic region.

Passive Drag Reduction via Bionic Hull Coatings

The resistance of ships is often dominated by friction between the hull and the water. This study explores possibilities of reducing skin-friction drag in a way inspired by dolphins. These possess a soft skin believed to diminish drag by delaying the transition from laminar to turbulent flow. The underlying mechanism builds on a stabilization of the laminar boundary layer by the compliant surface. To transfer this mechanism to ship hulls, coatings similar to dolphin skin have been designed numerically, made from polymeric materials, and tested in a water tunnel. For the best coating, a drag reduction by almost 3% has been predicted in the boundary layer along the hull model of a small search-and-rescue vessel. The trends of the numerical predictions have been confirmed in the experiments. 1. Overview Dolphins are able to sustain a swimming speed of about 9 m/sec across long distances (Carpenter et al. 2000). This fascinating endurance has stimulated the hypothesis that the pliable dolphin skin interacts with the surrounding flow so as to maintain low-drag laminar flow via a delay of transition to turbulence (Gad-el-Hak 1996). Although this hypothesis is under debate (Fish &amp; Lauder 2006), laboratory experiments have indeed demonstrated that soft coatings delay laminar-turbulent transition (Gaster 1987). According to boundary-layer linear stability theory (LST), compliant surfaces are able to mitigate the amplification of Tollmien-Schlichting (TS) waves, the forerunners of transition in low-disturbance environments (Carpenter &amp; Garrad 1985). However, wall compliance may also promote transition in that the flow may excite instability waves borne by the soft coating material. These waves are known as fluid-induced surface instabilities (FISI) (Carpenter &amp; Garrad 1985) or travelling-wave flutter (Gad-el-Hak 1996). FISI waves may in turn via nonlinear effects develop into quasi-steady surface corrugations ("divergence") (Carpenter &amp; Garrad 1986). This process is associated with an absolute instability mechanism (Tsigklifis &amp; Lucey 2015) that rapidly triggers transition via the bypass route. The boundary-layer instability over a compliant coating is hence more complex than that over a rigid surface owing to the presence of two wave-bearing media. Compliant-wall boundary layers also support non-modal transient growth inducing bypass transition (Tsigklifis &amp; Lucey 2015).

Spatiotemporal linear stability of viscoelastic free shear flows: Dilute regime

We report the temporal and spatiotemporal stability analyses of antisymmetric, free shear, viscoelastic flows obeying the Oldroyd-B constitutive equation in the limit of low to moderate Reynolds number (Re) and Weissenberg number (We). The resulting fourth-order Orr-Sommerfeld equation is reduced to a set of six auxiliary equations that are numerically integrated starting from the rescaled far-field conditions, i.e., via Compound Matrix Method. The temporal stability analysis indicates that with increasing We, (a) the entire range of the most unstable mode is shifted toward longer waves (i.e., the entire region of temporal instability is gradually concentrated near zero wavenumber), (b) the vorticity structure contours are dilated, and (c) the residual Reynolds stresses are diminished. All these analogous observations previously reported in the inertial limit [J. Azaiez and G. M. Homsy, “Linear stability of free shear flow of viscoelastic liquids,” J. Fluid Mech. 268, 37–69 (1994).] suggest a viscoelastic destabilization mechanism operating at low and moderate Re. The Briggs idea of analytic continuation is deployed to classify regions of temporal stability, absolute and convective instabilities, as well as evanescent modes. The main result is that the free shear flow of dilute polymeric liquids is either (absolutely/convectively) unstable for all Re or the transition to instability occurs at comparatively low Re, a finding attributed to the fact that viscoelasticity aggravates instabilities via shear-induced anisotropy and the slow relaxation effects.