Spatio-temporal Stability Analysis Research Articles

Spatiotemporal stability of a thin film in the presence of thermal and solutal Marangoni stresses

The stability analysis of a gravity-driven thin liquid film over a uniformly heated substrate is carried out, and the effect of an insoluble surfactant is studied. The heating induces a temperature variation at the liquid–gas interface that generates a gradient in surface tension and creates a thermocapillary flow at the liquid–gas interface. When the film is perturbed the thermocapillary stress leads to the growth of the perturbation. The governing equations for the evolution of the film thickness and surfactant concentration are simplified within the lubrication approximation. Four non-dimensional groups appear in the model that affect the film dynamics and stability, namely, Marangoni numbers, M and Σ, the surface Peclet number, Pes, and Biot number, Bi. The critical conditions are explored in terms of these non-dimensional parameters for the film to become temporally unstable. Further, spatiotemporal stability analysis is performed to characterise the instability as absolute and convective instability. A critical Marangoni number is found beyond which the system is absolutely unstable independent of the inclination angle. This critical Marangoni number seems to increase with the increase in solutocapillary effect.

Convective and absolute instabilities in inclined buoyancy layers

The linear instability of the buoyancy-driven flow adjacent to an inclined heated wall immersed in a thermally stratified medium is studied theoretically and numerically. For the temporally unstable system, spatiotemporal stability analysis is carried out to delineate the parameter space (Grashof number, Prandtl number, and tile angle) for convective/absolute instability. We provide an example of an absolute instability of the buoyancy layer on an inclined buoyancy layer. It is shown that the tile angle and Prandtl number have a dramatic influence on the spatial-temporal properties of the flow. For fixed Pr = 6.7, increasing tile angle decreases the domain of absolute instability, and when tile angle is greater than 20°, the absolute instability disappears. The flow will change from convectively unstable to absolutely unstable with the increase of Pr. Results from the direct numerical simulation are in agreement with the predictions of the linear temporal and spatial-temporal instabilities. These encouraging results should be helpful for understanding such a buoyancy-driven flow system.

Global oscillations in low-density round jets with parabolic velocity profiles

Experiments and spatiotemporal stability analysis are carried out to study the global oscillations in laminar low-density round jets with parabolic velocity profiles. The experimental results of laminar low-density jets with parabolic velocity profiles exhibit global axisymmetric oscillations. The spatiotemporal stability results based on base profiles from numerical simulations are consistent with the present experimental results. These results differ from the prediction of stability study by Coenen et al. (Phys. Fluids, vol. 20, 2008, p. 074104). They reported that the low-density jets with near parabolic velocity profiles show global helical oscillations. It is observed that the method used by Coenen et al. is not able to predict the nature of global oscillations observed in experiments for low-density jets with near parabolic profiles. The present spatiotemporal stability results demonstrate that the base flows from Navier–Stokes equations are required to predict the critical conditions observed in experiments for low-density jets with near parabolic velocity profiles. The breakdown distance of globally unstable low-density jets scales with $(Re-Re_c)^{-1/2}$ ( $Re_c$ is the critical Reynolds number), consistent with the scaling law obtained from the nonlinear global mode of the Ginzburg–Landau equation.

Buoyancy instabilities in a liquid layer subjected to an oblique temperature gradient

We investigate the temporal and spatio-temporal buoyancy instabilities in a horizontal liquid layer supported by a poorly conducting substrate and subjected to an oblique temperature gradient (OTG) with horizontal and vertical components, denoted as HTG and VTG, respectively. General linear stability analysis (GLSA) reveals a strong stabilizing effect of the HTG on the instabilities introduced by the VTG for Prandtl numbers $Pr>1$ via inducing an extra vertical temperature gradient opposing the VTG through energy convection. For $Pr<1$ , a new mode of instability arises as a result of a velocity jump in the liquid layer caused by cellular circulation. A long-wave weakly nonlinear evolution equation governing the spatio-temporal dynamics of the temperature perturbations is derived. Spatio-temporal stability analysis reveals the existence of a convectively unstable long-wave regime due to the HTG. Weakly nonlinear stability analysis reveals the supercritical type of bifurcation changing from pitchfork in the presence of a pure VTG to Hopf in the presence of the OTG. Numerical investigation of the spatio-temporal dynamics of the temperature disturbances in the layer in the weakly nonlinear regime reveals the emergence of travelling wave regimes propagating against the direction of the HTG and whose phase speed depends on $Pr$ . In the case of a small but non-zero Biot number, the wavelength of these travelling waves is larger than that of the fastest-growing mode obtained from GLSA.

Scaling Analysis on the Dynamic and Instability Characteristics of Isolated Wingtip Vortex

The isolated wingtip vortex is generated by an M6 wing under different angles of attack and Reynolds numbers , and is measured within 16 chord length downstream regions. Then, the spatiotemporal local linear stability analysis (LSA) is performed at the azimuthal wavenumber of to analyze the instability characteristics. Experimental results show that the radial circulation of wingtip vortex satisfies a scaling formula. Moreover, the LSA results demonstrate that the instability characteristics, including the stability curve, growth rate eigenvalue spectrum, and perturbation mode, are uniformed in time and space, laying the basis for scaling the wingtip vortex by considering them. In this situation, by plotting the circulation-based Reynolds number and dimensionless wandering amplitude in log scale, it is found that they are in the form of , where and . Particularly, along with the streamwise development, the swirl number of wingtip vortex asymptotically closes to the unstable region in the (, ) plane defined by Fabre and Jacquin (Journal of Fluid Mechanics, Vol. 500, Jan. 2004, pp. 239–262). By considering the role of growth rate in the vortex wandering, an alternative scaling formula , where and , is suggested to describe the wandering development.

Effects of confinement on absolute and convective instabilities for momentum-driven countercurrent shear layers

We present results from the first experimental realization of a two-dimensional confined countercurrent shear layer driven by momentum flows, without the use of suction. Distinct peaks are observed in the frequency spectrum of velocity fluctuations, suggesting the presence of global instabilities. The variation of these frequencies with injected mass flow rate is predicted reasonably well by a spatiotemporal linear stability analysis of local velocity profiles, which checks for the presence of absolute instability, and suggests that confinement destabilizes the shear layer over a broad range of ratio of shear layer thickness to channel width.

Absolute instability of an annular jet: local stability analysis

The paper presents parametric studies of the first and second azimuthal absolute modes in annular non-swirling and swirling jets. The spatio-temporal linear stability analysis is applied to investigate an influence of governing parameters including axial velocity gradients in inner and outer shear layers, back-flow velocity, swirl number and shape of the azimuthal velocity. A new base flow is formulated allowing a flexible variation of the shape of axial and azimuthal velocity profiles. It is shown that the first helical absolute mode is governed mainly by the back-flow velocity and swirl intensity. A steepness of the inner shear layer can control the absolute mode frequency. The velocity gradient in the outer shear layer and the shape of the azimuthal velocity have rather limited impact on the absolute mode characteristics. Finally, it is shown that the second helical absolute mode can dominate the flow with a stronger swirl intensity.

Instabilities and bifurcations of liquid films flowing down a rotating fibre

Abstract

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 $\bar {\varTheta }_{i}-{\varTheta }_{0}< 0$ .

Is Tollmien-Schlichting wave necessary for transition of zero pressure gradient boundary layer flow?

Comprehensive understanding of the routes of instability and transition for many flows is not complete yet. For a zero pressure gradient (ZPG) boundary layer, linear spatial theory predicted Tollmien-Schlichting (TS) waves, which have been experimentally verified by vortically exciting the flow by a monochromatic source. This is the well-known frequency response of dynamical system theory. Natural transition in real flows occurs due to polychromatic excitation, and to simulate such transition, the ZPG boundary layer has been excited via an impulse response in some of our recent direct numerical simulations. Such impulse responses cause transition even when TS waves are not excited. In the present exercise, we show the theoretical basis of natural transition by spatiotemporal stability analysis, as used in the work of Sengupta et al. [“Spatiotemporal growing wave fronts in spatially stable boundary layers,” Phys. Rev. Lett. 96(22), 224504 (2006)], by invoking finite start-up of the frequency response to wall excitation. There appear to be different instability mechanisms active for the frequency and the impulse responses to localized wall excitation. Here, we show that in both the frequency and impulse responses, the spatiotemporal wave-front (STWF) is the common element. Additionally, we also consider cases, where following different start-ups, the wall excitation remains constant, which also show the presence of the STWF. The presented results for the ZPG boundary layer show that the TS wave is not necessary for transition to turbulence and help us to re-evaluate our understanding of the transition mechanism for this canonical flow.

Spatio-temporal stability analysis of linear arrays of 2D density stratified wakes and jets

This paper investigates the spatio-temporal stability characteristics of multiple shear flow elements (wakes or jets) with density stratification. While the stability of single jets or wakes has been considered extensively, many applications exist where these canonical flow fields are aligned in multi-element configurations. A fundamental question is the relationship between the stability characteristics of a single element and the larger system. More fundamentally, this question involves the interaction of multiple regions of vorticity concentration and how they modify a system’s absolute stability as well as the manner in which density gradients influence the way these different regions of concentrated vorticity interact. This study presents a generalization of Yu and Monkewitz’s analysis [“The effect of nonuniform density on the absolute instability of two-dimensional inertial jets and wakes,” Phys. Fluids A 2, 1175 (1990)] for multiple jets and wakes, explicitly considering n = 2, 3, 4, and infinity elements. The velocity and density base profiles are parameterized by the density ratio, S, velocity shear ratio, λ, and the wake spatial separation parameter, L/D. The results show that the maximum absolute growth rate (ω0,i) exhibits a non-monotonic dependence on L/D. In addition, the most absolutely unstable mode switches between system-sinuous and system-varicose as the spatial separation parameter is varied, in agreement with prior experiments [I. Peschard and P. Le Gal, “Coupled wakes of cylinders,” Phys. Rev. Lett. 77, 3122 (1996)]. This transition in symmetry, as well as the specific L/D value at which a given mode dominates, can be approximately predicted using the resonant wave interaction model from Juniper et al. [“The effect of confinement on the stability of two-dimensional shear flows,” J. Fluid Mech. 565, 171 (2006)]. Furthermore, there are two distinct L/D regimes: a “near-wake regime” (L/D &amp;lt; ∼3) and a “far-wake regime.” In the near-wake regime, the system stability is a function of the number of elements, n, while in the far-wake regime, it is only the element spacing, not the number of elements that influences ω0,i.

Spatio-temporal instability of two superposed fluids in a channel with boundary slip

Spatio-temporal stability analysis of a viscosity and density stratified two-fluid flow in a channel with hydrophobic walls, which experience a finite tangential velocity slip, is considered for a range of parameters for which Squire’s theorem is not valid. The study restricted only to temporal stability analysis of two/three-dimensional infinitesimal disturbances reveals that three-dimensional disturbances are more unstable than the corresponding two-dimensional disturbances. We examine the role of different types of slip boundary conditions at the channel walls on the temporal growth rate. The influence of velocity slip at the upper wall on the nature of instability in the spatio-temporal framework for low density ratio is investigated, and the boundary that demarcates the convective/absolute instabilities are presented for both two and three-dimensional disturbances. In a configuration with a thinner lower layer away from the upper wall with slip, we found that the flow in the entire domain is absolutely unstable, except for two strips at low and high Reynolds numbers. For the parameters considered in the present study, it is found that the absolutely unstable region shrinks with decrease in velocity slip. The absolute growth rate exhibits a non-monotonic behavior with respect to velocity slip. The role of velocity slip in promoting absolute instability in some parameter regimes suggests that by designing the walls of the channel as hydrophobic/rough/porous surfaces, which can be modelled as smooth surfaces with tangential velocity slip, it may be possible to achieve early transition to turbulence in two-layer flow systems.

Absolute and convective instabilities of a film flow down a vertical fiber subjected to a radial electric field

We consider the motion of a gravity-driven flow down a vertical fiber subjected to a radial electric field. This flow exhibits rich dynamics including the formation of droplets, or beads, driven by a Rayleigh-Plateau mechanism modified by the presence of gravity as well as the Maxwell stress at the interface. A spatiotemporal stability analysis is performed to investigate the effect of electric field on the absolute-convective instability (AI-CI) characteristics. We performed a numerical simulation on the nonlinear evolution of the film to examine the transition from CI to AI regime. The numerical results are in excellent agreement with the spatiotemporal stability analysis. The blowup behavior of nonlinear simulation predicts the formation of touchdown singularity of the interface due to the effect of electric field. We try to connect the blowup behavior with the AI-CI characteristics. It is found that the singularities mainly occur in the AI regime. The results indicate that the film may have a tendency to form very sharp tips due to the enhancement of the absolute instability induced by the electric field. We perform a theoretical analysis to study the behaviors of the singularities. The results show that there exists a self-similarity between the temporal and spatial distances from the singularities.

Spatiotemporal stability and sensitivity analysis of a holling type-II prey predator system with Allee effect

In this article, an analytical investigation is carried out on two species prey predator model having Holling type II interaction with Allee effect. We analysed the local and global behaviour of the interior equilibrium point.The theoretical aspects such as the investigation of the existence and stability of the equilibriums as well as the oscillatory and sensitivity behaviours of the corresponding diffusive system have been pursued. Also, analysed the effect of white noise and diffusion analysis of the proposed system. Finally, all the analytical results are interpreted ecologically and compared with the numerical results generated by MATLAB.

Stability of a radially stretching disk beneath a uniformly rotating fluid

The steady radial stretching of a disk beneath a rigidly rotating flow with constant angular velocity is considered. The steady base flow is determined numerically for both a stretching and a shrinking disk. The convective instability properties of the flow are examined using temporal stability analysis of the governing Rayleigh equation, and typically for small to moderate radial wave numbers, the range of azimuthal wave numbers β , over which the flow is unstable increases for both a stretched and a shrinking disk, compared to the unstretched case. The inviscid absolute instability properties of the resulting base flows are also examined using spatiotemporal stability analysis. For suitably large stretching rates, the flow is absolutely unstable in only a small range of positive β . For small stretching rates there exists a second region of absolute instability for a range of negative β values. In this region the “effective” two-dimensional base flow, comprised of a linear combination of the radial and azimuthal velocity profiles that enter the Rayleigh equation calculation, has a critical point (unlike for β > 0 ) that can dominate the absolute instability growth rate contribution compared to the shear layer component. A similar behavior is found to occur for a radially shrinking disk, except these profiles have a strong shear layer structure and hence are more unstable than the stretching disk profiles. We thus find for a suitably large shrinking rate the absolute instability contribution from the critical point becomes subdominant to the shear layer contribution.

Thermocapillary effect on the absolute and convective instabilities of film flows down a fibre

We consider the motion of a gravity-driven flow down a uniformly heated vertical fibre. This flow exhibits rich dynamics including the formation of droplets, or beads, driven by a Rayleigh-Plateau mechanism modified by the presence of gravity as well as the thermocapillarity at the interface. A spatio-temporal stability analysis is performed to investigate the effect of thermocapillarity (Marangoni effect) on the convective/absolute instability (CI/AI) characteristics of the problem. We also performed a numerical simulation of Eq. (30) on the nonlinear evolution of the film to connect the breakup behaviours with the CI/AI characteristics. Our numerical results showed that for various Marangoni number (Ma), breakup of the film mainly occurs in the AI regime. With the increase of Ma, the film has a tendency to break up into droplets due to the enhancement of the absolute instability.

Transition and turbulence in horizontal convection: linear stability analysis

The linear instability mechanisms of horizontal convection in a rectangular cavity forced by a horizontal buoyancy gradient along its surface are investigated using local and global stability analyses for a Prandtl number equal to unity. The results show that the stability of the base flow, a steady circulation characterized by a narrow descending plume and a broad upwelling region, depends on the Rayleigh number, $Ra$. For free-slip boundary conditions at a critical value of $Ra\approx 2\times 10^{7}$, the steady base flow becomes unstable to three-dimensional perturbations, characterized by counter-rotating vortices originating within the plume region. A Wentzel–Kramers–Brillouin (WKB) method applied along closed streamlines demonstrates that this instability is of a Rayleigh–Taylor type and can be used to accurately reconstruct the global instability mode. In the case of no-slip boundary conditions, the base flow also becomes unstable to a self-sustained two-dimensional instability whose critical Rayleigh number is $Ra\approx 1.7\times 10^{8}$. Beyond this critical $Ra$, two-dimensional equilibrium stationary states of the Navier–Stokes equations are computed using the selective frequency damping method. The two-dimensional onset of instability is shown to be characterized by a family of modes also originating within the plume. A local spatio-temporal stability analysis shows that the flow becomes absolutely unstable at the origin of the plume. Taken together, these results illustrate the mechanisms behind the onset of turbulence that has been observed in horizontal convection.

Towards a quantitative comparison between global and local stability analysis

A methodology is proposed here to estimate the stability characteristics of bluff-body wakes using local analysis under the assumption of weakly non-parallel flows. In this connection, a generalisation of the classic spatio-temporal stability analysis for fully three-dimensional flows is first described. Secondly, an additional higher-order correction term with respect to the common saddle-point global frequency estimation is included in the analysis. The proposed method is first validated for the case of the flow past a circular cylinder and then applied to two fully three-dimensional flows: the boundary layer flow over a wall-mounted hemispherical body and the wake flow past a fixed sphere. In all the cases considered, both the estimated unstable eigenvalue and the spatial shape of the associated eigenmode are determined by local stability analysis, and they are compared with the reference counterparts obtained at a definitely higher computational cost by a fully three-dimensional global stability analysis. It is shown that the results of local stability analysis, when the higher-order correction term is included, are in excellent agreement with those obtained by global stability analysis. It is also shown that the retained correction term is of crucial importance in this perspective, leading to a remarkable improvement in accuracy with respect to the classical saddle-point estimation.

Simple and double microemulsions via the capillary breakup of highly stretched liquid jets

We present an exhaustive experimental and theoretical study of the shapes of simple and compound jets formed when one (simple) or two (compound) immiscible liquids are injected into another liquid. The viscosity of the co-flowing external liquid is chosen to vary the characteristic Reynolds number of the outer stream, $Re_{o}$, over a wide range of values. Our slender-body theory in Gordillo et al. (J. Fluid Mech., vol. 738, 2014, pp. 335–357) is extended to predict the shapes of simple jets when $Re_{o}$ is such that $Re_{o}\gg 1$ and also to predict the shapes of compound jets in the case of $Re_{o}\lesssim O(1)$. The validity of our theoretical results, applicable to describe the dynamics of simple or compound jets within an outer carrier fluid in a wide variety of practical situations, is tested using a set-up where the liquids flow from a pressurized chamber towards an extraction tube, finding a very good agreement between the predicted and the observed shapes. Moreover, when $Re_{o}\lesssim O(1)$, and thanks to the fact that the liquid jets produced using our method are highly stretched in the downstream direction, we find that the values of the critical capillary number above which a steady stretched jet is produced, with the capillary number defined here using the outer stream velocity and viscosity, is well below the corresponding critical values characterizing other similar procedures, like flow focusing. This experimental result, which is supported by a spatio-temporal stability analysis in which the axial gradients of the unperturbed solution are retained in the dispersion relation, imply a substantial saving of energy and of the volume of outer liquid necessary to generate a steady capillary jet from which drops are regularly produced. Additionally, making use of continuity arguments and of the fact that drops are formed as a consequence of the growth of a capillary instability, we provide closed expressions for the drop diameters and their production frequencies when the capillary number is above the critical one, in very good agreement with experiments. The simple or double microemulsions generated by the capillary disintegration of the type of simple or compound highly stretched steady jets described here might find applications in biotechnology, pharmacy, cosmetics or materials science.

Diffusion-flame flickering as a hydrodynamic global mode

The present study employs a linear global stability analysis to investigate buoyancy-induced flickering of axisymmetric laminar jet diffusion flames as a hydrodynamic global mode. The instability-driving interactions of the buoyancy force with the density differences induced by the chemical heat release are described in the infinitely fast reaction limit for unity Lewis numbers of the reactants. The analysis determines the critical conditions at the onset of the linear global instability as well as the Strouhal number of the associated oscillations in terms of the governing parameters of the problem. Marginal instability boundaries are delineated in the Froude number/Reynolds number plane for different fuel jet dilutions. The results of the global stability analysis are compared with direct numerical simulations of time-dependent axisymmetric jet flames and also with results of a local spatio-temporal stability analysis.