Onset Of Absolute Instability Research Articles

Absolute instability of the boundary-layer flows due to rotating a spheroid

This paper explores local absolute instability in the boundary layer flow over two distinct families of rotating spheroids (prolate and oblate). While convective instability was established in earlier work by Samad and Garrett [1], this study delves into the potential occurrence of local absolute instability. Some results of local convective instability under the assumption of stationary vortices are reproduced for a more comprehensive investigation. The analysis considers viscous and streamline curvature effects, demonstrating that the localized mean flow within the boundary layer over either family of the rotating spheroid is absolutely unstable for each fixed value of the eccentricity parameter e∈[0,0.8]. For certain combinations of Reynolds number Re and azimuthal wave number β, a third branch (Branch 3) of the dispersion relation intersects Branch 1 at a pinch point, indicating absolute instability. Neutral curves depict regions that are absolutely unstable, while below critical Reynolds numbers, the region is either convectively unstable or stable. The paper also illustrates the effect of increasing eccentricity on spatial branches within both convectively and absolutely unstable regions. From lower to moderate latitudes, the stabilizing effect of e on the onset of absolute instability is robust for the prolate family and almost negligible for the oblate family. At high latitudes of the prolate spheroid, the stabilizing effect of e is fainter but persists until close to the equator. Conversely, at high latitudes of the oblate spheroid, the stabilizing effect of e is more pronounced. The paper discusses the implications of the parallel flow assumption employed in the analyses.

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.

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.

On the impulse response and global instability development of the infinite rotating-disc boundary layer

Linear disturbance development in the von Kármán boundary layer on an infinite rotating-disc is investigated for an extensive range of azimuthal mode numbers$n$. The study expands upon earlier investigations that were limited to those values of$n$located near the onset of absolute instability (Lingwood,J. Fluid Mech., vol. 299, 1995, pp. 17–33), where disturbances to the genuine inhomogeneous flow were shown to be globally stable (Davies & Carpenter,J. Fluid Mech., vol. 486, 2003, pp. 287–329). Numerical simulations corresponding to azimuthal mode numbers greater than the conditions for critical absolute instability display a form of global linear instability that is characterised by a faster than exponential temporal growth, similar in appearance to that found on the rotating-disc with mass suction (Thomas & Davies,J. Fluid Mech., vol. 724, 2010, pp. 510–526) and other globally unstable flows (Huerre & Monkewitz,Annu. Rev. Fluid Mech., vol. 22, 1990, pp. 473–537). Solutions indicate that a change in the global behaviour arises for $n\in [80:100]$that is marginally greater than those disturbances studied previously. Furthermore, the Reynolds number associated with the larger azimuthal mode numbers coincides with the upper bound of experimental predictions for transition. Thus, the local–global linear stability of the infinite rotating-disc is similar to the scenario outlined by Huerre & Monkewitz (1990) that states a region of local absolute instability is necessary but not sufficient for global instability to ensue. Conditions are derived to predict the azimuthal mode number needed to bring about a change in global behaviour, based on solutions of the linearised complex Ginzburg–Landau equation coupled with numerical simulations of disturbances to the radially homogeneous flow. The long term response is governed by a detuning effect, based on radial variations of the temporal frequency and matching shifts in temporal growth that increases for larger$n$, eventually attaining values sufficient to engineer global linear instability. The analysis is extended to include mass transfer through the disc surface, with similar conclusions drawn for disturbances to large enough azimuthal mode numbers. Finally, we conclude that the high$n$modes are unlikely to have a strong influence on disturbance development and transition in the von Kármán flow, as they will be unable to establish themselves across an extended radial range before nonlinear effects are triggered by the huge growth associated with the wavepacket maxima of the lower$n$-valued convective instabilities.

Influence of optimally amplified streamwise streaks on the Kelvin–Helmholtz instability

The optimal energy amplifications of streamwise-uniform and spanwise-periodic perturbations of the hyperbolic-tangent mixing layer are computed and found to be very large, with maximum amplifications increasing with the Reynolds number and with the spanwise wavelength of the perturbations. The optimal initial conditions are streamwise vortices and the most amplified structures are streamwise streaks with sinuous symmetry in the cross-stream plane. The leading suboptimal perturbations have opposite (varicose) symmetry. When forced with finite amplitudes these perturbations modify the characteristics of the Kelvin–Helmholtz instability. Maximum temporal growth rates are reduced by optimal sinuous perturbations and are slightly increased by varicose suboptimal ones. In contrast, the onset of absolute instability is delayed by varicose suboptimal perturbations and is slightly promoted by sinuous optimal ones. We show that if, instead of the computed fully nonlinear basic-flow distortions, the stability analysis is based on a shape assumption for the flow distortions, then opposite effects on the flow stability are predicted in most of the considered cases. These strong differences are attributed to the spanwise-uniform component of the nonlinear basic-flow distortion which, we conclude, should be systematically included in sensitivity analyses of the stability of two-dimensional basic flows to three-dimensional basic-flow perturbations. We finally show that the leading-order quadratic sensitivity of the eigenvalues to the amplitude of the streaks is preserved if the effects of the mean flow distortion are included in the sensitivity analysis.

Global stability behaviour for the BEK family of rotating boundary layers

Numerical simulations were conducted to investigate the linear global stability behaviour of the Bödewadt, Ekman, von Kármán (BEK) family of flows, for cases where a disc rotates beneath an incompressible fluid that is also rotating. This extends the work reported in recent studies that only considered the rotating-disc boundary layer with a von Kármán configuration, where the fluid that lies above the boundary layer remains stationary. When a homogeneous flow approximation is made, neglecting the radial variation of the basic state, it can be shown that linearised disturbances are susceptible to absolute instability. We shall demonstrate that, despite this prediction of absolute instability, the disturbance development exhibits globally stable behaviour in the BEK boundary layers with a genuine radial inhomogeneity. For configurations where the disc rotation rate is greater than that of the overlying fluid, disturbances propagate radially outwards and there is only a convective form of instability. This replicates the behaviour that had previously been documented when the fluid did not rotate beyond the boundary layer. However, if the fluid rotation rate is taken to exceed that of the disc, then the propagation direction reverses and disturbances grow while convecting radially inwards. Eventually, as they approach regions of smaller radii, where stability is predicted according to the homogeneous flow approximation, the growth rates reduce until decay takes over. Given sufficient time, such disturbances can begin to diminish at every radial location, even those which are positioned outwards from the radius associated with the onset of absolute instability. This leads to the confinement of the disturbance development within a finitely bounded region of the spatial–temporal plane.

Absolute instability: A toy model and an application to the Rayleigh–Bénard problem with horizontal flow in porous media

The concept of absolute instability is surveyed and applied to the study of the Rayleigh–Bénard problem in a horizontal porous layer with longitudinal flow. The survey is aimed to provide a simple introduction to absolute instability by employing a toy model based on a one-dimensional Burgers’ equation. The method of analysis is based on the steepest descent approximation, for large times, of the Fourier integral expressing the wavepacket perturbation of the basic solution. The analysis of Burgers’ equation is a suitable arena for the illustration of the elementary features of absolute instability. Then, the onset of absolute instability in a horizontal porous layer with a prescribed wall temperature difference between the boundaries and subject to a longitudinal flow is analysed. The seepage flow is modelled through Darcy’s law by assuming a finite Darcy–Prandtl number. It is shown that the transition from convective to absolute instability occurs at supercritical conditions, except for the limiting case when the horizontal flow rate is vanishingly small. In this special case, corresponding to the Darcy–Bénard problem, the condition of convective instability yields also absolute instability. The effects of the governing parameters, the Péclet number and the Darcy–Prandtl number, on the onset of absolute instability are studied.

Spin-down in rotating Hagen–Poiseuille flow: a simple criterion to detect the onset of absolute instabilities

We conduct experiments in a circular pipe with rotating Hagen–Poiseuille flow (RHPF) to which we apply spin-down or impulsive spin-down to rest, in order to analyse the threshold between convective and absolute instabilities through flow visualisations in the inlet region of the pipe. For a constant value of the Reynolds number,$Re$, the finite-amplitude wave packets generated by the arbitrary perturbation that results by reducing the swirl parameter, propagate upstream or downstream depending on the initial value of the swirl parameter,$L_{0}$. In fact, the main characteristic of the flow is that the velocity front of these wave packets changes from negative to positive when absolutely unstable modes are present in the initial state. The experimental results show that spin-down becomes a precise, reliable procedure to detect the onset of absolute instabilities. In addition, we give evidence of a gradual transition for Reynolds numbers ranging from 300 to 500 where a mode shift from$n=-1$to$n=-2$appears in the absolutely unstable region.

Global stability of flow past a cylinder with centreline symmetry

Global absolute and convective stability analysis of flow past a circular cylinder with symmetry conditions imposed along the centreline of the flow field is carried out. A stabilized finite element formulation is used to solve the eigenvalue problem resulting from the linearized perturbation equation. All the computations carried out are in two dimensions. It is found that, compared to the unrestricted flow, the symmetry conditions lead to a significant delay in the onset of absolute as well as convective instability. In addition, the onset of absolute instability is greatly affected by the location of the lateral boundaries and shows a non-monotonic variation. Unlike the unrestricted flow, which is associated with von Kármán vortex shedding, the flow with centreline symmetry becomes unstable via modes that are associated with low-frequency large-scale structures. These lead to expansion and contraction of the wake bubble and are similar in characteristics to the low-frequency oscillations reported earlier in the literature. A global linear convective stability analysis is utilized to find the most unstable modes for different speeds of the disturbance. Three kinds of convectively unstable modes are identified. The ones travelling at very low streamwise speed are associated with large-scale structures and relatively low frequency. Shear layer instability, with relatively smaller scale flow structures and higher frequency, is encountered for disturbances travelling at relatively larger speed. For low blockage a new type of instability is found. It travels at relatively high speed and resembles a swirling flow structure. As opposed to the absolute instability, the convective instability appears at much lowerReand its onset is affected very little by the location of the lateral boundaries. Analysis is also carried out for determining the convective stability of disturbances that travel in directions other than along the free stream. It is found that the most unstable disturbances are not necessarily the purely streamwise travelling ones. Disturbances that move purely in the cross-stream direction can also be convectively unstable. The results from the linear stability analysis are confirmed by carrying out direct time integration of the linearized disturbance equations. The disturbance field shows transient growth by several orders of magnitude confirming that such flows act as amplifiers. Direct time integration of the Navier–Stokes equation is carried out to track the time evolution of both the large-scale low-frequency oscillations and small-scale shear layer instabilities. The criticalRefor the onset of convective instability is compared with earlier results from local analysis. Good agreement is found.

The absolute instability of the boundary layer on a rotating cone

This paper is concerned with the existence of local absolute instability in the boundary-layer flow over the outer surface of a rotating cone, thereby extending earlier work by Lingwood who considered the rotating disk. Both still outer fluids and non-zero axial flow are considered, viscous and streamline-curvature effects are included, and the analysis is conducted for a wide range of cone half-angles, ψ. In still outer fluid our predicted local Reynolds numbers at the onset of absolute instability is relatively insensitive to the value of ψ, and is in reasonable agreement with experimental data for the onset of turbulence when ψ > 50 ° . For ψ < 50 ° the discrepancy increases, suggesting that some other mechanism may be responsible for transition on more slender cones. The introduction of axial flow is found to significantly increase the Reynolds number for local absolute instability for each half-angle.

Absolute and convective instability of a radially expanding liquid sheet

A radially expanding liquid sheet of a finite radius can be formed by impacting a liquid jet on a circular disk [Savart, Ann. Chim. Phys. 54, 55 (1833); Ann. Chim. 54, 113 (1833)] or by impinging two jet heads on against each other [Savart, Ann. Chim. Phys. 55, 257 (1833)]. The breakup of the liquid sheet first observed by Savart at the outer rim of the sheet is explained from the point of view of absolute and convective instability. Whether the sheet is convectively or absolutely unstable depends on the local Weber number We=ρU2H/S, where ρ and S are, respectively, the liquid density and the surface tension, and H and U are, respectively, the local half-sheet thickness and the local average liquid velocity. It is shown that absolute instability occurs at We=1, and convective instability occurs in the region where We&amp;gt;1. In the radially expanding liquid sheet, H decreases inversely with the radial distance from the center of the sheet, but U remains constant. Thus, the local Weber number that is greater than one near the center must decrease radially, and eventually reaches one where the sheet encounters absolute instability. In absolute instability, the disturbance propagates upstream to terminate the liquid sheet. This is one of the two regimes of breakup. The onset of absolute instability in this regime always occurs at the broken free edge, where the local Weber number is 1, regardless of the sheet radius. Taylor [Proc. R. Soc. London, Ser. A 253, 296 (1959)] obtained the same criterion from a balance of the surface tension force with a time rate of change of momentum experienced by the fluid at the broken edge. In regime II, it is shown that the sheet may break up at local Weber numbers greater than 1, due to convective instability. The global implications of the local absolute instability are discussed in connection with the related phenomena of breakups of planar sheets, water bells and annular jets.

Secondary transverse instabilities in optical parametric oscillators

We investigate the effect of coupling between diffraction and walk-off on secondary instabilities in nondegenerate optical parametric oscillators. We show that traveling waves that propagate in the walk-off direction, which are generated at the onset of absolute instability, experience Eckhaus and zigzag phase instabilities. Each of these secondary instabilities splits into absolute and convective instabilities that modify the Eckhaus and zigzag instability boundaries. As a consequence, the stability domain of modulated traveling waves is enlarged and may coexist with uniform steady states. The predictions are consistent with the numerical solutions of the optical parametric oscillator model.

Fluid-loaded elastic plate with mean flow: point forcing and three-dimensional effects

The unsteady behaviour of an infinitely long fluid-loaded elastic plate subject to a single-frequency line forcing in the presence of a uniform mean flow is known to exhibit a number of interesting phenomena. These include the onset of absolute instability for non-dimensional flow speeds U in excess of some critical speed Uc, and various interesting propagation effects when U &lt; Uc. In the latter respect Crighton &amp; Oswell (1991) have shown that over a particular frequency range there exists an anomalous neutral mode with group velocity directed towards the driver, in violation of the usual Lighthill outgoing radiation condition. Similar results have been found by Peake (1997) when transverse curvature effects are included. In this paper we seek to extend these results and consider the substantially harder problem of a fluid-loaded elastic plate with uniform mean flow which is subject to a point forcing, thereby resulting in a two-dimensional structural problem. A systematic method for determining the absolute instability threshold is developed, and it is shown that the flow is absolutely unstable for flow speeds U &gt; Uc, where Uc is the one-dimensional value found by Crighton &amp; Oswell. At flow speeds U &lt; Uc the flow is marginally stable and convective growth is found to occur downstream of the driver, over a particular frequency range depending on the transverse Fourier wavenumber ky, within a wedge-shaped region. Outside this wedge-shaped region there is only neutral mode behaviour. Asymptotic forms are found for the dominant large-distance causal flexion response downstream of the driver inside and outside the wedge region, and the appropriate critical angle for the wedge region is identified. Within the convective instability wedge the flexion and critical angle take two different forms depending on whether the frequency ω is greater or less than U2/√5. In addition to this interesting behaviour, the flow also exhibits the usual anomalous neutral mode behaviour and, as with Peake's problem, we also find an extra stability (hoop) resulting in neutral mode behaviour over a small frequency range. Asymptotic forms are also found for the threshold frequencies which divide up the various regions of stability of the system (neutral, neutral anomalous, convectively unstable), as a function of ky, and are compared with the results of both Crighton &amp; Oswell and Peake.

Absolute instability of a liquid jet in a gas

The effect of the ambient gas density on the onset of absolute instability in a viscous liquid jet is examined. The critical Weber number, above which the instability is convective and below which the instability is absolute, is determined as a function of Reynolds number and the density ratio of gas to liquid. It is shown that the gas density has the effect of raising the critical Weber number. It also raises the cutoff wavenumber below which disturbances are spatially amplified and above which they are damped.

On the stability of the infinite dimensional fluid of hard hyperspheres: A statistical mechanical estimate of the density of closest packing of simple hypercubic lattices in spaces of large dimensionality

We report an analysis of the bifurcation of the solution to the nonlinear equation for the inhomogeneous singlet density in a system of hard hyperspheres; the instability examined corresponds to the liquid-to-simple hypercubic lattice transition. We propose that in the limit d→∞ the continuous bifurcation which occurs is at the maximum achievable density in a simple hypercubic lattice. Extension of this result to 1&amp;lt;d&amp;lt;∞ leads to estimates of the closest packing densities of simple hypercubic lattices in d dimensions. An examination of the liquid-to-simple hypercubic lattice transition for particles with a Gaussian pair repulsion leads to the identification of that transition with the onset of absolute instability, i.e., the spinodal of the liquid.

Gyrotron travelling wave amplifier: I. Analysis of oscillations

The operation of the gyrotron travelling wave amplifier is based on the convective cyclotron maser instability. It is found that this convective instability may become absolute (nonconvective) at a sufficiently high current level, resulting in oscillation instead of amplification. This threshold current for the transition depends sensitively on the applied magnetic field. The axial wavelength and the characteristic frequency of oscillation at the onset of absolute instability are given. It is found that momentum spread has virtually no effect on the threshold current. A small amount of resistive wall loss, however, raises the threshold current significantly. Oscillations due to partial reflection at the ends of the system are also examined. Preliminary experimental results on both types of oscillations are reported and are found to be in good agreement with the theory.