The shearing sheet and swing amplification revisited

ABSTRACT The principal results of the classic analysis of the shearing sheet and swing amplification by Julian and Toomre (JT) are re-derived in a more accessible way and used to gain a better quantitative understanding of the dynamics of stellar discs. The axisymmetric limit of the shearing sheet is derived and used to re-derive Kalnajs’ 1965 dispersion relation and Toomre’s 1964 stability criterion for axisymmetric disturbances. Using the shearing sheet to revisit Toomre’s important 1969 paper on the group velocity implied by the Lin–Shu–Kalnajs (LSK) dispersion relation, we discover that two wavepackets emerge inside corotation: one each side of the inner Lindblad resonance. An extended form of the JT equation is used to investigate the impact of there being a deficit or surplus of stars in a narrow range of angular momenta. Swing amplification of leading waves introduced by such a groove gives rise to transient trailing spirals that extend further in radius and live longer at smaller azimuthal wavenumbers. Although the LSK dispersion relation provides useful interpretations of wavepackets, the shearing sheet highlights the limitations of the LSK approach to disc dynamics. Disturbances do not avoid an annulus around corotation, as the LSK dispersion relation implies. While disturbances of the shearing sheet have a limited life in real space, they live on much longer in velocity space, which Gaia allows us to probe extensively. c++ code is provided to facilitate applications of winding spiral waves.

Evolution and stationarity of liquid toroidal drop in compressional Stokes flow

Dynamics of fluid tori in slow viscous flow is studied. Such tori are of interest as future carriers of biological and medicinal substances and are also viewed as potential building blocks towards more complex particles. In this study the immiscible ambient fluid is subject to a compressional flow (i.e., bi-extensional flow), and it comprises a generalization of our earlier report on the particular case with viscosity ratio$\unicode[STIX]{x1D706}=1$(see Zabarankinet al.,J. Fluid Mech., vol. 785, 2015, pp. 372–400), where$\unicode[STIX]{x1D706}$is the ratio between the torus viscosity and that of the ambient fluid. It is found that, for all viscosity ratios, the torus either collapses towards the axis of symmetry or expands indefinitely, depending on the initial conditions and the capillary number,Ca. During these dynamic patterns the cross-sections exhibit various forms of deformation. The collapse and expansion dynamic modes are separated by a limited deformation into a deformed stationary state which appears to exist in a finite interval of the capillary number,$0<Ca<Ca_{cr}(\unicode[STIX]{x1D706})$, and is unstable to axisymmetric disturbances, which eventually cause the torus either to collapse or to expand indefinitely. The characteristic dimensions and shapes of these unstable stationary tori and their dependence on the physical parametersCaand$\unicode[STIX]{x1D706}$are reported.

Pressure wave generation from perturbed premixed flames

Numerical simulations and perturbation analysis of a radially imploding laminar premixed flame are used to study the mechanisms responsible for the generation of pressure fluctuations at flame fronts for various Lewis numbers. The relative importance of mechanisms based on unsteady heat release and on vorticity is investigated using an optimization methodology. Particular attention is paid to the influence of non-axisymmetric conditions and local flame curvature. It is shown that vorticity-based noise generation prevails for high-wavenumber, non-axisymmetric disturbances at all curvatures, while heat-release-driven noise generation dominates the axisymmetric and low-wavenumber regimes. These results indicate that short-wavelength vorticity waves actively participate in flame acoustic activity and can surpass acoustic output mechanisms based on heat-release fluctuations in the vicinity of the flame front.

The decay of the flow in the end region of a suddenly blocked pipe

We consider the decay to rest of initially laminar flow within the end region of a suddenly blocked pipe. Here the flow is dominated by two temporally developing boundary layers, one on the pipe wall and one located at the blockage. The evolution and interaction of these boundary layers contributes to the creation and annihilation of toroidal vortices in the end-region flow, the number and extent growing with increasing Reynolds numbers. For larger Reynolds numbers, these nonlinear vortices delay the decay process within the end region, decaying at a slower rate than flow far downstream of the blockage. Our numerical simulations for pre-blockage Reynolds numbers up to 3000 indicate that the flow in this end region is stable to axisymmetric disturbances.

Stability of surfactant-laden core–annular flow and rod–annular flow to non-axisymmetric modes

The linear stability of core–annular fluid arrangements are considered in which two concentric viscous fluid layers occupy the annular region within a straight pipe with a solid rod mounted on its axis when the interface between the fluids is coated with an insoluble surfactant. The linear stability of this arrangement is studied in two scenarios: one for core–annular flow in the absence of the rod and the second for rod–annular flow when the rod moves parallel to itself along the pipe axis at a prescribed velocity. In the latter case the effect of convective motion on a quiescent fluid configuration is also considered. For both flows the emphasis is placed on non-axisymmetric modes; in particular their impact on the recent stabilization to axisymmetric modes at zero Reynolds number discovered by Bassom, Blyth and Papageorgiou (J. Fluid. Mech., vol. 704, 2012, pp. 333–359) is assessed. It is found that in general non-axisymmetric disturbances do not undermine this stabilization, but under certain conditions the flow may be linearly stable to axisymmetric disturbances but linearly unstable to non-axisymmetric disturbances.

Nonlinear stability of hypersonic flow over a cone with passive porous walls

This study investigates the nonlinear stability of hypersonic viscous flow over a sharp slender cone with passive porous walls. The attached shock and effect of curvature are taken into account. Asymptotic methods are used for large Reynolds number and large Mach number to examine the viscous modes of instability (first Mack mode), which may be described by a triple-deck structure. A weakly nonlinear stability analysis is carried out allowing an equation for the amplitude of disturbances to be derived. The coefficients of the terms in the amplitude equation are evaluated for axisymmetric and non-axisymmetric disturbances. The stabilizing or destabilizing effect of nonlinearity is found to depend on the cone radius. The presence of porous walls significantly influences the effect of nonlinearity, and results for three types of porous wall (regular, random and mesh microstructure) are compared.

Non-modal stability in sliding Couette flow

The problem of an incompressible flow between two coaxial cylinders with radii$a$and$b$subjected to a sliding motion of the inner cylinder in the axial direction is considered. The energy stability and the non-modal stability have been investigated for both axisymmetric and non-axisymmetric disturbances. For the non-modal stability, we focus on two problems: response to external excitations and response to initial conditions. The former is studied by examining the$\epsilon $-pseudospectrum, and the latter by examining the energy growth function$G(t)$. Unlike the results of the modal analysis in which the stability of sliding Couette flow is determined by axisymmetric disturbances, the energy analysis shows that a non-axisymmetric disturbance has a critical energy Reynolds number for all radius ratios$\eta = a/ b$. The results for non-modal stability show that rather large transient growth occurs over a wide range of azimuthal wavenumber$n$and streamwise wavenumber$\ensuremath{\alpha} $, even though the Reynolds number is far below its critical value. For the problem of response to external excitations, the response is sensitive to low-frequency external excitations. For all values of the radius ratio, the maximum response is achieved by non-axisymmetric and streamwise-independent disturbances when the frequency of external forcing$\omega = 0$. For the problem of response to initial conditions, the optimal disturbance is in the form of helical streaks at low Reynolds numbers. With the increase of$\mathit{Re}$, the optimal disturbance becomes very close to straight streaks. For each$\eta $, the maximum energy growth of streamwise-independent disturbances is of the order of${\mathit{Re}}^{2} $, and the optimal time is of the order of$\mathit{Re}$. This relation is qualitatively similar to that for plane Couette flow, plane Poiseuille flow and pipe Poiseuille flow. Direct numerical simulations are applied to investigate the transition of the streamwise vortex (SV) scenario at$\mathit{Re}= 1000$and 1500 for various$\eta $. The initial disturbances are the optimal streamwise vortices predicted by the non-modal analysis. We studied the streak breakdown phase of the SV scenarios by examining the instability of streaks. Moreover, we have investigated the sustainment of the energy of disturbances in the SV scenario.

The stability of developing pipe flow at high Reynolds number and the existence of nonlinear neutral centre modes

The high-Reynolds-number stability of unsteady pipe flow to axisymmetric disturbances is studied using asymptotic analysis. It is shown that as the disturbance amplitude is increased, nonlinear effects first become significant within the critical layer, which moves away from the pipe wall as a result. It is found that the flow stabilizes once the basic profile has become sufficiently fully developed. By tracing the nonlinear neutral curve back to earlier times, it is found that in addition to the wall mode, which arises from a classical upper branch linear stability analysis, there also exists a nonlinear neutral centre mode, governed primarily by inviscid dynamics. The centre mode problem is solved numerically and the results show the existence of a concentrated region of vorticity centred on or close to the pipe axis and propagating downstream at almost the maximum fluid velocity. The connection between this structure and the puffs and slugs of vorticity observed in experiments is discussed.

On the instability of a free viscous rim

This paper is devoted to the theoretical description of the dynamics of a rim formed by capillary forces at the edge of a free, thin liquid sheet. The rim dynamics are described using a quasi-one-dimensional approach accounting for the inertia of the liquid in the rim and for the liquid flow entering the rim from the sheet, surface tension and viscous stresses. The governing equations are derived from the mass, momentum and moment-of-momentum-balance equations of the rim. The theory provides a basis from which to analyse the linear stability of a straight line rim bounding a planar liquid sheet. The combined effect of the axisymmetric disturbances of the radius of the rim cross-section as well as of the transverse disturbances of the rim centreline is considered. The effect of the viscosity, relative film thickness and rim deceleration are investigated. The predicted wavelength of the most unstable mode is always very similar to the Rayleigh wavelength of the instability of an infinite cylindrical jet. This prediction is confirmed by various experimental data found in the literature. The maximum rate of growth of rim disturbances depends on all the parameters of the problem; however, the most pronounced effect can be attributed to the rim deceleration. This conclusion is confirmed by nonlinear simulations of rim deformation.