Sinuous Mode Research Articles

Linear stability of multilayer plane Poiseuille flows of Oldroyd B fluids

The linear stability of plane Poiseuille flows of two and three-symmetrical layers is studied by using both longwave and moderate wavelength analysis. The considered fluids follow Oldroyd-B constitutive equations and hence the stability is controlled by the viscous and elastic stratifications and the layer thicknesses. For the three symmetrical-layer Poiseuille flow, competition between varicose (symmetrical) and sinuous (antisymmetrical) mode is considered. In both cases (two and three symmetrical layers), the additive character of the longwave formula with respect to viscous and elastic terms is largely used to determine stable arrangements at vanishing Reynolds number. It is found that if the stability of such arrangements is due simultaneously to viscous and elastic stratification (the flow is stable for longwave disturbance and the Poiseuille velocity profile is convex), then the Poiseuille flow is also stable with respect to moderate wavelength disturbances and the critical thickness ratio around which the configurations becomes unstable is given by longwave analysis. Note that a convex velocity profile means a positive jump of shear rate at the interface. Finally, the destabilization due to a moderate increase in the Reynolds number is considered and two distinct behaviors are pointed according to the convexity of the Poiseuille velocity profile. Moreover, an important influence of the thickness ratio on the critical wavenumber is found for three symmetrical layer case (for two layer case, the critical wave number is of order one and depends weakly on the thickness ratio).

Absolute and convective instability of a viscous liquid curtain in a viscous gas

The linear instability of a viscous liquid flowing in a vertical sheet sandwiched between two viscous gases bounded externally by two vertical walls is investigated. The critical Weber number below which the flow is absolutely unstable and above which the flow is convectively unstable is found to be approximately equal to one and is weakly dependent on the rest of the parameters. The Weber number is defined asWe≡ρ1U20d/SwhereSis the surface tension,ρ1is the liquid density,U0is the centreline velocity of the liquid sheet, anddis the half-thickness of the uniform liquid sheet. The sinuous mode is found to have a greater amplification rate than the varicose mode in the convective instability regime. While absolute instability is caused by the surface tension, convective instability is caused by the amplification of disturbances near the liquid-gas interface. The surface tension, and viscosities of liquids and gases all suppress the amplification of the convectively unstable disturbances. An increase in the gravitational force or the gas density results in an enhancement of the amplification rate.

On an active resonant triad of mixed modes in symmetric shear flows: a plane wake as a paradigm

In this paper, we identify a new type of resonant triad which operates in a parallel or nearly parallel shear flow with a symmetric profile. The triad consists of a planar sinuous mode, an oblique sinuous mode and an oblique varicose mode, but is not of the usual subharmonic-resonance form. The development of the triad is studied in the non-equilibrium critical-layer régime. The equations governing the evolution of the modes are derived. We show that the quadratic resonance can significantly enhance the growth of both the oblique sinuous and varicose modes, and may cause them to grow super-exponentially. This can lead to a subsequent stage in which the oblique sinuous mode produces a back reaction on the oblique varicose mode through a phase-locked interaction, causing both oblique modes to evolve even more rapidly. We suggest that the resonant triad is a viable mechanism for the development of three-dimensional structures and varicose components observed in the later stage of plane wake transition.

Interaction of phase-locked modes: a new mechanism for the rapid growth of three-dimensional disturbances

In this paper, we have identified a new mechanism which can promote rapid growth of three-dimensional disturbances. The mechanism involves the interaction between a planar mode and an oblique mode, or a pair of oblique modes, which are phase-locked in the sense that they have the same phase speed. This allows a powerful nonlinear interaction to take place within the common critical layer(s). The disturbance is not required to form a subharmonic resonant triad, and hence the mechanism operates under much less restrictive conditions than does subharmonic resonance (although it is somewhat less powerful). We show that the quadratic interaction between the planar mode and the oblique modes drives an exceptionally large forced mode with the difference frequency, which in turn interacts with the planar mode to contribute the dominant nonlinear effect. This interaction can cause the oblique modes to grow super-exponentially provided that their magnitude is sufficiently small. As a result of the super-exponential growth, the oblique mode may soon become strong enough to produce a feedback effect on the planar mode, so that the interactions eventually become fully coupled. This subsequent stage takes slightly different forms depending on whether a single or a pair of oblique modes is present. Both cases are investigated. Particular attention is paid to symmetric plane shear layers, e.g. a planar wake or jet, for which subharmonic resonance of sinuous modes is inactive.

Electrohydrodynamic stability of a highly viscous jet

When a pendant drop of weakly conducting fluid is raised to a high electric potential, it frequently adopts the shape of a Taylor cone from whose apex a thin, charged jet is emitted. Such a jet can display surprising longevity, but eventually breaks up into fine droplets, a fact utilized in electro-spraying devices. This paper examines the linear stability of an incompressible cylindrical jet carrying surface charge q in a tangential electric field E, for various values of the permittivity ratio λ and the finite rate of charge relaxation, τ. The viscosity is assumed to be large. It is shown that all axisymmetric temporal modes can be stabilized for suitable values of (q, E), but sinuous modes with logarithmically large wavelengths are unstable. If these very long waves are excluded, the jet can sometimes be completely stabilized. It is also shown that an uncharged jet with low permittivity is unstable to sinuous waves for large E, contrary to previous belief.

Instability of an annular viscous liquid jet

A linear analysis has been carried out for the temporal instability of an annular viscous liquid jet moving in an inviscid gas medium, which includes three limiting cases of a round liquid jet, a gas jet and a plane liquid sheet. It is found that there exist two independent unstable modes, which become the well-known sinuous and varicose modes for plane liquid sheets as annular jet radii approach to infinity. Hence, they are named as para-sinuous and para-varicose. It is shown that an ambient gas medium always enhances the annular jet instability. The curvature effects in general increase the disturbance growth rate, and may not be neglected for the breakup process of an annular or conical liquid sheet. An annular jet with a sufficiently small thickness tends to break up much faster than the corresponding plane liquid sheet, in accordance with existing experimental observations. Liquid viscosity has complicated dual effects on the instability. It is also found that there exists a critical Weber number below which surface tension is the source of instability. Whereas above it, instability is suppressed by surface tension effect and it promoted by aerodynamic interaction between the liquid and gas phase. For the practical importance of large Weber numbers such as related to liquid atomization, the para-sinuous mode is always predominant.

The feedback loop in impinging two-dimensional high-subsonic and supersonic jets

The instabilities in a supersonic impinging jet are investigated by solving the two-dimensional Euler equations using the piecewise parabolic method (PPM) and Roe's linearized Riemann solver. The predicted shock cell spacing agrees well with the observed and theoretical values. The frequency and nature of the dominant instabilities are found to be a function of the impingement distance. Two instability modes are possible: a symmetric (or varicose) mode and an asymmetric (or sinuous) mode. For two given jet exit Mach numbers ( M = 0.98 and 1.29), the energy in, and frequency of, these modes are functions of impingement distance, leading to an integral staging due to an acoustic feedback loop. The predicted frequencies of the fundamental symmetric and asymmetric instabilities agree with the theoretically allowed values. The staging of predicted frequencies that occurs in experimental work is also predicted.

Temporal instability of plane gas sheets in a viscous liquid medium

This paper reports a linear temporal instability analysis of an incompressible plane gas sheet in a quiescent viscous liquid medium of infinite expanse. Results indicate that there exist two unstable modes of disturbance waves, sinuous and varicose, and surface tension always reduces, while the relative velocity between the gas and liquid phases and the gas density always enhance instability development. For both unstable modes, the presence of liquid viscosity increases the instability limit, which is however independent of the absolute value of viscosity. It is also shown that the sinuous mode becomes stable when the gas Weber number, defined as the ratio of aerodynamic forces to surface tension forces, is less than the critical value of one. At slightly larger gas Weber numbers, liquid viscosity exhibits dual effects—it may enhance or suppress the growth of unstable disturbances, depending on specific flow conditions. However, for sinuous mode at high Weber numbers and varicose mode at any Weber numbers, liquid viscosity always reduces disturbance growth rates and dominant wave numbers. Unlike the case for plane liquid sheets, varicose mode controls the instability process for all Weber number ranges and for both inviscid and viscous liquids, and only at high Weber numbers, do varicose and sinuous modes become almost equally important. It is further found that the wave propagation velocity for both unstable modes is much smaller than the gas velocity at the mode of maximum instability, implying that the disturbance waves appear almost stationary rather than travelling-wave type, in contrast with the plane liquid sheet results.

Sinuous Modes and Steady Warps of Polytropic Disks

Sinuous Modes and Steady Warps of Polytropic Disks

Fundamental and subharmonic secondary instabilities of Görtler vortices

The nonlinear development of stationary Görtler vortices leads to a highly distorted mean flow field where the streamwise velocity depends strongly not only on the wall-normal but also on the spanwise coordinates. In this paper, the inviscid instability of this flow field is analysed by solving the two-dimensional eigenvalue problem associated with the governing partial differential equation. It is found that the flow field is subject to the fundamental odd and even (with respect to the Görtler vortex) unstable modes. The odd mode, which was also found by Hall & Horseman (1991), is initially more unstable. However, there exists an even mode which has higher growth rate further downstream. It is shown that the relative significance of these two modes depends upon the Görtler vortex wavelength such that the even mode is stronger for large wavelengths while the odd mode is stronger for short wavelengths. Our analysis also shows the existence of new subharmonic (both odd and even) modes of secondary instability. The nonlinear development of the fundamental secondary instability modes is studied by solving the (viscous) partial differential equations under a parabolizing approximation. The odd mode leads to the well-known sinuous mode of break down while the even mode leads to the horseshoe-type vortex structure. This helps explain experimental observations that Görtler vortices break down sometimes by sinuous motion and sometimes by developing a horseshoe vortex structure. The details of these break down mechanisms are presented.

Non-axisymmetric instability of core–annular flow

Stability of core-annular flow of water and oil in a vertical circular pipe is studied with respect to non-axisymmetric disturbances. Results show that when the oil core is thin, the flow is most unstable to the asymmetric sinuous mode of disturbance, and the core moves in the form of corkscrew waves as observed in experiments. The asymmetric mode of disturbance is the most dangerous mode for quite a wide range of material and flow parameters. This asymmetric mode persists in vertical pipes with upward and downward flows and in horizontal pipes. The analysis also applies to the instability of freely rising axisymmetric cigarette smoke or a thermal plume. The study predicts a unique wavelength for the asymmetric meandering waves.

Instability of a high-speed submerged elastic jet

The linearized inertial instability of the parallel shear flow of a viscoelastic liquid is considered. An elastic Rayleigh equation is derived, for high Reynolds numbers and high Weissenberg numbers, and for a viscoelastic liquid whose first normal stress dominates other stresses. The equation is used to investigate the stability of a submerged jet, that may be planar or axisymmetric, having a parabolic velocity profile. The sinuous mode is found to be fully stabilized by sufficiently large elasticity. The varicose mode in the planar case is partially stabilized, being unstable only at longer wavelengths and with a reduced growth rate. An axisymmetric jet, which is stable to varicose perturbations at zero elasticity, is found to be unstable to short-wave disturbances for small non-zero elasticity. This novel instability involves elastic waves in the shear. It is also present in other modes but does not have the fastest growth rate.

Primary and secondary instabilities of the asymptotic suction boundary layer on a curved plate

The temporal growth of Görtler vortices and the associated secondary instability mechanisms are investigated numerically in the case of an asymptotic suction boundary layer on a curved plate. Highly inflectional velocity profiles are generated in both the spanwise and vertical directions. The inflectional velocity profile develops earlier in the spanwise direction. There exist two distinct modes of instability that eventually lead to the breakdown of Görtler vortices: the sinuous mode and the varicose mode. The temporal secondary instability analysis of the three-dimensional inflectional velocity profile reveals that the sinuous mode becomes unstable earlier than the varicose mode. The sinuous mode is shown to be primarily related to shear in the spanwise direction, ∂U/∂z, and the varicose mode to shear in the vertical direction, ∂U/∂y.

Observations of Gulf Stream Meander-Induced Disturbances

Abstract Using moored array data collected across the Gulf Stream at 55°W, the author has investigated the dynamic signatures of the velocity fluctuations at the thermocline (550 m) and near-bottom (4000 m) levels. The cross-stream amplitude and phase structures for motions with period from 64 to 12 days, in a coordinate frame aligned with the instantaneous stream, have a striking resemblance to those computed from small amplitude instability theory. It is the sinuous or asymmetric mode that appears to dominate the observed fluctuations as is predicted. In addition, the mean downstream velocity at the two levels is consistent with a two-layer model for the mean flow in which the lower layer has uniform potential vorticity. When fit to the data this model predicts a value for the rate of change of the Coriolis parameter appropriate to this latitude, but one would expect it to be augmented by a factor of 2 or so by the significant slope of the bottom in this region.

On the instability of plane liquid sheets in two gas streams of unequal velocities

This paper reports a linear stability analysis of plane liquid sheets in gas streams of unequal velocities. It is found that there exist two independent unstable modes, named herein as para-sinuous and para-varicose. They resemble, but certainly differ from, the well-known sinuous and varicose modes found for equal gas velocities. The gas velocity difference may increase or decrease the growth rate for the para-varicose mode, and for the para-sinuous mode at high Weber numbers. At Weber numbers slightly larger and less than unity, gas velocity enhances significantly the degree of instability for the para-sinuous mode. For the practical interest of large Weber numbers such as related to liquid atomization, the para-sinuous mode is always more unstable. Liquid viscosity has complicated dual effects on the instability, and surface tension is always stabilizing. Long and short wavelength approximations indicate that instability is due to the velocity jump across the two liquid-gas interfaces and from one gas stream to the other, hence is related to the classical Kelvin-Helmholtz instability, and from the disturbance energy equation that the mechanism of instability is due to the interfacial pressure fluctuations.

On the stability of vertical double-diffusive interfaces. Part 2. Two parallel interfaces

This is the second part of a three-part study of the stability of vertically oriented double-diffusive interfaces having an imposed vertical stable temperature gradient. In this study, flow is forced within a fluid of infinite extent by a prescribed excess of compositionally buoyant material between two parallel interfaces. Compositional diffusivity is ignored while thermal diffusivity and viscosity are finite. The stability of the interfaces is analysed first in the limit that they are close together (compared with the salt-finger lengthscale), then for general spacing. Attention is focused on whether the preferred mode of instability is varicose or sinuous and whether its wavevector is vertical or oblique.The interfaces are found to be unstable for some wavenumber for all values of the Prandtl number and interface spacing. The preferred mode of instability for closely spaced interfaces is varicose and vertical for Prandtl number less than about 9, sinuous oblique for Prandtl number between 9 and 15 and sinuous vertical for larger Prandtl number. For general spacing each of the four possible modes of instability is preferred for some range of Prandtl number and interface separation, with no clear pattern of preference, except that the sinuous oblique mode is preferred for widely separated interfaces. The growth rate of the preferred mode is largest for interfaces having separations of from 1 to 3 salt-finger lengths.

On the mechanism of sinuous and varicose modes in three-dimensional viscous secondary instability of nonlinear Görtler rolls

The present study begins with the solution of the linearized partial differential equations for viscous, secondary instabilities of nonlinearly developed primary Görtler rolls. A global energy balance verifies the amplification rates obtained from secondary instability analysis and confirms that the sinuous mode would dominate. This therefore explains the strong resemblance between the rms-streamwise u-velocity structure obtained by hot-wire measurements [Swearingen and Blackwelder, J. Fluid Mech. 182, 255 (1987)] in the cross-sectional yz plane and that of the sinuous mode. The u contours of the secondary disturbance have peaks at locations corresponding to the shoulder regions of the mushroom-like primary streamwise U-velocity structure and even more intense regions surrounding the mushroom stem near the wall. This structure is explained by detailed analysis of the energy-balancing mechanisms for u2/2 in yz plane: the Reynolds stress-conversion mechanism predominantly affects the primary spanwise rate of shear strain ∂U/∂z, whereas of less importance is the effect on the normal rate of shear strain ∂U/∂y; these are delicately balanced by the isotropizing mechanism of pressure rate of strain correlation and viscous dissipation. The appropriate pressure rates of strain correlation then provide the sources for the vertical v2/2 and spanwise w2/2 contributions to the kinetic energy. The sinuous mode can be identified with streamwise-propagating structures having spanwise waviness in flow visualization studies. The spanwise-antisymmetric w, which has intense regions in the upper and lower levels of the boundary layer, tends to wag the primary mushroom structure both at the upper domed region and in the stem region near the wall, not necessarily in phase, in the spanwise direction in the y,z plane, as shown in the flow-visualization work of Peerhossaini and Wesfreid [Intl. J. Heat Fluid Flow 9, 12 (1988)]. On the other hand, the infrequency of the varicose mode can be attributed to its relatively lower amplification rate for its most amplified mode. When observed, it is identifiable with the spanwise-symmetric streamwise propagating, repetitive ‘‘horseshoe’’ structures appearing in flow visualization. Energy balance analysis reveals that this mode is attributable mainly to the Reynolds stress-conversion mechanism associated with rate of strain ∂U/∂y in the high shear layer in the peak region, although the mechanism associated with ∂U/∂z is not dramatically smaller. Viscous dissipation, though not large enough to overcome the production mechanisms, is nevertheless prominent in the global balance as well as in the detailed structural balances in the yz plane. The secondary instability vorticity structure indicates that the sinuous mode is dominated by its vertical component ηs and the varicose mode by its spanwise component ζv.

Observations on transition in plane bubble plumes

Flow visualization studies of plane laminar bubble plumes have been conducted to yield quantitative data on transition height, wavelength and wave velocity of the most unstable disturbance leading to transition. These are believed to be the first results of this kind. Most earlier studies are restricted to turbulent bubble plumes. In the present study, the bubble plumes were generated by electrolysis of water and hence very fine control over bubble size distribution and gas flow rate was possible to enable studies with laminar bubble plumes. Present observations show that (a) the dominant mode of instability in plane bubble plumes is the sinuous mode, (b) transition height and wavelength are related linearly with the proportionality constant being about 4, (c) wave velocity is about 40% of the mean plume velocity, and (d) normalized transition height data correlate very well with a source Grashof number. Some agreement and some differences in transition characteristics of bubble plumes have been observed compared to those for similar single-phase flows.

Spatial instability of plane liquid sheets

Spatial instability of a thin moving plane liquid sheet is analyzed. It is shown that there exist two independent modes of instability, sinuous and varicose. For the sinuous mode, there is a critical Weber number of unity below which pseudo-absolute instability exists and above which convective instability appears. For the varicose mode, the instability is always convective. The convective mode is induced by the aerodynamic effects, and is related to the corresponding temporal instability through Gaster's relation at the sufficiently high liquid velocity. Perturbation analysis indicates that the spatial amplification rate depends strongly on the gas-to-liquid density ratio ϱ. For the convective instability of the sinuous mode, there exists a critical Weber number of slightly larger than unity, below which liquid viscosity exhibits both stabilizing and destabilizing effects and above which the viscous effects always reduce the growth rate and the dominant wavenumber. For situations where the Weber number, We ⪢ 1, but ϱ We ⩽ 0(1), the growth rate of the sinuous mode is larger than that of the varicose mode under the same flow conditions. If both We ⪢ 1 and ϱ We ⪢ 1, the spatial growth rate of both sinuous and varicose modes is almost the same, except at the small wavenumbers where the sinuous mode has slightly larger growth rate than the corresponding varicose mode. The viscous effects on the varicose mode is always stabilizing. The present results compare favorably with the existing experimental observations.

Non-parallel effects in the instability of Long's vortex

As shown in Foster & Smith (1989), at large flow force M, Long's self-similar vortex is in the form of a swirling ring-jet, whose axial velocity profile is of sech2 form. At azimuthal wavenumber n of comparable order to the axial wavenumber, linear helical modes of instability are essentially those of the Bickley jet varicose and sinuous modes. However, at small axial wavenumbers, the three-dimensionality of the vortex is important, and the instabilities depend heavily on the effects of the swirl. We explore here the effects of finite Reynolds number Re on these long-wave inertial modes. It is shown that, because the radial velocity scales with Re−1M, the non-parallelism of the flow is more important than the viscous terms in determining the finite-Re behaviour. The three-layer structure of the parallel-flow instability modes remains, but with a critical layer considerably modified by radial velocity. In investigating the critical range Re = O(M3), we find the following: for n > 1, the non-parallelism stabilizes the unstable inertial modes, leading to determination of neutral curves; for n < − 1, the non-parallel effects always destabilize the vortex to these helical modes. Determination of the unstable modes and neutral curves for the n > 1 case requires a computational scheme that accounts for the presence of viscosity. It turns out that the n < 1 (n > − 1) modes are prograde (retrograde) with respect to the rotation of the main vortex.