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NASA NTRS1994

Instability of streamwise vortices in plane channel flows

We present analysis and numerical experiments on the instability of streamwise vortices in 'minimal channel' flows and argue that this instability is a key feature in the observed intermittent cycle of formation, break-up, and re-formation of these structures. The base flow is a three-component, two-dimensional pair of counter-rotating rolls with axes aligned along the direction of the mean shear. While it is not a steady solution to the Navier-Stokes equations, we show numerically that this flow is unstable on a fast time scale to a secondary, three-dimensional Floquet mode. The growth of the secondary instability does not saturate in a new equilibrium, but continues until highly unstable local shear layers form and the entire flow breaks down into turbulence. Our analysis is motivated in part by the strong similarities between the intermittent turbulent cycle in minimal channel flows and one studied, both experimentally and in computations, in Couette-Taylor flow.

Coughlin, K.↗
NASA NTRS1995

Review of Combustion-acoustic Instabilities

Combustion-acoustic instabilities occur when the acoustic energy increase due to the unsteady heat release of the flame is greater than the losses of acoustic energy from the system. The problem of combustion-acoustic instability is a concern in many devices for various reasons, as each device may have a unique mechanism causing unsteady heat release rates and many have unique boundary conditions. To accurately predict and quantify combustion-acoustic stabilities, the unsteady heat release rate and boundary conditions need to be accurately determined. The present review brings together work performed on a variety of practical combustion devices. Many theoretical and experimental investigations of the unsteady heat release rate have been performed, some based on perturbations in the fuel delivery system particularly for rocket instabilities, while others are based on hydrodynamic processes as in ramjet dump combustors. The boundary conditions for rocket engines have been analyzed and measured extensively. However, less work has been done to measure acoustic boundary conditions in many other combustion systems.

Oyediran, Ayo↗
NASA NTRS1994

Testing the gravitational instability hypothesis?

We challenge a widely accepted assumption of observational cosmology: that successful reconstruction of observed galaxy density fields from measured galaxy velocity fields (or vice versa), using the methods of gravitational instability theory, implies that the observed large-scale structures and large-scale flows were produced by the action of gravity. This assumption is false, in that there exist nongravitational theories that pass the reconstruction tests and gravitational theories with certain forms of biased galaxy formation that fail them. Gravitational instability theory predicts specific correlations between large-scale velocity and mass density fields, but the same correlations arise in any model where (a) structures in the galaxy distribution grow from homogeneous initial conditions in a way that satisfies the continuity equation, and (b) the present-day velocity field is irrotational and proportional to the time-averaged velocity field. We demonstrate these assertions using analytical arguments and N-body simulations. If large-scale structure is formed by gravitational instability, then the ratio of the galaxy density contrast to the divergence of the velocity field yields an estimate of the density parameter Omega (or, more generally, an estimate of beta identically equal to Omega(exp 0.6)/b, where b is an assumed constant of proportionality between galaxy and mass density fluctuations. In nongravitational scenarios, the values of Omega or beta estimated in this way may fail to represent the true cosmological values. However, even if nongravitational forces initiate and shape the growth of structure, gravitationally induced accelerations can dominate the velocity field at late times, long after the action of any nongravitational impulses. The estimated beta approaches the true value in such cases, and in our numerical simulations the estimated beta values are reasonably accurate for both gravitational and nongravitational models. Reconstruction tests that show correlations between galaxy density and velocity fields can rule out some physically interesting models of large-scale structure. In particular, successful reconstructions constrain the nature of any bias between the galaxy and mass distributions, since processes that modulate the efficiency of galaxy formation on large scales in a way that violates the continuity equation also produce a mismatch between the observed galaxy density and the density inferred from the peculiar velocity field. We obtain successful reconstructions for a gravitational model with peaks biasing, but we also show examples of gravitational and nongravitational models that fail reconstruction tests because of more complicated modulations of galaxy formation.

Babul, Arif↗
NASA NTRS1994

The effects of Kelvin-Helmholtz instability on resonance absorption layers in coronal loops

One of the long-standing uncertainties in the wave-resonance theory of coronal heating is the stability of the resonance layer. The wave motions in the resonance layer produce highly localized shear flows which vary sinusoidally in time with the resonance period. This configuration is potentially susceptible to the Kelvin-Helmholtz instability (KHI), which can enhance small-scale structure and turbulent broadening of shear layers on relatively rapid ideal timescales. We have investigated numerically the response of a characteristic velocity profile, derived from resonance absorption models, to finite fluid perturbations comparable to photospheric fluctuations. We find that the KHI primarily should affect long (approximately greater than 6 x 10(exp 4) km) loops where higher velocity flows (M approximately greater than 0.2) exist in resonance layers of order 100 km wide. There, the Kelvin-Helmholtz growth time is comparable to or less than the resonance quarter-period, and the potentially stabilizing magnetic effects are not felt until the instability is well past the linear growth stage. Not only is the resonance layer broadened by the KHI, but also the convective energy transport out of the resonance layer is increased, thus adding to the efficiency of the wave-resonance heating process. In shorter loops, e.g., those in bright points and compact flares, the stabilization due to the magnetic field and the high resonance frequency inhibit the growth of the Kelvin-Helmholtz instability beyond a minimal level.

Karpen, Judith T.↗
NASA NTRS1994

Mixing by barotropic instability in a nonlinear model

A global, nonlinear, equivalent barotropic model is used to study the isentropic mixing of passive tracers by barotropic instability. Basic states are analytical zonal-mean jets representative of the zonal-mean flow in the upper stratosphere, where the observed 4-day wave is thought to be a result of barotropic, and possibly baroclinic, instability. As is known from previous studies, the phase speed and growth rate of the unstable waves is fairly sensitive to the shape of the zonal-mean jet; and the dominant wave mode at saturation is not necessarily the fastest growing mode; but the unstable modes share many features of the observed 4-day wave. Lagrangian trajectories computed from model winds are used to characterize the mixing by the flow. For profiles with both midlatitude and polar modes, mixing is stronger in midlatitude than inside the vortex; but there is little exchange of air across the vortex boundary. There is a minimum in the Lyapunov exponents of the flow and the particle dispersion at the jet maximum. For profiles with only polar unstable modes, there is weak mixing inside the vortex, no mixing outside the vortex, and no exchange of air across the vortex boundary. These results support the theoretical arguments that, whether wave disturbances are generated by local instability or propagate from other regions, the mixing properties of the total flow are determined by the locations of the wave critical lines and that strong gradients of potential vorticity are very resistant to mixing.

Bowman, Kenneth P.↗
NASA NTRS1994

Nonlinear instability of accelerating shock waves with application to supernovae

We consider the stability of a planar accelerating shock front in an exponential atmosphere, a situation that can be described by a self-similar solution. A previous linear instability analysis showed that there are three regimes, depending on the wavelength of the perturbation along the direction of the shock front: at long wavelengths the shock is unstable, at intermediate wavelengths it is overstable, and at short wavelenghts it is stable, where the characteristic length is set by the initial density scale height. We have carried out numerical simulations of such an accelerating shock front and have confirmed the results of the semianalytic linear analysis in the linear regime. In the nonlinear regime, the evolution again depends on the three wavelength domains. At long wavelengths, the instability continuously grows but the flow remains smooth, and at short wavelengths, the flow is stable. The intermediate wavelength, overstable regime shows more complex evolution. The growing linear oscillations saturate soon after entering the nonlinear regime and the oscillations continue with the same approximate period. The shock front develops moving points of intersection, which generate weak shock fronts and density and pressure structure in the immediate postshock flow. The density fluctuations, with a contrast of a factor of 2-3, become frozen into the downstream flow. The growth of the instability is slow, so substantial initial perturbations are needed; thesemay be present in Type II supernovae with red supergiant progenitor stars which have outer convective regions. The clumping in the outer supernova atmosphere may affect spectral line formation and may play a role in the formation of fast knots.

Luo, Ding↗
NASA NTRS1994

Hydrodynamic instability and coalescence of binary neutron stars

We study the importance of hydrodynamic effects on the evolution of coalescing binary neutron stars. Using an approximate energy functional constructed from equilibrium solutions for polytropic binary configuration, we incorporate hydrodynamic effects into the calculation of the orbital decay driven by gravitational wave emission. In particular, we follow the transition between the quasi-static, secular decay of the orbit at large separation and the rapid dynamical evolution of confirgurations approaching contact. We show that a purely Newtonian hydrodynamic instability can significantly accelerate the coalescence at small separation. Such an instability occurs in all close binary configurations containing sufficiently incompressible stars. Calculations are performed for various neutron star masses, radii, and spins. The influence of the stiffness of the equation of state is also explored by varying the effective polytopic index. Typically, we find that the radial infall velocity just prior to contact is about 10% of the tangential orbital velocity. Once the stability limit is reached, the final evolution only takes another orbit. Post-Newtonian effects can move the stability limit to a larger binary separation, and may induce an even larger radial velocity. We also consider the possibiltiy of mass transfer form one neutron star to the other. We show that stable mass transfer is unlikely except when the mass of one of the components is very small (M is less than or approximately 0.4 solar mass) and the viscosity is high enough to maintain corotation. Otherwise, either the two stars come into contact or the dynamical instability sets in before a Roche limit can be reached.

Lai, Dong↗
NASA NTRS1994

The whistler heat flux instability: Threshold conditions in the solar wind

Solar wind electrons are observed often to consist of two components: a core and a halo. The anisotropics and relative average speeds of these components correspond to a heat flux that has the potential to excite several different electromagetic instabilities; wave-particle scattering by the resulting enhanced fluctuations can limit this heat flux. This manuscript describes theoretical studies using the linear Vlasco dispersion equation for drifting bi-Maxwellian component distributions in a homogeneous plasma to examine the threshold of the whistler heat flux instability. Expressions for this threshold are obtained from two different parametric baselines: a local model that yields scalings as functions of local dimensionless plasma paramaters, and a global model based on average electron properties observed during the in-eliptic phase of the Ulysses mission. The latter model yields an expression for the heat flux at threshold of the whistler instability as a function of helisopheric radius that scales in the same way as the average heat flux observed form Ulysses and that provides an approximate upper bound for that same quantity. This theoretical scaling is combined with the observational results to yield a semi-empirical closure relation for the average electron heat flux in the solar wind between 1 and 5 AU.

Gary, S. Peter↗
NASA NTRS1994

Gravitational instabilities in a proto-planetary disk including the effects of magnetic fields

We investigate the gravitational instability of a thin, Keplerian protoplanetary disk including the effects of a largely azimuthal magnetic field. The model follows that of our previous work (Noh, Vishniac, & Cochran 1991) except for the inclusion of a magnetic field. The disk is assumed to consist of neutral and ionized gas and neutral dust which are coupled by gravity and friction. The growth rates and eigenfunctions are calculated numerically using nonaxisymmetric linear perturbation methods. The results show that the growth rate has a maximum at some intermediate azimuthal number m, but for each value of m it is reduced relative to the unmagnetized case. The effects of the magnetic field appear more strongly on small scales. As the strength of the equilibrium magnetic field increases the growth rates decrease, and the maximum instability occurs at a lower value on m due to the increasing magnetic pressure. The response of each component to the magnetic field is discussed using the behavior of the eigenfunctions in the radial direction. With the inclusion of the magnetic field, the effects of the ionization fraction and friction on the growth rates also appear to be important for high m modes. Increasing the ionization fraction or the friction suppresses instability, but only slightly changes the maximally unstable azimuthal scales. The enhanced growth rates due to a dust component for which thermal pressure is negligible are somewhat reduced by the inclusion of a magnetic field. The effects of different boundary conditions (reflecting and transmitting) on the growth rates are also shown.

Noh, Hyerim↗
NASA NTRS1995

Thermal instability in the inner coma of a comet

The spacecraft and ground based observations of comet Halley inner coma showed a localized ion density depletion region whose origin is not well understood. Although it has been linked to a thermal instability associated with negative ions, the photodetachment lifetime of negative ions (approximately 1 sec) is too short compared to the electron attachment time scale (approximately 100 sec) for this process to have a significant effect. A mechanism for the ion density depletion based on the thermal instability of the cometary plasma due to the excitation of rotational and vibrational levels of water molecules is proposed. The electron energy losses due to these processes peak near 4000 K (0.36 eV) and at temperatures higher than this value a localized cooling leads to further cooling (thermal instability) due to the increased radiation loss. The resulting increase in recombination leads to an ion density depletion and the estimates for this depletion at comet Halley agree with the observations.

Milikh, G. M.↗
NASA NTRS1995

The modulational instability for the TDNLS equations for weakly nonlinear dispersive MHD waves

In this paper we study the modulational instability for the TDNLS equations derived by Hada (1993) and Brio, Hunter, and Johnson to describe the propagation of weakly nonlinear dispersive MHD waves in beta approximately 1 plasmas. We employ Whitham's averaged Lagrangian method to study the modulational instability. This complements studies of the modulational instability by Hada (1993) and Hollweg (1994), who did not use the averaged Lagrangian approach.

Webb, G. M.↗
NASA NTRS1995

Electromagnetic instabilities in solar wind interaction with dusty cometary plasmas

Dusty plasmas contain charged dust grains which are much more massive than protons, carry high negative charges due to preferential capture of electrons, and do not have a fixed charge. Fluctuations in the grain charges due to liberation or capture of additional electrons and protons translate as mass and momentum losses or gains for these species, which can render linear modes unstable. On the other hand, many authors have addressed the pickup of ions of cometary origin by the solar wind, which for the parallel part is due to relative streaming between cometary and solar wind ions which excites low-frequency electromagnetic turbulence. In the present work we look again at those instabilities by including effects due to the presence of charged dust in the cometary environments. We have investigated several frequency regimes: nonresonant below the cometary watergroup gyrofrequency, nonresonant below the cometary charged dust gyrofrequency (new and interesting but highly unlikely!) and resonant with the cometary watergroup ions. For most parameter ranges either the existing instabilities are enhanced, showing that the presence of charged dust facilitates the cometary ion pickup by the solar wind, or new instabilities have been shown to exist. Similar conclusions might be relevant for other kinds of astrophysical and heliospheric plasmas containing charged dust, as in planetary rings.

Verheest, Frank↗
NASA NTRS1996

Nonlinear spatial evolution of inviscid instabilities on hypersonic boundary layers

The spatial development of an initially linear vorticity-mode instability on a compressible flat-plate boundary layer is considered. The analysis is done in the framework of the hypersonic limit where the free-stream Mach number M approaches infinity. Nonlinearity is shown to become important locally, in a thin critical layer, when sigma, the deviation of the phase speed from unity, becomes o(M(exp -8/7)) and the magnitude of the pressure fluctuations becomes 0(sigma(exp 5/2)M(exp 2)). The unsteady flow outside the critical layer takes the form of a linear instability wave but with its amplitude completely determined by the nonlinear flow within the critical layer. The coupled set of equations which govern the critical-layer dynamics reflect a balance between spatial-evolution, (linear and nonlinear) convection and nonlinear vorticity-generation terms. The numerical solution to these equations shows that nonlinear effects produce a dramatic reduction in the instability-wave amplitude.

Wundrow, David W.↗
NASA NTRS1996

Migrational Instabilities in Particle Suspensions

This work deals with an instability arising from the shear-induced migration of particles in dense suspensions coupled with a dependence of viscosity on particle concentration. The analysis summarized here treats the inertialess (Re = O) linear stability of homogeneous simple shear flows for a Stokesian suspension model of the type proposed by Leighton and Acrivos (1987). Depending on the importance of shear-induced migration relative to concentration-driven diffusion, this model admits short-wave instability arising from wave-vector stretching by the base flow and evolving into particle-depleted shear bands. Moreover, this instability in the time-dependent problem corresponds to loss of ellipticity in the associated static problem (Re = O, Pe = O). While the isotropic version of the Leighton-Acrivos model is found to be stable with their experimentally determined parameters for simple shear, it is known that the stable model does not give a good quantitative description of particle clustering in the core of pipe flow (Nott and Brady 1994). This leads to the conjecture that an appropriate variant on the above model could explain such clustering as a two-phase bifurcation in the base flow.

Goddard, Joe D.↗
NASA NTRS1996

Absolute and Convective Instability of a Liquid Jet in Microgravity

The transition from convective to absolute instability is observed in the 2.2 second drop tower of the NASA Lewis Research Center. In convective instability the disturbance grows spatially as it is convected downstream. In absolute instability the disturbance propagates both downstream and upstream, and manifests itself as an expanding sphere. The transition Reynolds numbers are determined for two different Weber numbers by use of Glycerin and a Silicone oil. Preliminary comparisons with theory are made.

Lin, Sung P.↗
NASA NTRS1996

Local and global effects of the cross-field current instability

The cross-field current instability (CCI) was proposed elsewhere as a plausible mechanism for the initiation and intensification of substorm expansions. This instability encompasses the modified two stream, the ion-Weibel and the lower hybrid drift modes. The work carried out in relation to this instability and its local and global effects is reviewed. Predicted local effects include current reduction, particle acceleration, the excitation of oblique whistlers and lower hybrid drift waves, and the breakdown of the frozen-in-field condition through anomalous dissipation. The predicted global effects of CCI include the offset of force equilibrium and the generation of field aligned currents at the disruption site, which allow the efficient large scale transportation of mass, momentum and energy within the magnetosphere.

Lui, A. T. Y.↗
NASA NTRS1997

Effects of Acoustically Lined Cylindrical Ducts on Instability Waves in Confined Supersonic Jets

The pressure disturbances generated by an instability wave in the shear layer of a supersonic jet are studied for an axisymmetric jet inside a lined cylindrical duct. For the supersonic jet, locally linear stability analysis with duct wall boundary conditions is used to calculate the eigenvalues and the eigenfunctions. These values are used to determine the growth rates and phase velocities of the instability waves and the radial pressure disturbance patterns. The study is confined to the dominant Kelvin-Helmholtz instability mode and to the region just downstream of the nozzle exit where the shear layer is growing but is still small in size compared to the radius of the duct. Numerical results are used to study the effects of changes in the outer flow, growth in the shear layer thickness, wall distance, wall impedance, and frequency. Results indicate that the effects of the duct wall on shear layer growth rates diminish as the outer flow increases. Also, wall reflections cause variations in growth rates depending on wall height and Strouhal number. These variations are due to the phase relationship between the outgoing and the reflected incoming pressure disturbances at the shear layer. The growth rate variations can be reduced and the maximum growth rate minimized by keeping the imaginary part of the impedance negative.

Dahl, Milo D.↗
NASA NTRS1996

[Kelvin-Helmholtz instability in protostellar jets]

NASA grant NAG 5 2866, funded by the Astrophysics Theory Program, enabled the study the Kelvin-Helmholtz instability in protostellar jets. In collaboration with co-investigator Philip Hardee, the PI derived the analytic dispersion relation for the instability in including a cooling term in the energy equation which was modeled as one of two different power laws. Numerical solutions to this dispersion relation over a wide range of perturbation frequencies, and for a variety of parameter values characterizing the jet (such as Mach number, and density ratio) were found It was found that the growth rates and wavelengths associated with unstable roots of the dispersion relation in cooling jets are significantly different than those associated with adiabatic jets, which have been studied previously. In collaboration with graduate student Jianjun Xu (funded as a research associate under this grant), hydrodynamical simulations were used to follow the growth of the instability into the nonlinear regime. It was found that asymmetric surface waves lead to large amplitude, sinusoidal distortions of the jet, and ultimately to disruption Asymmetric body waves, on the other hand, result in the formation of shocks in the jet beam in the nonlinear regime. In cooling jets, these shocks lead to the formation of dense knots and filaments of gas within the jet. For sufficiently high perturbation frequencies, however, the jet cannot respond and it remains symmetric. Applying these results to observed systems, such as the Herbig-Haro jets HH34, HH111 and HH47 which have been observed with the Hubble Space Telescope, we predicted that some of the asymmetric structures observed in these systems could be attributed to the K-H modes, but that perturbations on timescales associated with the inner disk (about 1 year) would be too rapid to cause disruption. Moreover, it was discovered that weak shock 'spurs' in the ambient gas produced by ripples in the jet surface due to nonlinear, modes of surface and/or body waves could accelerate the ambient gas to low velocity. This latter effect represents a new mechanism by which supersonic jets can accelerate low velocity outflows.

Stone, James↗