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NASA NTRS1983

Flare loop radiative hydrodynamics. IV - Dynamic evolution of unstable semiempirical loop models

The evolution of the unstable solar atmosphere into the nonlinear phase, in response to various perturbations, is followed. The initial dynamic evolution of the atmosphere follows the predictions of linear stability analysis. In the nonlinear phase, rapid changes are confined to the transition region; these changes are manifested as a propagation of the transition region through the plasma, i.e., chromospheric evaporation or condensation. Global evolution therefore proceeds on the coronal conductive time scale. The rate of propagation of the transition region is determined by the imbalance between the energy supplied by thermal conduction from the corona and radiative cooling within the transition region itself. Flow velocities in the lower corona during evaporation or condensation are, in the cases studied, of order 3 km/s. The observed dynamic evolution is consistent with the existence of relatively long-lived coronal loops whose brightnesses vary on the evaporative time scale.

An, C.-H.↗

NASA NTRS1983

A selection principle in Benard-type convection

In a Benard-type convection problem, the stationary flows of an infinite layer of fluid lying between two rigid horizontal walls and heated uniformly from below are determined. As the temperature difference across the layer increases beyond a certain value, other convective motions appear. These motions areoften cellular in character in that their streamlines are confined to certain well-defined cells having, for example, the shape of rolls or hexagons. A selection principle that explains why hexagonal cells seem to be preferred for certain ranges of the parameters is formulated. An operator-theoretical formulation of one generalized Bernard problem is given. The infinite dimensional problem is reduced to one of solving a finite dimensional system of equations, namely, the selection equations. These equations are solved and a linearized stability analysis of the resultant stationary flows is presented.

Knightly, G. H.↗

NASA NTRS1984

Theoretical analyses of baroclinic flows

Completed and ongoing research activities are discussed briefly, including a three-dimensional, linear stability analysis of the baroclinic Hadley cell and a numerical model of the baroclinic flow between two rotating concentric spheres. This model simulates axisymmetric flow in the Atmospheric General Circulation Experiment configuration. A computer code designed to solve the strongly nonlinear stability problem for the Eady basic state is mentioned.

Antar, B. N.↗

NASA NTRS1983

Electrostatic Kelvin-Helmholtz instability in a radially injected plasma cloud

An elecrostatic finite-sized particle simulation model is used to study the early time-scale phenomena asociated with an impulsively injected plasma, expanding radially, normal to a strong ambient magnetic field. Results of the simulation show the early formation of a radial polarization electric field due to ion-electron charge separation which causes electrons to drift in the azimuthal direction. Velocity shear within this motion gives rise to a Kelvin-Helmholtz (dioxotron) instability, creating a radial fluted pattern as the plasma expands. In the nonlinear stage of instability, azimuthal variations in the electric field are observed which cause electrons to drift across the magnetic field, thereby reducing the space charge created by energetic ion expansion. The results of a linear stability analysis reveal that for a decreasing amount of charge separation in the plasma, the number of unstable azimuthal modes at maximum growth rate increases. Also, as the thickness of the electron ring increases, a fewer number of unstable modes develops for a given amount of charge separation. It is concluded that electrons become unstable at the very early time scale, within an ion gyroperiod, and that unstable electrons lead to modification of the ion dynamics.

Sydora, R. D.↗

NASA NTRS1983

Three-dimensional baroclinic instability of a Hadley cell for small Richardson number

For the case of a baroclinic flow whose Richardson number, Ri, is of order unity, a three-dimensional linear stability analysis is conducted on the basis of a model for a thin, horizontal, rotating fluid layer which is subjected to horizontal and vertical temperature gradients. The Hadley cell basic state and stability analysis are both based on the Navier-Stokes and energy equations, and perturbations possessing zonal, meridional, and vertical structures are considered. An attempt is made to extend the previous theoretical work on three-dimensional baroclinic instability for small Ri to a more realistic model involving the Prandtl and Ekman numbers, as well as to finite growth rates and a wider range of the zonal wavenumber. In general, it is found that the symmetric modes of maximum growth are not purely symmetric, but have a weak zonal structure.

Antar, B. N.↗

NASA NTRS1985

Three-dimensional baroclinic instability of a Hadley cell for small Richardson number

A three-dimensional, linear stability analysis of a baroclinic flow for Richardson number, Ri, of order unity is presented. The model considered is a thin horizontal, rotating fluid layer which is subjected to horizontal and vertical temperature gradients. The basic state is a Hadley cell which is a solution of the complete set of governing, nonlinear equations and contains both Ekman and thermal boundary layers adjacent to the rigid boundaries; it is given in a closed form. The stability analysis is also based on the complete set of equations; and perturbation possessing zonal, meridional, and vertical structures were considered. Numerical methods were developed for the stability problem which results in a stiff, eighth-order, ordinary differential eigenvalue problem. The previous work on three-dimensional baroclinic instability for small Ri was extended to a more realistic model involving the Prandtl number, sigma, and the Ekman number, E, and to finite growth rates and a wider range of the zonal wavenumber.

Antar, B. N.↗

NASA NTRS1984

Morphological stability in the presence of fluid flow in the melt

Recent experiments have shown that the presence of a vertical buoyancy-driven flow adjacent to an initially cylindrical crystal-melt interface may produce a time-dependent helical deformation of the interface, with a rotation period ranging from several minutes to many hours, depending upon the width of the melt. The temperature distribution is such that the interface is expected to be morphologically stable in the absence of fluid flow. A linear stability analysis reveals that the instability is due to a coupling between a basic hydrodynamic instability in the buoyant flow and the deformable boundary separating the two phases. The crystal-melt interface lowers the critical Grashof number of an analogous rigid-walled system by an order of magnitude for succinonitrile with a Prandtl number P = 22.8; furthermore, the hydrodynamic mode that is actually destabilized by the interface is not the least stable mode in the rigid-walled system for P = 22.8. The results show that the instability may be regarded either as a rather large alteration of a basic hydrodynamic instability by the crystal-melt interface, or as a significant modification of the morphological stability of the interface by the presence of the buoyant flow.

Mcfadden, G. B.↗

NASA NTRS1984

Thermosolutal convection during directional solidification

During solidification of a binary alloy at constant velocity vertically upward, thermosolutal convection can occur if the solute rejected at the crystal-melt interface decreases the density of the melt. We assume that the crystal-melt interface remains planar and that the flow field is periodic in the horizontal direction. The time-dependent nonlinear differential equations for fluid flow, concentration, and temperature are solved numerically in two spatial dimensions for small Prandtl numbers and moderately large Schmidt numbers. For slow solidification velocities, the thermal field has an important stabilizing influence: near the onset of instability the flow is confined to the vicinity of the crystal-melt interface. Further, for slow velocities, as the concentration increases, the horizontal wavelength of the flow decreases rapidly - a phenomenon also indicated by linear stability analysis. The lateral inhomogeneity in solute concentration due to convection is obtained from the calculations. For a narrow range of solutal Rayleigh numbers and wavelengths, the flow is periodic in time.

Mcfadden, G. B.↗

NASA NTRS1985

Convective influence on the stability of a cylindrical solid-liquid interface

Experiments in which a long vertical, heated wire is surrounded by concentric annuli of a melt and its crystalline solid show that the convection state changes from a stable unicell surrounded by a stationary cylindrical solid-liquid interface, to a complex time-dependent flow surrounded by a rotating, helical solid-liquid interface. This transition occurs at a Grashof number of approximately 150, which is an order of magnitude less than the critical Grashof number calculated for a liquid annulus surrounded by rigid walls. A linear stability analysis has been carried out for an infinitely tall vertical annulus. When the deformable nature of the crystal-melt interface is taken into account in the boundary conditions, two new modes of instability arise. The most dangerous mode is asymmetrical and corresponds to helical waves travelling vertically upwards. The critical Grashof number and the scaling properties of the eigenstate agree with experiments. The results clearly demonstrate the coupling of convection with crystal-melt interfacial instabilities.

Fang, Q. T.↗

NASA NTRS1985

Magnetohydrodynamic thermal instabilities in cool inhomogeneous atmospheres

The stability of magnetic loops to current-driven filamentation instabilities is investigated. The unperturbed atmosphere is assumed to be composed of an (upper) isothermal optically thin low-density portion and a (lower) higher-density portion which is in radiative equilibrium; in both cases, the atmosphere is in hydrostatic equilibrium, so that gravitational stratification is taken into account. In order to provide specific equilibrium conditions for evaluation of the dispersion relation, conditions appropriate for the surface of a solar-type star are adopted; i.e., a fairly low temperature (T = 5000 K) appropriate for a 'precoronal' state associated, for example, with magnetic flux emerging from photospheric levels under the action of magnetic buoyancy. A linear stability analysis is performed, and numerical results show that physically plausible current densities, which would be generated by typical loop-footpoint motions, are effective in driving MHD instabilities in such a plasma. The instability growth rates are strongly dependent on the assumed current density distribution and on the density scale height.

Bodo, G.↗

NASA NTRS1985

A selection principle for Benard-type convection

In a Benard-type convection problem, the stationary flows of an infinite layer of fluid lying between two rigid horizontal walls and heated uniformly from below are determined. As the temperature difference across the layer increases beyond a certain value, other convective motions appear. These motions are often cellular in character in that their streamlines are confined to certain well-defined cells having, for example, the shape of rolls or hexagons. A selection principle that explains why hexagonal cells seem to be preferred for certain ranges of the parameters is formulated. An operator-theoretical formulation of one generalized Bernard problem is given. The infinite dimensional problem is reduced to one of solving a finite dimensional system of equations, namely, the selection equations. These equations are solved and a linearized stability analysis of the resultant stationary flows is presented.

Knightly, G. H.↗

NASA NTRS1985

Instabilities and transition in the wall boundary layers of low-disturbance supersonic nozzles

Linear stability analysis and experimental nozzles were employed in studying means to control Tollmein-Schlichting (TS) waves and Taylor-Goertler (TG) vortices at transition points along walls in a supersonic wind tunnel. It was hoped that control of the TS waves and TG vortices would cause the wall flows to remain laminar, thereby reducing the turbulence noise in the tunnel. The test nozzles injecting the waves and vertices had Mach numbers from 3 to 5. TG vortices triggered transition, while axisymmetric nozzle flows suppressed it. TS waves were controllable with favorable pressure graients at Mach 3.5. Finally, advances were made in the capability of predicting the locations of the transition points.

Chen, F.-J.↗

NASA NTRS1986

Separated flows near the nose of a body of revolution

The solution of the Navier-Stokes equations for the problem of cross-flow separataion about a deforming cylinder was achieved by iteration. It was shown that the separation starts at the rear stagnation point and the point of primary separation moves upstram along the cylinder surface. A general method of linear stability analysis for nonparallel external flows was constructed, which consists of representing the eigenfunctions with complete orthogonal sets and forms characteristic equations with the Galerkin method. The method was applied to the Kovasznay flow which is an exact solution of the Navier-Stokes equation. The results show that when the critical parameter is exceeded, there are only a few isolated unstable eigen-frequencies. Another exact solution is shown to be absolutely and monotonically stable with respect to infinitesimal disturbances of all frequencies. The flow is also globally, asymptotically, and monotonically stable in the mean with respect o three-dimensional disturbances. This result forms the sound foundation of rigorous stability analysis for nonparallel flows, and provides an invaluable test ground for future studies of nonparallel flows in which the basic states do not posses exact solutions. The application of this method to the study of the formation of spiral vorticies near the nose of a rotating body of revolution is underway. The same method will be applied to the stability analysis of reversed flow over a plate with suction.

Lin, S. P.↗

NASA NTRS1987

Mechanisms by which heat release affects the flow field in a chemically reacting, turbulent mixing layer

The mechanisms by which heat release affects the fluid dynamics in a turbulent reacting mixing layer are studied by direct numerical simulation. In agreement with previous laboratory experiments, the heat release is observed to lower the rate at which the mixing layer grows and to reduce the rate at which chemical products are formed. The baroclinic torque and thermal expansion in the mixing layer are shown to produce changes in the flame vortex structure that act to produce more diffuse vortices than in the constant density case, resulting in lower rotation rates of fluid elements. Previously unexplained anomalies observed in the mean velocity profiles of reacting jets and mixing layers are shown to result from vorticity generation by baroclinic torques. The density reductions also lower the generation rates of turbulent kinetic energy and the turbulent shear stresses, resulting in less turbulent mixing of fluid elements. Calculations of the energy in the various wave numbers show that the heat release has a stabilizing effect on the growth rates of individual modes. A linear stability analysis of a simlified model problem confirms this, showing that low density fluid in the mixing region will result in a shift of the frequency of the unstable modes to lower wave numbers (longer wavelengths). The growth rates of the unstable modes decrease, contributing to the slower growth of the mixing layer.

Riley, J. J.↗

NASA NTRS1986

Small-scale motions observed by aircraft in the tropical lower stratosphere - Evidence for mixing and its relationship to large-scale flows

Measurements of temperature and ozone from instrumented aircraft in the tropical lower stratosphere show the presence of small-scale disturbances generated by underlying cumulus convection and Kelvin-Helmholtz instability. The disturbances associated with underlying convection have peak-to-peak vertical parcel excursions of about 300 m. Flying conditions were smooth, suggesting an ensemble of gravity waves and little or no turbulent mixing. It is nevertheless possible that these waves break at other altitudes, leading to turbulent mixing and net fluxes of vertically stratified tracers. Disturbances attributed to KH instability implied vertical parcel excursions of 300-400 m. The disturbances coincided with rough flying conditions, suggesting turbulent mixing. A linear stability analysis of the atmospheric basic state defined by high-resolution radiosondes shows fastest growing waves with horizontal wavelengths of 1.4-1.8 km, consistent with the aircraft observations. The strong shears responsible for the KH instability are due to large-scale waves propagating into a region of small intrinsic frequency. Radiosonde observations show that the zonal length scale of these waves is about 1000 km.

Pfister, Leonhard↗

NASA NTRS1984

Effect of a forced Couette flow on coupled convective and morphological instabilities during unidirectional solidification

The effect of a forced Couette flow, parallel to a horizontal crystal-melt interface during directional solidification of an alloy of lead containing tin, on the onset of convective and morphological instabilities, is calculated numerically via a linear stability analysis. Such a flow does not affect perturbations with wave vectors perpendicular to the flow. For perturbations with wave vectors parallel to the flow, the onset of morphological instability is somewhat suppressed and thermosolutal convection is greatly suppressed. When instabilities occur, they are oscillatory and correspond to travelling waves. For values of the crystal growth velocity for which mixed morphological and convective modes occur, the presence of a forced flow produces sufficient decoupling to allow otherwise degenerate branches to be identified.

Coriell, S. R.↗

NASA NTRS1986

Boundary-layer stability and airfoil design

Several different natural laminar flow (NLF) airfoils have been analyzed for stability of the laminar boundary layer using linear stability codes. The NLF airfoils analyzed come from three different design conditions: incompressible; compressible with no sweep; and compressible with sweep. Some of the design problems are discussed, concentrating on those problems associated with keeping the boundary layer laminar. Also, there is a discussion on how a linear stability analysis was effectively used to improve the design for some of the airfoils.

Viken, Jeffrey K.↗

NASA NTRS1988

The effect of an electric field on the morphological stability of the crystal-melt interface of a binary alloy

A fully time-dependent linear stability analysis of the morphological stability of a planar interface during directional solidification of a binary alloy at constant velocity in the presence of an electric field, is performed. The electromigration of solute and the differing electrical conductivities of solid and liquid for a model in which the temperature gradient is constant are taken into account. The present results are compared with the constitutional supercooling criterion, and it is shown there may be substantial differences. A modified constitutional supercooling criterion which is valid over a large range of conditions is derived. It is also found under certain conditions that the onset of instability may be time dependent.

Wheeler, A. A.↗