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NASA NTRS1986

Adiabaticity and viscosity in deep mantle convection

A study has been conducted of steady convection with adiabatic and viscous heating for variable viscosity in the Boussinesq limit using the mean-field theory. A strong nonlinear coupling is found between the thermodynamic constants governing adiabatic heating and the rheological parameters. The range of rheological values for which adiabaticity would occur throughout the mantle has been established. Too large an activation volume, greater than 6 cu cm/mol for the cases examined, would produce unreasonably high temperature at the bottom of the mantle (greater than 6000 K) and superadiabatic gradients, especially in the lower mantle. Radiogenic heating plays a profound role in controlling dynamically mantle temperatures. Present values for the averaged mantle heat production would yield objectionably high temperatures in the lower mantle.

Quareni, F.↗

NASA NTRS1986

On cloud street development in three dimensions - Parallel and Rayleigh instabilities

Expected orientation angles and horizontal wavelengths of boundary layer rolls or cloud streets are determined from an analysis of a truncated spectral model of three dimensional shallow moist Boussinesq convection in a shearing environment. The nonlinear secondary circulations are organized into two dimensional forms by the height dependent wind field, and these rolls may develop from the combined effects of thermal stratification and mean wind shear. The associated thermal and parallel instability mechanisms are shown to be special cases of a single one. Only one mode is found when the stratification is unstable or neutral, but a second one is possible when the stratification is weakly stable. The first corresponds to relatively broadly spaced rolls having orientations for which the Fourier component of the roll perpendicular shear is nearly zero, but the second corresponds to relatively narrowly spaced rolls having orientations for which the Fourier coefficients of both the perpendicular and the parallel components of the shear are nearly equal.

Shirer, H. N.↗

NASA NTRS1986

Transitions to chaos in two-dimensional double-diffusive convection

The partial differential equations governing two-dimensional thermosolutal convection in a Boussinesq fluid with free boundary conditions have been solved numerically in a regime where oscillatory solutions can be found. A systematic study of the transition from nonlinear periodic oscillations to temporal chaos has revealed sequences of period-doubling bifurcations. Overstability occurs if the ratio of the solutal to the thermal diffusivity tau is less than 1 and the solutal Rayleigh number Rs is sufficiently large. Solutions have been obtained for two representative values of tau. For tau = 0.316, R(s) = 10,000, symmetrical oscillations undergo a bifurcation to asymmetry, followed by a cascade of period-doubling bifurcations leading to aperiodicity, as the thermal Rayleigh number R(T) is increased. At higher values of R(T), the bifurcation sequence is repeated in reverse, restoring simple periodic solutions. As R(T) is further increased more period-doubling cascades, followed by chaos, can be identified. Within the chaotic regions there are narrow periodic windows, and multiple branches of oscillatory solutions coexist. Eventually the oscillatory branch ends and only steady solutions can be found. The development of chaos has been investigated for tau = 0.1 by varying R(T) for several different values of R(s). When R(s) is sufficiently small there are periodic solutions whose period becomes infinite at the end of the oscillatory branch. As R(s) is increased, chaos appears in the neighborhood of these heteroclinic orbits. At higher values of R(s), chaos is found for a broader range in R(T). A truncated fifth-order model suggest that the appearance of chaos is associated with heteroclinic bifurcations.

Knobloch, E.↗

NASA NTRS1987

The thermal-vortex equations

The Boussinesq approximation is extended so as to explicitly account for the transfer of fluid energy through viscous action into thermal energy. Ideal and dissipative integral invariants are discussed, in addition to the general equations for thermal-fluid motion.

Shebalin, John V.↗

NASA NTRS1986

Boundary-layer computational model for predicting the flow and heat transfer in sudden expansions

Fully developed turbulent and laminar flows through symmetric planar and axisymmetric expansions with heat transfer were modeled using a finite-difference discretization of the boundary-layer equations. By using the boundary-layer equations to model separated flow in place of the Navier-Stokes equations, computational effort was reduced permitting turbulence modelling studies to be economically carried out. For laminar flow, the reattachment length was well predicted for Reynolds numbers as low as 20 and the details of the trapped eddy were well predicted for Reynolds numbers above 200. For turbulent flows, the Boussinesq assumption was used to express the Reynolds stresses in terms of a turbulent viscosity. Near-wall algebraic turbulence models based on Prandtl's-mixing-length model and the maximum Reynolds shear stress were compared.

Lewis, J. P.↗

NASA NTRS1986

High spatial resolution models of time-dependent, layered mantle convection

Time-dependent models of Boussinesq convection are analyzed in order to determine the potential existence of the transient phase of multiple layered convection initially described by Ross and Sacks (1982, 1984). The initial thermal profile for subsolidus mantle convection is examined in terms of an adiabatic or a conductive profile. The effects of varied spatial resolution, aspect ratio, Rayleigh number, initial buoyancy, and supersolidus temperature cutoffs on mantle convection are investigated. It is observed that the data supports the existence of the transient phase of multiple layered convection.

Boss, A. P.↗

NASA NTRS1987

Steady bimodal convection in a cylinder at large Prandtl numbers

Steady bimodal convection of an infinite Prandtl-number Boussinesq fluid in a cylinder is considered. An asymptotic analysis similar to the one used by Buell and Catton (1986) for axisymmetric convection yields a solvability condition that determines the radial wavenumber. The analysis is valid for convection far away from the origin, the lateral boundary, and any pattern dislocations. The azimuthal wave number is treated as a parameter, although in real systems it is dependent on the initial and boundary conditions. Results are presented for Rayleigh numbers between 14,000 and 60,000, and for azimuthal wave numbers between 5 and 7. It is shown that for increasing Rayleigh numbers, the selected radial wave number and the heat transfer tend to become independent of the azimuthal wave number. No quantitative experimental data are available, but one qualitative comparison is good.

Buell, Jeffrey C.↗

NASA NTRS1987

Instability of time-periodic flows

The instabilities of some spatially and/or time-periodic flows are discussed, in particular, flows with curved streamlines which can support Taylor-Gortler vortices are described in detail. The simplest flow where this type of instability can occur is that due to the torsional oscillations of an infinitely long circular cylinder. For more complicated spatially varying time-periodic flows, a similar type of instability can occur and is spatially localized near the most unstable positions. When nonlinear effects are considered it is found that the instability modifies the steady streaming boundary layer induced by the oscillatory motion. It is shown that a rapidly rotating cylinder in a uniform flow is susceptible to a related type of instability; the appropriate stability equations are shown to be identical to those which govern the instability of a Boussinesq fluid of Prandtl number unity heated time periodically from below.

Hall, Philip↗

NASA NTRS1985

Three-dimensional convection in horizontal cylinders - Numerical solutions and comparison with experimental and analytical results

The results of a three-dimensional numerical simulation of Boussinesq free convection in a horizontal differentially heated cylinder are presented. The computation was based on a Samarskii-Andreyev scheme (described by Leong, 1981) and a false-transient advancement in time, with vorticity, velocity, and temperature as dependent variables. Solutions for velocity and temperature distributions were obtained for Rayleigh numbers (based on the radius) Ra = 74-18,700, thus covering the core- and boundary-layer-driven regimes. Numerical solutions are compared with asymptotic analytical solutions and experimental data. The numerical results well represent the complex three-dimensional flows found experimentally.

Smutek, C.↗

NASA NTRS1987

Acoustic radiation-induced static strains in solids

The controversy surrounding the magnitude of the radiation-induced static strain accompanying acoustic wave propagation in solids is resolved by a consideration of the associated Boussinesq radiation stress. Experimental verification of the results is presented for waves propagating along the pure mode directions of crystalline silicon.

Cantrell, John H.↗

NASA NTRS1987

The effects of adiabatic and viscous heatings on plumes

The influences of viscous and adiabatic heatings on the transient development of plumes have been assessed by employing the time-dependent convection equations in the extended-Boussinesq limit. Both temperature-dependent and temperature- and pressure-dependent viscosities are employed. For constant thermal expansion of coefficient, the vigor of plume flows is diminished with larger magnitudes of the thermal expansivity. This effect is caused by the local domination of adiabatic decompression over viscous heating inside the developing plume for both the Cartesian and axisymmetric domains. For deep mantle flows a thermal expansivity decreasing with depth is required for thin plume structures to exist. Values of the thermal expansivity contrast, which can promote plume growth, are in accord with estimates from recent studies in seismic equation of state.

Zhao, Wuling↗

NASA NTRS1988

Geoid and topography for infinite Prandtl number convection in a spherical shell

Geoid anomalies and surface and lower-boundary topographies are calculated for numerically generated thermal convection for an infinite Prandtl number, Boussinesq, axisymmetric spherical fluid shell with constant gravity and viscosity, for heating both entirely from below and entirely from within. Convection solutions are obtained for Rayleigh numbers Ra up to 20 times the critical Ra in heating from below and 27 times critical for heating from within. Geoid parallels surface undulations, and boundary deformation generally increases with increasing cell wavelength. Dimensionless geoid and topography in heating from below are about 5 times greater than in heating from within. Values for heating from within correlate more closely with geophysical data than values from heating from below, suggesting a predominance of internal heating in the mantle. The study emphasizes that dynamically induced topography and geoid are sensitive to the mode of heating in the earth's mantle.

Bercovici, D.↗

NASA NTRS1988

Dynamics of two fluids under periodic acceleration

The evolution of the interface between two fluids confined in a rectangular cavity is investigated numerically to predict its transient behavior. These computations address mixing characteristics of fluids under microgravity conditions, particularly g-jitter conditions resulting from aircraft vibration or crew motion, and have applications in solution crystal growth. The two-dimensional formulation employs the Boussinesq approximation and treats the mixing of two fluids as an initial value problem with a prescribed concentration field. For fluid mixing of practical applications inside a cavity, it is shown that nonlinear convective transport can dominate over both viscous and molecular diffusion. However, viscous diffusion can become important for low Reynolds number or certain cavity sizes. A stacking phenomenon is shown to occur for aspect ratios (Ar) approximately between 5 to 10. For aspect ratios in the neighborhood of 0.1 to 0.2, inner cells evolve at the interface with increasing time. For a square cavity of Ar = 1, chaotic mixing of the fluid appears to occur for a Stokes-Reynolds number greater than about 5.

Duval, Walter M. B.↗

NASA NTRS1989

Computational study of 3-D Benard convection with gravitational modulation

In this numerical study the effects of a modulated gravitational field on three-dimensional Rayleigh-Benard convection with heating from above or from below is investigated. The full, nonlinear, time-dependent, Boussinesq Navier-Stokes equations and the energy equation are solved by a semiimplicit, pseudo-spectral procedure. This study has been motivated by the need to better understand the effects of vibration (G-Jitter) on fluids systems especially in the low gravity environment.

Biringen, S.↗

NASA NTRS1989

A simple dynamical model of a stratocumulus-topped boundary layer

The evolution of a planetary boundary layer topped by stratocumulus clouds is investigated theoretically. The derivation of a simulation model based on two-dimensional shallow moist Boussinesq convection is examined in detail, and the numerical results are presented in extensive tables and maps. The horizontal asymmetry of the circulation within a convective couplet is shown to increase with cloud depth and latent heating, so that the circulation becomes detached from that of the subcloud layer when the clouds fill one-third to one-half of the domain.

Laufersweiler, Mark J.↗

NASA NTRS1989

Blow-up of unsteady two-dimensional Euler and Navier-Stokes solutions having stagnation-point form

The time-dependent form of the classic, two-dimensional stagnation-point solution of the Navier-Stokes equations is considered. If the viscosity is zero, a class of solutions of the initial-value problem can be found in closed form using Lagrangian coordinates. These solutions exhibit singular behavior in finite time, because of the infinite domain and unbounded initial vorticity. Thus, the blow-up found by Stuart in three dimensions using the stagnation-point form, also occurs in two. The singularity vanishes under a discrete, finite-dimensional 'point vortex' approximation, but is recovered as the number of vortices tends to infinity. We find that a small positive viscosity does not arrest the breakdown, but does strongly alter its form. Similar results are summarized for certain Boussinesq stratified flows.

Childress, S.↗

NASA NTRS1989

Penetrative convective flows induced by internal heating and mantle compressibility

Penetrative convective flows induced in a spherical shell by combined effects of internal heating and mantle compressibility are investigated using mathematical and numerical formulations for compressible spherical shell convection. Isothermal stress-free boundary conditions applied at the top and the bottom of the shell are solved using a time-dependent finite difference code in a temperature, vorticity, stream function formulation for Rayleigh numbers ranging from the critical Rc up to 2000 Rc. Results indicate that compressibility, together with internal heating, could be a mechanism capable of generating spontaneously layered convection and local melting in the mantle and that non-Boussinesq effects must be considered in interpretations of geophysical phenomena.

Machetel, Philippe↗

NASA NTRS1990

On the dynamic and thermodynamic structures of marine stratocumulus

Latent heating effects on stratocumulus circulations were studied successfully with a nine-coefficient spectral model of two-dimensional shallow Boussinesq convection (Laufersweiler and Shirer, 1989). Further, more realistic investigations are being performed currently with a larger, 18-coefficient spectral model, in which the effects of cloud top radiational cooling and in-cloud radiational heating are also being represented. Because assuming a rigid lid at the inversion base may have affected previous results significantly, the domain top was raised to include the lower portion of the capping inversion. As in the previous model, a uniform cloud base is assumed and latent heating effects are included implicitly such that the motions in the sub- and above-cloud regions are dry adiabatic and the motions in the cloud region are moist adiabatic. The effects of forcing by radiational heating profiles that are tied to the cloud layer, such as the one used by Nicholls will be investigated, as will profiles measured during the FIRE experiment. One concern of using truncated spectral models is that the phenomena are so poorly represented that they can change dramatically as the number of spectral coefficients is increased. The efficacy of the nine-coefficient model results is checked by examining the steady state solutions of the 18-coefficient model for parameter values used by Laufersweiler and Shirer (1989), which corresponds to the case of a moderately deep cloud and no capping inversion. The horizontally asymmetric circulation patterns that have narrow downdraft areas and broad updraft areas are virtually the same as those found in the smaller spectral model. Also captured in the case of weaker heating is an elevated circulation centered at cloud base. Thus, the results of the smaller model are substantiated. Since one of the goals of studying the new model is to represent a more realistic domain, the second test of the model is to investigate whether the steady solutions are suppressed in the case of an inversion with no cloud.

Laufersweiler, Mark J.↗