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All sourcesNASA NTRSDOE OSTI
NASA NTRS1984

High Rayleigh number heat transfer in a horizontal cylinder with adiabatic wall

The present investigation is concerned with an experimentally guided approach to the estimation of Nusselt numbers (Nu) at high Rayleigh numbers (Ra) for a cylinder with an adiabatic side wall. The Rayleigh number dependence of the Nusselt number for a horizontal cylinder with an adiabatic wall is presented in a graph. The obtained data are compared with results reported by Shih (1981). Shih has extended a three-term expansion for velocity and temperature distributions reported by Bejan and Tien (1978).

Schiroky, G. H.↗
NASA NTRS1980

Stellar convection. III - Convection at large Rayleigh numbers

The results of a numerical, three-dimensional study of convection in a self-gravitating sphere of Boussinesq fluid with a Rayleigh number of 10-billion and a Prandtl number of 1 are presented. The velocity and temperature are computed by using spectral methods in the horizontal and finite-differencing in the radial directions. An eddy viscosity and diffusivity are needed to model the subresolution flow. For Rayleigh numbers much less than 10-billion the flows do not have well-defined inertial ranges, and an eddy viscosity and diffusivity cannot be assigned in this self-consistent manner. By computing the energy spectra as well as the detailed energy budgets as a function of wavenumber, it is shown that for Rs = 10-billion there is an inertial range for the modes corresponding to spherical harmonics with l greater than 6.

Marcus, P. S.↗
NASA NTRS1995

Localization of toroidal motion and shear heating in 3-D high Rayleigh number convection with temperature-dependent viscosity

We have applied spectral-transform methods to study three-dimensional thermal convection with temperature-dependent viscosity. The viscosity varies exponentially with the form exp(-BT), where B controls the viscosity contrast and T is temperature. Solutions for high Rayleigh numbers, up to an effective Ra of 6.25 x 10(exp 6), have been obtained for an aspect-ratio of 5x5x1 and a viscosity contrast of 25. Solutions show the localization of toroidal velocity fields with increasing vigor of convection to a coherent network of shear-zones. Viscous dissipation increases with Rayleigh number and is particularly strong in regions of convergent flows and shear deformation. A time-varying depth-dependent mean-flow is generated because of the correlation between laterally varying viscosity and velocity gradients.

Balachandar, S.↗
NASA NTRS1979

Stellar convection 3: Convection at large Rayleigh numbers

A three dimensional study of convection in a self gravitating sphere of Boussinesq fluid with a Rayleigh number of 10 to the 10th power and a Prandtl of 1 is presented. The velocity and temperature of the fluid are computed at the largest wavelengths using spectral methods. A confirmation that the fluid is anisotropic and that the energy spectra are not smooth functions of wavelength but have a large amount of fine structure is discussed. The parameterization of the transport properties of the unresolvable inertial subrange with eddy viscosities and diffusivities is described. The time dependent fluctuations in the energy spectra and how they cascade from large to small wavelengths is examined.

Marcus, P. S.↗
NASA NTRS1988

High Rayleigh number convection in rectangular enclosures with differentially heated vertical walls and aspect ratios between zero and unity

High Rayleigh number convection in a rectangular cavity with insulated horizontal surfaces and differentially heated vertical walls was analyzed for an arbitrary aspect ratio smaller than or equal to unity. Unlike previous analytical studies, a systematic method of solution based on linearization technique and analytical iteration procedure was developed to obtain approximate closed-form solutions for a wide range of aspect ratios. The predicted velocity and temperature fields are shown to be in excellent agreement with available experimental and numerical data.

Kassemi, Siavash A.↗
NASA NTRS1987

Surface deformation, gravity and the geoid from a three-dimensional convection model at low Rayleigh numbers

The two-dimensional surface deformation, gravity field, and geoid are calculated from the temperature fields of a number of numerical models of constant-viscosity three-dimensional convective flows, heated from within and from below. The surface deformation and the geoid are insensitive to the short-wavelength features of the temperature variation; the gravity field is less smooth, although it still does not clearly indicate the narrowness of the upwelling and downwelling regions at large Rayleigh numbers. The results indicate that the convective upwelling beneath regions showing positive geoid and residual depth anomalies is more localized than the horizontal extent of these features would suggest. The results are relevant to attempts to understand the behavior of the earth mantle.

Craig, Claire Harvey↗
NASA NTRS1993

A spectral method determination of the first critical Rayleigh number for a low-Prandtl number crystal melt in a cylindrical container

The onset of laminar axisymmetric Rayleigh-Benard convection is investigated for a low-Prandtl number liquid metal in a cylindrical container. All surfaces are considered to be solid and no-slip. Two separate cases are examined for the thermal boundary conditions at the side wall, one with conducting and the other with insulated surface. The governing Boussinesq system is first perturbed and then simplified by introducing a Stokes stream function. Subsequently, a Chebyshev Galerkin spectral model is employed to reduce the simplified system to a system of first-order nonlinear ordinary differential equations. A local stability analysis determines the two values of the first critical Rayleigh number, Ra(sub cl), for the insulated and conducting side walls. As expected, the conducting Ra(sub cl) value of 2882.5 obtained from the present approach exceeded the corresponding insulated Ra(sub cl) value of 2331.6. For the insulated case, an earlier study using a different numerical approach suggests that Ra(sub cl) = 2261.9, while an experimental study measured Ra(sub cl) = 2700.

Dietz, C. M., Jr.↗
NASA NTRS1988

A direct interaction approximation treatment of high Rayleigh number convective turbulence and comparison with experiment

A method is proposed for treating steady-state, thermally driven convection using the full direct interaction approximation to treat the nonlinear energy transfer and a prescribed instability function to model the eenrgy input. The instability function used is the growth rate of the mechanism generating the turbulence. This methodology can be easily generalized and applied to other driving mechanisms. The 1/3 power law form of the N vs. R relation for water is duplicated here and the coefficient is computed, using a two-point closure, to be less than about 0.08.

Hartke, Gregory J.↗
NASA NTRS1992

Convection treatment for high Rayleigh number, laminar, natural convection calculation

The problem of natural convection in a square cavity of a Boussinesq incompressible fluid is computed using a consistent central-difference approximation, a consistent second-order upwind scheme, and the hybrid scheme for the energy equation. It was found that, for this problem, the second-order upwind scheme yields satisfactory solutions and that an apparently low-order combination of schemes, with the second-order upwind scheme for the momentum equations and the hybrid scheme for the energy equation, can produce satisfactory predictions for this problem.

Shyy, Wei↗
NASA NTRS1992

Chemical differentiation of a convecting planetary interior: Consequences for a one-plate planet such as Venus

Chemically depleted mantle forming a buoyant, refractory layer at the top of the mantle can have important implications for the evolution of the interior and surface. On Venus, the large apparent depths of compensation for surface topographic features might be explained if surface topography were supported by variations in the thickness of a 100-200 km thick chemically buoyant mantle layer or by partial melting in the mantle at the base of such a layer. Long volcanic flows seen on the surface may be explained by deep melting that generates low-viscosity MgO-rich magmas. The presence of a shallow refractory mantle layer may also explain the lack of volcanism associated with rifting. As the depleted layer thickens and cools, it becomes denser than the convecting interior and the portion of it that is hot enough to flow can mix with the convecting mantle. Time dependence of the thickness of a depleted layer may create episodic resurfacing events as needed to explain the observed distribution of impact craters on the venusian surface. We consider a planetary structure consisting of a crust, depleted mantle layer, and a thermally and chemically well-mixed convecting mantle. The thermal evolution of the convecting spherical planetary interior is calculated using energy conservation: the time rate of change of thermal energy in the interior is equated to the difference in the rate of radioactive heat production and the rate of heat transfer across the thermal boundary layer. Heat transfer across the thermal boundary layer is parameterized using a standard Nusselt number-Rayleigh number relationship. The radioactive heat production decreases with time corresponding to decay times for the U, Th, and K. The planetary interior cools by the advection of hot mantle at temperature T interior into the thermal boundary layer where it cools conductively. The crust and depleted mantle layers do not convect in our model so that a linear conductive equilibrium temperature distribution is assumed. The rate of melt production is calculated as the product of the volume flux of mantle into the thermal boundary layer and the degree of melting that this mantle undergoes. The volume flux of mantle into the thermal boundary layer is simply the heat flux divided by amount of heat lost in cooling mantle to the average temperature in the thermal boundary layer. The degree of melting is calculated as the temperature difference above the solidus, divided by the latent heat of melting. A maximum degree of melting is prescribed corresponding to the maximum amount of basaltic melt that the mantle can initially generate. As the crust thickens, the pressure at the base of the crust becomes high enough and the temperature remains low enough for basalt to transform to dense eclogite.

Parmentier, E. M.↗
NASA NTRS1979

Subsolidus convective cooling histories of terrestrial planets

The subsolidus convective cooling histories of terrestrial planets evolving from hot initial states are investigated quantitatively. A simple analytic model simulating average heat flux from a vigorously convecting mantle and incorporating a mantle viscosity proportional to mantle temperature and a lithosphere which thickens as the planet cools is employed. Heat flux from the convecting mantle is calculated on the basis of a power law relation between Nusselt number and Rayleigh number. The temperature distribution in the lithosphere is assumed to be linear throughout the cooling history of the planet. Cooling histories have been determined for the earth, Mars, Mercury and the moon and the mantle temperature decreases, mantle viscosity increases and decreases of heat flux to the surface and to the base of the lithosphere and of Nusselt and Rayleigh numbers are illustrated for each planet. It is found that primordial heat can contribute substantially to the present surface heat flux of a planet.

Schubert, G.↗
NASA NTRS1980

Numerical analysis of natural convection in two-dimensional square and circular containers in low gravity

A numerical study of natural convection in circular cylinder and square enclosures shows that the analytic low Rayleigh number theory of previous investigators is valid for Rayleigh numbers up to 1000. For a Rayleigh number of 5000, steady state values of maximum fluid velocity differ by 20 percent. This deviation between analytic theory and numerical results increases for higher Rayleigh numbers. In addition, the low Rayleigh number theory is shown to be valid for higher Rayleigh numbers for a portion of the transient phase before significant deviation becomes apparent. It is also shown that square shaped experimental configurations may be analytically approximated with good accuracy by circular cylinders of equal cross sectional area for the prediction of convection velocities and flow patterns at low Rayleigh number.

Robertson, S. J.↗
NASA NTRS1986

An analysis of ullage heat transfer in the orbital refueling system

The Orbital Refueling System was an experiment flown on Shuttle Mission STS 41-G in October, 1984. Liquid hydrazine fuel was transferred back and forth from one spherical bladder tank to another using pressurized nitrogen as the driving force. Compressive heating of the ullage gas in the receiving tank could lead to a hazardous situation if any hydrazine leaked through to the ullage side of the bladder and was heated above about 175 F, where it can undergo spontaneous exothermic decomposition. Early analysis of the flight data indicated that the ullage compression process was much closer to an isothermal than an adiabatic one. In this study, a thorough review of the pertinent literature was used to make an a priori best-estimate for the ullage gas heat transfer coefficient (defining the Nusselt Number as a function of Reynolds and Rayleigh Numbers). Experimental data from the flight were analyzed in detail. It is evident that there is considerably more heat transfer than can be accounted for by conduction alone, but the observed increases do not correlate well with Reynolds Number, Rayleigh Number or vehicle acceleration. There are large gaps in the present understanding of convective heat transfer in closed containers with internal heat generation, especially in the presence of vibrations or other random disturbances. A program of experiments to fill in these gaps is suggested, covering both ground and orbital environments.

Kauffman, D.↗
NASA NTRS1992

Analysis of natural convection in a low gravity environment

Natural convection inside a spherical container was studied experimentally with two apparatuses at low buoyancy levels. The data generated by these experiments, plotted nondimensionally as the Nusselt versus Rayleigh numbers, give correlations for Rayleigh numbers between 1000 and 10 exp 8, a range previously untested. These results show that natural convection has significant effects at a Rayleigh number of 1000 and higher, although the behavior of the Nusselt number as the conduction limit is approached is still unknown for a spherical geometry.

Mattor, Ethan E.↗
NASA NTRS1969

Numerical results in the statistical theory of thermal turbulence

Several statistical theories of turbulence are applied to thermal convection problem, and are compared in a series of numerical calculations. Results are presented for both the case of very large and very small Prandtl numbers, and for moderate Rayleigh numbers. The theories compared are the direct-interaction approximation, the self-consistent field approximation, the quasi-linear approximation, the quasi-normal approximation, and a 'Markovianized' quasi-normal approximation. Results for these theories are then compared with some simple exact solutions to the statistical problem. The results indicate that the direct-interaction and the self-consistent field method give satisfactory results at all Rayleigh numbers investigated, whereas the procedures based on the quasi-normal approximation give sensible results only at small Rayleigh numbers.

Herring, J. R.↗
NASA NTRS2004

Convective Instability in Ice I with Non-Newtonian Rheology: Application to the Galilean Satellites

At the temperatures and stresses associated with the onset of convection in an ice I shell of the Galilean satellites, ice behaves as a non-Newtonian fluid with a viscosity that depends on both temperature and strain rate. The convective stability of a non-Newtonian ice shell can be judged by comparing the Rayleigh number of the shell to a critical value. Previous studies suggest that the critical Rayleigh number for a non-Newtonian fluid depends on the initial conditions in the fluid layer, in addition to the thermal, rheological, and physical properties of the fluid. We seek to extend the existing definition of the critical Rayleigh number for a non-Newtonian, basally heated fluid by quantifying the conditions required to initiate convection in an ice I layer initially in conductive equilibrium. We find that the critical Rayleigh number for the onset of convection in ice I varies as a power (-0.6 to -0.5) of the amplitude of the initial temperature perturbation issued to the layer, when the amplitude of perturbation is less than the rheological temperature scale. For larger-amplitude perturbations, the critical Rayleigh number achieves a constant value. We characterize the critical Rayleigh number as a function of surface temperature of the satellite, melting temperature of ice, and rheological parameters so that our results may be extrapolated for use with other rheologies and for a generic large icy satellite. The values of critical Rayleigh number imply that triggering convection from a conductive equilibrium in a pure ice shell less than 100 km thick in Europa, Ganymede, or Callisto requires a large, localized temperature perturbation of a few kelvins to tens of kelvins to soften the ice and therefore may require tidal dissipation in the ice shell.

Barr, A. C.↗