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NASA NTRS1990

Numerical simulation of 3-D Benard convection with gravitational modulation

In this numerical study, randomly and sinusoidally modulated gravitational fields imposed on three-dimensional Rayleigh-Benard convection are investigated in an effort to understand the effects of vibration (G-Jitter) on fluid systems. The time-dependent, Navier-Stokes equations and the energy equation with Boussinesq approximations are solved by a semi-implicit, pseudospectral procedure. An analysis of energy balances indicates that with increasing modulation amplitude, transition from synchronous to relaxation oscillation goes through the subharmonic response. Random modulations are found to be less stabilizing than sinusoidal and are shown to impose three-dimensionality on the flow for some parameter ranges both at terrestrial and zero base gravity conditions.

Biringen, S.↗

NASA NTRS1981

Basic lubrication equations

Lubricants, usually Newtonian fluids, are assumed to experience laminar flow. The basic equations used to describe the flow are the Navier-Stokes equation of motion. The study of hydrodynamic lubrication is, from a mathematical standpoint, the application of a reduced form of these Navier-Stokes equations in association with the continuity equation. The Reynolds equation can also be derived from first principles, provided of course that the same basic assumptions are adopted in each case. Both methods are used in deriving the Reynolds equation, and the assumptions inherent in reducing the Navier-Stokes equations are specified. Because the Reynolds equation contains viscosity and density terms and these properties depend on temperature and pressure, it is often necessary to couple the Reynolds with energy equation. The lubricant properties and the energy equation are presented. Film thickness, a parameter of the Reynolds equation, is a function of the elastic behavior of the bearing surface. The governing elasticity equation is therefore presented.

Hamrock, B. J.↗

NASA NTRS1983

Thermal elastohydrodynamic lubrication of line contacts

A numerical solution to the problem of thermal elastohydrodynamic lubrication of line contacts was obtained by using a finite difference formulation. The solution procedure consists of simultaneous solution of the thermal Reynolds equation, the elasticity equation, and the energy equation subject to appropriate boundary conditions. Pressure distribution, film shape, and temperature distribution were obtained for fully flooded conjunctions, a paraffinic lubricant, and various dimensionless speed parameters while the dimensionless load and materials parameters were held constant. Reduction in the minimum film thickness due to thermal effects (as a ratio of thermal to isothermal minimum film thickness) is given by a simple formula as a function of the thermal loading parameter Q: H(min)/H(min,I) = 10/10+ Q(0.4). Plots of pressure distribution, film shape, temperature distribution, and flow are shown for some representative cases.

Ghosh, M. K.↗

NASA NTRS1985

Thermal elastohydrodynamic lubrication of line contacts

A numerical solution to the problem of thermal elastohydrodynamic lubrication of line contacts was obtained by using a finite difference formulation. The solution procedure consists of simultaneous solution of the thermal Reynolds equation, the elasticity equation, and the energy equation subject to appropriate boundary conditions. Pressure distribution, film shape, and temperature distribution were obtained for fully flooded conjunctions, a paraffinic lubricant, and various dimensionless speed parameters while the dimensionless load and materials parameters were held constant. Reduction in the minimum film thickness due to thermal effects (as a ratio of thermal to isothermal minimum film thickness) is given by a simple formula as a function of the thermal loading parameter Q: H(min)/H(min, I) = 10/10 + Q(0.4). Plots of pressure distribution, film shape, temperature distribution, and flow are shown for some representative cases.

Ghosh, M. K.↗

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 NTRS1990

Combustion instability analysis

A theory and computer program for combustion instability analysis are presented. The basic theoretical foundation resides in the concept of entropy-controlled energy growth or decay. Third order perturbation expansion is performed on the entropy-controlled acoustic energy equation to obtain the first order integrodifferential equation for the energy growth factor in terms of the linear, second, and third order energy growth parameters. These parameters are calculated from Navier-Stokes solutions with time averages performed on as many Navier-Stokes time steps as required to cover at least one peak wave period. Applications are made for a 1-D Navier-Stokes solution for the Space Shuttle Main Engine (SSME) thrust chamber with cross section area variations taken into account. It is shown that instability occurs when the mean pressure is set at 2000 psi with 30 percent disturbances. Instability also arises when the mean pressure is set at 2935 psi with 20 percent disturbances. The system with mean pressures and disturbances more adverse that these cases were shown to be unstable.

Chung, T. J.↗

NASA NTRS1993

Time-dependent thermocapillary convection in a Cartesian cavity - Numerical results for a moderate Prandtl number fluid

The present numerical simulation explores a thermal-convective mechanism for oscillatory thermocapillary convection in a shallow Cartesian cavity for a Prandtl number 6.78 fluid. The computer program developed for this simulation integrates the two-dimensional, time-dependent Navier-Stokes equations and the energy equation by a time-accurate method on a stretched, staggered mesh. Flat free surfaces are assumed. The instability is shown to depend upon temporal coupling between large scale thermal structures within the flow field and the temperature sensitive free surface. A primary result of this study is the development of a stability diagram presenting the critical Marangoni number separating steady from the time-dependent flow states as a function of aspect ratio for the range of values between 2.3 and 3.8. Within this range, a minimum critical aspect ratio near 2.3 and a minimum critical Marangoni number near 20,000 are predicted below which steady convection is found.

Peltier, L. J.↗

NASA NTRS1996

Three-Dimensional Simulations of Marangoni-Benard Convection in Small Containers by the Least-Squares Finite Element Method

This paper reports a numerical study of the Marangoni-Benard (MB) convection in a planar fluid layer. The least-squares finite element method (LSFEM) is employed to solve the three-dimensional Stokes equations and the energy equation. First, the governing equations are reduced to be first-order by introducing variables such as vorticity and heat fluxes. The resultant first-order system is then cast into a div-curl-grad formulation, and its ellipticity and permissible boundary conditions are readily proved. This numerical approach provides an equal-order discretization for velocity, pressure, vorticity, temperature, and heat conduction fluxes, and therefore can provide high fidelity solutions for the complex flow physics of the MB convection. Numerical results reported include the critical Marangoni numbers (M(sub ac)) for the onset of the convection in containers with various aspect ratios, and the planforms of supercritical MB flows. The numerical solutions compared favorably with the experimental results reported by Koschmieder et al..

Yu, Sheng-Tao↗

NASA NTRS1972

Analysis of heat transfer in a porous cooled wall with variable pressure and temperature along the coolant exit boundary

Fluid from a reservior at constant pressure and temperature is forced through a porous wall of uniform thickness. The boundary through which the fluid exits has specified variations in pressure and temperature along it in one direction so that the flow and heat transfer are two-dimensional. The local fluid and matrix temperatures are assumed to be equal and therefore a single energy equation governs the temperature distribution within the wall. The solution is obtained by transforming this energy equation into potential plane coordinates, which results in a separable equation. A technique yielding an integral equation is used to adapt the general solution so that it satisfies the variable-pressure boundary condition. Analytical expressions are given for the normal exit velocity and heat flux along the exit boundary. Illustrative examples are carried out which indicate to what extent the solution is locally one-dimensional.

Siegel, R.↗

NASA NTRS1982

'Scaling' analysis of the ice accretion process on aircraft surfaces

A comprehensive set of scaling parameters is developed for the ice accretion process by analyzing the energy equations of the dynamic freezing zone and the already frozen ice layer, the continuity equation associated with supercooled liquid droplets entering into and impacting within the dynamic freezing zone, and energy equation of the ice layer. No initial arbitrary judgments are made regarding the relative magnitudes of each of the terms. The method of intrinsic reference variables in employed in order to develop the appropriate scaling parameters and their relative significance in rime icing conditions in an orderly process, rather than utilizing empiricism. The significance of these parameters is examined and the parameters are combined with scaling criteria related to droplet trajectory similitude.

Keshock, E. G.↗

NASA NTRS1995

The Development of a Nonequilibrium Radiative Heat Transfer Computational Model for High Altitude Entry Vehicle Flowfield Methods

This final report will attempt to concisely summarize the activities and accomplishments associated with NASA Grant and to include pertinent documents in an appendix. The project initially had one primary and several secondary objectives. The original primary objective was to couple into the NASA Johnson Space Center (JSC) nonequilibrium chemistry Euler equation entry vehicle flowfield code, INEQ3D, the Texas A&M University (TAMU) local thermodynamic nonequilibrium (LTNE) radiation model. This model had previously been developed and verified under NASA Langley and NASA Johnson sponsorship as part of a viscous shock layer entry vehicle flowfield code. The secondary objectives were: (1) to investigate the necessity of including the radiative flux term in the vibrational-electron-electronic (VEE) energy equation as well as in the global energy equation, (2) to determine the importance of including the small net change in electronic energy between products and reactants which occurs during a chemical reaction, and (3) to study the effect of atom-atom impact ionization reactions on entry vehicle nonequilibrium flowfield chemistry and radiation. For each, of these objectives, it was assumed that the code would be applicable to lunar return entry conditions, i.e. altitude above 75 km, velocity greater, than 11 km/sec, where nonequilibrium chemistry and radiative heating phenomena would be significant. In addition, it was tacitly assumed that as part of the project the code would be applied to a variety of flight conditions and geometries.

Carlson, Leland A.↗

NASA NTRS1990

Temperatures of individual ion species and heating due to charge exchange in the ionosphere of Venus

The coupled electron and multispecies ion energy equations were solved for daytime conditions in the Venus ionosphere. The heating rates due to charge exchange between hot oxygen atoms and thermal oxygen ions were calculated and incorporated into the energy equations. The combination of the traditional EUV heating and this hot oxygen energy source leads to calculated electron and individual ion temperatures significantly lower than the measured values during solar cycle maximum conditions. Calculations were also carried out for solar cycle minimum conditions, which led to considerably lower temperatures; no data are available which would allow direct comparisons of these results with measurements. In order to obtain calculated temperature values consistent with the observed ones, for solar cycle maximum conditions, topside heat inflows into the ion and electron gases have to be introduced or the thermal conductivity must be reduced by considering the effect of steady and fluctuating magnetic fields, as was done in previous studies. The addition of hot oxygen heating leads to minor increases in the calculated ion temperatures except for the case of reduced thermal conductivities. Separate temperatures were calculated for each ion species for a number of different conditions and in general the differences were found to be relatively small.

Kim, Jhoon↗

NASA NTRS1986

Explicit finite-volume time-marching calculations of total temperature distributions in turbulent flow

A method was developed which calculates two-dimensional, transonic, viscous flow in ducts. The finite volume, time-marching formulation is used to obtain steady flow solutions of the Reynolds-averaged form of the Navier-Stokes equations. The entire calculation is performed in the physical domain. This paper investigates the introduction of a new formulation of the energy equation which gives improved transient behavior as the calculation converges. The effect of variable Prandtl number on the temperature distribution through the boundary layer is also investigated. A turbulent boundary layer in an adverse pressure gradient (M = 0.55) is used to demonstrate the improved transient temperature distribution obtained when the new formulation of the energy equation is used. A flat plate turbulent boundary layer with a supersonic free-stream Mach number of 2.8 is used to investigate the effect of Prandtl number on the distribution of properties through the boundary layer. The computed total temperature distribution and recovery factor agree well with the measurements when a variable Prandtl number is used through the boundary layer.

Nicholson, Stephen↗

NASA NTRS1987

Explicit finite-volume time-marching calculations of total temperature distributions in turbulent flow

Nicholson, Stephen↗

NASA NTRS1982

Isothermal separation processes

The isothermal processes of membrane separation, supercritical extraction and chromatography were examined using availability analysis. The general approach was to derive equations that identified where energy is consumed in these processes and how they compare with conventional separation methods. These separation methods are characterized by pure work inputs, chiefly in the form of a pressure drop which supplies the required energy. Equations were derived for the energy requirement in terms of regular solution theory. This approach is believed to accurately predict the work of separation in terms of the heat of solution and the entropy of mixing. It can form the basis of a convenient calculation method for optimizing membrane and solvent properties for particular applications. Calculations were made on the energy requirements for a membrane process separating air into its components.

England, C.↗

NASA NTRS1989

Compact finite volume methods for the diffusion equation

An approach to treating initial-boundary value problems by finite volume methods is described, in which the parallel between differential and difference arguments is closely maintained. By using intrinsic geometrical properties of the volume elements, it is possible to describe discrete versions of the div, curl, and grad operators which lead, using summation-by-parts techniques, to familiar energy equations as well as the div curl = 0 and curl grad = 0 identities. For the diffusion equation, these operators describe compact schemes whose convergence is assured by the energy equations and which yield both the potential and the flux vector with second order accuracy. A simplified potential form is especially useful for obtaining numerical results by multigrid and alternating direction implicit (ADI) methods. The treatment of general curvilinear coordinates is shown to result from a specialization of these general results.

Rose, Milton E.↗

NASA NTRS1989

Compact finite volume methods for the diffusion equation

The paper describes an approach to treating initial-boundary-value problems by finite volume methods in which the parallel between differential and difference arguments is closely maintained. By using intrinsic geometrical properties of the volume elements, it is possible to describe discrete versions of the div, curl, and grad operators which lead, using summation-by-parts techniques, to familiar energy equations as well as the div curl = 0 and curl grad = 0 identities. For the diffusion equation, these operators describe compact schemes whose convergence is assured by the energy equations and which yield both the potential and the flux vector with second-order accuracy. A simplified potential form is especially useful for obtaining numerical results by multigrid and ADI methods.

Rose, Milton E.↗

NASA NTRS1967

Heat generation in a viscoelastic solid.

Temperature distributions and heat generation in viscoelastic isotropic solid resulting from mechanical deformations determined by energy equation

ENERGY EQUATION↗