A Rodrigues' formula for the Laguerre polynomials.
Rodrigues type formula for generalized Laguerre polynomials based upon semigroup property for solutions of heat equation and generating function related to source solution
HEAT EQUATION↗
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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.
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Some practical aspects of surface temperature measurement by optical and ratio pyrometers
Radiation heat equations and surface temperature measurements of opaque materials by optical and ratio pyrometers
HEAT EQUATION↗
Migration of the separation point on a deforming cylinder
An iterative scheme of solving the Navier-Stokes equations for unsteady two-dimensional flow about a deforming and translating cylinder is given. In the kth iteration, the convected vorticity appears as the source term in an equation of transient diffusion of vorticity. A novel integral transform is used to reduce this transient vorticity equation to a k-dimensional heat equation. The bounded solution of this equation is obtained with a general method of superposition for problems involving a moving boundary. An equation describing the migration of the separation point on a deforming cylinder in unsteady cross flows is derived from the analytically obtained velocity field. The radial expansion of the cylinder surface is shown to hasten the separation time and to increase the separation angle. The results imply that in the steady flow past a body that is moving forward and sinking at constant rates, the locus of points at which the azimuthal component of skin friction changes sign originates on the leeward ray at a point downstream of the front tip.
Lin, S. P.↗
Combustion response modeling for composite solid propellants
A computerized mathematical model of the combustion response function of composite solid propellants was developed with particular attention to the contributions of the solid phase heterogeneity. The one-dimensional model treats the solid phase as alternating layers of ammonium perchlorate and binder, with an exothermic melt layer at the surface. Solution of the Fourier heat equation in the solid provides temperature and heat flux distributions with space and time. The problem is solved by conserving the heat flux at the surface from that produced by a suitable model of the gas phase. An approximation of the BDP flame model is utilized to represent the gas phase. By the use of several reasonable assumptions, it is found that a significant portion of the problem can be solved in closed form. A method is presented by which the model can be applied to tetramodal particle size distributions. A computerized steady-state version of the model was completed, which served to validate the various approximations and lay a foundation for the combustion response modeling. The combustion response modeling was completed in a form which does not require an iterative solution, and some preliminary results were acquired.
Source record↗
Development of a laminar boundary layer behind a suction point
A theoretical investigation is made of the development of a laminar boundary layer behind a suction slot that is assumed to cut off part of the boundary layer without exerting any sink effect. The development, which is approximate, is based on the heat conduction equation. The heat conduction equation enters the analysis through a linearization of the Prandtl-Mises form of the boundary-layer equation.
EQUATIONS - BOUNDARY LAYER↗
Bistable direction switching in an off-axis pumped continuous wave ruby laser
A report is presented of the observation of hysteretic bistable direction switching in a single-mode CW ruby laser system. This effect is only observed when the pump beam which is focused into the ruby rod is misaligned with respect to the rod end faces. At low pump powers, the ruby lases in a mode nearly collinear with the pump axis. At a higher pump power the ruby switches to a mode that is collinear with the rod end faces and preserves the original polarization. The effect is large enough to switch the beam by an angle equal to twice the diffraction angle. The observations show that under steady-state pumping, a CW ruby laser can exhibit bistable operation in its output direction and power. A calculation using the heat equation with two concentric cylinders with one as a heat source (pump laser) and the outer wall of the other held at 77 K, gives an increase in core temperature of about 0.01 K. Therefore, the increase in temperature is not large enough to change the index of refraction to account for such large macroscopic effects.
Afzal, R. Sohrab↗
The Evolution of Finite Amplitude Wavetrains in Plane Channel Flow
We consider a viscous incompressible fluid flow driven between two parallel plates by a constant pressure gradient. The flow is at a finite Reynolds number, with an 0(l) disturbance in the form of a traveling wave. A phase equation approach is used to discuss the evolution of slowly varying fully nonlinear two dimensional wavetrains. We consider uniform wavetrains in detail, showing that the development of a wavenumber perturbation is governed by Burgers equation in most cases. The wavenumber perturbation theory, constructed using the phase equation approach for a uniform wavetrain, is shown to be distinct from an amplitude perturbation expansion about the periodic flow. In fact we show that the amplitude equation contains only linear terms and is simply the heat equation. We review, briefly, the well known dynamics of Burgers equation, which imply that both shock structures and finite time singularities of the wavenumber perturbation can occur with respect to the slow scales. Numerical computations have been performed to identify areas of the (wavenumber, Reynolds number, energy) neutral surface for which each of these possibilities can occur. We note that the evolution equations will breakdown under certain circumstances, in particular for a weakly nonlinear secondary flow. Finally we extend the theory to three dimensions and discuss the limit of a weak spanwise dependence for uniform wavetrains, showing that two functions are required to describe the evolution. These unknowns are a phase and a pressure function which satisfy a pair of linearly coupled partial differential equations. The results obtained from applying the same analysis to the fully three dimensional problem are included as an appendix.
Hewitt, R. E.↗
Progress on Quantitative Infrared Thermography at the NASA Langley Aerothermodynamic Laboratory
A quantitative infrared thermography technique is being developed for accurate and high-resolution measurements of surface temperature and heat flux in the hypersonic wind tunnels at NASA Langley Research Center. A hemisphere test campaign was carried out in the Langley Aerothermodynamic Laboratory 20-Inch Mach 6 Air Tunnel to assess the current performance of the technique and to identify remaining challenges. Measurements were obtained for three different model materials over a range of freestream Reynolds numbers. The raw infrared images were converted to temperature via a radiometric calibration of the camera sensor. Corrections were made for losses due to test article emissivity and window transmissivity. A preliminary uncertainty analysis yielded an estimated accuracy within 4 K at moderate viewing angles, similar to that of a standard thermocouple. Surface heat flux was obtained from the measured surface temperatures via a finite-difference solution of the one-dimensional heat equation that modeled thermal variation of the material properties. Comparisons of the measured heating to both computational and theoretical predictions demonstrated agreement within the expected uncertainty of the technique. The heat transfer measurement near the stagnation point was found to be highly sensitive to the test article emissivity, which establishes the need for improved characterization of the wind tunnel model materials for high temperature applications.
infrared thermography↗
Analytic Modeling of the Hydrodynamic, Thermal, and Structural Behavior of Foil Thrust Bearings
A simulation and modeling effort is conducted on gas foil thrust bearings. A foil bearing is a self acting hydrodynamic device capable of separating stationary and rotating components of rotating machinery by a film of air or other gaseous lubricant. Although simple in appearance these bearings have proven to be complicated devices in analysis. They are sensitive to fluid structure interaction, use a compressible gas as a lubricant, may not be in the fully continuum range of fluid mechanics, and operate in the range where viscous heat generation is significant. These factors provide a challenge to the simulation and modeling task. The Reynolds equation with the addition of Knudsen number effects due to thin film thicknesses is used to simulate the hydrodynamics. The energy equation is manipulated to simulate the temperature field of the lubricant film and combined with the ideal gas relationship, provides density field input to the Reynolds equation. Heat transfer between the lubricant and the surroundings is also modeled. The structural deformations of the bearing are modeled with a single partial differential equation. The equation models the top foil as a thin, bending dominated membrane whose deflections are governed by the biharmonic equation. A linear superposition of hydrodynamic load and compliant foundation reaction is included. The stiffness of the compliant foundation is modeled as a distributed stiffness that supports the top foil. The system of governing equations is solved numerically by a computer program written in the Mathematica computing environment. Representative calculations and comparisons with experimental results are included for a generation I gas foil thrust bearing.
Bruckner, Robert J.↗
Finite-element solution for a combined radiative-conductive analysis with mixed diffuse-specular surface characteristics
The finite-element method has been applied to solve a combined radiative-conductive heat transfer problem for a large space telescope similar to those used in orbiting satellites. The derivation of the underlying matrices and associated solution algorithm for a 2-D triangular element is presented in detail. The resulting expressions for this triangular element typify such an analysis, which yields constitutive matrices when the heat equation is cast in the matrix form. The relevant matrices include those pertaining to thermal conductance, internal heat generation, radiative exchanges, and all possible external thermal loadings. Emphasis is placed on the treatment of non-linear radiative interchange between surfaces in an enclosure having mixed diffuse-specular surface characteristics. Essential differences in governing equations describing these distinctive surface characteristics are identified. Concluding remarks are drawn from an example simulating a Cassegrainian space telescope.
Lee, H.-P.↗
Similarity solution of the boundary-layer equations for laser heated flows
The laminar boundary-layer equations with local similarity approximation are solved in order to estimate the heat transfer rate to the wall in a laser-heated rocket thruster using pure hydrogen, where the average temperature of the hot plasma core is about 14,000 K and the core Reynolds number based on nozzle throat diameter is about 2000. Under these conditions, the density-viscosity product at the wall can be 10 times the freestream value, and the hydrogen is completely dissociated. Hence the boundary layer equations with variable transport properties are solved by a quasi-linearization technique, and the equilibrium properties of hydrogen are used in the calculations. Velocity profiles are obtained, and the wall shear and wall stagnation enthalpy gradient are plotted against the pressure gradient parameter.
Wu, P. K.↗
An extension of A-stability to alternating direction implicit methods
An alternating direction implicit (ADI) scheme was constructed by the method of approximate factorization. An A-stable linear multistep method (LMM) was used to integrate a model two-dimensional hyperbolic-parabolic partial differential equation. Sufficient conditions for the A-stability of the LMM were determined by applying the theory of positive real functions to reduce the stability analysis of the partial differential equations to a simple algebraic test. A linear test equation for partial differential equations is defined and then used to analyze the stability of approximate factorization schemes. An ADI method for the three-dimensional heat equation is also presented.
Warming, R. F.↗
Convergence of Galerkin approximations for operator Riccati equations: A nonlinear evolution equation approach
An approximation and convergence theory was developed for Galerkin approximations to infinite dimensional operator Riccati differential equations formulated in the space of Hilbert-Schmidt operators on a separable Hilbert space. The Riccati equation was treated as a nonlinear evolution equation with dynamics described by a nonlinear monotone perturbation of a strongly coercive linear operator. A generic approximation result was proven for quasi-autonomous nonlinear evolution system involving accretive operators which was then used to demonstrate the Hilbert-Schmidt norm convergence of Galerkin approximations to the solution of the Riccati equation. The application of the results was illustrated in the context of a linear quadratic optimal control problem for a one dimensional heat equation.
Rosen, I. G.↗
Computational methods for optimal linear-quadratic compensators for infinite dimensional discrete-time systems
An abstract approximation theory and computational methods are developed for the determination of optimal linear-quadratic feedback control, observers and compensators for infinite dimensional discrete-time systems. Particular attention is paid to systems whose open-loop dynamics are described by semigroups of operators on Hilbert spaces. The approach taken is based on the finite dimensional approximation of the infinite dimensional operator Riccati equations which characterize the optimal feedback control and observer gains. Theoretical convergence results are presented and discussed. Numerical results for an example involving a heat equation with boundary control are presented and used to demonstrate the feasibility of the method.
Gibson, J. S.↗
Centrifugally driven winds from protostellar disks. I - Wind model and thermal structure
The thermal structure of a wind that is centrifugally driven from the surface of a protostellar disk is studied. A generalized version of the Blandford and Payne self-similar wind model is introduced, and the temperature and ionization distributions in the outflow are investigated. For the evolution of atomic winds, the heat equation and the rate equations that describe the ionization and excitation state of hydrogen are solved self-consistently. Ambipolar diffusion is found to be a robust mechanism for heating the gas. In the more powerful outflows, molecular hydrogen is collisionally dissociated close to the disk surface, and hydrogen is mainly atomic within a few astronomical units from the central source. It is also demonstrated that these outflows have enough momentum to lift dust grains from the disk surface.
Safier, Pedro N.↗
Pseudo-time algorithms for the Navier-Stokes equations
A pseudo-time method is introduced to integrate the compressible Navier-Stokes equations to a steady state. This method is a generalization of a method used by Crocco and also by Allen and Cheng. We show that for a simple heat equation that this is just a renormalization of the time. For a convection-diffusion equation the renormalization is dependent only on the viscous terms. We implement the method for the Navier-Stokes equations using a Runge-Kutta type algorithm. This permits the time step to be chosen based on the inviscid model only. We also discuss the use of residual smoothing when viscous terms are present.
Swanson, R. C.↗
Development of an Equation to Characterize Heat Flux Through IMLI and A Method to Implement in Thermal Desktop®
The goal is to create an empirical equation defining heat flux across IMLI for use in a system or vehicle level thermal model as a function of layer count, substrate temperature, and outer surface temperature.
IMLI↗
Intermediate boundary conditions for LOD, ADI and approximate factorization methods
A general approach to determining the correct intermediate boundary conditions for dimensional splitting methods is presented. The intermediate solution U is viewed as a second order accurate approximation to a modified equation. Deriving the modified equation and using the relationship between this equation and the original equation allows us to determine the correct boundary conditions for U*. This technique is illustrated by applying it to locally one dimensional (LOD) and alternating direction implicit (ADI) methods for the heat equation in two and three space dimensions. The approximate factorization method is considered in slightly more generality.
Leveque, R. J.↗