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All sourcesNASA NTRSDOE OSTI
NASA NTRS1973

Numerical marching techniques for fluid flows with heat transfer

The finite difference formulation and method of solution is presented for a wide variety of fluid flow problems with associated heat transfer. Only a few direct results from these formulations are given as examples, since the book is intended primarily to serve a discussion of the techniques and as a starting point for further investigations; however, the formulations are sufficiently complete that a workable computer program may be written from them. In the appendixes a number of topics are discussed which are of interest with respect to the finite difference equations presented. These include a very rapid method for solving certain sets of linear algebraic equations, a discussion of numerical stability, the inherent error in flow rate for confined flow problems, and a method for obtaining high accuracy with a relatively small number of mesh points.

Hornbeck, R. W.↗
NASA NTRS1981

Calculation of sensitivity derivatives in thermal problems by finite differences

The optimum design of a structure subject to temperature constraints is considered. When mathematical optimization techniques are used, derivatives of the temperature constraints with respect to the design variables are usually required. In the case of large aerospace structures, such as the Space Shuttle, the computation of these derivatives can become prohibitively expensive. Analytical methods and a finite difference approach have been considered in studies conducted to improve the efficiency of the calculation of the derivatives. The present investigation explores two possibilities for enhancing the effectiveness of the finite difference approach. One procedure involves the simultaneous solution of temperatures and derivatives. The second procedure makes use of the optimum selection of the magnitude of the perturbations of the design variables to achieve maximum accuracy.

Haftka, R. T.↗
NASA NTRS1986

Preconditioned conjugate residual methods for the solution of spectral equations

Conjugate residual methods for the solution of spectral equations are described. An inexact finite-difference operator is introduced as a preconditioner in the iterative procedures. Application of these techniques is limited to problems for which the symmetric part of the coefficient matrix is positive definite. Although the spectral equation is a very ill-conditioned and full matrix problem, the computational effort of the present iterative methods for solving such a system is comparable to that for the sparse matrix equations obtained from the application of either finite-difference or finite-element methods to the same problems. Numerical experiments are shown for a self-adjoint elliptic partial differential equation with Dirichlet boundary conditions, and comparison with other solution procedures for spectral equations is presented.

Wong, Y. S.↗
NASA NTRS1976

Numerical solution of nonlinear partial differential equations of mixed type

A review is presented of some recently developed numerical methods for the solution of nonlinear equations of mixed type. The methods considered use finite difference approximations to the differential equation. Central difference formulas are employed in the subsonic zone and upwind difference formulas are used in the supersonic zone. The relaxation method for the small disturbance equation is discussed and a description is given of difference schemes for the potential flow equation in quasi-linear form. Attention is also given to difference schemes for the potential flow equation in conservation form, the analysis of relaxation schemes by the time dependent analogy, the accelerated iterative method, and three-dimensional calculations.

Jameson, A.↗
NASA NTRS1996

Numerical Study of Multigrid Methods with Various Smoothers for the Elliptical Grid Generation Equations

A robust solver for the elliptic grid generation equations is sought via a numerical study. The system of PDEs is discretized with finite differences, and multigrid methods are applied to the resulting nonlinear algebraic equations. Multigrid iterations are compared with respect to the robustness and efficiency. Different smoothers are tried to improve the convergence of iterations. The methods are applied to four 2D grid generation problems over a wide range of grid distortions. The results of the study help to select smoothing schemes and the overall multigrid procedures for elliptic grid generation.

Golik, W. L.↗
NASA NTRS1989

Understanding transition and turbulence through direct simulations

Direct simulations consist in solving the full Navier-Stokes equations, without any turbulence model, and describing all the detailed features of the flow. Usually the flows are three-dimensional and time-dependent and contain both coarse and fine structures, which makes the numerical task very challenging in terms of both the algorithm and the computational effort. Most of the work until now has involved spectral methods, which are highly accurate but not very flexible in terms of geometry or complex equations. For that reason, future work will also rely on high-order finite-difference or other methods. Direct simulations complement experimental work, and both contribute to the theory and the empirical knowledge of turbulence. Once such a simulation has been shown to be accurate, the flow field is completely known in three dimensions and time, including the pressure, the vorticity and any other quantity. On the other hand, most simulations to date solved the incompressible equations in rather simple geometries, and direct simulations will always be limited to moderate Reynolds numbers. Extensive simulations have been conducted in homogeneous turbulence, channel flows, boundary layers, and mixing layers. Much effort is devoted to addressing flows with compressibility and chemical reactions, and to new geometries such as a backward-facing step.

Spalart, P. R.↗
NASA NTRS2000

Onset of Convection Due to Surface Tension Variations in Multicomponent and Binary Fluid Layers

Under certain conditions, such as in thin liquid films or microgravity, surface tension variations along a free surface can induce convection. Convection onset due to surface tension variation is important to many terrestrial technological processes in addition to microgravity materials processing applications. Examples include coating, drying crystallization, solidification, liquid surface contamination, and containerless processing. In double-diffusive and multicomponent systems, the spatial variations of surface tension are associated with two or more stratifying agencies, respectively. For example, both temperature and species (concentration) gradients are associated with convection in the solidification of binary alloys or salt ponds. The direction of the two (or more) gradients has a profound effect on the nature of the flow at or slightly beyond the onset of convection. Our recent work at the NASA Lewis Research Center focused on characterizing surface-tension-induced onset of convection, often referred to as Marangoni-Benard convection. Exact solutions for the stationary neutral stability of multicomponent fluid layers with interfacial deformation were derived. These solutions also permit the computation of a boundary curve that separates the long and finite wavelength instabilities. Computing points along this boundary using the exact solution (when possible) is more efficient than the typical numerical approaches, such as finite difference or spectral methods. Above the curve, a long wavelength instability was predicted, suggesting that convection would occur principally through one large flow cell in the layer, whereas below the curve, finite wavelength instabilities occur which suggest multiple finite-sized circulation cells. For many common liquids with layer depths greater than 100 mm, finite wave instability is predicted under terrestrial conditions; however, with little exception, long wavelength instability is predicted in microgravity for the identical fluid systems.

Skarda, J. Raymond Lee↗
NASA NTRS1994

The least-squares finite element method for low-mach-number compressible viscous flows

The present paper reports the development of the Least-Squares Finite Element Method (LSFEM) for simulating compressible viscous flows at low Mach numbers in which the incompressible flows pose as an extreme. Conventional approach requires special treatments for low-speed flows calculations: finite difference and finite volume methods are based on the use of the staggered grid or the preconditioning technique; and, finite element methods rely on the mixed method and the operator-splitting method. In this paper, however, we show that such difficulty does not exist for the LSFEM and no special treatment is needed. The LSFEM always leads to a symmetric, positive-definite matrix through which the compressible flow equations can be effectively solved. Two numerical examples are included to demonstrate the method: first, driven cavity flows at various Reynolds numbers; and, buoyancy-driven flows with significant density variation. Both examples are calculated by using full compressible flow equations.

Yu, Sheng-Tao↗
NASA NTRSDate not supplied

Realistic Thermographic Simulation of Impact Damage with Quadrupole Method

Flash thermography has been shown to be an effective method for detection of delaminations in carbon fiber reinforced polymer (CFRP) composite structures. Improved understanding of the limitations of the technique can be obtained by simulating the inspection process. Time domain finite difference and finite element methods are well suited for such simulations but can be computationally intensive. An alternate method is to solve the Laplace transform of the heat equation, then invert the Laplace transform to produce a time domain response. Often this is referred to as the thermal quadrupole method and is well suited for simulating flash thermography. The quadrupole method has been used extensively for simulating one-dimensional heat flow in multilayer systems with contact resistances at the interfaces. It is also applicable for three dimensional configurations, in particular for simulation of changes in thermographic response due to delaminations in composites. This presentation discusses three-dimensional simulations with realistic shapes and contact resistances at the interfaces between layers. The results of these simulations are compared to thermography measurements on composites specimens with impact induced delaminations.

thermography↗
NASA NTRSDate not supplied

Realistic Thermographic Simulation of Impact Damage with Quadrupole Method

Flash thermography has been shown to be an effective method for detection of delaminations in carbon fiber reinforced polymer (CFRP) composite structures. Improved understanding of thelimitations of the technique can be obtained by simulating the inspection process. Time domainfinite difference and finite element methods are well suited for such simulations but can be computationally intensive. An alternate method is to solve the Laplace transform of the heat equation, then invert the Laplace transform to produce a time domain response. Often this is referred to as the thermal quadrupole method and is well suited for simulating flash thermography. The quadrupole method has been used extensively for simulating one-dimensional heat flow in multilayer systems with contact resistances at the interfaces. It is also applicable for three dimensional configurations, in particular for simulation of changes in thermographic response due to delaminations in composites. This presentation discusses three-dimensional simulations with realistic shapes and contact resistances at the interfaces between layers. The results of these simulations are compared to thermography measurements on composites specimens with impact induced delaminations.

Thermography↗
NASA NTRS1991

A simple method for simulating gasdynamic systems

A simple method for performing digital simulation of gasdynamic systems is presented. The approach is somewhat intuitive, and requires some knowledge of the physics of the problem as well as an understanding of the finite difference theory. The method is explicitly shown in appendix A which is taken from the book by P.J. Roache, 'Computational Fluid Dynamics,' Hermosa Publishers, 1982. The resulting method is relatively fast while it sacrifices some accuracy.

Hartley, Tom T.↗
NASA NTRS1974

Rapid finite-difference computation of subsonic and transonic aerodynamic flows

Rapid iterative (or semidirect) computation methods are developed for the finite-difference solution of the nonlinear equations of subsonic and transonic aerodynamics. At each iteration, a fast, direct elliptic algorithm solves the entire computation field. In an application to subsonic flow over a lifting airfoil, the full nonlinear stream-function equation is solved. Finally, a direct Cauchy-Riemann solver is used for the nonlinear transonic small-disturbance equations for a biconvex airfoil. At M = 0.7, t/c = 0.1 (subcritical), three iterations on a 39 x 32 mesh (totaling 2.45 sec on an IBM 360/67 computer) obtain convergence within 0.1%. A slightly supercritical case requires seven iterations (6.75 sec) for convergence within 1%.

Martin, E. D.↗
NASA NTRS2015

Estimation of Surface Temperature and Heat Flux by Inverse Heat Transfer Methods Using Internal Temperatures Measured While Radiantly Heating a Carbon/Carbon Specimen up to 1920 F

The ability to solve the heat conduction equation is needed when designing materials to be used on vehicles exposed to extremely high temperatures; e.g. vehicles used for atmospheric entry or hypersonic flight. When using test and flight data, computational methods such as finite difference schemes may be used to solve for both the direct heat conduction problem, i.e., solving between internal temperature measurements, and the inverse heat conduction problem, i.e., using the direct solution to march forward in space to the surface of the material to estimate both surface temperature and heat flux. The completed research first discusses the methods used in developing a computational code to solve both the direct and inverse heat transfer problems using one dimensional, centered, implicit finite volume schemes and one dimensional, centered, explicit space marching techniques. The developed code assumed the boundary conditions to be specified time varying temperatures and also considered temperature dependent thermal properties. The completed research then discusses the results of analyzing temperature data measured while radiantly heating a carbon/carbon specimen up to 1920 F. The temperature was measured using thermocouple (TC) plugs (small carbon/carbon material specimens) with four embedded TC plugs inserted into the larger carbon/carbon specimen. The purpose of analyzing the test data was to estimate the surface heat flux and temperature values from the internal temperature measurements using direct and inverse heat transfer methods, thus aiding in the thermal and structural design and analysis of high temperature vehicles.

Pizzo, Michelle↗
NASA NTRS1986

A mixed pseudospectral/finite difference method for the axisymmetric flow in a heated, rotating spherical shell

For a Spacelab flight, a model experiment of the earth's atmospheric circulation has been proposed. This experiment is known as the Atmospheric General Circulation Experiment (AGCE). In the experiment concentric spheres will rotate as a solid body, while a dielectric fluid is confined in a portion of the gap between the spheres. A zero gravity environment will be required in the context of the simulation of the gravitational body force on the atmosphere. The present study is concerned with the development of pseudospectral/finite difference (PS/FD) model and its subsequent application to physical cases relevant to the AGCE. The model is based on a hybrid scheme involving a pseudospectral latitudinal formulation, and finite difference radial and time discretization. The advantages of the use of the hybrid PS/FD method compared to a pure second-order accurate finite difference (FD) method are discussed, taking into account the higher accuracy and efficiency of the PS/FD method.

Macaraeg, M. G.↗
NASA NTRS1978

Comparative study of the convergence rates of two numerical techniques

The paper examines the applicability of the three-step Stetter (1968) method to the problem of hypersonic viscous flow over a blunt axisymmetric body used for planetary entry probes at zero angle of attack. The flow-field results using the two-step finite-difference MacCormack (1969) method are reported by Kumar and Graves (1977). Only the computational efficiency of Stetter's method is compared with that of MacCormack's in terms of the iterative time steps and computing time required for the steady-state solution. Advantages of Stetter's method over MacCormack's are established.

Kumar, A.↗
NASA NTRS1990

Cell averaging Chebyshev methods for hyperbolic problems

A cell averaging method for the Chebyshev approximations of first order hyperbolic equations in conservation form is described. Formulas are presented for transforming between pointwise data at the collocation points and cell averaged quantities, and vice-versa. This step, trivial for the finite difference and Fourier methods, is nontrivial for the global polynomials used in spectral methods. The cell averaging methods presented are proven stable for linear scalar hyperbolic equations and present numerical simulations of shock-density wave interaction using the new cell averaging Chebyshev methods.

Wei, Cai↗
NASA NTRS1991

A comparison of numerical methods for the Rayleigh equation in unbounded domains

A second-order finite difference and two spectral methods, including a Chebyshev tau and a Chebyshev collocation method were implemented to determine the linear hydrodynamic stability of an unbounded shear flow. The velocity profile of the basic flow in the stability analysis mimicks that of a one-stream free mixing layer. Local and global eigenvalue solution methods are used to determine individual eigenvalues and the eigenvalue spectrum, respectively. The calculated eigenvalue spectrum includes a discrete mode, a continuous spectrum associated with the equation singularity and a continuous spectrum associated with the domain unboundedness. The efficiency and the accuracy of these discretization methods in the prediction of the eigensolutions of the discrete mode were evaluated by comparison with a conventional shooting procedure. Their capabilities in mapping out the continuous eigenvalue spectra are also discussed.

Liou, W. W.↗