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Martin, E. D.

Publications and source records attributed to Martin, E. D..

At least 19 records
All sourcesNASA NTRSDOE OSTI
NASA NTRS1990

Three-dimensional potential flows from functions of a 3D complex variable

Potential, or ideal, flow velocities can be found from the gradient of an harmonic function. An ordinary complex valued analytic function can be written as the sum of two real valued functions, both of which are harmonic. Thus, 2D complex valued functions serve as a source of functions that describe two-dimensional potential flows. However, this use of complex variables has been limited to two-dimensions. Recently, a new system of three-dimensional complex variables has been developed at the NASA Ames Research Center. As a step toward application of this theory to the analysis of 3D potential flow, several functions of a three-dimensional complex variable have been investigated. The results for two such functions, the 3D exponential and 3D logarithm, are presented in this paper. Potential flows found from these functions are investigated. Important characteristics of these flows fields are noted.

Kelly, Patrick↗
NASA NTRS1978

A split-recoupled-semidirect computational technique applied to transonic flow over lifting airfoils

A new version of the semidirect iterative method eliminates significant restrictions of previous versions of the method. A semidirect method solves finite-difference equations by a rapid globally implicit iterative process driven by a fast direct elliptic solver. The new approach can treat complex systems of equations in an efficient 'correction form', and allows the use of general, nonorthogonal, boundary-fitted coordinate transformations. These features are expected to lead to significant practical applications with conservation-equation systems in either two or three dimensions. The present application to the full potential equations for steady transonic flow over an airfoil at angle of attack illustrates the utility of the technique.

Martin, E. D.↗
NASA NTRS1977

Fast methods incorporating direct elliptic solvers for nonlinear applications in fluid dynamics

Semidirect methods are discussed, their present role, as well as some developments for their application in computational fluid dynamics. A semidirect method is a computational scheme that uses a fast, direct, elliptic solver as the driving algorithm for the iterative solution of finite difference equations. Specific subtopics include: (1) direct Cauchy Riemann solvers for first order elliptic equations; (2) application of the semidirect method to the mixed elliptic hyperbolic problem of steady, inviscid transonic flow; and (3) the treatment of interior conditions, such as those on an airfoil or wing, in semidirect methods.

Martin, E. D.↗
NASA NTRS1977

Variants and extensions of a fast direct numerical cauchy-riemann solver, with illustrative applications

Revised and extended versions of a fast, direct (noniterative) numerical Cauchy-Riemann solver are presented for solving finite difference approximations of first order systems of partial differential equations. Although the difference operators treated are linear and elliptic, one significant application of these extended direct Cauchy-Riemann solvers is in the fast, semidirect (iterative) solution of fluid dynamic problems governed by the nonlinear mixed elliptic-hyperbolic equations of transonic flow. Different versions of the algorithms are derived and the corresponding FORTRAN computer programs for a simple example problem are described and listed. The algorithms are demonstrated to be efficient and accurate.

Martin, E. D.↗
NASA NTRS1976

Advances in the application of fast semidirect computational methods in transonic flow

The paper uses finite-difference algorithms called 'fast direct elliptic solvers' within an iteration scheme for the rapid solution of the equations of inviscid transonic aerodynamics. The methods are called 'direct' (or semidirect) because the entire computational field is solved at once rather than in successive traverses over the field. These semidirect iterative methods have been limited here to the investigation of two-dimensional steady inviscid flow over airfoils in subsonic free stream.

Martin, E. D.↗
NASA NTRS1976

A technique for accelerating iterative convergence in numerical integration, with application in transonic aerodynamics

A technique is described for the efficient numerical solution of nonlinear partial differential equations by rapid iteration. In particular, a special approach is described for applying the Aitken acceleration formula (a simple Pade approximant) for accelerating the iterative convergence. The method finds the most appropriate successive approximations, which are in a most nearly geometric sequence, for use in the Aitken formula. Simple examples are given to illustrate the use of the method. The method is then applied to the mixed elliptic-hyperbolic problem of steady, inviscid, transonic flow over an airfoil in a subsonic free stream.

Martin, E. D.↗
NASA NTRS1975

A technique for accelerating iterative convergence in numerical integration, with application in transonic aerodynamics

A technique is described for the efficient numerical solution of nonlinear partial differential equations by rapid iteration. In particular, a special approach is described for applying the Aitken acceleration formula (a simple Pade approximant) for accelerating the iterative convergence. The method finds the most appropriate successive approximations, which are in a most nearly geometric sequence, for use in the Aitken formula. Simple examples are given to illustrate the use of the method. The method is then applied to the mixed elliptic-hyperbolic problem of steady, inviscid, transonic flow over an airfoil in a subsonic free stream.

Martin, E. D.↗
NASA NTRS1975

A fast semidirect method for computing transonic aerodynamic flows

A fast, semidirect, iterative computational method, previously introduced for finite-difference solution of subsonic and slightly supercritical flow over airfoils, is extended both to apply to strongly supercritical conditions and to include full second-order accuracy in computing inviscid flows over airfoils. The nonlinear small-disturbance equations are solved iteratively by a direct, linear, elliptic solver. General, fully conservative, type-dependent difference equations are formulated, including parabolic- and shock-point transition operators that provide consistency with the integral conservation laws. These equations specialize to either first-order or to fully second-order-accurate equations. Various free parameters are evaluated for rapid convergence of the first-order scheme. Resulting pressure distributions and computing times are compared with the improved Murman-Cole line-relaxation method.

Martin, E. D.↗
NASA NTRS1974

Fast direct numerical solution of the nonhomogeneous Cauchy-Riemann equations

A fast direct (noniterative) 'Cauchy-Riemann Solver' is developed for solving the finite-difference equations representing systems of first-order elliptic partial differential equations in the form of the nonhomogeneous Cauchy-Riemann equations. The method is second-order accurate and requires approximately the same computer time as a fast cyclic-reduction Poisson solver. The accuracy and efficiency of the direct solver are demonstrated in an application to solving an example problem in aerodynamics: subsonic inviscid flow over a biconvex airfoil. The analytical small-perturbation solution contains singularities, which are captured well by the computational technique. The algorithm is expected to be useful in nonlinear subsonic and transonic aerodynamics.

Lomax, H.↗
NASA NTRS1974

A generalized-capacity-matrix technique for computing aerodynamic flows

A numerical generalized-capacity-matrix technique is developed for application to aerodynamic flow computations. This technique allows the very fast direct (noniterative) numerical elliptic solvers to be used in problems with arbitrary internal boundaries and with a wide class of boundary conditions, including numerical application of the Kutta condition on an airfoil without iteration. Accuracy, speed, and usefulness of the technique are demonstrated with linear problems for potential flows over airfoil shapes. The method's main advantages, however, can be exploited within iterative procedures for a variety of complex flow problems governed by systems of equations not necessarily elliptic or linear.

Martin, E. D.↗
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 NTRS1973

Direct numerical solution of three-dimensional equations containing elliptic operators.

A direct three-dimensional elliptic solver is presented for application in a wide class of numerical methods for solving partial differential equations in physics and engineering. The derived algorithm and FORTRAN code implement Buzbee, Golub and Nielson's proposed extension of Buneman's Cyclic-Reduction Poisson solver to three dimensions. Both a 'most direct' cyclic reduction and a revised method (to eliminate roundoff error difficulties) are derived. Tests on an IBM 360/67 computer, using various optional combinations of subroutines, showed significant differences in accuracy and computing time, with the optimum subroutine combination depending on mesh size.

Martin, E. D.↗
NASA NTRS1972

Effects of circular geometry in simulation of convection in rotating spacecraft tanks

For computational simulation of the convection and mixing of stratified fluid in a rotating tank (such as used in Apollo flights) with time-dependent rotation, the Navier-Stokes convection problem was formulated for a circular tank configuration. The final equations results from a general approximate theory for combined forced and contained natural convection in a time-dependent rotating system. The equations are cast in terms of vorticity and stream function in a form convenient for computation, with a transformed coordinate system, and appropriate boundary conditions are derived. Accurate representations for the cryogenic supercritical oxygen thermodynamic properties are used in the computations, and an efficient numerical finite difference scheme and computational method are employed.

Martin, E. D.↗
NASA NTRS1972

A method for calculating laminar viscous, compressible flows with small pressure gradients

A method is presented for calculating laminar viscous, compressible flows in which pressure gradients are assumed to be neglibible in energy conservation and in developing property relations, but not in momentum conservation. The equations are developed from Navier-Stokes equations in a noninertial reference frame. The method is an extension of an approximate method developed for calculating combined forced and contained natural convection in a rotating tank. The restriction to small density and temperature variations in the previous method is removed. For illustration, a two-dimensional simulation of the rotating tank problem is formulated, and the numerical computation procedure to be used is indicated.

Martin, E. D.↗
NASA NTRS1971

Mixing effectiveness in the Apollo oxygen tanks of spin-up and rotation-reversal maneuvers

Two-dimensional simulations of stratified flows in the Apollo oxygen tanks have been used to estimate the mixing effectiveness of spin-up and rotation-reversal maneuvers. Calculations have been made for square and circular cylindrical tank geometries. Differences arising from heater position on the tank wall or near the center of the tank have been investigated. In the event of a prolonged period without normal maneuvers, the potential pressure decay (drop in pressure that would result from adiabatic mixing) can be suppressed by more than a factor of two through the use of spin-up and rotation-reversal maneuvers. Changes in rotation rate of order three revolutions per hour or greater are sufficient for this purpose.

Baldwin, B.↗