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At least 271 records · Page 15

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NASA NTRS1995

Three-dimensional aerodynamic shape optimization using discrete sensitivity analysis

An aerodynamic shape optimization procedure based on discrete sensitivity analysis is extended to treat three-dimensional geometries. The function of sensitivity analysis is to directly couple computational fluid dynamics (CFD) with numerical optimization techniques, which facilitates the construction of efficient direct-design methods. The development of a practical three-dimensional design procedures entails many challenges, such as: (1) the demand for significant efficiency improvements over current design methods; (2) a general and flexible three-dimensional surface representation; and (3) the efficient solution of very large systems of linear algebraic equations. It is demonstrated that each of these challenges is overcome by: (1) employing fully implicit (Newton) methods for the CFD analyses; (2) adopting a Bezier-Bernstein polynomial parameterization of two- and three-dimensional surfaces; and (3) using preconditioned conjugate gradient-like linear system solvers. Whereas each of these extensions independently yields an improvement in computational efficiency, the combined effect of implementing all the extensions simultaneously results in a significant factor of 50 decrease in computational time and a factor of eight reduction in memory over the most efficient design strategies in current use. The new aerodynamic shape optimization procedure is demonstrated in the design of both two- and three-dimensional inviscid aerodynamic problems including a two-dimensional supersonic internal/external nozzle, two-dimensional transonic airfoils (resulting in supercritical shapes), three-dimensional transport wings, and three-dimensional supersonic delta wings. Each design application results in realistic and useful optimized shapes.

Burgreen, Gregory W.↗

NASA NTRS1992

Progress in unstructured-grid methods development for unsteady aerodynamic applications

The development of unstructured-grid methods for the solution of the equations of fluid flow and what was learned over the course of the research are summarized. The focus of the discussion is on the solution of the time-dependent Euler equations including spatial discretizations, temporal discretizations, and boundary conditions. An example calculation with an implicit upwind method using a CFL number of infinity is presented for the Boeing 747 aircraft. The results were obtained in less than one hour CPU time on a Cray-2 computer, thus, demonstrating the speed and robustness of the capability. Additional calculations for the ONERA M6 wing demonstrate the accuracy of the method through the good agreement between calculated results and experimental data for a standard transonic flow case.

Batina, John T.↗

NASA NTRS1989

Numerical modeling of a vortex stabilized arcjet thruster

A numerical method to solve the equations governing a vortex stabilized arcjet thruster is being developed. The model will allow the effects of swirling flow and geometry on arcjet thruster performance to be determined. The propellant flow equations, i.e., the axisymmetric, thin layer, Navier-Stokes equations, are solved using a Gauss-Siedel line-relaxation procedure. An implicit FTCS method is used to solve the electromagnetic field equations. A grid generation scheme was developed for an arbitrary arcjet geometry. The model will allow the radial and axial components of velocity and current distributions to be determined from a region upstream of the cathode, through the constrictor, to the exit plane of the nozzle.

Keith, Theo G., Jr.↗

NASA NTRS1985

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.↗

NASA NTRS1992

An implicit multigrid scheme for hypersonic strong-interaction flowfields

An implicit multigrid method for the compressible Navier-Stokes equations is applied to a series of two-dimensional, steady, laminar hypersonic flows over a compression ramp, including both attached and separated flow conditions. The algorithm uses upwind-biased differencing for the convective and pressure terms and central differencing for the shear-stress and heat-transfer terms. An implicit spatially factored scheme is used to advance the solution in terms on a given mesh, and a V-cycle multigrid strategy is used to accelerate convergence. The multigrid algorithm provided an order of magnitude decrease in the computational time required to obtain a converged position of the separation location and enabled a substantially grid-independent result to be obtained for each of the flows considered.

Thomas, James L.↗

NASA NTRS1975

Study of viscous flow about airfoils by the integro-differential method

An integro-differential method was used for numerically solving unsteady incompressible viscous flow problems. A computer program was prepared to solve the problem of an impulsively started 9% thick symmetric Joukowski airfoil at an angle of attack of 15 deg and a Reynolds number of 1000. Some of the results obtained for this problem were discussed and compared with related work completed previously. Two numerical procedures were used, an Alternating Direction Implicit (ADI) method and a Successive Line Relaxation (SLR) method. Generally, the ADI solution agrees well with the SLR solution and with previous results are stations away from the trailing edge. At the trailing edge station, the ADI solution differs substantially from previous results, while the vorticity profiles obtained from the SLR method there are in good qualitative agreement with previous results.

Wu, J. C.↗

NASA NTRS1984

Finite-difference fluid dynamics computer mathematical models for the design and interpretation of experiments for space flight

Numerical methods are used to design a spherical baroclinic flow model experiment of the large scale atmosphere flow for Spacelab. The dielectric simulation of radial gravity is only dominant in a low gravity environment. Computer codes are developed to study the processes at work in crystal growing systems which are also candidates for space flight. Crystalline materials rarely achieve their potential properties because of imperfections and component concentration variations. Thermosolutal convection in the liquid melt can be the cause of these imperfections. Such convection is suppressed in a low gravity environment. Two and three dimensional finite difference codes are being used for this work. Nonuniform meshes and implicit iterative methods are used. The iterative method for steady solutions is based on time stepping but has the options of different time steps for velocity and temperature and of a time step varying smoothly with position according to specified powers of the mesh spacings. This allows for more rapid convergence. The code being developed for the crystal growth studies allows for growth of the crystal as the solid-liquid interface. The moving interface is followed using finite differences; shape variations are permitted. For convenience in applying finite differences in the solid and liquid, a time dependent coordinate transformation is used to make this interface a coordinate surface.

Roberts, G. O.↗

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 NTRS1984

Development of a dynamically adaptive grid method for multidimensional problems

An approach to solution adaptive grid generation for use with finite difference techniques, previously demonstrated on model problems in one space dimension, has been extended to multidimensional problems. The method is based on the popular elliptic steady grid generators, but is 'dynamically' adaptive in the sense that a grid is maintained at all times satisfying the steady grid law driven by a solution-dependent source term. Testing has been carried out on Burgers' equation in one and two space dimensions. Results appear encouraging both for inviscid wave propagation cases and viscous boundary layer cases, suggesting that application to practical flow problems is now possible. In the course of the work, obstacles relating to grid correction, smoothing of the solution, and elliptic equation solvers have been largely overcome. Concern remains, however, about grid skewness, boundary layer resolution and the need for implicit integration methods. Also, the method in 3-D is expected to be very demanding of computer resources.

Holcomb, J. E.↗

NASA NTRS1979

Calculation of three-dimensional turbulent subsonic flows in transition ducts

A method for computing three-dimensional turbulent subsonic flow in curved ducts is being developed. A set of tube-like surface oriented coordinates is employed for a general class of geometries applicable to subsonic diffusers with offset bends. The geometric formulation is complex and no previous treatment of this class of viscous flow problems is known to the authors. The duct centerline is a space curve specified by piecewise polynomials. A Frenet frame is located on the centerline at each axial location. The cross sections are described by superellipses imbedded in the Frenet frame. Duct surfaces are also coordinate surfaces, which greatly simplifies the boundary conditions. The resulting coordinates are nonorthogonal. An approximate set of governing equations is employed for viscous flows having strong flow in a primary flow direction. The derivation is coordinate invariant and the resulting equations are expressed in tensor form. These equations are solved by an efficient alternating direction implicit (ADI) method. This numerical method is generally stable and permits solution in difficult geometries using the general tensor formulation.

Levy, R.↗

NASA NTRS1992

Three-dimensional boundary layer calculation by a characteristic method

A numerical method for solving the three-dimensional boundary layer equations for bodies of arbitrary shape is presented. In laminar flows, the application domain extends from incompressible to hypersonic flows with the assumption of chemical equilibrium. For turbulent boundary layers, the application domain is limited by the validity of the mixing length model used. In order to respect the hyperbolic nature of the equations reduced to first order partial derivative terms, the momentum equations are discretized along the local streamlines using of the osculator tangent plane at each node of the body fitted coordinate system. With this original approach, it is possible to overcome the use of the generalized coordinates, and therefore, it is not necessary to impose an extra hypothesis about the regularity of the mesh in which the boundary conditions are given. By doing so, it is possible to limit, and sometimes to suppress, the pre-treatment of the data coming from an inviscid calculation. Although the proposed scheme is only semi-implicit, the method remains numerically very efficient.

Houdeville, R.↗

NASA NTRS2001

Additive Runge-Kutta Schemes for Convection-Diffusion-Reaction Equations

Additive Runge-Kutta (ARK) methods are investigated for application to the spatially discretized one-dimensional convection-diffusion-reaction (CDR) equations. First, accuracy, stability, conservation, and dense output are considered for the general case when N different Runge-Kutta methods are grouped into a single composite method. Then, implicit-explicit, N = 2, additive Runge-Kutta ARK2 methods from third- to fifth-order are presented that allow for integration of stiff terms by an L-stable, stiffly-accurate explicit, singly diagonally implicit Runge-Kutta (ESDIRK) method while the nonstiff terms are integrated with a traditional explicit Runge-Kutta method (ERK). Coupling error terms are of equal order to those of the elemental methods. Derived ARK2 methods have vanishing stability functions for very large values of the stiff scaled eigenvalue, z(exp [I]) goes to infinity, and retain high stability efficiency in the absence of stiffness, z(exp [I]) goes to zero. Extrapolation-type stage-value predictors are provided based on dense-output formulae. Optimized methods minimize both leading order ARK2 error terms and Butcher coefficient magnitudes as well as maximize conservation properties. Numerical tests of the new schemes on a CDR problem show negligible stiffness leakage and near classical order convergence rates. However, tests on three simple singular-perturbation problems reveal generally predictable order reduction. Error control is best managed with a PID-controller. While results for the fifth-order method are disappointing, both the new third- and fourth-order methods are at least as efficient as existing ARK2 methods while offering error control and stage-value predictors.

Kennedy, Christopher A.↗

NASA NTRS1971

Finite difference solution to the three-dimensional, incompressible thermal energy boundary-layer equation

An implicit numerical method has been adopted for the solution to the three-dimensional, energy boundary-layer equation. The energy equation is written in terms of a dimensionless temperature function, the relative stagnation-enthalpy difference, and transformed by the introduction of a Blasius-type transformation of coordinates as well as dimensionless stream functions. The method is applied to the problem consisting of an infinite cylinder joined with its axis perpendicular to a thin, flat, heated plate. A Prandtl number equal to one is simply considered.

Fillo, J. A.↗

NASA NTRS1975

Viscous flow solutions with a cubic spline approximation

A cubic spline approximation is used for the solution of several problems in fluid mechanics. This procedure provides a high degree of accuracy even with a nonuniform mesh, and leads to a more accurate treatment of derivative boundary conditions. The truncation errors and stability limitations of several typical integration schemes are presented. For two-dimensional flows a spline-alternating-direction-implicit (SADI) method is evaluated. The spline procedure is assessed and results are presented for the one-dimensional nonlinear Burgers' equation, as well as the two-dimensional diffusion equation and the vorticity-stream function system describing the viscous flow in a driven cavity. Comparisons are made with analytic solutions for the first two problems and with finite-difference calculations for the cavity flow.

Rubin, S. G.↗

NASA NTRS1975

Numerical study of supersonic turbulent flow over small protuberances

Supersonic turbulent boundary layers over two-dimensional protuberances are investigated, using the numerical finite difference alternating direction implicit (ADI) method. The turbulence is modeled mathematically. The turbulence is represented here by the eddy viscosity approach. The turbulent boundary layer structure as well as an interest in thick boundary layers and much larger protuberance heights than in the laminar case lead to new difficulties. The problems encountered and the means to remove them are discussed.

Polak, A.↗

NASA NTRS1976

Condensation of water vapor in rarefaction waves. I - Homogeneous nucleation

A detailed theoretical investigation has been made of the condensation of water vapor/carrier gas mixtures in the nonstationary rarefaction wave generated in a shock tube. It is assumed that condensation takes place by homogeneous nucleation. The equations of motion together with the nucleation rate and the droplet growth equations were solved numerically by the method of characteristics and Lax's method of implicit artificial viscosity. It is found that, for the case considered, the condensation wave formed by the collapse of the metastable nonequilibrium state is followed by a shock wave generated by the intersection of characteristics of the same family. The expansion is practically isentropic up to the onset of condensation. The condensation front accelerates in the x,t plane. The results of the computations for a chosen case of water vapor/nitrogen mixture are presented by plotting variations of pressure, nucleation rate, number density of critical clusters, and condensate mass-fraction along three particle paths. Some consideration is given to homogeneous condensation experiments conducted in a shock tube. Although a direct comparison of the present theoretical work and these experiments is not possible, several worthwhile interpretative features have resulted nevertheless.

Sislian, J. P.↗

NASA NTRS1978

Large eddy simulation of incompressible turbulent channel flow

The three-dimensional, time-dependent primitive equations of motion were numerically integrated for the case of turbulent channel flow. A partially implicit numerical method was developed. An important feature of this scheme is that the equation of continuity is solved directly. The residual field motions were simulated through an eddy viscosity model, while the large-scale field was obtained directly from the solution of the governing equations. An important portion of the initial velocity field was obtained from the solution of the linearized Navier-Stokes equations. The pseudospectral method was used for numerical differentiation in the horizontal directions, and second-order finite-difference schemes were used in the direction normal to the walls. The large eddy simulation technique is capable of reproducing some of the important features of wall-bounded turbulent flows. The resolvable portions of the root-mean square wall pressure fluctuations, pressure velocity-gradient correlations, and velocity pressure-gradient correlations are documented.

Moin, P.↗

NASA NTRS1982

A composite velocity procedure for the compressible Navier-Stokes equations

A new boundary-layer relaxation procedure is presented. In the spirit of the theory of matched asymptotic expansions, a multiplicative composite of the appropriate velocity representations for the inviscid and viscous regions is prescribed. The resulting equations are structured so that far from the surface of the body the momentum equations lead to the Bernoulli relation for the pressure, while the continuity equation reduces to the familiar compressible potential equation. Close to the body surface, the governing equations and solution techniques are characteristic of those describing interacting boundary-layers; although, the full Navier-Stokes equations are considered here. Laminar flow calculations for the subsonic flow over an axisymmetric boattail simulator geometry are presented for a variety of Reynolds and Mach numbers. A strongly implicit solution method is applied for the coupled velocity components.

Khosla, P. K.↗