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Results for “Finite-Difference Methods”

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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At least 163 records · Page 9
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NASA NTRS1975

Subgrid or Reynolds stress-modeling for three-dimensional turbulence computations

A review is given of recent advances in two distinct computational methods for evaluating turbulence fields, namely, statistical Reynolds stress modeling and turbulence simulation, where large eddies are followed in time. It is shown that evaluation of the mean Reynolds stresses, rather than use of a scalar eddy viscosity, permits an explanation of streamline curvature effects found in several experiments. Turbulence simulation, with a new volume averaging technique and third-order accurate finite-difference computing is shown to predict the decay of isotropic turbulence in incompressible flow with rather modest computer storage requirements, even at Reynolds numbers of aerodynamic interest.

Rubesin, M. W.↗
NASA NTRS1976

Improved interactive calculation procedure for supersonic flows

An interactive numerical procedure has been developed for supersonic viscous flows (either two-dimensional or axisymmetric configurations). The flow field is divided into two regions: (1) an inner region which is highly viscous and mostly subsonic, and (2) an outer region where the flow is supersonic and in which viscous effects are small, but not negligible. This paper presents a detailed description of: I. Outer Region - numerical solution obtained by applying the method of characteristics to a system of equations which includes viscous and conduction transport terms only normal to the streamlines; II. Inner Region - treated by a system of equations of the boundary-layer type that includes higher order effects, such as longitudinal and transverse curvature and normal pressure gradients (equations are coupled and solved simultaneously in physical coordinates, using an implicit finite-difference scheme); III. Interactive Procedure - in the interaction mode, the two regions are coupled iteratively along a matching line, where the Mach number is of the order of 1.2.

Tassa, Y.↗
NASA NTRS1972

Numerical solutions of reactive fluid flows during postignition transients in hybrid rocket systems.

A computational method has been developed for the study of the post-ignition transients in hybrid rocket systems. The particular system chosen consisted of a gaseous oxidizer flowing within a tube of solid fuel, resulting in heterogeneous combustion. With the appropriate assumptions, two-dimensional, time-dependent conservation equations were derived for the reacting gas phase, and for the solid phase, in a cylindrical coordinate system. These were then programmed for numerical computation, using two implicit finite-difference schemes, the Lax-Wendroff scheme for the gas phase, and the Crank-Nicolson scheme for the solid phase. Appropriate initial and boundary conditions were represented, including heat and mass conservation at the interface between gas and solid. Initially, no attempt was made to relate the recession rate at the surface to the surface temperature, or to include heat transfer by radiation. A simple case was selected for preliminary calculations, with aluminum and oxygen as fuel and oxidizer, and aluminum oxide as the product.

Hung, W. S. Y.↗
NASA NTRS1979

Analysis of Combustion Instability in Liquid Fuel Rocket Motors

The development of a technique to be used in the solution of nonlinear velocity-sensitive combustion instability problems is described. The orthogonal collocation method was investigated. It found that the results are heavily dependent on the location of the collocation points and characteristics of the equations, so the method was rejected as unreliabile. The Galerkin method, which has proved to be very successful in analysis of the pressure sensitive combustion instability was found to work very well. It was found that the pressure wave forms exhibit a strong second harmonic distortion and a variety of behaviors are possible depending on the nature of the combustion process and the parametric values involved. A one-dimensional model provides further insight into the problem by allowing a comparison of Galerkin solutions with more exact finite-difference computations.

Wong, K. W.↗
NASA NTRS1976

A numerical study of viscous flow around an airfoil

An integrodifferential method, previously formulated in terms of velocity and vorticity vectors, is reformulated in terms of stream function and vorticity for two-dimensional incompressible viscous flows. The reformulated integrodifferential method is shown to retain the distinguishing feature of the previous formulation in permitting the confinement of the solution field to the viscous region of the flow and consequently offers great computational advantages. The application of this procedure in a study of an incompressible flow around an impulsively started 9% thick symmetric Joukowski airfoil at an angle of attack of 15 deg and a Reynolds number of 1000 is discussed. Numerical results are presented and compared with available finite-difference results.

Wu, J. C.↗
NASA NTRS1978

Analysis of turbulent free jet hydrogen-air diffusion flames with finite chemical reaction rates

The nonequilibrium flow field resulting from the turbulent mixing and combustion of a supersonic axisymmetric hydrogen jet in a supersonic parallel coflowing air stream is analyzed. Effective turbulent transport properties are determined using the (K-epsilon) model. The finite-rate chemistry model considers eight reactions between six chemical species, H, O, H2O, OH, O2, and H2. The governing set of nonlinear partial differential equations is solved by an implicit finite-difference procedure. Radial distributions are obtained at two downstream locations of variables such as turbulent kinetic energy, turbulent dissipation rate, turbulent scale length, and viscosity. The results show that these variables attain peak values at the axis of symmetry. Computed distributions of velocity, temperature, and mass fraction are also given. A direct analytical approach to account for the effect of species concentration fluctuations on the mean production rate of species (the phenomenon of unmixedness) is also presented. However, the use of the method does not seem justified in view of the excessive computer time required to solve the resulting system of equations.

Sislian, J. P.↗
NASA NTRS1976

A two-dimensional finite difference solution for the transient thermal behavior of tubular solar collector

A numerical procedure was established using the finite-difference technique in the determination of the time-varying temperature distribution of a tubular solar collector under changing solar radiancy and ambient temperature. Three types of spatial discretization processes were considered and compared for their accuracy of computations and for selection of the shortest computer time and cost. The stability criteria of this technique was analyzed in detail to give the critical time increment to ensure stable computations. The results of the numerical analysis were in good agreement with the analytical solution previously reported. The numerical method proved to be a powerful tool in the investigation of the collector sensitivity to two different flow patterns and several flow control mechanisms.

Lansing, F. L.↗
NASA NTRS1977

A perturbation solution for interlaminar stresses in bidirectional laminates

The paper considers a thin elastic bidirectional symmetric composite laminate subjected to a uniform axial strain. All interlaminar stresses are determined by a zeroth-order perturbation analysis of the governing equations. The method of solution considers the laminate to be composed of an interior region (removed from the free edge) and a boundary layer region. The interior region is analyzed by dropping terms related to powers of the thickness-to-width ratio in the dimensionless elasticity equations. The boundary layer region is studied by introducing a stretching transformation to the governing equations. A uniform solution is then formed by satisfying Prandtl's matching principle of perturbation theory. Results for a four-layer bidirectional graphite-epoxy laminate are compared with existing finite-difference results. It is shown that the perturbation analysis provides more insight into the stress distribution near the free edge.

Hsu, P. W.↗
NASA NTRS1974

The numerical calculation of laminar boundary-layer separation

Iterative finite-difference techniques are developed for integrating the boundary-layer equations, without approximation, through a region of reversed flow. The numerical procedures are used to calculate incompressible laminar separated flows and to investigate the conditions for regular behavior at the point of separation. Regular flows are shown to be characterized by an integrable saddle-type singularity that makes it difficult to obtain numerical solutions which pass continuously into the separated region. The singularity is removed and continuous solutions ensured by specifying the wall shear distribution and computing the pressure gradient as part of the solution. Calculated results are presented for several separated flows and the accuracy of the method is verified. A computer program listing and complete solution case are included.

Klineberg, J. M.↗
NASA NTRS1974

On laminar boundary-layer separation

Iterative finite-difference techniques are developed for integrating the boundary-layer equations, without approximation, through a region of reversed flow. The numerical procedures are used to calculate incompressible laminar separated flows and to investigate the conditions for regular behavior at the point of separation. Regular flows are shown to be characterized by an integrable saddle-type singularity that makes it difficult to obtain numerical solutions which pass continuously into the separated region. The singularity is removed and continuous solutions ensured by specifying the wall shear distribution and computing the pressure gradient as part of the solution. Calculated results are presented for a number of separated flows and the accuracy of the method is verified.

Klineberg, J. M.↗
NASA NTRS1975

Time-dependent transonic flow solutions for axial turbomachinery

Three-dimensional unsteady transonic flow through an axial turbomachine stage is described in terms of a pair of two-dimensional formulations pertaining to orthogonal surfaces, namely, a blade-to-blade surface and a hub-to-casing surface. The resulting systems of nonlinear, inviscid, compressible equations of motion are solved by an explicit finite-difference technique. The blade-to-blade program includes the periodic interaction between rotor and stator blade rows. Treatment of the boundary conditions and of the blade slipstream motion by a characteristic type procedure is discussed in detail. Harmonic analysis of the acoustic far field produced by the blade row interaction, including an arbitrary initial transient, is outlined. Results from the blade-to-blade program are compared with experimental measurements of the rotating pressure field at the tip of a high-speed fan. The hub-to-casing program determines circumferentially averaged flow properties on a meridional plane. Blade row interactions are neglected in this formulation, but the force distributions over the entire blade surface for both the rotor and stator are obtained. Results from the hub-to-casing program are compared with a relaxation method solution for a subsonic rotor. Results are also presented for a quiet fan stage which includes transonic flow in both the rotor and stator and a normal shock in the stator.

Erdos, J.↗
NASA NTRS1972

Finite element analysis of large transient elastic-plastic deformations of simple structures, with application to the engine rotor fragment containment/deflection problem

Assumed-displacement versions of the finite-element method are developed to predict large-deformation elastic-plastic transient deformations of structures. Both the conventional and a new improved finite-element variational formulation are derived. These formulations are then developed in detail for straight-beam and curved-beam elements undergoing (1) Bernoulli-Euler-Kirchhoff or (2) Timoshenko deformation behavior, in one plane. For each of these categories, several types of assumed-displacement finite elements are developed, and transient response predictions are compared with available exact solutions for small-deflection, linear-elastic transient responses. The present finite-element predictions for large-deflection elastic-plastic transient responses are evaluated via several beam and ring examples for which experimental measurements of transient strains and large transient deformations and independent finite-difference predictions are available.

Wu, R. W.↗
NASA NTRS1974

Liquefaction dynamics of n-octadecane in cylindrical coordinates

The study presents one method of mathematically modeling the effects of gravity-induced free convection upon the melting phenomena of a finite paraffin slab. The theoretical model is based on the numerical computer solution of the solid phase and liquid phase energy equations, coupled with an imposed ideal-viscous flow approximation. An implicit alternating-direction technique is used as the finite-difference approximation for the basic energy equations in the numerical solution. The theoretical predictions are compared to experimental data from an annular test cell, in which n-octadecane, C18H38, was used as the test material. The results show that the model is a good first approximation to the solution of solid-liquid phase change involving gravity-induced free convection.

Bain, R. L.↗
NASA NTRS1975

A finite-difference program for stresses in anisotropic, layered plates in bending

The interlaminar stresses induced in a layered laminate that is bent into a cylindrical surface are studied. The laminate is modeled as a continuum, and the resulting elasticity equations are solved using the finite difference method. The report sets forth the mathematical framework, presents some preliminary results, and provides a listing and explanation of the computer program. Significant among the results are apparent symmetry relationships that will reduce the numerical size of certain problems and an interlaminar stress behavior having a sharp rise at the free edges.

Salamon, N. J.↗
NASA NTRS1975

An investigation of several numerical procedures for time-asymptotic compressible Navier-Stokes solutions

The status of an investigation of four numerical techniques for the time-dependent compressible Navier-Stokes equations is presented. Results for free shear layer calculations in the Reynolds number range from 1000 to 81000 indicate that a sequential alternating-direction implicit (ADI) finite-difference procedure requires longer computing times to reach steady state than a low-storage hopscotch finite-difference procedure. A finite-element method with cubic approximating functions was found to require excessive computer storage and computation times. A fourth method, an alternating-direction cubic spline technique which is still being tested, is also described.

Rudy, D. H.↗
NASA NTRS1978

Numerical solution of the Navier-Stokes equations for super-sonic flows with strong shocks

The numerical solution of the full Navier-Stokes Equations for viscous flows with high Mach numbers and a strong detached bow shock was obtained. Two dimensional flows around a circular cylinder, and a circular cylinder with an aft-body in the form of a fairing, were considered. The solution of the compressible N.S. equations was accomplished by the method of finite differences. An implicit scheme of solution, the S.O.R., was used with the optimum acceleration parameters determined by trial and error. The tensor notation was used in writing the N-S Equations transformed into general curvilinear coordinates. The equations for the generation of the coordinate system were solved, followed by the solution of the N.S. equations, at the end of a set of given number of time steps. "Wiggles", constituted the one major problem that needed to be overcome. These oscillations give rise to quantities such as negative temperatures, which ultimately caused the computational program to break down. Certain dissipative finite-difference schemes damped these oscillations.

Devarayalu, K.↗
NASA NTRS1978

Numerical method for boundary layers with blowing - The exponential box scheme

The paper describes a new numerical scheme based on exponential difference operator concepts combined with Keller's (1968) box scheme approach to produce a stable second-order accurate finite-difference scheme for convection-diffusion problems arising in boundary layer flows in the presence of massive injection through a porous surface. The technique is demonstrated by application to the self-similar boundary layer equations with massive blowing at the surface.

El-Mistikawy, T. M.↗