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At least 127 records · Page 7

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NASA NTRS1975

Supercritical wing sections 2, volume 108

A mathematical theory for the design and analysis of supercritical wing sections was previously presented. Examples and computer programs showing how this method works were included. The work on transonics is presented in a more definitive form. For design, a better model of the trailing edge is introduced which should eliminate a loss of fifteen or twenty percent in lift experienced with previous heavily aft loaded models, which is attributed to boundary layer separation. How drag creep can be reduced at off-design conditions is indicated. A rotated finite difference scheme is presented that enables the application of Murman's method of analysis in more or less arbitrary curvilinear coordinate systems. This allows the use of supersonic as well as subsonic free stream Mach numbers and to capture shock waves as far back on an airfoil as desired. Moreover, it leads to an effective three dimensional program for the computation of transonic flow past an oblique wing. In the case of two dimensional flow, the method is extended to take into account the displacement thickness computed by a semi-empirical turbulent boundary layer correction.

Bauer, F.↗

NASA NTRS1975

Numerical solution of the Navier-Stokes equations for arbitrary two-dimensional multi-element airfoils

Abstracts are presented on a method of numerical solution of the Navier-Stokes equation for the flow about arbitrary airfoils, using a numerically generated curvilinear coordinate system having a coordinate line coincident with the body contour. Results of continuing research are reported and include: application of the Navier-Stokes solution in the vorticity-stream function formulation to a number of single airfoils at Reynolds numbers up to 2000; programming of the Navier-Stokes solution for multiple airfoils in the primitive variable formulation; testing of the potential flow solution of multiple bodies; and development of a generalized coordinate system program.

Thompson, J. F.↗

NASA NTRS1975

Numerical solution of the Navier-Stokes equations for arbitrary two-dimensional airfoils

A method for numerical solution of the Navier-Stokes equations for the flow about arbitrary airfoils or other bodies is presented. This method utilizes a numerically generated curvilinear coordinate system having a coordinate line coincident with the body contour. Streamlines, velocity profiles, and pressure and force coefficients for several airfoils and an arbitrary rock are given. Potential flow solutions are also presented. The procedure capable of treating multiple-element airfoils, and potential flow results are presented.

Thames, F. C.↗

NASA NTRS1975

Numerical calculation of the internal flow field in a centrifugal compressor impeller

An iterative numerical method has been developed for the calculation of steady, three-dimensional, viscous, compressible flow fields in centrifugal compressor impellers. The computer code, which embodies the method, solves the steady three dimensional, compressible Navier-Stokes equations in rotating, curvilinear coordinates. The solution takes place on blade-to-blade surfaces of revolution which move from the hub to the shroud during each iteration.

Walitt, L.↗

NASA NTRS1977

Geometrical meaning of the curl operation when A dot curl A does not equal 0

We examine the geometrical interpretation of the vector operation curl for vector fields A such that A dot curl A does not equal 0. It is not possible to construct even locally a surface normal to field lines of A at all points; an explicit geometrical demonstration of this fact is given. Such fields do not admit the construction of orthogonal curvilinear coordinates in which one set of coordinate curves lies along field lines of A.

Barnes, A.↗

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

The computation of incompressible, three-dimensional flows in containers with extensible walls

The theoretical study of flow in biological vascular systems is made very difficult in connection with local and temporal changes of the cross section. Experimental investigations with the aid of model tests are, however, not enough for a solution of the problems, and numerical solutions are more and more employed for a description of such flows. A description is presented of a difference procedure for the solution of the complete Navier-Stokes equations with curvilinear coordinates for three-dimensional flows in containers. The integration of the Navier-Stokes equations for flows in containers with rigid and moving walls is considered.

Liu, N. S.↗

NASA NTRS1979

A computational model for three-dimensional incompressible wall jets with large cross flow

A computational model for the flow field of three dimensional incompressible wall jets prototypic of thrust augmenting ejectors with large cross flow is presented. The formulation employs boundary layer equations in an orthogonal curvilinear coordinate system. Simulation of laminar as well as turbulen wall jets is reported. Quantification of jet spreading, jet growth, nominal separation, and jet shrink effects due to corss flow are discussed.

Murphy, W. D.↗

NASA NTRS1980

A numerical technique for calculation of the noise of high-speed propellers with advanced blade geometry

A numerical technique and computer program were developed for the prediction of the noise of propellers with advanced geometry. The blade upper and lower surfaces are described by a curvilinear coordinate system, which was also used to divide the blade surfaces into panels. Two different acoustic formulations in the time domain were used to improve the speed and efficiency of the noise calculations: an acoustic formualtion with the Doppler factor singularity for panels moving at subsonic speeds and the collapsing sphere formulation for panels moving at transonic or supersonic speeds. This second formulation involves a sphere which is centered at the observer position and whose radius decreases at the speed of sound. The acoustic equation consisted of integrals over the curve of intersection for both the sphere and the panels on the blade. Algorithms used in some parts of the computer program are discussed. Comparisons with measured acoustic data for two model high speed propellers with advanced geometry are also presented.

Nystrom, P. A.↗

NASA NTRS1980

Numerical method for predicting flow characteristics and performance of nonaxisymmetric nozzles. Part 2: Applications

A computer implemented numerical method for predicting the flow in and about an isolated three dimensional jet exhaust nozzle is summarized. The approach is based on an implicit numerical method to solve the unsteady Navier-Stokes equations in a boundary conforming curvilinear coordinate system. Recent improvements to the original numerical algorithm are summarized. Equations are given for evaluating nozzle thrust and discharge coefficient in terms of computed flowfield data. The final formulation of models that are used to simulate flow turbulence effect is presented. Results are presented from numerical experiments to explore the effect of various quantities on the rate of convergence to steady state and on the final flowfield solution. Detailed flowfield predictions for several two and three dimensional nozzle configurations are presented and compared with wind tunnel experimental data.

Thomas, P. D.↗

NASA NTRS1980

A diagonal form of an implicit approximate-factorization algorithm with application to a two dimensional inlet

A modification of an implicit approximate-factorization finite-difference algorithm applied to the two dimensional Euler and Navier-Stokes equations in general curvilinear coordinates is presented for supersonic free stream flow about and through inlets. The modification transforms the coupled system of equations into an uncoupled diagonal form which requires less computation work. For steady-state applications the resulting diagonal algorithm retains the stability and accuracy characteristics of the original algorithm. Solutions are given for inviscid and laminar flow about a two dimensional wedge inlet configuration. Comparisons are made between computed results and exact theory.

Chaussee, D. S.↗

NASA NTRS1980

Forebody and base region real-gas flow in severe planetary entry by a factored implicit numerical method. I - Computational fluid dynamics

A new code for the simulation of full (forebody and base region) flowfields about bluff bodies in the hypersonic regime of severe planetary entry is described. The present 'maximally conservative, maximally differenced' formulation of the unsteady compressible Navier-Stokes equations for 2-D axisymmetric 3-D flow is contrasted for stability with previous formulations of Viviand, Kutler, et al, and Thomas and Lombard. Discrete metric relations peculiar to the axisymmetric finite volume formulation are presented along with a general discussion of their relations to and consequences of failure to close computational cells. A computational mesh of curvilinear coordinate topology singular in the flow regime is presented that permits aligned capturing of the major physical features of the complex flowfield.

Lombard, C. K.↗

NASA NTRS1980

Effect of pressure gradients on Gortler instability

Gortler instability for boundary-layer flows over generally curved walls is considered. The full linearized disturbance equations are obtained in an orthogonal curvilinear coordinate system. A perturbation procedure to account for second-order effects is used to determine the effects of the displacement thickness and the variation of the streamline curvature on the neutral stability of the Blasius flow. The pressure gradient in the mean flow is accounted for by solving the nonsimilar boundary-layer equations. Growth rates are obtained for the actual mean flow and the Falkner-Skan flows. The results demonstrate the strong influence of the pressure gradient and the nonsimilarity of the basic flow on the stability characteristics.

Ragab, S. A.↗

NASA NTRS1982

Numerical Solution for Navier-Stokes Equations

Carefully selected blend of computational techniques solves complete set of equations for viscous, unsteady, hypersonic flow in general curvilinear coordinates. New algorithm has tested computation of axially directed flow about blunt body having shape similar to that of such practical bodies as wide-body aircraft or artillery shells. Method offers significant computational advantages because of conservation-law form of equations and because it reduces amount of metric data required.

Warsi, Z. U. A.↗

NASA NTRS1980

A quadrilateralized spherical cube Earth data base

A quadrilateralized spherical cube was constructed to form the basis for the rapid storage and retrieval of high resolution data obtained of the Earth's surface. The structure of this data base was derived from a spherical cube, which was obtained by radially projecting a cube onto its circumscribing sphere. An appropriate set of curvilinear coordinates were chosen such that the resolution cells on the spherical cube were of equal area and were also of essentially the same shape. The main properties of the Earth data base were that the indexing scheme was binary and telescopic in nature, the resolution cells were strung together in a two dimensional manner, the cell addresses were easily computed, and the conversion from geographic to data base coordinates was comparatively simple. It was concluded that this data base structure was perhaps the most viable one for handling remotely sensed data obtained by satellites.

Chan, F. K.↗

NASA NTRS1980

On the solution of the unsteady Navier-Stokes equations for hypersonic flow about axially-symmetric blunt bodies

A formulation of the complete Navier-Stokes problem for a viscous hypersonic flow in general curvilinear coordinates is presented. This formulation is applicable to both the axially symmetric and three dimensional flows past bodies of revolution. The equations for the case of zero angle of attack were solved past a circular cylinder with hemispherical caps by point SOR finite difference approximation. The free stream Mach number and the Reynolds number for the test case are respectively 22.04 and 168883. The whole algorithm is presented in detail along with the preliminary results for pressure, temperature, density and velocity distributions along the stagnation line.

Warsi, Z. U. A.↗

NASA NTRS1980

Grid generation using differential systems techniques

The errors in approximating the derivatives of a function by traditional central differences at grid points of a curvilinear coordinate system were examined. The implications concerning the accuracy of the numerical solution of a partial differential equation are explained by considering several numerical examples. Although this study only considers the two dimensional case, the techniques and implications are equally valid for three dimensional grids. An interesting feature of the error analysis is its simplicity. Most of the results follow by merely working with the truncation terms of some power series expansion. These series expansions also give rise to higher order difference approximations which can significantly reduce error when the grid spacing changes rapidly, as might be the case in problems with shock waves or thin boundary layers.

Thompson, J. F.↗

NASA NTRS1981

A diagonal form of an implicit approximate-factorization algorithm

A modification of an implicit approximate-factorization finite-difference algorithm applied to partial differential equations is presented. This algorithm is applied to the two- and three-dimensional Euler equations in general curvilinear coordinates. The modification transforms the coupled system of equations into an uncoupled diagonal form that requires less computational work. For steady-state applications, the resulting diagonal algorithm retains the stability and accuracy characteristics of the original algorithm. The diagonal algorithm reduces the storage requirement of the implicit solution process and therefore has an important effect on the application of implicit finite-difference schemes to vector processors. Results are presented for realistic two-dimensional transonic flow fields about airfoils. Computation costs are reduced to 24-34%.

Pulliam, T. H.↗