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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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NASA NTRS1976

A systematic method for computer design of supercritical airfoils in cascade

A computer code has been developed for the direct calculation of shockless transonic airfoils whose pressure distributions can be assigned within reasonable limits. The partial differential equations of two-dimensional inviscid gas dynamics are solved by analytic continuation into the domain of two independent complex characteristic coordinates. The domain of integration is mapped conformally onto the unit circle in the hodograph plane of one of these coordinates. It is possible to formulate a boundary value problem on this circle for the stream function that is well posed in the case of transonic flow. This enables the formulation of a procedure for the calculation of an airfoil on which the speed is prescribed as a function of the arc length

Garabedian, P.↗

NASA NTRS1977

Numerical calculation of the transonic flow past a swept wing

A numerical method is presented for analyzing the transonic potential flow past a lifting, swept wing. A finite difference approximation to the full potential equation is solved in a coordinate system which is nearly conformally mapped from the physical space in planes parallel to the symmetry plane, and reduces the wing surface to a portion of one boundary of the computational grid. A coordinate invariant, rotated difference scheme is used, and the difference equations are solved by relaxation. The method is capable of treating wings of arbitrary planform and dihedral, although approximations in treating the tips and vortex sheet make its accuracy suspect for wings of small aspect ratio. Comparisons of calculated results with experimental data are shown for examples of both conventional and supercritical transport wings. Agreement is good for both types, but it was found necessary to account for the displacement effect of the boundary layer for the supercritical wing, presumably because of its greater sensitivity to changes in effective geometry.

Jameson, A.↗

NASA NTRS1975

Design of high lift airfoils with a Stratford distribution by the Eppler method

Airfoils having a Stratford pressure distribution, which has zero skin friction in the pressure recovery area, were investigated in an effort to develop high lift airfoils. The Eppler program, an inverse conformal mapping technique where the x and y coordinates of the airfoil are developed from a given velocity distribution, was used.

Thomson, W. G.↗

NASA NTRS1977

Two-body coordinate system generation using body-fitted coordinate system and complex variable transformation

Attempts are made to generate acceptable coordinate systems for two-body configurations. The first method to be tried was to use the body-fitted coordinate system technique to obtain the best system. This technique alone did not produce very good results, so another approach was investigated. This new approach involved using a combination of the body fitted coordinate system procedure and a complex variable transformation method that was used successfully in conformal mapping.

Long, W. S.↗

NASA NTRS1978

An efficient, direct finite difference method for computing sound propagation in arbitrarily shaped two-dimensional and axisymmetric ducts without flow

An efficient, direct finite difference method is presented for computing sound propagation in non-stepped two-dimensional and axisymmetric ducts of arbitrarily varying cross section without mean flow. The method is not restricted by axial variation of acoustic impedance of the duct wall linings. The non-uniform two-dimensional or axisymmetric duct is conformally mapped numerically into a rectangular or cylindrical computational domain using a new procedure based on a method of fast direct solution of the Cauchy-Riemann equations. The resulting Helmholtz equation in the computational domain is separable. The solution to the governing equation and boundary conditions is expressed as a linear combination of fundamental solutions. The fundamental solutions are computed only once for each duct shape by means of the fast direct cyclic reduction method for the discrete solution of separable elliptic equations. Numerical results for several examples are presented to show the applicability and efficiency of the method.

Chakravarthy, S.↗

NASA NTRS1978

An exact derivation of contact resistance to planar devices

A mixed boundary-value problem is formulated for an imperfect rectangular contact to a semiconducting sheet (layer). An exact model for derivation of contact resistance is developed and its solution by conformal mapping and eigenfunction expansion is presented. The expression for contact resistance is similar to that for a perfect (lossless) contact but includes an additional term containing the lowest-order coefficient in the infinite series expansion of the complex potential function. The calculated contact resistances are compared with those obtained from three approximate models: lossless contact, transmission line, and extended transmission line models. The extended transmission line model appears to be a very satisfactory approximation provided the ratio of contact length to sheet thickness is no less than 0.5.

Schuldt, S. B.↗

NASA NTRS1979

New dominance characteristics for the multivariable Nyquist array method

Three new dominance characteristics are introduced to the multivariable Nyquist array methods. The first characteristic utilizes a conformal mapping procedure to establish a new set of bands in the 'image' plane to assist in feedback gain selection and stability assessment. The second characteristic uses the 'image' band concept to provide a theoretical foundation for finite frequency dominance considerations, thus removing the restrictive requirement of dominance for all s on the Nyquist D contour. The third characteristic provides for the sharing of system dominance among the feedback control loops. The dominance sharing concept may be used to establish dominance and/or to improve the dominance condition in prespecified feedback loops.

Leininger, G. G.↗

NASA NTRS1978

Analysis of solidification interface shape during continuous casting of a slab

An analysis was made of the two-dimensional interface shape of a slab ingot being cast continuously by withdrawing it from a mold. The sides of the ingot are being cooled and the upper boundary of the ingot is in contact with a pool of molten metal. The solidification interface shape was determined from the analysis of the heat flow, utilizing the condition that the solidification interface is at constant temperature and must be normal to the lines of heat flow carrying away latent heat of fusion from the interface. The analysis includes the effect of interface curvature which, for a constant rate of withdrawing the cast ingot from the mold, causes the solidification to be nonuniform along the interface. The analysis was carried out by a conformal mapping procedure.

Siegel, R.↗

NASA NTRS1981

The design and analysis of low-speed airfoils

PROFILE program solves diverse and inverse airfoil-flow problems. It combines conformational mapping method for design of airfoils with prescribed velocity-distribution characteristics, panel method for potential-flow analysis, and boundary-layer method. PROFILE is written in FORTRAN IV for implementation on CDC 6000-series computer.

Eppler, R.↗

NASA NTRS1980

Numerical simulation of shear flows over wavy boundaries

The full time-dependent two dimensional Navier-Stokes equations are solved using spectral methods to achieve high spatial accuracy and high order time-splitting methods together with conformal mapping methods to allow simulation of flow over steep waves, including general turbulent Reynolds stresses. The programs are used to study the properties of laminar and turbulent flow over a flat plate and in various wavy geometries. Results are compared with both available theory and experiment. For laminar flow, good agreement is achieved. For turbulent flow, the agreement is less satisfactory and may be explained by the simplified turbulent modelling use in the calculations.

Orszag, S. A.↗

NASA NTRS1980

CAS2D: FORTRAN program for nonrotating blade-to-blade, steady, potential transonic cascade flows

An exact, full-potential-equation (FPE) model for the steady, irrotational, homentropic and homoenergetic flow of a compressible, homocompositional, inviscid fluid through two dimensional planar cascades of airfoils was derived, together with its appropriate boundary conditions. A computer program, CAS2D, was developed that numerically solves an artificially time-dependent form of the actual FPE. The governing equation was discretized by using type-dependent, rotated finite differencing and the finite area technique. The flow field was discretized by providing a boundary-fitted, nonuniform computational mesh. The mesh was generated by using a sequence of conforming mapping, nonorthogonal coordinate stretching, and local, isoparametric, bilinear mapping functions. The discretized form of the FPE was solved iteratively by using successive line overrelaxation. The possible isentropic shocks were correctly captured by adding explicitly an artificial viscosity in a conservative form. In addition, a three-level consecutive, mesh refinement feature makes CAS2D a reliable and fast algorithm for the analysis of transonic, two dimensional cascade flows.

Dulikravich, D. S.↗

NASA NTRS1980

A computer program for the design and analysis of low-speed airfoils

A conformal mapping method for the design of airfoils with prescribed velocity distribution characteristics, a panel method for the analysis of the potential flow about given airfoils, and a boundary layer method have been combined. With this combined method, airfoils with prescribed boundary layer characteristics can be designed and airfoils with prescribed shapes can be analyzed. All three methods are described briefly. The program and its input options are described. A complete listing is given as an appendix.

Eppler, R.↗

NASA NTRS1980

Oblique-incidence secondary emission from charged dielectrics

Secondary electron emission coefficients were measured on FEP-Teflon for normal and oblique incidence in the presence of a normal electric field. Such measurements require knowledge of the electrostatic environment surrounding the specimen, and they require calculation of particle trajectories such that particle impact parameters can be known. A simulation using a conformal mapping, a Green's integral, and a trajectory generator provides the necessary mathematical support for the measurements, which were made with normal fields of 1.5 and 2.7 kV/mm. When incidence is normal and energy exceeds the critical energy, the coefficient is given by (V sub 0/V) to the .58 power, and for oblique incidence this expression may be divided by the cosine of the angle. The parameter V sub 0 is a function of normal field.

Robinson, J. W.↗

NASA NTRS1980

Numerical grid generation techniques

The state of the art in topology and flow geometry is presented. Solution techniques for partial differential equations are reviewed and included developments in coordinate transformations, conformal mapping, and invariant imbeddings. Applications of these techniques in fluid mechanics, flow geometry, boundary value problems, and fluidics are presented.

Source record↗

NASA NTRS1980

Generation of C-type cascade grids for viscous flow computation

A rapid procedure for generating C-type cascade grids suitable for viscous flow computations in turbomachinery blade rows is presented. The resulting mesh is periodic from one blade passage to the next, nearly orthogonal, and continuous across the wake downstream of a blade. The procedure employs a pair of conformal mappings that take the exterior of the cascade into the interior of an infinite strip with curved boundaries. The final transformation to a rectangular computational domain is accomplished numerically. The boundary values are obtained from a panel solution of an integral equation and the interior values by a rapid ADI solution of Laplace's equation. Examples of C-type grids are presented for both compressor and turbine blades and the extension of the procedure to three dimensions is briefly outlined.

Sockol, P. M.↗

NASA NTRS1980

Grid generation for two-dimensional finite element flowfield computation

The finite element method for fluid dynamics was used to develop a two dimensional mesh generation scheme. The method consists of shearing and conformal maps with upper and lower surfaces handled independently to allow sharp leading edges. The method also generates meshes of triangular or quadrilateral elements.

Tatum, K. E.↗

NASA NTRS1981

Studies of compressible shear flows and turbulent drag reduction

Compressible shear flows and drag reduction were examined and three methods are addressed: (1) the analytical and numerical aspects of conformal mapping were summarized and a new method for computation of these maps is presented; (2) the computer code SPECFD for solution of the three dimensional time dependent Navier-Stokes equations for compressible flow on the CYBER 203 computer is described; (3) results of two equation turbulence modeling of turbulent flow over wavy walls are presented. A modified Jones-Launder model is used in two dimensional spectral code for flow in general wavy geometries.

Orszag, S. A.↗

NASA NTRS1981

Computation of separated flows by a vortex-tracing algorithm

Numerical solutions for two-dimensional, time-dependent, separated flows around bodies are obtained, using a new version of the vortex method. This method provides an efficient representation of flows involving large regions of separation. The modifications incorporated in the new version, which improve its accuracy, versatility, and computing speed, are described. The computer cost is only of the order of the 3/2 power on N, instead of N-squared, for each step with N vortices. Arbitrary shapes can be treated; a conformal mapping is not required. Special attention is paid to the viscous character of the solution and to the accurate computation of the pressure distribution at the body surface. The vortex solution for the outer flow is coupled to an inner solution for the attached part of the boundary layer. Numerical results are presented for several bluff bodies exhibiting dependence on Reynolds number, for stationary airfoils under steady or transient conditions and for oscillating airfoils, including dynamic stalls. These results are compared with other available results, analytical or experimental, and demonstrate the enhanced reliability and accuracy of the improved method.

Spalart, P. R.↗