Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “curvilinear coordinates”

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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 145 records · Page 8
All sourcesNASA NTRSDOE OSTI
NASA NTRS1981

A PANSONIC Navier-Stokes solver

A finite-difference formulation of the full Navier-Stokes equations which demonstrates a capability to economically solve two-dimensional problems has been developed. The basic algorithm was derived from the full, Reynolds-averaged, conservative, Navier-Stokes equations expressed in curvilinear coordinates. Eddy viscosity was determined by the Baldwin and Lomax algebraic turbulence model. This non-iterative, second-order accurate, implicit, numerical algorithm is based on the approximate factorization finite-difference scheme of Beam and Warming. Results indicate a facility for solving subsonic, transonic, and supersonic (hence PANSONIC) flows about arbitrary airfoils for a wide range of Reynolds numbers, Mach numbers, and angles of attack. Current computations demonstrate that vectorized implementations of this algorithm can solve steady-state, two-dimensional problems in five to ten minutes of computer time.

Cooper, G. K.↗
NASA NTRS1981

Numerical method for gas dynamics combining characteristic and conservation concepts

An efficient implicit numerical method that solves the compressible Navier-Stokes equations in arbitrary curvilinear coordinates by the finite-volume technique is presented. An intrinsically dissipative difference scheme and a fully implicit treatment of boundary conditions, based on characteristic and conservation concepts, are used to improve stability and accuracy. Efficiency is achieved by using a diagonal form of the implicit algorithm and spatially varying time-steps. Comparisons of various schemes and methods are presented for one- and two-dimensional flows, including transonic separated flow past a thick circular-arc airfoil in a channel. The new method is equal to or better than a version of MacCormack's hybrid method in accuracy and it converges to a steady state up to an order of magnitude faster.

Coakley, T. J.↗
NASA NTRS1981

Goertler instability

Goertler 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 streamwise 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 compared with those for the Blasius flow and the Falkner-Skan flows. The results demonstrate the strong influence of the streamwise pressure gradient and the nonsimilarity of the basic flow on the stability characteristics.

Ragab, S. A.↗
NASA NTRS1982

Numerical solution of potential flow about arbitrary 2-dimensional multiple bodies

A procedure for the finite-difference numerical solution of the lifting potential flow about any number of arbitrarily shaped bodies is given. The solution is based on a technique of automatic numerical generation of a curvilinear coordinate system having coordinate lines coincident with the contours of all bodies in the field, regardless of their shapes and number. The effects of all numerical parameters involved are analyzed and appropriate values are recommended. Comparisons with analytic solutions for single Karman-Trefftz airfoils and a circular cylinder pair show excellent agreement. The technique of application of the boundary-fitted coordinate systems to the numerical solution of partial differential equations is illustrated.

Thompson, J. F.↗
NASA NTRS1982

Computations of transonic flow over an oscillating airfoil with shock-induced separation

A numerical study is presented on unsteady transonic flow over an airfoil in pitching oscillation at a moderately high incidence. The study was focused on the effects of unsteady shock-wave/boundary-layer interactions and subsequent flow separations. An implicit finite-difference scheme was used to solve the full time-dependent compressible Navier-Stokes equations in conservation-law form. The algorithm adopted for this study is the Beam-Warming spatially factored scheme. It is, however, constructed in a body-fitted curvilinear coordinate system for efficient unsteady flow computations. The computational results are compared with experimental data measured in the Ames 11- by 11-Foot Transonic Wind Tunnel.

Chyu, W. J.↗
NASA NTRS1982

Approximate factorization with an elliptic pressure solver for incompressible flow

Two-dimensional curvilinear coordinates are used to solve the incompressible Navier-Stokes equations, in conjunction with approximate factorization for the solution of the momentum equation and the successive overrelaxation by lines method for the solution of a Poisson equation for the pressure. The combined algorithm, although not fully explicit, is marginally stable at Reynolds numbers lower than 10,000 and time increments of 0.01. Pressure distributions calculated for attack angles of zero and 6 deg are of the same shape as the experimental curves, but are shifted to one side.

Bernard, R. S.↗
NASA NTRS1982

Implicit marching solution of compressible viscous subsonic flow in planar and axisymmetric ducts

A new streamwise marching procedure was developed and coded for compressible viscous subsonic flow in planar or axisymmetric ducts with or without centerbodies. The continuity, streamwise momentum, cross-flow momentum, and energy equations are written in generalized orthogonal curvilinear coordinates. To allow the use of a marching procedure, second derivatives in the streamwise momentum equation are written as the sum of a known two dimensional imposed pressure field and an unknown one dimensional viscous correction. For turbulent flow, the Reynolds stress and heat flux terms are modeled using two-layer eddy viscosity turbulence models.

Towne, C. E.↗
NASA NTRS1983

Transformation of two and three-dimensional regions by elliptic systems

Finite difference methods for composite grids were analyzed. It was observed that linear interpolation between grids would suffice only where low order accuracy was required. In the context of fluid flow, this would be in regions where the flow was essentially free stream. Higher order interpolation schemes were also investigated. The well known quadratic and cubic interpolating polynomials would increase the formal accuracy of the overall numerical algorithm. However, it can also be shown that the stability of the algorithm may be adversely affected. Further numerical results are needed in order to assess the nature of this instability induced by the interpolation procedure. Finally, error analysis and the order of difference expressions on general curvilinear coordinates are discussed.

Thompson, J. F.↗
NASA NTRS1982

Goertler instability in compressible boundary layers along curved surfaces with suction and cooling

The Goertler instability of the laminar compressible boundary layer flows along concave surfaces is investigated. The linearized disturbance equations for the three-dimensional, counter-rotating streamwise vortices in two-dimensional boundary layers are presented in an orthogonal curvilinear coordinate. The basic approximation of the disturbance equations, that includes the effect of the growth of the boundary layer, is considered and solved numerically. The effect of compressibility on critical stability limits, growth rates, and amplitude ratios of the vortices is evaluated for a range of Mach numbers for 0 to 5. The effect of wall cooling and suction of the boundary layer on the development of Goertler vortices is investigated for different Mach numbers.

El-Hady, N.↗
NASA NTRS1982

Solution of viscous internal flows on curvilinear grids generated by the Schwarz-Christoffel transformation

The combination of an orthogonal, curvilinear coordinate generation procedure with a stable forward marching viscous flow solution technique is presently employed in the solution of flow fields for arbitrary, axisymmetric ducts. Coordinate generation is accomplished by means of both potential lines and plane potential flow streamlines. Since the coordinate streamlines approximate actual ones, the equations of motion for viscous compressible flow can be parabolized in order to solve for both the boundary layer and the core flow in a single streamwise pass. The method's versatility is demonstrated by two examples of viscous compressible swirling flow through complex radial gas turbine passages.

Anderson, O. L.↗
NASA NTRS1983

A zonal approach for the steady transonic simulation of inviscid rotational flow

A finite difference zonal method is developed to compute steady inviscid transonic flow by coupling a semi-flux split form of the Euler equations in a vorticity producing zone with a zone of scalar and vector (i.e., dual) potential equations. The dual potential equations permit vorticity convection, but not production, and are efficiently solved as an iteratively decoupled set of scalar equations. Zonal results presented for a nonlifting biconvex airfoil on a stretched Cartesian grid show substantial savings in CPU time compared to solving the semi-flux split Euler equations alone. The dual potential equations also provide an alternate way of treating potential flows with circulation. This has been demonstrated by computing a subcritical flow over a lifting airfoil using generalized curvilinear coordinates.

Chaderjian, N. M.↗
NASA NTRS1983

Three-dimensional implicit lambda methods

This paper derives the three dimensional lambda-formulation equations for a general orthogonal curvilinear coordinate system and provides various block-explicit and block-implicit methods for solving them, numerically. Three model problems, characterized by subsonic, supersonic and transonic flow conditions, are used to assess the reliability and compare the efficiency of the proposed methods.

Napolitano, M.↗
NASA NTRS1983

Three-dimensional unsteady Euler equation solutions using flux vector splitting

A method for numerically solving the three dimensional unsteady Euler equations using flux vector splitting is developed. The equations are cast in curvilinear coordinates and a finite volume discretization is used. An explicit upwind second-order predictor-corrector scheme is used to solve the discretized equations. The scheme is stable for a CFL number of two and local time stepping is used to accelerate convergence for steady-state problems. Characteristic variable boundary conditions are developed and used in the far field and at surfaces. No additional dissipation terms are included in the scheme. Numerical results are compared with results from an existing three dimensional Euler code and experimental data.

Whitfield, D. L.↗
NASA NTRS1983

Numerical solution of the Navier-Stokes equations for arbitrary 2-dimensional multi-element airfoils

Numerical solutions of the Navier-Stokes equations, with an algebraic turbulence model, for time-dependent two dimensional flow about multi-element airfoils were developed. Fundamental to these solutions was the use of numerically-generated boundary-conforming curvilinear coordinate systems to allow bodies of arbitrary shape to be treated. A general two dimensional grid generation code for multiple-body configuration was written as a part of this project and made available through the COSMIC code library.

Thompson, J. F.↗
NASA NTRS1984

A conservative treatment of zonal boundaries for Euler equation calculations

Finite-difference calculations require the generation of a grid for the region of interest. A zonal approach, wherein the given region is subdivided into zones and the grid for each zone is generated independently, makes the grid-generation process for complicated topologies and for regions requiring selective grid refinement a fairly simple task. This approach results in new boundaries within the given region, that is, zonal boundaries at the interfaces of the various zones. The zonal-boundary scheme (the integration scheme used to update the points on the zonal boundary) for the Euler equations must be conservative, accurate, stable, and applicable to general curvilinear coordinate systems. A zonal-boundary scheme with these desirable properties is developed in this study. The scheme is designed for explicit, first-order-accurate integration schemes but can be modified to accommodate second-order-accurate explicit and implicit integration schemes. Results for inviscid flow, including supersonic flow over a cylinder, blast-wave diffraction by a ramp, and one-dimensional shock-tube flow are obtained on zonal grids. The conservative nature of the zonal-boundary scheme permits the smooth transition of the discontinuities associated with these flows from one zone to another. The calculations also demonstrate the continuity of contour lines across zonal boundaries that can be achieved with the present zonal scheme.

Rai, M. M.↗
NASA NTRS1984

A Newton multigrid method for the Euler equations

A multigrid method is used to apply Newton's method to the Euler equations in a two dimensional curvilinear coordinate system. The objective is to obtain rapid convergence for steady state problems. Solutions computed with the method evolve in a non-time-like manner. Stable pressure distributions typically develop in eight to ten Newton-multigrid steps, which is equivalent to the computational work of about 70 iterations with a factored implicit algorithm.

Childs, R. E.↗
NASA NTRS1983

Static shape forming for an electrostatically controlled membrane mirror

A design concept is described permitting the static construction of a flexible membrane reflector for use in a radiometer spacecraft. The scheme utilizes electrostatic forces to actively shape the membrane toward a specified reflector surface with a required degree of precision. The formulations of the required electrostatic attractive forces, and stress and strain distributed in the membrane are presented in this paper in terms of curvilinear coordinates. An example is given to illustrate a variety of spherical membrane characteristics associated with different boundary conditions.

Juang, J. N.↗
NASA NTRS1984

Three-dimensional unsteady Euler equations solution using flux vector splitting

A method for numerically solving the three-dimensional unsteady Euler equations using flux vector splitting is developed. The equations are cast in curvilinear coordinates and a finite volume discretization is used. An explicit upwind second-order predictor-corrector scheme is used to solve the discretized equations. The scheme is stable for a CFL number of 2 and local time stepping is used to accelerate convergence for steady-state problems. Characteristic variable boundary conditions are developed and used in the far-field and at surfaces. No additional dissipation terms are included in the scheme. Numerical results are compared with results from an existing three-dimensional Euler code and experimental data.

Whitfield, D. L.↗