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NASA NTRS1993

A general theory of two- and three-dimensional rotational flow in subsonic and transonic turbomachines

This report represents a general theory applicable to axial, radial, and mixed flow turbomachines operating at subsonic and supersonic speeds with a finite number of blades of finite thickness. References reflect the evolution of computational methods used, from the inception of the theory in the 50's to the high-speed computer era of the 90's. Two kinds of relative stream surfaces, S(sub 1) and S(sub 2), are introduced for the purpose of obtaining a three-dimensional flow solution through the combination of two-dimensional flow solutions. Nonorthogonal curvilinear coordinates are used for the governing equations. Methods of computing transonic flow along S(sub 1) and S(sub 2) stream surfaces are given for special cases as well as for fully three-dimensional transonic flows. Procedures pertaining to the direct solutions and inverse solutions are presented. Information on shock wave locations and shapes needed for computations are discussed. Experimental data from a Deutsche Forschungs- und Versuchsanstalt fur Luft- und Raumfahrt e.V. (DFVLR) rotor and from a Chinese Academy of Sciences (CAS) transonic compressor rotor are compared with the computed flow properties.

Wu, Chung-Hua↗

NASA NTRS1995

The National Grid Project: A system overview

The National Grid Project (NGP) is a comprehensive numerical grid generation software system that is being developed at the National Science Foundation (NSF) Engineering Research Center (ERC) for Computational Field Simulation (CFS) at Mississippi State University (MSU). NGP is supported by a coalition of U.S. industries and federal laboratories. The objective of the NGP is to significantly decrease the amount of time it takes to generate a numerical grid for complex geometries and to increase the quality of these grids to enable computational field simulations for applications in industry. A geometric configuration can be discretized into grids (or meshes) that have two fundamental forms: structured and unstructured. Structured grids are formed by intersecting curvilinear coordinate lines and are composed of quadrilateral (2D) and hexahedral (3D) logically rectangular cells. The connectivity of a structured grid provides for trivial identification of neighboring points by incrementing coordinate indices. Unstructured grids are composed of cells of any shape (commonly triangles, quadrilaterals, tetrahedra and hexahedra), but do not have trivial identification of neighbors by incrementing an index. For unstructured grids, a set of points and an associated connectivity table is generated to define unstructured cell shapes and neighboring points. Hybrid grids are a combination of structured grids and unstructured grids. Chimera (overset) grids are intersecting or overlapping structured grids. The NGP system currently provides a user interface that integrates both 2D and 3D structured and unstructured grid generation, a solid modeling topology data management system, an internal Computer Aided Design (CAD) system based on Non-Uniform Rational B-Splines (NURBS), a journaling language, and a grid/solution visualization system.

Gaither, Adam↗

NASA NTRS1995

A comparative study of upwind and MacCormack schemes for CAA benchmark problems

In this study, upwind schemes and MacCormack schemes are evaluated as to their suitability for aeroacoustic applications. The governing equations are cast in a curvilinear coordinate system and discretized using finite volume concepts. A flux splitting procedure is used for the upwind schemes, where the signals crossing the cell faces are grouped into two categories: signals that bring information from outside into the cell, and signals that leave the cell. These signals may be computed in several ways, with the desired spatial and temporal accuracy achieved by choosing appropriate interpolating polynomials. The classical MacCormack schemes employed here are fourth order accurate in time and space. Results for categories 1, 4, and 6 of the workshop's benchmark problems are presented. Comparisons are also made with the exact solutions, where available. The main conclusions of this study are finally presented.

Viswanathan, K.↗

NASA NTRS1995

Computational aeroacoustics on massively parallel computers

A computational fluid dynamics (CFD) technique is employed to solve aeroacoustics problems on massively parallel computers. The algorithm is based on a 4th order accurate central finite difference and 4th order accurate 4-stage Runge-Kutta time integration method that solves the 3-D full Navier-Stokes/Euler equations on a curvilinear coordinate system. The code has been developed to predict noise radiation from ducted fans. A specific application of this code is made here to the oscillating circular piston problem (Workshop Category 4, Problem 2) by solving the full Euler equations with nonreflecting boundary conditions in the axisymmetric mode. The features of the code that apply to this particular problem are described in this paper.

Oezyoeruek, Yusuf↗

NASA NTRS1995

Numerical Studies of Boundary-Layer Receptivity

Direct numerical simulations (DNS) of the acoustic receptivity process on a semi-infinite flat plate with a modified-super-elliptic (MSE) leading edge are performed. The incompressible Navier-Stokes equations are solved in stream-function/vorticity form in a general curvilinear coordinate system. The steady basic-state solution is found by solving the governing equations using an alternating direction implicit (ADI) procedure which takes advantage of the parallelism present in line-splitting techniques. Time-harmonic oscillations of the farfield velocity are applied as unsteady boundary conditions to the unsteady disturbance equations. An efficient time-harmonic scheme is used to produce the disturbance solutions. Buffer-zone techniques have been applied to eliminate wave reflection from the outflow boundary. The spatial evolution of Tollmien-Schlichting (T-S) waves is analyzed and compared with experiment and theory. The effects of nose-radius, frequency, Reynolds number, angle of attack, and amplitude of the acoustic wave are investigated. This work is being performed in conjunction with the experiments at the Arizona State University Unsteady Wind Tunnel under the direction of Professor William Saric. The simulations are of the same configuration and parameters used in the wind-tunnel experiments.

Reed, Helen L.↗

NASA NTRS1996

Multigrid, Fractional-Step Computation Of Flow

Speed of computer code solving three-dimensional Navier-Stokes equations of flow of incompressible fluid by fractional-step method increased significantly by use of multigrid procedures. In method, equations solved on general nonorthogonal curvilinear coordinate grid, using volume fluxes. At each time step, computations performed in two fractional steps. In first step, equations of conservation of momentum solved by use of gradient of pressure from previous time step via explicit approximate-factorization method, yielding approximate flow field that does not satisfy equation of conservation of mass. In second step, discrete Poisson-like equation with Neumann-type boundary conditions, formed by combining equations of conservation of momentum and mass, solved iteratively.

Kwak, Dochan↗

NASA NTRS1996

Multigrid Method for Modeling Multi-Dimensional Combustion with Detailed Chemistry

A highly accurate and efficient numerical method is developed for modeling 3-D reacting flows with detailed chemistry. A contravariant velocity-based governing system is developed for general curvilinear coordinates to maintain simplicity of the continuity equation and compactness of the discretization stencil. A fully-implicit backward Euler technique and a third-order monotone upwind-biased scheme on a staggered grid are used for the respective temporal and spatial terms. An efficient semi-coarsening multigrid method based on line-distributive relaxation is used as the flow solver. The species equations are solved in a fully coupled way and the chemical reaction source terms are treated implicitly. Example results are shown for a 3-D gas turbine combustor with strong swirling inflows.

Zheng, Xiaoqing↗

NASA NTRS1997

Low-Dispersion Scheme for Nonlinear Acoustic Waves in Nonuniform Flow

The linear dispersion-relation-preserving scheme and its boundary conditions have been extended to the nonlinear Euler equations. This allowed computing, a nonuniform flowfield and a nonlinear acoustic wave propagation in such a medium, by the same scheme. By casting all the equations, boundary conditions, and the solution scheme in generalized curvilinear coordinates, the solutions were made possible for non-Cartesian domains and, for the better deployment of the grid points, nonuniform grid step sizes could be used. It has been tested for a number of simple initial-value and periodic-source problems. A simple demonstration of the difference between a linear and nonlinear propagation was conducted. The wall boundary condition, derived from the momentum equations and implemented through a pressure at a ghost point, and the radiation boundary condition, derived from the asymptotic solution to the Euler equations, have proven to be effective for the nonlinear equations and nonuniform flows. The nonreflective characteristic boundary conditions also have shown success but limited to the nonlinear waves in no mean flow, and failed for nonlinear waves in nonuniform flow.

Baysal, Oktay↗

NASA NTRS1997

Algorithmic Extensions of Low-Dispersion Scheme and Modeling Effects for Acoustic Wave Simulation

Accurate computation of acoustic wave propagation may be more efficiently performed when their dispersion relations are considered. Consequently, computational algorithms which attempt to preserve these relations have been gaining popularity in recent years. In the present paper, the extensions to one such scheme are discussed. By solving the linearized, 2-D Euler and Navier-Stokes equations with such a method for the acoustic wave propagation, several issues were investigated. Among them were higher-order accuracy, choice of boundary conditions and differencing stencils, effects of viscosity, low-storage time integration, generalized curvilinear coordinates, periodic series, their reflections and interference patterns from a flat wall and scattering from a circular cylinder. The results were found to be promising en route to the aeroacoustic simulations of realistic engineering problems.

Kaushik, Dinesh K.↗

NASA NTRS1996

Analytic Solutions of the Vector Burgers Equation

The well-known analytical solution of Burgers' equation is extended to curvilinear coordinate systems in three dimensions by a method that is much simpler and more suitable to practical applications than that previously used. The results obtained are applied to incompressible flow with cylindrical symmetry, and also to the decay of an initially linearly increasing wind.

Nerney, Steven↗

NASA NTRS1997

Computation of Reacting Flows in Combustion Processes

The main objective of this research was to develop an efficient three-dimensional computer code for chemically reacting flows. The main computer code developed is ALLSPD-3D. The ALLSPD-3D computer program is developed for the calculation of three-dimensional, chemically reacting flows with sprays. The ALL-SPD code employs a coupled, strongly implicit solution procedure for turbulent spray combustion flows. A stochastic droplet model and an efficient method for treatment of the spray source terms in the gas-phase equations are used to calculate the evaporating liquid sprays. The chemistry treatment in the code is general enough that an arbitrary number of reaction and species can be defined by the users. Also, it is written in generalized curvilinear coordinates with both multi-block and flexible internal blockage capabilities to handle complex geometries. In addition, for general industrial combustion applications, the code provides both dilution and transpiration cooling capabilities. The ALLSPD algorithm, which employs the preconditioning and eigenvalue rescaling techniques, is capable of providing efficient solution for flows with a wide range of Mach numbers. Although written for three-dimensional flows in general, the code can be used for two-dimensional and axisymmetric flow computations as well. The code is written in such a way that it can be run in various computer platforms (supercomputers, workstations and parallel processors) and the GUI (Graphical User Interface) should provide a user-friendly tool in setting up and running the code.

Keith, Theo G., Jr.↗

NASA NTRS1981

Two-Dimensional Inlet Simulation Using a Diagonal Implicit Algorithm

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 freestream 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 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 24-34%.

Pulliam, T. H.↗

NASA NTRS1999

A Mixed Multi-Field Finite Element Formulation for Thermopiezoelectric Composite Shells

Analytical formulations are presented which account for the coupled mechanical, electrical, and thermal response of piezoelectric composite shell structures. A new mixed multi-field laminate theory is developed which combines "single layer" assumptions for the displacements along with layerwise fields for the electric potential and temperature. This laminate theory is formulated using curvilinear coordinates and is based on the principles of linear thermopiezoelectricity. The mechanics have the inherent capability to explicitly model both the active and sensory responses of piezoelectric composite shells in thermal environment. Finite element equations are derived and implemented for an eight-noded shell element. Numerical studies are conducted to investigate both the sensory and active responses of piezoelectric composite shell structures subjected to thermal loads. Results for a cantilevered plate with an attached piezoelectric layer are com- pared with corresponding results from a commercial finite element code and a previously developed program. Additional studies are conducted on a cylindrical shell with an attached piezoelectric layer to demonstrate capabilities to achieve thermal shape control on curved piezoelectric structures.

Lee, Ho-Jun↗

NASA NTRS1986

A Conservation 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, Man Mohan↗

NASA NTRS1987

Nonlinear Beam Kinematics by Decomposition of the Rotation Tensor

A simple matrix expression is obtained for the strain components of a beam in which the displacements and rotations are large. The only restrictions are on the magnitudes of the strain and of the local rotation, a newly-identified kinematical quantity. The local rotation is defined as the change of orientation of material elements relative to the change of orientation of the beam reference triad. The vectors and tensors in the theory are resolved along orthogonal triads of base vectors centered along the undeformed and deformed beam reference axes, so Cartesian tensor notation is used. Although a curvilinear coordinate system is natural to the beam problem, the complications usually associated with its use are circumvented. Local rotations appear explicitly in the resulting strain expressions, facilitating the treatment of beams with both open and closed cross sections in applications of the theory. The theory is used to obtain the kinematical relations for coupled bending, torsion extension, shear deformation, and warping of an initially curved and twisted beam.

Danielson, D. A.↗

NASA NTRS1999

Implicit and Multigrid Method for Ideal Multigrid Convergence: Direct Numerical Simulation of Separated Flow Around NACA 0012 Airfoil

Numerical investigation of flow separation over a NACA 0012 airfoil at large angles of attack has been carried out. The numerical calculation is performed by solving the full Navier-Stokes equations in generalized curvilinear coordinates. The second-order LU-SGS implicit scheme is applied for time integration. This scheme requires no tridiagonal inversion and is capable of being completely vectorized, provided the corresponding Jacobian matrices are properly selected. A fourth-order centered compact scheme is used for spatial derivatives. In order to reduce numerical oscillation, a sixth-order implicit filter is employed. Non-reflecting boundary conditions are imposed at the far-field and outlet boundaries to avoid possible non-physical wave reflection. Complex flow separation and vortex shedding phenomenon have been observed and discussed.

Liu, Chao-Qun↗

NASA NTRS2000

Flow Analysis of X-34 Main Propulsion System Feedlines

This viewgraph presentation reviews the flow analysis of the propulsion feedlines in the X-34. The objective of the analysis was to determine the flow development at the engine interface, and to predict the pressure drop in the feedline. The presentation shows the design of the feedlines, and the computational grids. It reviews the numerical method, which included solving the curvilinear coordinate transport equations. Line graphs review the computational fluid dynamic results. The results and conclusions are also presented.

Vu, Bruce↗