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NASA NTRS1995

Flow simulations about steady-complex and unsteady moving configurations using structured-overlapped and unstructured grids

The limiting factor in simulating flows past realistic configurations of interest has been the discretization of the physical domain on which the governing equations of fluid flow may be solved. In an attempt to circumvent this problem, many Computational Fluid Dynamic (CFD) methodologies that are based on different grid generation and domain decomposition techniques have been developed. However, due to the costs involved and expertise required, very few comparative studies between these methods have been performed. In the present work, the two CFD methodologies which show the most promise for treating complex three-dimensional configurations as well as unsteady moving boundary problems are evaluated. These are namely the structured-overlapped and the unstructured grid schemes. Both methods use a cell centered, finite volume, upwind approach. The structured-overlapped algorithm uses an approximately factored, alternating direction implicit scheme to perform the time integration, whereas, the unstructured algorithm uses an explicit Runge-Kutta method. To examine the accuracy, efficiency, and limitations of each scheme, they are applied to the same steady complex multicomponent configurations and unsteady moving boundary problems. The steady complex cases consist of computing the subsonic flow about a two-dimensional high-lift multielement airfoil and the transonic flow about a three-dimensional wing/pylon/finned store assembly. The unsteady moving boundary problems are a forced pitching oscillation of an airfoil in a transonic freestream and a two-dimensional, subsonic airfoil/store separation sequence. Accuracy was accessed through the comparison of computed and experimentally measured pressure coefficient data on several of the wing/pylon/finned store assembly's components and at numerous angles-of-attack for the pitching airfoil. From this study, it was found that both the structured-overlapped and the unstructured grid schemes yielded flow solutions of comparable accuracy for these simulations. This study also indicated that, overall, the structured-overlapped scheme was slightly more CPU efficient than the unstructured approach.

Newman, James C., III↗

NASA NTRS1981

Recent advances in the development of implicit schemes for the equations of fluid dynamics

Innovations and extensions of implicit schemes for equations of fluid dynamics are presented. The notation and theory for linear multistep methods are reviewed, and extensions of work by Beam and Warming (1979) include the implementation of one-leg methods, ADI methods for equations with mixed derivatives, flux vector splitting, the P-dimensional wave equation, and boundary conditions. Numerical experiments indicate that implicit treatment of the boundary conditions is necessary for unconditional stability, and the improvement and implementation of the boundary condition theory should improve the implicit algorithms for gas dynamic equations.

Warming, R. F.↗

NASA NTRS1992

Navier-Stokes analysis of three-dimensional unsteady flows inside turbine stages

This study presents a numerical method for solving the 3D Navier-Stokes equations for unsteady, viscous flow through multiple turbomachinery blade rows. The method solves the fully 3D Navier-Stokes equations with an implicit scheme which is based on a control volume approach. A two-equation turbulence model with a low Reynolds number modification is employed. A third-order accurate upwinding scheme is used to approximate convection terms, while a second order accurate central difference scheme is used for the discretization of viscous terms. A second-order accurate scheme is employed for the temporal discretization. The numerical method is applied to study the unsteady flowfield of the High Pressure Fuel side Turbo-Pump (HPFTP) of the Space Shuttle Main Engine (SSME). The stage calculation is performed by coupling the stator and the rotor flowfields at each time step through an over-laid grid. Numerical results for the complete geometry with the vane trailing edge cutback are presented and compared with the available experimental data.

Hah, C.↗

NASA NTRS1982

Conservative supra-characteristics method for splitting the hyperbolic systems of gasdynamics for real and perfect gases

A conservative flux difference splitting is presented for the hyperbolic systems of gasdynamics. The stable robust method is suitable for wide application in a variety of schemes, explicit or implicit, iterative or direct, for marching in either time or space. The splitting is modeled on the local quasi one dimensional characteristics system for multi-dimensional flow similar to Chakravarthy's nonconservative split coefficient matrix method; but, as the result of maintaining global conservation, the method is able to capture sharp shocks correctly. The embedded characteristics formulation is cast in a primitive variable the volumetric internal energy (rather than the pressure) that is effective for treating real as well as perfect gases. Finally the relationship of the splitting to characteristics boundary conditions is discussed and the associated conservative matrix formulation for a computed blown wall boundary condition is developed as an example. The theoretical development employs and extends the notion of Roe of constructing stable upwind difference formulae by sending split simple one sided flux difference pieces to appropriate mesh sites. The developments are also believed to have the potential for aiding in the analysis of both existing and new conservative difference schemes.

Lombard, C. K.↗

NASA NTRS1983

A finite element method for diffusion dominated unsteady viscous flows

A general conforming finite element scheme for computing viscous flows is presented which is of second-order accuracy in space and time. Viscous terms are treated implicitly and advection terms are treated explicitly in the time marching segment of the algorithm. A method for solving the algebraic equations at each time step is given. The method is demonstrated on two test problems, one of them being a plane vortex flow for which asymptotic methods are used to obtain suitable numerical boundary conditions at each time step.

Gunzburger, M. D.↗

NASA NTRS1990

Boundary-integral method for calculating aerodynamic sensitivities with illustration for lifting-surface theory

The technique of implicit differentiation has been used in combination with linearized lifting-surface theory to derive analytical expressions for aerodynamic sensitivities (i.e., rates of change of lifting pressures with respect to general changes in aircraft geometry, including planform variations) for steady or oscillating planar or nonplanar lifting surfaces in subsonic, sonic, or supersonic flow. The geometric perturbation is defined in terms of a single variable, and the user need only provide simple expressions or similar means for defining the continuous or discontinuous global or local perturbation of interest. Example expressions are given for perturbations of the sweep, taper, and aspect ratio of a wing with trapezoidal semispan planform. The present process appears to be readily adaptable to more general surface-panel methods.

Yates, E. Carson, Jr.↗

NASA NTRS1984

Implicit treatment of the unsteady full potential equation in conservation form

An implicit, conservative treatment for the unsteady full potential equation in two-dimensions is presented. The method employs a local time linearization for density, and introduces flux biasing concepts based on sonic conditions for the generation of artificial viscosity to capture shocks without any overshoots. The boundary condition is treated implicitly using a splitting procedure consistent with the approximate factorization scheme. This allows for extremely large Courant numbers, even for nonorthogonal grid at the body. The method has application not only to unsteady problems, but also to generate the starting blunt body solution for a supersonic full potential marching code. Results are presented for flows over cylinders, spheres and airfoils. Comparisons are made with available Euler and full potential results, and are in excellent agreement.

Shankar, V.↗

NASA NTRS1987

Simple numerical method for solving the steady Euler equations

A numerical method for solving the isenthalpic form of the governing equations for compressible inviscid flows is developed. The method is based on the concept of flux vector splitting in its implicit form and is tested on several demanding configurations. Time marching to steady state is accelerated by the implementation of the multigrid procedure which very effectively increases the rate of convergence. Steady-state results are obtained for various test cases. Only short computational times are required due to the relative efficiency of the basic method.

Von Lavante, E.↗

NASA NTRS1987

Multigrid acceleration of the isenthalpic form of the compressible flow equations

A numerical method for solving the isenthalpic form of the governing equations for compressible inviscid flows was developed. The method is based on the concept of flux vector splitting in its implicit form and was tested on several demanding configurations. Time marching to steady state was accelerated by the implementation of the multigrid procedure which very effectively increased the rate of convergence. High quality steady-state results were obtained for various test cases and required only short computational times due to the relative efficiency of the basic method.

Melson, N. Duane↗

NASA NTRS1987

Multigrid acceleration of the isenthalpic form of the compressible flow equations

Melson, N. Duane↗

NASA NTRS2015

Lattice Boltzmann Method for Spacecraft Propellant Slosh Simulation

A scalable computational approach to the simulation of propellant tank sloshing dynamics in microgravity is presented. In this work, we use the lattice Boltzmann equation (LBE) to approximate the behavior of two-phase, single-component isothermal flows at very low Bond numbers. Through the use of a non-ideal gas equation of state and a modified multiple relaxation time (MRT) collision operator, the proposed method can simulate thermodynamically consistent phase transitions at temperatures and density ratios consistent with typical spacecraft cryogenic propellants, for example, liquid oxygen. Determination of the tank forces and moments relies upon the global momentum conservation of the fluid domain, and a parametric wall wetting model allows tuning of the free surface contact angle. Development of the interface is implicit and no interface tracking approach is required. Numerical examples illustrate the method's application to predicting bulk fluid motion including lateral propellant slosh in low-g conditions.

Orr, Jeb S.↗

NASA NTRS1978

A split-recoupled-semidirect computational technique applied to transonic flow over lifting airfoils

A new version of the semidirect iterative method eliminates significant restrictions of previous versions of the method. A semidirect method solves finite-difference equations by a rapid globally implicit iterative process driven by a fast direct elliptic solver. The new approach can treat complex systems of equations in an efficient 'correction form', and allows the use of general, nonorthogonal, boundary-fitted coordinate transformations. These features are expected to lead to significant practical applications with conservation-equation systems in either two or three dimensions. The present application to the full potential equations for steady transonic flow over an airfoil at angle of attack illustrates the utility of the technique.

Martin, E. D.↗

NASA NTRS1980

Computational aerodynamics on large computers

Three examples of advances in computational aerodynamics; (1) three-dimensional inviscid transonic analysis, (2) design calculations for wings, and (3) the computation of viscous-induced aileron buzz, are reviewed. Attention is given to wing surface pressures, design optimization, computer memory, speed and advanced solution methods on parallel computer architecture. It is determined that many implicit approximate-factorization schemes, that have been developed for Navier-Stokes equations, can be coded to run efficiently on microprocessors.

Ballhaus, W. F.↗

NASA NTRS1982

Numerical experimentation on spherically symmetric one-dimensional magnetohydrodynamic /MHD/ wave propagation

Radial propagation of one-dimensional magnetohydrodynamic (MHD) waves are analyzed numerically on the basis of the Implicit-Continuous-Fluid-Eulerian (ICE) scheme. Accuracy of the numerical method and other properties are tested through the study of MHD wave propagation. The three different modes of MHD waves (i.e., fast-, slow- and Alfven (transverse) mode) are generated by applying physically consistent boundary perturbations derived from MHD compatibility relations. It is shown that the resulting flow following these waves depend upon the relative configurations of the initial magnetic field and boundary perturbations.

Han, S. M.↗

NASA NTRS1986

Advances in numerical and applied mathematics

This collection of papers covers some recent developments in numerical analysis and computational fluid dynamics. Some of these studies are of a fundamental nature. They address basic issues such as intermediate boundary conditions for approximate factorization schemes, existence and uniqueness of steady states for time dependent problems, and pitfalls of implicit time stepping. The other studies deal with modern numerical methods such as total variation diminishing schemes, higher order variants of vortex and particle methods, spectral multidomain techniques, and front tracking techniques. There is also a paper on adaptive grids. The fluid dynamics papers treat the classical problems of imcompressible flows in helically coiled pipes, vortex breakdown, and transonic flows.

South, J. C., Jr.↗

NASA NTRS1988

A diagonally inverted LU implicit multigrid scheme

A new Diagonally Inverted LU Implicit scheme is developed within the framework of the multigrid method for the 3-D unsteady Euler equations. The matrix systems that are to be inverted in the LU scheme are treated by local diagonalizing transformations that decouple them into systems of scalar equations. Unlike the Diagonalized ADI method, the time accuracy of the LU scheme is not reduced since the diagonalization procedure does not destroy time conservation. Even more importantly, this diagonalization significantly reduces the computational effort required to solve the LU approximation and therefore transforms it into a more efficient method of numerically solving the 3-D Euler equations.

Yokota, Jeffrey W.↗

NASA NTRS1988

Numerical computation of space shuttle orbiter flow field

A new parabolized Navier-Stokes (PNS) code has been developed to compute the hypersonic, viscous chemically reacting flow fields around 3-D bodies. The flow medium is assumed to be a multicomponent mixture of thermally perfect but calorically imperfect gases. The new PNS code solves the gas dynamic and species conservation equations in a coupled manner using a noniterative, implicit, approximately factored, finite difference algorithm. The space-marching method is made well-posed by special treatment of the streamwise pressure gradient term. The code has been used to compute hypersonic laminar flow of chemically reacting air over cones at angle of attack. The results of the computations are compared with the results of reacting boundary-layer computations and show excellent agreement.

Tannehill, John C.↗

NASA NTRS1988

A diagonally inverted LU implicit multigrid scheme

A new Diagonally Inverted LU Implicit scheme is developed within the framework of the multigrid method for the three-dimensional unsteady Euler equations. The matrix systems that are to be inverted in the LU scheme are treated by local diagonalizing transformations that decouple them into systems of scalar equations. Unlike the Diagonalized ADI method, the time accuracy of the LU scheme is not reduced since the diagonalizing procedure does not destroy time conservation. Even more importantly, this diagonalization significantly reduces the computational effort required to solve the LU approximation and therefore transforms it into a more efficient method of numerically solving the three-dimensional Euler equations.

Yokota, J. W.↗