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NASA NTRS1979

An implicit algorithm for the conservative, transonic full-potential equation with effective rotated differencing

A new differencing scheme for the conservative full potential equation which effectively simulates rotated differencing is presented. The scheme was implemented by an appropriate upwind bias of the density coefficient along coordinate directions. A fast, fully implicit, approximate factorization iteration scheme was then used to solve the resulting difference equations. Solutions for a number of traditionally difficult transonic airfoil test cases are presented.

Holst, T. L.↗

NASA NTRS1994

Trellis Coding of Non-coherent Multiple Symbol Full Response M-ary CPFSK with Modulation Index 1/M

This paper introduces a trellis coded modulation (TCM) scheme for non-coherent multiple full response M-ary CPFSK with modulation index 1/M. A proper branch metric for the trellis decoder is obtained by employing a simple approximation of the modified Bessel function for large signal to noise ratio (SNR). Pairwise error probability of coded sequences is evaluated by applying a linear approximation to the Rician random variable.

trellis coded modulation TCM↗

NASA NTRS1989

A dual potential formulation of the Navier-Stokes equations

A dual potential formulation for numerically solving the Navier-Stokes equations is developed and presented. The velocity field is decomposed using a scalar and vector potential. Vorticity and dilatation are used as the dependent variables in the momentum equations. Test cases in two dimensions verify the capability to solve flows using approximations from potential flow to full Navier-Stokes simulations. A three-dimensional incompressible flow formulation is also described. An interesting feature of this approach to solving the Navier-Stokes equations is the decomposition of the velocity field into a rotational part (vector potential) and an irrotational part (scalar potential). The Helmholtz decomposition theorem allows this splitting of the velocity field. This approach has had only limited use since it increases the number of dependent variables in the solution. However, it has often been used for incompressible flows where the solution scheme is known to be fast and accurate. This research extends the usage of this method to fully compressible Navier-Stokes simulations by using the dilatation variable along with vorticity. A time-accurate, iterative algorithm is used for the uncoupled solution of the governing equations. Several levels of flow approximation are available within the framework of this method. Potential flow, Euler and full Navier-Stokes solutions are possible using the dual potential formulation. Solution efficiency can be enhanced in a straightforward way. For some flows, the vorticity and/or dilatation may be negligible in certain regions (e.g., far from a viscous boundary in an external flow). It is possible to drop the calculation of these variables then and optimize the solution speed. Also, efficient Poisson solvers are available for the potentials. The relative merits of non-primitive variables versus primitive variables for solution of the Navier-Stokes equations are also discussed.

Gegg, S. G.↗

NASA NTRS1990

Three-dimensional Euler time accurate simulations of fan rotor-stator interactions

A numerical method useful to describe unsteady 3-D flow fields within turbomachinery stages is presented. The method solves the compressible, time dependent, Euler conservation equations with a finite volume, flux splitting, total variation diminishing, approximately factored, implicit scheme. Multiblock composite gridding is used to partition the flow field into a specified arrangement of blocks with static and dynamic interfaces. The code is optimized to take full advantage of the processing power and speed of the Cray Y/MP supercomputer. The method is applied to the computation of the flow field within a single stage, axial flow fan, thus reproducing the unsteady 3-D rotor-stator interaction.

Boretti, A. A.↗

NASA NTRS1985

A Lagrangian method for the shallow water equations based on a Voronoi mesh - Flows on a rotating sphere

A Lagrangian scheme using the Voronoi mesh is applied to study shallow water flow on a sphere. Discrete approximations to the shallow water equations are obtained for the surfaces of a nonrotating and a rotating sphere, and discrete differential operators are defined for the gradient and the divergence on the sphere. Dissipation is put into the model, when needed, by merging fluid points when they get too close to each other. The full numerical scheme is described and results of numerical computations on various test cases are given, including zonal flow and the Riemann problem.

Augenbaum, J. M.↗

NASA NTRS1988

Second order accurate finite difference approximations for the transonic small disturbance equation and the full potential equation

New shock-capturing finite difference approximations for solving two scalar conservation law nonlinear partial differential equations describing inviscid, isentropic, compressible flows of aerodynamics at transonic speeds are presented. A global linear stability theorem is applied to these schemes in order to derive a necessary and sufficient condition for the finite element method. A technique is proposed to render the described approximations total variation-stable by applying the flux limiters to the nonlinear terms of the difference equation dimension by dimension. An entropy theorem applying to the approximations is proved, and an implicit, forward Euler-type time discretization of the approximation is presented. Results of some numerical experiments using the approximations are reported.

Mostrel, M. M.↗

NASA NTRS1991

Center for Modeling of Turbulence and Transition (CMOTT). Research briefs: 1990

Brief progress reports of the Center for Modeling of Turbulence and Transition (CMOTT) research staff from May 1990 to May 1991 are given. The objectives of the CMOTT are to develop, validate, and implement the models for turbulence and boundary layer transition in the practical engineering flows. The flows of interest are three dimensional, incompressible, and compressible flows with chemistry. The schemes being studied include the two-equation and algebraic Reynolds stress models, the full Reynolds stress (or second moment closure) models, the probability density function models, the Renormalization Group Theory (RNG) and Interaction Approximation (DIA), the Large Eddy Simulation (LES) and Direct Numerical Simulation (DNS).

Povinelli, Louis A.↗

NASA NTRS1986

Incompressible viscous flow simulations of the NFAC wind tunnel

The capabilities of an existing 3-D incompressible Navier-Stokes flow solver, INS3D, are extended and improved to solve turbulent flows through the incorporation of zero- and two-equation turbulence models. The two-equation model equations are solved in their high Reynolds number form and utilize wall functions in the treatment of solid wall boundary conditions. The implicit approximate factorization scheme is modified to improve the stability of the two-equation solver. Applications to the 3-D viscous flow inside the 80 by 120 feet open return wind tunnel of the National Full Scale Aerodynamics Complex (NFAC) are discussed and described.

Champney, Joelle Milene↗

NASA NTRS1984

Entropy condition satisfying approximations for the full potential equations of transonic flow

A class of conservative difference approximations for the steady full potential equation was presented. They are, in general, easier to program than the usual density biasing algorithms, and in fact, differ only slightly from them. Rigorous proof indicated that these new schemes satisfied a new discrete entropy inequality, which ruled out expansion shocks, and that they have sharp, steady, discrete shocks. A key tool in the analysis is the construction of a new entropy inequality for the full potential equation itself. Results of some numerical experiments using the new schemes are presented.

Osher, S.↗

NASA NTRS1985

Entropy condition satisfying approximations for the full potential equation of transonic flow

Osher, S.↗

NASA NTRS1984

An efficient approximate factorization implicit scheme for the equations of gasdynamics

An efficient implicit finite-difference algorithm for the gas dynamic equations utilizing matrix reduction techniques is presented. A significant reduction in arithmetic operations is achieved while maintaining the same favorable stability characteristics and generality found in the Beam and Warming approximate factorization algorithm. Steady-state solutions to the conservative Euler equations in generalized coordinates are obtained for transonic flows about a NACA 0012 airfoil. The theoretical extension of the matrix reduction technique to the full Navier-Stokes equations in Cartesian coordinates is presented in detail. Linear stability, using a Fourier stability analysis, is demonstrated and discussed for the one-dimensional Euler equations. It is shown that the method offers advantages over the conventional Beam and Warming scheme and can retrofit existing Beam and Warming codes with minimal effort.

Barth, T. J.↗

NASA NTRS1984

Application of a full potential method for computation of three-dimensional supersonic flows

A nonlinear aerodynamic analysis technique based on the full potential equation in conservative form has been modified to permit treatment of supersonic flows with embedded subsonic regions (typically near the fuselage-canopy juncture and the wing leading edge). Solution procedures for the equations do not require any specific form of geometry or physical grid system. This results in the capability to analyze easily very complex geometries provided the posed problem lies within the isentropic restrictions of the full potential theory. Characteristic signal propagation theory is used to monitor the type dependent flow and a conservative switching scheme is employed to transition from the supersonic marching algorithm to a subsonic relaxation procedure and vice versa. An implicit approximate factorization scheme is used to solve the finite-difference equations. These modifications now permit analysis of fully three-dimensional flowfields including the interference effects due to lifting surface wakes. Improved grid generation capability allows analysis of complete complex aircraft geometries (fuselage, wing, tail, wing wake, and tail wake). Results are presented showing very good correlations with experimental surface pressure data and aerodynamic force data at both design and off-design operating points. Configurations examined include several waverider concepts, an arrow wing-body with wake, an advanced tactical fighter concept, and a fighter forebody-canard configuration.

Jones, K. M.↗

NASA NTRS1972

Channel coding/decoding alternatives for compressed TV data on advanced planetary missions.

The compatibility of channel coding/decoding schemes with a specific TV compressor developed for advanced planetary missions is considered. Under certain conditions, it is shown that compressed data can be transmitted at approximately the same rate as uncompressed data without any loss in quality. Thus, the full gains of data compression can be achieved in real-time transmission.

Rice, R. F.↗

NASA NTRS1992

Navier-Stokes simulations of unsteady transonic flow phenomena

Numerical simulations of two classes of unsteady flows are obtained via the Navier-Stokes equations: a blast-wave/target interaction problem class and a transonic cavity flow problem class. The method developed for the viscous blast-wave/target interaction problem assumes a laminar, perfect gas implemented in a structured finite-volume framework. The approximately factored implicit scheme uses Newton subiterations to obtain the spatially and temporally second-order accurate time history of the blast-waves with stationary targets. The inviscid flux is evaluated using either of two upwind techniques, while the full viscous terms are computed by central differencing. Comparisons of unsteady numerical, analytical, and experimental results are made in two- and three-dimensions for Couette flows, a starting shock-tunnel, and a shock-tube blockage study. The results show accurate wave speed resolution and nonoscillatory discontinuity capturing of the predominantly inviscid flows. Viscous effects were increasingly significant at large post-interaction times. While the blast-wave/target interaction problem benefits from high-resolution methods applied to the Euler terms, the transonic cavity flow problem requires the use of an efficient scheme implemented in a geometrically flexible overset mesh environment. Hence, the Reynolds averaged Navier-Stokes equations implemented in a diagonal form are applied to the cavity flow class of problems. Comparisons between numerical and experimental results are made in two-dimensions for free shear layers and both rectangular and quieted cavities, and in three-dimensions for Stratospheric Observatory For Infrared Astronomy (SOFIA) geometries. The acoustic behavior of the rectangular and three-dimensional cavity flows compare well with experiment in terms of frequency, magnitude, and quieting trends. However, there is a more rapid decrease in computed acoustic energy with frequency than observed experimentally owing to numerical dissipation. In addition, optical phase distortion due to the time-varying density field is modelled using geometrical constructs. The computed optical distortion trends compare with the experimentally inferred result, but underpredicts the fluctuating phase difference magnitude.

Atwood, C. A.↗

NASA NTRS2001

Multigrid approaches to non-linear diffusion problems on unstructured meshes

The efficiency of three multigrid methods for solving highly non-linear diffusion problems on two-dimensional unstructured meshes is examined. The three multigrid methods differ mainly in the manner in which the nonlinearities of the governing equations are handled. These comprise a non-linear full approximation storage (FAS) multigrid method which is used to solve the non-linear equations directly, a linear multigrid method which is used to solve the linear system arising from a Newton linearization of the non-linear system, and a hybrid scheme which is based on a non-linear FAS multigrid scheme, but employs a linear solver on each level as a smoother. Results indicate that all methods are equally effective at converging the non-linear residual in a given number of grid sweeps, but that the linear solver is more efficient in cpu time due to the lower cost of linear versus non-linear grid sweeps.

Mavriplis, Dimitri J.↗

NASA NTRS1982

An efficient full potential implicit method based on characteristics for analysis of supersonic flows

A nonlinear aerodynamic prediction technique based on the full potential equation in conservation form has been developed for the treatment of supersonic flows. The method uses the theory of characteristic signal propagation to accurately simulate the flow structure, which includes shock waves and mixed elliptic-hyperbolic crossflow. An implicit approximate factorization scheme is employed to solve the finite-differenced equation. The necessary body-fitted grid system in every marching plane is generated numerically, using an elliptic grid solver. Results are shown for conical and nonconical wing-body combinations and compared with experimental data and Euler calculations. The method demonstrates an enormous savings in execution time and memory requirements over Euler methods.

Shankar, V.↗

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 NTRS1988

An improved version of NCOREL: A computer program for 3-D nonlinear supersonic potential flow computations

A computer code called NCOREL (for Nonconical Relaxation) has been developed to solve for supersonic full potential flows over complex geometries. The method first solves for the conical at the apex and then marches downstream in a spherical coordinate system. Implicit relaxation techniques are used to numerically solve the full potential equation at each subsequent crossflow plane. Many improvements have been made to the original code including more reliable numerics for computing wing-body flows with multiple embedded shocks, inlet flow through simulation, wake model and entropy corrections. Line relaxation or approximate factorization schemes are optionally available. Improved internal grid generation using analytic conformal mappings, supported by a simple geometric Harris wave drag input that was originally developed for panel methods and internal geometry package are some of the new features.

Siclari, Michael J.↗