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NASA NTRS1998

Discretization and Preconditioning Algorithms for the Euler and Navier-Stokes Equations on Unstructured Meshes

Several stabilized demoralization procedures for conservation law equations on triangulated domains will be considered. Specifically, numerical schemes based on upwind finite volume, fluctuation splitting, Galerkin least-squares, and space discontinuous Galerkin demoralization will be considered in detail. A standard energy analysis for several of these methods will be given via entropy symmetrization. Next, we will present some relatively new theoretical results concerning congruence relationships for left or right symmetrized equations. These results suggest new variants of existing FV, DG, GLS, and FS methods which are computationally more efficient while retaining the pleasant theoretical properties achieved by entropy symmetrization. In addition, the task of Jacobean linearization of these schemes for use in Newton's method is greatly simplified owing to exploitation of exact symmetries which exist in the system. The FV, FS and DG schemes also permit discrete maximum principle analysis and enforcement which greatly adds to the robustness of the methods. Discrete maximum principle theory will be presented for general finite volume approximations on unstructured meshes. Next, we consider embedding these nonlinear space discretizations into exact and inexact Newton solvers which are preconditioned using a nonoverlapping (Schur complement) domain decomposition technique. Elements of nonoverlapping domain decomposition for elliptic problems will be reviewed followed by the present extension to hyperbolic and elliptic-hyperbolic problems. Other issues of practical relevance such the meshing of geometries, code implementation, turbulence modeling, global convergence, etc, will. be addressed as needed.

Barth, Timothy J.↗

NASA NTRS1998

Euler Technology Assessment for Preliminary Aircraft Design-Unstructured/Structured Grid NASTD Application for Aerodynamic Analysis of an Advanced Fighter/Tailless Configuration

This study supports the NASA Langley sponsored project aimed at determining the viability of using Euler technology for preliminary design use. The primary objective of this study was to assess the accuracy and efficiency of the Boeing, St. Louis unstructured grid flow field analysis system, consisting of the MACGS grid generation and NASTD flow solver codes. Euler solutions about the Aero Configuration/Weapons Fighter Technology (ACWFT) 1204 aircraft configuration were generated. Several variations of the geometry were investigated including a standard wing, cambered wing, deflected elevon, and deflected body flap. A wide range of flow conditions, most of which were in the non-linear regimes of the flight envelope, including variations in speed (subsonic, transonic, supersonic), angles of attack, and sideslip were investigated. Several flowfield non-linearities were present in these solutions including shock waves, vortical flows and the resulting interactions. The accuracy of this method was evaluated by comparing solutions with test data and Navier-Stokes solutions. The ability to accurately predict lateral-directional characteristics and control effectiveness was investigated by computing solutions with sideslip, and with deflected control surfaces. Problem set up times and computational resource requirements were documented and used to evaluate the efficiency of this approach for use in the fast paced preliminary design environment.

Michal, Todd R.↗

NASA NTRS1986

Finite element solver for 3-D compressible viscous flows

The space shuttle main engine (SSME) has extremely complex internal flow structure. The geometry of the flow domain is three-dimensional with complicated topology. The flow is compressible, viscous, and turbulent with large gradients in flow quantities and regions of recirculations. The analysis of the flow field in SSME involves several tedious steps. One is the geometrical modeling of the particular zone of the SSME being studied. Accessing the geometry definition, digitalizing it, and developing surface interpolations suitable for an interior grid generator require considerable amount of manual labor. There are several types of grid generators available with some general-purpose finite element programs. An efficient and robust computational scheme for solving 3D Navier-Stokes equations has to be implemented. Post processing software has to be adapted to visualize and analyze the computed 3D flow field. The progress made in a project to develop software for the analysis of the flow is discussed. The technical approach to the development of the finite element scheme and the relaxation procedure are discussed. The three dimensional finite element code for the compressible Navier-Stokes equations is listed.

Reddy, K. C.↗

NASA NTRS1997

Discretization and Preconditioning Algorithms for the Euler and Navier-Stokes Equations on Unstructured Meshes

Several stabilized discretization procedures for conservation law equations on triangulated domains will be considered. Specifically, numerical schemes based on upwind finite volume, fluctuation splitting, Galerkin least-squares, and space discontinuous Galerkin discretization will be considered in detail. A standard energy analysis for several of these methods will be given via entropy symmetrization. Next, we will present some relatively new theoretical results concerning congruence relationships for left or right symmetrized equations. These results suggest new variants of existing FV, DG, GLS and FS methods which are computationally more efficient while retaining the pleasant theoretical properties achieved by entropy symmetrization. In addition, the task of Jacobian linearization of these schemes for use in Newton's method is greatly simplified owing to exploitation of exact symmetries which exist in the system. These variants have been implemented in the "ELF" library for which example calculations will be shown. The FV, FS and DG schemes also permit discrete maximum principle analysis and enforcement which greatly adds to the robustness of the methods. Some prevalent limiting strategies will be reviewed. Next, we consider embedding these nonlinear space discretizations into exact and inexact Newton solvers which are preconditioned using a nonoverlapping (Schur complement) domain decomposition technique. Elements of nonoverlapping domain decomposition for elliptic problems will be reviewed followed by the present extension to hyperbolic and elliptic-hyperbolic problems. Other issues of practical relevance such the meshing of geometries, code implementation, turbulence modeling, global convergence, etc. will be addressed as needed.

Barth, Timothy↗

NASA NTRS1989

Numerical algorithms for steady and unsteady incompressible Navier-Stokes equations

The numerical analysis of the incompressible Navier-Stokes equations are becoming important tools in the understanding of some fluid flow problems which are encountered in research as well as in industry. With the advent of the supercomputers, more realistic problems can be studied with a wider choice of numerical algorithms. An alternative formulation is presented for viscous incompressible flows. The incompressible Navier-Stokes equations are cast in a velocity/vorticity formulation. This formulation consists of solving the Poisson equations for the velocity components and the vorticity transport equation. Two numerical algorithms for the steady two-dimensional laminar flows are presented. The first method is based on the actual partial differential equations. This uses a finite-difference approximation of the governing equations on a staggered grid. The second method uses a finite element discretization with the vorticity transport equation approximated using a Galerkin approximation and the Poisson equations are obtained using a least squares method. The equations are solved efficiently using Newton's method and a banded direct matrix solver (LINPACK). The method is extended to steady three-dimensional laminar flows and applied to a cubic driven cavity using finite difference schemes and a staggered grid arrangement on a Cartesian mesh. The equations are solved iteratively using a plane zebra relaxation scheme. Currently, a two-dimensional, unsteady algorithm is being developed using a generalized coordinate system. The equations are discretized using a finite-volume approach. This work will then be extended to three-dimensional flows.

Hafez, Mohammed↗

DOE OSTI2026

Development of MOSCATO: A CFD-Level Electrochemistry and Corrosion Simulator for Molten Salt Systems

For both coolant and fueled variants of molten salt reactors (MSRs), the corrosion of structural materials is a significant challenge. The corrosion stems from chemical and electrochemical reactions initiated by fissile material, fission products, and impurities in the salt. Lower-fidelity models rely on empirical correlations for mass transfer, simplified lumped temperature profiles, and similar assumptions. They do not capture detailed spatial variations in complex geometries, creating the need for high-fidelity modeling to bridge this gap.As we approach the demonstration and possible deployment of MSRs in this decade, the development of a high-fidelity, high-performance simulator becomes imperative. To simulate the complex electrochemical environment and corrosion within molten salt systems, we have developed the Molten Salt Chemistry And TranspOrt (MOSCATO) code. This endeavor is comprised of three essential components. First, mass transfer equations are coupled with the Navier-Stokes equations in order to account for the transport of species in the salt. Second, the diffusion of alloy constituents, such as Cr, Fe, Ni, etc. is simulated within the structural metals. Third, the alloy and salt domains are coupled to account for the heterogeneous chemical and electrochemical reactions that occur at the salt-alloy interface.MOSCATO manages all three components within the framework of the highly scalable, open-source spectral element method computational fluid dynamics code Nek5000/NekRS. This integration enables MOSCATO to harness the immense computational power of modern high-performance computing resources, ensuring both high fidelity and computational speed.In addition to code development, we have initiated a comprehensive verification and validation campaign, utilizing data from diverse sources. First, MOSCATO's electrochemical solver was verified with reference numerical data. Then validation occurred against experiments: one of a thermal galvanic cell and the other for corrosion in flowing molten salt of FLiNaK (LiF-NaF-KF). This campaign verified and validated MOSCATO as a reliable tool for simulating electrochemical environments and corrosion in molten salt systems.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

NASA NTRS1997

MFLOP to GFLOP: The Impact on High Fidelity Based Computational Aeroelasticity

Aeroelasticity which involves strong coupling of fluids, structures and controls is an important element in designing an aircraft. Computational aeroelasticity using low fidelity methods such as the linear aerodynamic flow equations coupled with the modal structural equations are well advanced. Though these low fidelity approaches are computationally less intensive, they are not adequate for the analysis of modern aircraft which can experience complex flow/structure interactions. Even at moderate angles of attack supersonic aircraft can experience vortex induced aeroelastic oscillations. Near transonic speeds buffet associated structural oscillations are possible. Aircraft flying in transonic regime may experience a dip in the flutter speed. For accurate aeroelastic computations at these complex fluid/structure interaction situations, high fidelity equations such as the Navier-Stokes for fluids and the finite-elements for structures are needed. Computations using these high fidelity equations require large computational resources both in memory and speed. Current conventional supercomputers have reached their limitations both in memory and speed. As a result, parallel computers have evolved to overco me the limitations of conventional computers. This paper will address the transition that is taking place in computational aeroelasticity from conventional computers to parallel computers. The paper will address special techniques needed to take advantage of the architecture of new parallel computers. Results will be illustrated from computations made on iPSC/860 and IBM SP2 computer by using ENSAERO code that directly couples the Euler/Navier-Stokes flow equations with high resolution finite-element structural equations. Modifications required in both fluids and structural solvers in order to run efficiently on parallel computers will be discussed. Implementation of moving grids and fluid/structural interface on parallel computers will be discussed.

Guruswamy, Guru P.↗

NASA NTRS1987

Wall jet analysis for circulation control aerodynamics. Part 1: Fundamental CFD and turbulence modeling concepts

An overview of parabolic and PNS (Parabolized Navier-Stokes) methodology developed to treat highly curved sub and supersonic wall jets is presented. The fundamental data base to which these models were applied is discussed in detail. The analysis of strong curvature effects was found to require a semi-elliptic extension of the parabolic modeling to account for turbulent contributions to the normal pressure variations, as well as an extension to the turbulence models utilized, to account for the highly enhanced mixing rates observed in situations with large convex curvature. A noniterative, pressure split procedure is shown to extend parabolic models to account for such normal pressure variations in an efficient manner, requiring minimal additional run time over a standard parabolic approach. A new PNS methodology is presented to solve this problem which extends parabolic methodology via the addition of a characteristic base wave solver. Applications of this approach to analyze the interaction of wave and turbulence processes in wall jets is presented.

Dash, S. M.↗

NASA NTRS1985

Development of a boundary-layer-type solver based on simultaneous iteration technique for axisymmetric separated flows

A boundary-layer-type solver is developed for the numerical solution of axisymmetric separated flows. A new fully implicit coupling scheme for the viscous and inviscid regions is demonstrated. This fully implicit coupling technique is similar to the work of Carter, Veldman, and an extension of an earlier work of Halim and Hafez. A comparison is made for the convergence rate using this new fully implicit coupling technique and the semiimplicit coupling of Halim and Hafez. Numerical results using the fully implicit coupling are obtained for laminar incompressible separated flows, including a boattail and a series of trough geometries. Also, the near-wake flow problem is considered using the present formulation. A clear conclusion of this investigation is that the present scheme using the fully implicit coupling method converges at a faster rate than the semiimplicit coupling and the partially parabolized Navier-Stokes (PPNS) procedures.

Halim, A. A. M.↗

NASA NTRS1992

Development of a fractional-step method for the unsteady incompressible Navier-Stokes equations in generalized coordinate systems

A fractional step method is developed for solving the time-dependent three-dimensional incompressible Navier-Stokes equations in generalized coordinate systems. The primitive variable formulation uses the pressure, defined at the center of the computational cell, and the volume fluxes across the faces of the cells as the dependent variables, instead of the Cartesian components of the velocity. This choice is equivalent to using the contravariant velocity components in a staggered grid multiplied by the volume of the computational cell. The governing equations are discretized by finite volumes using a staggered mesh system. The solution of the continuity equation is decoupled from the momentum equations by a fractional step method which enforces mass conservation by solving a Poisson equation. This procedure, combined with the consistent approximations of the geometric quantities, is done to satisfy the discretized mass conservation equation to machine accuracy, as well as to gain the favorable convergence properties of the Poisson solver. The momentum equations are solved by an approximate factorization method, and a novel ZEBRA scheme with four-color ordering is devised for the efficient solution of the Poisson equation. Several two- and three-dimensional laminar test cases are computed and compared with other numerical and experimental results to validate the solution method. Good agreement is obtained in all cases.

Rosenfeld, Moshe↗

NASA NTRS1991

Implicit solution of three-dimensional internal turbulent flows

The scalar form of the approximate factorization method was used to develop a new code for the solution of three dimensional internal laminar and turbulent compressible flows. The Navier-Stokes equations in their Reynolds-averaged form were iterated in time until a steady solution was reached. Evidence was given to the implicit and explicit artificial damping schemes that proved to be particularly efficient in speeding up convergence and enhancing the algorithm robustness. A conservative treatment of these terms at the domain boundaries was proposed in order to avoid undesired mass and/or momentum artificial fluxes. Turbulence effects were accounted for by the zero-equation Baldwin-Lomax turbulence model and the q-omega two-equation model. The flow in a developing S-duct was then solved in the laminar regime in a Reynolds number (Re) of 790 and in the turbulent regime at Re equals 40,000 by using the Baldwin-Lomax model. The Stanitz elbow was then solved by using an invicid version of the same code at M sub inlet equals 0.4. Grid dependence and convergence rate were investigated, showing that for this solver the implicit damping scheme may play a critical role for convergence characteristics. The same flow at Re equals 2.5 times 10(exp 6) was solved with the Baldwin-Lomax and the q-omega models. Both approaches show satisfactory agreement with experiments, although the q-omega model was slightly more accurate.

Michelassi, V.↗

NASA NTRS1990

Implicit solution of three-dimensional internal turbulent flows

The scalar form of the approximate factorization method was used to develop a new code for the solution of three-dimensional internal laminar and turbulent compressible flows. The Navier-Stokes equations in their Reynolds-averaged form are iterated in time until a steady solution is reached. Evidence is given to the implicit and explicit artificial damping schemes that proved to be particularly efficient in speeding up convergence and enhancing the algorithm robustness. A conservative treatment of these terms at domain boundaries is proposed in order to avoid undesired mass and/or momentum artificial fluxes. Turbulence effects are accounted for by the zero-equation Baldwin-Lomax turbulence model and the q-omega two-equation model. For the first, an investigation on the model behavior in case of multiple boundaries is performed. The flow in a developing S-duct is then solved in the laminar regime at Reynolds number (Re) 790 and in the turbulent regime at Re=40,000 using the Baldwin-Lomax model . The Stanitz elbow is then solved using an inviscid version of the same code at M(sub inlet)=0.4. Grid dependence and convergence rate are investigated showing that for this solver the implicit damping scheme may play a critical role for convergence characteristics. The same flow at Re=2.5x10(exp 6) is solved with the Baldwin-Lomax and the q-omega models. Both approaches showed satisfactory agreement with experiments, although the q-omega model is slightly more accurate.

Michelassi, V.↗

NASA NTRS1992

Development of a 3-D upwind PNS code for chemically reacting hypersonic flowfields

Two new parabolized Navier-Stokes (PNS) codes were developed to compute the three-dimensional, viscous, chemically reacting flow of air around hypersonic vehicles such as the National Aero-Space Plane (NASP). The first code (TONIC) solves the gas dynamic and species conservation equations in a fully coupled manner using an implicit, approximately-factored, central-difference algorithm. This code was upgraded to include shock fitting and the capability of computing the flow around complex body shapes. The revised TONIC code was validated by computing the chemically-reacting (M(sub infinity) = 25.3) flow around a 10 deg half-angle cone at various angles of attack and the Ames All-Body model at 0 deg angle of attack. The results of these calculations were in good agreement with the results from the UPS code. One of the major drawbacks of the TONIC code is that the central-differencing of fluxes across interior flowfield discontinuities tends to introduce errors into the solution in the form of local flow property oscillations. The second code (UPS), originally developed for a perfect gas, has been extended to permit either perfect gas, equilibrium air, or nonequilibrium air computations. The code solves the PNS equations using a finite-volume, upwind TVD method based on Roe's approximate Riemann solver that was modified to account for real gas effects. The dissipation term associated with this algorithm is sufficiently adaptive to flow conditions that, even when attempting to capture very strong shock waves, no additional smoothing is required. For nonequilibrium calculations, the code solves the fluid dynamic and species continuity equations in a loosely-coupled manner. This code was used to calculate the hypersonic, laminar flow of chemically reacting air over cones at various angles of attack. In addition, the flow around the McDonnel Douglas generic option blended-wing-body was computed and comparisons were made between the perfect gas, equilibrium air, and the nonequilibrium air results.

Tannehill, J. C.↗

NASA NTRS1997

External Boundary Conditions for Three-Dimensional Problems of Computational Aerodynamics

We consider an unbounded steady-state flow of viscous fluid over a three-dimensional finite body or configuration of bodies. For the purpose of solving this flow problem numerically, we discretize the governing equations (Navier-Stokes) on a finite-difference grid. The grid obviously cannot stretch from the body up to infinity, because the number of the discrete variables in that case would not be finite. Therefore, prior to the discretization we truncate the original unbounded flow domain by introducing some artificial computational boundary at a finite distance of the body. Typically, the artificial boundary is introduced in a natural way as the external boundary of the domain covered by the grid. The flow problem formulated only on the finite computational domain rather than on the original infinite domain is clearly subdefinite unless some artificial boundary conditions (ABC's) are specified at the external computational boundary. Similarly, the discretized flow problem is subdefinite (i.e., lacks equations with respect to unknowns) unless a special closing procedure is implemented at this artificial boundary. The closing procedure in the discrete case is called the ABC's as well. In this paper, we present an innovative approach to constructing highly accurate ABC's for three-dimensional flow computations. The approach extends our previous technique developed for the two-dimensional case; it employs the finite-difference counterparts to Calderon's pseudodifferential boundary projections calculated in the framework of the difference potentials method (DPM) by Ryaben'kii. The resulting ABC's appear spatially nonlocal but particularly easy to implement along with the existing solvers. The new boundary conditions have been successfully combined with the NASA-developed production code TLNS3D and used for the analysis of wing-shaped configurations in subsonic (including incompressible limit) and transonic flow regimes. As demonstrated by the computational experiments and comparisons with the standard (local) methods, the DPM-based ABC's allow one to greatly reduce the size of the computational domain while still maintaining high accuracy of the numerical solution. Moreover, they may provide for a noticeable increase of the convergence rate of multigrid iterations.

Tsynkov, Semyon V.↗

NASA NTRS1997

Efficient Gradient-Based Shape Optimization Methodology Using Inviscid/Viscous CFD

The formerly developed preconditioned-biconjugate-gradient (PBCG) solvers for the analysis and the sensitivity equations had resulted in very large error reductions per iteration; quadratic convergence was achieved whenever the solution entered the domain of attraction to the root. Its memory requirement was also lower as compared to a direct inversion solver. However, this memory requirement was high enough to preclude the realistic, high grid-density design of a practical 3D geometry. This limitation served as the impetus to the first-year activity (March 9, 1995 to March 8, 1996). Therefore, the major activity for this period was the development of the low-memory methodology for the discrete-sensitivity-based shape optimization. This was accomplished by solving all the resulting sets of equations using an alternating-direction-implicit (ADI) approach. The results indicated that shape optimization problems which required large numbers of grid points could be resolved with a gradient-based approach. Therefore, to better utilize the computational resources, it was recommended that a number of coarse grid cases, using the PBCG method, should initially be conducted to better define the optimization problem and the design space, and obtain an improved initial shape. Subsequently, a fine grid shape optimization, which necessitates using the ADI method, should be conducted to accurately obtain the final optimized shape. The other activity during this period was the interaction with the members of the Aerodynamic and Aeroacoustic Methods Branch of Langley Research Center during one stage of their investigation to develop an adjoint-variable sensitivity method using the viscous flow equations. This method had algorithmic similarities to the variational sensitivity methods and the control-theory approach. However, unlike the prior studies, it was considered for the three-dimensional, viscous flow equations. The major accomplishment in the second period of this project (March 9, 1996 to March 8, 1997) was the extension of the shape optimization methodology for the Thin-Layer Navier-Stokes equations. Both the Euler-based and the TLNS-based analyses compared with the analyses obtained using the CFL3D code. The sensitivities, again from both levels of the flow equations, also compared very well with the finite-differenced sensitivities. A fairly large set of shape optimization cases were conducted to study a number of issues previously not well understood. The testbed for these cases was the shaping of an arrow wing in Mach 2.4 flow. All the final shapes, obtained either from a coarse-grid-based or a fine-grid-based optimization, using either a Euler-based or a TLNS-based analysis, were all re-analyzed using a fine-grid, TLNS solution for their function evaluations. This allowed for a more fair comparison of their relative merits. From the aerodynamic performance standpoint, the fine-grid TLNS-based optimization produced the best shape, and the fine-grid Euler-based optimization produced the lowest cruise efficiency.

Baysal, Oktay↗

NASA NTRS1991

An Explicit Upwind Algorithm for Solving the Parabolized Navier-Stokes Equations

An explicit, upwind algorithm was developed for the direct (noniterative) integration of the 3-D Parabolized Navier-Stokes (PNS) equations in a generalized coordinate system. The new algorithm uses upwind approximations of the numerical fluxes for the pressure and convection terms obtained by combining flux difference splittings (FDS) formed from the solution of an approximate Riemann (RP). The approximate RP is solved using an extension of the method developed by Roe for steady supersonic flow of an ideal gas. Roe's method is extended for use with the 3-D PNS equations expressed in generalized coordinates and to include Vigneron's technique of splitting the streamwise pressure gradient. The difficulty associated with applying Roe's scheme in the subsonic region is overcome. The second-order upwind differencing of the flux derivatives are obtained by adding FDS to either an original forward or backward differencing of the flux derivative. This approach is used to modify an explicit MacCormack differencing scheme into an upwind differencing scheme. The second order upwind flux approximations, applied with flux limiters, provide a method for numerically capturing shocks without the need for additional artificial damping terms which require adjustment by the user. In addition, a cubic equation is derived for determining Vegneron's pressure splitting coefficient using the updated streamwise flux vector. Decoding the streamwise flux vector with the updated value of Vigneron's pressure splitting improves the stability of the scheme. The new algorithm is applied to 2-D and 3-D supersonic and hypersonic laminar flow test cases. Results are presented for the experimental studies of Holden and of Tracy. In addition, a flow field solution is presented for a generic hypersonic aircraft at a Mach number of 24.5 and angle of attack of 1 degree. The computed results compare well to both experimental data and numerical results from other algorithms. Computational times required for the upwind PNS code are approximately equal to an explicit PNS MacCormack's code and existing implicit PNS solvers.

Korte, John J.↗

NASA NTRS1994

A Navier-Strokes Chimera Code on the Connection Machine CM-5: Design and Performance

We have implemented a three-dimensional compressible Navier-Stokes code on the Connection Machine CM-5. The code is set up for implicit time-stepping on single or multiple structured grids. For multiple grids and geometrically complex problems, we follow the 'chimera' approach, where flow data on one zone is interpolated onto another in the region of overlap. We will describe our design philosophy and give some timing results for the current code. A parallel machine like the CM-5 is well-suited for finite-difference methods on structured grids. The regular pattern of connections of a structured mesh maps well onto the architecture of the machine. So the first design choice, finite differences on a structured mesh, is natural. We use centered differences in space, with added artificial dissipation terms. When numerically solving the Navier-Stokes equations, there are liable to be some mesh cells near a solid body that are small in at least one direction. This mesh cell geometry can impose a very severe CFL (Courant-Friedrichs-Lewy) condition on the time step for explicit time-stepping methods. Thus, though explicit time-stepping is well-suited to the architecture of the machine, we have adopted implicit time-stepping. We have further taken the approximate factorization approach. This creates the need to solve large banded linear systems and creates the first possible barrier to an efficient algorithm. To overcome this first possible barrier we have considered two options. The first is just to solve the banded linear systems with data spread over the whole machine, using whatever fast method is available. This option is adequate for solving scalar tridiagonal systems, but for scalar pentadiagonal or block tridiagonal systems it is somewhat slower than desired. The second option is to 'transpose' the flow and geometry variables as part of the time-stepping process: Start with x-lines of data in-processor. Form explicit terms in x, then transpose so y-lines of data are in-processor. Form explicit terms in y, then transpose so z-lines are in processor. Form explicit terms in z, then solve linear systems in the z-direction. Transpose to the y-direction, then solve linear systems in the y-direction. Finally transpose to the x direction and solve linear systems in the x-direction. This strategy avoids inter-processor communication when differencing and solving linear systems, but requires a large amount of communication when doing the transposes. The transpose method is more efficient than the non-transpose strategy when dealing with scalar pentadiagonal or block tridiagonal systems. For handling geometrically complex problems the chimera strategy was adopted. For multiple zone cases we compute on each zone sequentially (using the whole parallel machine), then send the chimera interpolation data to a distributed data structure (array) laid out over the whole machine. This information transfer implies an irregular communication pattern, and is the second possible barrier to an efficient algorithm. We have implemented these ideas on the CM-5 using CMF (Connection Machine Fortran), a data parallel language which combines elements of Fortran 90 and certain extensions, and which bears a strong similarity to High Performance Fortran. We make use of the Connection Machine Scientific Software Library (CMSSL) for the linear solver and array transpose operations.

Jespersen, Dennis C.↗

NASA NTRS1993

A concurrent hybrid Navier-Stokes/Euler approach to fluid dynamic computations

We present a methodology for the numerical simulation of flow fields by the simultaneous application of two distinct approaches to computational aerodynamics. We compute the three dimensional flow field of a missile at moderate angle of attack by dividing the flow field into two regions: a region near the surface where we use a structured grid and a Navier Stokes solver, and a region farther away from the surface where we utilize an unstructured grid and an Euler solver. The two solvers execute as independent UNIX processes either on the same machine or on two machines. The solvers communicate data across their common interfaces within the same machine or over the network. The computations indicate that extensively separated flow fields can be computed without significant distortion by combining viscous and inviscid solvers.

Tavella, Domingo A.↗