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The development and applications of a parallel, time-dependent, three-dimensional incompressible Navier-Stokes flow solver and a parallel multigrid elliptic kernel are described.
We discuss our implementation strategies for finite difference multigrid partial differential equation (PDE) solvers on message-passing systems. Our target parallel architecture is Intel parallel computers: the Delta and Paragon system.
The interaction between a high-enthalpy flow and a thermal protection material is inherently multiscale and multiphysics. In conventional aerothermal analyses, the external flow and material response are generally modeled using separate computational domains coupled through boundary conditions at the material surface. Although this approach has supported many practical applications, it requires assumptions about the location and behavior of the interface and may become difficult to apply when material decomposition, internal reactions, and surface recession substantially alter the porous structure. This report presents the development of a one-domain formulation in which the free-fluid and porous-material regions are represented within a single computational domain. The formulation is based on the volume-averaged Navier–Stokes (VANS) equations, derived from the governing equations for reacting, compressible flow and condensed material. Volume averaging transfers the influence of the unresolved material microstructure to the macroscale equations through effective transport properties, interfacial source terms, and dispersion fluxes. Particular attention is given to regions in which porosity and permeability vary rapidly, including the diffuse transition between a porous material and the surrounding fluid. The resulting equations are implemented in the Porous-material Analysis Toolbox based on OpenFOAM (PATO). The report describes the pressure–velocity coupling strategy used by the solver, examines spatial filtering techniques for deriving effective properties, and evaluates the influence of a smoothly varying interface permeability. Numerical demonstrations include canonical porous-flow configurations, a flow-tube configuration representative of FiberForm® permeability experiments, and the oxidation of a porous carbon material. The purpose of this work is to establish a mathematical and computational foundation for a unified treatment of flow and thermal protection material response. The present formulation is intended to support the progressive inclusion of additional physical processes, including multicomponent transport, finite-rate gas–surface chemistry, pyrolysis, internal oxidation, and material recession. It also provides a framework for connecting pore-scale simulations and microstructural characterization with macroscale aerothermal-response calculations. This report is intended for researchers and engineers working in computational fluid dynamics, porous-media transport, material response, and thermal protection system modeling. It documents both the theoretical development and the initial numerical assessment of the one-domain approach, while identifying the closure of effective and dispersion terms as an important subject for continued investigation.
Three parallel algorithms were compared for the direct solution of tridiagonal linear systems of equations. The algorithms are suitable for computers such as ILLIAC 4 and CDC STAR. For array computers similar to ILLIAC 4, cyclic odd-even reduction has the least operation count for highly structured sets of equations, and recursive doubling has the least count for relatively unstructured sets of equations. Since the difference in operation counts for these two algorithms is not substantial, their relative running times may be more related to overhead operations, which are not measured in this paper. The third algorithm, based on Buneman's Poisson solver, has more arithmetic operations than the others, and appears to be the least favorable. For pipeline computers similar to CDC STAR, cyclic odd-even reduction appears to be the most preferable algorithm for all cases.
The use of a fast elliptic solver in combination with relaxation is presented as an effective way to accelerate the convergence of transonic flow calculations, particularly when a marching scheme can be used to treat the supersonic zone in the relaxation process.
The inhomogeneous Cauchy-Riemann equations in a rectangle are discretized by a finite difference approximation. Several different boundary conditions are treated explicitly, leading to algorithms which have overall second-order accuracy. All boundary conditions with either u or v prescribed along a side of the rectangle can be treated by similar methods. The algorithms presented here have nearly minimal time and storage requirements and seem suitable for development into a general-purpose direct Cauchy-Riemann solver for arbitrary boundary conditions.
A new technique is described for solving supersonic fluid dynamic problems containing multiple regions of continuous flow, each bounded by a permeable or impermeable surface. Region boundaries are, in general, arbitrarily shaped and time dependent. Discretization of such a region for solution by conventional finite difference procedures is accomplished using an elliptic solver which alleviates the dependence on a particular base coordinate system. Multiple regions are coupled together through the boundary conditions. The technique has been applied to a variety of problems including the shock diffraction and pointed wedge with detached bow shock.
A three-dimensional elliptic solver technique is utilized to generate surface-fitted coordinates about wing/wing-tip configurations. The method is applicable to wings of arbitrary section profile and camber, leading-edge sweep, taper ratio, and spanwise thickness variation. The basic theory of three-dimensional elliptic mappings is developed along with a method to compute interior coordinate control functions. Examples of grids generated about several wing/wing-tip geometries are given. A 49 x 33 x 17 grid requires about 3 minutes of CPU time on a CYBER 203 computer.
An iterative procedure is presented for solving steady inviscid 3-D subsonic rotational flow problems. The procedure combines concepts from classical secondary flow theory with an extension to 3-D of a novel semi-direct Cauchy-Riemann solver. It is developed for generalized coordinates and can be exercised using standard finite difference procedures. The stability criterion of the iterative procedure is discussed along with its ability to capture the evolution of inviscid secondary flow in a turning channel.
Massively parallel computers, as considered in this investigation, are not yet available. However, a large-scale parallel computer cannot usefully be designed before the hypothetical algorithms which will employ it are studied. Most of the studies of parallel partial differential equations (PDE) solvers are based on solution techniques much slower (on sequential machines) than multigrid methods. Multigrid methods are highly parallelizable. Each of their processes can simultaneously be performed at all grid points. The present investigation is concerned with a preliminary exploration of the potential of multigrid, or, more generally, Multi-Level Adaptive Techniques (MLAT) on computers with many processors. Basic processes are considered, taking into account coarse-grid approximation, relaxation, coarse-grid corrections, full multigrid algorithms, nonlinear problems and eigenvalue problems, fine-to-coarse correction, and chains of problems. Details of parallel multigrid processing are also examined.
An iterative procedure is presented for solving steady inviscid 3-D subsonic rotational flow problems. The procedure combines concepts from classical secondary flow theory with an extension to 3-D of a novel semi-direct Cauchy-Riemann solver. It is developed for generalized coordinates and can be exercised using standard finite difference procedures. The stability criterion of the iterative procedure is discussed along with its ability to capture the evolution of inviscid secondary flow in a turning channel.
A two-dimensional elliptic grid solver is presented and its application to various three-dimensional configurations, both internal and external, is demonstrated. The method uses proper forcing terms to cluster grid points near boundaries with a specified grid spacing and allows grid lines to intersect the boundaries at a specified angle. By segmenting the region, grid results are generated for sharp leading edged configurations and wing-vertical tail combinations.
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.
Godunov's method and several other methods for computing solutions to the equations of gas dynamics use Riemann solvers to resolve discontinuities at the interface between cells. A new method is proposed here for solving the Riemann problem based on a global existence proof for the solution to the Riemann problem. The method is found to be very reliable and computationally efficient.
The current paper develops a constrained hierarchical least square nonlinear equation solver. The procedure can handle the response behavior of systems which possess indefinite tangent stiffness characteristics. Due to the generality of the scheme, this can be achieved at various hierarchical application levels. For instance, in the case of finite element simulations, various combinations of either degree of freedom, nodal, elemental, substructural, and global level iterations are possible. Overall, this enables a solution methodology which is highly stable and storage efficient. To demonstrate the capability of the constrained hierarchical least square methodology, benchmarking examples are presented which treat structure exhibiting highly nonlinear pre- and postbuckling behavior wherein several indefinite stiffness transitions occur.
The patched mesh CSCM flow solver is used with a fast adaptive grid technique to compute complex two dimensional and axisymmetric, inviscid and viscous flows accurately and efficiently. The patched grid technique allows the user to generate the initial grid around complex regions with minimal effort. The adaptive grid strategy permits the grid points to adjust to the solution in accordance with the flow structure and user specified grid constraints and thus improve the accuracy of the solution. Solutions to challenging flow problems are obtained with adapted grids and the numerical examples and the results presented here show clearly the effectiveness of the adaptive grid solution strategy to obtain highly accurate numerical solutions with minimal human effort and computational resources.
A new Navier-Stokes solver is developed by combining the efficiency of the LU-SSOR scheme and the accuracy of the flux-limited dissipation scheme. Application to laminar and turbulent flows and hypersonic flows proves the reliability of the new algorithm.
In this study a multigrid solver for Euler equations (FLO52R) was examined to determine its performance potential on a hypothetical computer using a data flow architecture. The proposed computer would require massive parallelism to realize its design performance. On the other hand this parallelism would be more easily realized than with a conventional vector processor such as the Cray-1S. Several changes to the proposed design substantially alleviated most of the remaining bottlenecks to parallel processing. Other changes allowed clearer definition of memory access and disk I/O. Finally, a portion of the algorithm was rewritten to improve parallel performance. With these changes, performance levels approaching that of a Cray-1S may be possible for a computer costing far less. Estimates are given for overall speed, memory, and network bandwidth, and for instruction memory requirements.