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NASA NTRS1999

A Pseudo-Temporal Multi-Grid Relaxation Scheme for Solving the Parabolized Navier-Stokes Equations

A multi-grid, flux-difference-split, finite-volume code, VULCAN, is presented for solving the elliptic and parabolized form of the equations governing three-dimensional, turbulent, calorically perfect and non-equilibrium chemically reacting flows. The space marching algorithms developed to improve convergence rate and or reduce computational cost are emphasized. The algorithms presented are extensions to the class of implicit pseudo-time iterative, upwind space-marching schemes. A full approximate storage, full multi-grid scheme is also described which is used to accelerate the convergence of a Gauss-Seidel relaxation method. The multi-grid algorithm is shown to significantly improve convergence on high aspect ratio grids.

White, J. A.↗

NASA NTRS1981

Implicit calculations of transonic flows using monotone methods

Implicit approximate-factorization algorithms have been developed that use monotone methods for the calculation of steady and unsteady transonic flows governed by the small-disturbance-potential equation. These algorithms use the new Engquist-Osher switch in the type-dependent differencing in place of the standard Murman-Cole switch. The resulting algorithms are more stable; hence, calculations can be done more efficiently. For steady flows, the convergence rate is about 35% faster, and for unsteady flows the allowable time step is about 10 times larger. These improvements are achieved with no increase in computer storage and with only minor modifications in codes that use the Murman-Cole switch. Also an implicit algorithm has been developed for the steady full-potential equation in one-dimension, which uses monotone methods.

Goorjian, P. M.↗

NASA NTRS2015

CFD Modeling of the Multipurpose Hydrogen Test Bed (MHTB) Self-Pressurization and Spray Bar Mixing Experiments in Normal Gravity: Effect of the Accommodation Coefficient on the Tank Pressure

A CFD model for simulating the self-pressurization of a large scale liquid hydrogen storage tank is utilized in this paper to model the MHTB self-pressurization experiment. The kinetics-based Schrage equation is used to account for the evaporative and condensi ng interfacial mass flows in this model. The effect of the accommodation coefficient for calculating the interfacial mass transfer rate on the tank pressure during tank selfpressurization is studied. The values of the accommodation coefficient which were considered in this study vary from 1.0e-3 to 1.0e-1 for the explicit VOF model and from 1.0e-4 to 1.0e-3 for the implicit VOF model. The ullage pressure evolutions are compared against experimental data. A CFD model for controlling pressure in cryogenic storage tanks by spraying cold liquid into the ullage is also presented. The Euler-Lagrange approach is utilized for tracking the spray droplets and for modeling the interaction between the droplets and the continuous phase (ullage). The spray model is coupled with the VOF model by performing particle tracking in the ullage, removing particles from the ullage when they reach the interface, and then adding their contributions to the liquid. Droplet-ullage heat and mass transfer are modeled. The flow, temperature, and interfacial mass flux, as well as droplets trajectories, size distribution and temperatures predicted by the model are presented. The ul lage pressure and vapor temperature evolutions are compared with experimental data obtained from the MHTB spray bar mixing experiment. The effect of the accommodation coefficient for calculating the interfacial and droplet mass transfer rates on the tank pressure during mixing of the vapor using spray is studied. The values used for the accommodation coefficient at the interface vary from 1.0e-5 to 1.0e-2. The droplet accommodation coefficient values vary from 2.0e-6 to 1.0e-4.

cryogenic fluid storage↗

NASA NTRS1984

Universal single level implicit algorithm for gasdynamics

A single level effectively explicit implicit algorithm for gasdynamics is presented. The method meets all the requirements for unconditionally stable global iteration over flows with mixed supersonic and supersonic zones including blunt body flow and boundary layer flows with strong interaction and streamwise separation. For hyperbolic (supersonic flow) regions the method is automatically equivalent to contemporary space marching methods. For elliptic (subsonic flow) regions, rapid convergence is facilitated by alternating direction solution sweeps which bring both sets of eigenvectors and the influence of both boundaries of a coordinate line equally into play. Point by point updating of the data with local iteration on the solution procedure at each spatial step as the sweeps progress not only renders the method single level in storage but, also, improves nonlinear accuracy to accelerate convergence by an order of magnitude over related two level linearized implicit methods. The method derives robust stability from the combination of an eigenvector split upwind difference method (CSCM) with diagonally dominant ADI(DDADI) approximate factorization and computed characteristic boundary approximations.

Lombard, C. K.↗

NASA NTRS1975

An investigation of several numerical procedures for time-asymptotic compressible Navier-Stokes solutions

The status of an investigation of four numerical techniques for the time-dependent compressible Navier-Stokes equations is presented. Results for free shear layer calculations in the Reynolds number range from 1000 to 81000 indicate that a sequential alternating-direction implicit (ADI) finite-difference procedure requires longer computing times to reach steady state than a low-storage hopscotch finite-difference procedure. A finite-element method with cubic approximating functions was found to require excessive computer storage and computation times. A fourth method, an alternating-direction cubic spline technique which is still being tested, is also described.

Rudy, D. H.↗

NASA NTRS1993

Faulty behavior of asynchronous storage elements

It is often assumed that the faults in storage elements (SE's) can be modeled as output/input stuck-at-faults of the element. They are implicitly considered equivalent to the stuck-at faults in the combinational logic surrounding the SE cells. A more accurate higher level fault model for elementary SE's used in asynchronous circuits is presented. This model offers better representation of the physical failures. It is shown that the stuck-at model may be adequate if only modest fault coverage is desired. The enhanced model includes some common fault behaviors of SE's that are not covered by the stuck-at model. These include data-feed-through behaviors that cause the SE to be combinational. Fault models for complex SE cells can be obtained without a significant loss of information about the structure of the circuit.

Al-Assadi, Waleed K.↗

NASA NTRS1993

Upwind MacCormack Euler solver with non-equilibrium chemistry

A computer code, designated UMPIRE, is currently under development to solve the Euler equations in two dimensions with non-equilibrium chemistry. UMPIRE employs an explicit MacCormack algorithm with dissipation introduced via Roe's flux-difference split upwind method. The code also has the capability to employ a point-implicit methodology for flows where stiffness is introduced through the chemical source term. A technique consisting of diagonal sweeps across the computational domain from each corner is presented, which is used to reduce storage and execution requirements. Results depicting one dimensional shock tube flow for both calorically perfect gas and thermally perfect, dissociating nitrogen are presented to verify current capabilities of the program. Also, computational results from a chemical reactor vessel with no fluid dynamic effects are presented to check the chemistry capability and to verify the point implicit strategy.

Sherer, Scott E.↗

NASA NTRS1991

Asynchronous, macrotasked relaxation strategies for the solution of viscous, hypersonic flows

A point-implicit, asynchronous macrotasked relaxation of the steady, thin-layer, Navier-Stokes equations is presented. The method employs multidirectional, single-level storage Gauss-Seidel relaxation sweeps, which effectively communicate perturbations across the entire domain in 2n sweeps, where n is the dimension of the domain. In order to enhance convergence the application of relaxation factors to specific components of the Jacobian is examined using a stability analysis of the advection and diffusion equations. Attention is also given to the complications associated with asynchronous multitasking. Solutions are generated for hypersonic flows over blunt bodies in two and three dimensions with chemical reactions, utilizing single-tasked and multitasked relaxation strategies.

Gnoffo, Peter A.↗

NASA NTRS1990

Implicit application of polynomial filters in a k-step Arnoldi method

The Arnoldi process is a well known technique for approximating a few eigenvalues and corresponding eigenvectors of a general square matrix. Numerical difficulties such as loss of orthogonality and assessment of the numerical quality of the approximations as well as a potential for unbounded growth in storage have limited the applicability of the method. These issues are addressed by fixing the number of steps in the Arnoldi process at a prescribed value k and then treating the residual vector as a function of the initial Arnoldi vector. This starting vector is then updated through an iterative scheme that is designed to force convergence of the residual to zero. The iterative scheme is shown to be a truncation of the standard implicitly shifted QR-iteration for dense problems and it avoids the need to explicitly restart the Arnoldi sequence. The main emphasis of this paper is on the derivation and analysis of this scheme. However, there are obvious ways to exploit parallelism through the matrix-vector operations that comprise the majority of the work in the algorithm. Preliminary computational results are given for a few problems on some parallel and vector computers.

Sorensen, D. C.↗

NASA NTRS1985

The science of computing - The evolution of parallel processing

The present paper is concerned with the approaches to be employed to overcome the set of limitations in software technology which impedes currently an effective use of parallel hardware technology. The process required to solve the arising problems is found to involve four different stages. At the present time, Stage One is nearly finished, while Stage Two is under way. Tentative explorations are beginning on Stage Three, and Stage Four is more distant. In Stage One, parallelism is introduced into the hardware of a single computer, which consists of one or more processors, a main storage system, a secondary storage system, and various peripheral devices. In Stage Two, parallel execution of cooperating programs on different machines becomes explicit, while in Stage Three, new languages will make parallelism implicit. In Stage Four, there will be very high level user interfaces capable of interacting with scientists at the same level of abstraction as scientists do with each other.

Denning, P. J.↗

NASA NTRS1978

Finite element dynamic analysis on CDC STAR-100 computer

Computational algorithms are presented for the finite element dynamic analysis of structures on the CDC STAR-100 computer. The spatial behavior is described using higher-order finite elements. The temporal behavior is approximated by using either the central difference explicit scheme or Newmark's implicit scheme. In each case the analysis is broken up into a number of basic macro-operations. Discussion is focused on the organization of the computation and the mode of storage of different arrays to take advantage of the STAR pipeline capability. The potential of the proposed algorithms is discussed and CPU times are given for performing the different macro-operations for a shell modeled by higher order composite shallow shell elements having 80 degrees of freedom.

Noor, A. K.↗

NASA NTRS1989

Three dimensional PNS solutions of hypersonic internal flows with equilibrium chemistry

An implicit procedure for solving parabolized Navier-Stokes equations under the assumption of a general equation of state for a gas in chemical equilibrium is given. A general and consistent approach for the evaluation of Jacobian matrices in the implicit operator avoids the use of unnecessary auxiliary quantities and approximations, and leads to a simple expression. Applications to two- and three-dimensional flow problems show efficiency in computer time and economy in storage.

Liou, May-Fun↗

NASA NTRS1990

Improved boundary layer heat transfer calculations near a stagnation point

A thermal design of a solar receiver has been developed for the solutions of problems involving phase-change thermal energy storage and natural convection loss. Two dimensional axisymmetrical solidification and melting of materials contained between two concentric cylinders of finite length has been studied for thermal energy storage analysis. For calculation of free convection loss inside receiver cavity, two dimensional axisymmetrical, laminar, transient free convection including radiation effects has been studied using integral/finite difference method. Finite difference equations are derived for the above analysis subject to constant or variable material properties, initial conditions, and boundary conditions. The validity of the analyses has been substantiated by comparing results of the present general method with available analytic solutions or numerical results reported in the literature. Both explicit and implicit schemes are tested in phase change analysis with different number of nodes ranging from 4 to 18. The above numerical methods have been applied to the existing solar receiver analyzing computer code as additional subroutines. The results were computed for one of the proposed Brayton cycle solar receiver models running under the actual environmental conditions. Effect of thermal energy storage on the thermal behavior of the receiver has been estimated. Due to the thermal energy storage, about 65% reduction on working gas outlet temperature fluctuation has been obtained; however, maximum temperature of thermal energy storage containment has been increased about 18%. Also, effect of natural convection inside a receiver cavity on the receiver heat transfer has been analyzed. The finding indicated that thermal stratification occurs during the sun time resulting in higher receiver temperatures at the outlet section of the gas tube, and lower temperatures at the inlet section of the gas tube when compared with the results with no natural convection. Due to heat supply from the air during the shade time, minimum temperature has been increased, while maximum temperature has been reduced due to convection loss to air. Consequently, cyclic temperature fluctuation has been reduced 29% for working gas and 16% for thermal energy storage containment. On the other hand, despite the presence of the natural convection the time-averaged temperatures for receiver components were found to be similar for two cases with/without natural convection (maximum difference was 1.8%).

Ahn, Kyung Hwan↗

NASA NTRS1987

An L-U implicit multigrid algorithm for the three-dimensional Euler equations

An LU implicit multigrid scheme is developed for the calculation of three-dimensional transonic flow through rotating cascades. This numerical method solves the unsteady Euler equations of gas dynamics in a finite-volume form. The implicit scheme makes it possible to take a much larger time step than is normally permitted in most explicit schemes, while the multigrid method is incorporated to accelerate the convergence rate for steady state calculations. Using this method, computational storage requirements are comparable to those of explicit schemes, while operation counts are considerably less than those found in the more widely-used ADI schemes.

Yokota, Jeffrey W.↗

NASA NTRS1975

Calculation of three-dimensional compressible laminar and turbulent boundary layers. An implicit finite-difference procedure for solving the three-dimensional compressible laminar, transitional, and turbulent boundary-layer equations

An implicit finite-difference procedure is presented for solving the compressible three-dimensional boundary-layer equations. The method is second-order accurate, unconditionally stable (conditional stability for reverse cross flow), and efficient from the viewpoint of computer storage and processing time. The Reynolds stress terms are modeled by (1) a single-layer mixing length model and (2) a two-layer eddy viscosity model. These models, although simple in concept, accurately predicted the equilibrium turbulent flow for the conditions considered. Numerical results are compared with experimental wall and profile data for a cone at an angle of attack larger than the cone semiapex angle. These comparisons clearly indicate that the numerical procedure and turbulence models accurately predict the experimental data with as few as 21 nodal points in the plane normal to the wall boundary.

Harris, J. E.↗

NASA NTRS1997

Improving Discrete-Sensitivity-Based Approach for Practical Design Optimization

In developing the automated methodologies for simulation-based optimal shape designs, their accuracy, efficiency and practicality are the defining factors to their success. To that end, four recent improvements to the building blocks of such a methodology, intended for more practical design optimization, have been reported. First, in addition to a polynomial-based parameterization, a partial differential equation (PDE) based parameterization was shown to be a practical tool for a number of reasons. Second, an alternative has been incorporated to one of the tedious phases of developing such a methodology, namely, the automatic differentiation of the computer code for the flow analysis in order to generate the sensitivities. Third, by extending the methodology for the thin-layer Navier-Stokes (TLNS) based flow simulations, the more accurate flow physics was made available. However, the computer storage requirement for a shape optimization of a practical configuration with the -fidelity simulations (TLNS and dense-grid based simulations), required substantial computational resources. Therefore, the final improvement reported herein responded to this point by including the alternating-direct-implicit (ADI) based system solver as an alternative to the preconditioned biconjugate (PbCG) and other direct solvers.

Baysal, Oktay↗

NASA NTRS1984

Application of the implicit MacCormack scheme to the parabolized Navier-Stokes equations

MacCormack's implicit finite-difference scheme was used to solve the two-dimensional parabolized Navier-Stokes (PNS) equations. This method for solving the PNS equations does not require the inversion of block tridiagonal systems of algebraic equations and permits the original explicit MacCormack scheme to be employed in those regions where implicit treatment is not needed. The advantages and disadvantages of the present adaptation are discussed in relation to those of the conventional Beam-Warming scheme for a flat plate boundary layer test case. Comparisons are made for accuracy, stability, computer time, computer storage, and ease of implementation. The present method was also applied to a second test case of hypersonic laminar flow over a 15% compression corner. The computed results compare favorably with experiment and a numerical solution of the complete Navier-Stokes equations.

Lawrence, J. L.↗

NASA NTRS1991

A globally convergent matrix-free algorithm for implicit time-marching schemes arising in finite element analysis in fluids

A solution procedure for solving nonlinear time-marching problems is presented. The nonsymmetric systems of equations arising from a Newton-type linearization of these time-marching problems are solved using an iterative strategy based on the generalized minimal residual (GMRES) algorithm. Matrix-free techniques leading to reduction in storage are presented. Incorporation of a linesearch algorithm in the Newton-GMRES scheme is discussed. An automatic time-increment control strategy is developed to increase the stability of the time-marching process. High-speed flow computations demonstrate the effectiveness of these algorithms.

Johan, Zdenek↗