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At least 289 records · Page 16
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NASA NTRS1992

An effective multigrid method for high-speed flows

The use is considered of a multigrid method with central differencing to solve the Navier-Stokes equations for high speed flows. The time dependent form of the equations is integrated with a Runge-Kutta scheme accelerated by local time stepping and variable coefficient implicit residual smoothing. Of particular importance are the details of the numerical dissipation formulation, especially the switch between the second and fourth difference terms. Solutions are given for 2-D laminar flow over a circular cylinder and a 15 deg compression ramp.

Swanson, R. C.↗
NASA NTRS1997

Comparison of Several Numerical Methods for Simulation of Compressible Shear Layers

An investigation is conducted on several numerical schemes for use in the computation of two-dimensional, spatially evolving, laminar variable-density compressible shear layers. Schemes with various temporal accuracies and arbitrary spatial accuracy for both inviscid and viscous terms are presented and analyzed. All integration schemes use explicit or compact finite-difference derivative operators. Three classes of schemes are considered: an extension of MacCormack's original second-order temporally accurate method, a new third-order variant of the schemes proposed by Rusanov and by Kutier, Lomax, and Warming (RKLW), and third- and fourth-order Runge-Kutta schemes. In each scheme, stability and formal accuracy are considered for the interior operators on the convection-diffusion equation U(sub t) + aU(sub x) = alpha U(sub xx). Accuracy is also verified on the nonlinear problem, U(sub t) + F(sub x) = 0. Numerical treatments of various orders of accuracy are chosen and evaluated for asymptotic stability. Formally accurate boundary conditions are derived for several sixth- and eighth-order central-difference schemes. Damping of high wave-number data is accomplished with explicit filters of arbitrary order. Several schemes are used to compute variable-density compressible shear layers, where regions of large gradients exist.

Kennedy, Christopher A.↗
NASA NTRSDate not supplied

Explicit Discontinuous Galerkin Methods for Conservation Laws

The two explicit DG methods in this study are based on a ‘predictor-corrector’ formulation, the first introduced by Lörcher, Gassner, and Munz (2007, 2008) called space–time expansion discontinuous Galerkin or STE-DG scheme, and the second, introduced independently by the author (Huynh 2006, 2013) called the upwind moment scheme. The predictor step of the two methods is essentially identical using a Cauchy-Kovalevsky (CK) procedure, which involves no interaction of the data among neighboring cells. The corrector step also shares the same space-time integration formulation and is where interaction of the data among neighboring cells takes place; the difference, however, is in how the resulting space-time volume integral is estimated. As a consequence of the different estimates, for the case of advection in one spatial dimension (1D), the moment scheme has a CFL (Courant-Friedrichs-Lewy) condition of 1 for all p and is accurate to order 2p+1, i.e., it possesses the super accuracy property, whereas the STE-DG method has a more restrictive CFL condition and is accurate to the expected order of p+1. For 1D advection, compared with the CFL conditions of 1/(2p+1) of standard RK-DG (Runge-Kutta) scheme where space and time discretization are of the same order, the moment scheme allows a significantly larger time step size. It also turns out that the scheme yields a result identical to Van Leer’s scheme III (1977), which amounts to shifting the data a distance of advection corresponding to the time step and projecting the result onto the space of polynomial solutions. Contrary to Van Leer’s approach, however, the space-time ‘predictor-corrector’ formulation facilitates extensions to the case of systems of equations. Concerning 2D extensions, in the case of advection, when the flow is along the diagonal direction, the CFL conditions for the moment schemes become restrictive as will be shown by Fourier (Von Neumann) stability and accuracy analyses. Since the moment scheme employs the right Radau points as collocation points in time, the method is closely related to the implicit Radau IIA scheme, which is stable for any time step size. The role of Radau IIA in relieving stability restriction for these explicit DG schemes remains to be explored

Discontinuous Galerkin↗
NASA NTRS1994

Some considerations on velocity vector accuracy in dust trajectory analysis

The relative contributions of comets and asteroids to the reservoir of dust in the interplanetary medium is not known. There are direct observations of dust released from comets and there is evidence to associate the IRAS dust bands with possible collisions of asteroids in the main belt. A means towards sorting out the parent sources has been proposed in the establishment of a dust collector in orbit about the Earth. The purpose of such a facility would be to collect not only cosmic dust particles intact but also the state vectors, as they arrive at the detector, the idea being that one may combine analytical laboratory analysis of the physics and chemistry of the captured particles with orbital data in order to help distinguish between bodies and identify parent bodies. The theoretical study of dust particle orbits in the solar system takes on greatly more importance if we use collected trajectory data. The orbital motion of dust when radiation and forces alone are acting is well understood. When gravitational forces due to the planets are included, the motion can become quite complex. In order to characterize the orbits of particles as they crossed the Earth's orbits, a study of the long-time dust orbital evolution was undertaken. We have considered various parameters associated with these dust orbits to see if one may in a general way discriminate between particles evolved from comets and asteroids. We proceed in this study as we have done previously. That is, we considered the dust particles as ideal black bodies, of density 1 gm/cc, spherical, with radii 10-100 microns. Particles of this size are affected by radiation forces, photon pressure, and Poynting-Robertson drag. Account was also taken of solar wind drag, which amounts to about 30 percent of the Poynting-Robertson drag negligible. The gravitational forces due to the planets are included, unlike in our previous study; the planetary orbits are those of true n-body interaction so that the possibility of secular resonance is included. Our method was to calculate explicitly by a numerical procedure the orbits of dust particles after they left their parent bodies. The motion is determined numerically with the implicit Runge-Kutta integrator using Gauss-Radau spacings.

Jackson, A. A.↗
NASA NTRS1993

Computational techniques for high-speed flows with viscous and chemical effects

Algorithms for solving the Euler and the Navier-Stokes equations in conjunction with chemical kinetic equations are presented. The convective flux is estimated from a quasi one dimensional interpolation procedure. Shock, contact, and expansion waves and thermochemical nonequilibrium phenomena are captured by the Lax-Friedrichs technique. Relaxation techniques were developed to enhance their effectiveness in dealing with spatial and temporal stiffness associated with the physical problems. Both explicit and implicit smoothers were implemented into the standard multigrid time stepping method. Unsteady and steady scalar problems are discussed. A perfect gas and equilibrium air shock tube problem is investigated. Numerical schemes and techniques are compared for the problems of shock and boundary layer interaction and three dimensional viscous, nonequilibrium flow encompassing an aerobrake. The results are comparable in accuracy against other high order non-oscillatory techniques. The multigrid methods are assessed using a Mach 8 flow over a complete planar body, a sphere, and a blunt delta wing at 20 deg incidence. Applying an implicit multigrid method on a nested grid of 128 by 64 nodes the reduction factor is 0.25. The central processing unit reduction factor is 2.2 after both the single and multigrid Runge-Kutta solutions converged to machine zero on a grid of 37 by 41 by 73 nodes.

Li, C. P.↗
NASA NTRS1993

Computation of propulsion-related flowfields using unstructured adaptive meshes

In this paper, we describe a computational fluid dynamics (CFD) technique based on unstructured triangular/tetrahedral meshes. A finite-volume scheme is used in conjunction with a multi-stage Runge-Kutta algorithm. Convergence enhancements in the form of dual time-stepping and time-derivative preconditioning are used to overcome the limitations of conventional multi-stage schemes. The method is applied to propulsion-related flows and shown to perform satisfactorily.

Weiss, Jonathan M.↗
NASA NTRSDate not supplied

Discretization Error Estimation and Control for Farfield Acoustic Signatures

We investigate the utility of adjoint-based error estimates for sonic boom farfield simulations governed by solutions of the augmented Burgers’ equation. Solution of this nonlinear system uses operator splitting with a second-order finite volume discretization in space and second-order Runge-Kutta time marching, while the absorption and molecular relaxation are solved using second-order central differencing. The discretization error in selected ground sonic boom cost functionals is estimated using the method of adjoint-weighted residuals. Key elements of the implementation process are emphasized with details provided on the practical aspects as appliedto the sonic boom farfield propagation. We establish the accuracy of the adjoint solutions usingcomplex step and finite difference approaches, and examine the accuracy of the error estimates using analytical N-wave solutions. We then apply it to a pressure waveform corresponding to the X-59 research aircraft. The investigations demonstrate that the method of adjoint-weighted residuals accurately predicts the level of discretization error present in sonic boom farfield simulations while offering insight into which features of the near field signal are the primary drivers of ground noise metrics. The numerical results indicate that at sampling frequencies as low as50kHz, discretization error in the propagation is under 0.01 dB[A] for realistically complex examples.

CST↗
NASA NTRS1985

SSME Turbopump Turbine Computations

A two-dimensional viscous code was developed to be used in the prediction of the flow in the SSME high-pressure turbopump blade passages. The rotor viscous code (RVC) employs a four-step Runge-Kutta scheme to solve the two-dimensional, thin-layer Navier-Stokes equations. The Baldwin-Lomax eddy-viscosity model is used for these turbulent flow calculations. A viable method was developed to use the relative exit conditions from an upstream blade row as the inlet conditions to the next blade row. The blade loading diagrams are compared with the meridional values obtained from an in-house quasithree-dimensional inviscid code. Periodic boundary conditions are imposed on a body-fitted C-grid computed by using the GRAPE GRids about Airfoils using Poisson's Equation (GRAPE) code. Total pressure, total temperature, and flow angle are specified at the inlet. The upstream-running Riemann invariant is extrapolated from the interior. Static pressure is specified at the exit such that mass flow is conserved from blade row to blade row, and the conservative variables are extrapolated from the interior. For viscous flows the noslip condition is imposed at the wall. The normal momentum equation gives the pressure at the wall. The density at the wall is obtained from the wall total temperature.

Jorgenson, P. G. E.↗
NASA NTRS1992

Efficient computation of inviscid flow fields around complex configurations using a multiblock multigrid method

The applicability of a multigrid technique to block-structured, body-fitted meshes is examined focusing on three different strategies. In the first strategy data are exchanged between blocks in each stage of a five-stage Runge-Kutta time-stepping scheme which keeps a possible time lag between blocks to a minimum, but requires a large amount of I/O operations and storage. The second strategy is based on performing a complete Runge-Kutta cycle within a block before switching to the next. In the third strategy both a complete Runge-Kutta cycle and the residual evaluation for the restriction operator are done within a block, allowing a minimum of I/O and storage. The inviscid flow around a wing-body/engine-pylon configuration was computed on a mesh consisting of 11 computational blocks. It was found that both the first and the second strategies delivered converged results, but the third failed due to larger time lag between blocks.

Rossow, C.-C.↗
NASA NTRS2011

Rapid Calculation of Spacecraft Trajectories Using Efficient Taylor Series Integration

A variable-order, variable-step Taylor series integration algorithm was implemented in NASA Glenn's SNAP (Spacecraft N-body Analysis Program) code. SNAP is a high-fidelity trajectory propagation program that can propagate the trajectory of a spacecraft about virtually any body in the solar system. The Taylor series algorithm's very high order accuracy and excellent stability properties lead to large reductions in computer time relative to the code's existing 8th order Runge-Kutta scheme. Head-to-head comparison on near-Earth, lunar, Mars, and Europa missions showed that Taylor series integration is 15.8 times faster than Runge- Kutta on average, and is more accurate. These speedups were obtained for calculations involving central body, other body, thrust, and drag forces. Similar speedups have been obtained for calculations that include J2 spherical harmonic for central body gravitation. The algorithm includes a step size selection method that directly calculates the step size and never requires a repeat step. High-order Taylor series integration algorithms have been shown to provide major reductions in computer time over conventional integration methods in numerous scientific applications. The objective here was to directly implement Taylor series integration in an existing trajectory analysis code and demonstrate that large reductions in computer time (order of magnitude) could be achieved while simultaneously maintaining high accuracy. This software greatly accelerates the calculation of spacecraft trajectories. At each time level, the spacecraft position, velocity, and mass are expanded in a high-order Taylor series whose coefficients are obtained through efficient differentiation arithmetic. This makes it possible to take very large time steps at minimal cost, resulting in large savings in computer time. The Taylor series algorithm is implemented primarily through three subroutines: (1) a driver routine that automatically introduces auxiliary variables and sets up initial conditions and integrates; (2) a routine that calculates system reduced derivatives using recurrence relations for quotients and products; and (3) a routine that determines the step size and sums the series. The order of accuracy used in a trajectory calculation is arbitrary and can be set by the user. The algorithm directly calculates the motion of other planetary bodies and does not require ephemeris files (except to start the calculation). The code also runs with Taylor series and Runge-Kutta used interchangeably for different phases of a mission.

Scott, James R.↗
NASA NTRS1990

A multiblock multigrid three-dimensional Euler equation solver

Current aerodynamic designs are often quite complex (geometrically). Flexible computational tools are needed for the analysis of a wide range of configurations with both internal and external flows. In the past, geometrically dissimilar configurations required different analysis codes with different grid topologies in each. The duplicity of codes can be avoided with the use of a general multiblock formulation which can handle any grid topology. Rather than hard wiring the grid topology into the program, it is instead dictated by input to the program. In this work, the compressible Euler equations, written in a body-fitted finite-volume formulation, are solved using a pseudo-time-marching approach. Two upwind methods (van Leer's flux-vector-splitting and Roe's flux-differencing) were investigated. Two types of explicit solvers (a two-step predictor-corrector and a modified multistage Runge-Kutta) were used with multigrid acceleration to enhance convergence. A multiblock strategy is used to allow greater geometric flexibility. A report on simple explicit upwind schemes for solving compressible flows is included.

Cannizzaro, Frank E.↗
NASA NTRS1993

Development of an unstructured solution adaptive method for the quasi-three-dimensional Euler and Navier-Stokes equations

A general solution adaptive scheme based on a remeshing technique is developed for solving the two-dimensional and quasi-three-dimensional Euler and Favre-averaged Navier-Stokes equations. The numerical scheme is formulated on an unstructured triangular mesh utilizing an edge-based pointer system which defines the edge connectivity of the mesh structure. Jameson's four-stage hybrid Runge-Kutta scheme is used to march the solution in time. The convergence rate is enhanced through the use of local time stepping and implicit residual averaging. As the solution evolves, the mesh is regenerated adaptively using flow field information. Mesh adaptation parameters are evaluated such that an estimated local numerical error is equally distributed over the whole domain. For inviscid flows, the present approach generates a complete unstructured triangular mesh using the advancing front method. For turbulent flows, the approach combines a local highly stretched structured triangular mesh in the boundary layer region with an unstructured mesh in the remaining regions to efficiently resolve the important flow features. One-equation and two-equation turbulence models are incorporated into the present unstructured approach. Results are presented for a wide range of flow problems including two-dimensional multi-element airfoils, two-dimensional cascades, and quasi-three-dimensional cascades. This approach is shown to gain flow resolution in the refined regions while achieving a great reduction in the computational effort and storage requirements since solution points are not wasted in regions where they are not required.

Jiang, Yi-Tsann↗
NASA NTRS1993

Development of an unstructured solution adaptive method for the quasi-three-dimensional Euler and Navier-Stokes equations

A general solution adaptive scheme-based on a remeshing technique is developed for solving the two-dimensional and quasi-three-dimensional Euler and Favre-averaged Navier-Stokes equations. The numerical scheme is formulated on an unstructured triangular mesh utilizing an edge-based pointer system which defines the edge connectivity of the mesh structure. Jameson's four-stage hybrid Runge-Kutta scheme is used to march the solution in time. The convergence rate is enhanced through the use of local time stepping and implicit residual averaging. As the solution evolves, the mesh is regenerated adaptively using flow field information. Mesh adaptation parameters are evaluated such that an estimated local numerical error is equally distributed over the whole domain. For inviscid flows, the present approach generates a complete unstructured triangular mesh using the advancing front method. For turbulent flows, the approach combines a local highly stretched structured triangular mesh in the boundary layer region with an unstructured mesh in the remaining regions to efficiently resolve the important flow features. One-equation and two-equation turbulence models are incorporated into the present unstructured approach. Results are presented for a wide range of flow problems including two-dimensional multi-element airfoils, two-dimensional cascades, and quasi-three-dimensional cascades. This approach is shown to gain flow resolution in the refined regions while achieving a great reduction in the computational effort and storage requirements since solution points are not wasted in regions where they are not required.

Jiang, Yi-Tsann↗
NASA NTRS1990

Adaptive quadrilateral and triangular finite-element scheme for compressible flows

The development of an adaptive mesh refinement procedure for analyzing high-speed compressible flows using the finite-element method is described. This new adaptation procedure, which uses both quadrilateral and triangular elements, was implemented with two explicit finite-element algorithms - the two-step Taylor-Galerkin and the multistep Galerkin-Runge-Kutta schemes. A von Neumann stability analysis and a rotating 'cosine hill'problem demonstrate the instability of the Taylor-Galerkin scheme when coupled with the adaptation procedure. For the same adaptive refinement scheme, the Galerkin-Runge-Kutta procedure yields stable solutions within its explicit stability limit. The utility of this new adaptation procedure for the prediction of compressible flow features is illustrated for inviscid problems involving strong shock interactions at hypersonic speeds.

Ramakrishnan, R.↗
NASA NTRS1990

3D Euler analysis of ducted propfan flowfields

A numerical method is presented for predicting the steady inviscid flow through a ducted propfan based on a time-marching solution of the three-dimensional Euler equations. A four-stage multiple-block Runge-Kutta finite volume numerical technique, utilizing implicit residual smoothing and a blended second and fourth difference dissipation, is applied to predict the transonic flowfield about both single-rotation and counter-rotations ducted rotors. Counter-rotation predictions are based on an average-passage system of equations approach. Calculations are performed for both a single sheared H-type grid system and a multiple-block grid system incorporating a C-type grid about the cowl. Numerical results are compared with experimental data for two cases: a low speed ducted propeller and a 1.15 pressure ratio fan stage.

Hall, Edward J.↗
NASA NTRS1985

Convergence acceleration of iterative solutions of Euler equations for transonic flow computations

In this paper, two acceleration techniques for Euler calculations are investigated. The first technique is an extrapolation procedure based on the Power Method; it is applicable when the iterative matrix has dominant eigenvalues. Both real and complex conjugate roots are allowed. The second technique is a generalization of the Minimal Residual Method, where the extrapolation step consists of a weighted combination of the corrections at different iteration levels and the weights are chosen to minimize the Lz norm of the residual. Numerical results, using Jameson's Runge-Kutta Multigrid Code, are presented. The extra computational work to apply either technique is negligible and the extra storage is not a problem on current supercomputers.

Hafez, M.↗
NASA NTRS1995

Application of traditional CFD methods to nonlinear computational aeroacoustics problems

This paper describes an implementation of a high order finite difference technique and its application to the category 2 problems of the ICASE/LaRC Workshop on Computational Aeroacoustics (CAA). Essentially, a popular Computational Fluid Dynamics (CFD) approach (central differencing, Runge-Kutta time integration and artificial dissipation) is modified to handle aeroacoustic problems. The changes include increasing the order of the spatial differencing to sixth order and modifying the artificial dissipation so that it does not significantly contaminate the wave solution. All of the results were obtained from the CM5 located at the Numerical Aerodynamic Simulation Laboratory. lt was coded in CMFortran (very similar to HPF), using programming techniques developed for communication intensive large stencils, and ran very efficiently.

Chyczewski, Thomas S.↗