Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “Runge Kutta Methods”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 379 records · Page 21
All sourcesNASA NTRSDOE OSTI
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.↗
NASA NTRS1988

Application of advanced grid generation techniques for flow field computations about complex configurations

In the computation of flowfields about complex configurations, it is very difficult to construct a boundary-fitted coordinate system. An alternative approach is to use several grids at once, each of which is generated independently. This procedure is called the multiple grids or zonal grids approach; its applications are investigated. The method conservative providing conservation of fluxes at grid interfaces. The Euler equations are solved numerically on such grids for various configurations. The numerical scheme used is the finite-volume technique with a three-stage Runge-Kutta time integration. The code is vectorized and programmed to run on the CDC VPS-32 computer. Steady state solutions of the Euler equations are presented and discussed. The solutions include: low speed flow over a sphere, high speed flow over a slender body, supersonic flow through a duct, and supersonic internal/external flow interaction for an aircraft configuration at various angles of attack. The results demonstrate that the multiple grids approach along with the conservative interfacing is capable of computing the flows about the complex configurations where the use of a single grid system is not possible.

Kathong, Monchai↗
NASA NTRS1991

Use of finite volume schemes for transition simulation

The use of finite-volume methods in the study of spatially and temporally evolving transitional flows over a flat plate is investigated. Schemes are developed with both central and upwind differencing. The compressible Navier-Stokes equations are solved with a Runge-Kutta time stepping scheme. Disturbances are determined using linear theory and superimposed at the inflow boundary. Time accurate integration is then used to allow temporal and spatial disturbance evolution. Characteristic-based boundary conditions are employed. The requirements of using finite-volume algorithms are studied in detail. Special emphasis is placed on difference schemes, grid resolution, and disturbance amplitudes. Moreover, comparisons are made with linear theory for small amplitude disturbances. Both subsonic and supersonic flows are considered, and it is shown that the locations of branch 1 and branch 2 of the neutral stability curve are well predicted, given sufficient resolution.

Fenno, Charles C., Jr.↗
NASA NTRSDate not supplied

Wall-Modeled Large Eddy Simulation Method for Unstructured-Grid Navier-Stokes Solvers

This paper reports on the implementation and assessment of a Wall-Modeled Large-Eddy Simulation (WMLES) methodology in an unstructured-grid, node-centered flow solver, FUN3D that is developed and supported at the NASA Langley Research Center. Finite-volume (FV) and finite-element (FE) discretization schemes considered in the study provide formal second-order spatial accuracy. Large-Eddy Simulations (LES) resolve large-scale turbulent-flow features and filter out small-scale effects using the Vreman subgrid-scale model. At solid-wall boundaries, a shear-stress model is employed to provide a proper boundary-flux closure. The nonlinear equations are integrated in time using either an optimized backward difference formula or an implicit multistage Runge-Kutta temporal scheme. The implicit equations at each time step are solved by strong nonlinear iteration schemes. WMLES demonstrations are shown for two high-lift configurations, namely, the McDonnell Douglas 30P30N multielement airfoil and a NASA High-Lift Common Research Model. Results show that the WMLES approaches implemented in the FV and FE discretization methods produce consistent solutions and are capable of capturing key aerodynamic characteristics and flow structures for high-lift configurations at a wide range of angles of attack including maximum-lift conditions. In the 30P30N example, correct trends in the variations of integrated aerodynamic forces and moments, surface pressure distributions, and boundary-layer profiles are captured as the Reynolds number is increased.

CFD; turbulence modeling; High-Lift flow simulatio↗
DOE OSTI2025

High order interpolation of magnetic fields with vector potential reconstruction for particle simulations

We propose a method for interpolating divergence-free continuous magnetic fields via vector potential reconstruction using Hermite interpolation, which ensures high-order continuity for applications requiring adaptive, high-order ordinary differential equation (ODE) integrators, such as the Dormand-Prince method. The method provides C(m) continuity and achieves high-order accuracy, making it particularly suited for particle trajectory integration and Poincaré section analysis under optimal integration order and timestep adjustments. Through numerical experiments, we demonstrate that the Hermite interpolation method preserves volume and continuity, which are critical for conserving toroidal canonical momentum and magnetic moment in guiding center simulations, especially over long-term trajectory integration. Furthermore, we analyze the impact of insufficient derivative continuity on Runge-Kutta schemes and show how it degrades accuracy at low error tolerances, introducing discontinuity-induced truncation errors. Lastly, we demonstrate performant Poincaré section analysis in two relevant settings of field data collocated from finite element meshes.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
NASA NTRS1994

Fourth-order 2N-storage Runge-Kutta schemes

A family of five-stage fourth-order Runge-Kutta schemes is derived; these schemes required only two storage locations. A particular scheme is identified that has desirable efficiency characteristics for hyperbolic and parabolic initial (boundary) value problems. This scheme is competitive with the classical fourth-order method (high-storage) and is considerably more efficient and accurate than existing third-order low-storage schemes.

Carpenter, Mark H.↗
NASA NTRS1988

Essentially non-oscillatory shock capturing methods applied to turbulence amplification in shock wave calculations

ENO (essentially non-oscillatory) schemes can provide uniformly high order accuracy right up to discontinuities while keeping sharp, essentially non-oscillatory shock transitions. Recently, an efficient implementation of ENO schemes was obtained based on fluxes and TVD Runge-Kutta time discretizations. The resulting code is very simple to program for multi-dimensions. ENO schemes are especially suitable for computing problems with both discontinuities and fine structures in smooth regions, such as shock interaction with turbulence, for which results for 1-D and 2-D Euler equations are presented. Much better resolution is observed by using third order ENO schemes than by using second order TVD schemes for such problems.

Osher, Stanley↗
NASA NTRS1988

A conservative approach for flow field calculations on multiple grids

In the computation of flow fields about complex configurations, it is very difficult to construct body-fitted coordinate systems. An alternative approach is to use several grids at once, each of which is generated independently. This procedure is called the 'multiple grids' or 'zonal grids' approach and its applications are investigated in this study. The method follows the conservative approach and provides conservation of fluxes at grid interfaces. The Euler equations are solved numerically on such grids for various configurations. The numerical scheme used is the finite-volume technique with a three-stage Runge-Kutta time integration. The code is vectorized and programmed to run on the CDC VPS-32 computer. Some steady state solutions of the Euler equations are presented and discussed.

Kathong, M.↗
NASA NTRS1988

A conservative approach for flow field calculations on multiple grids

In the computation of flow fields about complex configurations, it is very difficult to construct body-fitted coordinate systems. An alternative approach is to use several grids at once, each of which is generated independently. This procedure is called the multiple grids or zonal grids approach and its applications are investigated in this study. The method follows the conservative approach and provides conservation of fluxes at grid interfaces. The Euler equations are solved numerically on such grids for various configurations. The numerical scheme used is the finite-volume technique with a three-state Runge-Kutta time integration. The code is vectorized and programmed to run on the CDC VPS-32 computer. Some steady state solutions of the Euler equations are presented and discussed.

Kathong, Monchai↗