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NASA NTRS1993

Effects of spatial order of accuracy on the computation of vortical flowfields

The effect of the order-of-accuracy, used for the spatial discretization, on the resolution of the leading edge vortices over sharp-edged delta wings is investigated. The flowfield is computed using a viscous/inviscid zonal approach. The viscous flow in the vicinity of the wing is computed using the conservative formulation of the compressible, thin-layer Navier-Stokes equations. The leeward-side vortical flowfield and the other flow regions away from the surface are computed as inviscid. The time integration is performed with both an explicit fourth-order Runge-Kutta scheme and an implicit, factorized, iterative scheme. High-order-accurate inviscid fluxes are computed using both a conservative and a non-conservative (primitive variable) formulation. The nonlinear, inviscid terms of the primitive variable form of the governing equations are evaluated with a finite-difference numerical scheme based on the sign of the eigenvalues. High-order, upwind-biased, finite difference formulas are used to evaluate the derivatives of the nonlinear convective terms. Computed results are compared with available experimental data, and comparisons of the flowfield in the vicinity of the vortex cores are presented.

Ekaterinaris, J. A.↗

NASA NTRS1995

Flow simulations about steady-complex and unsteady moving configurations using structured-overlapped and unstructured grids

The limiting factor in simulating flows past realistic configurations of interest has been the discretization of the physical domain on which the governing equations of fluid flow may be solved. In an attempt to circumvent this problem, many Computational Fluid Dynamic (CFD) methodologies that are based on different grid generation and domain decomposition techniques have been developed. However, due to the costs involved and expertise required, very few comparative studies between these methods have been performed. In the present work, the two CFD methodologies which show the most promise for treating complex three-dimensional configurations as well as unsteady moving boundary problems are evaluated. These are namely the structured-overlapped and the unstructured grid schemes. Both methods use a cell centered, finite volume, upwind approach. The structured-overlapped algorithm uses an approximately factored, alternating direction implicit scheme to perform the time integration, whereas, the unstructured algorithm uses an explicit Runge-Kutta method. To examine the accuracy, efficiency, and limitations of each scheme, they are applied to the same steady complex multicomponent configurations and unsteady moving boundary problems. The steady complex cases consist of computing the subsonic flow about a two-dimensional high-lift multielement airfoil and the transonic flow about a three-dimensional wing/pylon/finned store assembly. The unsteady moving boundary problems are a forced pitching oscillation of an airfoil in a transonic freestream and a two-dimensional, subsonic airfoil/store separation sequence. Accuracy was accessed through the comparison of computed and experimentally measured pressure coefficient data on several of the wing/pylon/finned store assembly's components and at numerous angles-of-attack for the pitching airfoil. From this study, it was found that both the structured-overlapped and the unstructured grid schemes yielded flow solutions of comparable accuracy for these simulations. This study also indicated that, overall, the structured-overlapped scheme was slightly more CPU efficient than the unstructured approach.

Newman, James C., III↗

NASA NTRS1985

Spectral methods for modeling supersonic chemically reacting flow fields

A partial implicit numerical algorithm has been developed for solving the equations describing chemically reacting supersonic flows. The algorithm employs a two-stage Runge-Kutta method for integrating the equations in time and a Chebyshev spectral method for integrating the equations in space. The accuracy and efficiency of the new technique have been assessed by comparison with an existing implicit finite-difference procedure for modeling chemically reacting flows. The comparison showed that the new procedure yielded equivalent accuracy on much coarser grids as compared to the finite-difference procedure with resultant significant gains in computational efficiency.

Drummond, J. P.↗

NASA NTRS1989

An unconditionally stable Runge-Kutta method for unsteady flows

A quasi-three-dimensional analysis was developed for unsteady rotor-stator interaction in turbomachinery. The analysis solves the unsteady Euler or thin-layer Navier-Stokes equations in a body-fitted coordinate system. It accounts for the effects of rotation, radius change, and stream surface thickness. The Baldwin-Lomax eddy viscosity model is used for turbulent flows. The equations are integrated in time using a four-stage Runge-Kutta scheme with a constant time step. Implicit residual smoothing was employed to accelerate the solution of the time accurate computations. The scheme is described and accuracy analyses are given. Results are shown for a supersonic through-flow fan designed for NASA Lewis. The rotor:stator blade ratio was taken as 1:1. Results are also shown for the first stage of the Space Shuttle Main Engine high pressure fuel turbopump. Here the blade ratio is 2:3. Implicit residual smoothing was used to increase the time step limit of the unsmoothed scheme by a factor of six with negligible differences in the unsteady results. It is felt that the implicitly smoothed Runge-Kutta scheme is easily competitive with implicit schemes for unsteady flows while retaining the simplicity of an explicit scheme.

Jorgenson, Philip C. E.↗

NASA NTRS1989

An unconditionally stable Runge-Kutta method for unsteady rotor-stator interaction

A quasi-three-dimensional analysis has been developed for unsteady rotor-stator interaction in turbomachinery. The analysis solves the unsteady Euler or thin-layer Navier-Stokes equations in a body-fitted coordinate system. It accounts for the effects of rotation, radius change, and stress-surface thickness. The Baldwin-Lomax eddy-viscosity model is used for turbulent flows. The equations are integrated in time using an explicit four-stage Runge-Kutta scheme with a constant time step. Implicit residual smoothing is used to increase the stability limit of the time-accurate computations. The scheme is described, and stability and accuracy analyses are given.

Chima, Rodrick V.↗

NASA NTRS2002

Higher Order Time Integration Schemes for the Unsteady Navier-Stokes Equations on Unstructured Meshes

The efficiency gains obtained using higher-order implicit Runge-Kutta schemes as compared with the second-order accurate backward difference schemes for the unsteady Navier-Stokes equations are investigated. Three different algorithms for solving the nonlinear system of equations arising at each timestep are presented. The first algorithm (NMG) is a pseudo-time-stepping scheme which employs a non-linear full approximation storage (FAS) agglomeration multigrid method to accelerate convergence. The other two algorithms are based on Inexact Newton's methods. The linear system arising at each Newton step is solved using iterative/Krylov techniques and left preconditioning is used to accelerate convergence of the linear solvers. One of the methods (LMG) uses Richardson's iterative scheme for solving the linear system at each Newton step while the other (PGMRES) uses the Generalized Minimal Residual method. Results demonstrating the relative superiority of these Newton's methods based schemes are presented. Efficiency gains as high as 10 are obtained by combining the higher-order time integration schemes with the more efficient nonlinear solvers.

Jothiprasad, Giridhar↗

NASA NTRS1988

A Navier-Stokes solver for cascade flows

A computer code for solving the Reynolds averaged full Navier-Stokes equations has been developed and applied using sheared H-type grids. The Baldwin-Lomax eddy-viscosity model is used for turbulence closure. The integration in time is based on an explicit four-stage Runge-Kutta scheme. Local time stepping, variable coefficient implicit residual smoothing, and a full multigrid method have been implemented to accelerate steady state calculations. Comparisons with experimental data show that the code is an accurate viscous solver and can give very good blade-to-blade predictions for engineering applications in less than 100 multigrid cycles on the finest mesh.

Arnone, A.↗

NASA NTRS1993

A Navier-Stokes solver for turbomachinery applications

A computer code for solving the Reynolds-averaged full Navier-Stokes equations has been developed and applied using H- and C-type grids. The Baldwin-Lomax eddy-viscosity model is used for turbulence closure. The integration in time is based on an explicit four-stage Runge-Kutta scheme. Local time stepping, variable coefficient implicit residual smoothing, and a full multigrid method have been implemented to accelerate steady-state calculations. A grid independence analysis is presented for a transonic rotor blade. Comparisons with experimental data show that the code is an accurate viscous solver and can give very good blade-to-blade predictions for engineering applications.

Arnone, A.↗

NASA NTRS1977

An ILLIAC program for the numerical simulation of homogeneous incompressible turbulence

An algorithm and ILLIAC computer program, developed for the simulation of homogeneous incompressible turbulence in the presence of an applied mean strain, are described. The turbulence field is represented spatially by a truncated triple Fourier series (spectral method) and followed in time using a fourth-order Runge-Kutta algorithm. These include: (1) transformation of variables suggested by Taylor's sudden-distortion theory; (2) implicit viscous diffusion by use of an integrating factor; (3) implicit pressure calculation suggested by Taylor's sudden-distortion theory, and (4) inexpensive control of aliasing by random and phased coordinate shifts.

Rogallo, R. S.↗

NASA NTRS1993

Development of an explicit multiblock/multigrid flow solver for viscous flows in complex geometries

A new computer program is being developed for doing accurate simulations of compressible viscous flows in complex geometries. The code employs the full compressible Navier-Stokes equations. The eddy viscosity model of Baldwin and Lomax is used to model the effects of turbulence on the flow. A cell centered finite volume discretization is used for all terms in the governing equations. The Advection Upwind Splitting Method (AUSM) is used to compute the inviscid fluxes, while central differencing is used for the diffusive fluxes. A four-stage Runge-Kutta time integration scheme is used to march solutions to steady state, while convergence is enhanced by a multigrid scheme, local time-stepping and implicit residual smoothing. To enable simulations of flows in complex geometries, the code uses composite structured grid systems where all grid lines are continuous at block boundaries (multiblock grids). Example results are shown a flow in a linear cascade, a flow around a circular pin extending between the main walls in a high aspect-ratio channel, and a flow of air in a radial turbine coolant passage.

Steinthorsson, E.↗

NASA NTRS1993

Development of an explicit multiblock/multigrid flow solver for viscous flows in complex geometries

A new computer program is being developed for doing accurate simulations of compressible viscous flows in complex geometries. The code employs the full compressible Navier-Stokes equations. The eddy viscosity model of Baldwin and Lomax is used to model the effects of turbulence on the flow. A cell centered finite volume discretization is used for all terms in the governing equations. The Advection Upwind Splitting Method (AUSM) is used to compute the inviscid fluxes, while central differencing is used for the diffusive fluxes. A four-stage Runge-Kutta time integration scheme is used to march solutions to steady state, while convergence is enhanced by a multigrid scheme, local time-stepping, and implicit residual smoothing. To enable simulations of flows in complex geometries, the code uses composite structured grid systems where all grid lines are continuous at block boundaries (multiblock grids). Example results shown are a flow in a linear cascade, a flow around a circular pin extending between the main walls in a high aspect-ratio channel, and a flow of air in a radial turbine coolant passage.

Steinthorsson, E.↗

NASA NTRS1976

Conservation properties of numerical integration methods for systems of ordinary differential equations

If a system of ordinary differential equations represents a property conserving system that can be expressed linearly (e.g., conservation of mass), it is then desirable that the numerical integration method used conserve the same quantity. It is shown that both linear multistep methods and Runge-Kutta methods are 'conservative' and that Newton-type methods used to solve the implicit equations preserve the inherent conservation of the numerical method. It is further shown that a method used by several authors is not conservative.

Rosenbaum, J. S.↗

NASA NTRS1985

Spectral methods for modeling supersonic chemically reacting flow fields

A numerical algorithm was developed for solving the equations describing chemically reacting supersonic flows. The algorithm employs a two-stage Runge-Kutta method for integrating the equations in time and a Chebyshev spectral method for integrating the equations in space. The accuracy and efficiency of the technique were assessed by comparison with an existing implicit finite-difference procedure for modeling chemically reacting flows. The comparison showed that the procedure presented yields equivalent accuracy on much coarser grids as compared to the finite-difference procedure with resultant significant gains in computational efficiency.

Drummond, J. P.↗

NASA NTRS1993

Multigrid time-accurate integration of Navier-Stokes equations

Efficient acceleration techniques typical of explicit steady-state solvers are extended to time-accurate calculations. Stability restrictions are greatly reduced by means of a fully implicit time discretization. A four-stage Runge-Kutta scheme with local time stepping, residual smoothing, and multigridding is used instead of traditional time-expensive factorizations. Some applications to natural and forced unsteady viscous flows show the capability of the procedure.

Arnone, Andrea↗

NASA NTRS1993

Multigrid time-accurate integration of Navier-Stokes equations

Arnone, Andrea↗

NASA NTRS1982

Computer program for solving laminar, transitional, or turbulent compressible boundary-layer equations for two-dimensional and axisymmetric flow

A numerical algorithm and computer program are presented for solving the laminar, transitional, or turbulent two dimensional or axisymmetric compressible boundary-layer equations for perfect-gas flows. The governing equations are solved by an iterative three-point implicit finite-difference procedure. The software, program VGBLP, is a modification of the approach presented in NASA TR R-368 and NASA TM X-2458, respectively. The major modifications are: (1) replacement of the fourth-order Runge-Kutta integration technique with a finite-difference procedure for numerically solving the equations required to initiate the parabolic marching procedure; (2) introduction of the Blottner variable-grid scheme; (3) implementation of an iteration scheme allowing the coupled system of equations to be converged to a specified accuracy level; and (4) inclusion of an iteration scheme for variable-entropy calculations. These modifications to the approach presented in NASA TR R-368 and NASA TM X-2458 yield a software package with high computational efficiency and flexibility. Turbulence-closure options include either two-layer eddy-viscosity or mixing-length models. Eddy conductivity is modeled as a function of eddy viscosity through a static turbulent Prandtl number formulation. Several options are provided for specifying the static turbulent Prandtl number. The transitional boundary layer is treated through a streamwise intermittency function which modifies the turbulence-closure model. This model is based on the probability distribution of turbulent spots and ranges from zero to unity for laminar and turbulent flow, respectively. Several test cases are presented as guides for potential users of the software.

Harris, J. E.↗

NASA NTRS1992

Investigation of upwind, multigrid, multiblock numerical schemes for three dimensional flows. Volume 1: Runge-Kutta methods for a thin layer Navier-Stokes solver

A state-of-the-art computer code has been developed that incorporates a modified Runge-Kutta time integration scheme, upwind numerical techniques, multigrid acceleration, and multi-block capabilities (RUMM). A three-dimensional thin-layer formulation of the Navier-Stokes equations is employed. For turbulent flow cases, the Baldwin-Lomax algebraic turbulence model is used. Two different upwind techniques are available: van Leer's flux-vector splitting and Roe's flux-difference splitting. Full approximation multi-grid plus implicit residual and corrector smoothing were implemented to enhance the rate of convergence. Multi-block capabilities were developed to provide geometric flexibility. This feature allows the developed computer code to accommodate any grid topology or grid configuration with multiple topologies. The results shown in this dissertation were chosen to validate the computer code and display its geometric flexibility, which is provided by the multi-block structure.

Cannizzaro, Frank E.↗

NASA NTRS1994

Interlaminar stress analysis of dropped-ply laminated plates and shells by a mixed method

A mixed method of approximation based on Reissner's variational principle is developed for the linear analysis of interlaminar stresses in laminated composites, with special interest in laminates that contain terminated internal plies (dropped-ply laminates). Two models are derived, one for problems of generalized plane deformation and the other for the axisymmetric response of shells of revolution. A layerwise approach is taken in which the stress field is assumed with an explicit dependence on the thickness coordinate in each layer. The dependence of the stress field on the thickness coordinate is determined such that the three-dimensional equilibrium equations are satisfied by the approximation. The solution domain is reduced to one dimension by integration through the thickness. Continuity of tractions and displacements between layers is imposed. The governing two-point boundary value problem is composed of a system of both differential and algebraic equations (DAE's) and their associated boundary conditions. Careful evaluation of the system of DAE's was required to arrive at a form that allowed application of a one-step finite difference approximation. A two-stage Gauss implicit Runge-Kutta finite difference scheme was used for the solution because of its relatively high degree of accuracy. Patch tests of the two models revealed problems with solution accuracy for the axisymmetric model of a cylindrical shell loaded by internal pressure. Parametric studies of dropped-ply laminate characteristics and their influence on the interlaminar stresses were performed using the generalized plane deformation model. Eccentricity of the middle surface of the laminate through the ply drop-off was found to have a minimal effect on the interlaminar stresses under longitudinal compression, transverse tension, and in-plane shear. A second study found the stiffness change across the ply termination to have a much greater influence on the interlaminar stresses.

Harrison, Peter N.↗