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First- and second-order explicit difference schemes are derived for a three-dimensional hyperbolic system of conservation laws, without recourse to dimensional factorization. All schemes are upwind biased and optimally stable.
This paper presents a computational and experimental investigation of supersonic flow past a cavity in a flat plate. The source of the particular interest in this problem is the ongoing study of the aerodynamic interference effects between a separating store and its bay in the parent body. An upwind relaxation scheme, utilizing flux vector splitting and line-Gauss-Seidel iterations, is used to solve Reynolds-averaged Navier-Stokes equations. Spatial discretizations of this two-dimensional analysis are based on implicit and finite-volume methods. Turbulence is modeled and shocks are captured. The flowfield of the symmetry plane at the half-width is computationally visualized and all flow properties are computed. Experimental tests are conducted in the Langley Unitary Plan Wind Tunnel to measure wall pressures and to capture schlieren photographs. Qualitative as well as quantitative data of computations and experiments agree very well. These two vehicles of investigation are merged to show open, closed and transitional cavity flow behaviors.
An Euler code designed for computing the unsteady, three-dimensional, transonic flow about single-rotating and counter-rotating propfans using dynamic blocked-grids is presented. The algorithm is a finite volume, flux-split, upwind, implicit scheme and solves the equations which have been written in a time-dependent curvilinear coordinate system. Relative motion of the blades for counter-rotating configurations is handled by requiring that grid lines be aligned after each discrete rotation of fore and aft rotor grid blocks. The method by which information is passed across block interfaces, as well as how downstream characteristic outflow boundary conditions which enforce simple radial equilibrium are implemented, is discussed. Comparisons of computed flow-field parameters and propfan performance with experimental data indicate good overall agreement between predictions and measurements.
The two-dimensional flow over a blunt leading-edge plate is simulated on the basis of an Euler/Navier-Stokes zonal scheme. The scheme uses an implicit upwind finite-volume scheme, which is based on the van Leer flux-vector splitting. It is shown that the Euler/Navier-Stokes zonal scheme with downstream boundary-layer compatibility conditions is accurate and efficient.
A shock-on-shock interaction near the cowl lip of an engine inlet is studied numerically by solving the compressible full Navier-Stokes equations. The present study focuses on Edney's type III shock interference pattern in which the turbulent free shear layer impinges on the cowl lip (modeled as a cylinder). Van Leer's flux-vector splitting upwind MUSCL scheme is used in the solver with algebraic turbulence models. Turbulent solutions of surface pressures and heat fluxes along the cowl lip are in reasonably good agreement with experiment except in the local shear layer impingement region. The present study showed some indications that higher order turbulent models may be required to improve the solution near the free shear layer impingement region.
The long-term goal is to develop the capability to predict chemically-reacting, multi-stream nozzle and plume flow fields. Two basic Navier-Stokes solvers, including the widely used F-3D code, are upgraded to include several upwind difference schemes and portable chemistry packages. Current computational capabilities for solving equilibrium single-stream and multi-stream, frozen gas, and finite rate chemistry problems are described. A variety of complex nozzle and plume flows were computed. Solutions presented include axisymmetric plume flow for ideal and equilibrium air, 3-D NASP nozzle/afterbody flow, and an internal nozzle calculation comparing various finite-rate chemistry packages.
A Navier-Stokes equations solver based on a pressure correction method with a pressure-staggered mesh and calculations of separated three-dimensional flows are presented. It is shown that the velocity pressure decoupling, which occurs when various pressure correction algorithms are used for pressure-staggered meshes, is caused by the ill-conditioned discrete pressure correction equation. The use of a partial differential equation for the incremental pressure eliminates the velocity pressure decoupling mechanism by itself and yields accurate numerical results. Example flows considered are a three-dimensional lid driven cavity flow and a laminar flow through a 90 degree bend square duct. For the lid driven cavity flow, the present numerical results compare more favorably with the measured data than those obtained using a formally third order accurate quadratic upwind interpolation scheme. For the curved duct flow, the present numerical method yields a grid independent solution with a very small number of grid points. The calculated velocity profiles are in good agreement with the measured data.
Recent progress in computational vortex flow aerodynamics at the Langley Research Center is reviewed. Emphasis is placed on Navier-Stokes methodology, both for compressible and incompressible flows, and results are presented from central and upwind-biased schemes for fully laminar, transitional, and fully turbulent flows. In addition, results are presented from selected potential-based methods to address the hierarchy of formulations presently available for the computational analysis of aerodynamic vortex flows. Some comparisons among this hierarchy of methods are shown.
A finite difference, three dimensional incompressible Navier-Stokes formulation to calculate the flow through turbopump components is utilized. The solution method is based on the pseudo compressibility approach and uses an implicit upwind differencing scheme together with the Gauss-Seidel line relaxation method. Both steady and unsteady flow calculations can be performed using the current algorithm. Here, equations are solved in steadily rotating reference frames by using the steady state formulation in order to simulate the flow through a turbopump inducer. Eddy viscosity is computed by using an algebraic mixing-length turbulence model. Numerical results are compared with experimental measurements and a good agreement is found between the two.
A finite-difference, three-dimensional incompressible Navier-Stokes formulation to calculate the flow through turbopump components is utilized. The solution method is based on the pseudocompressibility approach and uses an implicit-upwind differencing scheme together with the Gauss-Seidel line relaxation method. Both steady and unsteady flow calculations can be performed using the current algorithm. In this work, the equations are solved in steadily rotating reference frames by using the steady-state formulation in order to simulate the flow through a turbopump inducer. Eddy viscosity is computed by using an algebraic mixing-length turbulence model. Numerical results are compared with experimental measurements and a good agreement is found between the two. Included in the appendix is a paper on incompressible viscous flow through artificial heart devices with moving boundaries. Time-accurate calculations, such as impeller and diffusor interaction, will be reported in future work.
The extension of computational fluid dynamics techniques to artificial heart flow simulations is illustrated. Unsteady incompressible Navier-Stokes equations written in 3-D generalized curvilinear coordinates are solved iteratively at each physical time step until the incompressibility condition is satisfied. The solution method is based on the pseudo compressibility approach and uses an implicit upwind differencing scheme together with the Gauss-Seidel line relaxation method. The efficiency and robustness of the time accurate formulation of the algorithm are tested by computing the flow through model geometries. A channel flow with a moving indentation is computed and validated with experimental measurements and other numerical solutions. In order to handle the geometric complexity and the moving boundary problems, a zonal method and an overlapping grid embedding scheme are used, respectively. Steady state solutions for the flow through a tilting disk heart valve was compared against experimental measurements. Good agreement was obtained. The flow computation during the valve opening and closing is carried out to illustrate the moving boundary capability.
The objective of the proposed paper is to develop a computational procedure that solves incompressible Navier-Stokes equations for pump flows. The solution method is based on the pseudo-compressibility approach and uses an implicit-upwind differencing scheme together with the Gauss-Seidel line relaxation method. The equations are solved in steadily rotating reference frames and the centrifugal force and the Coriolis force are added to the equation of motion. As a benchmark problem, the flow through the Rocketdyne inducer is numerically simulated. A coarse grid solution is obtained with a single zone by using an algebraic turbulence model. In multi-zone fine grid computation, a one-equation Baldwin-Barth turbulence model is utilized. Numerical results are compared with experimental measurements and a good agreement is found between the two. The resulting computer code is then applied to the flow analysis inside a two-stage fuel pump impeller operating at 80 percent, 100 percent, and 120 percent of design flow.
A finite-difference, three-dimensional, incompressible Navier-Stokes formulation for calculating the flow through turbopump components is presented. The solution method is based on the pseudocompressibility approach and uses an implicit-upwind differencing scheme together with the Gauss-Seidel line-relaxation method. Both steady and unsteady flow calculations can be performed using the presented algorithm. In this paper, the equations are solved in steadily rotating reference frames by using the steady-state formulation in order to simulate the flow through a turbopump inducer. Eddy viscosity is computed by using the Baldwin-Lomax model. Numerical results are compared with experimental measurements and good agreement is found between the two. Time-accurate calculations will be reported in future publications.
Incompressible viscous turbulent flows over single- and multiple-element airfoils are numerically simulated in an efficient manner by solving the incompressible Navier-Stokes equations. The solution algorithm uses the method of pseudocompressibility with an upwind-differencing scheme for the convective fluxes and an implicit line-relaxation scheme to study high-lift take-off and landing configurations and to compute lift and drag at various angles of attack up to stall. Two different turbulence models are tested in computing the flow over an NACA 4412 airfoil. The approach used for multiple-element airfoils involves the use of multiple zones of structured grids fitted to each element. Two different approaches are compared: a patched system of grids and an overlaid Chimera system of grids. Computational results are presented for two-element, three-element, and four-element airfoil configurations. Excellent agreement with experimental surface-pressure coefficients is seen. The code converges in less than 200 iterations, requiring on the order of one minute of CPU time on a CRAY YMP per element in the airfoil configuration.
Compressible jet plumes were studied using a two-equation turbulence model. A space marching procedure based on an upwind numerical scheme was used to solve the governing equations and turbulence transport equations. The computed results indicate that extending the space marching procedure for solving supersonic/subsonic mixing problems can be stable, efficient and accurate. Moreover, a newly developed correction for compressible dissipation has been verified in fully expanded and underexpanded jet plumes. For a sonic jet plume, no improvement in results over the standard two-equation model was seen. However for a supersonic jet plume, the correction due to compressible dissipation successfully predicted the reduced spreading rate of the jet compared to the sonic case. The computed results were generally in good agreement with the experimental data.
A fully nonlinear 3-dimensional numerical model (GEOSIM), previously developed and validated for several cases of geophysical fluid flow, has been used to investigate the dynamical behavior of laboratory experiments of fluid flows similar to those of the Earth's atmosphere. The phenomena investigated are amplitude vacillation, and the response of the fluid system to uneven heating and cooling. The previous year's work included hysteresis in the transition between axisymmetric and wave flow. Investigation is also continuing of the flows in the Geophysical Fluid Flow Cell (GFFC), a low-gravity Spacelab experiment. Much of the effort in the past year has been spent in validation of the model under a wide range of external parameters including nonlinear flow regimes. With the implementation of a 3-dimensional upwind differencing scheme, higher spectral resolution, and a shorter time step, the model has been found capable of predicting the majority of flow regimes observed in one complete series of baroclinic annulus experiments of Pfeffer and co-workers. Detailed analysis of amplitude vacillation has revealed that the phase splitting described in the laboratory experiments occurs in some but not all cases. Through the use of animation of the models output, a vivid 3-dimensional view of the phase splitting was shown to the audience of the Southeastern Geophysical Fluid Dynamics Conference in March of this year. A study on interannual variability was made using GEOSIM with periodic variations in the thermal forcing. Thus far, the model has not predicted a chaotic behavior as observed in the experiments, although there is a sensitivity in the wavenumber selection to the initial conditions. Work on this subject, and on annulus experiments with non-axisymmetric thermal heating, will continue. The comparison of GEOSIM's predictions will result from the Spacelab 3 GFFC experiments continued over the past year, on a 'back-burner' basis. At this point, the study (in the form of a draft of a journal article) is nearly completed. The results from GEOSIM compared very well with the experiments, and the use of the model allows the demonstration of flow mechanics that were not possible with the experimental data. For example, animation of the model output shows that the forking of the spiral bands is a transient phenomenon, due to the differential east-west propagation of convection bands from different latitudes.
Flow through pump components, such as an inducer and an impeller, is efficiently simulated by solving the incompressible Navier-Stokes equations. The solution method is based on the pseudocompressibility approach and uses an implicit-upwind differencing scheme together with the Gauss-Seidel line relaxation method. the equations are solved in steadily rotating reference frames and the centrifugal force and the Coriolis force are added to the equation of motion. Current computations use a one-equation Baldwin-Barth turbulence model which is derived from a simplified form of the standard k-epsilon model equations. The resulting computer code is applied to the flow analysis inside a generic rocket engine pump inducer, a fuel pump impeller, and SSME high pressure fuel turbopump impeller. Numerical results of inducer flow are compared with experimental measurements. In the fuel pump impeller, the effect of downstream boundary conditions is investigated. Flow analyses at 80 percent, 100 percent, and 120 percent of design conditions are presented.
The incompressible, viscous, turbulent flow over single and multi-element airfoils is numerically simulated in an efficient manner by solving the incompressible Navier-Stokes equations. The computer code uses the method of pseudo-compressibility with an upwind-differencing scheme for the convective fluxes and an implicit line-relaxation solution algorithm. The motivation for this work includes interest in studying the high-lift take-off and landing configurations of various aircraft. In particular, accurate computation of lift and drag at various angles of attack, up to stall, is desired. Two different turbulence models are tested in computing the flow over an NACA 4412 airfoil; an accurate prediction of stall is obtained. The approach used for multi-element airfoils involves the use of multiple zones of structured grids fitted to each element. Two different approaches are compared: a patched system of grids, and an overlaid Chimera system of grids. Computational results are presented for two-element, three-element, and four-element airfoil configurations. Excellent agreement with experimental surface pressure coefficients is seen. The code converges in less than 200 iterations, requiring on the order of one minute of CPU time (on a CRAY YMP) per element in the airfoil configuration.