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The combined use of an implicit, upwind finite-volume scheme with an 'equivalent' gamma formulation (for real gas effects) and with a three-dimensional conservative patched grid formulation is discussed. Results are presented over a wide Mach number range on both single and patched grids.
General difference approximations to the fluid dynamic equations require an artificial viscosity in order to converge to a steady state. This artificial viscosity serves two purposes. One is to suppress high frequency noise which is not damped by the central differences. The second purpose is to introduce an entropy-like condition so that shocks can be captured. These viscosities need a coefficient to measure the amount of viscosity to be added. In the standard scheme, a scalar coefficient is used based on the spectral radius of the Jacobian of the convective flux. However, this can add too much viscosity to the slower waves. Hence, it is suggested that a matrix viscosity be used. This gives an appropriate viscosity for each wave component. With this matrix valued coefficient, the central difference scheme becomes closer to upwind biased methods.
Two algorithms for the solution of the time-dependent Euler equations are presented for unsteady aerodynamic analysis of oscillating airfoils. Both algorithms were developed for use on an unstructured grid made up of triangles. The first flow solver involves a Runge-Kutta time-stepping scheme with a finite-volume spatial discretization that reduces to central differencing on a rectangular mesh. The second flow solver involves a modified Euler time-integration scheme with an upwind-biased spatial discretization based on the flux-vector splitting of Van Leer. The paper presents descriptions of the Euler solvers and dynamic mesh algorithm along with results which assess the capability.
A numerically efficient method is presented for solving the three-dimensional governing equations of the viscous compressible flow about complex configurations with topologically different components. The physical domain is decomposed into regions for which the grid generation is relatively simple and virtually with no significant restrictions. The Navier-Stokes equations are solved by an implicit, approximately factored, upwind, finite-volume scheme. The block inversions and the diagonalized scalar inversions of the coefficient matrices are modified to allow the holes created in the computational domain by the embedded and overlapped grids. The convergence is accelerated by a multigrid algorithm despite the existence of such holes. The solution for s supersonic flow past a blunt-nose-cylinder at high angle-of-attack is obtained using a C-O grid embedded in a global Cartesian grid.
Scramjet simulant-gas exhaust flows are computed using two-dimensional Navier-Stokes (NS) equations. The NS equations are solved using an implicit, upwind, finite-volume scheme, and the Reynolds stresses are modeled algebraically. The solutions are compared with experimental data where the freestream and the nozzle exhaust fluids are air. The NS equations for the flow, where the scramjet exhaust simulant gas is a mixture of Freon-12 and argon, are solved by an explicit, finite difference scheme. The results are compared with experimental data, and certain discrepancies are explained.
Navier-Stokes computations of three-dimensional vortical flows over a round-edged double-delta wing and a tangent-ogive forebody are performed using an implicit upwind finite-difference scheme. Computed particle traces are compared with experimental oil-flow streaks.
Supersonic flow about cone calculated accurately. Report presents theoretical basis of computer code solving parabolized Navier-Stokes equations of supersonic and hypersonic flow. For increased accuracy in resolution of details of strong aerodynamic shocks, code incorporates implicit, finite-volume, upwind numerical-integration scheme. Performs well in numerical simulations of flows around simple bodies.
An upwind cell-centered scheme for the solution of steady laminar viscous high-speed flows is implemented on unstructured two-dimensional meshes. The first-order implementation employs Roe's (1981) approximate Riemann solver, and a higher-order extension is produced by using linear reconstruction with limiting. The procedure is applied to the solution of inviscid subsonic flow over an airfoil, inviscid supersonic flow past a cylinder, and viscous hypersonic flow past a double ellipse. A detailed study is then made of a hypersonic laminar viscous flow on a 24-deg compression corner. It is shown that good agreement is achieved with previous predictions using finite-difference and finite-volume schemes. However, these predictions do not agree with experimental observations. With refinement of the structured grid at the leading edge, good agreement with experimental observations for the distributions of wall pressure, heating rate and skin friction is obtained.
The unsteady, compressible, double and single thin-layer Navier-Stokes equations are used to numerically study for the passive control of steady and unsteady, supersonic asymmetric flows around circular and noncircular cones. The main computational scheme of the present study is the implicit upwind, flux-difference splitting, finite-volume scheme. Passive control of flow asymmetry is studied by using a vertical fin in the leeward plane of geometric symmetry and side strakes with and without thickness at different orientations. The study is focused on circular-section cones since they are the most potential section-shapes for strong flow asymmetry. Side-strakes passive control has been shown to be more efficient and practical than the vertical-fin passive control.
A new two-dimensional computer code was developed to analyze the viscous flow around unconventional airfoils at various Mach numbers and angles of attack. The Navier-Stokes equations are solved using an implicit, upwind, finite-volume scheme. Both laminar and turbulent flows can be computed. A new nonequilibrium turbulence closure model was developed for computing turbulent flows. This two-layer eddy viscosity model was motivated by the success of the Johnson-King model in separated flow regions. The influence of history effects are described by an ordinary differential equation developed from the turbulent kinetic energy equation. The performance of the present code was evaluated by solving the flow around three airfoils using the Reynolds time-averaged Navier-Stokes equations. Excellent results were obtained for both attached and separated flows about the NACA 0012 airfoil, the RAE 2822 airfoil, and the Integrated Technology A 153W airfoil. Based on the comparison of the numerical solutions with the available experimental data, it is concluded that the present code in conjunction with the new nonequilibrium turbulence model gives excellent results.
A data parallel mesh-vertex upwind finite-volume scheme for solving the Euler equations on triangular unstructured meshes is described. A novel vertex-based partitioning of the problem is introduced which minimizes the computation and communication costs associated with distributing the computation to the processors of a massively parallel computer. Finally, the performance of this unstructured computation on 8K processors of the Connection Machine CM-2 is compared with one processor of a Cray-YMP. The experiments show that 8K processors of the CM-2 achieve approximately 70 percent of the performance of one processor of the Cray-YMP on the unstructured mesh computations described here.
The unsteady, compressible, thin-layer Navier-Stokes equations are used to numerically study the passive control of steady and unsteady supersonic asymmetric flows around circular and noncircular cones. The main computational scheme of the present study is an implicit upwind, flux-difference splitting, finite-volume scheme. Passive control of flow asymmetry is studied by using a vertical fin in the leeward plane of geometric symmetry and side strakes with and without thickness at different orientations. The study focuses on circular-section cones since they are the most likely section-shapes for strong flow asymmetry. Side-strake passive control is shown to be more efficient and practical than vertical-fin passive control.
This paper presents recent improvements in high-order K-exact reconstruction on unstructured meshes. The new reconstruction procedures are incorporated into a basic upwind finite-volume scheme suitable for solving scalar advection-diffusion equations as well as the Euler and Navier-Stokes equations. Numerical calculations are performed comparing the present method with lower order accurate reconstruction procedures (piecewise constant and piecewise linear) and various competing technologies such as the fluctuation splitting method of Roe (1987) and Deconinck et al. (1992) and a system-variant of the streamline diffusion Petrov-Galerkin method developed by Hansbo (1991) and Hansbo and Johnson (1991). Five test problems are used in the numerical comparisons: scalar circular advection, transonic and supersonic Euler flow, laminar boundary-layer flow, and general compressible Navier-Stokes flow.
The primary objective of the present study is to assess the limitations and capabilities of RPLUS in predicting the entrainment, mixing, and burning characteristics of a high subsonic planar shear layer; these computations are performed in conjunction with an experiment being conducted at NASA Lewis Research Center. Turbulence is modeled by a two-equation k-epsilon closure modified for compressibility effects. Combustion is modeled by two different models: one in which a finite-rate laminar chemistry model is used for H2-O2 oxidation, and the other based on a composition joint pdf approach to account for the turbulence-chemistry interaction effects. Between upwind and centered differencing schemes, minimal differences are observed for the flow properties of a nonreacting shear layer case studied. In comparison with experimental data, computed growth rates are underpredicted in both reacting and nonreacting cases of a shear layer, but more severely for the reacting flows. However, the qualitative agreement between computation and experiment is reasonably good for the other observable characteristics, and indicative of potential for further improvements.
One of the major design tasks involved in reducing aircraft drag is the integration of the engine nacelles and airframe. With this impetus, nacelle shapes with and without the presence of a flat-plate wing nearby were optimized. This also served as a demonstration of the 3-D version of the recently developed aerodynamic design optimization methodology using sensitivity analysis, ADOS. The required flow analyses were obtained by solving the three-dimensional, compressible, thin-layer Navier-Stokes equations using an implicit, upwind-biased, finite volume scheme. The sensitivity analyses were performed using the preconditioned version of the SADD scheme (sensitivity analysis on domain decomposition). In addition to demonstrating the present method's capability for automatic optimization, the results offered some insight into two important issues related to optimizing the shapes of multicomponent configurations in close proximity. First, inclusion of the mutual interference between the components resulted in a different shape as opposed to shaping an isolated component. Secondly, exclusion of the viscous effects compromised not only the flow physics but also the optimized shapes even for isolated components.
Six months of funding was received for the proposed three year research program (funding for the period from March 1, 1997 to August 31, 1997). Although the official starting date for the project was March 1, 1997, no funding for the project was received until July 1997. In the funded research period, considerable progress was made on Phase I of the proposed research program. The initial research efforts concentrated on applying the 10-, 20-, and 35-moment Gaussian-based closures to a series of standard two-dimensional non-reacting single species test flow problems, such as the flat plate, couette, channel, and rearward facing step flows, and to some other two-dimensional flows having geometries similar to those encountered in chemical-vapor deposition (CVD) reactors. Eigensystem analyses for these systems for the case of two spatial dimensions was carried out and efficient formulations of approximate Riemann solvers have been formulated using these eigenstructures. Formulations to include rotational non-equilibrium effects into the moment closure models for the treatment of polyatomic gases were explored, as the original formulations of the closure models were developed strictly for gases composed of monatomic molecules. The development of a software library and computer code for solving relaxing hyperbolic systems in two spatial dimensions of the type arising from the closure models was also initiated. The software makes use of high-resolution upwind finite-volumes schemes, multi-stage point implicit time stepping, and automatic adaptive mesh refinement (AMR) to solve the governing conservation equations for the moment closures. The initial phase of the code development was completed and a numerical investigation of the solutions of the 10-moment closure model for the simple two-dimensional test cases mentioned above was initiated. Predictions of the 10-moment model were compared to available theoretical solutions and the results of direct-simulation Monte Carlo (DSMC) calculations. The first results of this study were presented at a meeting last year.
A detailed comparison of numerical results obtained by solving the Euler equations for the inviscid flow over delta wings in reentry configurations is presented. The investigation involves a side-by-side comparison between independently developed upwind Euler solvers at VKI (M3D) and NASA Johnson Space Center (E3D) using identical grids. In both solvers the governing equations are integrated by means of time marching finite volume shock capturing methods, based on a cell centered upwind evaluation of the cell face fluxes and nonlinear limiters. High resolution schemes are obtained via MUSCL characteristic variable extrapolation to ensure total variation diminishing (TVD) properties and therefore monotonic discontinuity capturing. Further comparisons are made with several published results by other authors based on both upwind and central discretizations. The present results contribute in asserting the high resolution upwind TVD schemes as the most reliable numerical technique to handle the strong discontinuities typical of high speed flows. The performances of the two upwind solvers are satisfactory and the numerical results in good agreement. However, the important issue of reaching a grid converged solution for the present complex three-dimensional inviscid flow problems was not achieved.
Several algorithms for introducing artificial dissipation into a central difference approximation to the Euler and Navier Stokes equations are considered. The focus of the paper is on the convective upwind and split pressure (CUSP) scheme, which is designed to support single interior point discrete shock waves. This scheme is analyzed and compared in detail with scalar and matrix dissipation (MATD) schemes. Resolution capability is determined by solving subsonic, transonic, and hypersonic flow problems. A finite-volume discretization and a multistage time-stepping scheme with multigrid are used to compute solutions to the flow equations. Numerical results are also compared with either theoretical solutions or experimental data. For transonic airfoil flows the best accuracy on coarse meshes for aerodynamic coefficients is obtained with a simple MATD scheme.