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An unsteady TVD Navier-Stokes solver was developed and applied to the problem of shock reflection on a circular cylinder. The obtained numerical results were compared with the Schlieren photos from an experimental study. These results show that the present computer code has the ability of capturing moving shocks.
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Improved algorithms for the solution of the three-dimensional, time-dependent Euler equations are presented for aerodynamic analysis involving unstructured dynamic meshes. The improvements have been developed recently to the spatial and temporal discretizations used by unstructured-grid flow solvers. The spatial discretization involves a flux-split approach that is naturally dissipative and captures shock waves sharply with at most one grid point within the shock structure. The temporal discretization involves either an explicit time-integration scheme using a multistage Runge-Kutta procedure or an implicit time-integration scheme using a Gauss-Seidel relaxation procedure, which is computationally efficient for either steady or unsteady flow problems. With the implicit Gauss-Seidel procedure, very large time steps may be used for rapid convergence to steady state, and the step size for unsteady cases may be selected for temporal accuracy rather than for numerical stability. Steady flow results are presented for both the NACA 0012 airfoil and the Office National d'Etudes et de Recherches Aerospatiales M6 wing to demonstrate applications of the new Euler solvers. The paper presents a description of the Euler solvers along with results and comparisons that assess the capability.
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Solutions have been computed and results are presented for Problem 1, the case of Mach 9 transitional flow past a 7 deg half-angle cone at zero incidence. The solutions were computed using a code developed for the integration of the parabolized Navier-Stokes equations. The algorithm employed in the code is based on a Roe-type flux-difference-splitting scheme applied following a finite-volume approach. The basic algorithm has been modified to make it implicit and second-order accurate in the crossflow directions. Results are presented in terms of surface pressure and heat transfer as well as boundary layer profiles of pitot pressure, Mach number, and tangential velocity. The case was recalculated several times in an effort to determine sensitivities to such parameters as grid density, wall temperature, turbulence model parameters, as well as freestream expansion. Comparisons with the experimental data are presented and discussed.
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Steady flow over the leading portion of a multicomponent airfoil section is studied using computational fluid dynamics (CFD) employing an unstructured grid. To simplify the problem, only the inviscid terms are retained from the Reynolds-averaged Navier-Stokes equations - leaving the Euler equations. The algorithm is derived using the finite-volume approach, incorporating explicit time-marching of the unsteady Euler equations to a time-asymptotic, steady-state solution. The inviscid fluxes are obtained through either of two approximate Riemann solvers: Roe's flux difference splitting or van Leer's flux vector splitting. Results are presented which contrast the solutions given by the two flux functions as a function of Mach number and grid resolution. Additional information is presented concerning code verification techniques, flow recirculation regions, convergence histories, and computational resources.
We present a new approach towards the construction of a genuinely multidimensional high-resolution scheme for computing steady-state solutions of the Euler equations of gas dynamics. The unique advantage of this approach is that the Gauss-Seidel relaxation is stable when applied directly to the high-resolution discrete equations, thus allowing us to construct a very efficient and simple multigrid steady-state solver. This is the only high-resolution scheme known to us that has this property. The two-dimensional scheme is presented in detail. It is formulated on triangular (structured and unstructured) meshes and can be interpreted as a genuinely two-dimensional extension of the Roe scheme. The quality of the solutions obtained using this scheme and the performance of the multigrid algorithm are illustrated by the numerical experiments. Construction of the three dimensional scheme is outlined briefly as well.
A floating-shock fitting method for the Euler equations has been developed that uses one-sided spatial differences along and across streamlines. The coordinate-invariant formulation of the spatial differences permits automatic capture of shears. Results are presented for unsteady shocked flow in a duct with a ramp, for supercritical flow over a circular cylinder, and for subsonic, transonic, and supersonic (0.3 is less than or equal to M(sub infinity) is less than 1.5) flow over airfoils. In flows with strong shears, the coordinate-invariant differencing concept appears to yield some gains in accuracy over Euler methods that rely on coordinate-aligned differencing concepts. In applications to transonic airfoils, fitted shocks have a tendency to be predicted upstream of captured shocks, regardless of whether coordinate-invariant or coordinate-aligned differencing is used. The coordinate-invariant differencing method requires between 2 and 3.5 times as much computing time as its coordinate-aligned counterpart.
A 3-D compressible Navier-Stokes solver has been developed and applied to 3-D viscous flow over clean and iced wings. This method uses a third order accurate finite volume scheme with flux difference splitting to model the inviscid fluxes, and second order accurate symmetric differences to model the viscous terms. The effects of turbulence are modeled using a Kappa-epsilon model. In the vicinity of the sold walls the kappa and epsilon values are modeled using Gorski's algebraic model. Sampling results are presented for surface pressure distributions, for untapered swept clean and iced wings made of NACA 0012 airfoil sections. The leading edge of these sections is modified using a simulated ice shape. Comparisons with experimental data are given.
The magnetosphere contains a significant amount of ionospheric O{+}, particularly during geomagnetically active times. The presence of ionospheric plasma in the magnetosphere has a notable impact on magnetospheric composition and processes. We present a new multifluid MHD version of the BATS-R-US model of the magnetosphere to track the fate and consequences of ionospheric outflow. The multi-fluid MHD equations are presented as are the novel techniques for overcoming the formidable challenges associated with solving them. Our new model is then applied to the May 4, 1998 and March 31, 2001 geomagnetic storms. The results are juxtaposed with traditional single- fluid MHD and multispecies MHD simulations from a previous study, thereby allowing us to assess the benefits of using a more complex model with additional physics. We find that our multi-fluid MHD model (with outflow) gives comparable results to the multi-species MHD model (with outflow), including a more strongly negative Dst, reduced CPCP, and a drastically improved magnetic field at geosynchronous orbit, as compared to single-fluid MHD with no outflow. Significant differences in composition and magnetic field are found between the multi-species and multi-fluid approach further away from the Earth. We further demonstrate the ability to explore pressure and bulk velocity differences between H{+} and O(+}, which is not possible when utilizing the other techniques considered.
Presentation topics include background and motivation; physical modeling including governing equations and thermochemistry; finite element formulation; results of inviscid thermal nonequilibrium chemically reacting flow and viscous thermal equilibrium chemical reacting flow; and near-term effort.