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NASA NTRS1991

A fast implicit upwind solution algorithm for three-dimensional unstructured dynamic meshes

A fast implicit upwind algorithm for the solution of the time-dependent Euler equations is presented for aerodynamic analysis involving unstructured dynamic meshes. The spatial discretization of the scheme is based on the upwind approach of Roe, referred to as flux-difference splitting (FDS). The FDS approach is naturally dissipative and captures shock waves and contact discontinuities sharply. The temporal discretization of the scheme involves an implicit time-integration using a two-sweep Gauss-Seidel relaxation procedure. The procedure is computationally efficient for either steady or unsteady flow problems. A detailed description is given of the implicit upwind solution algorithm along with results which assess the capability. The results are presented for the NACA 0012 airfoil and for the Boeing 747 aircraft. The 747 geometry includes the fuselage, wing, horizontal and vertical tails, under-wing pylons, and flow-through engine nacelles. Euler solutions for the 747 aircraft on an unstructured tetrahedral mesh containing approximately 100,000 cells were obtained to engineering accuracy in less than one hour CPU time on a Cray-2 computer.

Batina, John T.↗

NASA NTRS1991

Supersonic propulsion simulation by incorporating component models in the large perturbation inlet (LAPIN) computer code

An approach to simulating the internal flows of supersonic propulsion systems is presented. The approach is based on a fairly simple modification of the Large Perturbation Inlet (LAPIN) computer code. LAPIN uses a quasi-one dimensional, inviscid, unsteady formulation of the continuity, momentum, and energy equations. The equations are solved using a shock capturing, finite difference algorithm. The original code, developed for simulating supersonic inlets, includes engineering models of unstart/restart, bleed, bypass, and variable duct geometry, by means of source terms in the equations. The source terms also provide a mechanism for incorporating, with the inlet, propulsion system components such as compressor stages, combustors, and turbine stages. This requires each component to be distributed axially over a number of grid points. Because of the distributed nature of such components, this representation should be more accurate than a lumped parameter model. Components can be modeled by performance map(s), which in turn are used to compute the source terms. The general approach is described. Then, simulation of a compressor/fan stage is discussed to show the approach in detail.

Cole, Gary L.↗

NASA NTRS1992

A compressible Navier-Stokes solver with two-equation and Reynolds stress turbulence closure models

This report outlines the development of a general purpose aerodynamic solver for compressible turbulent flows. Turbulent closure is achieved using either two equation or Reynolds stress transportation equations. The applicable equation set consists of Favre-averaged conservation equations for the mass, momentum and total energy, and transport equations for the turbulent stresses and turbulent dissipation rate. In order to develop a scheme with good shock capturing capabilities, good accuracy and general geometric capabilities, a multi-block cell centered finite volume approach is used. Viscous fluxes are discretized using a finite volume representation of a central difference operator and the source terms are treated as an integral over the control volume. The methodology is validated by testing the algorithm on both two and three dimensional flows. Both the two equation and Reynolds stress models are used on a two dimensional 10 degree compression ramp at Mach 3, and the two equation model is used on the three dimensional flow over a cone at angle of attack at Mach 3.5. With the development of this algorithm, it is now possible to compute complex, compressible high speed flow fields using both two equation and Reynolds stress turbulent closure models, with the capability of eventually evaluating their predictive performance.

Navier-Stoke solver↗

NASA NTRS1992

Upwind schemes and bifurcating solutions in real gas computations

The area of high speed flow is seeing a renewed interest due to advanced propulsion concepts such as the National Aerospace Plane (NASP), Space Shuttle, and future civil transport concepts. Upwind schemes to solve such flows have become increasingly popular in the last decade due to their excellent shock capturing properties. In the first part of this paper the authors present the extension of the Osher scheme to equilibrium and non-equilibrium gases. For simplicity, the source terms are treated explicitly. Computations based on the above scheme are presented to demonstrate the feasibility, accuracy and efficiency of the proposed scheme. One of the test problems is a Chapman-Jouguet detonation problem for which numerical solutions have been known to bifurcate into spurious weak detonation solutions on coarse grids. Results indicate that the numerical solution obtained depends both on the upwinding scheme used and the limiter employed to obtain second order accuracy. For example, the Osher scheme gives the correct CJ solution when the super-bee limiter is used, but gives the spurious solution when the Van Leer limiter is used. With the Roe scheme the spurious solution is obtained for all limiters.

Suresh, Ambady↗

NASA NTRS1992

A fast implicit upwind solution algorithm for three-dimensional unstructured dynamic meshes

Batina, John T.↗

NASA NTRS1992

Finite element Euler calculations of unsteady transonic cascade flows

A Galerkin finite element procedure incorporating an explicit Runge-Kutta time-stepping scheme has been developed in this work to solve unsteady transonic flow in cascades. The computational domain is discretized by a globally unstructured but locally structured blade-fitted deformable mesh. The Galerkin approximation is applied to the unsteady Euler equations based on a mixed Eulerian-Lagrangian description. The semi-discretized equations are integrated forward in time using a multistage Runge-Kutta scheme. An artificial dissipation operator of the type proposed by Jameson is adapted in the current scheme to capture shocks and suppress nonphysical oscillations. Phase-shifted boundary conditions are used to reduce the computational domain to a single reference passage. Results for both steady and unsteady transonic flows through cascades are presented and compared to existing finite volume solutions.

Hsiao, Chingteng↗

NASA NTRS1992

On central-difference and upwind schemes

A class of numerical dissipation models for central-difference schemes constructed with second- and fourth-difference terms is considered. The notion of matrix dissipation associated with upwind schemes is used to establish improved shock capturing capability for these models. In addition, conditions are given that guarantee that such dissipation models produce a Total Variation Diminishing (TVD) scheme. Appropriate switches for this type of model to ensure satisfaction of the TVD property are presented. Significant improvements in the accuracy of a central-difference scheme are demonstrated by computing both inviscid and viscous transonic airfoil flows.

Swanson, R. C.↗

NASA NTRS1992

A general explicit or semi-explicit algorithm for compressible and incompressible flows

A unified finite element algorithm is developed which is applicable to a wide range of problems of fluid mechanics without recourse to artificial, empirically determined factors. In its explicit form, the algorithm is similar to the Taylor-Galerkin scheme and is easily adopted to standard codes. The scheme proposed here possesses sufficient natural balancing diffusion and thus reduces and sometimes eliminates the need for special 'shock capturing' diffusion. The efficiency of the algorithm is demonstrated using several examples ranging from incompressible through transonic regions to supersonic flows.

Zienkiewicz, O. C.↗

NASA NTRS1992

A contribution to the great Riemann solver debate

The aims of this paper are threefold: to increase the level of awareness within the shock capturing community to the fact that many Godunov-type methods contain subtle flaws that can cause spurious solutions to be computed; to identify one mechanism that might thwart attempts to produce very high resolution simulations; and to proffer a simple strategy for overcoming the specific failings of individual Riemann solvers.

Quirk, James J.↗

NASA NTRS1993

PDF approach for compressible turbulent reacting flows

The objective of the present work is to develop a probability density function (pdf) turbulence model for compressible reacting flows for use with a CFD flow solver. The probability density function of the species mass fraction and enthalpy are obtained by solving a pdf evolution equation using a Monte Carlo scheme. The pdf solution procedure is coupled with a compressible CFD flow solver which provides the velocity and pressure fields. A modeled pdf equation for compressible flows, capable of capturing shock waves and suitable to the present coupling scheme, is proposed and tested. Convergence of the combined finite-volume Monte Carlo solution procedure is discussed, and an averaging procedure is developed to provide smooth Monte-Carlo solutions to ensure convergence. Two supersonic diffusion flames are studied using the proposed pdf model and the results are compared with experimental data; marked improvements over CFD solutions without pdf are observed. Preliminary applications of pdf to 3D flows are also reported.

Hsu, A. T.↗

NASA NTRS1993

Development of high-accuracy convection schemes for sequential solvers

An exploration is conducted of the applicability of such high resolution schemes as TVD to the resolving of sharp flow gradients using a sequential solution approach borrowed from pressure-based algorithms. It is shown that by extending these high-resolution shock-capturing schemes to a sequential solver that treats the equations as a collection of scalar conservation equations, the speed of signal propagation in the solution has to be coordinated by assigning the local convection speed as the characteristic speed for the entire system. A higher amount of dissipation is therefore needed to eliminate oscillations near discontinuities.

Thakur, Siddharth↗

NASA NTRS1993

A finite-volume high-order ENO scheme for two-dimensional hyperbolic systems

The finite-volume approach is presently used to obtain a 2D, high-order accurate and basically nonoscillatory shock-capture method whose high-order spatial accuracy is obtained by means of a piecewise polynomial approximation of the solution from cell averages. Attention is given to a high-order spatial operator that is able to both retain high-order accuracy in smooth regions and avoid the oscillations that are associated with interpolations across steep gradients. The operator is extended to hyperbolic systems of equations and curvilinear meshes.

Casper, Jay↗

NASA NTRS1991

A staggered mesh finite difference scheme for the computation of hypersonic Euler flows

A shock capturing finite difference method for systems of hyperbolic conservation laws is presented which avoids the need to solve Riemann problems while being competitive in performance with other current methods. A staggered spatial mesh is employed, so that complicated nonlinear waves generated at cell interfaces are averaged over cell interiors at the next time level. The full method combines to form a conservative version of the modified method of characteristics. The advantages of the method are discussed, and numerical results are presented for the two-dimensional double ellipse problem.

Sanders, Richard↗

NASA NTRS1993

A comparison of two formulations for high-order accurate essentially non-oscillatory schemes

The finite-volume and finite-difference implementations of high-order accurate essentially nonoscillatory shock-capturing schemes are discussed and compared. Results obtained with fourth-order accurate algorithms based on both formulations are examined for accuracy, sensitivity to grid irregularities, resolution of waves that are oblique to the mesh, and computational efficiency. Some algorithm modifications that may be required for a given application are suggested. Conclusions that pertain to the relative merits of both formulations are drawn, and some circumstances for which each might be useful are noted.

Casper, J.↗

NASA NTRS1993

Conservative high-order-accurate finite-difference methods for curvilinear grids

Two fourth-order-accurate finite-difference methods for numerically solving hyperbolic systems of conservation equations on smooth curvilinear grids are presented. The first method uses the differential form of the conservation equations; the second method uses the integral form of the conservation equations. Modifications to these schemes, which are required near boundaries to maintain overall high-order accuracy, are discussed. An analysis that demonstrates the stability of the modified schemes is also provided. Modifications to one of the schemes to make it total variation diminishing (TVD) are also discussed. Results that demonstrate the high-order accuracy of both schemes are included in the paper. In particular, a Ringleb-flow computation demonstrates the high-order accuracy and the stability of the boundary and near-boundary procedures. A second computation of supersonic flow over a cylinder demonstrates the shock-capturing capability of the TVD methodology. An important contribution of this paper is the dear demonstration that higher order accuracy leads to increased computational efficiency.

Rai, Man M.↗

NASA NTRS1993

PNS predictions of axisymmetric hypersonic blunt-body and afterbody flowfields

A new space-marching full-body PNS algorithm capable of treating the complete blunt-body and afterbody flowfields over typical wide-bodied configurations is developed and demonstrated. A hybrid differencing scheme involving Flux-Vector Splitting (FVS) across embedded shocks and flowfield discontinuities, and central differencing in smooth (shock-free) regions is used. It is demonstrated that this new full-body PNS scheme can be marched from the spherical stagnation point over the entire body using bow-shock capturing, and provides an efficient and effective way for predicting blunt-body flowfields for various reentry application. The Mach 20 flow over a 30-deg sphere cone is predicted and comparisons are made with an existing VSL scheme for the blunt-body region and an existing PNS scheme for the conical afterbody region. The predicted flowfield and surface-measurable quantities are in excellent agreement, and demonstrate the accuracy and efficiency of the new full-body PNS scheme.

Bhutta, Bilal A.↗

NASA NTRS1990

A comparison of optimization-based approaches for solving the aerodynamic design problem

Three optimization-based methods for solving aerodynamic design problems are compared. The Euler equations for one-dimensional duct flow was used as a model problem, and the three methods are compared for efficiency, robustness, and implementation difficulty. The smoothness of the design problem with respect to different shock-capturing finite difference schemes, and in the presence of grid refinement, is investigated.

Frank, Paul D.↗

NASA NTRS1993

A comparison of high resolution upwind solvers on 3-D inviscid hypersonic flows

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.

Manna, M.↗