The geometry of solution trajectories for three first‐order coupled linear differential equations can be related and classified using three matrix invariants. This provides a generalized approach to the classification of elementary three‐dimensional flow patterns defined by instantaneous streamlines for flow at and away from no‐slip boundaries for both compressible and incompressible flow. Although the attention of this paper is on the velocity field and its associated deformation tensor, the results are valid for any smooth three‐dimensional vector field. For example, there may be situations where it is appropriate to work in terms of the vorticity field or pressure gradient field. In any case, it is expected that the results presented here will be of use in the interpretation of complex flow field data.

A general classification of three-dimensional flow fields

https://fun3d.larc.nasa.gov/papers/compfluid.pdf

An implicit upwind algorithm for computing turbulent flows on unstructured grids

Introduction E is an honor and challenge to present the Dryden Lecture ..i Research for 1979. Since my topic concerns a new trend in fluid mechanics, it should not be surprising that some aspects of this paper involve basic mechanics of turbulence, a field enriched by numerous contributions of Dr. Hugh L. Dryden. Having worked in related fields of fluid mechanics during past years, and long respected both his professional contributions and personal integrity, it is a special pleasure to present this Dryden lecture. The field of computational fluid dynamics during recent years has developed sufficiently to initiate some changes in traditional methods of aerodynamic design. Both computer power and numerical algorithm efficiency are simultaneously improving with time, while the energy resource for driving large wind tunnels is becoming progressively more valuable. Partly for these reasons it has been advocated that the impact of computational aerodynamics on future methods of aircraft design will be profound. ' Qualitatively, the changes taking place are not foreign to past experience in other fields of engineering. For example, trajectory mechanics and neutron transport mechanics already have been largely revolutionized by the computer. Computations rather than experiments now provide the principal source of detailed information in these fields. The amount of reactor experimentation required has been much reduced over former years; experiments now are performed mainly on clear, physically describable arrays of elements aimed at further confirmation of computational techniques; and better designs are achieved than with former experimental methods alone. Similar changes in the relative roles of experimental and computational aerodynamics are anticipated in the future. There are three compelling motivations for vigorously developing computational aerodynamics. One is to provide important new technological capabilities that cannot be provided by experimental facilities. Because of their fundamental limitations, wind tunnels have rarely been able to simulate, for example, Reynolds numbers of aircraft flight, flowfield temperatures around atmosphere entry vehicles, aerodynamics of probes entering planetary atmospheres, aeroelastic distortions present in flight, or the propulsiveexternal flow interaction in flight. In addition, transonic wind tunnels are notoriously limited by wall and support interference; and stream nonuniformities of wind tunnels severely affect laminar-turbulent transition. Moreover, the dynamic-aerodynamic interaction between vehicle motion in flight and transition-dependent separated flow also is inaccessible to wind-tunnel simulation. In still different ways ground facilities for turbomachinery experiments are limited in their ability, for example, to simulate flight inlet-flow nonuniformities feeding into a compressor stage, or to determine detailed flowfields between rotating blades. Numerical flow simulations, on the other hand, have none of these fundamental limitations, but have their own: computer speed and memory. These latter limitations are fewer, but previously have been much more restrictive overall because the full Navier-Stokes equations are of such great complexity that only highly truncated and approximate forms could be handled in the past. In recent years the Navier-Stokes equations have begun to yield under computational attack with the largest current computers. Since the fundamental limitations of computational speed and memory are rapidly decreasing with time, whereas the fundamental limitations of experimental facilities are not, numerical simulations offer the potential of mending many ills of wind-tunnel and turbomachinery experiments, and of providing thereby important new technical capabilities for the aerospace industry. A second compelling motivation concerns energy conservation. The large developmental wind tunnels require large amounts of energy, whereas computers require comparatively

Computational Aerodynamics Development and Outlook

The primary objective of this work is to provide accurate numerical solutions for selected flow fields and to compare and evaluate the performance of selected turbulence models with experimental results. Four popular turbulence models have been tested and validated against experimental data often turbulent flows. The models are: (1) the two-equation k-epsilon model of Wilcox, (2) the two-equation k-epsilon model of Launder and Sharma, (3) the two-equation k-omega/k-epsilon SST model of Menter, and (4) the one-equation model of Spalart and Allmaras. The flows investigated are five free shear flows consisting of a mixing layer, a round jet, a plane jet, a plane wake, and a compressible mixing layer; and five boundary layer flows consisting of an incompressible flat plate, a Mach 5 adiabatic flat plate, a separated boundary layer, an axisymmetric shock-wave/boundary layer interaction, and an RAE 2822 transonic airfoil. The experimental data for these flows are well established and have been extensively used in model developments. The results are shown in the following four sections: Part A describes the equations of motion and boundary conditions; Part B describes the model equations, constants, parameters, boundary conditions, and numerical implementation; and Parts C and D describe the experimental data and the performance of the models in the free-shear flows and the boundary layer flows, respectively.

Turbulence Modeling Validation, Testing, and Development

Boundary-fitted coordinate systems for numerical solution of partial differential equations—A review

Results from the Viscous Transonic Airfoil Workshop held at the AIAA 25th Aerospace Sciences Meeting at Reno, NV in January 1987, are compared with each other and with experimental data. Test cases used in this workshop include attached and separated transonic flows for three different airfoils: the NACA 0012 airfoil, the RAE 2822 airfoil, and the Jones airfoil. A total of 23 sets of numerical results from 15 different author groups are included. The numerical methods used vary widely and include: 16 Navier-Stokes methods, 2 Euler/boundary-layer methods, and 5 full-potential/boundary-layer methods. The results indicate a high degree of sophistication among the numerical methods with generally good agreement between the various computed and experimental results for attached or moderately-separated cases. The agreement for cases with larger separation is only fair and suggests additional work is required in this area.

Viscous transonic airfoil workshop compendium of results

A new, implicit approximate factorization (AF) algorithm designed to solve the conservative full-potential equation for the transonic flow past arbitrary airfoils has been developed. The new algorithm uses an upwind bias of the density coefficient to provide stability in supersonic regions. This allows the simple two- and three-banded matrix form of the AF scheme to be retained over the entire flow field, even in regions of supersonic flow. A numerical transformation is used to establish an arbitrary body-fitted finite-difference mesh. Airfoil pressure distributions have been computed and are in good agreement with independent results.

Implicit Algorithm for the Conservative Transonic Full-Potential Equation Using an Arbitrary Mesh

Implicit approximate factorization techniques (AF) are investigated for the solution of matrix equations resulting from finite-difference approximations to the full potential equation in conservation form. For transonic flows, an artificial viscosity, required to maintain stability in supersonic regions, is introduced by an upwind bias of the density. Two implicit AF procedures are presented, and their convergence performance is compared with that of the standard transonic solution procedure: successive line overrelaxation (SLOR). Subcritical and supercritical test cases are considered. Results indicate a substantial improvement in convergence rate for AF schemes relative to SLOR.

Fast, Conservative Schemes for the Full Potential Equation Applied to Transonic Flows

A computationally efficient and versatile technique for use in the design of advanced transonic wing configurations has been developed. A reliable and fast transonic wing flow-field analysis program, TWING, has been coupled with a modified quasi-Newton method, unconstrained optimization algorithm, QNMDIF, to create a new design tool. Fully three-dimensional wing designs utilizing both specified wing pressure distributions and drag-to-lift ration minimization as design objectives are demonstrated. Because of the high computational efficiency of each of the components of the design code, in particular the vectorization of TWING and the high speed of the Cray X-MP vector computer, the computer time required for a typical wing design is reduced by approximately an order of magnitude over previous methods. In the results presented here, this computed wave drag has been used as the quantity to be optimized (minimized) with great success, yielding wing designs with nearly shock-free (zero wave drag) pressure distributions and very reasonable wing section shapes.

Numerical optimization design of advanced transonic wing configurations

A fast, fully implicit approximate factorization algorithm designed to solve the conservative, transonic, full-potential equation in either two or three dimensions is described. The algorithm uses an upwind bias of the density coefficient for stability in supersonic regions. This provides an effective upwind difference of the streamwise terms for any orientation of the velocity vector (i.e., rotated differencing), thereby greatly enhancing the reliability of the present algorithm. A numerical transformation is used to establish an arbitrary body-fitted, finite-difference mesh. Computed results for both airfoils and simplified wings demonstrate substantial improvement in convergence speed for the new algorithm relative to standard successive-line over-relaxation algorithms.

/pdf/fast-conservative-algorithm-for-solving-the-transonic-full-37xw1xl403.pdf

Terry L. Holst

Papers

Viscous transonic airfoil workshop compendium of results

Implicit Algorithm for the Conservative Transonic Full-Potential Equation Using an Arbitrary Mesh

Fast, Conservative Schemes for the Full Potential Equation Applied to Transonic Flows

Numerical optimization design of advanced transonic wing configurations

Fast, Conservative Algorithm for Solving the Transonic Full-Potential Equation