Computational fluid dynamics drag prediction: Results from the Viscous Transonic Airfoil Workshop

Results from the Viscous Transonic Airfoil Workshop are compared with each other and with experimental data. Test cases used include attached and separated transonic flows for the NACA 0012 airfoil. A total of 23 sets of numerical results from 15 different author groups are included. The numerical method used vary widely and include: 16 Navier-Stokes methods, 2 Euler boundary layer methods, and 5 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

Numerical computation of transonic flow governed by the full-potential equation

Numerical solution techniques for solving transonic flow fields governed by the full potential equation are discussed. In a general sense relaxation schemes suitable for the numerical solution of elliptic partial differential equations are presented and discussed with emphasis on transonic flow applications. The presentation can be divided into two general categories: An introductory treatment of the basic concepts associated with the numerical solution of elliptic partial differential equations and a more advanced treatment of current procedures used to solve the full potential equation for transonic flow fields. The introductory material is presented for completeness and includes a brief introduction (Chapter 1), governing equations (Chapter 2), classical relaxation schemes (Chapter 3), and early concepts regarding transonic full potential equation algorithms (Chapter 4)

The relative merits of several numerical techniques for solving the compressible Navier-Stokes equations

Four explicit finite difference techniques designed to solve the time-dependent, compressible Navier Stokes equations are compared. These techniques are: (1) MacCormack, (2) modified Du Fort-Frankel, (3) modified hopscotch, and (4) Brailovskaya. The comparison was made numerically by solving the quasi-one dimensional Navier Stokes equations for the flow in a converging-diverging nozzle. Solutions with and without standing normal shock waves were computed for unit Reynolds numbers (based on total conditions) ranging from 45374 to 2269. The results indicate that all four techniques are comparable in accuracy; however, the modified hopscotch scheme is two to three times faster than the Brailovskaya and MacCormack schemes and three to six times faster than the modified Du Fort-Frankel scheme

Numerical solution of the Navier-Stokes equations about three-dimensional configurations: A survey

The numerical solution of the Navier-Stokes equations about three-dimensional configurations is reviewed. Formulational and computational requirements for the various Navier-Stokes approaches are examined for typical problems including the viscous flow field solution about a complete aerospace vehicle. Recent computed results, with experimental comparisons when available, are presented to highlight the presentation. The future of Navier-Stokes applications in three-dimensions is seen to be rapidly expanding across a broad front including internal and external flows, and flows across the entire speed regime from incompressible to hypersonic applications. Prospects for the future are described and recommendations for areas of concentrated research are indicated

Field weed population models: a review of approaches and application domains

We evaluated models of weed population dynamics based on an analysis of their assumptions, biological rationale, flexibility, documentation, accessibility, demand for parameter estimation and documented validity. We arrived at general recommendations regarding which modelling approach should be applied in order to address different application domains

A new solution-adaptive grid generation method for transonic airfoil flow calculations

The clustering algorithm is controlled by a second-order, ordinary differential equation which uses the airfoil surface density gradient as a forcing function. The solution to this differential equation produces a surface grid distribution which is automatically clustered in regions with large gradients. The interior grid points are established from this surface distribution by using an interpolation scheme which is fast and retains the desirable properties of the original grid generated from the standard elliptic equation approach

An implicit algorithm for the conservative, transonic full-potential equation with effective rotated differencing

A new differencing scheme for the conservative full potential equation which effectively simulates rotated differencing is presented. The scheme was implemented by an appropriate upwind bias of the density coefficient along coordinate directions. A fast, fully implicit, approximate factorization iteration scheme was then used to solve the resulting difference equations. Solutions for a number of traditionally difficult transonic airfoil test cases are presented

Numerical optimization design of advanced transonic wing configurations

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

A consistent spatial differencing scheme for the transonic full-potential equation in three dimensions

A full-potential steady transonic wing flow solver has been modified so that freestream density and residual are captured in regions of constant velocity. This numerically precise freestream consistency is obtained by slightly altering the differencing scheme without affecting the implicit solution algorithm. The changes chiefly affect the fifteen metrics per grid point, which are computed once and stored. With this new method, the outer boundary condition is captured accurately, and the smoothness of the solution is especially improved near regions of grid discontinuity

Conservative implicit schemes for the full potential equation applied to transonic flows

Implicit approximate factorization techniques (AF) were 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, was 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. The results indicate that the AF schemes are substantially faster than SLOR