· · · Aerospace Engineering reports

The technical concepts of a method for the generation of 3D block-structured grids are presented. The generation process is decomposed in four more or less sequential steps. 1. Topological block decomposition. 2. Geometrical modeling of known and unknown coordinate functions. 3. Grid initialisation by multi-linear transfinite interpolation. 4. Elliptic cell partitioning. The elliptic method is based on simple partial differential equations representing locally decoupled ID stretchings followed by 3D rotations. Mesh-size and smoothness control occurs with user-provided weight functions that are inversely proportional to desired mesh sizes. Test results are shown.

The application of multigrid relaxation to transonic potential-flow calculation was investigated. Fully conservative potential flows around aerofoils were taken as test problems. The solution algorithm was based on Newton iteration. In each Newton iteration step, multigrid relaxation was used to calculate correction potentials. It was found that the iteration to the circulation has to be kept outside the multigrid algorithm. In order to obtain meaningful norms of residuals (to be used in termination tests of loops), difference formulas with asymptotic scaling were introduced. Nonlinear instability problems were solved by upwind differencing using mass-flux-vector splitting instead of artificial viscosity or artificial density. It was also found that the multigrid method cannot efficiently update shock positions due to the (mainly) linear character of individual multigrid relaxation cycles. For subsonic flows, the algorithm is quite efficient. For transonic flows, the algorithm was found robust; its efficiency should be increased by improving the iteration on the shock positions; this is a highly nonlinear process.

The calculation of transonic flows with shocks with multigrid fastsolver algorithms is known to be not so efficient as in elliptic problems, because the appropriate treatment of shocks is a problem. As a first step to a solution, a few ideas concerning the design of finite-difference equations over shocks and the rapid calculation of such flows with a fast solver are presented. The fast solver studied here consists of a combination of Newton iteration and full multigrid relaxation. The ideas were applied to calculate steady transonic potential flows in a one-dimensional channel, and around an aerofoil. It was found that the fast solver has full convergence speed if shockwave positions are updated by a simple special subalgorithm in the fast solver; this subalgorithm eliminates discontinuity jumps that arise at shocks during iterations. Results of numerical experiments illustrate the findings.

The application of a combination of Newton iteration and multigrid line relaxation to transonic potential-flow calculations was investigated. Fully conservative potential flows around aerofoils were used as test problems. The multigrid process was used in each Newton iteration step to calculate correction potentials. It was found that the iteration to the circulation has to be kept outside the multigrid algorithm in order to achieve good calculation efficiency. Meaningful residual norms, to be used in termination tests of loops, could be designed using finite-difference formulas with asymptotic scaling. Mass-flux-vector splitting was found to solve instability problems encountered at sonic lines with artificial-viscosity terms. The multigrid algorithms seem not to be suited for the efficient update of shock positions. A special sub algorithm outside the multigrid algorithm was used to update shock positions; this sub algorithm will have to be replaced by a more efficient sub algorithm, however. The combined Newton/multigrid algorithm turned out to be very efficient for subsonic flows around aerofoils with lift. For transonic flows, the algorithm is convergent.

In Euler-flow calculations based on cell-centered schemes, the number of equations required to determine the flow-state evaluation at grid points half a mesh outside the flow domain usually exceeds the number of boundary-condition equations provided by characteristic theory. Here, the required additional equations are obtained by a physically well-defined numerical procedure, consisting of three steps. First, a layer of auxiliary cells on flow boundaries is introduced, and semi-discrete conservation equations for these cells are defined. Subsequently, the time variations of the state in these auxiliary cells at the boundary are transformed into characteristic form, and time variations of characteristic variables corresponding to incoming information from the boundary into the flow are replaced by boundary conditions for these time variations. In the third step, the boundary equations so obtained are mapped back into a form with primitive variables, and numerically integrated in time. The characteristic boundary conditions are first-order differential equations for time variations at boundary points of characteristic variables. These equations may be chosen to express that given functions of the flow state on the boundary should as3nnptotically tend with time to prescribed steady-state values. The procedure for modeling such equations is explained with four examples. The procedure has been succesfully tested numerically with ID Euler channel flows. The extension to 3D Euler flows is discussed. The characteristic boundary conditions are derived using a numerically and physically useful eigenvector-eigenvalue decomposition of the Jacobian matrices that appear in characteristic theory.

An introductory description of a hodograph method for the Computation of advanced shock-free aerofoils is presented. The method is based on the approximate solution of Tricomi boundary value problems for the hodograph equations. The approximate solution is a sum of a basic solution having the required singularity on the hodograph surface corresponding to free-stream conditions and a finite linear combination of regular additional solutions. The basic and additional solutions are constructed with function-theoretic methods developed by Lighthill, Goldstein, Nieuwland and others. The coefficients in the finite linear combination of solutions are determined in such a way that approximate solutions of Tricomi boundary value problems are obtained. A special error functional is introduced and minimized in the process of determining aerodynamically acceptable approximate solutions. A few computational results give an idea of the performance of the method.

The hodograph method and its application as described in reference 1 allow the construction of steady transonic potential flows around aerofoil sections. This report gives results of the application of this method as far as non-lifting symmetrical sections are concerned. The results show that with a restricted use of the possibihties of the method an already remarkable variety of practically interesting sections and corresponding pressure distributions can be obtained. The sections can be computed with a precision sufficient to meet accuracy requirements of model manufacturing.

Within the scope of a study of efficient algortihms for the calculation of transonic potential flows, an interactive research computation system has . been developed, which permits numerical experimentation with a multigrid fast solver. This research system must be replaced by a production version having improved modularity. This concerns the internal breakdown of the algorithm as well of the data structure. The readibility of the code must also be improved. The replacement is also necessary in order to allow for future extensions and improvements. The functional requirements, general layout, and computation processes, that are specific to a flow calculation, are presented in this memorandum.

TRAFS is a new fast-solver program for the calculation of steady subsonic and transonic inviscid flows around aerofoils. It uses a potential-flow model representing mass conservation in the flow. Potential users of TRAFS are given an impression of the user capabilities. These are illustrated by analyzing sixteen flow cases covering incompressible, subsonic, and transonic (supercritical) flows with and without shocks, with and without lift. The control and inspection of the level of accuracy of the calculation results is discussed in detail. Some of the advanced plotting options to analyse the calculation results are illustrated (isobar plots, overplotting of pressure distributions).

Na inleidende opmerkingen betreffende de state-of-the-art van het berekenen van stromingen op basis van de Euler vergelijkingen wordt voorgesteld rekenprogramma's te ontwikkelen voor de simulatie van propellor slipstroom-effecten over motorgondel-vleugel configuraties. Verder zal aandacht worden geschonken aan de berekening van stromingen in het inlaatkanaal achter de propellor. De berekeningen zullen op het Euler stromingsmodel worden gebaseerd,

In dit literatuviroverzicht wordt een bespreking gegeven van de gelineariseerde transsone theorieën voor vlakke symmetrische en aaiaal symmetrische stromingen, zoals die door Oswatitsch en Keune, Maeder en Thommen, en Spreiter en Alksne zijn ontwikkeld. Een vereenvoudigde versie van de theorie van de z.g. gelineariseerde transsone potentiaalvergelijking wordt gegeven door een directe definitie van elementaire oplossingen ("transsone bronnen") van deze vergelijking. Dit rapport is samengesteld met het doel een inzicht te krijgen in de fysische en mathematische geldigheid van de in deze theorieën gebruikte benaderingen.

Dit memo heeft tot doel bij te dragen in de discussies over het ontwikkelen van nieuwe rekenprogramma's voor drie-dimensionale stationaire transsone stromingen om romp-vleugel combinaties. Gepresenteerd wordt: a. een inventarisatie van thans bekende snelle oplossers voor randwaarde problemen van partiële differentiaal vergelijkingen; b. een samenvatting van thans bekende ervaringen met deze snelle oplossers bij transsone stationaire stromingsberekeningen met differentiemethoden. Er wordt geconcludeerd, dat rekenprogramma's voor drie-dimensionale stationaire transsone potentiaal stromingen ontwikkeld kunnen worden, die ruim een half uur rekentijd op de NLR Cyber T2 computer zullen vergen. Daarbij wordt uitsluitend gebruik gemaakt van een combinatie van thans beschikbare numerieke methoden. Er wordt een indicatie gegeven van mogelijkheden die deze rekentijden nog verder kunnen bekorten.

Nieuwland's hodograph theory for Ufting quasi-elliptical aerofoils has been applied to compute a number of profiles with supercritical shockfree flow at the design condition. These profiles appear to have nose camber only. The pressure distribution at the design condition is of the peaky type. The minimum radius of curvature at the nose is of the order of 0.2 % to 1 1/2% of the chord length. It is possible to compute the coordinates with sufficient precision for engineering applications. The shapes of the profiles computed depend upon seven parameters. It appears, that four of these parmeters have to be chosen carefully in order to avoid limit lines or branch points in that part of the physical plane, which is of interest. One of the profiles has been tested in the NLR Pilot tunnel to investigate: (a) the effect of viscosity on the theoretical results; (b) the off-design behaviour. The pressure distribution in the experimental design condition (this is the condition with the weakest shocks) indicates practically shock-free flow. A 20% loss in lift is found at this condition. By deflecting a trailing-edge flap the lift loss could be reduced to 10% without worsening the drag characteristics. The margin between rapid drag rise boundary and buffet boundary at high Mach numbers appeared to be small.

A robust fast-solver for the calculation of transonic potential flows around lifting airfoils is presented. The solver is a combination of Newton iteration and CS (correction scheme) multigrid relaxation. This combination allows a simpler analysis of convergence properties than the PAS (full approximation storage) multigrid relaxation that is usually applied, and was selected for this reason. The solver has been implemented in the NLR code TRAFS and has been extensively tested numerically.

It is known that design of arbitrary shock-free transonic aerofoils is possible by solving general boundary value problems for the mixed elliptichyperbolic hodograph equations. Based on this approach a design method for shock-free transonic aerofoils has been developed. Some problems of the mathematical theory underlying the method are discussed. Several examples of computed aerofoils are presented. These examples demonstrate that the method is sufficiently flexible and powerful for the design of a very large class of aerodynamically interesting shock-free transonic aerofoils.

An approxinate m ethod is presented for the mapping of simply - connected closed regions bounded by a smooth ourve onto a circular disk using harmonic polymials. Due attention is being paid to the handling of the singularities that usually disturb the regularity of the mapping near the boundary of the region. Some numerical examples illustrate the effectiveness of the method.