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Reprint H. Sobieczky, S. I. Choudhry DLR Inst. f. Fluid Mechanics Göttingen Th. Eggers DLR Inst. f. Design Aerodynamics Braunschweig, Germany Parameterized Supersonic Transport Configurations Generic HSCT conﬁguration and Euler CFD analysis of Wing-body conﬁguration at Mach = 2.4, Isomach fringes: 2.3 < M < 2.5 Paper presented at 7th European Aerospace Conference ‘The Supersonic Transport of Second Generation’ (EAC ‘94) Toulouse, France, 25-27 Oct. 1994 1 this page intentionally blank 2 Reprint of 7th European Aerospace Conference EAC’94, Toulouse, France PARAMETERIZED SUPERSONIC TRANSPORT CONFIGURATIONS H. Sobieczky, S. I. Choudhry DLR, Institute for Fluid Mechanics Göttingen Th. Eggers DLR, Institute for Design Aerodynamics Braunschweig modern CAD methods, but CFD data preprocess- Abstract: ing calls for more directly coupled software which should be handled interactively by the designer ob- Design tools for high speed design aerodynamics serving computational results quickly and thus en- are developed using a set of mathematical func- abling him to develop his own intuition for the tions to create curves and surfaces in 3D space, relative importance of the several used and varied steady or moving for unsteady phenomena, adap- shape parameters. tation and optimization. Added is the knowledge The requirements of transonic aerodynamics of base of designing supersonic waveriders by in- transport aircraft for high subsonic Mach numbers verse methods. Coupled with fast grid generation, as well as recent activities in generic hypersonics preliminary design variations are studied by an Eul- for aerospace plane design concepts have guided er CFD method and analyzed with a powerful inter- our previous activities in the development of dedi- active visualization tool. The geometry generator is cated geometry generation [1, 2]. Based on experi- a preprocessor for new developments in CAD ence with the definition of test cases for transonic methods. aerodynamics [3] and with fast optimization tools for hypersonic configurations [4], as well as taking into account new developments in interactive Introduction graphics, some fast and efficient tools for aerody- namic shape design are presently under develop- Renewed interest in Supersonic Civil Transport ment. The concept seems well suited for (SCT) or High Speed Civil Transport (HSCT) calls application to various design tasks in high speed for extensive computational simulation of nearly ev- aerodynamics and fluid mechanics of SCT aircraft ery aspect of design and development of the whole projects. system. CAD methods are available presently for This paper, after a brief illustration of the basic many applications in the design phase. Neverthe- shape definition concept, shows examples for ge- less, work in early aerodynamic design lacks com- neric SCT aircraft and its variations for CFD analy- putational tools which enable the engineer to sis and design modifications. The use of a powerful perform quick comparative calculations with gradu- interactive fluid mechanics visualization software ally varying configurations or their components. To system [5] greatly adds to the efficiency of the pro- perform aerodynamic optimization, surface model- posed shape design method. ling is needed which allows parametric variations of wing sections, planforms, leading and trailing edg- es, camber, twist and control surfaces, to mention Geometry Tools only the wing. The same is true for fuselage, em- pennage, engine and integration of these compo- The geometry tools used here for high speed appli- nents. This can be supported in principle by cations are adapted to contain some of the most 3 important parameters of supersonic configuration design, to be varied in numerical early stage design Curves and optimization studies and finally yield a suitably dense set of data needed as an input for industrial The next step is the composition of curves by a CAD/CAM systems. piecewise scaled use of these functions. Figure 1 il- Focusing on surfaces of aerodynamically efficient lustrates this for an arbitrary set of support points, aircraft components, we realize that the goal of sur- with slopes prescribed in the supports and curva- face generation requires much control over contour ture or other desired property of each interval de- quality like slopes and curvature, while structural termining the choice of function identifiers G. The constraints require also corners, flat parts and other difference to using spline fits for the given supports compromises against otherwise idealized shapes. is obvious: for the price of having to prescribe the When familiarity is gained with a set of simple ana- function identifier and up to four parameters for lytic functions and the possibility is used to occa- each interval we have a strong control over the sionally extend the existing collection of 1D curve. The idea is to use this control for a more functions, ground is laid to compose these functions dedicated prescription of special aerodynamically suitably to yield complex 2D curves and surfaces in relevant details of airframe geometry, hoping to 3D space. This way we intend to develop tools to minimize the number of optimization parameters as define data of airospace vehicles with a nearly un- well as focusing on problem areas in CFD flow limited variety within conventional, new and exotic analysis code development. configurations. A brief illustration of the principle to Characteristic curves (“keys”) distinguish between start with 1D functions, define curves in 2D planes a number of needed curves, the example shows and vary them in 3D space to create surfaces is giv- two different curves and their support points. Below en: the graphs a table of input numbers is depicted, il- lustrating the amount of data required for these Function Catalog curves. Nondimensional function slopes a, b are calculated from input dimensional slopes s1 and s2, A set of functions Y(X) is suitably defined within the as well as the additional parameters eG, fG are interval 0 < X < 1, with end values at X,Y = (0, 0) found by suitable transformation of e1 and f2. and (1, 1), see Fig. 1, sketches above. We can A variation of only single parameters allows dra- imagine a multiplicity of algebraic and other explicit matic changes of portions of the curves, observing functions Y(X) fulfilling the boundary requirement certain constraints and leaving the rest of the curve and, depending on their mathematical structure, al- unchanged. This is the main objective of this ap- lowing for the control of certain properties especially proach, allowing strong control over specific shape at the interval ends. Four parameters or less were variations during optimization and adaptation. chosen to describe end slopes (a, b) and two addi- tional properties (eG, fG) depending on a function Surfaces identifier G. The squares shown depict some alge- braic curves where the additional parameters de- Aerospace applications call for suitable mathemati- scribe exponents in the local expansion (G=1), zero cal description of components like wings, fuselag- curvature without (G=2) or with (G=20) straight ends es, empennages, pylons and nacelles, to mention added, polynomials of fifth order (G=6, quintics) and just the main parts which will have to be studied by with square root terms (G=7) allowing curvatures parameter variation. Three-view geometries of being specified at interval ends. Other numbers for wings and bodies are defined by planforms, crown G yield splines, simple Bezier parabolas, trigono- lines and some other basic curves, while sections metric and exponential functions. For some of them or cross sections require additional parameters to eG and/or fG do not have to be specified because of place surfaces fitting within these planforms and simplicity, like G=4 which yields just a straight line. crown lines. The more recently introduced functions like G=20 Figure 2 shows a surface element defined by suit- give smooth connections as well as the limiting cas- able curves (generatrices) in planes of 3D space, it es of curves with steps and corners. Implementation can be seen that the strong control which has been of these mathematically explicit relations to the established for curve definition, is maintained here computer code allows for using functions plus their for surface slopes and curvature. first, second and third derivatives. It is obvious that this library of functions is modular and may be ex- Sections and cross sections tended for special applications, the new functions fit into the system as long as they begin and end at So far the geometry definition tool is quite general (0,0) and (1,1), a and b describe the slopes and two and may be used easily for solid modelling of near- additional parameters are permitted. ly any device if a parametric variation of its shape is 4 intended. In aerodynamic applications we want to exact surface. make use of knowledge bases from hydrodynamics and gasdynamics, i. e. classical airfoil theory and Bodies and wing-body connections basic supersonics should determine choice of func- tions and parameters. In the case of wing design we Body axis is basically parallel to the x axis in the will need to include airfoil shapes as wing sections, main flow direction, again some characteristic with data resulting from previous research. Such curves are a function of this independent variable. data will be useful if they are either describing the Here upper and lower crown line, side extent and airfoil with many spline supports, or defining the suitable superelliptic parameters of the cross sec- shape by a low number of carefully selected sup- tion are one possibility to shape a fuselage. Other, ports, which can be used for spline interpolation in a more complicated bodies are defined by optional suitably blown-up scale (Fig. 3a). For such few sup- other shape definition subprograms. Here we show ports each point takes the role of a parameter, wavy that it is useful to define the body’s horizontal coor- spline interpolation may be avoided. An early ver- dinates because this allows an easy shaping of the sion of this geometry tool [1] was used to optimize wing root toward the body. Fig. 5 shows that this wing shapes in transonic flow [6] by moving single can be applied generally to two components F1 and wing section spline supports. Other local deforma- F2 with the condition that for the first component tions may be the addition of bumps and additional one coordinate (here the spanwise y) needs to be camber functions to given airfoil data, modelling defined by an explicit function y = F1(x,z), while the adaptive wing sections (Fig. 3b, c). Finally, com- other component F2 may be given as a dataset for pletely analytical airfoils seem useful especially for a number of surface points. Using a blending func- supersonic applications, where sharp leading edges tion for a portion of the spanwise coordinate, all of wedge - type sections are allowing control of surface points of F2 within this spanwise interval shock- and expansion waves but also may have to may be moved toward the surface F1 depending on meet practical constraints like minimum leading the local value of the blending function. Fig. 5 edge radii and trailing edge thickness (Fig. 3d). shows that this way the wing root (F2) emanates from the body (F1), wing root fillet geometry can be Wings designed as part of the wing prior to this wrapping process. Several refinements to this simple projec- Aerodynamic performance of aircraft mainly de- tion technique have been used successfully. pends on the quality of its wing, design focuses therefore on optimizing this component. Using the Waverider wings present shape design method, we illustrate the amount of needed “key curves” along wing span Our present gasdynamic knowledge base includes which is inevitably needed to describe and vary the the design of waverider delta wings which exploit wing shape, Fig. 4. The key numbers are just identi- known 2D (plane or conical) supersonic flow fields fication names: span of the wing yo in the wing coor- with shocks and expansion waves (Fig. 6), in such dinate system is a function of a first independent a way that non-trivial 3D shapes are found which variable 0 < p < 1, the curve yo(p) is key 20. All fol- generate such flow fields. Recently we developed a lowing parameters are functions of this wing span: concept to extend this inverse method to design planform and twist axis (keys 21-23), dihedral (24) more general planforms by prescribing a more gen- and actual 3D space span coordinate (25), section eral shock surface of constant strength, suitably us- twist (26) and a spanwise section thickness distribu- ing ‘osculating cone’ flows to determine wing shape tion function (27). Finally we select a suitably small and flow parameters between wing and shock number of support airfoils to form sections of this wave. Using a graphic workstation, a very fast and wing. Key 28 defines a blending function 0 < r < 1 flexible optimization method [4] has been devel- which is used to define a mix between the given air- oped to arrive at such waverider wings (Fig. 7), the foils, say, at the root, at some main section and at known flow field provides lift over drag as objective the tip. The graphics in Fig. 4 shows how the role of function. Using this tool and an Euler code for off the main airfoil may be dominating across this design analysis, we investigate the use of waverid- swept wing. Practical designs may require a larger er configurations in other than the operating condi- number of input airfoils and a careful tailoring of the tions they have been designed for, for instance at section twist αo to arrive at optimum lift distribution, the Mach numbers where an SCT would operate. It for a given planform. has been shown [7] that aerodynamic performance Because of a completely analytic description of in off-design conditions is high even for relatively each wing surface point without any interpolation low supersonic Mach numbers despite the waverid- and iteration, other than sectional data arrays may er having been designed for hypersonic Mach num- be obtained with the same accuracy describing the bers. This makes waverider wings or some 5 elements of such configurations useful for direct wake emanates from the wing trailing edge and the shape definition, most of the inverse nature of the whole wing-body configuration is defined here by a design approach can be converted to direct geome- cross section surface grid. Boundary conditions are try input parameters and this way guides us how given this way for CFD aerodynamic analysis, but wing sections should be shaped for given leading also for aeroacoustic investigations and, with engine edges as long as they are supersonic leading edges. exhaust modelling included, for investigating jet con- Integration of subsonic parts of the wing and of trails. The latter tasks are especially of interest for course fuselages is most effectively carried out with research on the environmental impact of SCT air- the present direct approach. craft. A first series of design/analysis runs is carried out on Example: Generic SCT aircraft the wing-body configuration cut off at the wing trail- ing edge using a simple algebraic grid with 33 x 81 x Case studies for new generation supersonic trans- 33 meshpoints (Fig. 11) and short runs with the DLR port aircraft have been carried out through the past Euler code [9]. Visualization of the pressure distribu- years in research institutions and the aircraft indus- tion with the HIGHEND graphic system [5] shows try. Our present tool to shape such configurations isobar patterns in color or zebra graphics and select- needs to be tested by trying to model the basic fea- ed cross section pressure checks (Figs. 12, 13) al- tures of various investigated geometries. Knowing low an assessment of chosen airfoils and twist that the fine-tuning of aerodynamic performance distributions before refined grids and longer Euler must be done by careful selection of support airfoils runs are executed. Refining grids near the leading and wing twist distribution, our initial exercise is try- edge is necessary but basic information about need- ing to geometrically model some of the published ed airfoil changes is already provided by the present configurations, generate CFD grids around them and runs; the refinement of geometry and CFD analysis develop optimization strategies to find suitable sec- may begin. tion and twist distributions. This is still a difficult task but tackling its solution greatly contributes to building Visualizing Shock Waves up a knowledge base for supersonic design. A case study is illustrated next, the purpose of generating Visualization of the shock waves system emanating this geometry is the definition of a test case for CFD from the body tip and the wing is shown in Fig. 14. code development: A new visualization technique [10] allows for analyz- ing shock waves in 3D space: their quality near the Mach 2.4 HSCT aircraft, as shown, or with refined CFD analysis in the farfield to investigate sonic boom propagation. The following figures illustrate generation and pre- The figure shows a cut-off domain of the shock sur- liminary CFD analysis of a configuration generated faces: A shock strength threshold allows analysis of from a Boeing HSCT design case for Mach 2.4 [8]. local sonic boom quantities. Finally the geometry of Fig. 8 shows a three-view and a shaded graphics vi- the shock isosurfaces was imported into Alias Studio sualization. The configuration consists of 6 compo- software for final rendering. nents plus their symmetric images, engine pylons are not yet included. The wing has a subsonic lead- ing edge in the inner portion and a supersonic lead- Towards an Aerodynamics Workbench ing edge on the outer portion. We try to use a minimum of support airfoils (Fig. 9) to get a reason- In this paper we have shown some portions of a full able pressure distribution: a rounded leading edge design cycle, which involves configuration design, section in most of the inner wing and an almost grid generation, computation of a numerical flow field wedge-sharp section in the outer wing portion define solution and finally the analysis of the solution using the basic shape of the wing. Wing root fillet blending, modern visualization tools. After we have analysed the smooth transition between rounded and sharp the solution we can go back and improve the design. leading edge and the tip geometry are effectively This design cycle is still quite tedious, since we have shaped by the previously illustrated keys 27 and 28, to use several different tools, and can’t interactively while lift distribution along span is of course con- switch between them on the fly, since all the tools trolled by wing twist, key 26. are standalone. For this particular example the cycle time is about two hours, including a 45 minute wait in Fig. 10 shows an extension of the shape generation the queue to get the solution from a supercomputer. tool particularly useful for supersonic applications: A So for the future we are working on an ‘Aerodynam- computational far field boundary is generated just ics Workbench’, an interactive tool for graphics work- like a fuselage in cross sections, a computational stations which will integrate all of the above 6 mentioned tools in the design process, and make it a References lot faster and easier for an aerospace engineer to go through the design cycle. Our goal is to bring the de- 1. Sobieczky, H., ‘Geometry Generation for Transon- sign cycle time down into the range of minutes. This ic Design’, Recent Advances in Numerical Methods speedup will become even of more significance in Fluids, Vol. 4, Ed. W. G. Habashi, Swansea: Pin- when we want to optimize configurations by varying eridge Press, pp. 163 - 182. (1985). parameters in the design and understand its effects in the solutions. 2. Sobieczky, H., Stroeve, J. C., ‘Generic Supersonic and Hypersonic Configurations’, AIAA 9. Applied Aerodynamics Conf. Proc., AIAA 91-3301CP, (1991) Novel configurations 3. Sobieczky, H., ‘DLR-F5: Test Wing for CFD and The already operational and the projected tools stim- Applied Aerodynamics’, Case B-5 in: Test Cases for ulate us to study innovative aircraft concepts. The CFD Evaluation, AGARD FDP AR-303, (1994) development of conventional configurations like the one modelled above, may still face crucial technolo- 4. Center, K. B., Jones, K. D., Dougherty, F. C., See- gy problems resulting in reduced chances to operate bass, A. R., Sobieczky, H., ‘Interactive Hypersonic economically [11]. An elegant concept avoiding Waverider Design and Optimization’, Proc. 18th some of these problems is the Oblique Flying Wing ICAS Congress, Beijing, (1992) for supersonic transport. Its high aerodynamic effi- ciency calls for ongoing in-depth investigation, con- 5. Pagendarm, H. G., ‘Unsteady Phenomena, Hy- tinuing the work already done through the past years personic Flows and Co-operative Flow Visualization and more recently [12]. Our tools seem ideally suited in Aerospace Research’, in: G.M. Nielson, D. Berg- to aid such work by parametric shape variation and eron, Proceedings Visualization ‘93, pp. 370-373, implementation of inverse design methods based on IEEE Computer Society Press, Los Alamitos, CA gasdynamic modelling. 1993 6. Cosentino, G. B., Holst, T. L., ‘Numerical Optimi- Conclusion zation Design of Advanced Transonic Wing Configu- rations’, Journal of Aircraft, Vol. 23, pp. 192-199, Software for supersonic generic configurations has (1986) been developed to support the design requirements in high speed aerodynamics and which should allow 7. Eggers, T., ‘Untersuchungen zum Off-Design-Ver- extensions for multidisciplinary design consider- halten von Wellenreitern im Überschallbereich’, Ta- ations. Based on simple, explicit algebra a set of gungsband Deutscher Luft- u. Raumfahrtkongress / flexible model functions is used for curve and sur- DGLR Jahrestagung (1994) face design which is tailored to create realistic air- planes or their components with various surface grid 8. Kulfan, R.,’High Speed Civil Transport Opportuni- metrics. The explicit and non-iterative calculation of ties, Challenges and Technology Needs’, Lecture at surface data sets make this tool extremely rapid and Taiwan IAA 34th National Conference, (1992) this way suitable for generating series of configura- tions in optimization cycles. The designer has control 9. Kroll, N., Radespiel, R., ‘An Improved Flux Vector over parameter variations and builds up a knowl- Split Discretization Scheme for Viscous Flows’. DLR- edge base about the role of these parameters for FB 93-53, (1993) flow quality and aerodynamic performance coeffi- cients. Some basic gasdynamic relations describing 10. Pagendarm, H. G., Choudhry, S. I., ‘Visualization supersonic flow phenomena in 2 or 3 dimensions of Hypersonic Flows - Exploring the Opportunities to have become guides to select key functions in the Extend AVS’, 4th Eurographics Workshop on Visual- shape design; these and other model functions allow ization in Scientific Computing, (1993) for the gradual development of our design experi- ence if the generic configurations are used as 11. Seebass, A. R., ‘The Prospect of Commercial boundary conditions for numerical analysis. With a Transport at Supersonic Speeds’, AIAA paper 94- number of efficient tools available now, the combina- 0017, (1994) tion to an interactive design system for not only aero- dynamic but also multidisciplinary optimization 12. Van der Velden, A. J. M., ‘Aerodynamic Design seems feasible. and Synthesis of the Oblique Flying Wing Superson- ic Transport’, PHD Thesis, Stanford University, (1992). 7 1 x3 x2 YG(a,b,eG,fG,X) fG f2 b G = 2; G=1 G = 20 eG a X c 0 1 f1 G=6 G=7 f3 x1 Fig. 2: Surface deﬁnition by cross sections c in plane (x1, x3) determined by generatrices fi along x2 xj 2 and in planes (x1, x2), (x2, x3). 1 xi a 0 1 2 3 4 5 6 7 -1 -2 b key u F(u) s1 G s2 e1 f2 1 0.0 0.0 0. 7 0.25 4. 0. 1 0.5 0.5 0. 4 c 1 1.7 0.8 0.25 6 0. 0. -0.2 1 5.0 2.0 0. 6 -0.8 -0.2 -0.2 1 7.5 1.0 2 0.0 0.0 0. 7 -0.5 4. 0. d 2 1.0 -0.7 -0.5 20 0. 2. 2 7.5 -1.3 Fig. 3: Airfoils given as datasets either with few sup- Fig. 1: Some basic functions YG in nondimensional ports in blown-up scale (a) or from external data- unit interval (above). Construction of arbitrary, di- base; with additional parameters for local mensional curves in plane (xi, xj) by peacewise use deformation (b → c) and as analytical functions, for of scaled basic functions. Parameter input list (be- standard airfoils or guided by known ﬂow ﬁeld solu- low), example with 2 parameters changed, resulting tions (d). in dashed curves. 8 z 1 p 20 y yo y = F1(x, z) F2(x, y, z) = 0 x 22 z 23 21 y yo y, z 25 Fig. 5: Kombination of two components by a blend- ed projection technique β 24 yo x’ αο MachWR 26 z yo y τ 1 27 x’ yo 1 r 28 x Fig. 6: Exploiting known supersonic ﬂow ﬁelds to Fig. 4: Wing parameters and respective key numbers design wing sections (above) and using osculating for section distribution, planform,an/dihedral, twist, cones concept to design waverider wings (below). thickness distribution and airfoil blending. 9 tip η supersonic leading edge kink subsonic root Fig. 9: Airfoil support shapes along span. Fig. 7: Waverider wing conﬁguration designed from given leading edge, Mach number and ob- lique shock wave angle. Supersonic leading edge and a completely integrated body are trademarks of waverider conﬁgurations. Fig. 10: Cross sections of wing-body, far ﬁeld boundary and a computational wake model. Fig. 8: “Conﬁguration 950” Generic HSCT conﬁgu ration derived from Boeing Mach 2.4 test case Fig. 11: Algebraic grid (33 x 81 x 33) for preliminary CFD analysis near design conditions. 10 Fig. 12: Euler analysis Mach = 2.4: Isobar fringes Fig. 14: Visualization of the shock system emanat- ing from body tip and wing. -cp x y Fig. 13: Quality check of cross section pressure di tributions (33 x 81 x 33 grid, 330 time steps) 11

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Scientific Computing, Hans- Georg, G. Pagendarm, Interner Bericht, Supersonic Transport, Shock Waves, citation index, Information Superhighway, IEEE Computer Graphics and Applications, International Symposium

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