Lecture Notes in Engineering The Springer-Verlag Lecture Notes provide rapid (approximately six months), refereed publication of topical items, longer than ordinary journal articles but shorter and less formal than most monographs and textbooks. They are published in an attractive yet economical format; authors or editors provide manuscripts typed to specifications, ready for photo-reproduction. The Editorial Board Managing Editors C. A Brebbia S.AOrszag Wessex Institute of Technology Applied and Computational Mathematics Ashurst Lodge, Ashurst 218 Fine Hall Southampton S04 2AA (UK) Princeton, NJ 08544 (USA) Consulting Editors Materials Science and Computer Simulation: S. Yip Chemical Engineering: Dept. of Nuclear Engg., MIT J. H. Seinfeld Cambridge, MA 02139 (USA) Dept. of Chemical Engg., Spaulding Bldg. Calif. 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Mechanics Cambridge, MA 02139 (USA) The University of Arizona W. Wunderlich Tucson, AZ 85721 (USA) Inst fUr Konstruktiven Ingenieurbau Ruhr-Universitat Bochum Hydrology: Universitatsstr. 150, G.Pinder D-4639 Bochum-Ouerenburg (FRG) School of Engineering, Dept. of Cjvil Engg. Princeton University Structural Engineering, Fluids and Princeton, NJ 08544 (USA) Thermodynamics: J. Argyris Laser Fusion - Plasma: Inst fOr Statik und Dynamik der R. McCrory Luft-und Raumfahrtkonstruktion Lab. for Laser Energetics, University of Rochester Pfaffenwaldring 27 Rochester, NY 14627 (USA) D-7000 Stuttgart 80 (FRG) Lecture Notes in Engineering Edited by C. A. Brebbia and S. A. Orszag 54 T. J. Mueller (Editor) Low Reynolds Number Aerodynamics Proceedings of the Conference Notre Dame, Indiana, USA, 5-7 June 1989 • Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Series Editors C. A. Brebbia . S. A. Orszag Consulting Editors J. Argyris . K.-J. Bathe· A. S. Cakmak . J. Connor' R. McCrory C. S. Desai· K.-P. Holz . F. A. Leckie· G. Pinder· A. R. S. Pont J. H. Seinfeld . P. Silvester· P. Spanos' W. Wunderlich· S. Yip Editor Thomas J. Mueller Dept. of Aerospace and Mechanical Engineering University of Notre Dame Notre Dame, IN 46556 USA ISBN-13: 978-3-540-51884-6 e-ISBN-13: 978-3-642-84010-4 001: 10.1007/978-3-642-84010-4 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re·use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9,1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. © Springer-Verlag Berlin, Heidelberg 1989 The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. 2161/3020-543210 Printed on acid-free paper. PREFACE Continuing interest in a wide variety of low Reynolds number applications has focused attention on the design and evaluation of airfoil sections at chord Reynolds numbers below 500,000. These applications include remotely or robotically piloted vehicles at high altitudcs as well as ultra-light and human powered vehicles and mini-RPVs at low altitudes. Other examples include small axial-flow fans used to cool electronic equipment in the unpressurized sections of high-altitude aircraft and gas turbine blades. High Reynolds number airfoil design strategies attempt to control the onset and development of turbulent boundary layers. This is difficult at low Reynolds numbers because of the increased stability of attached laminar boundary layers. Therefore, laminar separation is common even at small angles of attack at low Reynolds numbers. Under these conditions, the development of a turbulent boundary layer usually depends on the foonation of a transitional separation bubble. This volume is the collection of papers presented at the Conference on Low Reynolds Number Aerodynamics held June 4-7, 1989 at the University of Notre Dame. The Conference was sponsored by the Department of Aerospace and Mechanical Engineering and the College of Engineering at Notre Dame. Over fifty active researchers in this field from Europe, Canada, and the United States were present. This Conference followed the 1986 International Conference in London by about three years and the first Notre Dame Conference of 1985 by four years. It is clear from the papers in this volume that a great deal of progress has been made in understanding the occurrence and behavior of laminar separation and transition as well as their overall effect on the performance of airfoils at low chord Reynolds numbers. The ultimate goals of this understanding arc improved analytical methods for the design and evaluation of a variety of practical applications. Significant progress has been made in the achicvcmcnt of these goals. I would like to thank the participants for their contributions and the staff of Springer Verlag for putting together this volume. Thomas 1. Mueller Notre Dame, IN July 1989 CONTENTS XFOIL: An Analysis and Design System for Low Reynolds Number Airfoils 1 M.Drela Prediction of Aerodynamic Performance of Airfoils in Low Reynolds Number Flows 13 D.P. Coiro and C. deNicola A Fast Method for Computation of Airfoil Characteristics 24 A. Bertelrud Low Reynolds Number Airfoil Design and Wind Tunnel Testing 39 at Princeton University J.F. Donovan and M.S. Selig Study of Low-Reynolds Number Separated Flow Past the 58 Wortmann FX 63-137 Airfoil K.N. Ghia, G. Osswald and U. Ghia An Interactive Boundary-Layer Stability-Transition Approach for Low-Reynolds 70 Number Airfoils T. Cebeci and M. Mcllvaine The Instability of Two-Dimensional Laminar Separation 82 LL. Pauley, P. Main and W.C. Reynolds Bursting in Separating Flow and in Transition 93 F.T. Smith A Review of Low Reynolds Number Aerodynamic Research at 104 The University of Glasgow RA. MCD. Galbraith and FN. Caton Experimental Aerodynamic Characteristics of the Airfoils LA 5055 115 and DU 86-084/18 at Low Reynolds Numbers L.M.M. Boermans, FJ. Danker Duyvis, J.L. van Ingen and W.A. Timmer Performance Measurements of an Airfoil at Low Reynolds Numbers 131 R.J. McGhee and B.S. Walker Correlation of Theory to Wind-Tunnel Data at Reynolds Numbers Below 500,000 146 R. Evangelista, R.J. McGhee and B.S. Walker An Experimental Study of Low-Speed Single-Surface Airfoils 161 with Faired Leading Edges J.D. DeLaurier A Computationally Efficient Modelling of Laminar Separation Bubbles 174 P. Dini and M.D. Maugluner A Comparison Between Boundary Layer Measurements in a Laminar Separation 189 Bubble Flow and Linear Stability Theory Calculations P. LeBlanc, R. Blackwelder and R. Liebeck v Unsteady Aerodynamics of Wortmann FX63-137 Airfoil at Low Reynolds Numbers 206 A.M. Wo and E.E. Covert A Method to Detennine the Perfonnance of Low-Reynolds-Number Airfoils Under 218 Off-Design Unsteady Freestream Conditions H.L. Reed and B.A. Toppe! An Unsteady Model of Animal Hovering 231 P. Freymuth Control of Low-Reynolds-Number Airfoils: A Review 246 M. Gad-eL-Hak The Low Frequency Oscillation in the Flow Over a NACA 0012 Airfoil 271 with an "Iced" Leading Edge KB.M.Q. Zaman and M.G. Potapezuk Detachment of Turbulent Boundary Layers with Varying Free-Stream 283 Turbulence and Lower Reynolds Numbers J.L. Potter, R.I. Barnett, C.E. Koukousakis and C.E. Fisher Wind-Tunnel Investigations of Wings with Serrated Sharp Trailing Edges 295 P.MH.W. Vijgen, C. P. van Dam, B.I. HoLmes and F.G. Howard Low Reynolds Number Airfoil Design for Subsonic Compressible Flow 314 RH. Liebeek Computation of Viscous Unsteady Compressible Flow About Profiles 331 K.Dortmann Compressible Navier-Stokes Solutions Over Low Reynolds Number Airfoils 343 Z. Alsalihi Shockffurbulent Boundary Layer Interaction in Low Reynolds Number 358 Supercritical Flows GR. Inger Summary of Experimental Testing of a Transonic Low Reynolds Number Airfoil 369 P.L. Toot The Design of a Low Reynolds Number RPV 381 S. Siddiqi, R. Evangelista and T.S. Kwa Captive Carry Testing of Remotely Piloted Vehicles 394 A. Cross Flight Testing Navy Low Reynolds Number (LRN) Unmanned Aircraft 407 R.I. Foeh and P.L. Toot Vortex Lock-On and Flow Control in Bluff Body Ncar-Wakes 418 O.M. Griffin Wake Studies on Yawed, Stranded Cables 433 l.v. Nebres, S.M. Balill and R.C. Nelson XFOIL: An Analysis and Design System for Low Reynolds Number Airfoils Mark Drela MIT Dept. of Aeronautics and Astronautics, Cambridge, Massachusetts Abstract Calculation procedures for viscous/inviscid analysis and mixed-inverse design of sub critical airfoils are presented. An inviscid linear-vorticity panel method with a Karman-Tsien compressiblity correction is developed for direct and mixed-inverse modes. Source distributions superimposed on the airfoil and wake permit modeling of viscous layer influence on the potential flow. A two-equation lagged dissipation integral method is used to represent the viscous layers. Both laminar and turbulent layers are treated, with an e9-type amplification formulation determinining the transition point. The boundary layer and transition equations are solved simultaneously with the inviscid flowfield by a global Newton method. The procedure is especially suitable for rapid analysis of low Reynolds number airfoil flows with transi tional separation bubbles. Surface pressure distributions and entire polars are calculated and compared with experimental data. Design procedure examples are also presented. 1 Introduction Effective airfoil design procedures require a fast and robust analysis method for on-design and off design performance evaluation. Of the various airfoil analysis algorithms which have been developed to date, only the interacted viscous/inviscid zonal approaches have been fast and reliable enough for routine airfoil design work. For low Reynolds number airfoils (Re < 1/2 million), the demands on the analysis method become especially severe. Not only must the complex physics of transitional separation bubbles be captured, but the solution algorithm must be able to handle the very strong and nonlinear coupling between the viscous, transition, and inviscid formulations at a separation bubble. Of the various calculation methods currently in use (GBK code [lJ , GRUMFOIL code [2]), only the ISES code [3,4,5J can routinely predict low Reynolds number airfoil flowfields. Its fully compatible laminar and turbulent viscous formulations, a reliable transition formulation, and a global Newton solution method represent the necessary ingredients for prediction of such flows. ISES has been successfully applied to the design of low Reynolds number airfoils for human-powered aircraft [6J, analysis of established airfoils [7], and the design of high Reynolds number transonic transport airfoils, even though it can be demanding in terms of computer time. About two minutes are required to calculate an entire 20-point polar on a dedicated supercomputer. For users limited to a microVAX-ciass (0.1 MFLOP) machine, this may require several hours, which severely hinders the inherently iterative design process. A major goal in the development of the present XFOIL code was to significantly reduce these computational requirements while retaining the ability to predict low Reynolds number flows. The analysis formulation was also embedded in an interactive driver which also allows the designer to exercise an inverse solver and a geometry-manipulation facility. The overall goal is to improve the productivity of the designer. The present paper will outline the basic inviscid and viscous formulations of XFOIL, and demonstrate its performance on a number of airfoil cases. The mixed-inverse formulation and associated user interface will also be described. Finally, the code's overall design/analysis environment will be discussed. 2 '>;.1 y, --r-t- ~:~' \.~ N+2 N-1 N 5 (j Figure 1: Airfoil and wake paneling with vorticity and source distributions, with trailing edge detail. 2 Inviscid Formulation Numerous two-dimensional panel methods have been developed in the past [8,9,10], all being more or less successful for inviscid analysis of arbitrary airfoils. The present linear-vorticity streamfunction for mulation is designed specifically for compatibility with an inverse mode, and for a natural incorporation of viscous displacement effects. A general two-dimensi0I1;al inviscid airfoil flowfield is constructed by the superposition of a freestream flow, a vortex sheet of streD.'gth "Y on the airfoil surface, and a source sheet of strength a on the airfoil surface and wake. The streamfunction of this configuration is given by 2./ 2./ w(x,y) = u=y-v=x + 1(8) Inr(8; x,y) d8 + a(8)O(8;X,Y)d8 (1) 211" 211" where 8 is the coordinate along the vortex and source sheets, r is the magnitude of the vector between the point at s and the field point x, y , 0 is the vector's angle, and u= = q= cos a: , V= = q= sin a: are the freestream velocity components. The airfoil contour and wake trajectory are discretized into flat panels, with N panel nodes on the airfoil, and N., nodes on the wake as shown in Figure 1. Each airfoil panel has a linear vorticity distribution defined by the node values "Yi (1::; i ::; N). Each airfoil and wake panel also has a constant source strength ai (1::; i::; N+N.,-l) associated with it. These source strengths will be later related to viscous layer quantities. A panel of uniform source strength aTE and vortex strength "YTE must be also be placed across the airfoil trailing edge gap if it has a finite thickness. For smooth flow off the trailing edge, the trailing bY edge panel strengths aTE, 1TE , must be related to the local airfoil surface vorticity (2) where s is the unit vector bisecting the trailing edge angle, and t is the unit vector along the trailing edge panel as shown in Figure 1. For the airfoil with flat panels, equation (1) evaluates to the following expression for the streamfunc tion at any field point x, y. u=y - V=x + -1 N+LNm -l wj(X,y) 2ai 411" i=1 1 Nf;- l + 411" wj+(X,y) bj+l +"Yi) + wr<x,y) bi+l-"Yi) to + 4~ (wj:,,(X,y) Is, tl + w;/(X,y) Is x bl - "YN) (3) 3 L ' A /V ~~ 8, r, field point 8. /' r. ----./.- - x,Y panel-iV~'Y _ / LY x Figure 2: Local panel coordinates. The unit streamfunctions in equation (3) are readily defined in terms of local panel coordinates x - ti as shown in Figure 2. Wr(X,y) Xl In rl - xzln rz + Xz - Xl + ti( 01 - Oz) (4) wnx,y) [(Xl + xz)wr + ri Inr2 - r; InrI + ~(x; - xi) ti] ~ (5) 2 Xl - Xz X-z 0z -X-I0I + Y- In r-l (6) r2 By requiring that the streamfunction be equal to some constant value Wo at each node on the airfoil, the following linear system results from the above relations. N N+N.,-l L a;j "'Ij - Wo = -UooYi + V_Xi - L bijuj (7) j=l j=l The coefficient matrices a;j and bij are fully determined from the unit streamfunctions (4-6) if all the airfoil panel nodes Xi, Yi and the wake nodes are known. Combining the linear system (7) with a Kutta condition, "'11 + "'IN = 0 (8) gives a linear (N+l) x (N+I) system for the N node values "'Ii and the airfoil surface streamfunction wo. A special treatment is required for an airfoil with a sharp trailing edge. In this case the nodes i = 1 and i = N coincide, and hence their corresponding equations in (7) are identical. The result is a singular system which cannot be solved for "'Ii. To circumvent this problem, the i = N equation in (7) is discarded and replaced by an extrapolation of the mean "'I (between top and bottom) to the trailing edge. (9) 2.1 Inviscid analysis procedure For an analysis problem where the geometry is known, the linear system formed by the matrix equation (7) and Kutta condition (8) can be readily solved by Gaussian elimination. This gives the solution for the airfoil surface vorticity values as N+N.,-l "'Ii = "'10; cos a + "'190; sina + L b:jO"j j l~i~N (10) j=l where "'10 and "'190 are the vorticity distributions corresponding to a freestream a of O· and 90·, and b:; = -ai/bi; is the source-influence matrix. By setting O"i = 0 in the surface vorticity expression (10) and specifying an angle of attack, an inviscid solution is immediately obtained. For viscous flows the source strengths O"i are not known a priori, 50 the equation set (10) must be supplanted with the boundary layer equations to obtain a solvable closed system. This will be dealt with in the viscous analysis section. 4 JIJUK 1-.1.0.11 IIlfA • q.OOO CCL" ·• 1-0.l. nlS I CPU Requirements on MicroVAX II c, • ElIACT N CPU (s) CL Error 40 4.0 0.766 % 60 7.5 0.340 % 100 20.0 0.175 % 160 54.0 0.085 % Figure 3: Joukowsky airfoil test case. N = 120 Cp distribution shown. I_~:1f,:l . . . , , , il tR Figure 4: Mixed-inverse problem Figure 3 shows the comparison between the calculated and exact pressure distributions on a Joukowsky airfoil. Since the flow inside the airfoil is stagnant, the surface velocity is equal to the surface vorticity, and hence the surface pressure coefficient is C = 1 - h / q=)2. Also shown is the accuracy and CPU p requirements as functions of the number of panels. The results are typical of most panel methods. 2.2 Inviscid mixed-inverse procedure A mixed-inverse problem results when the geometry is prescribed over a part of the airfoil surface, and the surface vorticity (or equivalently, speed), is prescribed over the remainder. The local unknown associated with any node i is then either the vorticity 1i as in the analysis case, or the normal geometric displacement ni of the node from a seed airfoil geometry as shown in Figure 4. The surface vorticity at node i is specified in the form (11) which introduces two free parameters Ai , A2 into the specified vorticity distribution qspi The free parameters weight the two specified shape functions lli , 12i , which are specified to be quadratic over the inverse segment of the airfoil in ::; i ::; iL . It is necessary to add the two degrees of freedom to the specified vorticity qspi in (11) to allow geometric continuity to be enforced at the two joining points between the inverse segment and the fixed part of the airfoil (Figure 4). (12) This is consistent with the Lighthill constraints [10] which do not allow a totally arbitrary speed distri bution on the airfoil. The inverse formulation of Kennedy and Marsden [9] does not address this issue, and hence cannot perfectly satisfy the necessary streamfunction constraint (7) at each panel node. Since the governing streamfunction constraint (7) is nonlinear in the geometry, a Newton-Raphson procedure is used to solve the overall system. Eliminating the source distribution (only the inviscid in verse problem is treated), the equation system (7), the Kutta condition (8), and the geometry-continuity