A VISCOUS FLOW BASED MEMBRANE WING MODEL By RICHARD W. SMITH A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNrVERSrTY OF FLORIDA 1994 ACKNOWLEDGEMENTS I wouldfirstlike tothankmy advisor, ProfessorWei Shyy, forallowingme to follow my own research interests and for providing a productive working environment for all his students. Without his support and tolerant attitude this thesis could not have been written. It is with the same sense ofgratitude and good fortune that I acknowledge here the support and encouragement my family has continuously provided over the years. I would also like to thank all my colleagues in the AeMES Department at the University of Florida. In particular, Dr. Jeffrey Wright and Dr. SiddharthThakurwere very generous with their time. Their willingness to share their knowledge and experience made my work much easier. Finally, I would like to thankEglin Air Force Base and the Waterways Experiment Station Supercomputing Center for the use oftheir Cray computers. 111 TABLE OF CONTENTS page ACKNOWLEDGEMENTS ii ABSTRACT v i CHAPTERS INTRODUCTION 1 1 1.1 Overview ofthe Sail Analysis Problem 1 1.2 Review ofthe Literature 2 1.2.1 Two-Dimensional Potential Flow Based Models 2 1.2.2 Two-Dimensional Viscous Flow Based Models 4 1.2.3 Three-Dimensional Potential Flow Based Models 5 1.3 Focus ofthe Present Work 6 2 GOVERNING EQUATIONS 9 2. Membrane Equilibrium 9 2.2 Fluid Dynamic Conservation Laws 11 2.3 Nondimensionalization ofthe Governing Equations 12 NUMERICAL METHOD 17 3 3. Membrane Equilibrium 17 3.2 Fluid Dynamic Conservation Laws 17 3.3 Consistent Implementation ofthe Continuity Equation 26 3.4 The Aeroelastic Computational Procedure 27 3.5 A Potential Flow Model for Thin Wings 29 4 ELEMENTARY TEST CASES 31 4. Rotated Channel Flow 31 4.2 Uniform Flow Using a Moving Grid 33 4.3 Elastic Membrane Under a Uniform Pressure Load 34 MEMBRANE WINGS IN STEADY FLOW 37 5 5.1 Rigid Wing in a Viscous Fluid 37 5.1.1 Effect ofOuter Boundary Location 37 5.1.2 Effect ofGrid Refinement 39 in 1 5.2 Flexible Membrane Wing in a Viscous Fluid 45 5.2.1 Elastic Membrane Case 46 5.2.2 Constant Tension Membrane Case 50 5.2.3 Inextensible Membrane Case 51 6 MEMBRANE WINGS IN UNSTEADY FLOW 56 6. Constant Tension Membrane Case 58 6.2 Elastic Membrane Case 59 6.3 Inextensible Membrane Case 66 7 A THREE-DIMENSIONAL MEMBRANE WING MODEL 70 7.1 The Elastic Membrane Problem in Three-Dimensions 70 7.2 The Aerodynamic Problem in Three-Dimensions 77 7.3 The Aeroelastic Problem 81 7.4 Three-Dimensional Test Cases 82 7.5 Application to an Elliptical Planform Elastic Wing 86 SUMMARY AND CONCLUSIONS 90 8 REFERENCES 92 BIOGRAPHICAL SKETCH 95 IV Abstract ofDissertation Presented to the Graduate School ofthe University ofFlorida in Partial Fulfillment ofthe Requirements for the Degree ofDoctor of Philosophy A VISCOUS FLOW BASED MEMBRANE WING MODEL By Richard W. Smith December 1994 Chairman: Wei Shyy Major Department: Aerospace Engineering, Mechanics and Engineering Science Inthisworkanumericalmodel simulatingthe aeroelasticcharacteristicsofaflexible membrane wing is presented. The use ofthe Navier-Stokes equations as the fluid dynamic model in the present membrane wing theory is a substantial departure from previous work on sail aerodynamics which has, almost universally, adopted a potential based description ofthe flow field. The two-dimensional, viscous aeroelastic problem is nondimensionalized and a set of six basic dimensionless parameters is derived which govern the physical problem. An additional parameter, the frequency ratio, is proposed as a meaningful A parameter for characterizing the harmonically driven unsteady aeroelastic problem. numerical procedure is developed for the solution of the coupled aeroelastic problem and is shown to yield results that are in agreement with available analytic solutions for several appropriate limiting cases. The numerical procedure is also shown to satisfy certain identities as dictated by the fundamental fluid dynamic conservation laws. The role of viscosity in membrane wing aerodynamics is investigated using the numerical model for both steady and unsteady flows. These investigations are facilitated by distinguishing three distinct classes ofproblems which are associated with limiting cases of the dimensionless parameter set. The aerodynamic characteristics at Reynolds numbers between 103 and 104 are shown to differ substantially from those predicted by a potential based membrane wing theory. The role ofviscosity is shown to be preeminent in the harmonically forced unsteady flow about a membrane wing. In thiscase in the influence ofviscosity is enhanced since the acceleration and deceleration ofthe freestream velocity strongly influences the separation and reattachment ofthe flow. The periodic appearance and collapse ofrecirculation zones, along with an attendant adjustment in the membrane configuration, result in an aeroelastic response which may not be characterized as a simple harmonic response at the freestream forcing frequency. A three-dimensional potential based membrane wing element is also derived and is shown to yield results that, for several limiting cases, are in agreement with available analytic solutions. VI CHAPTER 1 INTRODUCTION 1.1 Overview ofthe Sail Analysis Problem Marchaj, in his second book on the science of sailing (Marchaj 1979), begins the section on sail design with the following comments. Despite the fact that mathematics, computers and wind tunnel testing are playing an increasing part in the designing ofsails, sailmaking as well as sail tuning are still strongholds of art based on a hit-or-miss technique rather than on science. . . .Afterall,unliketheaeroplanewing,whichcanberegardedasarigidstructurewhose shapeisunaffectedbyvariationinincidenceandspeed,sailshapeisafunctionofboth; inwhichtheshapeofthesailaffectsthepressuredistributionandviceversa,inarather unpredictable manner, (p. 500) These comments by Marchaj suggest two things concerning the analysis ofmarine sails: first, the behavior and performance ofa sail is governed by the aeroelastic interaction ofafluiddynamicfieldandadeformablesurface andsecondly,asaresultofthisinteraction, ; as well as the presence of other complexities, analytic and computational methods which have enjoyed considerable success in the design ofaircraft have not as yet proven useful to sail designers. It is likely that the second observation concerning the usefulness ofanalytic methods to sail designers will not always hold true. Asanillustrationofthepotentialusefulnessofcomputationalmethodsintheanalysis of sails consider the membrane wing ofFig. 1, which is shown operating near a boundary such as the surface ofthe sea. Since the sail is located very near the boundary it is immersed inthe shearlayeradjacenttothe free surface. The kinematic inflowconditions tothe sail are consequently nonuniform and may also be unsteady due to the presence of wind gusts. Furthermore, sincethesailisnotstationarybutratherhas some forwardvelocity,the relative 1 inflow velocity far upstream ofthe sail will vary in both magnitude and direction along the sail span as well as possibly varying in time. Computational methods provide a method of simulating such a complex flow environment which would otherwise be nearly impossible to reproduce experimentally. membrane wing inflow velocity free surface Figure 1.1. Schematic ofamembranewingoffinitespanoperatingnearafree surface. 1.2 Review ofthe Literature 1.2.1 Two-Dimensional Potential Flow Based Models The vast majority ofthe published works related to membrane wing aerodynamics have made several simplifying approximationsconcerningboththeelastic characteristicsof the membrane itself as well as the nature of the surrounding flow field. Perhaps the most significant of these simplifying assumptions is that the fluid dynamics can be adequately described by apotential based model ofthe flow field. In addition to the almost universally adopted potential flow assumption, the additional approximations associated with thin airfoil theory - small camber and incidence angle - are also often made and the membrane itself is generally considered to be inextensible. A comprehensive review of the work published prior to 1987 related to membrane wing aerodynamics is given by Newman (1987). The analysis ofmembrane wings begins with the historical works ofVoelz (1950), Thwaites (1961) and Nielsen (1963). These works considered the steady, two dimensional, irrotational flow over an inextensible membrane with slack. As a consequence of the inextensible assumption andthe additional assumptionofsmallcamberandincidence angle the membrane wing boundary value problem is linearized and maybe expressedcompactly in nondimensional integral equation form as > dHy/a) o where y(x) defines the membrane profile as a function the x coordinate, a is the flow incidence angle and Cj is the tension coefficient. Equation (1.1) has been referred to as the 'Thwaites sail equation' by Chambers (1966) and simply as the 'sail equation' by Newman (1987) andGreenhalghetal. (1984).Thisequation,togetherwithadimensionless geometric parameter e which specifies the excess length of the membrane, completely defines the linearized theory ofan inextensible membrane wing in an steady, inviscid flow field. Different analytic and numerical procedures have been applied to the sail equation inorderto determine the membrane shape, aerodynamic properties, and membrane tension in terms of the angle of attack and excess length. In particular, Thwaites (1961) obtained eigensolutions ofthe sail equation which are associated with awing atthe ideal (singularity free) angle of incidence. Nielsen (1963) obtained solutions to the same equation using a Fourier series approach which is valid for wings at angles ofincidence other than the ideal angle. Other more recent but similar works are those by Chambers (1966), Vanden-Broeck and Keller (1981), Greenhalgh et al. (1984) and Sugimoto and Sato (1988). Variousextensionsofthe lineartheory have appearedin the literature overtheyears. Vanden-Broeck(1982) aswellasMurai andMaruyama(1980)developednonlineartheories valid for large camber and incidence angle. The effect ofelasticity has been included in the membrane wing theories ofJackson (1983) and Sneyd (1984) and the effects ofmembrane porosity have been investigated by Murata andTanaka (1989). In apaperby de Matteis and de Socoi (1986)experimentallydeterminedseparationpointswereusedto modify thelifting potential flow problem in an attempt to model flow separation near the trailing edge. The effect ofelasticity and porosity were also considered in this work. Comparisons of the various potential flow based membrane wing theories with experimental datahavebeen reportedby several authors includingGreenhalghetal. (1984), Sugimoto and Sato (1988) and Newman and Low (1984). In general, there has been considerable discrepancy between the measurements made by the different authors which have all been in the turbulent flow regime at Reynolds numbers between 105 and 106 (Jackson and Fiddes 1993) As a result ofthe discrepancies in the reported data-primarily . duetodifferencesinReynoldsnumberandexperimentalprocedure-theagreementbetween the potential based membrane theories and the data has been mixed. In particular, the measuredliftisinfairagreementwiththepredictedvaluewhentheexcesslengthratiois less than .01 and the angle of attack of attack is less than 5°. However, even for this restricted range of values the measured tension is significantly less than predicted by theory. Furthermore, forlargerexcess lengths and incidence angles, both lift and tension are poorly predicted by the theory. Flow visualization studies indicate the main reason for the disagreement is the existence of a thick boundary layer or region of separated flow on the membrane, typically near the trailing edge. It has been noted by several authors that the presence of viscous effects such as thick boundary layers and separation regions will overshadow any implications associated with the linearizing approximations made by thin wing theory (Newman 1987). 1.2.2 Two-Dimensional Viscous Flow Based Models Although the importance of viscous effects on membrane wing aerodynamics has been recognized for quite some time (Nielsen 1963, Newman and Low 1984), very few