Viscous flow based membrane wing model

104 Pages·1994·2.9 MB·English

by  Smith, Richard W.

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