Low Reynolds Number Aerodynamics: Proceedings of the Conference Notre Dame, Indiana, USA, 5–7 June 1989

455 Pages·1989·26.52 MB·English

by  Mark Drela (auth.)

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54
T. J. Mueller (Editor)
Low Reynolds
Number Aerodynamics
Proceedings of the Conference
Notre Dame, Indiana, USA, 5-7 June 1989
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Thomas J. Mueller
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Notre Dame, IN 46556
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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