A Theory of Rectangular Wing Aerodynamics
Shiang-yu Lee, Ph. D.*
The Boeing Company, Seattle, Wa, 98124-2207
A kinematic model for the formation of circulation on a rectangular plate was proposed
recently by the author based on the reactive flow and its dependency on the presence of side
edges. The resulting analytical expressions appear to yield reasonable results consistent with
finite wing experimental behavior. In this paper, further results including the general
expression of circulatory flow over the entire plate and a derivation of the drag component
are presented. In addition, an empirical relation representing the non-linear force is
included to evaluate the total aerodynamic coefficients for comparison to experimental data.
The results demonstrate excellent agreement with experiments. Built on these observations,
a new interpretation of the classical Prandtl “Lifting-line Theory” based on the current
formulation is presented.
Nomenclature
A
= wing aspect ratio
b, c
= wing span and chord
CLP, CDC, CNC, CNN = Potential lift, cross flow drag, cross flow and non-linear normal force coefficients respectively
V
= main stream velocity
w
= velocity component normal to the wing surface
x, y, z
= Cartesian coordinates
= angle of attack
= circulation or vortex strength
= air density
I.
Introduction
he basic concepts embodying the Prandtl “Lifting Line Theory1” (LLT) has long been accepted by the
aerodynamics discipline as the underlying principle for finite wings. In brief, the Biot-Savart Law in the general
field theory is utilized to interpret the wing and wake vortex system, providing a mathematical evaluation of the
induced downwash flow. The total downwash at a chosen location counteracts the normal component of free stream
flow, satisfying the no flow boundary condition on the wing surface. The downwash produced by the trailing
vortices is attributed to reducing the effective angle of attack and the lift as well as producing the “lift induced drag”.
An analytical solution is available for the case of wings with elliptic lift distribution, which demonstrates that the
down wash effect becomes more pronounced with decreasing aspect ratio, to such an extend that when the span
approaches zero, no lift can be produced. It is commonly accepted that while this trend in lift degradation is in
general accord with physical observations, the values are not accurate for engineering purposes.
Prandtl’s theory also predicts that the lift induced drag of a finite wing is proportional to the square of the lift
force and inversely to the aspect ratio. This relationship, commonly known as the “Drag Polar” of airplane wings,
matches experiments very well for moderate and high aspect ratios and is utilized as a primary parameter in classical
aircraft design. An extension of the theory, due to Munk2, indicates that the induced drag would be at a minimum if
the lift has an elliptic distribution over the span, an assertion that influenced the design of many aircrafts during
Word War II, as exemplified by the famous English Spitfire fighter. However, experiments3 evidently has disproved
this assertion. There are several improved variations of the vortex model which provides better interpretation of the
flow and boundary conditions as summarized by, for example, Schlichting4.
T
*
Engineer Scientist, Boeing Information Technology, PO Box 3707, Seattle WA, 98124-2207/MS 2R-97, AIAA
Senior Member.
1
American Institute of Aeronautics and Astronautics
For narrow wings, the Slender Wing Theory5,6 (SLW) indicates that the lift producing circulation is determined
by satisfying the boundary condition solely with the induced flow produced by the stream-wise vortex system. The
dismissal of the effective angle of attack creates a notable point of contradiction between the LLT and SWL
theories. Another important conclusion of the SLW is that the wing is only effective in producing lift in the portions
with an expanding span. It is also interesting to note that, for small aspect ratios, the LLT predicts the drag to be the
projection of lift in the stream direction, CL, whereas the SLW gives only half of that value, again presenting major
disagreements.
The more sophisticated lifting surface theorem as well as modern computer based computational methods were
built on the same concept of vortex flow interactions. These results appear, generally, in agreement with the SLW in
the narrow wing range.
Thwaites7 collected the work of
investigators in a volume summarizing the
significant contributions up to 1960 and
Kuchemann8 filled in many later work
involving high speed and supersonic flows.
Modern numerical analyses are based on
the same kinematic model of the analytical
approach but delegated the grunt work of
computation to digital computers, as
described in Plotkin9, et al.
In the earlier theories, the contribution
due to nonlinear forces are largely ignored.
Bollay10established a trailing vortex effect
Figure 1. Reactive Flow on a Moving Plate
model for the narrow rectangular wings
11
and Gersten followed with a lifting
surface method for analyzing arbitrary
planforms. Both theories indicated that the nonlinear forces were included and their solutions converge to Newton’s
theory in the narrow wing limiting case. However, neither paper presented an explicit analytical expressions that can
be evaluated or extended. Polhamus12 introduced the idea of “Leading Edge Suction Analogy” where the notion of
forward suction is adopted and used to calculate the nonlinear lift force. The results appeared to match experiments
quite well in low aspect ration ranges for delta wings as well as in cases of other extended geometry investigated by
Lamar13. These results, however, at no where approach the Newtonian non-linear force.
Recently, Lee14 presented an alternate kinematical model and mathematical representation for the flows of
rectangular wings and an ensuing paper15 provided an analytical result for delta wings, both of which indicated a
good match with experimental data. In this paper, the discussions on rectangular wings is further refined with
expressions for drag and non-linear forces and a direct comparison to experimental data is presented.
II.
Representation of Rectangular Wings
In the historical developments of aerodynamics, the existence of circulation on a wing has been taken as an
accepted phenomenon and not very much reasoning were devoted to explain exactly how it occurred. The KuttaJoukowski theory is a mathematical description of an observed state but gives no causal relationships. It has been
popular to attribute the development of wing vortex and circulation to the viscous effects at the trailing edge. These
arguments are offered without recognizing that a planar vortex flow actually posses a field of very high shear strain
rate and any vortical flows would dissipate quickly. This is evident in that investigations into the viscosity argument,
as summarized by Sears16, has only produce the conclusion that higher viscosity could result in the lowering of
circulation strength. On the other hand, modern theoretical representations of wing circulation remain solely as
functions of geometry, independent of viscosity.
Lee14 presented the concept that the formation of circulations is the consequence of fluid motion attempting to
circumvent a solid obstacle. It is well established in potential flow theory that, when an infinite flat strip is placed in
a stream of uniform flow normal to its surface, since no fluid particle can penetrate the solid surface, the plate can be
represented as a system of doublets, Fig 1. It was shown that, for a steady relative motion, if the flow remains
attached to the plate, the reactive flow forms a cylinder of circulatory flows around each the edges of the plate. Since
the reactive flow is everywhere equal and opposite to the impinging flow, the circulations therefore vary in strength
in accordance to its distance from the edge, with the maximum circulation occurring at the centerline.
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American Institute of Aeronautics and Astronautics
In real flow, the vortical flows can not stay in such closed loops and separations must occur. However, if the
plate is brought into motion also in the lateral direction, then the circulations of the trailing edge side can not stay
attached to the plate and must drift into the wake, thus forming the “starting vortex”, whereas the forward circulation
would wrap around the wing, forming the bound circulation. When a steady state is reached after shedding of the
starting vortex, the forward circulating flow essentially forms a cylindrical core of rotational flow while only the
largest circulation value becomes relevant and the lift producing circulation, as indicated in Fig.1, reproduces the
Kutta-Joukowski theory.
Figure 2. Reactive Flow on Finite Plate
Figure 3. Finite Plate Vortex Ring
From the forgoing discussions, we can conclude that the circulation on a wing is the consequence of the reactive
flow of the wing surface against the on-coming flow and that the circulation is determined only at the centerline
where the resulting circulation is highest. Implied by this assertion is that the circulation on a wing is determined by
the flow solution at the on-set of motion, before the trailing circulations drift into the wake.
For a finite rectangular plate, when the plate is set into motion normal to its surface, the reaction flow can also be
represented by a system of doublets distributed over the two dimensional planform. Figure 2 depicts a plate with an
aspect ratio slightly larger then unity. Now, because the presence of lateral edges, some of the circulatory flow must
flow around these edges as well. That is to say that the circulatory streamline circuits must follow a trajectory
slanted toward the corner in a more or less radial direction. This slanting in orientation means that when the plate
moves forward, the full lifting capability of the chord could not be realized as the forward circulation component is
only a fraction of the maximum possible strength.
Now, visualizing the plate moving forward, Fig.3, we can examine the trailing “vortex ring” configuration. Since
the mid-chord line separates the forward and backward wrapping circulations, it is natural to assume that it is also
the separation line for the trailing wake system. As only the forward component of the circulation would remain
attached and wrap around the wing, likewise, only the lateral component would circulate the trailing vortices. At
each span location, the lateral component would form the vortex “core” for the trailing vortex. Here we can invoke
a special form of the Helmholtz circulation conservation law, namely that the lateral circulation at each span
location must equal the forward component as it represents the extension of the forward component in the wake. In
other words, the conservation law mandates a constant value of circulation at each span location of the vortex core
manifested in a complete vortex ring loop which includes the wing attached circulation, the trailing vortex and the
shed starting vortex system.
These discussions entail a reactive flow system around the plate at any location, at the on-set of motion, to be,
wx = (b / 2
y ) /[(c / 2 x) + (b / 2
y )]
(1)
Which yields the circulation distribution,
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x
= w(c 2 x)(b 2 y ) /[(c 2 x) + (b 2 y )] =
(2)
y
For the maximum circulation at mid-chord, Eq. 2 can be written as,
y
= wc(b 2 y ) /(c + b 2 y ) =
x
(3)
The circulation distribution and corresponding reactive flow components are shown in Fig. 4. It is interesting to
note that the forward, or x component of reactive flow and the circulation do assume a near elliptic distribution,
consistent with experimental observations. It is noted that the Prandtl LLT analysis assumes that the elliptic lift as
the consequence of an elliptic planform in conjunction with wing twist or other geometrical effects, and there is no
indication that it could be a natural effect of a rectangular wing.
Figure 4. Reactive Flow and
Circulation Distribution at Mid-Chord
Figure 5. Circulation Flow Geometry
It is further observed that the forward wrapping circulations are bound by the bifurcation lines AB and CD as
indicated in Fig. 5. In this case, the forward circulation in the region near the edge is that issued from the bisect line.
Base on these observations, Lee14 proceeded to derive the lift coefficients for rectangular wing as, for A > 1,
C LP =
2
b/2
0
y
( y )Vdy
(1 / 2) V 2 S
= (4 sin / b){
(b c ) / 2
0
[(b 2 y ) /(c + b 2 y )]dy +
b/2
(b / 2
y )dy}
y )dy = ( A / 2) sin
(5)
(b c ) / 2
(4)
= 2 sin {[ A 1 ln(1 + A) + ln 2] + 1 / 4} / A
And for A < 1,
C LP = (4 sin / b)
b/2
0
(b / 2
It is noted that, in theory, the lift component indicated by Eg. 4 and 5 are the pure “potential lift” force as it is
the product of the free stream velocity and the circulation on the wing and therefore perpendicular to the flow.
Proceeding with the evaluations, the linear drag component is determined by the cross flow component, wy,
which is complementary to the axial component, wx, as also shown in Fig.4. The cross flow drag coefficient can,
therefore, be derived as,
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C DC =
2
b/2
0
y
( y ) w y dy
2
(1 / 2) V S
= 2 A sin 2 (
A
+ ln( A + 1) ln A 1) (6)
A +1
Now, by assuming that the drag force to be the projection of a normal force on the plate,
C DC = C NC sin
(7)
We can write then, instead of (6),
C NC = 2 A sin (
A
+ ln( A + 1) ln A 1) = C NC sin
A +1
(8)
Where C NC is recognized as the slope of the “cross flow normal force coefficient”.
III.
Comparison to Experimental Data
Figure 6 provides an illustration of the accuracy of the potential lift coefficient of the current theory as given in
Equations 4 and 5. It is seen that the values fit right with the Prandtl-Betz17 experiments and is consistent with the
classical slender wing theory or lifting surface results. However, the physical implications of Eq. 5 is entirely
different. As shown in Fig. 7, the theoretical result is the consequence that the lift generating circulation being
confined to the forward triangle region of the plate as bounded by the bisect lines from the wing tip corners. This is
very reasonable as beyond this region, the circulations would be flowing around the lateral edges, producing cross
flow vortices and not a potential lift.
Figure 7. Narrow Wing Circulation
Figure 6. Potential Lift Coefficient
To evaluate the overall theory, the force and angle of attack relationships are compared to the original
experiments by Winter18. For wings of small aspect ratio, since the non-linear force also makes significant
contribution to the overall lift, they must be properly accounted for as well. There is, however, to this date no
commonly accepted representation of the non-linear lift and drag forces. In his study for delta wings, Lee15
introduced an empirical relation representing the non-linear normal force component. For the current case of
rectangular wings, we modify the formula slightly as guided by the experimental drag value for square plates,
C NN =C NN sin 2
=
2
sin 2
1.77
( A + 1)
(9)
Although extremely simple, the formula does converge to the Newtonian non-linear force in the extreme slender
configuration and the force diminishes quickly as aspect ratio increases. For the square plate, the equation would
yield half of the test drag value3 of 1.17, as in an inclined lifting configuration, only the lateral edges would have
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separations and contributing to the normal non-linear force. This representation is used here for our comparison to
experimental results with the lift coefficient calculated as,
C L = C LP sin
+ C NC sin cos
+ C NN sin 2
cos
(10)
Whereas the total drag is evaluated from,
C D = C NC sin 2
+ C NN sin 3
(11)
Since we are comparing our theory against the Winter test data, which are primarily recorded in terms of normal
force, we utilize the following expression to summarize all force contributions;
C N = C LP sin cos
+ C NC sin
+ C NN sin 2
(12)
The accuracy of the current theory is quite remarkable as demonstrated in Fig. 8 through 11. The normal force
coefficients are slightly higher for small angles of attack but match experiments exactly at higher angles in every
case.
Figure 8. Lift and Drag Coefficients, A = 0.35
Figure 9. Lift and Drag Coefficients, A = 0.66
Figure 10. Lift and Drag Coefficients, A = 1.0
Figure 11. Lift and Drag Coefficients, A = 2.0
Figure 12 shows the drag polar comparison for the case of A = 0.35. As demonstrated, the current theory predicts
a drag to lift relationship lower then the experiment. On the other hand, the Prandtl TTL theory over-predicts the
drag with similar error margins.
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In the low aspect region, Hoerner3 indicated that the Winter data demonstrated an extraordinary rise in non-linear
“extra lift” above the Newtonian non-linear force. The maximum value he quoted is of the order of 3.6. Since his
comparison is based on the SLW theoretical values, it is equivalent to the sum of the second and third terms in Eq.
11. To compare with Hoerner’s assertion, we express both terms as non-linear parameters,
C N = C LP cos + (C NC / sin
+ C NN ) sin 2
(13)
The first term in the parenthesis is divided by the sine of 30 degrees for comparison in Fig. 13. It is seen that the
current model prediction agrees very well with the Hoerner representation. With this verification, we may conclude
that the Hoerner observation might have misconstrued to attribute all forces above the potential flow lift as
contributed by the non-linear force. In fact the majority contribution could be from a linear normal force resulting
from the cross flow vortex momentum which also produces the “induced drag”.
Figure 12. Drag Polar Comparison A = 0.35
IV.
Figure 13. Drag and Non-Linear Lift Coefficients
Discussions of the Lifting Line Theory
We have demonstrated that experimental data confirms that the circulation formation approach introduced in this
paper provides very reasonable predictions for aerodynamic forces on rectangular wings. The reasoning supporting
the formulation is somewhat different from traditional approach but provides a realistic physical representation. We
can summarize the findings as follows.
By using the model based on the reactive flow of a plate at onset of motion, we are able to determine the
orientations of the circulation flows which would persist once translational motion is introduced. The reactive flows
are inclined toward the side edges of the wing, thereby reducing the stream-wise circulation and the lift. In addition,
the slanted circulation also dictates a cross flow component which results in the formation of trailing vortices.
Instead of being an “induced flow” due to trailing vortices, we view the reactive flows as forming the cylindrical
“core” of the vortices with distribution known along the span. The sum of this cross flow and that of the forward
circulation component offsets the impinging free stream flow, thereby satisfying the boundary condition everywhere
on the wing. However, only the reactive flow at the centerline, which entails largest circulation, is significant for
calculating the overall circulation at any span location.
The forward flowing circulation is also bounded by the bisect lines issuing from the leading edge corners,
beyond which the circulations can not traverse around the leading edge. This geometrical constraint results in a
circulation distribution equivalent to the slender wing theory, matching early experiments on lift coefficients. This
lifting force is labeled as the pure “potential lift” that produces no drag.
The cross flow component is used to compute the drag force on the wing, which in turn is regarded to be a
projection component of a wing normal force. This normal force, being a linear (sine) function of the angle of
attack, is additive to the potential lift. We identify this force component as the “cross flow vortex lift” of the trailing
vortices.
With these reasoning and by adding a nonlinear component guided by Newton’s law and experimental data, a
mathematical model that agrees with experiments has been established.
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The reactive flow components, wx and wy, as illustrated in Fig. 4, can be viewed as equivalent to the “induced
angle” and “effective angle of attack” in the traditional LLT theory. However, the distribution of the cross flow
component, wy, is far from being a constant as in the LLT theory.
The induced flow concept and application of the Biot-Savart law could be subject to debate. Although both the
electro-magnetic field theory and the flow theory are governed by the same Laplace’s equation, the original BiotSavart law is specified to describe the field distribution around a conducting wire which causes a magnetic field
around the circuit. Therefore, an entire magnetic field is the “induced” consequence of the electrical current. On the
other hand, in the flow situation, the trailing vortex is the result of some other forces but it does not contain within
itself a field source of energy for causing an “induced flow”. If the LLT usage of Biot-Savart law holds, it means
then that the trailing vortices would grow in strength after it drifts away from the trailing edge, as it has been
assumed that the induced flow at the wing is only half the strength of an infinite vortex. This is obviously not an
observed phenomenon but on the contrary, the trailing vortices usually decay. In our current theory, since the flow
model is different and not relying on the Biot-Savart law, we do not face the same dilemma.
The current theory is further justified by the Hoerner observation that the pure potential lift due to the slender
wing theory falls far short of the measured actual total lift. By adjusting the cross flow linear normal force with a sin
value, we are able to create a matching value for Hoerner’s “extra lift”. The cross flow normal force component is
only significant at very small aspect ratios and diminishes as wings become wider, consistent with the “induced
drag” observation. We also observe that the cross flow force is concentrated near the wing tips.
In all, the current theoretical model has generated a new set of representation for flow configuration and
aerodynamic forces; the results appear to be very accurate and realistic. Based on these observations, the current
theory could be considered as a candidate alternative interpretation of the classical lifting line theory of wing
aerodynamics.
References
1
Prantl, L , “Uber Flussigkeitsbewegung bei sehr kleiner Reibung”, Verh. 3, int. Math. Kongr., Heidelber, 1914
Munk, M. M., “Isoperimetriche Aufgaben aus der Theorie des Fluges”, Dissertation, Gottingen University, 1919
3
Hoerner, S. F. “Fuild-Dynamic Drag”, 3rd ed., Hoerner, Midland Park, N.J., 1965
4
Schlichting, H, E. Truckenbrodt and H. Ramm, “Aerodynamics of the Airplane”, MacGrow-Hill, New York, 1979.
5
Jones, R. T., “Properties of low-aspect ratio pointed wings at speeds below and above the speed of sound”, NACA Report
1340, Washington, D.C., 1946.
6
De Young, N, “Spanwise Loading for Wings and Control Surfaces of Low Aspect Ratio”, NACA Tech Note 2011, 1950.
7
Twaites, B., “Incompressible Aerodynamics”, Dover Publication, New York, 1987
8
Kuchemann, L. M., “Theoretical Aerodynamics”, 4th Edition, Constable and Company, Ltd., London, G.B., 1973; Printed by
Dover Publications, Inc., New York, N.Y., 10014, 1982.
9
Katz, J. and A. Plotkin, “Low Speed Aerodynamics,” Cambridge University Press, Cambridge, UK, 2001.
10
Bollay, W, “A Non-linear Wing Theory and its Application to Rectangular Wings of Small Aspect Ratio“, Z. Angew.
Math. Mech., 19, 21, 1939
11
Gersten, K., “A Nonlinear Lifting-Surface Theory Especially for Low-Aspect Ratio Wings“, AIAA J. 1, 1963.
12
Polhamus, E. C., “Predictions of Vortex-Lift Characteristics by a Leading-Edge Suction Analogy”, AIAA J. Aircraft, 8,
1971.
13
Lamar, J. E., “Extension of Leading Edge-Suction Analogy to Wings with Separated Flow Around Leading Edges at
Subsonic Speed”, NASA TR R-428, October 1974
14
Lee, S., “ The Vortex Impulse Theory of Wing Aerodynamics”, AIAA Paper AIAA-2004-4733, Presented at the 12th AIAA
Applied Aerodynamics Conference, Providence RI, 2004.
15
Lee, S., “An Analytical Representation of Delta Wing Aerodynamics”, AIAA Paper AIAA-2005-5192, Pesented at the 4th
AIAA Theoretical Fluid Dynamics Conference, Toronto, Canada, 2005
16
Sears, W. R., “Some Recent Developments on Airfoil Theory”, J. Aer. Sci., 23, 1956
17
Prandtl, L and Betz, A, “ Experimentelle Prufung der Umbrechnungs-formeln”, Ergebn. Aerodyn. VersAnst. Gottingen 1,
50, 1920
18
Winter, H., “Flow Phenomena on Plates and Airfoils of Short Span”, NACA, TM 798, 1936.
2
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