PointedTip Wings at
Low Reynolds Numbers
Tara Chklovski
When thou seest an eagle, thou seest a portion of genius; lift up thy head!
William Blake
1. Introduction and Motivation
1.2.1. Significance of Low Speed
1.2.2. Complex and Variable Geometry
2. Basic Principles of the Standard Aerodynamic Model
2.1. The Standard Model Roadmap
2.2. Flow Assumptions of the Standard Model
2.3. Calculating lift forces on an airplane
2.3.1. Calculating lift force on a twodimensional airfoil using KuttaJoukowski Theorem
2.3.2. Finite wing modeled by single line in Lifting Line Theory
2.3.3. Spanwise Lift Distribution on different planforms
2.3.4. Dependence of Lift on Camber and Reynolds number
2.3.5. Summary of principles used to calculate lift on a wing
2.5. Total Drag  sum of pressure and frictional drag
3. Application of Standard Aerodynamic Model to Bird Flight
3.1. Characteristics of Low Reynolds numbers flows
3.2. Airfoil Behaviour at Low Reynolds Numbers
3.4. Effect of Variable and Complex Geometry
3.5. Significance of Unsteady effects
4. Critical Review of Biological Literature
4.1.1. Variation of Aspect Ratio and Wing Area among different species
4.1.2. Variation of Aspect Ratio in Procellariiformes
4.1.3. Comparison of wing tip shapes in soaring birds
4.1.4. Summarizing Aerodynamic Studies of Structural Wing Features
4.2. Studies of Varying Wing Geometry during Flight
4.2.1. Summarizing Studies of Active Flow Control
4.3. Investigating Lift Distribution
4.4. Experimenting with dead birds
4.4.1. Summarizing Aerodynamic Studies of Dead Birds
5. Critical Review of Literature on Low Reynolds Number Flows
5.1. Low Reynolds Number Experiments with Airfoils and Wings
5.1.1. Thin Airfoil Experiments
5.1.2. Wings with Low Aspect Ratios and Different Planforms
5.1.3. Measuring Profile Drag at Low Reynolds Number
5.2. Challenges in Experimental Setup
5.2.2. Complicated Wake Structure
5.2.3. Aerodynamic Force Measurements at Low Reynolds Numbers
5.3. Summary of Low Reynolds Number Airfoil Studies
6. Identification of a Finite, Tractable Problem
6.2.1. Geometry of Wing Models
6.2.2. Wind Tunnel Measurements
6.2.3. Study of Pointed Tip Wings with Numerical Methods
7. Numerical study of pointed tip wings
7.1. Introduction to Potential Based Panel Method  PMARC
7.1.2. Resultant Force Coefficients
7.3. Sensitivity to Number of Panels
7.5. Variation in aerodynamic parameters for Elliptical wing with Aspect Ratio
7.6. Wing Geometry effects on Force Coefficients
7.7. Effect of Angle of Attack on Aerodynamic Parameters
7.8.2. Lift and drag coefficients obtained from surface pressure integration
Bird flight is an extremely complex subject and even after a century of manned flight its intricacies have not been completely understood. This may be due to the rapid progress in aviation and increase in flying speeds that in turn led to the disappearance of obvious parallels between general aviation and birds. Thus, many recent advances in aviation have been made without the study of bird flight. But general aviation is founded on the work of pioneers such as Leonardo da Vinci, Cayley, Lilienthal and the Wright brothers who deduced key principles of flight from their studies of birds. For instance, Leonardo da Vinci made detailed observations of birds executing turns, noting wing and tail postures (Figure 1‑1).
Figure 1‑1 Leonardo da Vinci’s sketches of bird flight maneuvers. Taken from (Ruppell 1977).
Through his studies of bird flight, Cayley recognized that the lift and thrust function of bird wings were distinct and hence could be separately reproduced in fixed wing aircraft. He also observed that a bird feather set at a particular angle in moving air could produce lift (Ruppell 1977).
Lilienthal was the first to make numerous successful gliding flights over long distances. He wrote a book, “Bird flight as the basis for aviation” founded on his observations of bird flight. Some examples of his studies can be seen in Figure 1‑2.
Figure 1‑2 Study of feathers in a stork and analysis of wingbeats. Taken from (Ruppell 1977).
The Wright brothers also studied birds and noticed that birds change their wing shape in order to maneuver. This observation led them to obtain roll control in their aircraft by warping a part of the wing (About).
In 1910, Igo Etrich designed and flew the Taube (Figure 1‑3) whose unique structure and wing shape were inspired by the dove. It was a slow, reconnaissance plane and represented the transition between bulky experimental aircraft and the nimble fighters of WWI (Naughton 2002).
Figure 1‑3 Taube – an aircraft with a distinct birdinspired structure.
After the world wars, there was even greater progress in aviation and recognizable points of commonality between general aviation and birds decreased. Thus while great advances have been made in understanding general aviation, many questions pertaining to bird flight remain unanswered and pose exciting challenges. The aim of this research is to study one aspect of bird aerodynamics, and further understanding of avian flight.
The problem at hand deals with why many oceanic birds have pointed wing tips. In Figure 1‑4 it can be seen that the larger birds, albatrosses and the larger petrels have pointed wing tips. As size decreases, wing tips become less pointed and more emarginated or notched.
Figure 1‑4 Progression in wing tip shapes with size in albatrosses[1] and petrels (Pennycuick 1982).
In case of pointed wing tips, classical aerodynamic theory predicts high local lift coefficients and low Reynolds numbers leading to tip stall and flow separation. This increases profile and induced drag and is very undesirable. This is in contrast to modern aircraft that usually have wings with reasonable taper to prevent tip stall. Some examples of wing tips found in different aircraft can be seen in Figure 1‑5.
Figure 1‑5 Wing tips. Taken from (Raymer 1999).
Wings with low sweep back usually have a taper ratio of 0.40.5, while highly swept back wings have a taper ratio of about 0.20.3 (Raymer 1999). Taper also enables the lift distribution along the wing to approximate that of an ellipse[2], as seen in Figure 1‑6.
Figure 1‑6 Relationship of local lift distribution and taper ratio. Taken from (Raymer 1999).
Oceanic birds provide good testing ground for theories as they fly in uncluttered environments and thus do not have to make drastic compromises between optimum flight and suitable landing and takeoff requirements. An example of such a compromise can be seen in land soaring birds such as eagles, vultures and buzzards. They need to be able to exploit thermals and air currents and thus require large lifting surfaces with low induced drag. They should also be able to take off steeply amongst areas crowded with trees and other obstacles. This constraint requires that their wings be smaller than the optimum soaring size. The compromise evolved was emarginated wing tips that effectively reduce induced drag, while allowing smaller wings. Oceanic birds on the other hand, do not have such stringent taking off constraints and can develop very long wings (within structural limits).
The question then arises as to why they have consistently developed pointed wing tips. However, it needs to kept in mind that birds are not pure flying machines. They are living, multiplying creatures that need to fulfill many other functions. Thus, there may be manifold reasons for pointed wing tips that are completely independent of aerodynamic efficiency.
Bird flight has some unique features that differentiate it from conventional aerodynamics. Their wings generate both lift and thrust, and consequently, have complex wing geometry and wing kinematics. They also fly at low speeds and are under the influence of unsteady flow field effects. Thus in some cases it might be harder for direct application of standard aerodynamic principles, while in others these principles might not be applicable at all. The following paragraphs will briefly deal with three of these features.
Birds are smaller (wing span for a hummingbird is 0.09 m and that of a wandering albatross is 3 m) and slower (with average speeds ranging from 9 m/s to 22 m/s) than general aircraft. This difference can be seen quite clearly in Figure 1‑7. The diagram plots insects, birds and aircraft (human powered, ultralights, sailplanes, small engine powered airplanes and large commercial airliners). The smallest bird is the Ruby throated hummingbird with a wing span of 0.09m flying at 8.1 m/s. The largest bird is the Wandering Albatross with a wing span of 3.03 m (34 times greater than the hummingbird) flying at 19.4 m/s. The smallest engine powered airplane in this diagram is the Piper Warrior that has a wing span 3.5 times larger than the albatross. However, the Boeing 747400 has a wing span of 65m (21 times greater than the albatross) and a cruising speed that is almost 10 times that of the albatross.
Figure 1‑7 The Great Flight Diagram comparing wing loading and cruising speeds of the largest and fastest airliners to the smallest insects. Taken from (Tennekes 1996).
Because of low speed and small length scales, birds fly in the low Reynolds number range, 10^{4} to 10^{5}. In Figure 1‑8, Mach number has been plotted against Reynolds number and different flying vehicles are grouped according to size and speed. Insects occupy the lower end of the scale. Large model airplanes closely resemble birds in size and flying speeds. General aviation falls within high Reynolds numbers = 10^{6} to 10^{7 }while jet aircraft have Reynolds numbers = 10^{7} to 10^{8}.
Figure 1‑8 Variation of Reynolds numbers with speed across insects, microair vehicles, birds, Model airplanes, Human powered vehicles, aircraft, hanggliders and lighter than air airships. Adapted from (Lissaman 1983).
The Low Reynolds number regime is usually characterized by laminar flows with a tendency towards separation. Much work remains to be done in determining the precise effects of low Reynolds number on flow around bird’s wings.
Bird’s wings can be twisted, emarginated, pointed, or lunate with nonstraight leading and trailing edges. During the course of a wingbeat cycle, a bird may vary its wingspan, wing area (sweep), twist or camber. With variation in flight speed, the bird may also optimize its morphology to suit the flight speed. Most aircraft on the other hand, have simple, wellstudied shapes (elliptical, rectangular or modestly tapered) that remain fixed in flight. There is therefore a need for much experimentation, exploring different planforms and wing profiles at relevant Reynolds numbers, to quantify the effect of geometry.
Due to flapping and flexing of the wings, the lift and thrust components vary with time, leading to unsteady flow field effects. Viscous effects, such as turbulence and vortex shedding can also result in fluctuating or periodic effects. This is a vastly complex field of new problems. Current particle image velocimetry techniques can do much towards understanding such time varying phenomena. However, this research work will not focus on unsteady effects.
This research aims to further understanding of avian aerodynamics by studying two aspects of bird flight – low Reynolds numbers and complex geometry. Specifically, this research will determine the following:
(i) Aerodynamics of rigid, fixed, wings in Low Reynolds Numbers flows.
(ii) Effect of pointed wing tips on aerodynamic characteristics of wings.
The following chapters will discuss classical aerodynamic theory, its application to bird flight, complexities of bird flight and studies conducted by biologists dealing with bird wing geometry. Studies of Low Reynolds numbers flows will also be reviewed. Finally, Chapter 6 will deal in detail with the proposed research plan.
The standard aerodynamic model is of interest because general aviation is founded on it and it has been very successful in characterizing and predicting flow behaviour for general aircraft. The main principles of this model need to be analyzed to determine whether the model applies to bird flight as well. This chapter will focus primarily on the main principles of the standard aerodynamic model and the following chapter will deal with the areas where bird flight is significantly different from general aviation.
The aim of the model is to describe flow around a wing and to calculate forces acting on it.
Flow around a wing can be very complex. Therefore, key simplifications have been introduced in the standard model. These simplifying assumptions have been very effective in accurately characterizing flow around a wing. Based on these assumptions, various theories have been developed that characterize flow around a wing and that are used to calculate forces acting on the wing.
The following sections will briefly deal with the simplifying assumptions and then review principles used to calculate lift and drag on airplanes.
The standard model assumes the flow around a body is steady, inviscid, irrotational and incompressible.
Steady Flow  In steady flow, all parameters necessary to describe the flow do not vary with time anywhere in the flow. Thus, parameters such as pressure, density, magnitude and direction of velocity are functions of space, but not time.
Inviscid Flow – Flow is assumed to have zero or negligible viscosity. Viscosity is the process by which molecular motions transfer momentum. For example, if there were two layers of fluid with different velocities aligned with the flow, there would be an exchange of molecules between the two layers that would equalize the velocities. The rate at which this exchange of crossstream transfer of momentum is effected by the molecules per unit area, is called shearing stress or viscous force per unit area (Kuethe and Chow 1950). Stress _{} is thus directly proportional to viscosity as represented by the following equation.
_{} Eq 2‑1
where _{} is the coefficient of viscosity, dV is the relative speed of the layers and ds is the distance between the layers.
Irrotational Flow – An irrotational flow has zero vorticity, where vorticity is the curl of V (the velocity vector of the flow field).
_{} Eq 2‑2
Incompressible Flow  Compressibility is the amount by which a substance can be compressed. It is proportional to the change in density with pressure changes. Thus, an incompressible fluid has constant density. Compressibility effects are observed in supersonic flows and hence are inapplicable to the bird flight regime.
As mentioned earlier, an important goal of an aerodynamic model is to determine forces acting on a body moving through a fluid. The following sections will first examine various principles used to calculate lift on wings and then drag on complete airplanes.
The KuttaJoukowski theorem is fundamental in determining lift and drag forces on a two dimensional body. The theorem states that the lift force acting on any cylindrical body is equal to the product of density (of the fluid around the body), freestream velocity and circulation around the body. The crosssectional shape of the body does not affect the calculation and so this theorem can be applied to a twodimensional airfoil. The proof of this theorem is as follows:
In an ideal fluid, there are no viscous forces acting on the body, and consequently pressure forces can be separated into lift and drag. The forces acting on area dS (Figure 2‑1) can be given by:
_{} Eq 2‑3
Figure 2‑1 Lift and Drag forces acting on a two dimensional airfoil. Taken from (Panton 1984).
A complex representation of the forces is
Pressure on the surface can be determined using Bernoulli’s equation
_{} Eq 2‑5
where W (complex velocity) = _{} , F (complex potential) = _{}[3]
Using this representation for pressure in Eq 2‑4
_{} Eq 2‑6
The surface contour of the airfoil is a constant streamline thereby enabling _{} to be real and _{}. Integrating Eq 2‑6 the Blasius Theorem is obtained:
_{} can be calculated using the Residue Theorem that states _{}
where _{} are the residues of _{}.
The complex potential F for any streaming motion can be represented as a sum of potentials for uniform flow, source of strength m, vortex of circulation_{}, a doublet and higher order singularities.
_{} Eq 2‑8
Complex velocity W is obtained by differentiating F with respect to z.
_{} Eq 2‑9
The residue for W^{2} is obtained by taking the coefficient of the 1/z term, giving:
_{} Eq 2‑10
Substituting the value of _{}into Eq 2‑7 an expression for lift and drag forces acting on the airfoil are obtained.
_{} Eq 2‑11
Prandtl developed the Lifting Line Model where an entire wing was replaced by one single line called the Lifting Line. Lift forces were assumed to act on this line instead of on the whole wing.
The following points briefly outline Prandtl’s lifting line theory.
1. Representation of a finite wing  A finite wing in uniform flow is replaced by vortex that is in part a vortex sheet bound to the wing and a free vortex sheet as seen in Figure 2‑2.
Figure 2‑2 Bound and free vortex sheets on finite wing. Taken from (Karamcheti 1966).
For a wing of large aspect ratio, the bound vortex is further reduced to a single bound vortex line (of varying strength). This single bound vortex line is known as a Lifting Line (Figure 2‑3).
Figure 2‑3 Wing replaced by a Lifting line. Taken from (Karamcheti 1966).
2. Trailing Vortices  Circulation around any wing section is equal to the amount of vorticity flowing out of that section. As a consequence, change in circulation along the wing span results in shedding of vorticity from the wing. Thus, wherever the lift changes on the wing, a trailing vortex is generated, of strength equal to the circulation change about the wing at that point.
3. Vortex System – According to Helmholtz laws the bound vortex on the wing cannot end at the tips and thus a starting vortex and two trailing vortices exist to complete the circuit.
4. Simplification of vortex system  Velocity induced (_{}) by a given vortex is inversely proportional to distance from the vortex. Hence with time as the starting vortex recedes from the wing, induced velocity at the wing becomes negligible compared with velocities induced by the trailing vortices near the wing. The vortex loop can thus be represented by a horseshoe vortex. The trailing vortex system becomes a vortex sheet of zero total strength since it consists of superimposed horseshoe vortices, the trailing edges of which are vortex pairs of equal and opposite strength.
5. Downwash  At any point on the lifting line, the velocity is a resultant of _{}and induced velocity_{}. _{} has a spanwise component and a downwards directed component (downwash _{}) normal to the lifting line and _{} (Figure 2‑4) . The spanwise component of _{} is assumed to be small and hence neglected. However, this assumption is not valid near wing tips or for sections of low aspect ratio (Kuethe and Chow 1950; Karamcheti 1966).
Figure 2‑4 Resultant velocities acting at the lifting line. Taken from (Karamcheti 1966).
From the KuttaJoukowski theorem it is known that the force acting on unit span of a right cylinder (of any crosssection) is _{} and acts perpendicularly to _{}. Hence, to calculate lift force on the wing, circulation _{} needs to be first obtained. Circulation around an infinite wing is proportional to _{} and the angle of attack _{} (if small and measured from the zerolift direction, Figure 2‑5).
_{} Eq 2‑12
where K(y) is a constant dependent on form and size of the wing section (Karamcheti 1966). Figure 2‑5 illustrates the relation between the induced angle of attack and the zero lift angle_{}.
_{} Eq 2‑13
Figure 2‑5 Induced angle of attack. Taken from (Karamcheti 1966).
where _{} (as seen in Figure 2‑4)
If _{} then _{} giving
_{} Eq 2‑14
The BiotSavart law is used to get increment in downwash at _{} induced by element dx of vortex filament of strength_{}.
_{} Eq 2‑15
Figure 2‑6 Calculating downwash on lifting line. Induced by trailing vortex. Taken from (Kuethe and Chow 1950).
The entire vortex filament at y contributes
_{} Eq 2‑16
Total downwash at _{} is given by
_{} Eq 2‑17
Using values for total downwash and induced angle of attack, an expression for circulation is obtained. The underlying assumption is that as _{} hence_{}.
_{} Eq 2‑18
This equation for circulation is subject to the constraint that
_{}
Using the KuttaJoukowski theorem, spanwise distribution of lift forces can be obtained as follows:
Figure 2‑7 Lift and drag forces on wing section. Taken from (Karamcheti 1966).
_{} Eq 2‑19
where _{} is the lift for a slice dy of the wing. Lift distribution across the entire wing span is given by
_{} Eq 2‑20
Thus to calculate lift, the circulation distribution is required. First computing for an elliptical lift distribution (which is a simple case), the circulation can be given by
_{} Eq 2‑21
where _{} is a constant. Using this circulation distribution, total lift across the wing span can be calculated analytically and is given by the following expression:
_{} Eq 2‑22
For the lift distribution across any wing, a Fourier sine series can be used to represent the circulation distribution.
_{} Eq 2‑23
A_{n}’s, in this equation must satisfy Prandtl’s lifting line theory (Eq 2‑18). Thus, N equations can be obtained with unknown A_{n}’s, corresponding to N spanwise locations on the wing. These equations can be solved numerically to obtain the lift distribution across the wing. An example is shown in Figure 2‑8, which presents lift force and lift coefficients[4] for different planforms. Note that the lift distribution for the elliptical planform is also elliptical.
Figure 2‑8 Lift force and lift coefficient distribution across an elliptical, rectangular and triangular wing planforms. Taken from (Hoerner and Borst 1975).
Some variations of the elliptical planform (with no twist) can be seen in Figure 2‑9.
Figure 2‑9 Planforms with elliptical chord distribution resulting in elliptical lift distribution. Taken from (Mises 1945).
One of the aims of a well designed airfoil is to have large lift values. Lift generated by a flat plate can be increased by introducing camber. In Figure 2‑10, it is observed that the airfoil with positive camber has a higher maximum lift coefficient than the uncambered airfoil. For a symmetric or uncambered airfoil, lift is zero when the angle of attack is zero, while the cambered airfoil has a negative zerolift angle. For a two dimensional airfoil (infinite wing), the theoretical lift curve slope is_{}.
Figure 2‑10 Coefficient of Lift versus angle of attack until stall. Taken from (Raymer 1999).
In Figure 2‑11, the experimental lift curve slope matches the theoretical lift curve slope closely till stall (Abbot and Doenhoff 1959). Note that lift coefficient depends on Reynolds number. The airfoil at the higher Reynolds number (8.9 x 10^{6}) has a higher maximum lift coefficient (_{}) than the airfoil at Reynolds number = 3.1 x 10^{6}. This is because flow at higher Reynolds number is more resistant to strong adverse pressure gradients. This phenomenon will be discussed in greater detail in the following sections.
Figure 2‑11 Experimental lift curve slope compared with theoretical. Taken from (Abbot and Doenhoff 1959).
The Prandtl lifting line theory successfully models flow behavior around the wing and predicts forces acting on it. Trailing vortices are a byproduct of replacing an infinite wing with a finite wing.
Lift distribution can be obtained for simple shapes where analytical expressions describing circulation distribution exist. In cases where the wing planform shape is complex, numerical methods based on the standard aerodynamic model are equally accurate.
The preceding section studied calculation of lift forces on a wing. Another force acting on a body moving in a fluid is drag. Certain types of drag forces do not exist in an inviscid fluid. Thus viscosity and its accompanying effects play a key role in determining drag on a body.
The next sections will first review the Reynolds number  a measure of the balance of viscous and inertial forces acting on a body and then review a characteristic feature of viscous fluids – boundary layers.
Body characteristics like shape, size and orientation influence relative motion between the body and the surrounding fluid medium (Kuethe and Chow 1950). To incorporate body and fluid properties into one relation, force experienced by the body can be defined as a function of:
where F = force experienced by the body and l = characteristic dimension of the body. Eq 2‑24 can be restated as
_{} Eq 2‑25
where the dimensions of the product _{} are the same as that of force i.e. _{}.
Equation the exponents of the dimensions of force with those of _{} and expressing them in terms of a, a combination of terms _{}and _{}is obtained (Mises 1945).
_{} Eq 2‑26
Force can then be determined from the following expression
_{} Eq 2‑27
where_{} = force coefficient, _{} has been replaced by A = area and _{}is the dynamic pressure.
Using _{}, the force acting on a body (of certain area) moving in a fluid at a particular speed can be calculated.
is known as the Reynolds number (Re No) and is the ratio of inertial forces to viscous forces. If the Re No of a fluid flow is small, viscous forces are larger than inertial forces in the flow. Re No range from 10 for the smallest flying insects to greater than 10^{7} for large, high speed aircraft. Reynolds numbers for birds usually range from 10^{3} for passerines to 10^{5} for larger birds such as Frigatebirds. If the Re No is infinite, the flow becomes an ideal fluid flow (with zero viscosity). Flows about bodies that are geometrically similar (and similar in roughness) are completely same if their Reynolds numbers are the same. This similitude law forms the basis of wind tunnel testing. Forces acting on an aircraft can be obtained by testing a small scale model of that aircraft in a wind tunnel. A suitable combination of density and velocity can be used to obtain the same Re No for the model as for the full size aircraft. This enables the flow characteristics and force coefficients for the model to be the same as for the actual aircraft.
Thus, the Re No is an important parameter used to distinguish the bird flight regime from that of general aircraft.
Viscosity is a physical property that affects stresses in a fluid present due to fluid motion. For instance, when a viscous fluid flows past a body, it adheres to the body surface and frictional forces retard a thin layer of fluid adjacent to the surface. As seen in Figure 2‑12, velocity is a function of distance from the surface and at a certain distance, it is equal to the freestream velocity. The distance required by the fluid to reach 99% of the freestream velocity, is known as the boundary layer.
Figure 2‑12 Boundary layer thickness _{} and velocity profile i.e variation in velocity with y. Redrawn from (Schlichting 1958).
As seen from Figure 2‑12, the thickness of the boundary layer depends on the distance from the leading edge of the surface. The thickness is also directly proportional to the viscosity.
Model for complete flow at high Reynolds number
The shearing stresses in the boundary layer are large, even in low viscous fluids. This is because the change in velocity is much greater than the thickness of the boundary layer. Fluid velocity at the surface is zero, while at the edge of the boundary layer it is almost equal to the freestream velocity. However, outside the boundary layer, velocities gradients are very small.
In case of high Re No flows, due to the thinness of the boundary layer, a model can be used that views the flow as two different flows. One thin boundary layer confines the viscous effects while the other outer layer can be effectively viewed as inviscid.
Total drag, also known as profile drag, is the sum of frictional and pressure drag. The total drag coefficient is given by
Where _{} is the minimum profile drag occurring at zero lift and _{} is the induced drag coefficient.
Pressure drag is the integral of all the normal forces acting on a body (Schlichting 1958). It can be categorized into boundary layer drag and induced drag. Boundary layer drag is a direct consequence of viscosity whereas induced drag is a an ideal fluid effect and is independent of viscosity (Mises 1945).
In an inviscid fluid, pressures acting on the front of a body would be the same as those acting on the rear and would be equal to the freestream pressure p_{0}. Thus, there would be no boundary layer drag. In viscous fluids, a boundary layer exists and its thickness depends on the viscosity of the fluid. The thicker the boundary layer, the more modified and enlarged is the “effective body” shape as seen by the fluid. This results in an increase in pressure drag. If there is separation of the boundary layer, a turbulent wake is created which lowers pressures acting on the rear of the body. There is thus a pressure defect which results in a further increase in pressure or boundary layer drag.
To understand boundary layer separation, a rollercoaster analogy can be used (Hoerner 1965). As seen in Figure 2‑13, the car starts rolling down the hill from point A, loses its potential energy and gains kinetic energy. It then starts climbing the hill towards C and if there were no energy losses, it would reach the same height as point A. However, as energy losses are always present, each successive hill would have to be lower than the previous one to keep the car rolling and ascending.
Figure 2‑13 Boundary layer comparison with rollercoastercar. Taken from (Hoerner 1965).
Airfoils usually have high velocity, lowpressure regions on the upper surface. At the trailing edge, this highspeed, lowpressure flow has to return to the freestream conditions (at a higher pressure). Thus, there is an adverse pressure gradient acting on the fluid particles. Particles closest to the body continually lose kinetic energy due to viscous losses. They come to a halt when they lose all their kinetic energy and lack sufficient energy to overcome the adverse pressure gradient (at s_{2} in Figure 2‑14). At s_{3}, the fluid element starts moving backwards under the effect of the adverse pressure gradient.
Figure 2‑14 Reversed flow in presence of an adverse pressure gradient. Redrawn from (Anderson 1985).
This results in the flow separating from the surface creating a large wake of recirculating flow downstream of the surface. The turbulent boundary layer might reattach with the airfoil surface again forming a separation bubble as seen in Figure 2‑15.
Figure 2‑15 Transition, boundary layer separation and reattachment on an airfoil. Taken from (Thwaites 1960).
The primary flow no longer conforms to the shape of the surface, but is affected by the large separation region. The outer, undisturbed flow sees an enlarged “effective airfoil” and pressure distributions are modified in comparison with those in inviscid flow (Figure 2‑16).
Figure 2‑16 Inviscid approximation of pressure distribution around airfoil in separated flow at increasing angles of attack compared with experimental observation. Taken from (Thwaites 1960).
Due to turbulent motion in the wake, pressures in the rear of a body will be much lower than freestream pressure. For instance in Figure 2‑17, two sets of experimental pressure values at the cylinder rear have been compared with the ideal, inviscid flow values. The coefficient of pressure at Re No = 2 – 10 x 10^{4} is roughly 60% lower than the inviscid values, while that at Re No = 2 – 10 x 10^{5} is roughly 33% lower than that for ideal flow. This pressure defect contributes to the increase in pressure drag resulting in the drag coefficient at Re No = 2 – 10 x 10^{4} being almost 70% higher than that at Re No = 2 – 10 x 10^{5}.
Figure 2‑17 Different experimental pressure distribution at the cylinder rear at Re = 210 x 10^{4} and 210 x 10^{5} compared with ideal, inviscid flow. Taken from (Hoerner 1965).
In Figure 2‑18 flow patterns around cylinders in fluids of decreasing viscosity can be seen. In the Reynolds Number range 10^{4}10^{5} periodic shedding of vortices occurs until a constant flow pattern is reached at above Re No = 4x10^{5} (critical Re No for the cylinder). For inviscid flows, the flow pattern is symmetric about both axes and hence there is zero pressure drag.
Figure 2‑18 Variation in flow pattern and drag coefficients for cylinders with increase in Reynolds number. Taken from (Hoerner 1965).
Effect of body shape on pressure drag
Different body shapes have varying pressure gradients and different boundary layer transition and separation points. Boundary layer, wake thickness and consequently pressure drag thus depend on the shape of the body. In Figure 2‑19 this dependence of pressure drag on body shape can be seen where both bodies have approximately the same drag at high Reynolds number.
Figure 2‑19 Drag of an airfoil of chord c is relatively the same as the drag of a cylinder of diameter 0.005c. Taken from (Thwaites 1960).
Effect of body shape can also be observed in Figure 2‑20 when the drag coefficient reduces by roughly 80% on streamlining the cylinder (Hoerner 1965).
Figure 2‑20 Variation in flow pattern and drag coefficients for two dimensional bodies of different shapes at Re No greater than 4 x 10^{5}. Taken from (Hoerner 1965).
The second component of pressure drag is induced drag. This is the drag force acting on a finite wing seen in Prandtl’s Lifting Line theory (Figure 2‑7). The drag acting on a wing section dy can be given by
_{} Eq 2‑29
Zero drag exists when there is no downwash along the wing span (occurring only on infinite wings). This drag that exists due to the induced velocity field is called induced drag. Total induced drag _{} is therefore a function of lift and is expressed by
_{} Eq 2‑30
Induced drag can also be viewed as the work done in creating the trailing vortices behind a finite wing (Karamcheti 1966).
As induced drag is a function of lift, drag forces can be calculated either analytically or numerically depending on the circulation distribution. In case of the elliptical planform, circulation distribution is also elliptical and is given by Eq 2‑21. Total induced drag across the wing is constant (as seen in Figure 2‑21) and is given by:
_{} Eq 2‑31
Figure 2‑21 Induced angle of attack or downwash distribution across an elliptical, rectangular and triangular wing planforms. Taken from (Hoerner and Borst 1975).
The induced drag coefficient for the elliptical planform can be expressed as a function of aspect ratio and lift
_{} Eq 2‑32
Note that induced drag is minimum for the elliptical circulation distribution.
For an arbitrary wing shape, using numerical values for circulation, the coefficient of induced drag is given by
Eq 2‑33
where e ≤ 1 and is the airfoil efficiency factor. The maximum value of e occurs when spanwise circulation distribution is an elliptic function.
The previous sections reviewed one part of total drag – pressure drag. The second component of total drag is friction drag, which is the integral of all shearing stresses taken over the entire body surface. It is due to the contact of body surface particles with fluid particles as the body moves through the fluid. This friction generates tangential, viscous shearing stresses over the body surface exposed to the fluid flow (wetted area) and consequently skin friction drag does not exist in inviscid flows. This drag depends on the properties of the body surface and viscosity of the fluid through which it moves.
Friction drag in laminar and turbulent flows
For a flat plate moving through a fluid in its own plane, there is negligible pressure drag and mostly skin friction drag acting on the surface. As seen in Figure 2‑22, drag coefficient for laminar flow over the plate is lower than for turbulent flow. This is because in turbulent flow, the agitating molecular motion causes more molecules to come into contact with the surface in comparison with laminar flow.
The flat plate skin friction coefficient _{} is defined as
_{} Eq 2‑34
Knowing the velocity profile for laminar flow past a plate, Blasius obtained an exact solution for _{}(White 1999).
_{} Eq 2‑35
For turbulent flow, using Karman’s momentum relation[5] an approximation for the flat plate skin friction coefficient is obtained (White 1999).
_{} Eq 2‑36
Figure 2‑22 Drag coefficient for smooth and rough plates with laminar and turbulent boundary layers. Taken from (White 1999).
As mentioned earlier, skin friction drag depends on Re No of the fluid in which a body is moving and on the properties of the body surface. Surface roughness can be represented in terms of grain size k which is the diameter of a surface grain. As seen in Figure 2‑22, drag coefficient increases with increase in _{} (where _{} is the roughness height and L is the length of the plate).
As seen earlier, the total drag coefficient of an airplane is a sum of pressure and skin friction drag. The induced drag coefficient is a function of lift and thus Eq 2‑28 can be restated as:
_{} Eq 2‑37
where _{}
Drag force can then be expressed in terms of velocity as follows:
_{} Eq 2‑38
where A and B are constants depending on the geometry and weight of the airplane.
The variation of induced drag zero lift drag with increase in velocity can be seen in Figure 2‑23. A clearly defined minimum drag can also be observed. This minimum drag occurs when the induced drag is equal to_{}.
Figure 2‑23 Variation in induced drag, skin friction drag and total drag with increase in speed. Taken from (Houghton and Brock 1960).
The standard aerodynamic model assuming inviscid, incompressible, steady and irrotational flow has been very successful in determining aerodynamic characteristics of the complete airplane. For instance, in Figure 2‑24, theoretical and experimental lift forces acting on different parts of the Boeing 737 wing match closely (Kuethe and Chow 1950).
Figure 2‑24 Comparison of experimental and numerical results (based on standard aerodynamic theory) for the Boeing 737 wing body model. Taken from (Kuethe and Chow 1950).
Using various theoretical, experimental or numerical methods, it is now possible to determine forces acting on the entire airplane. For example, in Figure 2‑25, variation in experimental zerolift drag coefficients for complete aircraft is seen with increase in Mach number (Raymer 1999).
Figure 2‑25 Increase of _{} with increase in Mach number for different aircraft. Taken from (Raymer 1999).
Thus standard aerodynamic theory has proven adequate in modeling flows around general aircraft.
As seen in the previous chapter, aerodynamic theory has been very successful in characterizing flow around general aircraft. However, the bird flight regime greatly differs from that of general aviation in Reynolds number, geometry and unsteady effects. This discrepancy affects the standard model assumptions of inviscid and steady state flow.
The question then arises as to what extent the standard aerodynamic model is suited to bird flight. The following sections will discuss the above mentioned factors with this question in view.
Flows with Reynolds numbers less than 30,000 are characterized by laminar boundary layers (Lissaman 1983). Particles in such a boundary layer move along lines parallel to each other with locally constant velocities (Hoerner 1965). If the pressure gradient is not too steep, as in cases with low angles of attack and low lift, the boundary layer can transit from the low pressure region above the airfoil to freestream conditions without separating. When the angle of attack or lift is increased, the decelerations are severe. The laminar boundary layer does not have sufficient kinetic energy, and separates, thereby increasing pressure drag.
If a laminar boundary layer is energized, it will transit into a turbulent boundary layer. The particles in a turbulent boundary layer are in agitated motion and a continuous mass and momentum exchange takes place between each sheet of the boundary layer and its neighbours. Thus there is a continuous transport of mass and momentum from the outer flow towards the body surface. Consequently a turbulent boundary layer has more energy than a laminar one and is more resistant to adverse pressure gradients.
An example of this phenomenon can be seen in Figure 3‑1, where the experimental drag coefficient for a sphere decreases with increase in Reynolds number. At a particular Re No (4x10^{5}), laminar separation terminates and the boundary layer becomes turbulent. The sheets next to the surface receive an increase in momentum and are subsequently better equipped to negotiate the adverse pressure gradient. The separated flow reattaches and there is a decrease in pressure drag from 0.47 to 0.1 (Hoerner 1965).
Figure 3‑1 Variation of experimental drag coefficient for a sphere with Reynolds number. Taken from (Hoerner 1965).
The transition from a laminar to turbulent boundary layer can be triggered by energy supplying methods such as increasing the Re No or surface roughness. For instance, in Figure 3‑2 the rough airfoils have a higher liftdrag ratio than the smooth ones in the Re No range (10^{4} 10^{5}). Also, the smooth airfoils experience an impressive increase in liftdrag ratio by more than an order of magnitude (Figure 3‑2), while that of the rough airfoils steadily improves. This indicates the existence of a turbulent boundary layer on the rough airfoil while the smooth airfoil has a laminar boundary layer. Increasing the Re No from 10^{4} – 10^{5}, causes the laminar boundary layer on the smooth airfoil to transit to a turbulent one, thereby delaying separation and decreasing drag.
Figure 3‑2 Low Re No performance of smooth and rough airfoils. Taken from (McMasters and Henderson 1980).
Once the flow separates, there could be three possibilities for the separated boundary layer. It could either separate at some point on the airfoil or extend over the entire airfoil or reattach at some point on the airfoil forming a separation bubble.
Despite a vast body of experimental and theoretical studies done in the lower Re No regime there is no method that accurately predicts boundary layer transition or formation of separation bubbles. A rough rule presented by Carmichael (Carmichael 1981) is that the distance from separation to reattachment (bubble length) can be described by a chord Re No of approximately 50,000. If the chord Re No for an airfoil is about 50,000, it would be too short for flow reattachment. This can be generalized to another rough rule that if the Re No is less than 70,000, then reattachment might not happen (Lissaman 1983).
The size of the separation bubble also affects airfoil performance. In Re No flows of about 10^{5}, the bubble covers 2030 % of the airfoil (Lissaman 1983). This greatly affects pressure distributions, as the outer flow sees a much thicker airfoil profile. In flows of higher Reynolds numbers, the separation bubble is shorter, covering a distance that is a few percent of the airfoil chord. Thus, the pressure distributions are not altered as significantly as in the case of lower Re No flows.
Airfoils can behave in unpredictable and surprising ways at low Reynolds number. For instance in Figure 3‑3, at 4 x 10^{4} a flat plate outperforms a smooth airfoil, while a curved plate performs better than the flat plate. However, at 1.2 x 10^{5} the smooth airfoil outperforms the cambered thin plate and the flat plate. This is in contrast to behaviour at high Reynolds numbers where a cambered airfoil usually performs better than the uncambered one (Figure 2‑10).
Figure 3‑3 Comparison of liftdrag polars for smooth airfoils and flat plates at Re = 4 x 10^{4} and Re = 1.2 x 10^{5}. Taken from (Jones 1990).
Another example of interesting airfoil behaviour at low Re No can be seen in the hysteresis loops in Figure 3‑4. If the angle of attack is increased, the lift increases until it reaches the first peak. Separation then occurs and the lift value drops. However, if the angle of attack is decreased, lift values are different from the former ones. The lower the Reynolds number, the larger is the range of available angles, for which lift has two (or more) possible values.
Figure 3‑4 Lift versus angle of attack for Re No = 10^{5}. Taken from (Jones 1990).
The Re No for a bird with chord = 0.1 m, flying at 10 m/s is 7 x 10^{4}, neatly falling within the regime of complex flow behaviour (section 3.2). In case of birds, sufficient, quantified, experimental data confirming flow characteristics over the wing is absent. It is quite possible that flow over a bird’s wing is usually turbulent due to environmental conditions or surface irregularities (Spedding 1992). There is therefore a need to ascertain and quantify the following issues: thickness of the boundary layer, existence of laminar or turbulent flow, transition point (if any) from laminar to turbulent flow, presence of separation and location of separation point, presence, position, size and behaviour of separation bubbles.
The standard model assumes that wings are rigid, fixed in span, shape, twist and camber. A bird however, may vary its wingspan, wing area (sweep), wing angle of attack, twist, camber, wing tip shape, rotational velocity of wings, flapping plane, tail spread or tail angle of attack during the course of one wing beat cycle. With variation in flight speed, birds can also optimize their morphology to suit speed by varying the abovementioned parameters. For example, while gliding at high speeds, birds have been observed to flex their elbow and wrist joints to reduce their wingspan and wing area, thereby reducing drag. In some species, such as the albatross, low speed performance is optimized by varying the camber. This is done by flexing the patagial tendon seen in Figure 3‑6. At low speeds, the wing is fully extended and the tendon is tightly stretched. This pulls the secondary feathers downwards thereby increasing camber. In some albatross the patagial tendon is thickened and it then serves as a gliding lock (Pennycuick 1982).
Figure 3‑6
Figure 3‑6 Mechanism of extending the wing. Taken from (Burton 1990).
Besides inflight variations in wing geometry, there is also variance in structural features, such as wing twist. In Figure 3‑7, twist in a pigeon’s wing changes from 0° at the root to 13° at the tip(Nachtigall and Wieser 1966).
Figure 3‑7 Variation in camber and twist in a pigeon’s wing (Nachtigall and Wieser 1966).
In his study of wing shapes of 48 procellariiform species, Warham (Warham 1977) found that with increase in mass, the ratio of the humerus:ulna:(manus + primaries) generally ranged from 1:0.9:4.5 in the Sooty Black Petrel, Hydrobates pelagicus (mass = 28 g) to 1:1:1.4 in the Snowy Albatross, Diomedea exulans chionoptera (mass = 8677 g). Thus, the distal segment forms a major part in the wing of smaller birds. Warham suggests that in the case of the petrel, the large flexible distal wing section contributes to its great maneuverability. In the case of the albatross, the inner wing segments contribute more to the wing area, forming a stiffer, thicker, wing suited for soaring. The propatagium (narrow fold of skin stretched between the shoulder and the manus) comes out to almost 55% of the semispan and covers a chord area of 30% in the elbow joint region (Pennycuick 1982).
Feathers also contribute greatly towards controlling and smoothing the flow around wings. In a study of feather transmissivity it was found that transmissivity from dorsal (upper) to ventral (lower) is 10% higher than in the reverse direction (Muller and Patone 1998). Secondly, marginal feathers (Figure 3‑8) were more transmissive than the primaries or secondaries. This may be due to their positioning, which is at the thickest part of the wing where low energy flow is prone to separate. By virtue of being transmissive, the marginal feathers might make the boundary layer more turbulent, thereby preventing separation.
Figure 3‑8 Different kinds of feathers covering a wing (Burton 1990).
In the same study of feathers, it was further found that the outer vane of a feather is more transmissive than the inner vane. The outer vane overlaps the inner vane of the previous feather as seen in Figure 3‑9 and as it is more transmissive, it allows pressure on its dorsal side to be conducted to its ventral side. The volume of trapped air in the zone of overlap is relatively small and thus even a small flow is sufficient to influence it. A pressure gradient is developed from the less transmissive inner vane towards the outer vane. This results in the feathers being smoothly pressed together.
Figure 3‑9 Zone of overlap between inner vane and outer vane of two feathers (Muller and Patone 1998).
Changing morphology such as variation in wing area results in varying circulation and consequently, lift values. Variations in wingspan, tail spread or feet positioning could vary skin friction drag, form drag and induced drag values. For instance, some birds spread their tails and flare their feet while landing to increase wetted area and skin friction drag. Other birds, such as storks raise the alula while landing[6].
Thus, it can be seen that bird wing geometry is extremely complex and variable. Many studies have been conducted investigating different aspects of varying wing geometry in birds, but many more unexplored areas exist. Chapter 4 deals with relevant experiments, their conclusions and possible areas of investigation.
Flapping motion results in the cyclic varying of forces and produces unsteadiness over the bird’s wing. Flow field variables such as speed, direction and pressure are constantly changing with respect to time. Thus, boundary and initial conditions have three dimensional factors – velocity U, length scale L and a frequency f that may vary independently, in addition to inertial, pressure and viscous forces. The ‘reduced frequency’ parameter can be used to characterize unsteady aerodynamic effects during bird flight (Walker 1925). The reduced frequency is given by the following expression
_{} Eq 3‑1
where f = frequency of oscillation of the wing, c = reference chord length, U = velocity of the wing in the forward direction. k is thus the ratio of the motion of the wing due to oscillation to the motion of the wing due to the translation of the body (Spedding 1992).
The reduced frequency can be used to determine the balance between unsteady, periodic motions and the average forward motion. k can range from zero (in gliding flight) to infinity (in hovering flight, takeoff and landing). Generally animal flight takes place at k values ranging between 1 and 4 (Spedding and Lissaman 1998). A typical reduced frequency value for birds is 0.6, when forward velocity is 10 m/s, flapping frequency is 10 Hz and the wing chord is 10 cm in length (Spedding, Hedenstrom et al. 2003). If the reduced frequency is small, that implies forward flight velocity is greater than flapping velocity of the wings. In this case, unsteady effects are small and the quasisteady assumption[7] can be used. However as k increases from 0.1 to 0.4, timeaveraged forces, moments and torque calculations are underpredicted by more than 10% (Spedding 1992). If the reduced frequency is large, then forward velocity is low and the wing is affected by unsteadiness caused by the flapping motion. In this case, the unsteady effects might be too large to be neglected.
Values of ‘k’ give a rough estimate of unsteady effects. However, magnitude of unsteadiness caused by each of the three phenomena  flapping, changing wing morphology and viscous effects need to be determined and quantified. Three categories need to be determined – namely cases when unsteady effects can be neglected, when quasisteady aerodynamics is appropriate and cases when unsteadiness plays a significant role in shaping flow over birds’ wings.
Biologists have conducted many experimental and theoretical investigations to determine effect of wing geometry on a bird’s aerodynamic performance. In this chapter, these studies have been categorized into four classes:
· Salient, structural features such as aspect ratio and their effect on flying performance
· Wing geometry variation in flight
· Lift distribution on birds’ wings
· Experiments with dead birds
Pennycuick studied the effect of aspect ratio and wing area on flight performance in seabirds (Pennycuick 1987). Flight characteristics of a Magnificent Frigatebird, Brown Pelican, Razorbill and Blueeyed Shag to the Whitechinned Petrel[8] were compared. From Figure 4‑1 it can be seen that five configurations are obtained on increasing or decreasing aspect ratio and wing area in comparison to the Petrel.
Figure 4‑1 Adjustments of aspect ratio and wing area leading to five different wing shapes. Taken from (Pennycuick 1987).
Effect of changes in Aspect ratio and wing area on flying ability can be seen in Table 1.
Adaptation 
Bird 
Advantage 
Disadvantage 
Shorter wings (same AR) 
1. Razorbill 
Dual medium (air, water) ability 
Loss in efficiency – twice as much power needed. 
Shorter wings (same AR) 
Macaroni Penguin 

Loss of flying ability 
Longer wings (same AR) 
2. Frigatebird 
Highly maneuverable, with soaring ability at low speeds. Also has flapping ability. 
Cannot take off from level surfaces 
Smaller AR (same area) 
3. Cormorant 
Can still slopesoar and occasionally soar in thermals. 
Higher cruising speed, continuous flapping flight. 
Larger area (same span) 
4. Pelican 
Can flap and glide and can take from level surfaces. 

Table 1 Comparison of flying ability between birds with varying aspect ratios and wing spans (Pennycuick 1987).
A systematic change in aspect ratio is observed with variation in mass, (Figure 4‑2 (Pennycuick 1982). The Wandering Albatross is the largest in this group, weighing 8.73 kg and an aspect ratio of 15, and the smallest is the Wilson’s Petrel weighing 0.038 kg with an aspect ratio of 8. The variation in mass is more clearly seen in Figure 4‑2. Aspect ratio varied with the 0.12 power of mass. The larger species proceed primarily by gliding whereas the three smallest species proceed primarily by flapgliding.
Figure 4‑2 Nine study species shown with constant wing span, depicting systematic change of aspect ratio with size. Taken from (Pennycuick 1982).
Figure 4‑3 Variation of Aspect ratio (and wing tip shape) with mass amongst Procellariiformes. Taken from (Pennycuick 1982).
The effect of aspect ratio on the flight performance of a bird is well understood. High aspect ratio wings enable birds such as albatrosses (AR = 15), terns and swifts to fly at low speeds and high lift coefficients. From classical aerodynamic theory it is known that long, narrow wings lower induced drag costs as seen from the relation. A combination of low wing loading values and high L/D ratios enables these birds to glide at low speeds and glide angles for sustained periods.
The lower aspect ratio of the Wilson’s Petrel (AR = 8) forces it to proceed primarily by flapping or flapgliding. Figure 4‑4 compares lift coefficients of different species calculated while they were flying over the sea. The Wandering albatross had a mean lift coefficient of 1.0 and the Wilson’s Petrel had a lift coefficient of 0.28. This corroborates the relation between low aspect ratio and high flying speeds and also demonstrates the relationship between lift coefficients and size (Pennycuick 1982).
Figure 4‑4 Lift coefficient values calculated from field observations of different Procellariiformes flying over the sea. Taken from (Pennycuick 1982).
As seen in Figure 4‑5, the frigate bird has the largest wing span (2.29m) and aspect ratio while the black vulture has the smallest (Pennycuick 1983). However, unlike in the Procellariiform study Figure 4‑4, the lift coefficients do not vary as drastically Table 2. This is in part due smaller variations in mass and also due to presence of emarginated wing tips on the Pelican and Black Vulture. This emargination compensates for the lower aspect ratio.
Figure 4‑5 Silhouettes from photographs of Fregata magnificens, Pelecanus occidentalis and Coragyps atratus (Pennycuick 1983).
The frigatebird has the lowest wing loading (36.5 N/m^{2}) and consequently has smallest circling radii. However, it cannot take off from horizontal surfaces like the pelican. The black vulture is most suited to land soaring and maneuvering through crowded environs.

Frigatebird 
Brown Pelican 
Black vulture 
Wing Span (m) 
2.29 
2.10 
1.38 
Aspect Ratio 
12.8 
9.8 
5.8 
Wing Loading (N/m^{2}) 
36.5 
57.8 
54.7 
C_{L} 
1.33 
1.45 
1.35 
Circling radius (m) 
12 
18 
17 
Table 2 Comparison of flight performance between three soaring species (Pennycuick 1983).
From these wing geometry studies, it is observed that mainly first order effects and characteristics were focused upon. Most results and conclusions are in accordance with standard aerodynamic theory. These studies thus serve as a yardstick for future experiments. However, the negative points of these studies are large measurement errors and lack of reproducible data. This was largely due to field experiments having many variable, uncontrollable factors such as wind speed, heading, attitude and location (of the bird).
Hankin (Hankin 1913) studied different land soaring birds as early as 1913. He observed that the sequence of birds taking off depended on their weight and the strength of thermals. He traced their flight by means of mirrors placed on the ground. Figure 4‑6 shows some of his observations regarding the buzzard. During slow soaring (below 8 m/s), tip slots are kept open (as induced drag is higher at low speeds and slots serve to decrease this drag). Camber is increased to increase lift. The wings are also swept forward, thereby moving the wing’s center of pressure ahead of the bird’s center of gravity. This results in a nose  up moment that counteracts the nosedown pitching moment caused by the wing’s high camber. As flying speeds increase, tip slots start closing and the wings are progressively swept back.
Figure 4‑6 Variation in wing sweep for a Buzzard (Otogyps calvus) with flight speed (Hankin 1913).
The profile drag of hawk’s wing was measured by placing a Pitot tube array behind the wing (Pennycuick, C.E et al. 1992). The wing area was divided into four sections Figure 4‑7. Profile drag was obtained by assuming that it equaled loss of momentum in the flow.
Figure 4‑7 Camber and planview of a Hawk’s wing (Pennycuick, C.E et al. 1992).
From Figure 4‑8 it can be seen that zone 2 has the highest deviation in measurements. This corresponds to the shoulder region where active control movements are centered.
Figure 4‑8 Variation in Profile drag coefficient across wing span of a Hawk’s wing (Pennycuick, C.E et al. 1992).
Another study of active flow control movements was conducted by Pennycuick (Pennycuick 1968) on a gliding pigeon. Figure 4‑9 shows progressive decrease in wingspan with increase in speed. Mean chord increased from 10.2 cm at low speeds (8.6 m/s) to 20.5 cm at 22 m/s. At the lowest speed and full wing (630 cm^{2}) and tail spread (100 cm^{2}), a maximum lift coefficient of 1.3 was calculated. At the highest speed of 22 m/s, wings were swept back and the lift coefficient was 0.25.
Figure 4‑9 Variable wing span of a pigeon during gliding flight at different speeds. Taken from (Pennycuick 1968)
The reduction in wingspan results in reduced profile drag, but increase in induced drag. However, the continued decrease in total drag measurements over a wide range of speeds (1018 m/s) suggests that reduction in profile drag is greater than increase in induced drag (Figure 4‑10). The solid line indicates induced drag values that would be obtained with constant span. Induced drag was calculated from the expression _{} that assumes an elliptical lift distribution. This assumption might hold for the low speed case, but could result in errors at high speeds, when the leading edge of the wing is almost parallel to the airflow. Thus, at high speeds, induced drag might be underestimated. This error would then travel to profile drag values as they are calculated by subtracting induced drag from total drag measurements. Thus at higher speeds profile drag is lower than expected whilst induced drag is higher.
Figure 4‑10 Induced and Profile drag measurements for speeds between 8 m/s and 22 m/s. Taken from (Pennycuick 1968).
A few instructional conclusions can be drawn from the variable wing geometry studies. Firstly, even small, almost indiscernible variations in geometry have strong implications in flight performance (for instance, the hawk wing experiment). Secondly, due to the combined effect of low Re No and complex geometry, experimental results might not correspond with classical aerodynamic predictions (as seen in the gliding pigeon experiment). Thirdly, many flow control movements that take place in birds have no parallels in general aviation. Thus, it becomes even harder to isolate features that cause certain effects.
In a gliding experiment with a kestrel, Spedding (Spedding 1987) calculated circulation in the two trailing vortices using photogrammetry (flow visualization particle image velocimetry) techniques.
Figure 4‑11 Two trailing wing tip vortices shed by a gliding kestrel. Taken from (Spedding 1987).
The wake was found to be similar to that measured behind an elliptically loaded airfoil of the same span. As a result, classical airfoil theory for an elliptically loaded wing was used to calculate parameters such as lift coefficients and efficiency factors. The airfoil efficiency factor for the kestrel was found to be 0.96 with a lift coefficient of 1.16 when calculated from the relation
_{} Eq 4‑1
The assumption here is that the kestrel produces enough lift to exactly balance its weight.
The lift coefficient was also calculated using the principle that an elliptically loaded wing and its wake vortices can be replaced by a horseshoe vortex that lengthens by dX in time dt. The circulation around the wing can be related to rate of change of momentum _{} using the following relation:
Eq 4‑2
where _{} is the spanwise spacing between tip vortices and b = wing semispan. Weight is assumed to be balanced by rate of change of momentum and the equation is solved for circulation. The lift coefficient is then calculated using the measured circulation and is equal to 1.25. The lift coefficient values obtained through two different sets of calculations are close enough to conclude that the lift distribution over a gliding kestrel is elliptical.
Oehme (Oehme 1971) conducted a theoretical study of wing twist and lift distribution for a starling and Collared turtle dove. An elliptical lift distribution was assumed and wing twist angles were calculated to match the lift distribution Figure 4‑12.
Figure 4‑12 Variation in wing twist across wing span of Starling and Collared turtle dove (Oehme 1971).
As seen in Table 3, the theoretical twist angles for the Starling are quite close to those actually measured, but the theoretical twist values are almost twice the measured values.
Profile No. 
Starling (actual) 
Starling (theoretical) 
Dove (actual) 
Dove (theoretical) 
3 
+6 
+4.5 
+3 
+6.5 
5 
+8.5 
+8.5 
+2 
+7.5 
Table 3 Comparison of measured and calculated wing twist angles for a Starling and Collared turtle dove (Oehme 1971).
Experimenting with dead birds is easier than either field studies or live bird experiments as it avoids the complications mentioned in Section 4.1.4. The drawback is that skin friction drag is usually higher than that of a live bird. This is because a live bird actively controls each feather[9] and thus the body and wing contours are smooth. This difference in friction drag can be clearly seen in the study of body drag of a dead Teal (Pennycuick, Klaassen et al. 1996). The coefficient of body drag was 0.4, whereas for a live teal in a wind tunnel it was measured as 0.08. In another experiment, the body of a snow goose was sprayed with hair spray and its body drag coefficient was observed to reduce by 15% (from 0.33 to 0.28) (Pennycuick, III et al. 1988).
Withers (Withers 1981) studied the aerodynamic properties of isolated fixed bird wings at Reynolds Numbers ranging from 1 x 104  5 x 104. The wings were wind tunnel tested and were observed to have high minimum drag coefficients (0.03 – 0.13), low maximum lift coefficients (0.81.2) Table 4. If unreasonable values are ignored ( >1), efficiency factors are low (0.20.8) in comparison with conventional airfoils (0.9 – 0.95).
Species 
C_{Dpro} 
C_{Lmax} 
C_{L} /C_{D} 
Efficiency factor 
Aspect Ratio 






Petrel 
0.007 
0.88 
4 
0.24 
4.1 
Nighthawk 
0.051 
1.15 
9.0 
0.97 
4.1 
Swift 
0.030 
0.8 
17 
4.08 
3.9 
Wood duck 
0.096 
0.90 
3.8 
0.64 
3.1 
Starling 
0.125 
1.00 
3.3 
0.46 
3.0 
Hawk 
0.074 
1.0 
3.8 
0.39 
3.0 
Wood cock 
0.082 
0.90 
3.5 
0.51 
1.9 
Quail 
0.055 
1.10 
6 
0.84 
1.8 
Table 4 Comparison of lift and drag coefficients measured on different wings (Withers 1981).
High drag coefficients were observed firstly due to high skin friction drag caused by ruffled wing feathers. Secondly, due to the low Reynolds number, the flow had a tendency to separate immediately after the thickest wing section and increasing pressure drag. Lastly, due to the absence of an endplate at the wing root, vorticity was shed from that region, thereby increasing induced drag.
Low maximum lift coefficients were observed at Reynolds numbers less than 10^{5}. This was primarily due to increased boundary layer thickness along the wing chord that altered the effective wing shape. However, the lift coefficient did not decrease drastically at high angles of attack. This could be due to the turbulent boundary layer separating at high angles of attack and then reattaching – forming a separation bubble. Withers suggested that this wing behaviour would prove useful to the bird while landing at high angles of attack.
As seen from the three reviewed experiments, lift and drag coefficients can be quite unrealistic (in comparison to measurements from field experiments). This is in part due to surface roughness but also due to the unnatural positions in which the bird wing is dried. As seen from the studies of variable wing geometry, live birds are always making adjustments to best exploit the air flow. Therefore, studies with dead birds can give a certain idea of what type of results to expect, but they may be unreliable in isolating particular flow features around the wing (such as position of separation bubbles, reattachment of flow and so on).
A number of studies of airfoil and wing performance at low Re No have been conducted to date. A few of these studies that investigate airfoils similar to those of birds in thinness and profile will be discussed here. Experiments comparing performance of various planforms and wings of different aspect ratio are also reviewed.
A significant challenge in measuring performance characteristics of airfoils or wings at low Re No is reproducibility and reliability of data. The second part of this chapter deals with various factors that affect accuracy of experimental data.
A group at the University of Illinois at UrbanaChampaign (UIUC) has tested 36 wind tunnel airfoil models which were designed primarily for model aircraft (Selig, Guglielmo et al. 1995). Out of these 36 airfoils, the GM15 most resembles a thin, cambered avian profile. The GM15 is a low speed, Modelplane airfoil that combines good climb, glide and endurance. It also has the highest lift coefficient of all the F1C airfoils. Figure 5‑2 shows lift and drag coefficients at Reynolds numbers ranging from 40,000 to 300,000. At around 10˚ angle of attack, the maximum lift coefficient is observed to be around 1.3 for Re No = 60,000.
Figure 5‑1 Lift and drag coefficients for GM15 at Reynolds numbers 40,000 – 300,000 (Selig, Guglielmo et al. 1995)
Another set of 37 airfoils comprising of airfoils used in R/C soaring competitions, powered R/C aircraft and small wing turbines was tested in the UIUC wind tunnel (Lyon, Broeren et al. 1997). The BW3 airfoil was primarily designed for small wind turbine systems. shows the lift and drag coefficients at Reynolds numbers = 60,000, 100,000, 200,000, 300,000 and 400,000. It can be seen that with decrease in Reynolds number, the range of lift coefficients for minimum drag progressively decreases. There is also an absence of large separation bubbles Figure 5‑2 at Reynolds numbers = 60,000 and 100,000 that were seen in thicker airfoils. Hysteresis[10] was not observed at any of the Reynolds numbers, further confirming the absence of separation bubbles. The lift coefficients appear to coincide for all Reynolds numbers except Re = 60,000, with a maximum lift coefficient of 1.4 at 12˚ angle of attack.
Figure 5‑2 Lift and drag coefficients for the BW3 airfoil (Lyon, Broeren et al. 1997)
Laitone (Laitone 1997) conducted some wind tunnel tests of rectangular wings of aspect ratio = 6 at Re No = 20,700. As seen in Figure 5‑3, the wing with 5% camber (1.3 % thick) had higher lift coefficients than either the thin wedge or the NACA 0012. The 5% camber wing also had a L/D ratio of 13.3 while the thin wedge had 8.15 and the NACA 0012 had 7.4. The 5% camber wing also had a lift curve slope of 0.098 that is 20% greater than that predicted by ideal potential flow theory for the same wing. This experiment implies that a rounded leading edge is not as crucial to an airfoil’s performance at low Reynolds numbers as it is at higher Reynolds numbers. This superior lift performance of sharp leading edges was seen again when the trailing edge of the NACA 0012 was placed into the flow Figure 5‑4. The sharp leading edge thus obtained caused an increase in lift coefficients from 0.041 to 0.064.
Figure 5‑3 Comparison of lift coefficients for 5% camber, thin wedge and NACA 0012. The dashed line indicates effect on lift performance of the NACA 0012 with increase of wind tunnel turbulence from 0.02% to 1.06%. Taken from (Laitone 1997).
Figure 5‑4 Effect on lift coefficients after placing the reversed NACA 0012 into the flow. Taken from (Laitone 1997).
On almost doubling the Re No (20,700 to 42,100), the initial lift curve slope for the 5% camber wing increased 1.4 times (from 0.098 to 0.138) Figure 5‑5. This value is about 26% higher than the lift curve slope _{} obtained from two dimensional potential flow theory. This is a further indication of classical aerodynamic theory failing to predict airfoil performance in the low Re No regime.
Figure 5‑5 Comparing variation in lift coefficients with increase in Re No from 20,700 to 42,100 for the NACA 0012 (Laitone 1997).
To further investigate the occurrence of lift coefficients greater than one for the 5% camber wing Figure 5‑3, Laitone conducted an experiment where aspect ratio for the wing was varied from 2.18 to 8 (Re No = 20,700). As seen from Figure 5‑6, wings of aspect ratio = 6 and 8 developed lift coefficients greater than one (after a stall at around 12.7˚). However, the wings of aspect ratio = 2.18 and 4 did not display the same behaviour. Laitone suggested that the high lift increase with angle of attack might be due to the curved lower surface of the airfoil behaving as a turbine blade.
Figure 5‑6 Lift coefficient variation with angle of attack for rectangular planform wings of 5% camber (1.3% thickness) at Re No = 20,700 (Laitone 2001).
In another set of experiments, the effect of planform and low aspect ratio on L/D was investigated (Laitone 2001). As seen in Figure 5‑7, best performance (L/D = 8) was obtained with the rectangular wing of aspect ratio 6. The elliptical wing had a L/D of only 6.6. In the low aspect ratio planforms, maximum L/D (5.4) was obtained with the square plate. The circular and delta planforms had similar L/D values (5.7 and 5.37) respectively. At high angles of attack highest lift coefficients were obtained by the square plate (C_{L} = 1.9 at_{}), but drag coefficient values were also the highest (C_{D} = 1.53). The circular disk had a lower lift coefficient (C_{L} = 1.44 at _{}), but also a lower drag coefficient (C_{D} = 1.04). The delta wing had a (C_{L} = 1.72 at _{}), and the lowest lower drag coefficient (C_{D} = 0.87).
Figure 5‑7 L/D variation with angle of attack for a circular disk, rectangular planform (AR = 6), elliptical planform (AR = 6), flat plate (AR =1) and a delta planform (AR = 1) (Laitone 2001).
As interest in micro air vehicles has burgeoned, experiments have focused on low aspect ratio planforms. Torres and Mueller (Torres and Mueller 2004) studied four different planforms with aspect ratio varying from 0.5 to 2.0 at Reynolds numbers ranging from 7 x 10^{4} to 2 x 10^{5 }(Figure 5‑8). It was observed that aspect ratio was the most defining characteristic of such low aspect ratio wings and planform the next.
Figure 5‑8 Various low aspect ratio wings with different planforms (Torres and Mueller 2004).
Low aspect ratio wings have large C_{Lmax }as seen in Figure 5‑9. A transition zone is also observed between aspect ratio 1.25 and 1.5. This may have been due to the large wing tip vortices energizing flow over the entire upper wing and thus delaying separation. With increase in aspect ratio, wing tip vortices decrease in strength (according to classical aerodynamic theory) and as a result, can not sufficiently energize the flow. Thus maximum lift coefficient values are lowered.
Figure 5‑9 Variation in C_{Lmax} with variation in aspect ratio and planform (Torres and Mueller 2004).
Spanwise variation in profile drag for a SD6060 airfoil was measured by a wake at Re No 200,000 to 500,000 (Guglielmo and Selig 1996). In Figure 5‑10, significant oscillations in drag are seen at angles of attack below 1.1 degrees. These oscillations vary in magnitude from 540%, but for_{}, profile drag increases from 0.006 to 0.016 within 2 inches of the span. At the lower angles of attack, drag might be high due to the presence of a laminar bubble on the undersurface of the airfoil. At higher angles of attack, the laminar separation bubble on the top surface of the airfoil might not develop uniformly, thus causing the asymmetric spanwise variation in drag. These phenomena clearly indicate the three dimensional nature of a flow that is expected to be twodimensional.
Figure 5‑10 Spanwise variations in profile drag for SD6060 at Re No 200,000. Taken from (Guglielmo and Selig 1996).
Increase in Re No also affects profile drag as seen in Figure 5‑11. Average profile drag coefficients at Re No = 200,000 are 0.010 and this reduces to 0.007 at Re No = 500,000.
Figure 5‑11 Variation in profile drag for SD6060 with increase in Re No from 200,000 to 500,00. Angles of attack are 0.03, 0.09, 0.06 and 0.05 deg (Guglielmo and Selig 1996).
Profile drag measurements were found to be dependent on distance of the wake rake from the trailing edge (Figure 5‑12). Large oscillations in drag values indicate that the wake has three dimensional structures and thus has a significant impact on profile drag.
Figure 5‑12 Spanwise profile drag measurements taken at three different downstream locations behind the SD6060. Re No = 200,000 and angles of attack are 0.02,0.05 and 0.03 deg (Guglielmo and Selig 1996).
One of the main challenges in measuring forces at low Re No is the great sensitivity of the flow to exterior disturbances and turbulence levels. Usually, every wind tunnel has its particular environmental disturbance levels and turbulence. This implies that results from different wind tunnels may differ widely as can be seen in Figure 5‑13. The results from NotreDame were almost 50% higher than those obtained from Delft and Stuttgart (Grundy, Keefe et al. 2001).
Figure 5‑13 C_{L} vs C_{D} values measured at four different locations for the Eppler 61 at Re No = 50,000 (Grundy, Keefe et al. 2001).
Turbulence levels at Delft, Stuttgart and NotreDame were ≤ 0.1. It has been reported by Mueller that lift and drag measurements of thin, flat and cambered airfoils are insensitive to turbulence variations if they are less than 1%. The hypothesis that acoustic disturbances affected flow behaviour at low Re No was proven through a series of experiments using environment disturbances and single frequency tonal noise. An E61 airfoil tested at Re No 25,000 (in turbulence levels of 0.1%) was observed to have a long separation bubble at positive incidence angles. As angle of incidence is increased, the bubble breaks into a shorter bubble (thereby increasing lift and decreasing drag). It was observed that if background noise levels were high (100300 Hz at 50dB), bubble breakdown was hastened as seen in Figure 5‑14.
Figure 5‑14 Variation in lift coefficient with angle of incidence and background noise levels. Taken from (Grundy, Keefe et al. 2001).
As seen in the spanwise profile drag experiment, placement of the wake rake affects profile drag measurements (Guglielmo and Selig 1996). For the E374 airfoil, the profile drag coefficient decreases in amplitude with increase in distance of the wake rake from the trailing edge (Figure 5‑15). This effect of the wake on measurements may be an additional reason for inconsistencies in data obtained from different facilities.
Figure 5‑15 Spanwise profile drag measurements taken at three different downstream locations behind the E374. Re No = 200,000 and angles of attack are 6.4, 6.4 and 6.3 deg (Guglielmo and Selig 1996).
At Reynolds numbers of 20,000, minimum drag values can be as low as 2 g (0.02 N) (Pelletier and T.J 1999). To accurately understand flow behavior at low Reynolds numbers, it is very important to be able to detect small variations in forces. Laitone (Laitone 1997) had a force balance with a sensitivity of ± 0.01 g and claimed that sensitivities of ± 1g were insufficient to obtain reliable data.
In contrast to biological studies, the low Re No airfoil studies have some degree of standardization due to the existence of specific airfoils. It is thus easier to obtain repeatable, reliable results. Furthermore, effect of low Re No is more clearly identified and isolated. However, complications in the shape of flow sensitivity still exist.
As seen earlier birds have many variable and complex wing features aimed at improving aerodynamic efficiency. It is therefore difficult to isolate the effect of a single feature on the bird’s aerodynamic performance. A simple example is the existence of twist on a birds’ wing such as that seen in Figure 4‑12. Twist enables a triangular wing to have an elliptical lift distribution. Thus, if twist were not taken into account, one might wrongly conclude that lift distribution over the wing was not optimum.
There are many studies that aim to prove general concepts or explain general phenomena. In providing such generality, they lose accuracy in explaining particular effects or features. For instance, the American turkey vulture (Cathartes aura) and the Great blue heron (Ardea herodias) have approximately same body mass, wing span and wing area. However, their necks, legs and body shapes are quite different, as seen in Figure 6‑1. Thus a model taking into account only body mass, wing span and wing area would predict similar flight performance for both (Pennycuick 1995). In actuality the turkey vulture is a land soaring bird and the great blue heron is a migratory bird.
Figure 6‑1 American Turkey Vulture and Great Blue Heron.
On the other hand, results from studies that deal with specifics such as a particular bird in a particular circumstance, may not be applicable even to the same bird if it is in a different situation.
In light of these points, a problem has been identified that allows for a certain degree of generality and specificity. The present research will focus on nature’s best endurance fliers – pelagic birds. These birds are aerodynamically efficient with high lift coefficients. The Wandering Albatross is a representative example with a lift coefficient of 1.15 (Pennycuick 1982). There are many reasons for choosing this particular category of birds as a research subject:
1. One complex feature of bird flight is unsteadiness due to flapping. Pelagic birds are primarily gliding birds and thus the unsteady flapping effect is eliminated in studies of these birds.
2. Oceanic birds exclusively have high aspect ratio wings (>8) that are not accompanied with complications such as alulas, wing slots or pronounced emargination. Thus effects of complex wing geometry are also minimized.
3. Pelagic birds fly in uncluttered environs and consequently do not need to compromise on aerodynamic efficiency to execute challenging takeoffs. Their wings develop to the fullest extent in size and are only constrained by structural limitations. For instance, the Wandering Albatross has a wing span of 3.03 m and an aspect ratio of 15 (Pennycuick 1982). This adaptive evolution is consistent with standard aerodynamic theory that predicts low induced drag for high aspect ratio wings.
4. A systematic change in wing planform can be seen in the Procellariiformes which allows parameterization of certain wing features. This contributes greatly towards a quantified and rigorous study.
The peculiar feature of the pointed wing tip in pelagic birds sets these birds apart from their man made counterparts – performance gliders. As seen in Chapter 2, a pointed wing tip will be deleterious to the wing’s aerodynamic performance. Due to the small surface area of the tip, local lift coefficients will be very high leading to tip stall (_{} and Figure 2‑10). This in turn will increase pressure drag (Section 2.5.1). Thus viewed in the light of standard aerodynamic theory, the pointed wing tip might be disadvantageous to the aerodynamic performance of oceanic birds (Lissaman 2004). On the other hand these birds are the most efficient fliers in nature (Section 4.1.2) and so an interesting conflict arises.
To determine the performance of pointed wings at mean chord R_{e} of about 60,000 through stall, and compare with elliptical and Kuchemann type planforms. To determine if this tip degrades performance or if there is some as yet undiscovered effect that makes the pointed shape desirable.
The focus of this research will be the aerodynamic characteristics of pointed wing tips, such as those found in Albatrosses and Petrels (Figure 1‑4). Defining characteristics of a standard representative of these birds are given in Table 5.
Species 
Mass (kg) 
Span b (m) 
Wing Area S (m^{2}) 
Wing Loading (Nm^{2}) 
Aspect Ratio (b^{2}/S) 
Gliding Speed (m/s) 
Reynolds Number 
Standard Sea Bird 
0.7 
1.09 
0.103 
66.6 
11.5 
1014 
65000 92000 
Table 5 Characteristics for standard sea bird as defined in (Pennycuick 1987).
Based on the aspect ratio and Re No of the standard sea bird, the half  wings would be 0.3 m in length and 0.05m in root chord length. The aim of this investigation is to isolate the effect of pointed wing tips by comparing their aerodynamic characteristics and behaviour with those of well studied planforms such as elliptical and crescent. Therefore, the different planforms (Figure 6‑2) being tested will be:
1) elliptical
2) Crescent shaped wing with straight trailing edge.
3) Pointed tip wing with the leading edge swept back by _{}and straight trailing edge.
Figure 6‑2 Three model planforms of Aspect Ratio =11.5.
A variant wing geometry for the pointed tip wing planform (with straight trailing edge) will be made by setting a 5 ˚ constant angle washout of the 15% tip region of the wing.
Time permitting, further investigations will be carried out determining the effect of increasing angles of backward sweep for the leading and trailing edges (Figure 6‑3). Thus, a family of wings will be obtained based on the varying parameter _{}. Such configurations are of interest as oceanic birds often vary sweep of the distal wing segment (Section 3.4) with changes in speed.
Figure 6‑3 Two Models for studying the effect of variation in backward sweep in leading and trailing edges.
A comparison study will be conducted between a flat plate with a pointed wing tip planform and one with an elliptical planform. The effect of camber will then be determined by studying all the planforms (Figure 6‑2) with the Eppler 387 airfoil profile (Figure 6‑4).
Figure 6‑4 Eppler 387. C_{mo} = 0.085 and _{}=3.1°
This particular airfoil was chosen as it was designed for thermal soaring sailplanes and is well studied (Figure 6‑5). Extensive low Re No. experiments have also been conducted in the Dryden wind tunnel using this airfoil (Figure 6‑6) (Spedding, Browand et al. 2005).
Figure 6‑5 Cl vs Cd for the Eppler 387 at Re No = 151600, 202300 and 303000. Angle of attack was varied from 5 to 10 deg (Selig, Guglielmo et al. 1995).
Figure 6‑6 Timeaveraged aerodynamic performance measurements in the Dryden wind tunnel of three wing shapes. Each data point is from at least 4 independent experiments on at least 2 days,. Error bars represent standard deviations over all experiments. Re No = 23000 (Spedding, Browand et al. 2005).
Measurements of lift and drag at 10 m/s and 20 m/s at angles of attack till stall will be made in the Dryden wind tunnel. Wind tunnel turbulence measurements showed that turbulence levels below 0.03% could be maintained in the Re No range 20,000 – 70,000. All measurements will be made using a strain gauge having ± 0.2 gm accuracy.
PIV is a measurement technique that uses a pulsed laser sheet to illuminate the flow field being studied. The position of the flow field particles is captured by cameras at every instant the laser light sheet is pulsed. Processing this PIV data yields particle displacement values between laser pulses. Particle velocity can be computed by using the time interval value between laser pulses.
This research will use PIV techniques to measure separated flow development and surface streamlines at 4 spanwise stations. PIV data will also provide wholefield velocity, spanwise vorticity measurements and instantaneous boundary layer characteristics along the entire wing. Some results from PIV experiments conducted in the Dryden wind tunnel are shown in Figure 6‑7 (Spedding, Browand et al. 2005).The flow over the flat plate upper surface separates, causing an irregular wake downstream. The Eppler wake is larger in size than the flat and cambered plate, which is indicative of the thicker boundary layer at the trailing edge.
Figure 6‑7 Spanwise vorticity at (L/D)_{max} for flat plate, cambered plate and Eppler 387 at Re No = 1.1 x 10^{4}. The disorderly flow behind the flat plate supports the L/D data in the previous figure(Spedding, Browand et al. 2005).
The potential based panel method PMARC_12 will be used to understand flow characteristics around pointed tip wings compared with elliptical and crescent shaped wings. Separation will be estimated using airfoil experimental 2D data. This will generate theoretical lift/drag polars as functions of angle of attack into the stalled range, predicting the stall development. Six wing cases, at speeds 7.5, 10, 15, 20 m/s will be analyzed.
Results from the Numerical study and experiment will be compared with regard to performance and separation development. If experiment indicates more favorable performance than the numerical study, then a more detailed analysis will be made to explain this. The results will show the performance difference between the wing planforms, and tests may indicate some favorable aerodynamic performance feature exhibited by pointed ornithic wings that is not predicted by numerical methods and standard theory. This will present an important avenue for continued research.
This study uses PMARC (Panel Method Ames Research Center), a firstorder, potential based panel method to understand aerodynamic behaviour of the pointed tip wing at Re No = 2 x 10^{6}. Comparisons are made with elliptical and crescent shaped wings.
The approach broadly consists of distributing the unknown quantity across the body’s boundary surface, instead of in the entire volume surrounding the body (as in finite difference methods). This results in faster computations. But, it applies only to inviscid flow or attached flow situations .
The external flow potential _{} at any point in the external flow is obtained by using Green’s theorem:
_{}_{ }= disturbance potential from surface distribution of doublets of strength _{} per unit area +
disturbance potential from surface distribution of sources of strength _{} per unit area
where _{} = internal flow potential (Katz and Plotkin 2001).
A Dirichlet boundary condition is applied on the boundary, satisfying the zero normal velocity condition at the surface (_{}). This results in solutions for the source strengths.
The wake is assumed to be thin, so that there is no entrainment and consequently there is no source contribution. Doublet values for the wake can be obtained as functions of doublet values on the body surface by applying the Kutta condition. The doublet strength on the first row of wake panels is set equal to the difference in doublet strengths of the upper and lower rows of surface panels that form the trailing edge. This results in a finite velocity at the trailing edge. Thus the only unknowns are the surface doublet strengths (Ashby, Dudley et al. 1988).
Once the doublet strengths are found, velocities and consequently pressure coefficients on the panels are determined. This leads to forces and moments for each panel that can be summed to get total forces acting on the entire body.
Induced drag is calculated by Trefftz plane analysis. The wing is surrounded by a large control volume such that the perturbation velocity components are zero everywhere except at the wake. In inviscid flow the wake is parallel to the freestream at a plane (known as the Trefftz plane) normal to the freestream velocity vector. The velocity components in the Trefftz plane exist only in the y and z directions. Induced drag is obtained by integrating pressure and momentum over the control volume.
_{}
where _{}is the local wake span (Katz and Plotkin 2001).
The three wings studied were elliptical (with straight quarter chord line), crescent (with straight trailing edge) and pointed (Figure 7‑1). Each wing was of aspect ratio = 7, untwisted with a NACA 0012 airfoil.
Figure 7‑1 Wing models studied using PMARC. The elliptical wing had N_{s} x N_{c} = 75 x 60, crescent wing had N_{s} x N_{c} = 110 x 110 and the pointed wing had N_{s} x N_{c} = 110 x 30.
The elliptical wing was cropped at the tip at a span fraction of 0.9983, while the crescent wing was cropped at a span fraction of 0.9999. The pointed wing had a tip chord that was 0.05% of the root chord. The pointed wing is made of a rectangular part that is 1/3 the span fraction.
Paneling on the elliptical wing was halfcosine spaced along the spanwise and chordwise direction. There was greater concentration of panels at the leading edge and tip. However, in case of the crescent and pointed wings, the paneling along the spanwise direction had to be equally spaced. This was because the area of spanwise columns progressively decreased, until it became almost negligible (in case of halfcosine spacing). The number of panels in all three cases was chosen based on computation time and variation in span efficiency with number of spanwise panels.
Figure 7‑2. Variation in span efficiency with number of spanwise panels for elliptical and crescent wings.
As seen in Figure 7‑2, span efficiency for the elliptical wing with 75 x 60 panels is within 0.12% of the value with 86 x 72 panels. The total lift coefficient value for the 75 x 60 panel case is within 0.2 % of the case with 86 x 72 panels. The 75 x 60 panel case took approximately 10 minutes to run, whereas the 86 x 72 panel case took approximately 20 minutes to run.
In case of the crescent wing, number of panels used was 110 x 110. This case took about 2 hours to run. However, this combination of panels was used for further computations, as the spacing was equal and spanwise column area at the tip was larger in comparison to those on the elliptical wing with halfcosine spacing.
In the pointed wing case, the upper limit for the number of panels was determined by the spanwise column area at the tip. The column area became negligible for a panel combination more than 110 x 30.
Based on results from a higher order panel method – PANAIR (Table 6), a straight wake was used instead of a rolled up wake.
Elliptical Wing 
CL 
E 
Straight wake 
0.30356 
0.9847 
Force free wake 
0.299965 
0.9691 
Crescent Wing 
CL 
E 
Straight wake 
0.30666 
0.9915 
Force free wake 
0.30353 
0.9897 
Table 6 Comparison of CL and span efficiency values for a straight wake and forcefree wake computed from Trefftz – plane integration (Smith and Kroo 1993).
The wake from the paneled geometry develops over a series of times steps and its shape reflects the trailing edge shape of the paneled geometry.
For the PMARC study, it was found that different placements of the Trefftz plane did not affect the lift coefficient and span efficiency values (Table 7). However, increasing the length of the wake by a factor of 10 caused the lift coefficient to increase by 2% and the coefficient of induced drag by 4.5%. The span efficiency remains unchanged.
Trefftz Plane Pos. (chord lengths) 
Wake length (chord lengths) 
CL 
CD 
CD_{i} 
E 






1 
10 
0.3537 
0.0063 
0.0044 
1.0083 
1 
50 
0.3612 
0.0062 
0.0046 
1.0083 
10 
50 
0.3612 
0.0062 
0.0046 
1.0083 
10 
100 
0.3623 
0.0062 
0.0046 
1.0083 






Table 7 PMARC results for elliptical wing. Force coefficients and span efficiencies are compared for different Trefftz plane placements and different wake lengths.
Based on variation in span efficiency with wake length, and computation time, a combination of wake length = 10 chord lengths and location of Trefftz plane behind the trailing edge were taken for further computations.
To validate the PMARC results, some comparisons were made with NACA 0012 experimental airfoil data and with results from the second order panel method  PANAIR (Smith and Kroo 1993).
In the elliptical wing case, two different aspect ratios were compared with experimental airfoil data. In Figure 7‑3, variation in lift coefficient with alpha can be seen for the different aspect ratios. The wing case of AR = 20 is closest to the airfoil data. As seen in Section 2.5.1, lift coefficients for the AR = 7 case is lower than that of the airfoil due to induced drag. The superior performance of the wing of AR = 20 can be seen again in Figure 7‑4 and Figure 7‑5 where it has a higher lift coefficient for the same drag coefficient and lower induced drag than the wings with lower aspect ratios.
Figure 7‑3 Section lift coefficient vs. angle of attack for NACA0012 AR = 7 and 20, compared with 2D experimental data (Abbot and Doenhoff 1959). Re No for 2D airfoil data was 9 x 10^6, for PMARC was 2 x 10^6.
Figure 7‑4 Total lift coefficient vs. total drag coefficients for elliptical wings of AR = 7,9,12 and 20.
Figure 7‑5 Variation in induced drag with angle of attack and aspect ratio.
Span efficiency for the elliptical wing of AR = 7 as computed by PMARC was then compared to results from the PANAIR program. Variation in span efficiency (from PMARC) at _{} = 2˚ and _{} = 16 ˚ is less than 1%. However it is 2% greater than the value computed by PANAIR.
Figure 7‑6. Variation in span efficiency with angle of attack. PMARC results for the elliptical wing are compared with those from PANAIR (Smith and Kroo 1993).
In Figure 7‑7, lift distribution for the three wing geometries is compared with the lift distribution predicted by PANAIR. The PANAIR results for the elliptical and crescent wing were very close to the true elliptical loading. However, the crescent wing produced less lift and more drag (Figure 7‑9) on the inboard region of the wing, in comparison to the elliptical wing.
Lift for the crescent wing from the PMARC program is about 10% higher than that produced by the elliptical wing at the root. However, unlike the PANAIR results, lift for the crescent wing remains higher than the elliptical wing across the span. The solid lines represent true elliptical loading and both elliptical and crescent wings are close to the ideal loading.
The pointed wing tip does not show an elliptical loading. There is a marked decrease in lift at 1/3 of the span fraction. This is due to the sharp joining of the rectangular and triangular parts of the wing. The lift coefficient at the wing root is 3.4 % lower than that for the elliptical wing.
Figure 7‑7 Lift distribution for crescent, elliptical and pointed wings compared with PANAIR results for elliptical wing (Smith and Kroo 1993).
Figure 7‑8 and Figure 7‑9 show drag distribution for the pointed, elliptical and crescent wings. PMARC results are compared with PANAIR results. According to PANAIR, the crescent wing has more drag in the inboard region of the wing than the elliptical wing. At the tip, the crescent wing has lesser drag than the elliptical wing.
The PMARC program predicts the elliptical wing has higher drag than the crescent wing across the entire span. At the root, the elliptical wing has 23 % higher drag than the crescent wing.
The pointed tip wing had the highest drag in the inboard portion of the wing. The peak was at the sharp joint between the rectangular and triangular parts of the wing. At the root, the drag was 34% higher than the elliptical wing. However after the joint section, the pointed tip wing had the lowest drag.
Figure 7‑8 Drag distribution for pointed, elliptical and crescent wings, compared with PANAIR elliptical and crescent wing results (Smith and Kroo 1993).
Figure 7‑9 Drag distribution for pointed, elliptical and crescent wings, compared with PANAIR elliptical and crescent wing results. The negative drag values have been omitted to aid distinguishing the drag distribution values for all cases (Smith and Kroo 1993).
As seen in Figure 7‑7 and Figure 7‑9, the sharp joint region where the rectangular part of the wing starts to taper, causes high drag. This could be improved by smoothening the leading edge as seen in Figure 7‑10.
The lift distribution is also smoother, without the sudden drop in lift at 1/3 span region Figure 7‑11. The drag distribution continues to exhibit a peak but the peak is halfway towards the tip.
Figure 7‑10 Coefficient of pressure distribution over a pointed tip wing with a curved leading edge. TNPC = No. of chordwise panels = 30 and TNPS = No. of spanwise panels = 110.
Figure 7‑11 Lift distribution for pointed tip wing with smooth leading edge.
Figure 7‑12 Drag distribution for pointed tip wing with smooth leading edge.
The following plots show the variation in lift and drag coefficients with angle of attack. In Figure 7‑13, the crescent wing has the highest lift (slope = 0.095) for each angle of attack in comparison to the elliptical (slope = 0.086) and pointed wings (slope = 0.082). The crescent wing also has the highest lift for a particular drag coefficient value (Figure 7‑14) and lowest induced drag (Figure 7‑15). The C_{L }vs. C_{D} curve for the elliptical wing is almost the same as the pointed tip wing, although the elliptical wing has higher induced drag than the pointed tip wing.
Figure 7‑13 Variation in section lift coefficient with angle of attack for pointed, elliptical and crescent shaped wings, compared with experimental NACA0012 data (Abbot and Doenhoff 1959). Slope of the airfoil data = 0.093.
Figure 7‑14 Total lift and drag coefficients for pointed, crescent and elliptical wings.
Figure 7‑15 Variation in induced drag with angle of attack for pointed, elliptical and crescent wings.
Figure 7‑16 compares span efficiencies for the three wing cases. The PANAIR results for the crescent wing exhibit greater span efficiency (0.85%) than the elliptical (1.0064) wing. However, according to the PMARC results the elliptical wing has a span efficiency of 0.25% greater than the crescent wing. The pointed tip wing has a span efficiency of 7.8% lower than the elliptical wing.
Figure 7‑16 Span efficiencies for different wings computed by PMARC compared to those from PANAIR (Smith and Kroo 1993).
There is a discrepancy in results obtained from the PMARC program and those obtained from PANAIR. This discrepancy could be attributed to many factors:
As seen in Error! Reference source not found., there is considerable upwash at the tip for all three wing cases. Thus, greater spanwise resolution is needed at the wing tips to correctly capture flow characteristics at the tip. The elliptical wing had halfcosine spanwise panel spacing, but the crescent and pointed wings had equal spacing.
PMARC computes total lift and profile drag coefficients by integrating the surface pressure. The combined effect of low panel resolution and large pressure gradients across the span might affect accuracy of lift and profile drag coefficients. However, the coefficient of induced drag is obtained from the Trefftz plane analysis which is less sensitive to spanwise resolution (Smith and Kroo 1993).
In this study, the wake was extended to 10 chord lengths behind the wing. This was done to allow faster computing time as some cases required more than 2 hours to complete. A better wake length would be 20 chord lengths (increasing with aspect ratio) (Katz and Plotkin 2001).
PMARC is a first order method with a constant source and constant doublet distribution over flat panels. PANAIR is a second order method with linear source distribution and quadratic doublet distribution. This leads to greater accuracy in modeling the flow field around a body, but also results in greater computational time.
In low order methods, the doublet strengths are not exactly matched between panels. This sometimes leads to inaccurate modeling and results as seen in Figure 7‑17 (Carmichael and Erickson 1981).
Figure 7‑17 Incorrect line vortex effect due to discontinuity in doublet strengths across panels. (Carmichael and Erickson 1981)
Based on this preliminary numerical study of different wing planforms, it can be seen that more validating studies need to be conducted before predictions for the pointed tip wing case can be trusted. Some errors can also be reduced by increasing the wake length.
Abbot, I. H. and A. E. v. Doenhoff (1959). Theory of Wing Sections (including a summary of airfoil data). New York, Dover Publications.
About, I. The History of the Airplane, Part 3: The Wright Brothers  First Flight.
Anderson, J. D. (1985). Fundamentals of Aerodynamics. New York, McGrawHill.
Ashby, D. L., M. Dudley, et al. (1988). Development and Validation of an Advanced LowOrder Panel Method. NASA Technical Memorandum 101024. Ames Research Center, Moffett Field, CA.
Burton, R. (1990). Bird Flight, Facts on File. New York.
Carmichael, B. H. (1981). "Low Reynolds Number Airfoil survey." National Aeronautics and Space Administration Contractor Report 165803 1.
Carmichael, R. L. and L. L. Erickson (1981). "PANAIR A Higher Order Panel Method for Predicting Subsonic or Supersonic Linear Potential flows about Arbitrary Configurations." AIAA Paper 81(1255).
Ellington, C. P. (1984). "The aerodynamics of hovering insect flight. I. The quasisteady analysis." Philosophical Transactions of the Royal Society of London B 305(1122): 115.
Grundy, T. M., G. P. Keefe, et al. (2001). Effects of Acoustic Disturbances on Low Re Aerofoil Flows. Fixed and Flapping Wing Aerodynamics for Micro Air Vehicle Applications. T. J. Mueller. Virginia, American Institute of Aeronautics and Astronautics, Inc. 195: 91113.
Guglielmo, J. J. and M. Selig (1996). "Spanwise Variations in Profile drag for Airfoils at Low Reynolds number." Journal of Aircraft 33(4): 699707.
Hankin, E. H. (1913). Animal Flight. A record of observation. London:Iliffe.
Hoerner, S. F. (1965). Fluiddynamic drag; practical information on aerodynamic drag and hydrodynamic resistance.
Hoerner, S. F. and H. V. Borst (1975). Fluiddynamic lift: practical information on aerodynamic and hydrodynamic lift. Brick Town, N.J, L.A Hoerner.
Houghton, E. L. and A. E. Brock (1960). Aerodynamics for Engineering Students. London, Edward Arnold (Publishers) Ltd.
Jones, R. T. (1990). Wing Theory. Princeton.
Karamcheti, K. (1966). Principles of Ideal Fluid Aerodynamics, Stanford University.
Katz, J. and A. Plotkin (2001). LowSpeed Aerodynamics, Cambridge University Press.
Kuethe, A. M. and C.Y. Chow (1950). Foundations of Aerodynamics: Bases of Aerodynamic Design, John Wiley & Sons.
Laitone, E. V. (1997). "Wind tunnel tests of wings at Reynolds numbers below 70,000." Experiments in Fluids 23: 405409.
Laitone, E. V. (2001). Wind Tunnel Tests of Wings and Rings at Low Reynolds Numbers. Fixed and Flapping Wing Aerodynamics for Micro Air Vehicle Applications. T. J. Mueller. Virginia, American Institute of Aeronautics and Astronautics, Inc. 195: 8390.
Lissaman, P. B. S. (1983). "LowReynoldsnumber airfoils." Annual Review of Fluid Mechanics 15: 223239.
Lissaman, P. B. S. (2004). Pointed Wing Tip Project: Personal Communication.
Lyon, C. A., A. P. Broeren, et al. (1997). Summary of LowSpeed Airfoil Data. Virginia Beach, VA, Soartech Publications.
McMasters, J. H. and M. L. Henderson (1980). "Low Speed Single Element Airfoil Synthesis." Technical Soaring 6(2): 121.
Mises, R. V. (1945). Theory of Flight. New York, Dover Publications.
Muller, W. and G. Patone (1998). "Air Transmittivity of Feathers." Journal of Experimental Biology 201: 25912599.
Nachtigall, H. v. W. and J. Wieser (1966). "Profilmessungen am Taubenflugel." Zeit. Vergl. Physiol. 52: 333346.
NaturalHistoryMuseum Feathers: An Engineering Marvel.
Naughton, R. (2002). Flying Wings.
Oehme, V. H. (1971). "on the geometrical twist of the Avian wing." Biol. Zentralbl 90: 14556.
Panton, R. L. (1984). Incompressible Flow, WileyInterscience.
Pelletier, A. and M. T.J (1999). Aerodynamic Force Moment measurements at very Low Reynolds number. Proceedings of the 46th Annual Conference of the Canadian Aeronautics and Space Institute, Montreal.
Pennycuick, C. J. (1968). "A WindTunnel Study of Gliding Flight in the Pigeon Columba Livia." Journal of Experimental Biology 49: 509526.
Pennycuick, C. J. (1982). "The Flight of Petrels and Albatrosses (Procellariiformes), observed in South Georgia and its vicinity." Philosophical Transactions of the Royal Society of London B300: 75106.
Pennycuick, C. J. (1983). "Thermal soaring compared in three dissimilar tropical bird species, Fregata Magnificens, Pelecanus Occidentalis and Coragyps Atratus." Journal of Experimental Biology 102: 307325.
Pennycuick, C. J. (1987). Flight of seabirds. Seabirds: feeding biology and role in marine ecosystems, Cambridge University Press, Great Britain: 4362.
Pennycuick, C. J. (1995). "The use and misuse of mathematical flight models." Israel Journal of Zoology 41: 307319.
Pennycuick, C. J., H. C.E, et al. (1992). "The profile drag of a Hawk's wing, measured by wake sampling in a wind tunnel." Journal of Experimental Biology 165: 119.
Pennycuick, C. J., H. H. O. III, et al. (1988). "Empirical estimates of body drag of large waterfowl and raptors." Journal of Experimental Biology 135: 253264.
Pennycuick, C. J., M. Klaassen, et al. (1996). "Wingbeat frequency and the body drag anomaly: wind tunnel observations on a thrush nightingale (Luscinia Luscinia) and a teal (Anas Crecca)." Journal of Experimental Biology 199: 27572765.
Raymer, D. P. (1999). Aircraft Design: A conceptual approach, AIAA, Inc. 1801 Alexander Bell Drive, Reston, VA 20191.
Ruppell, G. (1977). Bird flight. New York, Van Nostrand Reinhold Limited.
Schlichting, D. H. (1958). Boundary Layer Theory.
Selig, M. S., J. F. Donovan, et al. (1998). Airfoils at Low Speeds. Virginia 23451, H.A Stokely.
Selig, M. S., J. J. Guglielmo, et al. (1995). Summary of LowSpeed Airfoil Data. Virginia Beach, VA, Soartech Publications.
Smith, S. C. and I. Kroo (1993). "Computation of Induced Drag for Elliptical and CrescentShaped Wings." Journal of Aircraft 30(4): 446452.
Spedding, G., F. Browand, et al. (2005). An Experimental Program for Improving MAV Aerodynamic Performance, University of Southern California.
Spedding, G. R. (1987). "The wake of a kestrel (Falco tinnunculus) in gliding flight." Journal of Experimental Biology 127: 4557.
Spedding, G. R. (1992). The Aerodynamics of Flight. Mechanics of Animal Locomotion. A. R. McNeill. Berlin, SpringerVerlag: 51111.
Spedding, G. R., A. Hedenstrom, et al. (2003). "Quantitative studies of the wakes of freely flying birds in a lowturbulence wind tunnel." Experiments in Fluids 34: 291303.
Spedding, G. R. and P. B. S. Lissaman (1998). "Technical aspects of microscale flight systems." Journal of Avian Biology 29(4): 458468.
Tennekes, H. (1996). The Simple Science of Flight: from Insects to Jumbo jets. MA, MIT Press.
Thwaites, B. (1960). Incompressible Aerodynamics, Dover.
Torres, G. E. and T. J. Mueller (2004). "Low Aspect Ratio Wing Aerodynamics at Low Reynolds Numbers." AIAA Journal 42(5): 865874.
Walker, S. G. T. (1925). "The Flapping Flight of Birds." The Journal of the Royal Aeronautical Society 29: 590594.
Warham, J. (1977). "Wing loadings, wing shapes and flight capabilities of Procellariiformes." New Zealand Journal of Zoology 4: 7383.
White, F. M. (1999). Fluid Mechanics, WCB McGrawHill.
Withers, P. C. (1981). "An aerodynamic analysis of bird wings as fixed aerofoils." Journal of Experimental Biology 90: 143162.
[1] The species are Wandering albatross, Blackbrowed albatross, Greyheaded albatross, Lightmantled sooty albatross, Giant petrel, Whitechinned petrel, Cape pigeon, Dove Prion and Wilson’s petrel.
[2] An elliptical lift distribution results in the lowest induced drag as will be seen in Section 2.5.1.
[3] _{} .is the velocity potential defined by (v = velocity of the fluid flow) and _{} is the stream function.
[4] Coefficient of Lift _{} where S = wing area.
[5] The Karman integral relation is based on the theory that the rate of change of momentum in any direction within a closed loop (or surface) in a moving fluid is equal to the sum of all force components acting in that direction on the fluid (Abbot, I. H. and A. E. v. Doenhoff (1959). Theory of Wing Sections (including a summary of airfoil data). New York, Dover Publications.
[6] The alula functions as a high lift device and can increase lift by about 25% Ellington, C. P. (1984). "The aerodynamics of hovering insect flight. I. The quasisteady analysis." Philosophical Transactions of the Royal Society of London B 305(1122): 115.
[7] The quasisteady assumption assumes that the instantaneous forces on a flapping wing are the same as those on a wing in steady motion at the same instantaneous linear velocity and attitude Ibid.
.
[8] The White Chinned Petrel (Procellaria aequinoctialis) represents the ‘standard seabird’ with m = 700g, b = 1.09 m and area = 0.103 m^{2}.
[9] Up to 40 tiny muscles can be attached to the base of a single featherNaturalHistoryMuseum Feathers: An Engineering Marvel.
.
[10] Hysteresis in airfoil aerodynamics refers to the difference in lift, drag or moment coefficients at a particular angle of attack when this angle of attack is achieved by decreasing the angles of attack and then by increasing them Selig, M. S., J. F. Donovan, et al. (1998). Airfoils at Low Speeds. Virginia 23451, H.A Stokely.
.