Aerodynamic i

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DEPARTMENT OF AERONAUTICAL ENGINEERING STUDY MATERIALS
S.RAJASUDHAKAR,.B.E-AERO M.SHANMUGANATHAN,.B.E-AERO
AERODYNAICS I
UNIT-I
INTRODUCTION TO LOW SPEED FLOW
Euler equation, incompressible bernoulli’s equation. circulation and vorticity, green’s lemma and
stoke’s theorem, barotropic flow, kelvin’s theorem, streamline, stream function, irrotational flow,
potential function, equipontential lines, elementary flows and their combinations.
Energy equation
By conservation of energy, which can only be transferred and not created or destroyed leads to a
continuity equation, an alternative mathematical statement of energy conservation to the
thermodynamic laws.
Continuity equation
When a fluid is in motion, it must move in such a way that mass is conserved.
Flow through converging passage
Measurement of Flow Rate through Pipe Flow rate through a pipe is usually measured by providing a
coaxial area contraction within the pipe and by recording the pressure drop across the contraction.
Therefore the determination of the flow rate from the measurement of pressure drop depends on the
straight forward application of Bernoulli’s equation. Three different flow meters operate on this
principle.

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 Venturimeter
 Orifice meter
 Flow nozzle.
Continuity Equation
When a fluid is in motion, it must move in such a way that mass is conserved. To see how mass
conservation places restrictions on the velocity field, consider the steady flow of fluid through a duct
(that is, the inlet and outlet flows do not vary with time). The inflow and outflow are one-
dimensional, so that the velocity V and density rho are constant over the area A.
Figure 1.1 One-dimensional duct showing control volume.
Now we apply the principle of mass conservation. Since there is no flow through the side walls of the
duct, what mass comes in over A_1 goes out of A_2, (the flow is steady so that there is no mass
accumulation). Over a short time interval Delta t,

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This is a statement of the principle of mass conservation for a steady, one-dimensional flow,
with one inlet and one outlet. This equation is called the continuity equation for steady one-
dimensional flow. For a steady flow through a control volume with many inlets and outlets, the net
mass flow must be zero, where inflows are negative and outflows are positive.
Momentum Equation
Let us now derive the momentum equation resulting from the Reynolds Transport theorem, Now we
have = where is the momentum. Note that momentum is a vector quantity and that it has a
component in every coordinate direction. Thus,
Consider the left hand side, We have which is proportional to the applied force as per Newton's
Second Law of motion. Thus,
Where is again a vector. It is necessary to include both body forces, and surface forces,
Thus,

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Now we substitute for in the right hand side of Eqn. 3.27 giving,
Writing this as three equations, one for each coordinate direction we have,
term represents the u momentum that is convected in/out by the surfaceThe
in a direction normal to it. In fact momentum in other direction can also be convected out from the
same area. These are given by and.
As stated before the term is replaced by .
The equation thus derived finds immense application in fluid dynamic calculations such as force at the
bending of a pipe, thrust developed at the foundation of a rocket nozzle, drag about an immersed body
etc. We consider some of these later.
ENERGY EQUATION

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By conservation of energy, which can only be transferred and not created or destroyed leads to a
continuity equation, an alternative mathematical statement of energy conservation to the
thermodynamic laws.
Letting
 u = local energy density (energy per unit volume),
 q = energy flux (transfer of energy per unit cross-sectional area per unit time) as a vector,
The continuity equation is:
It seems that in the elliptical nature of orbits, the Inverse Square Law of Force, and the Third
Law were sequentially shown to be natural and valid via a priori methods - thus accomplishing the
primary aim of the book. But then in the four chapters leading up to this one, instead of halting, there
is a shift in emphasis, still in a priori mode, to the derivation of a tool that allows the prediction of
orbits and the analysis of what is happening to the planet at any given moment. In other words there is
a shift from descriptive mode to analytical mode. The four previous chapters culminate here in the a
priori proof of the Energy Equation, the relevant analytical tool. Specifically, it is shown in these five
chapters that contained in the hodograph is a right triangle representing a relationship between total
velocity and tangential velocity in terms of two variables - the distance to the Sun and the length of the
semi major axis. It is these relationships, inherent in the right triangle of the hodograph, that give us
the Energy Equation. - a useful tool. And so, instead of simply ending, this book will end with the
application of the Energy Equation to various orbital situations in Chapter 42 and with a subsequent
philosophical statement.
We will allow the empirical finding of Galileo that unequal masses fall to the ground at the same rate
but otherwise we will have gained this all from a priori methods. We have Galileo to thank, in that
sense, for giving us the term GM to use in order to turn many of our proportions into equations.

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We would have been valid without it but many of our equations including the Energy Equation that we
will derive in this chapter would have remained as a proportion instead of an equation.
This chapter's theme contrasts with the conventional Energy Equation, found in texts related to
celestial mechanics, which analyses the energy of a planet in terms of its kinetic energy and its
negative potential energy. This requires wrestling with the concept of negative potential energy
related to free fall from the distance infinity- a concept and situation I have difficulty envisioning.
where we demonstrated the formula for force, having learned about force from the hodograph,
2
R
GMm
F  .
We were able to get to 2
1
R
F  using only a priori methods but turned that proportion into the
equation using Galileo's empirical finding.
In the force equation to derive the formula for the velocity of a planet in a circular orbit. Galileo's
empiricism led to the transformation of the finding in the context of learning how to scale circular
hodographs, of the proportion
R
Vcircle
1
 into the equation
R
GM
R
GM
Vcircle  . In this chapter
we will use the equation for the velocity of a planet in a circular orbit for the demonstration of our
Energy Equation. Before proceeding, note that we showed via scaling methods for hodographs that
escape velocity for an orbit when the planet is at a certain distance from the Sun is 2 times greater
than the velocity of a planet in a circular orbit at that same distance: circleescape VV  2 .
Plugging in the equation for the velocity of a planet in a circular orbit :
R
GM
VV circleescape  22
Now let's square the above equation. The reason will become evident:

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R
GM
V escape
22

This has a familiar appearance. We saw a similar mathematical terms in pertaining to the total
velocity squared.







aR
GMV
12
3
2
Multiply the above equation into separate terms:
a
GM
R
GM
V 
2
3
2
Now look at what happens when we subtract total velocity squared from escape velocity squared. The
result is a constant:
a
GM
a
GM
R
GM
R
GM
a
GM
R
GM
R
GM
VV totalescape 






222222
Abbreviated fashion, recalling that we refer to total velocity on the hodograph as 3V :
a
GM
VV esc  3
22
This is our Energy Equation for elliptical orbits. It is a proud achievement using a priorimethods.
We would have arrived at the proportion,
a
VV esc
1
3
22
 without using any empirical findings but
allowing for the ground based observations of Galileo, we have obtained a full equation.
Now what is the meaning of this Energy Equation? The answer lies in the concept of kinetic energy,
the energy of motion, which is traditionally expressed in terms of mass and velocity squared. In fact,
as a matter of review, the standard formula in texts for kinetic energy is 2
2
1
mv .

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Now the mass of our planet does not change as we compare its present total velocity to its theoretical
escape velocity and so we can ignore mass in the kinetic energy formula when we evaluate the
difference between the two energy states- present versus theoretical escape. And we could fairly
divide the Energy Equation by 2 to make the terms resemble kinetic energy formula more closely if
we want to but that is not necessary.
Simply knowing that we are dealing with energy since our terms are velocity squared tells us the
meaning of our Energy Equation. The Energy Equation states that for every instant and every
position of the planet in its orbit the difference between the energy of motion required for escape and
the energy of motion actually present is a constant. In other words the planet is always lacking a
certain amount of energy that would allow it to escape no matter where it is in its orbit. When the
planet is close to the Sun it is moving fast and has a lot of energy of motion to propel it far away form
the Sun, but not enough to escape. When the planet is far from the Sun it has gained a lot of distance
toward the feat of escape but has slowed down and has very little energy left to propel it further away.
In both positions, the planet obeys the Energy Equation exactly.
Recall that earlier in this chapter we showed escape velocity to be 2 times the velocity of a planet in
a circular orbit. And if we square that equation we get circleesc VV 22
2 In other words the escape
velocity squared is equal to twice the square of velocity of the planet in a circular orbit, both velocities
pertaining to initial conditions at a specified radius to the Sun.
So we could further embellish upon the Energy Equation to say that it states that twice the circular
velocity squared minus the actual velocity squared is always a constant. It is only mentioned here in
terms of interest.
The importance of the Energy Equation goes well beyond its meaning. Since we have the constant
GM in the numerator on the left side of the equation, we can predict what effect changes in velocity
of our orbiting planet or spaceship would have on the denominator a .
In other words, we have acquired a tool that will tell us what the new semimajor axis will be when we
change the planet or spaceship velocity. This is an important tool in navigation from one orbit to the
next desired orbit for space travel.

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We can further put it to use in predicting things like how far a planet would travel at any point in its
orbit if the velocity were turned directly away from the Sun and - we can show using logic that it
would fly away as far as the distance a2 and then fall back to crash into the Sun.
Another use would be to predict, using extremely eccentric orbits how long it would take an object to
free fall towards a central body form an extreme distance in a gravitational field where the force varies
inversely with the square of the distance - an otherwise difficult calculation.
Barotropic fluid
In fluid dynamics, a barotropic fluid is a fluid whose density is a function of only pressure.Barotropic
fluids are useful model for fluid behavior in a wide variety of scientific fields, from meteorology to
astrophysics.
Most liquids have a density which varies weakly with pressure or temperature, i. e., the density of a
liquid is nearly constant, so to a first approximation liquids are barotropic. To greater precision, they
are not barotropic. For example, the density of seawater depends on temperature, salinity, and
pressure, but only by a few percent at most.
A barotropic flow is a generalization of the barotropic atmosphere. It is a flow in which the pressure is
a function of the density only and vice versa. In other words, it is a flow in which isobaric surfaces are
isopycnic surfaces and vice versa. One may have a barotropic flow with a non-barotropic fluid, but a
barotropic fluid must always follow a barotropic flow. Examples include barotropic layers of the
oceans, an isothermal ideal gas or an isentropic ideal gas.
It is expressible as a function of only, that is, , the [
The flow velocity of a fluid effectively describes everything about the motion of a fluid. Many
physical properties of a fluid can be expressed mathematically in terms of the flow velocity. Some
common examples follow:
Steady flow
The flow of a fluid is said to be steady if does not vary with time. That is if

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Incompressible flow
If a fluid is incompressible the divergence of is zero:
That is, if is a solenoidal vector field.
Irrotational flow
A flow is irrotational if the curl of is zero:
That is, if is an irrotational vector field.
A flow in a simply-connected domain which is irrotational can be described as a potential flow, through the
use of a velocity potential with If the flow is both irrotational and incompressible,
the Laplacian of the velocity potential must be zero:
Vorticity
The vorticity, , of a flow can be defined in terms of its flow velocity by
Thus in irrotational flow the vorticity is zero.
The velocity potential
If an irrotational flow occupies a simply-connected fluid region then there exists a scalar field such that
The scalar field is called the velocity potential for the flow. (See Irrotational vector field.)
Kelvin Circulation Theorem
According to the Kelvin circulation theorem, which is named after Lord Kelvin (1824-1907), the
circulation around any co-moving loop in an inviscid fluid is independent of time. The proof is as
follows. The circulation around a given loop is defined

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However, for a loop that is co-moving with the fluid, we have . Thus,
However, by definition, for a co-moving loop . Moreover, the equation of motion of
an incompressible inviscid fluid can be written
Since is a constant. Hence,
Since and is obviously a single-valued function.

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Figure 22: A vortex tube.
One corollary of the Kelvin circulation theorem is that the fluid particles that form the walls of a
vortex tube at a given instance in time continue to form the walls of a vortex tube at all subsequent
times. To prove this, imagine a closed loop that is embedded in the wall of a vortex tube but does
not circulate around the interior of the tube. See Figure 22. The normal component of the vorticity over
the surface enclosed by is zero, since all vorticity vectors are tangential to this surface. Thus, from ,
the circulation around the loop is zero. By Kelvin's circulation theorem, the circulation around the loop
remains zero as the tube is convected by the fluid. In other words, although the surface enclosed by
deforms, as it is convected by the fluid, it always remains on the tube wall, since no vortex filaments
can pass through it.
Another corollary of the circulation theorem is that the intensity of a vortex tube remains constant as it
is convected by the fluid. This can be proved by considering the circulation around the loop
pictured in Figure.
Prandtl equation and rankine-hugonoit relation
Rankine–Hugoniot relations, relate to the behavior of shock waves traveling normal to the prevailing
flow.
This is known as prandtl equation.
Oblique shock equation
It will occur when a supersonic flow encounters a corner that effectively turns the flow into itself and
compresses. The upstream streamlines are uniformly deflected after the shock wave.
Normal shock equation
If the shock wave is perpendicular to the flow direction it is called a normal shock. A normal shock
occurs in front of a supersonic object if the flow is turned by a large amount and the shock cannot
remain attached to the body.

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Shock polars
The minimum angle, , which an oblique shock can have is the Mach angle , where
is the initial Mach number before the shock and the greatest angle corresponds to a normal shock.
The range of shock angles is therefore . To calculate the pressures for
this range of angles, the Rankine-Hugoniot equations are solved for pressure:
Hodograph
The true air speed v, of an airplane in a glide has a horizontal and a vertical component. The vc was
called the rate of decent, RD. a graph which relates the horizontal component of velocity to the rate of
decent RD is referred to as a speed polar or hodograph.
Rayleigh &fanno flow
Rayleigh flow refers to adiabatic flow through a constant area duct where the effect of heat
addition or rejection is considered. Compressibility effects often come into consideration, although the
Rayleigh flow model certainly also applies to incompressible flow.Fanno flow refers to adiabatic flow
through a constant area duct where the effect of friction is considered. Compressibility effects often come
into consideration, although the Fanno flow model certainly also applies to incompressible flow
Source & Sink flow

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Figure . Source flow & Sink flow
Potential function for 2D source of strength m at r = 0:
It satisfies (check Laplace's equation in polar coordinate in the keyword
search utility), except (so must exclude r = 0 from flow)
If sink. Source with
located atstrength
:

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3D source - Spherical coordinates
A spherically symmetric solution: (verify except at )
Define 3D source of strength located at :
Solution: Equations for the velocity field for the 2D source flow.

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Solution: Net outward volume flux for 2D source.
Solution: Net outward flux for a 3D source.
The Net outward flux is
2D point vortex
Another particular solution: (verify except at )
Potential function for a point vortex of circulation at :

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Stream function:
FREE AND FORCED VORTEX FLOWS
 Pressure distribution in a vortex flow is usually found out by integrating the equation of motion
in the r direction. The equation of motion in the radial direction for a vortex flow can be
written as
 Integrating Eq. (14.12) with respect to dr, and considering the flow to be incompressible we
have,
 For a free vortex flow,
 Hence Equ becomes

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 If the pressure at some radius r = ra, is known to be the atmospheric pressure patm then equation
(14.14) can be written as
where z and za are the vertical elevations (measured from any arbitrary datum) at r and ra.
 Equation can also be derived by a straight forward application of Bernoulli’s equation between
any two points at r = ra and r = r.
 In a free vortex flow total mechanical energy remains constant. There is neither any energy
interaction between an outside source and the flow, nor is there any dissipation of mechanical
energy within the flow. The fluid rotates by virtue of some rotation previously imparted to it or
because of some internal action.
 Some examples are a whirlpool in a river, the rotatory flow that often arises in a shallow vessel
when liquid flows out through a hole in the bottom (as is often seen when water flows out from
a bathtub or a wash basin), and flow in a centrifugal pump case just outside the impeller.
Cylindrical Free Vortex
 A cylindrical free vortex motion is conceived in a cylindrical coordinate system with axis z
directing vertically upwards where at each horizontal cross-section, there exists a planar free
vortex motion with tangential velocity.

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 The total energy at any point remains constant and can be written as
 The pressure distribution
along the radius can be
found from Eq. by
considering z as
constant; again, for any
constant pressure p,
values of z, determining
a surface of equal
pressure, can also be
found
 If p is measured in gauge
pressure, then the value
of z, where p = 0 determines
the free surface if one exists.

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Fig 2.3 Cylindrical Free Vortex
 Flows where streamlines are concentric circles and the tangential velocity is directly
proportional to the radius of curvature are known as plane circular forced vortex flows.
The flow field is described in a polar coordinate system as,
and
 All fluid particles rotate with the same angular velocity ω like a solid body. Hence a forced
vortex flow is termed as a solid body rotation.
 The vorticity Ω for the flow field can be calculated as
 Therefore, a forced vortex motion is not irrotational; rather it is a rotational flow with a
constant vorticity 2ω. Equation is used to determine the distribution of mechanical energy
across the radius as
 Integrating the equation between the two radii on the same horizontal plane, we have,
 Thus, we see from Equ that the total head (total energy per unit weight) increases with an
increase in radius. The total mechanical energy at any point is the sum of kinetic energy, flow
work or pressure energy, and the potential energy.
 Therefore the difference in total head between any two points in the same horizontal plane can
be written as,

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 Substituting this expression of H2-H1 in Equ we get
 The same equation can also be obtained by integrating the equation of motion in a radial
direction as
 To maintain a forced vortex flow, mechanical energy has to be spent from outside and thus an
external torque is always necessary to be applied continuously.
 Forced vortex can be generated by rotating a vessel containing a fluid so that the angular
velocity is the same at all points

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UNIT-II
TWO DIMENSIONAL INVISCID INCOMPRESSIBLE FLOW
Ideal Flow over a circular cylinder, D’Alembert’s paradox, magnus effect, Kutta joukowski’s theorem,
starting vortex, kutta condition, real flow over smooth and rough cylinder
PRESSURE DISTRIBUTION ON BODIES WITH IDEAL & REAL FLUID FLOW
The pressure coefficient is a dimensionless number which describes the relative pressures
throughout a flow field in fluid dynamics. The pressure coefficient is used in aerodynamics and
hydrodynamics. Every point in a fluid flow field has its own unique pressure coefficient, .
In many situations in aerodynamics and hydrodynamics, the pressure coefficient at a point near
a body is independent of body size. Consequently an engineering model can be tested in a wind tunnel
or water tunnel, pressure coefficients can be determined at critical locations around the model, and
these pressure coefficients can be used with confidence to predict the fluid pressure at those critical
locations around a full-size aircraft or boat.
INCOMPRESSIBLE FLOW

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The pressure coefficient is a parameter for studying the flow of incompressible fluids such as
water, and also the low-speed flow of compressible fluids such as air.
The relationship between the dimensionless coefficient and the dimensional numbers
where:
is the pressure at the point at which pressure coefficient is being evaluated
is the pressure in the freestream (i.e. remote from any disturbance)
is the freestream fluid density (Air at sea level and 15 °C is 1.225 )
is the freestream velocity of the fluid, or the velocity of the body through the fluid

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Using Bernoulli's Equation, the pressure coefficient can be further simplified for incompressible,
lossless, and steady flow
where V is the velocity of the fluid at the point at which pressure coefficient is being evaluated.
This relationship is also valid for the flow of compressible fluids where variations in speed and
pressure are sufficiently small that variations in fluid density can be ignored. This is a reasonable
assumption when the Mach Number is less than about 0.3.
 of zero indicates the pressure is the same as the free stream pressure.
 of one indicates the pressure is stagnation pressure and the point is a stagnation point.
 of minus one is significant in the design of gliders because this indicates a perfect location
for a "Total energy" port for supply of signal pressure to the Variometer.
In the fluid flow field around a body there will be points having positive pressure coefficients up to
one, and negative pressure coefficients including coefficients less than minus one, but nowhere will
the coefficient exceed plus one because the highest pressure that can be achieved is the stagnation
pressure. The only time the coefficient will exceed plus one is when advanced boundary layer control
techniques, such as blowing, is used.
PRESSURE DISTRIBUTION

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An airfoil at a given angle of attack will have what is called a pressure distribution. This
pressure distribution is simply the pressure at all points around an airfoil. Typically, graphs of these
distributions are drawn so that negative numbers are higher on the graph, as the for the upper
surface of the airfoil will usually be farther below zero and will hence be the top line on the graph.
AND RELATIONSHIP
The coefficient of lift for an airfoil with strictly horizontal surfaces can be calculated from the
coefficient of pressure distribution by integration, or calculating the area between the lines on the
distribution.
This expression is not suitable for direct numeric integration using the panel method of lift
approximation, as it does not take into account the direction of pressure-induced lift.

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where:
is pressure coefficient on the lower surface
is pressure coefficient on the upper surface
is the leading edge
is the trailing edge
When the lower surface is higher (more negative) on the distribution it counts as a negative area as
this will be producing down force rather than lift.
PRESSURE DISTRIBUTION ON BODIES WITHOUT IDEAL & REAL FLUID
FLOW

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Some closed-form solutions for the potential flow over bodies of revolution are available and are
useful as reference results.
We noted that in 2-D the maximum velocity on an ellipse was given by:
umax/U = 1 + t/c
In 3-D the surface velocity over an ellipsoid of revolution is given by:
x is the distance from the midpoint in units so that the length is 2.0 and r is the radius (or in these
units, the ratio of diameter to length.)
The maximum velocity is given by:
The figure below (from Schlichting) illustrates the pressure distribution on bodies of revolution.
D/L=0.1

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Note that perturbation velocities are much smaller than in 2-D. These velocities may be estimated by
superimposing point sources. In this case for an ellipsoid:
Note that the maximum velocity is sensitive to the actual shape, with a paraboloid having about 50%
larger perturbations. The results from such a distribution of sources on the axis, slightly underpredicts
the velocity perturbations.
The figure below shows the pressure distribution on a typical fuselage shape with D/L = 0.09
computed by a source distribution on the x-axis. Note how the pressure falls near the center of the
cylindrical portion of the fuselage.
As indicated below, fuselages in inviscid flow produce a nose-up pitching moment when the angle of
attack is increased. This effect is destabilizing and is an important consideration in the sizing of the
horizontal tail.

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Although the inviscid picture suggests that no lift is produced, the viscous flow actually separates at
the back of the fuselage, making the moment somewhat smaller and the lift larger than predicted by
inviscid theory. This lift produces induced drag and the fuselage behaves as a low aspect ratio wing.
The figure below shows the effect of angle of attack on fuselage lift, drag, and moment based
on experimental data. Also shown is the estimated moment based on inviscid theory.
Kutta-Joukowski Lift Theorem

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Two early aerodynamicists, Kutta in Germany and Joukowski in Russia, worked to quantify the lift
achieved by an airflow over a spinning cylinder. The lift relationship is
Lift per unit length = L = ρGV
where ρ is the air density, V is the velocity of flow, and G is called the "vortex strength". The vortex
strength is given by
G = 2ρωr2
where ω is the angular velocity of spin of the cylinder.
Like all aerodynamic lift, this seems a bit mysterious, but it can be looked at in terms of a redirection
of the air motion. If the cylinder traps some air in a boundary layer at the cylinder surface and carries it
around with it, shedding it downward, then it has given some of the air a downward momentum. That
can act to give the cylinder an upward momentum in accordance with the principle of conservation of
momentum. Another approach is to say that you have exerted a downward component of force on the air
and by Newton's 3rd law there must be an upward force on the cylinder. Yet another approach is to say
that the top of the cylinder is assisting the airstream, speeding up the flow on the top of the cylinder.
Then by the Bernoulli equation, the pressure on the top of the cylinder is diminished, giving an effective
lift

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The Kutta–Joukowski theorem is a fundamental theorem of aerodynamics. It is named after the German
Martin Wilhelm Kutta and the Russian Nikolai Zhukovsky (or Joukowski) who first developed its key ideas in
the early 20th century. The theorem relates the lift generated by a right cylinder to the speed of the
cylinder through the fluid, the density of the fluid, and the circulation. The circulation is defined as the
line integral, around a closed loop enclosing the cylinder or airfoil, of the component of the velocity of
the fluid tangent to the loop. The magnitude and direction of the fluid velocity change along the path.
The flow of air in response to the
presence of the airfoil can be
treated as the superposition of a
translational flow and a
rotational flow. It is, however,
incorrect to think that there is a
vortex like a tornado encircling
the cylinder or the wing of an airplane in flight. It is the integral's path that encircles the cylinder, not a
vortex of air. (In descriptions of the Kutta–Joukowski theorem the airfoil is usually considered to be a
circular cylinder or some other Joukowski airfoil.)

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The theorem refers to two-dimensional flow around a cylinder (or a cylinder of infinite span) and
determines the lift generated by one unit of span. When the circulation is known, the lift per unit
span (or ) of the cylinder can be calculated using the following equation.
(1)

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where and are the fluid density and the fluid velocity far upstream of the cylinder, and is
the circulation defined as the line integral,
around a path (in the complex plane) far from and enclosing the cylinder or airfoil. This path must
be in a region of potential flow and not in the boundary layer of the cylinder. The is the
component of the local fluid velocity in the direction of and tangent to the curve and is an
infinitesimal length on the curve, . Equation (1) is a form of the Kutta–Joukowski theorem.
Kuethe and Schetzer state the Kutta–Joukowski theorem as follows:[3]
The force per unit length acting
on a right cylinder of any cross section whatsoever is equal to , and is perpendicular to the
direction of
The result derived above, namely, is a very general one and is valid for any closed body placed in a
uniform stream. It is named the Kutta-Joukowsky theorem in honour of Kutta and Joukowsky who
proved it independently in 1902 and 1906 respectively. The theorem finds considerable application in
calculating lift around aerofoils.
3.1.1 Magnus Effect
Magnus effect is produced when circulation is imposed upon a cylinder placed in uniform flow. This
force is nothing but the lift. This effect is called Magnus Effect in honour of the scholar Heinrich
Magnus (1802 - 1870). Sports involving balls, such as golf, baseball, tennis see this effect in action.

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KUTTA CONDITION
The Kutta condition is a principle in steady flow fluid dynamics, especially aerodynamics, that is
applicable to solid bodies which have sharp corners such as the trailing edges of airfoils. It is named for
German mathematician and aerodynamicist Martin Wilhelm Kutta.
Kuethe and Schetzer state the Kutta condition as follows:
A body with a sharp trailing edge which is moving through a fluid will create about itself a circulation
of sufficient strength to hold the rear stagnation point at the trailing edge.

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In fluid flow around a body with a sharp corner the Kutta condition refers to the flow pattern in which
fluid approaches the corner from both directions, meets at the corner and then flows away from the
body. None of the fluid flows around the corner remaining attached to the body.
The Kutta condition is significant when using the Kutta–Joukowski theorem to calculate the lift generated
by an airfoil. The value of circulation of the flow around the airfoil must be that value which would
cause the Kutta condition to exist.
THE KUTTA CONDITION APPLIED TO AIRFOILS
When a smooth symmetric body, such as a cylinder with oval cross-section, moves with zero angle of
attack through a fluid it generates no lift. There are two stagnation points on the body - one at the front
and the other at the back. If the oval cylinder moves with a non-zero angle of attack through the fluid
there are still two stagnation points on the body - one on the underside of the cylinder, near the front
edge; and the other on the topside of the cylinder, near the back edge. The circulation around this
smooth cylinder is zero and no lift is generated,
despite the positive angle of attack.

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If an airfoil with a sharp trailing edge begins to move with a positive angle of attack through air, the
two stagnation points are initially located on the underside near the leading edge and on the topside
near the trailing edge, just as with the cylinder. As the air passing the underside of the airfoil reaches
the trailing edge it must flow around the trailing edge and along the topside of the airfoil toward the
stagnation point on the topside of the airfoil. Vortex flow occurs at the trailing edge and, because the
radius of the sharp trailing edge is zero, the speed of the air around the trailing edge should be
infinitely fast! Real fluids cannot move at infinite speed but they can move very fast. The very fast
airspeed around the trailing edge causes strong viscous forces to act on the air adjacent to the trailing
edge of the airfoil and the result is that a strong vortex accumulates on the topside of the airfoil, near
the trailing edge. As the airfoil begins to move it carries this vortex, known as the starting vortex, along
with it. Pioneering aerodynamicists were able to photograph starting vortices in liquids to confirm
their existence.
The vorticity in the starting vortex is matched by the vorticity in the bound vortex in the airfoil, in
accordance with Kelvin's circulation theorem. As the vorticity in the starting vortex progressively increases
the vorticity in the bound vortex also progressively increases and causes the flow over the topside of
the airfoil to increase in speed. The starting vortex is soon cast off the airfoil and is left behind,
spinning in the air where the airfoil left it.
The stagnation point on the topside of the
airfoil then moves until it reaches the
trailing edge.[6]
The starting vortex
eventually dissipates due to viscous
forces.

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As the airfoil continues on its way, there is a stagnation point at the trailing edge. The flow over the
topside conforms to the upper surface of the airfoil. The flow over both the topside and the underside
join up at the trailing edge and leave the airfoil travelling parallel to one another. This is known as the
Kutta condition.
When an airfoil is moving with a positive angle of attack, the starting vortex has been cast off and the
Kutta condition has become established, there is a finite circulation of the air around the airfoil. The
airfoil is generating lift, and the magnitude of the lift is given by the Kutta–Joukowski theorem.
One of the consequences of the Kutta condition is that the airflow over the topside of the airfoil travels
much faster than the airflow under the underside. A parcel of air which approaches the airfoil along
the stagnation streamline is cleaved in two at the stagnation point, one half traveling over the topside
and the other half traveling along the underside. The flow over the topside is so much faster than the
flow along the underside that these two halves never meet again. They do not even re-join in the wake
long after the airfoil has passed. This is sometimes known as "cleavage". There is a popular fallacy
called the equal transit-time fallacy that claims the two halves rejoin at the trailing edge of the airfoil. This
fallacy is in conflict with the phenomenon of cleavage that has been understood since Martin Kutta's
discovery.
Whenever the speed or angle of attack of an airfoil changes there is a weak starting vortex which
begins to form, either above or below the trailing edge. This weak starting vortex causes the Kutta

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condition to be re-established for the new speed or angle of attack. As a result, the circulation around the
airfoil changes and so too does the lift in response to the changed speed or angle of attack.
Inviscid Flow Past a Cylindrical Obstacle
Consider the steady flow pattern produced when an impenetrable rigid cylindrical obstacle is placed in
a uniformly flowing, incompressible, inviscid fluid, with the cylinder orientated such that its axis is
normal to the flow. For instance, suppose that the radius of the cylinder is , and that its axis
corresponds to the line . Furthermore, let the unperturbed fluid velocity be of magnitude
, and be directed parallel to the -axis. Now, we expect the flow pattern to remain unperturbed very
far away from the cylinder. In other words, we expect as , which corresponds
to a boundary condition on the stream function of the form
Given that the fluid velocity field a large distance upstream of the cylinder is irrotational (since we
have already seen that the flow pattern associated with uniform flow is irrotational, it follows from the
Kelvin circulation theorem that the velocity field remains irrotational as it is convected past the
cylinder. Hence, according to, the stream function of the flow satisfies Laplace's equation,
The appropriate boundary condition at the surface of the cylinder is simply that the normal fluid
velocity there be zero, since the fluid must stay in contact with the cylinder, but cannot penetrate its
surface. Hence, , which implies that
Since is undetermined to an arbitrary additive constant.

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where
and is the circulation of the flow around the cylinder. (Note that the velocity field can be
irrotational, but still possess nonzero circulation around the cylinder, because a loop that encloses the
cylinder cannot be spanned by a surface lying entirely within the fluid. Thus, zero fluid vorticity does
not necessarily imply zero circulation around such a loop from Stokes' theorem.) Let us assume that
, for the sake of definiteness.
Figure : Streamlines of the flow generated by a cylindrical obstacle of radius , whose axis runs
along the -axis, placed in the uniform flow field . The normalized circulation is .

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Figure show streamlines of the flow calculated for various different values of the normalized
circulation, . For there exist a pair of points on the surface of the cylinder at which the flow
speed is zero. These are known as stagnation points, and can be located in Figures as the points at
which streamlines intersect the surface of the cylinder at right-angles. Now, the tangential fluid
velocity at the surface of the cylinder is
The stagnation points correspond to the points at which (since the normal velocity is
automatically zero at the surface of the cylinder). Thus, the stagnation points lie at .
When the stagnation points coalesce and move off the surface of the cylinder, as illustrated in
Figure (the stagnation point corresponds to the point at which two streamlines cross at right-angles).
Figure: Streamlines of the flow generated by a cylindrical obstacle of radius , whose axis runs
along the -axis, placed in the uniform flow field . The normalized circulation is .

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The irrotational form of Bernoulli's theorem, can be combined with the boundary condition as ,
as well as the fact that is constant in the present case, to give
Where is the constant static fluid pressure a large distance from the cylinder. In particular, the fluid
pressure on the surface of the cylinder is
Where . The net force per unit length exerted on the cylinder by the
fluid has the Cartesian components
Thus, it follows from that
Now, the component of the force which a moving fluid exerts on an obstacle, placed in its path, in a
direction parallel to that of the unperturbed flow is usually called drag. On the other hand, the
component of the force which the fluid exerts in a direction perpendicular to that of the unperturbed
flow is usually called lift. Hence, the above equations imply that if a cylindrical obstacle is placed in a
uniformly flowing inviscid fluid then there is zero drag. On the other hand, as long as there is net

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circulation of the flow around the cylinder, the lift is non-zero. Now, lift is generated because
(negative) circulation tends to increase the fluid speed directly above, and to decrease it directly
below, the cylinder. Thus, from Bernoulli's theorem, the fluid pressure is decreased above, and
increased below, the cylinder, giving rise to a net upward force (i.e., a force in the -direction).
Figure 30: Streamlines of the flow generated by a cylindrical obstacle of radius , whose axis runs
along the -axis, placed in the uniform flow field . The normalized circulation is
Suppose that the cylinder is placed in a fluid which is initially at rest, and that the fluid's uniform flow
velocity, , is then very slowly ramped up (in such a manner that no vorticity is induced in the
upstream flow at infinity). Since the flow pattern is initially irrotational, and since the flow pattern
well upstream of the cylinder is assumed to remain irrotational, the Kelvin circulation theorem
indicates that the flow pattern around the cylinder also remains irrotational. Consider the time
evolution of the circulation, , around some fixed curve that lies entirely within the
fluid, and encloses the cylinder. We have

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where use has been made of (with assumed constant). However, in two-dimensional
flow, and , where is an outward surface element of a unit depth (in the -direction)
surface whose normal lies in the - plane, and that cuts the - plane at . In other words,
We, thus, conclude that the rate of change of the circulation around is equal to minus the flux of the
vorticity across [assuming that vorticity is convected by the flow, which follows from (, the fact
that , and the fact that in two-dimensional flow]. However, we have already
seen that the flow field surrounding the cylinder is irrotational (i.e., such that ). It follows that
is constant in time. Moreover, since originally, because the fluid surrounding the cylinder was
initially at rest, we deduce that at all subsequent times. Hence, we conclude that, in an
inviscid fluid, if the circulation of the flow around the cylinder is initially zero then it remains zero. It
follows, from the above analysis, that, in such a fluid, zero drag force and zero lift force are exerted on
the cylinder as a consequence of the fluid flow. This result is known as d'Alembert's paradox, after the
French scientist Jean-Baptiste le Rond d'Alembert (1717-1783). D'Alembert's result is paradoxical
because it would seem, at first sight, to be a reasonable approximation to neglect viscosity alltogether
in high Reynolds number flow. However, if we do this then we end up with the nonsensical prediction
that a high Reynolds number fluid is incapable of exerting any force on an obstacle placed in its path.

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UNIT-III
AIRFOIL THEORY
Cauchy-riemann relations, complex potential, methodology of conformal transformation,
kuttajoukowski transformation and its applications, thin airfoil theory and its applications.
Introduction
This chapter discusses the use of complex analysis to simplify calculations in two-dimensional,
incompressible, inviscid, irrotational, fluid dynamics. Incidentally, incompressible, inviscid,
irrotational flow is usually referred to a potential flow, since the velocity field can be represented in
terms of a velocity potential that satisfies Laplace's equation. In the following, all flow patterns are
assumed to be such that the -coordinate is ignorable. In other words, the fluid velocity is everywhere
parallel to the - plane, and . It follows that all line sources and vortex filaments run parallel to
the -axis. Moreover, all solid surfaces are of infinite extent along the -axis, and have uniform
cross-sections. Hence, it is only necessary to specify the locations of line sources, vortex filaments,
and solid surfaces in the - plane
Complex Functions
The complex variable is conventionally written
where represents the square root of minus one. Here, and are both real, and are identified with
the corresponding Cartesian coordinates. (Incidentally, should not be confused with a -
coordinate: this is a strictly two-dimensional discussion.) We can also write
Where and are termed the modulus and argument of , respectively, but
can also be identified with the corresponding plane polar coordinates. Finally, DeMoivre's theorem,
implies that

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We can define functions of the complex variable, , just like we would define functions of a real
variable. For instance,
For a given function, , we can substitute and write
Where and are real two-dimensional functions. Thus, if
then
giving

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Cauchy-Riemann Relations
We can define the derivative of a complex function in just the same manner that we would define the
derivative of a real function: i.e.,
However, we now have a problem. If is a ``well-behaved'' function (i.e., finite, single-valued,
and differentiable) then it should not matter from which direction in the complex plane we approach
the point when taking the limit in Equation There are, of course, many different possible approach
directions, but if we look at a regular complex function, (say), then
is perfectly well-defined, and is, therefore, completely independent of the details of how the limit is
taken in Equation .
The fact that Equation has to give the same result, no matter from which direction we approach ,
means that there are some restrictions on the forms of the functions and in
Equation . Suppose that we approach along the real axis, so that . We obtain
Suppose that we now approach along the imaginary axis, so that . We get

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But, if is a well-behaved function then its derivative must be well-defined, which implies that
the above two expressions are equivalent. This requires that
These expressions are called the Cauchy-Riemann relations, and are, in fact, sufficient to ensure that
all possible ways of taking the limit give the same result.
Complex Velocity Potential
Note that Equations are identical to Equations. This suggests that the real and imaginary parts of a
well-behaved function of the complex variable can be interpreted as the velocity potential and stream
function, respectively, of some two-dimensional, irrotational, incompressible flow pattern. For
instance, suppose that
Where is real. It follows that
It can be seen, by comparison with the analysis of Section, that the complex velocity potential
corresponds to uniform flow of speed directed along the -axis. Furthermore, as is easily
demonstrated, the complex velocity potential associated with uniform flow of speed whose
direction subtends a (counter-clockwise) angle with the -axis is .
Suppose that

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Where is real. Since , it follows that
Thus, according to the analysis of Section, the complex velocity potential corresponds to the flow
pattern of a line source, of strength , located at the origin. (See Figure.) As a simple generalization
of this result, the complex potential of a line source, of strength , located at the point , ,
is , where . Note, from , that the complex velocity potential of
a line source issingular at the location of the source.
Suppose that
where is real. It follows that

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Thus, according to the analysis of Section the complex velocity potential corresponds to the flow
pattern of a vortex filament of intensity located at the origin. (See Figure.) Note, from , that the
complex velocity potential of a vortex filament is singular at the location of the filament.
Suppose, finally, that
Where , , and , are real. It follows that

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Thus, according to the analysis of Section, the complex velocity potential corresponds to uniform
inviscid flow of unperturbed speed , running parallel to the -axis, around an impenetrable
cylinder of radius , centered on the origin. (See Figures) Here, is the circulation of the flow about
the cylinder. Note that on the surface of the cylinder ( ), which ensures that the normal
velocity is zero on this surface, as must be the case if the cylinder is impenetrable.
Complex Velocity
It follows from Equations ,
Consequently, is termed the complex velocity. Note that
Where is the flow speed.
A stagnation point is defined as a point in a flow pattern where the flow speed, , falls to zero. It
follows, from the previous expression, that

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at a stagnation point. For instance, the stagnation points of the flow pattern produced when a
cylindrical obstacle of radius , centered on the origin, is placed in a uniform flow of speed ,
directed parallel to the -axis, and the circulation of the flow around is cylinder is , are found by
setting the derivative of the complex potential to zero. It follows that the stagnation points satisfy the
quadratic equation
The solutions are
where , with the proviso that , since the region is occupied by the
cylinder. Thus, if then there are two stagnation points on the surface of the cylinder at
and . On the other hand, if then there is a single stagnation point below the cylinder at
and .
Now, according to, Bernoulli's theorem in an steady, irrotational, incompressible fluid takes the form
Where is a uniform constant, and where gravity (or any other body force) has been neglected.
Thus, the pressure distribution in such a fluid can be written
Method of Images

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Let and be complex velocity potentials
corresponding to distinct, two-dimensional, irrotational, incompressible flow patterns whose stream
functions are and , respectively. It follows that both stream functions satisfy
Laplace's equation: i.e., . Suppose that . Writing
, it is clear that . Moreover, , so also
satisfies Laplace's equation. We deduce that two complex velocity potentials, corresponding to
distinct, two-dimensional, irrotational, incompressible flow patterns, can be superposed to produce a
third velocity potential that corresponds to another two-dimensional, irrotational, incompressible flow
pattern. As described below, this idea can be exploited to determine the flow patterns produced by line
sources and vortex filaments in the vicinity of rigid boundaries.
Figure : Two line sources.
As an example, consider a situation in which there are two line sources of strength located at the
points . See Figure . The complex velocity potential of the resulting flow pattern is the sum of
the complex potentials of each source taken in isolation. Hence,
Thus, the stream function of the flow pattern (which is the imaginary part of the complex potential) is

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Note that , which implies that there is zero flow normal to the plane . Hence, in the
region , we could interpret the above stream function as that generated by a single line source of
strength , located at the point , , in the presence of a planar rigid boundary at . This
follows because the stream function satisfies everywhere in the region , has the requisite
singularity (corresponding to a line source of strength ) at , , and satisfies the physical
boundary condition that the normal velocity be zero at the rigid boundary. Moreover, as is well-
known, the solutions of Poisson's equation are unique. The streamlines of the resulting flow pattern are
shown in Figure . Incidentally, we can think of the two sources in Figure as the ``images'' of one
another in the boundary plane. Hence, this method of calculation is usually referred to as the method of
images.
Figure: Stream lines of the 2D flow pattern due to a line source at , in the presence of a rigid
boundary at .
Now, the complex velocity associated with the complex velocity potential is

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Hence, the flow speed at the boundary is
It follows from (and the fact that the flow speed at infinity is zero) that the excess pressure on the
boundary, due to the presence of the source, is
Thus, the excess force per unit length (in the -direction) acting on the boundary in the -direction
is
The fact that the force is positive implies that the boundary is attracted to the source, and vice versa.

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Figure: Two vortex filaments.
As a second example, consider the situation, illustrated in Figure , in which there are two vortex
filaments of intensities and situated at , . As before, the complex velocity potential of the
resulting flow pattern is the sum of the complex potentials of each filament taken in isolation. Hence,
Thus, the stream function of the flow pattern is
As before, , which implies that there is zero flow normal to the plane . Hence, in the
region , we could interpret the above stream function as that generated by a single vortex
filament of intensity , located at the point , , in the presence of a planar rigid boundary at
. The streamlines of the resulting flow pattern are shown in Figure . We conclude that a vortex
filament reverses its sense of rotation (i.e., ) when ``reflected'' in a boundary plane.

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Figure: Stream lines of the 2D flow pattern due to a vortex filament at , in the presence of a
rigid boundary at .
As a final example, consider the situation, illustrated in Figure , in which there is an impenetrable
cylinder of radius , centered on the origin, and a line source of strength located at , , where
. Consider the so-called analog problem, also illustrated in Figure , in which the cylinder is replaced by
a source of strength , located at , , where , and a source of
strength , located at the origin. We can think of these two sources as the ``images'' of the external
source in the cylinder. Moreover, given that the solutions of Poisson's equation are unique, if the
analog problem can be adjusted in such a manner that is a streamline then the flow in the region
will become identical to that in the actual problem. Now, the complex velocity potential in the analog
problem is simply
Hence, writing , the corresponding stream function takes the form

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Now, we require the surface of the cylinder, , to be a streamline: i.e., constant. This is
easily achieved by setting . Thus, the stream function becomes
The corresponding streamlines in the region external to the cylinder are shown in Figure .
Complex Line Integrals
Consider the line integral of some function of the complex variable taken (counter-clockwise)
around a closed curve in the complex plane:
Since , and writing , where and are real
functions, it follows that , where
However, we can also write the above expressions in the two-dimensional vector form

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Where , , , , and , . Now, according to Stokes' theorem
Where is the region of the - plane enclosed by . Hence, we obtain
Let
Where is a closed curve in the complex plane that completely surrounds the smaller curve .
Consider

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Writing , a direct generalization of the previous analysis reveals that
Where is now the region of the - plane lying between the curves and . Suppose that
is well-behaved (i.e., finite, single-valued, and differentiable) throughout . It immediately follows
that its real and imaginary components, and , respectively, satisfy the Cauchy-Riemann relations,
throughout . However, if this is the case then it is apparent, from the previous two expressions, that
. In other words, if is well-behaved throughout then .
The circulation of the flow about some closed curve in the - plane is defined
Where is the complex velocity potential of the flow, and use has been made of Equation Thus,
the circulation can be evaluated by performing a line integral in the complex -plane. Moreover, as is
clear from the previous discussion, this integral can be performed around any loop that can be
continuously deformed into the loop whilst still remaining in the fluid, and not passing over a
singularity of the complex velocity,

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Figure: A line source in the presence of an impenetrable cylinder.
Figure: Stream lines of the 2D flow pattern due to a line source at , in the presence of a rigid
cylinder of radius centered on the origin.
Theorem of Blasius
Consider some flow pattern in the complex -plane that is specified by the complex velocity potential
. Let be some closed curve in the complex -plane. The fluid pressure on this curve is determined
from Equation, which yields
Let us evaluate the resultant force (per unit length), and the resultant moment (per unit length), acting
on the fluid within the curve as a consequence of this pressure distribution.

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Figure : Force acting across a short section of a curve.
Consider a small element of the curve , lying between , and , , which is
sufficiently short that it can be approximated as a straight line. Let be the local fluid pressure on the
outer (i.e., exterior to the curve) side of the element. As illustrated in Figure , the pressure force (per
unit length) acting inward (i.e., toward the inside of the curve) across the element has a component
in the minus -direction, and a component in the plus -direction. Thus, if and are the
components of the resultant force (per unit length) in the - and -directions, respectively, then
The pressure force (per unit length) acting across the element also contributes to a moment (per unit
length), , acting about the -axis, where
Thus, the - and -components of the resultant force (per unit length) acting on the of the fluid
within the curve, as well as the resultant moment (per unit length) about the -axis, are given by

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respectively, where the integrals are taken (counter-clockwise) around the curve . Finally, given that
the pressure distribution on the curve takes the , and that a constant pressure obviously yields zero
force and zero moment, we find that
Now, , and , where indicates a complex conjugate. Hence, , and
. It follows that
However,

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Where and . Suppose that the curve corresponds to a streamline
of the flow, in which case on . Thus, on , and so . Hence, on ,
which implies that
This result is known as the Blasius theorem.
Now, . Hence,
or, making use of an analogous argument to that employed above,
Note, finally, that Equations hold even when is not constant on the curve , as long as can be
continuously deformed into a constant - curve without leaving the fluid or crossing over a
singularity of .

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Figure 47: Source in the presence of a rigid boundary.
As an example of the use of the Blasius theorem, consider again the situation, in which a line source of
strength is located at , , and there is a rigid boundary at . As we have seen, the
complex velocity in the region takes the form
Suppose that we evaluate the Blasius integral, about the contour shown in Figure. This contour runs
along the boundary, and is completed by a semi-circle in the upper half of the -plane. As is easily
demonstrated, in the limit in which the radius of the semi-circle tends to infinity, the contribution of
the curved section of the contour to the overall integral becomes negligible. In this case, only the
straight section of the contour contributes to the integral. Note that the straight section corresponds to
a streamline (since it is coincident with a rigid boundary). In other words, the contour corresponds
to a streamline at all constituent points that make a finite contribution to the Blasius integral, which
ensures that is a valid contour for the application of the Blasius theorem. In fact, the Blasius
integral specifies the net force (per unit length) exerted on the whole fluid by the boundary. Note,
however, that the contour can be deformed into the contour , which takes the form of a small circle
surrounding the source, without passing over a singularity of . See Figure . Hence, we can
evaluate the Blasius integral around without changing its value. Thus,

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or
Writing , , and taking the limit , we find that
In other words, the boundary exerts a force (per unit length) on the fluid. Hence,
the fluid exerts an equal and opposite force on the boundary. Of course, this
result is consistent with Equation . Incidentally, it is easily demonstrated from that there is zero
moment (about the -axis) exerted on the boundary by the fluid, and vice versa.
Consider a line source of strength placed (at the origin) in a uniformly flowing fluid whose
velocity is , . From the complex velocity potential of the net flow is
The net force (per unit length) acting on the source (which is calculated by performing the Blasius
integral around a large loop that follows streamlines, and then shrinking the loop to a small circle
centered on the source)
Note that the force acts in the opposite direction to the flow. Thus, an external force , acting in the
same direction as the flow, must be applied to the source in order for it to remain stationary. In fact,
the above result is valid even in a non-uniformly flowing fluid, as long as is interpreted as the fluid
velocity at the location of the source (excluding the velocity field of the source itself).
Finally, consider a vortex filament of intensity placed at the origin in a uniformly flowing fluid
whose velocity is , . From, the complex velocity potential of the net flow is

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The net force (per unit length) acting on the filament (which is calculated by performing the Blasius
integral around a small circle centered on the filament)
Note that the force is directed at right-angles to the direction of the flow (in the sense obtained by
rotating through in the opposite direction to the filament's direction of rotation). Again, the
above result is valid even in a non-uniformly flowing fluid, as long as is interpreted as the fluid
velocity at the location of the filament (excluding the velocity field of the filament itself).
Joukowsky transform
In applied mathematics, the Joukowsky transform, named after Nikolai Zhukovsky is a conformal
map historically used to understand some principles of airfoil design.
The transform is
where is a complex variable in the new space and is a complex variable
in the original space. This transform is also called the Joukowsky transformation, the Joukowski
transform, the Zhukovsky transform and other variations.
In aerodynamics, the transform is used to solve for the two-dimensional potential flow around a class
of airfoils known as Joukowsky airfoils. A Joukowsky airfoil is generated in the z plane by applying
the Joukowsky transform to a circle in the plane. The coordinates of the centre of the circle are
variables, and varying them modifies the shape of the resulting airfoil. The circle encloses the
point = −1 (where the derivative is zero) and intersects the point = 1. This can be achieved for
any allowable centre position by varying the radius of the circle.
Joukowsky airfoils have a cusp at their trailing edge. A closely related conformal mapping,
the Kármán–Trefftz transform, generates the much broader class of Kármán–Trefftz airfoils by
controlling the trailing edge angle. When a trailing edge angle of zero is specified, the Kármán–Trefftz
transform reduces to the Joukowsky transform.

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General Joukowsky transform
The Joukowsky transform of any complex number to is as follows
So the real (x) and imaginary (y) components are:

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Sample Joukowsky airfoil
The transformation of all complex numbers on the unit circle is a special case.
So the real component becomes and the imaginary component
becomes
Thus the complex unit circle maps to a flat plate on the real number line from −2 to +2.
Transformation from other circles make a wide range of airfoil shapes.
Velocity field and circulation for the Joukowsky airfoil[edit]
The solution to potential flow around a circular cylinder is analytic and well known. It is the
superposition of uniform flow, a doublet, and a vortex.
The complex velocity around the circle in the plane is
where
 is the complex coordinate of the centre of the circle
 is the freestream velocity of the fluid
 is the angle of attack of the airfoil with respect to the freestream flow
 R is the radius of the circle, calculated using
 is the circulation, found using the Kutta condition, which reduces in this case to

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The complex velocity W around the airfoil in the z plane is, according to the rules of conformal
mapping and using the Joukowsky transformation:
Here with and the velocity components in the and directions,
respectively ( with and real-valued). From this velocity, other properties of interest
of the flow, such as the coefficient of pressure or lift can be calculated.
A Joukowsky airfoil has a cusp at the trailing edge.
The transformation is named after Russian scientist Nikolai Zhukovsky. His name has historically
been romanized in a number of ways, thus the variation in spelling of the transform.
Kármán–Trefftz transform
Example of a Kármán–Trefftz transform. The circle above in the ς-plane is transformed into the
Kármán–Trefftz airfoil below, in the z-plane. The parameters used are: μx = –0.08, μy = +0.08
and n = 1.94. Note that the airfoil in the z-plane has been normalised using the chord length.
The Kármán–Trefftz transform is a conformal map closely related to the Joukowsky transform.
While a Joukowsky airfoil has a cusped trailing edge, a Kármán–Trefftz airfoil—which is the result
of the transform of a circle in the ς-plane to the physicalz-plane, analogue to the definition of the
Joukowsky airfoil—has a non-zero angle at the trailing edge, between the upper and lower airfoil

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surface. The Kármán–Trefftz transform therefore requires an additional parameter: the trailing-edge
angle α. This transform is equal to
(A)
with n slightly smaller than 2. The angle α, between the tangents of the upper and lower airfoil surface,
at the trailing edge is related to n by:
The derivative , required to compute the velocity field, is equal to:
Background
First, add and subtract two from the Joukowsky transform, as given above:
Dividing the left and right hand sides gives:
The right hand side contains (as a factor) the simple second-power law from potential flow theory,
applied at the trailing edge near From conformal mapping theory this quadratic map is
known to change a half plane in the -space into potential flow around a semi-infinite straight line.
Further, values of the power less than two will result in flow around a finite angle. So, by changing the
power in the Joukowsky transform—to a value slightly less than two—the result is a finite angle
instead of a cusp. Replacing 2 by n in the previous equation gives:[1]

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which is the Kármán–Trefftz transform. Solving for z gives it in the form of equation (A).
Conformal mapping is a method used to extend the application of potential flow theory to practical
aerodynamics. Standard potential flow theory begins with an ideal flow to show that lift on a body is
proportional to the circulation about a closed path encompassing an object. Potential flows start with
flows over cylinders since the mathematics is more tractable. However, to use potential flow theory on
usable airfoils one must rely on conformal mapping to show a relation between realistic airfoil shapes
and the knowledge gained from flow about cylinders.
Brief review of complex numbers:
Conformal mapping relies entirely on complex mathematics. Therefore, a brief review is undertaken at
this point.
A complex number z is a sum of a real and imaginary part; z = real + iimaginary
The term i, refers to the complex number
so that;
Complex numbers can be presented in a graphical format. If the real portion of a complex number is
taken as the abscissa, and the imaginary portion as the ordinate, a two-dimensional plane is formed.
L =  V 
Lift = ?
1i
1,,1,1
432
 iiiii

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z = real + iimaginary = x + iy
A complex number can be written in polar form using Euler's equation;
z = x + iy = rei
= r(cos + isin)
where r2
= x2
+ y2
Complex multiplication: z1z2 = (x1+iy1)(x2+iy2) = (x1x2 - y1y2) + i(x1y2 + y1x2)
Complex representation of potential flows
The basic flows used in potential flow theory such as uniform flow, source, sink, doublet and vortex,
can all be represented using complex numbers. For example, if a complex number w with both real
and imaginary parts represents a potential flow, then the form of the number is;
w(z) =  + i = (velocity potential) + i(stream function)
Here, both velocity potential and stream function are themselves complex numbers. As an example,
the uniform flow can be written;
Uniform flow: w(z) = Vz =  + i = V(x+iy) = Vx + Vy
as seen previously,  = Vx = V rcos  = Vy = V rsin
Source flow:
y, imaginary
x, real
)(
2121
2121  

iii
errerer







2
)ln(
2
))(ln(
2
)ln(
2
)ln(
2









 irirreizw
i

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Vortex flow:
Doublet flow:
In complex terms the flow past a cylinder with lift is written:
Velocity Components:
When a potential flow is represented in complex form, the velocity components can be found using
one of two methods;
1. Re-write the expression from the complex variable z form into its separate real and complex
components. The form of this expression will be w =  + i. The individual velocity
components are found by completing the appropriate differentiation on  or . to obtain u or
v. As an example consider the complex form of the source flow;
0
2
2
)ln(
2
)ln(
2
















V
rr
V
irizw
r
2. An alternative method would be to differentiate on the complex expression directly and then
separate the real and complex portions to obtain the velocity components according to;
ivu
dz
dw 
Conformal Mapping
)ln(
22
))(ln(
2
)ln(
2
)ln(
2
riirireiiziw
i







 









)sin(cos
1
2
1
2
1
2
1
2






i
r
k
e
r
k
re
k
i
z
k
w
i
i







)ln(
2
)(
2
zi
z
R
zVzw








 

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A conformal mapping is performed through the transformation of a complex function from one
coordinate system to another. A transformation function is applied to the original function to perform
the mapping. For aerodynamics applications the Joukowski transform is the most commonly used
function;
Here, b is a constant. Graphically, a conformal mapping will transform a complex plane in z (z = x+iy)
into a complex plane in a new variable w (w = +i).
In the diagram a uniform flow in the z plane is transformed into an equivalent form in the w plane
using a transform of the form w = f(z). As an example consider a circle drawn in the z plane, z = bei
.
The Joukowski transform maps the circle into a flat plate,
z
bzw
2

0)cos(2
2
ibbebe
be
bbew ii
i
i
 



 = C1
 = C2
 = C3
 = K1  = K3


z plane
x
y
w plane
 = C3
 = C2
 = K3
z
bzw
2


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A circle of radius b is mapped into a straight line in the w plane entirely on the real axis between -2b
and 2b. If a uniform flow had been drawn over the circle, the transform would have mapped that flow
into the flow over a flat plate in the w plane. If the circle originally had a radius slightly larger than the
transform constant b, z = aei
, with a > b, the circle would have formed an ellipse instead of the flat
plate.
iyx
a
bai
a
ba
ae
bae
z
bzw i
i












 )sin()cos(
2222


Which can be written,
12
2
2
2
2
2
















a
ba
y
a
ba
x
If the flow over a cylinder had been transformed it would have created the flow over an ellipse.
Joukowski Airfoils
x
z
bzw
2

y
b



z plane w plane
2b-2b
x
y
2b-2b

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From and aerodynamics point of view, the most interesting application of the Joukowski transform is
to an offset circle. If we consider a circle slightly offset from the origin along the negative real axis,
one obtains a symmetric Joukowski airfoil.
The equation of the offset circle is z = aei
-eb where the constant e is a small number. If the cylinder is
displaced slightly along the complex axis as well, one obtains a cambered airfoil shape.
Here, the points A and B are the intercepts of the displaced circle on the real axis and their
corresponding points in the transformed plane. The angle  is the angle formed by the line joining the
point A (or B) and the origin with the real axis. If lifting flow about the original circle had been
imposed, the Joukowski transformation would have generated a lifting flow about the Joukowski
airfoil;
w plane
x
y


z plane
z
bzw
2

b
a
eb
-2b 2bx
y


z plane w plane
z
bzw
2

b
a
eb -2b 2bx
y


z plane w plane
z
bzw
2

A

B BA

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Although such a flow is mathematically possible, in reality it may not be realistic. The stagnation
points on the cylinder map to stagnation points that are not always realistic. For instance the stagnation
point on the top surface of the airfoil cannot exist is steady flight since the velocity would tend to
infinity as one moves very close to the trialing edge. The only means of making a realistic flow is to
impose the Kutta condition where the stagnation point is forced to exist at the trailing edge thus
making the streamlines flow smoothly from this point. This is done by adjusting the value of vorticity
strength , such that the stagnation points on the cylinder reside at the cylinder’s intercepts of the real
axis. In this case, when the cylinder is transformed, one stagnation point will be forced to the trailing
edge.
The lift force generated by the lifting flow over the cylinder is proportional to the circulation about the
cylinder imposed by the added vortex flow according to the Kutta-Joukowski relation, L’ = V .
The lifting force on the resulting Joukowski airfoil is not clear. To evaluate the lift, the circulation is
needed and therefore the velocity field. The velocity fields in each plane can be related to each other
through the chain rule of differentiation. If the lifting flow about the cylinder is defined as function Q
where Q = Q(z) in the z plane and Q = Q(w) in the w plane, the velocities in each plane are;
w
Q
V
z
Q
V wz






By chain rule:
z
w
w
Q
z
Q







z
wVV wz


Using the Joukowski transformation;
w plane
x
y


z plane
z
bzw
2


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2
22
z
bz
z
w 


Clearly, the velocity field very close to the cylinder and its transformed counterpart are dissimilar as
one would expect. However, farther away from these objects the velocity fields become identical as
the magnitude of z becomes larger than the constant value of b. Since the circulation can be calculated
about any closed path, including paths very far from the object surface, the circulations must be the
same in both planes.
Joukowskicylinder VV   
Vortex strength
The appropriate vortex strength to impose the Kutta condition must be determined. Consider the lifting
flow about a cylinder. The velocity in the  direction is,





  
R
VV


2
)sin(2
Here, R is the radius of the cylinder surface.
This velocity is zero on the surface of the cylinder at the
stagnation points. At the these points  =-.
R
V


2
)sin(20  
)sin(4  RV
If the field is rotated by  to simulate an angle of attack,
)sin(4   RV
Since the cord length of the Joukowski airfoil is 4b, the lift coefficient can be written,
bV
RV
bVbV
V
cV
LCL 2
2
222 2
)sin(4
24
2
1
2
1 












Making the assumption that b  R,
)(2)sin(2  LC
Thin airfoil theory
x
y
z plane
-

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Thin airfoil theory is a simple theory of airfoils that relates angle of attack to lift for incompressible,
inviscid flows. It was devised by German-American mathematician Max Munk and further refined by
British aerodynamicist Hermann Glauert and others in the 1920s. The theory idealizes the flow around
an airfoil as two-dimensional flow around a thin airfoil. It can be imagined as addressing an airfoil of
zero thickness and infinite wingspan.
Thin airfoil theory was particularly notable in its day because it provided a sound theoretical basis for
the following important properties of airfoils in two-dimensional flow:
(1) on a symmetric airfoil, the center of pressure and aerodynamic center lies exactly one quarter of
the chord behind the leading edge
(2) on a cambered airfoil, the aerodynamic center lies exactly one quarter of the chord behind the
leading edge
(3) the slope of the lift coefficient versus angle of attack line is units per radian
As a consequence of (3), the section lift coefficient of a symmetric airfoil of infinite wingspan is:
where is the section lift coefficient,
is the angle of attack in radians, measured relative to the chord line.
(The above expression is also applicable to a cambered airfoil where is the angle of attack measured
relative to the zero-lift line instead of the chord line.)
Also as a consequence of (3), the section lift coefficient of a cambered airfoil of infinite wingspan is:
where is the section lift coefficient when the angle of attack is zero.
Thin airfoil theory does not account for the stall of the airfoil, which usually occurs at an angle of
attack between 10° and 15° for typical airfoils.
Derivation of thin airfoil theory

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The airfoil is modeled as a thin lifting mean-line (camber line). The mean-line, y(x), is considered to
produce a distribution of vorticity along the line, s. By the Kutta condition, the vorticity is zero
at the trailing edge. Since the airfoil is thin, x (chord position) can be used instead of s, and all angles
can be approximated as small.
From the Biot–Savart law, this vorticity produces a flow field where
where is the location where induced velocity is produced, is the location of the vortex element
producing the velocity and is the chord length of the airfoil.
Since there is no flow normal to the curved surface of the airfoil, balances that from the
component of main flow , which is locally normal to the plate – the main flow is locally inclined to
the plate by an angle . That is:
This integral equation can by solved for , after replacing x by

SRS/MKS
,
as a Fourier series in with a modified lead term
That is
(These terms are known as the Glauert integral).
The coefficients are given by
and
By the Kutta–Joukowski theorem, the total lift force F is proportional to
and its moment M about the leading edge to
The calculated Lift coefficient depends only on the first two terms of the Fourier series, as
The moment M about the leading edge depends only on and , as
The moment about the 1/4 chord point will thus be,
.
From this it follows that the center of pressure is aft of the 'quarter-chord' point 0.25 c, by

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The aerodynamic center, AC, is at the quarter-chord point. The AC is where the pitching moment M'
does not vary with angle of attack, i.e.,
Thin Airfoil Theory Applications
The basic equations derived from thin airfoil theory are repeated below:
Several important results are derived from these expressions and are described in the following
sections.
Ideal angle of attack
The constants An just depend on the airfoil shape -- except for A0 which depends also on the angle of
attack. The A0 term is a strange term in the Fourier series for g since it leads to a singularity at the
leading edge (g -> infinity). Thus, there is one angle of attack, called the ideal angle of attack, at which
A0 = 0 and the vorticity goes to 0 at the leading edge. This angle is called the ideal angle of attack.
Of course in real flows, the vorticity would not become infinite (why?), but the concept of ideal angle
of attack is still important, identifying the flow conditions for which leading edge pressure peaks are
avoided.
Lift curve slope
The rate of change of lift coefficient with angle of attack, dCL/da can be inferred from the expressions
above.

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The result, that CL changes by 2p per radian change of angle of attack (.1096/deg) is not far from the
measured slope for many airfoils. The effects of thickness and viscosity which are ignored here cancel
each other out to some extent with the result that most airfoils have a lift curve slope within 10% of
the 2p value given by thin airfoil theory.
Aerodynamic center and pitching moment at L = 0
The expression for pitching moment coefficient measured about the leading edge is given above. If we
measure the moment about another reference center at a position x0/c, the expression becomes:
Note that if we choose the point x0 = 0.25c, then the lift dependence drops out and the moment
coefficient measured about this point is independent of the angle of attack*. The point about which
dCm/dCL = 0 is called the aerodynamic center and according to thin airfoil theory it is the quarter
chord point of the airfoil. Experiments show this to be quite close.
UNIT-IV
SUBSONIC WING THEORY
Vortex filament, biot and savart law, bound vortex and trailing vortex, horse shoe vortex, lifting line
theory and its limitations.
Vortex filament
A 2-D vortex, which we have examined previously, can be considered as a 3-D vortex which is
straight and extending to ±∞. Its velocity field is Vθ = Γ /2πr Vr = 0 Vz = 0 (2-D vortex) In contrast, a
general 3-D vortex can take any arbitrary shape. However, it is subject to the Helmholtz Vortex
Theorems: 1) The strength Γ of the vortex is constant all along its length 2) The vortex cannot end
inside the fluid. It must either a) extend to ±∞, or b) end at a solid boundary, or c) form a closed loop.
Proofs of these theorems are beyond scope here. However, they are easy to apply in flow modeling
situations.
Biot and savart law
The Biot–Savart law is also used in aerodynamic theory to calculate the velocity induced by vortex lines.

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In the aerodynamic application, the roles of vorticity and current are reversed in comparison to the
magnetic application.
In Maxwell's 1861 paper 'On Physical Lines of Force',[7]
magnetic field strength H was directly
equated with pure vorticity (spin), whereas Bwas a weighted vorticity that was weighted for the density
of the vortex sea. Maxwell considered magnetic permeability μ to be a measure of the density of the
vortex sea. Hence the relationship,
1. Magnetic induction current
was essentially a rotational analogy to the linear electric current relationship,
2. Electric convection current
where ρ is electric charge density. B was seen as a kind of magnetic current of vortices aligned in their
axial planes, with H being the circumferential velocity of the vortices.
The electric current equation can be viewed as a convective current of electric charge that involves
linear motion. By analogy, the magnetic equation is an inductive current involving spin. There is no

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linear motion in the inductive current along the direction of the B vector. The magnetic inductive
current represents lines of force. In particular, it represents lines of inverse square law force.
In aerodynamics the induced air currents are forming solenoidal rings around a vortex axis that is
playing the role that electric current plays in magnetism. This puts the air currents of aerodynamics
into the equivalent role of the magnetic induction vector B in electromagnetism.
In electromagnetism the B lines form solenoidal rings around the source electric current, whereas in
aerodynamics, the air currents form solenoidal rings around the source vortex axis.
Hence in electromagnetism, the vortex plays the role of 'effect' whereas in aerodynamics, the vortex
plays the role of 'cause'. Yet when we look at the B lines in isolation, we see exactly the aerodynamic
scenario in so much as that B is the vortex axis and H is the circumferential velocity as in Maxwell's
1861 paper.
In two dimensions, for a vortex line of infinite length, the induced velocity at a point is given by
where Γ is the strength of the vortex and r is the perpendicular distance between the point and the
vortex line.
This is a limiting case of the formula for vortex segments of finite length:
where A and B are the (signed) angles between the line and the two ends of the segment.
Starting vortex
The starting vortex is a vortex which forms in the air adjacent to the trailing edge of an airfoil as it is
accelerated from rest in a fluid.[1]
It leaves the airfoil (which now has an equal but opposite "bound
vortex" around it), and remains (nearly) stationary in the flow.[2][3][4]
It rapidly decays through the
action of viscosity.

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The initial (and quite brief) presence of a starting vortex as an airfoil begins to move was predicted by
early aerodynamicists, and eventually photographed.[5][6][7]
Whenever the speed or angle of attack of an airfoil changes there is a corresponding amount of vorticity
deposited in the wake behind the airfoil, joining the two trailing vortices. This vorticity is a continuum of
mini-starting-vortexes. The wake behind an aircraft is a continuous sheet of weak vorticity, between
the two trailing vortices, and this accounts for the changes in strength of the trailing vortices as the
airspeed of the aircraft and angle of attack on the wing change during flight.[8]
(The strength of
a vortex cannot change within the fluid except by the dissipative action of viscosity. Vortices either form
continuous loops of constant strength, or they terminate at the boundary of the fluid - usually a solid
surface such as the ground.)
The starting vortex is significant to an understanding of the Kutta condition and its role in
the circulation around any airfoil generating lift.
The starting vortex has certain similarities with the "starting plume" which forms at the leading edge
of a slug of fluid, when one fluid is injected into another at rest. See plume (hydrodynamics).
Bound Vortex
a vortex that is considered to be tightly associated with the body around which a liquid or gas flows, a
nd equivalent with respect to themagnitude of speed circulation to the real vorticity that forms in the b
oundary layer owing to viscosity.
In calculations of the lift of a wing of infinite span, the wing can be replaced by a bound vortex that h
as a rectilinear axis and generates in thesurrounding medium the same circulation as that generated by
the real wing. In the case of a wing of finite span, the bound vortex continuesinto the surrounding med
ium in the form of free vortices. Knowledge of the vortex system of a wing permits calculation of the a
erodynamicforces acting upon the wing. In particular, the interaction between bound and free vortices
gives rise to the induced drag of the wing. The ideaof the bound vortex was made use of by N. E. Zhuk
ovskii in the theory of the wing and the screw propeller.
Generation of trailing vortices

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Euler computation of a tip vortex rolling up from the trailed vorticity sheet.
When a wing generates aerodynamic lift the air on the top surface has lower pressure relative to the
bottom surface. Air flows from below the wing and out around the tip to the top of the wing in a
circular fashion. An emergent circulatory flow pattern named vortex is observed, featuring a low-
pressure core.
Three-dimensional lift and the occurrence of wingtip vortices can be approached with the concept
of horseshoe vortex and described accurately with the Lanchester–Prandtl theory. In this view, the trailing
vortex is a continuation of the wing-bound vortex inherent to the lift generation.
If viewed from the tail of the airplane, looking forward in the direction of flight, there is one wingtip
vortex trailing from the left-hand wing and circulating clockwise, and another one trailing from the
right-hand wing and circulating anti-clockwise. The result is a region of downwash behind the aircraft,
between the two vortices.
The two wingtip vortices do not merge because they are circulating in opposite directions. They
dissipate slowly and linger in the atmosphere long after the airplane has passed. They are a hazard to
other aircraft, known as wake turbulence.
Wingtip vortices
Wingtip vortices are circular patterns of rotating air left behind a wing as it generates lift. One
wingtip vortex trails from the tip of each wing. Wingtip vortices are sometimes named trailing or lift-

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