Drag and Lift Force
INTRODUCTION
Fluid flow over solid bodies frequently
occurs in practice, and it is responsible for numerous physical phenomena. When
a fluid flows over a stationary body, a force is exerted by the fluid on the
body. Similarly, when a body is moving in a stationary fluid, a force is
exerted by the fluid on the body. And these are referred to as external flow. Some
of the examples of the fluids flowing over stationary bodies or bodies moving
in a stationary fluid are:
1.flow of air over buildings
2.the drag force acting on automobiles,
power lines, trees, and underwater pipelines
3.the lift developed by airplane wings
4.flow of water over bridges
Force Exerted by a flowing fluid on a
stationary body
A body meets some resistance when it is
forced to move through a fluid, especially a liquid. This Fluid will exert a
force on the body.
This figure shows Force on a
stationary body.
The total force (Fr) exerted by the fluid on the body is perpendicular
to the surface of the body. Thus, the total force is inclined to the direction
of motion. The force can be resolved in two components, one in the direction of
motion and the other perpendiculars to the direction of motion.
1.
DRAG: The force
exerted by the fluid in direction of motion is known as a Drag force. In this
figure (Fd) represents Drag force. Drag is usually an undesirable effect, like friction,
and we do our best to minimize it. But in some case drag produces a very
beneficial effect and we try to maximize it (e.g., automobile brakes)
2.
LIFT: The component of the total force in
a direction perpendicular to the direction of motion is Lift. It is represented
by (Fl). Lift only occurs when the axis of the body is inclined to the
direction of fluid flow. If the axis of the body is parallel to the direction
of fluid flow, the lift force is zero
Drag and lift
expressions:
Lift is created by the flow of air over an airfoil. The shape of an airfoil causes air to flow faster on top than on bottom. The fast flowing air decreases the surrounding air pressure. Because the air pressure is greater below the airfoil than above, a resulting lift force is created. To further understand how an airfoil creates lift, it is necessary to use two important equations of physical science. The pressure variations of flowing air is best represented by Bernoulli's equation.
To
understand this equation, one must first understand another important principle
of physical science, the continuity equation. It simply states that in any
given flow, the density (ρ) times the cross-sectional area (A) of the flow,
times the velocity (V) is constant. The continuity equation is written as:
ρAV = Constant
The shape of a typical airfoil is
asymmetrical - its surface area is greater on the top than on the bottom. As
the air flows over the airfoil, it is displaced more by the top surface than
the bottom. According to the continuity law, this displacement, or loss of flow
area, must lead to an increase in velocity. Consider an airfoil in a pipe with
flowing water. Water will flow faster in a narrow section of the pipe. The
large area of the top surface of the airfoil narrows the pipe more than the
bottom surface does. Thus, water will flow faster on top than on bottom. The
flow velocity is increased some by the bottom airfoil surface, but considerably
less than the flow on top. The Bernoulli equation states that an increase in
velocity leads to decrease in pressure. Thus higher the velocity of the flow,
lower the pressure. Air flowing over an airfoil will decrease in pressure. The
pressure loss over the top surface is greater than that of the bottom surface.
The result is a net pressure force in the upward (positive) direction. This
pressure force is lift. There is no predetermined shape for a wing airfoil,
it is designed based on the function of the aircraft it will be used for. To
aid the design process, engineers use the lift coefficient to measure the
amount of lift obtained from a particular airfoil shape. Lift is proportional
to dynamic pressure and wing area. The lift equation is written as:
Like
lift, drag is proportional to dynamic pressure and the area on which it acts.
The drag coefficient, analogous to the lift coefficient, is a measure of the
amount of dynamic pressure gets converted into drag. Unlike the lift
coefficient however, engineers usually design the drag coefficient to be as low
as possible. Low drag coefficients are desirable because an aircraft's
efficiency increases as drag decreases.
Frictional drag and pressure drag:
A body
moving through fluid experiences a drag force, which is usually divided into
two components: frictional drag and pressure drag. Frictional drag comes from
friction between the fluid and the surfaces over which it is flowing. This
friction is associated with the development of boundary layers and it scales
with the Reynolds number. Pressure drag comes from the eddying motions that are set
up in the fluid by the passage of the body. This drag is associated with the
formation of awake and it is usually less sensitive to Reynolds number than
frictional drag.
Drag on spheres:
Stokes obtained the solution for the pressure and velocity field for the slow motion of a viscous fluid past a sphere. In his analysis, Stokes neglected the inertia terms of Navier-Stokes equations. Avoiding details, integrating the pressure distribution and the shearing stress over the the surface of a sphere of radius R, Stokes found that the drag D of the sphere, which is placed in a parallel stream of uniform velocity is given by
This is Stokes' equation for the drag of a sphere. It can be shown
that one-third of the total drag is due to pressure distribution and the
remaining two-third arises from frictional forces. If the drag coefficient is
defined according to the relation.
where 
is the frontal area of the sphere,
then
or 
Drag of a sphere:
The aerodynamic drag on an object depends on several factors,
including the shape, size, inclination, and flow conditions. All of these factors
are related to the value of the drag through the drag equation.
D
= Cd * .5 * rho * V^2 * A
Where D is equal to the drag, rho is the air density, V is the
velocity, A is a reference area, and Cd is the drag coefficient.
A drag coefficient is a dimensionless number that characterizes
all of the complex factors that affect drag. The drag coefficient is usually
determined experimentally using a model in a wind tunnel. In the tunnel, the
velocity, density, and size of the model are known. Measuring the drag then
determines the value of the drag coefficient as given by the above equation.
The drag coefficient and the drag equation can then be used to determine the
drag on a similarly shaped object at different flow conditions as long as several
flow similarity parameters are matched. In particular, Mach number similarity
ensures that the compressibility effects are correctly modeled, and Reynolds
numbers similarity ensures that the viscous effect is correctly modeled. The
Reynolds number is the ratio of the inertia forces to the viscous forces and is
given by:
Re
= V * rho * l / mu
where l is a reference length, and mu is the viscosity
coefficient. For most aerodynamic objects, the drag coefficient has a nearly
constant value across a large range of Reynolds numbers.
But for a simple sphere, the value of the drag coefficient varies
widely with the Reynolds number. To understand these variations, we are going to
look in some detail at the flow past a cylinder. The two-dimensional flow past
a cylinder is very similar to the three-dimensional flow past a sphere but is
a little easier to compute and understand because of the reduced
dimensionality.
In all of the cases presented on this figure, the density,
viscosity, and diameter of the ball are the same. The flow velocity is
gradually increased from the left to increase the Reynolds number.
1. Case 1 shows very slow flow in which we have neglected
viscosity entirely. We have an ideal flow with no boundary level along the
surface, completely attached flow and no viscous wake downstream of the
cylinder. Because the flow is symmetric from upstream to downstream.Neglecting
viscosity simplifies the analysis, but this type of flow does not occur in
nature where there is always some small amount of viscosity present in any
fluid.
2. Case 2 illustrates what actually occurs for low velocity. A
stable pair of vortices are formed on the downwind side of the cylinder. The
flow is separated but steady and the vortices generate a high drag on the
cylinder or sphere.
3.Case 3 shows the flow as velocity is increased. The downstream
vortices become unstable, separate from the body, and are alternately shed
downstream. The wake is very wide and generates a large amount of drag. The
alternate shedding is called the Karman vortex street. This type of flow is
periodic, it is unsteady but repeats itself at some time interval. The pressure
variation associated with the velocity changes produces a sound called an aeolian
tone. This is the sound you hear when the wind blows over high-power wires or
past tree limbs in the fall or winter. It is a low frequency, haunting tone.
4.Case 4 shows the flow as the velocity is increased even more.
The periodic flow breaks down into a chaotic wake. The flow in the boundary
layer on the windward side of the cylinder is laminar and orderly and the
chaotic wake is initiated as the flow turns onto the leeward side of the
cylinder. The wake is not quite as wide as for Case 3, so the drag is slightly
less.
5. Case 5 shows the flow at even higher velocity. The boundary
layer transitions to chaotic turbulent flow with vortices of many different
scales being shed in a turbulent wake from the body. The separation point is
initially slightly downstream from the laminar separation point, so the wake is
initially slightly smaller and the drag is less than the corresponding laminar
drag. Increasing velocity eventually brings the turbulent drag up to and even
higher than the laminar drag value, but there is a range of Reynolds numbers,
during the transition from laminar to full turbulent, for which the turbulent drag
is less than the laminar drag.
But since drag depends on the flow in the boundary layer, we can
expect some changes with surface roughness. It is observed experimentally that a
roughened cylinder or ball will transition to turbulent flow at a lower
Reynolds number than a smooth cylinder or ball. The size and speed of a golf
ball fall within this Reynolds number range. That is why a golf ball has
dimples; the roughened surface causes a transition to turbulence that would not
occur yet on a smooth golf ball. The lower drag on the dimpled golf ball allows
the ball to fly farther than a smooth ball of the same speed, diameter, and
weight.
Terminal velocity:
An object which is falling through the atmosphere is subjected to
two external forces. One force is the gravitational force, expressed as the
weight of the object. The other force is the air resistance or drag of the
object. If the mass of an object remains constant, the motion of the object can
be described by Newton's second law of motion, force F equals mass m times
acceleration a:
F = m * a
which can be solved for the acceleration of the object in terms of
the net external force and the mass of the object:
a = F / m
Weight and drag are forces which
are vector quantities. The net external force F is then equal to the difference of the weight W and the drag D
F = W - D
The acceleration of a falling object then becomes:
a = (W - D) /
m
The magnitude of the drag is given by the drag equation.
Drag D depends on a drag coefficient
Cd, the atmospheric density r, the square of the air velocity V, and some reference area A of the object.
D = Cd * r * V
^2 * A / 2
Drag increases with the square of the speed. So as an object
falls, we quickly reach conditions where the drag becomes equal to the weight if the weight is small. When drag is equal to the weight, there is no net external
force on the object and the vertical acceleration goes to zero. With no
acceleration, the object falls at a constant velocity as described by Newton's
first law of motion. The constant vertical velocity is called the terminal
velocity.
D
= W
Cd * r * V ^2
* A / 2 = W
V = sqrt ( (2
* W) / (Cd * r * A)
By this we can find the terminal velocity of
object.
DEVELOPMENT
OF LIFT ON A CIRCULAR CYLINDER
When a body
is placed in a fluid in such a way that its access is parallel to the direction
of fluid flow and the body is symmetrical, the resultant force acting on the
body will be in the direction of the flow. There is no force component on the
body perpendicular to the direction of flow. But as we know that the component
of force that is perpendicular to the direction of flow is known as lift. But in
this case lift will be 0.
Therefore,
lift will be acting on the body when the axis of the symmetrical body is
inclined to the direction of the flow or when the body is unsymmetrical. In
this case of circular cylinder, the body is symmetrical and the axis is
parallel to the direction of the flow when cylinder is stationary and hence the
lift will be ZERO. But if the cylinder is rotated, the axis of the cylinder is
no longer parallel to the direction of flow and hence there will be lift acting
on the rotating cylinder.
·
Flow of ideal fluid over a stationary cylinder:
Now if we consider the flow of an
ideal fluid over a stationary cylinder.
As you can
see from the figure above that the velocity distribution over the upper half
and the lower half of the cylinder from the axis AB of the cylinder are
identical therefore the pressure distributions will also be same. Hence the
lift that will be acting on the cylinder will be 0.
·
Flow Pattern around the cylinder when a constant circulation is imparted
to the cylinder:
Circulation is defined as the flow along a closed curve. Circulation is
obtained if the product of the velocity component along the curve at any point
and the length of the small element containing that point is integrated around
the curve.
Circulation for the flow field in a free-vortex :
The figure below shows the flow pattern for a free vertex flow
consists of streamlines which are a series of concentric circles.
In case of free vortex flow, the stream velocity at any point on a circle of radius is equal to the tangential velocity or at that point please stop this means that angle between the streamlines and tangent on the stream is 0.
· Flow over Cylinder due to Constant Circulation:
This figure represents the flow pattern over a rotating cylinder.
For the upper half portion of the cylinder, θ varies from 0 ° to 180°. But for the lower half portion of this cylinder, θ varies from 180
° to 360 °.This means that the velocity on the
upper half portion will be more than the lower half portion of the cylinder. But
from Bernoulli’s theorem, we know that at a surface where velocity is less,
pressure will be more over there. So due to the pressure difference between the
two portions of cylinder, a force will be acting on the cylinder in a direction
perpendicular to the direction of flow. This force is known as lift force.
·
Drag Force acting on rotating cylinder:
From the figure above, we can see that the resultant flow pattern is symmetrical about the vertical axis of the cylinder. Therefore, the velocity distribution and pressure distribution is symmetrical about vertical axis and hence there will be no drag on the cylinder.
·
Magnus Effect:
When a cylinder is rotated in a uniform flow, a lift force is produced on
the cylinder. This phenomenon of the lift force produced by rotating cylinder
in uniform flow is known as Magnus Effect. This fact was investigated by German
Physicist H.G. Magnus. Hence the name is given as Magnus Effect.
The
planar incompressible potential flow past a spinning cylinder (radius = R) is
constructed by superposition of the velocity potentials for (a) a uniform
stream, φ = Ux (b) a doublet, φ = UR2 cos θ/r at the center of the cylinder and
(c) a potential vortex with circulation, Γ, such that φ = Γθ/2π.
Here the
circulation, Γ, is defined as positive in the anticlockwise direction. This
simulates the flow due to anticlockwise rotation of the cylinder and we will
proceed to find the velocity and pressure on the surface of the cylinder as a
function of angular position, θ, and subsequently, the lift force, L, acting on
the cylinder. The incompressible planar potential flow past a spinning cylinder
(radius = R) in a uniform stream of velocity, U, in the positive x direction is
given by the velocity potential:
where
r, θ are polar coordinates in which θ = 0 corresponds to the positive x
direction and Γ is the circulation defined as positive in the anticlockwise
direction. It follows that the velocities, ur and uθ in the r and θ directions,
are given by
Consequently
the velocities on the surface of the cylinder are (ur)r=R = 0
To
find the pressure on the surface, (p)r=R, we now use Bernoulli’s theorem to
obtain
where
p∞ is the pressure in the uniform stream. Integrating this pressure over the
surface in order to obtain the drag, the result for the drag is zero as
expected from D’Alembert’s paradox. Integrating the pressure over the surface
in order to obtain the lift, L, per unit depth normal to the planar flow,
Which leads to
L = −ρUΓ
This is called the Magnus effect. The lift is due to the fact that the
cylinder rotation in the anticlockwise direction decreases the surface velocity
on the upper side and increases the velocity on the lower side. By Bernoulli’s
equation, this increases the pressure on the upper side and decreases the
pressure on the lower side thus causing the negative lift on the cylinder. This
effect also occurs with three-dimensional objects such as spheres and is
particularly evident in the flight of a well-struck golf ball.
DEVELOPMENT
OF LIFT IN AEROFOIL
Flight is a
phenomenon that has long been a part of the natural world. Birds fly not only
by flapping their wings, but by gliding with their wings outstretched for long
distances. Smoke, which is composed of tiny particles, can rise thousands of
feet into the air. Both these types of flight are possible because of the
principles of physical science. Likewise, man-made aircraft rely on these
principles to overcome the force of gravity and achieve flight. Lighter-than-air craft, such as the hot air balloon,
work on a buoyancy principle. They float on air much like rafts float on water.
The density of a raft is less than that of water, so it floats. Although the
density of water is constant, the density of air decreases with altitude. The
density of hot air inside a balloon is less than that of the air at sea level,
so the balloon rises. It will continue to rise until the air outside of the
balloon is of the same density as the air inside. Smoke particles rise on a plume
of hot air being generated by a fire. When the air cools, the particles fall
back to Earth. Heavier-than-air flight is made
possible by a careful balance of four physical forces: lift, drag, weight, and thrust. For flight, an aircraft's lift must
balance its weight, and its thrust must exceed its drag. A plane uses its wings
for lift and its engines for thrust. Drag is reduced by a plane's smooth shape
and its weight is controlled by the materials it is constructed of. In order
for an aircraft to rise into the air, a force must be created that equals or
exceeds the force of gravity. This force is called lift. In heavier-than-air
craft, lift is created by the flow of air over an airfoil. The shape of an
airfoil causes air to flow faster on top than on bottom. The fast flowing air
decreases the surrounding air pressure. Because the air pressure is greater
below the airfoil than above, a resulting lift force is created. The wings provide lift by creating a situation where the
pressure above the wing is lower than the pressure below the wing. Since the
pressure below the wing is higher than the pressure above the wing, there is a
net force upwards.
To create this pressure difference, the surface of the wing must satisfy one or both of the following conditions. The wing surface must be:
1. Cambered (curved); and/or
- Inclined relative to the airflow direction.
Viscosity is essential in generating lift. The effects of
viscosity lead to the formation of the starting vortex , which, in turn is
responsible for producing the proper conditions for lift. The starting vortex
rotates in a counter-clockwise direction. To satisfy the conservation of
angular momentum, there must be an equivalent motion to oppose the vortex
movement. This takes the form of circulation around the wing. The velocity
vectors from this counter circulation add to the free flow velocity vectors,
thus resulting in a higher velocity above the wing and a lower velocity below
the wing.
The following presents two of several ways to
show that there is a lower pressure above the wing than below.
One method is with the Bernoulli Equation,
which shows that because the velocity of the fluid below the wing is lower than
the velocity of the fluid above the wing, the pressure below the wing is higher
than the pressure above the wing.
A second approach uses Euler's Equations (which
the Bernoulli equation is derived from) across the streamlines.
Due to the curvature of the wing, the higher velocities and acceleration over
the top of the wing requires a pressure above the wing lower than the ambient
pressure.Thus, using either of the two methods, it is shown that the pressure
below the wing is higher than the pressure above the wing. This pressure
difference results in an upward lifting force on the wing, allowing the
airplane to fly in the air. Unfortunately Bernoulli's
Principle does not explain how an aeroplane can fly upside down. Nor does it
explain how aircraft and other structures with flat plate wings or even kites
and paper aeroplanes can fly or remain airborne. This is where Newton's Laws
come to the rescue. Isaac Newton did not propose a theory of flight but he did
provide Newton's Laws of Motion the
physical laws which can be used to explain aerodynamic lift.
Newton's Second Law states that:
The force on an object is equal to its mass times its acceleration
or equivalently to its rate of change of momentum. In this case, since momentum
is a vector quantity, the change in direction of the airflow around the wing
must be associated with a force on the volume of air involved.
Newton's Third Laws states that:
To every action there is an equal and opposite reaction.This means
that the force of the aerofoil pushing the air downwards, creating the downwash, is accompanied by an equal
and opposite force from the air pushing the aerofoil upwards and hence
providing the aerodynamic lift.It is thus the turning of the air flow which creates the lift. Aircraft are kept in the air by
the forward thrust of the wings or aerofoils, through the air. The thrust
driving the wing forward is provided by an external source, in this case by
propellers or jet engines.The
result of the movement of the wing through stationary air is a lift force
perpendicular to the motion of the wing, which is greater than the downwards
gravitational force on the wing and so keeps the aircraft airborne. There
is no predetermined shape for a wing airfoil, it is designed based on the
function of the aircraft it will be used for. To aid the design process,
engineers use the lift coefficient to measure the amount of lift obtained from
a particular airfoil shape. Lift is proportional to dynamic pressure and wing
area. The lift equation is
written as:
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