3

Chapter 1

Introduction

1.1 Opening remarks

The normal operation of a civil transport airplane involves take-off, climb to cruise altitude, cruising, descent, loiter and landing (Fig.1.1).

In addition, the airplane may also carry out glide (which is descent with power off), curved flights in horizontal and vertical planes and other flights involving accelerations.

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Fig 1.1 Typical flight path of a passenger airplane

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Apart from the flights during controlled operations, an airplane may also be subjected to disturbances which may cause changes in its flight path and produce rotations about its axes.

The study of these motions of the airplane – either intended by the pilot or those following a disturbance–forms the subject of Flight Dynamics.

Flight Dynamics: It is a branch of dynamics dealing with the forces acting and the motion of an object moving in the earth’s atmosphere.

In this course our attention is focused on motion of the airplane. Helicopters, rockets and missiles are not covered .

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At this stage it may be helpful to recapitulate the anatomy of the airplane . Fig.1.2a and b show the major components of an airplane.

(From Ref.1.10, chapter 2)Fig 1.2a Major components of an airplane

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Fig 1.2b Control surfaces and flaps on an airplane(From Ref.1.10, chapter 2)

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The features that make flight dynamics a separate subject are :

i. The motion of an object in flight can take place along three axes and about three axes. This is more complicated than the motions of machinery and mechanisms which are restrained by kinematic constraints, or those of land based or water based vehicles which are confined to move on a surface.

ii. The special nature of the forces, like aerodynamic forces, acting on the object (Fig 1.3) whose magnitude and direction changes with the orientation of the airplane , relative to its flight path.

9Fig 1.3 Forces on an airplane

iii. The system of aerodynamic controls used in flight

(aileron, elevator, rudder).

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To formulate and solve a problem in dynamics we need a system of axes. To define such a system we note that an airplane is nearly symmetric in geometry and mass distribution about a plane which is a called the plane of symmetry. This plane is used for defining the body axes system. Figure 1.4b shows a system of axes (OXbYbZb) fixed on the airplane (body axes system) which moves with the airplane. The origin ‘O’ of the body axes system is the center of gravity (c.g.) of the body which, by assumption of symmetry , lies in the plane of symmetry (Fig.1.4a) . The axis OXb is taken positive in the forward direction. The axis OZb is perpendicular to OXb in the plane of symmetry , positive downwards .

1.2 Body axes system

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Fig 1.4 a Plane of symmetry and body axis system

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Fig 1.4b. Body axes system, forces , moments and linear and angular velocities

(Adapted from Ref.1.2d, chapter 1)

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The axis OYb is perpendicular to the plane of symmetry such that OXbYbZb is a right handed system.

Figure 1.4b shows the forces and moments acting on the airplane and the components of linear and angular velocities. The quantity V is the velocity vector. The quantities X,Y,Z are the components of the resultant aerodynamic force, along OXb, OYb and OZb axes. L’ , M, N are the rolling moment, pitching moment and yawing moment respectively about OXb, OYb and OZb; the rolling moment is denoted by L’ to distinguish it from lift (L) . u,v,w are the components , along OXb, OYb and OZb of the velocity vector (V). The angular velocity components are indicated by p,q,r.

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1.3 Forces acting on an airplane

During the analysis of its motion the airplane will be considered as a rigid body. The forces acting on an object in flight are

– Gravitational forces

– Aerodynamic forces

– Propulsive forces.

The aerodynamic forces and moments arise due to motion of airplane relative to air. The aerodynamic forces are the drag, the lift and the side force. The moments are the rolling moment, the pitching moment and the yawing moment.

The propulsive force is the thrust produced by

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the engine or the engine propeller combination.

In the case of an airplane, the gravitational force is mainly due to the attraction of the earth.

The magnitude of the gravitational force is the weight of the airplane (in Newtons).

W = mg; where W is the gravitational force, m is the mass of the airplane and g is the acceleration due to gravity.

The line of action of the gravitational force is along the line joining the centre of gravity (c.g.) of the airplane and the center of the earth. It is directed towards the center of earth (see next section for further discussion).

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The value of the acceleration due to gravity (g) decreases with increase in altitude (h) . It can be calculated based on it’s value at sea level (g0), and using the following formula:

(g/g0) = [R / (R + h)]2 ( 1.1)

Where R is the radius of the earth,

R = 6400 km (approx.) and g0 = 9.81ms-2

However for typical airplane flights (h<20 km) , g is generally taken to be constant.

17Fig.1.5.Flat Earth Model

1.4 Flat earth and spherical earth models

In flight mechanics, there are two ways of dealing with the gravitational force, namely the flat earth model and the spherical earth model.

In the flat earth model, the gravitational acceleration is taken to act vertically downwards (Fig 1.5).

When the distance over which the flight takes place is small, the flat earth model is adequate. See Miele (Ref 1.1) for details.

Flight path

Gravitational forceW=mg

18Fig 1.6. Spherical earth model

In the spherical earth model, the gravitational force is taken to act along the line joining the center of earth and c.g. of the airplane. It is directed towards the center of the earth (Fig. 1.6).

The spherical earth model is used for accurate analysis of flights involving very long distances.

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Remark :

In this course we use the flat earth model. This is adequate for the following reasons.

(a) The distances involved in flights with acceleration are small and the gravitational force can be considered in the vertical direction by proper choice of axes.

(b) In un-accelerated flights like level flight we consider the forces at the chosen instant of time and obtain the distance covered etc. by integration. This procedure is accurate as long as we understand that the altitude means

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height of the airplane above the surface of the earth and the distance is measured on a sphere of radius equal to the sum of the radius of earth plus the altitude of airplane. This type of analysis is also called point performance analysis

1.5 Approach

The approach used in flight mechanics is to apply Newton’s laws to the motion of objects in flight.

Let us recall these laws:

Newton’s first law states that every object at rest or in uniform motion continues to be in that state unless acted upon by an external force.

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Newton’s second law states that the force acting on a body is equal to its rate of change of momentum.

Newton’s third law states that to every action, there is an equal and opposite reaction.

Newton’s second law can be written as:

F = ma ; a = dV/dt; V = dr/dt.

Where F = sum of all forces acting on the body, m= mass, a= acceleration, V= velocity, r= theposition vector of the object and t= time.

(quantities in bold are vectors)

(1.2)

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Acceleration is the rate of change of velocity and velocity is the rate of change of position vector.

To prescribe the position vector, we need to have a co-ordinate system with reference to which the position vector/displacement is measured.

1.6 Frame of reference

A frame of reference (coordinate system) in which Newton’s laws of motion are valid is known as a Newtonian frame of reference.

Since Newton’s laws deal with acceleration, a frame of reference moving with uniform velocity with respect to a Newtonian frame is also a Newtonian frame or inertial frame.

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However, if the reference frame is rotating with some angular velocity ( ), then, additional accelerations like centripetal acceleration { x ( x r)} and coriolis acceleration (v x )will come into picture.

For further details on non-Newtonian reference frame, see Ref 1.2a.

In flight mechanics, a co-ordinate system attached to the earth approximates a Newtonian frame (Fig.1.7).

The effects of the rotation of earth around itself and around the sun on this approximation can be estimated as follows.

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Fig 1.7 Earth fixed and body fixed co-ordinate systems

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We know that the earth rotates about itself once per day. Hence

= 2 / (3600x24) = 7.27x10-5s-1;

Since r equals 6400 km; the maximum centripetal acceleration ( 2r) equals 0.034 ms-2.

The earth also goes around the sun and completes one orbit in approximately 365 days. Hence

= 2 / (365x3600x24) = 1.99x10-7s-1;In this case the radius would be the mean distance between the sun and the earth which is 1.5x1011m. Consequently

2R = 0.006 ms-2.

Thus we note that the centripetal accelerations due

to rotation of earth about itself and around the sun are small as compared to acceleration due to gravity.

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These rotational motions would also bring about coriolis acceleration (v x ). However its magnitude, which depends on the flight velocity, would be much smaller than the acceleration due to gravity in flights up to Mach number of 3. Hence the influence can be neglected.

Thus, taking a reference frame attached to the surface of the earth as a Newtonian frame is adequate for the analysis of airplane flight. Figure 1.7 shows such a coordinate system.

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1.7 Equilibrium

The above three types of forces (aerodynamic, propulsive and gravitational) and moments govern the motion of an airplane in flight.

If the sums of all these forces and moments are zero, then the airplane is said to be in equilibrium and will move along a straight line with constant velocity (see Newton's first law). If any of the forces is unbalanced , then the airplane will have a linear acceleration in the direction of a unbalanced force. If any of the moments is unbalanced, then the airplane will have an angular acceleration about the axis of the unbalanced moment.

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The relationship between the unbalanced forces and the linear accelerations and those between unbalanced moments and angular accelerations are provided by Newton’s second law of motion. These relationships are called equations of motion.

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1.8 Equations of motion

To derive the equations of motion we need to know the acceleration of a particle on the body. The acceleration is the rate of change of velocity and the velocity is the rate of change of position vector with respect to the chosen a frame of reference.

The minimum number of coordinates required to prescribe the motion is called the number of degrees of freedom. The number of equations governing the motion equals the degrees of freedom. As an example we may recall that the motion of a particle moving in a plane is prescribed by the x- and y-coordinates of the particle at various instants of time and this motion is described by two equations.

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Similarly the position of any point on a rigid pendulum is describe by just one coordinate namely the angular position ( ) of the pendulum (Fig.1.8). In this case we have only one equation to describe the motion . In yet another example , if a particle is constrained to move on a sphere, then its position is prescribed by the longitude and latitude . This motion has only two degrees of freedom.

To describe its motion we treat the airplane as a rigid body. It may be recalled that in a rigid body the distance between any two points is fixed. Thus r inFig. 1.9 does not change during the motion. To decide the minimum number of coordinates needed to prescribe the position of a point on a rigid body which

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Fig. 1.8 Motion of a single degree of freedom system

32Fig 1.9 Position of a point on a rigid airplane

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is translating and rotating, we proceed as follows.

A rigid body with N particles may appear to have 3N degrees of freedom, but the constraint of rigidity reduces this number . To arrive at the minimum number of coordinates we approach the problem in a different way. Following Ref.1.2b, we can state that to fix the location of a point on a rigid body we do not need to prescribe its distance from all the points, but only need to prescribe its distance from three points which do not lie on the same line (Fig.1.10a). Thus if the positions of these three points are prescribed with respect to a reference frame , then the position of any point on the body is known. This may indicate nine

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degrees of freedom . This number is reduced to sixbecause the distances r12, r23 and r13 in Fig.1.10a are constants .

Another way of looking at the problem is to consider that we prescribe the three coordinates of point 1 with respect to the reference frame. Now the point 2 is constrained, because of rigid body assumption, to move on a sphere centered on point 1 and needs only two coordinates to prescribe its motion. Once the points 1 and 2 are determined, the point 3 is constrained , again due to rigid body assumption, to move on a circle about the axis joining points 1 and 2. Hence only one independent coordinate is needed to prescribe the position of point 3. Thus the number of independent coordinates is six (3+2+1).

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Fig 1.10a Position of a point with respect to three reference points

(Adapted from Ref.1.2b, Chapter 4)

C

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From the above discussion it is clear that the coordinates could be lengths or angles.

In mechanics the six degrees of freedom associated with a rigid body, consists of the three coordinates of the origin of the body with respect to the chosen frame of reference and the three angles which describe the angular position of a coordinate system fixed on the body (OXbYbZb) with respect to the fixed frame of reference (EXeYeZe) as shown in Fig.1.10b. These angles are known as Eulerian angles. These will be discussed in ch.4 of flight dynamics- II . See also Ch.4 of Ref.1.2b.

Remarks:

i) The derivation of the equations of motions in a general case with six degrees of freedom (see Ref 1.1)

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Fig 1.10b Coordinates of a point on a rigid body

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1.9 Subdivisions of flight dynamics

The subject of flight dynamics is generally divided into two main branches viz.

Performance Analysis and Stability and control.

In performance analysis, we generally consider the equilibrium of forces only. It is assumed that by proper deflections of the controls, the moments can be made zero and that the changes in aerodynamic forces due to deflection of controls are small. The motions considered in performance analysis are steady and accelerations,

is rather involved and would be out of place here.ii) Herein, we consider various cases separately andwrite down the equations of motion in each case.

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when involved, do not change rapidly with time.

The following flights are included in performance analysis

-Unaccelerated flights

• Steady level flight

• Climb, glide and descent

-Accelerated Flights

• Accelerated level flight and climb

• Take-off and landing

• Turn, loop and other flights along curved paths which are called maneuvers.

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Roughly speaking, the stability analysis is concerned with the motion following a disturbance. Stability analysis tells us whether an airplane, after being disturbed, will return to it’s original flight path or not.

Control analysis deals with the forces that the deflection of the controls must produce to bring to zero the three moments (rolling, pitching and yawing) and achieve a desired flight condition. It also deals with design of control surfaces and the forces on control wheel/stick /pedals.

Stability and control are linked together and are generally studied under a common heading.

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Flight dynamics I of this course deals with performance analysis. By carrying out this analysis we can obtain variation of performance items such as maximum level speed, minimum level speed, rate of climb, angle of climb, distance covered , with a given amount of fuel called ‘Range’ , time of flight called ‘Endurance’ , minimum radius of turn, maximum rate of turn, take off distance, landing distance etc. The effects of flight conditions namely the weight , altitude and flight velocity of the airplane can also be examined. This study would also help in solving design problems of deciding the

power required, thrust required , fuel required etc. for given specifications like maximum speed,

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maximum rate of climb, range, endurance etc.

Remark:

Alternatively, the performance analysis can be considered as the analysis of the motion of flight vehicle considered as a point mass, moving under the influence of applied forces. The stability analysis similarly can be considered as motion of a vehicle of finite size, under the influence of applied forces and moments.

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1.10 General Remarks

i) Attitude :

As mentioned in section 1.8 the instantaneous position of the airplane , with respect to the earth fixed axes system (EXe Ye Ze) , is given by the coordinates of the c.g. at that instant of time. The attitude of the airplane is described by the angular orientation of the OXbYbZb

system with respect to OXeYeZe system or the Euler angles mentioned in section 1.8 (see Ref.1.2c, chapter 10 for details) . Let us consider simpler cases. When an airplane climbs along a straight line its attitude is given by the angle ‘ ’ between the axis OXb and

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and the horizontal (Fig.1.11a ). When an airplaneexecutes a turn, the projection of OXb axis , in the horizontal plane , makes an angle with reference to a fixed horizontal axis (Fig.1.11b) . When an airplane is banked the axis OYb makes an angle with respect to the horizontal (Fig.1.11c).

ii) Flight path:

In the subsequent sections, the flight path, also called the trajectory, means the path or theline along which the c.g. of the airplane moves. The tangent to this curve at a point gives the direction of flight velocity at that point on the flight path. The relative wind is in a direction opposite to that of the flight velocity .

45Fig 1.11a Airplane in a climb

46Fig 1.11b Airplane in a turn-view from top

(Adapted from Ref.1.10, chapter 6)

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Fig 1.11c Angle of bank ( )

(Adapted from Ref. 1.11, chapter 3)

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iii) Angle of attack and side slip

While discussing the forces acting on an airfoil, we take the chord of the airfoil as the reference line and the angle between the chord line and the relative wind as the angle of attack( ). The aerodynamic forces viz lift (L) and drag (D) , produced by the airfoil, depend on the angle of attack ( ) and are respectively perpendicular and parallel to relative wind direction (Fig.1.11 d).

In the case of an airplane the flight path, as mentioned earlier , is the line along which c.g. of the airplane moves . The tangent to the flight path is the direction of flight velocity (V). The relative wind is in a direction opposite to the flight velocity. If the flight path is confined to the

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Fig 1.11d Angle of attack and forces on a airfoil

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plane of symmetry, then the angle of attack would be the angle between the relative wind direction and the fuselage reference line (FRL) or OXb axis(see Fig.1.11e) . However in a general case the velocity vector (V) will have components both along and perpendicular to the plane of symmetry. The component perpendicular to the plane of symmetry is denoted by ‘v’ . The projection of the velocity vector in the plane of symmetry would have components u and w along OXb and OZb axes(Fig.1.11f) . With this background we define the angle of sideslip and angle of attack .

51Fig 1.11e Flight path in the plane of symmetry

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Fig 1.11f Velocity components in a general case and definition of angle of attack and sideslip

(Adapted from Ref.1.2d , chapter 1)

o

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The angle of sideslip ( )is the angle between the velocity vector (V) and the plane of symmetry i.e.

= sin-1 (v/ |V|); where |V| is the magnitude of V.

The angle of attack ( ) is the angle between the projection of velocity vector (V) in the XB-ZB planeand the OXb axis or

1 1 1

2 2 2 2

w w wtan sin sinu | | v u wV

Remark:

It is easy to show that , if V denotes magnitude of velocity (V) , thenu = V cos cos , v= V sin ; w= V sin cos .

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iv) By definition, the aerodynamic drag (D) is parallel to the relative wind direction. The lift force lies in the plane of symmetry of the airplane and is perpendicular to the direction of flight velocity.v) Simplified treatment in performance analysis

In a steady flight, there is no acceleration along the flight path and in a level flight, the altitude of the flight remains constant. A steady, straight and level flight, generally means a flight along a straight line at a constant velocity and constant altitude.

Sometimes, this flight is also referred to as unaccelerated level flight. To illustrate the simplified

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treatment in performance analysis, we consider the case of unaccelerated level flight.

The forces acting on an airplane in unaccelerated level flight are shown in the Fig.1.12. They are:

Lift (L)Thrust (T)Drag (D) andWeight (W) of the airplane.

It may be noted that the point of action of the thrust and it’s direction depend on the engine location. However, the direction of the thrust can be taken parallel to the airplane reference axis.

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Fig 1.12 Forces acting in steady level flight

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The lift and drag, being perpendicular to the relative wind, are in the vertical and horizontal directions respectively, in this flight.

The weight acts at the c.g. in a vertically downward direction.

In an unaccelerated level flight, the components of acceleration in the horizontal and vertical directions are zero.

Hence, the sums of the components of all the forces in these directions are zero. Resolving the forces along and perpendicular to the flight path

(see Fig.1.12.), we get the following equations of force equilibrium:

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T cos – D = 0

T sin + L – W = 0

Apart from these equations, equilibrium demands that the moment about the y-axis to be zero, i.e.,

Mcg = 0

Unless the moment condition is satisfied, the airplane will begin to rotate about the c.g.

Let us now examine how the moment is balanced in an airplane.

The contributions to Mcg come from all the components of the airplane.

As regards the wing , the point where the resultant vector of the lift and drag intersects the plane of symmetry is known as the centre of

(1.3)(1.4)

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pressure. This resultant force produces a moment about the c.g. However, the location of the center of pressure depends on the lift coefficient and hence the moment contribution of wing changes with the angle of attack as the lift coefficient depends on the angle of attack. For convenience, the lift and the drag are transferred to the aerodynamic center along with a moment (Mac).Recall that moment coefficient about the a.c. (Cmac) is, by definition, constant with change in angle of attack.

Similarly, the moment contributions of the fuselage and the horizontal tail change with the angle of attack. The engine thrust also produces a moment about the c.g. which depends on the thrust required.

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Hence, the sum of the moments about the c.g. contributed by the wing, fuselage, horizontal tail and engine changes with the angle of attack. By appropriate choice of the horizontal tail setting (i.e. incidence of horizontal tail with respect to fuselage central line ) , one may be able to make the sum of these moments to be zero in a certain flight condition, which is generally the cruise flight condition. Under other flight conditions, generation of corrective aerodynamic moment is facilitated by suitable deflection of elevator (See Fig.1.2b for location of elevator). By deflecting the elevator , the lift on the horizontal tail surface can be varied and the moment produced by the horizontal tail balances the moments produced by all other components.

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Example 1.1

A jet aircraft weighing 60,000 N has it’s line of thrust 0.15 m below the line of drag. When flying at a certain speed, the thrust required is 12,000 N and the center of pressure of the wing lift is 0.45 m aft of the airplane c.g.. What is the lift on the wing and the load on the tail plane whose center of pressure is 7.5 m behind the c.g.? Assume unaccelerated level flight and the angle of attack to be small during the flight.

The above points will now be illustrated with the help of an example.

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Solution:

The various forces and dimensions are presented in Fig.1.13. The lift on the wing is LW and the lift on the tail is LT. Since the angle of attack ( ) is small, one may take cos = 1 and sin = 0. Thus, from the force equilibrium (Eqs. 1.3 and 1.4), we get:

T – D = 0

LW + LT – W = 0

i.e. D = T = 12000 N and LT + LW = 60000 N

From Fig.1.13., the moment equilibrium about the c.g. gives:

T (zd + 0.15) – D.zd – 0.45.LW – 7.5.LT = 0

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Fig. 1.13 Forces acting on an airplane in steady level

flight

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where zd is the distance of drag below the c.g..

Solving these equations, we get,

LW = 63574.47 N and LT = -3574.47 N

It is seen that

A) The lift on the wing is about 63.6 kN while the lift on the tail is only 3.6 kN, in the downward direction.

B) The contribution of tail to the total lift is thus small, in this case, about 6% and negative. This negative contribution necessitates the wing lift to be more than the weight of the airplane. This increase the lift results in additional drag called trim drag.

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D) Generally, the angle of attack ( ) is small. Hence, sin is small and cos is nearly equal to unity. Thus, the equations of force equilibrium reduce to

T – D = 0 and L – W = 0.

E) It is assumed that the pitching moment equilibrium i.e. Mcg=0 is achieved by appropriate deflection of the elevator. The changes in the lift and drag due to elevator deflections are generally small and in performance analysis, as stated earlier, these changes are ignored and one

C) The distance zd is of no significance in this problem as the drag and thrust form a couple whose moment is equal to the thrust multiplied by the distance between them.

(1.5)

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Fig.1.14. Simplified picture of the forces acting on an airplane in level flight.

considers the simplified picture as shown in Fig.1.14.

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Some background material is required for performance analysis. We know that:

L = (1/2) V2 S CL

D = (1/2) V2 S CD

Where CL and CD are the lift and drag coefficients.

S is the area of the wing.

CL and CD depend on , Mach number (M = V / a) and Reynolds number (Re = V l /μ ) i.e.

CD = f(CL,M, Re)

The relation between CLand CD at given M and Re is known as the drag polar of the airplane.

1.11 Course content

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Similarly, the density of air ( ) depends on the flight altitude. Further the Mach number depends on the speed of sound, which in turn depends on the ambient air temperature. Thus, for performance analysis, we need to know the variations of pressure, temperature, density, viscosity etc. with altitude in earth’s atmosphere.

For evaluation of performance we also need to know the engine characteristics such as, variations of thrust/ power and fuel consumption with the flight speed and altitude.

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3.1. Introduction

As mentioned in chapter 1, to evaluate the performance of an airplane we need to know as to what will be the drag coefficient of the airplane (CD) when the lift coefficient (CL) and Mach number are given.

The relationship between the drag coefficient and the lift coefficient is called drag polar.

The usual method to estimate the drag of an airplane is to add the drags of the major components of the airplane and then apply correction for the interference effects.

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The major components of the airplane which contribute to drag are wing, fuselage, horizontal tail, vertical tail, nacelles and landing gear.

Thus,

D = Dwing + Dfuse + Dht + Dvt + Dnac + Dlg +

Detc + Dint

where Dwing, Dfuse , Dht, Dvt and Dlg denote drag due to wing, fuselage, horizontal tail, vertical tail and landing gear respectively.

Detc includes the drag of items like external fuel tanks, bombs, struts etc..

(3.1)

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Dint is the drag due to interference. This arises due to the following reasons.

While estimating the drag of wing, fuselage and other components we consider the drag of the component when it is free from the influence of any other components. Whereas in an airplane the wing, fuselage, and tails lie in close proximity of each other and flow past one component is influenced by that past the other. As an illustration let us consider an airfoil kept in a stream of velocity V . Let the drag be 5 N. Now consider a small plate whose drag at the same speed of be 2 N.

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Then the drag of the airfoil and the plate as a combination (Fig. 3.1) would, in general, be higher than the sum of individual drags. i.e.

D airfoil+plate> (5+2)=(5+2)+Dint

It is evident that Dint will also depend on the place where the plate is located on the airfoil.

Remarks

i) Ways to reduce interference drag

A large number of studies have been carried out on interference drag and it is found that Dint canbe brought down to 5 to 10% of the sum of the drags of all components, by giving proper fillets at the junctions of wing and fuselage and tails and fuselage ( Fig 3.2 ).

6Fig 3.1 Interference drag

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Fig 3.2 Reduction of interference drag using fillets(Adapted from Ref.3.1, pp. 181)

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ii) Favorable interference effect

The interference effects need not always increase the drag . The drag of the airfoil plus the plate can be lower than the drag of the airfoil when a thin plate is attached to the trailing edge of the airfoil which is called splitter plate. The birds flying in formation flight experience lower drag than when flying individually.

(iii) Contributions to airplane lift

The main contribution to the lift comes from wing-fuselage combination and a small contribution from the horizontal tail i.e. :

L = Lwing + fuselage + Lht (3.2)

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For airplanes with wings having aspect ratio greater than six, the lift due to the wing-fuselage combination is roughly equal to the lift produced by the gross wing area. The gross wing area (S) is the planform area of the wing, extended into the fuselage, up to the plane of the symmetry.

iv) Contributions to airplane pitching moment

The pitching moment of the airplane is taken about its center of gravity and denoted by Mcg.

Main contributions to Mcg are from wing, fuselage, nacelle and horizontal tail i.e.

Mcg = Mwing + Mfuselage + Mht + Mnac (3.3)

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(v) Non-dimensional quantities

To obtain the non-dimensional quantities namely drag coefficient (CD), lift coefficient (CL) and pitching moment coefficient (Cmcg) the reference quantities are the free stream dynamic pressure (½ V 2) ,the gross wing area (S) and the mean aerodynamic chord of the wing ( c ). Consequently ,

_

D L mcg2 2 21 1 12 2 2

C = ; C = ; C = cgMD LV S V S V Sc

(3.4)

However, the drag coefficient and lift coefficient of the individual components are based on their own reference areas i.e.

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(a) For wing, horizontal tail and vertical tail thereference area is their planform area.

(b) For fuselage, nacelle, fuel tanks, bombs and such other bodies the reference area is either the wetted area or the frontal area. The wetted area is the area of the surface of the body in contact with the fluid. The frontal area is the maximum cross-sectional area of the body.

(c) For other components like landing gear the reference area is given along with the definition of CD.

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Note:(I) The reference area, on which the CD and CL of an

individual component is based, is also called proper area and denoted by S ; the drag coefficient based on S is denoted by CD .

(II)The reference areas for different components are different for the following reasons. The aim of using non-dimensional quantities like CD is to be able to predict characteristics of many similar shapes by conducting computations or tests on a few models. For this to be effective, the phenomena causing the drag must be taken into account. In this context the drag of streamline shapes like wing and slender bodies is mainly due

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to skin friction and depends on the wetted area. Whereas the drag of bluff bodies like the fuselage of a piston-engined airplane , is mainly the pressure drag and depends on the frontal area. It may be added that for wings, the usual practice is to take the reference area as the planform area because it is proportional to the wetted area.

(III) At this stage the reader is advised to the revise the background on aerodynamics (see for examples references 1.7 & 1.8 ).

Following the above remarks we can express the total drag of the airplane as :

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lg int

2 21 12 2

2 2 21 1 12 2 2

2 21 1lg2 2

wing fuse

nac ht vt

etc

D fuse D

nac D ht D vt D

D etc D

D V S C V S C

V S C V S C V S C

V C S V S C D

lg in t

212

lg

fu se h t v t

n a c e tc

D

fu se h t v tD w in g D D D

na c e tcD D D D

DCV S

S S SC C C CS S SSS SC C C C

S S S

(3.5)

(3.6)

Or

It may be recalled that Setc and CDetc referred to areas and drag coefficients of other items like external fuel tanks , bombs , struts etc..

15

The data on drag lift and pitching moment, compiled from various sources, is available in references 1.7,1.8,1.9 and 3.1a to 3.7.

16

3.2. Estimation of Drag Polar – Low Speed Case

As mentioned in the previous section, the drag polar of an airplane can be obtained by summing-up the drags of individual components and then adding 5 to 10% for interference drag. This exercise has to be done at different angles of attack. A few remarks are mentioned before obtaining the drag polar.

Remarks

i) Angles of Attack:

For defining the angle of attack of an airplane, the fuselage reference line is taken as the airplane reference line (Figs. 1.9,3.3). However the angles of attack of the wing and tail are not the same as that

17

of the fuselage.

The wing is fixed on the fuselage such that it makes an angle, iw, to the fuselage reference line (Fig 3.3). The angle iw is generally chosen such that during the cruising flight the wing can produce enough lift when fuselage is at zero angle of attack. This is done because the fuselage produces least drag when it is at zero angle of attack and that is what one would like to have during cruising flight, i.e. during cruise the wing produces the lift required to balance the weight whereas the fuselage being at zero angle of attack produces least drag.

The tail is set on fuselage at an angle it (Fig. 3.3) such that during cruise the lift required from the tail,

18

Fig 3.3 Wing setting and tail setting

19

to make the airplane pitching moment zero, is produced by the tail without elevator deflection. This is because, the drag, at low angles of attack, is least when the required lift is produced without elevator deflection.

ii) Drag coefficient of wing

The drag coefficient of a wing consist of the (a) profile drag due to airfoil (Cd) and (b) the induced drag due to the finite aspect ratio of the wing (CDi). The profile drag of the airfoil consists of the skin friction drag and the pressure drag. It may be recalled that an element of airfoil in a flow experiences shears stress tangential to the surface and pressure normal to it . The shear stress

20

multiplied by the area of the element gives the tangential force. The component of this tangential force in the free stream direction when integrated over the profile gives the skin friction drag. Similarly the pressure distribution results in normal force on the element whose component in the free stream direction, integrated over the profile gives the pressure drag. The pressure drag is also called form drag. The sum of the skin friction drag and the pressure drag is called profile drag. The profile drag depends on the airfoil shape, Reynolds number, angle of attack and surface roughness.

21

The chord of the wing varies along the span and further the shapes of the profiles may also change along it (span). Hence for the purpose of calculation of profile drag of the wing , a representative airfoil may be chosen with chord equal to the average chord ( ); where S is the wing area and b is the wing span.

As regards the generation of induced drag it may be recalled that a wing has a finite span. This results in a system of trailing vertices and induced angle due to these vertices tilts the aerodynamic force rearwards. This results in a component in the free stream direction which is called induced drag. The induced drag

avgSCb

22

2(1 )L

DiCC A

coefficient is given by :

(3.7)

Where A is the wing aspect ratio (A=b2/S) andis a factor which depends on wing aspect ratio, taper ratio and sweep.When a flap is deflected, there will be increments in lift and both profile drag and induced drag.

A similar procedure can be used to estimate drags of horizontal and vertical tails. Howevercontributions to induced drag can be neglected for the tail surfaces.

23

iii) Drag coefficient of fuselage

The drag coefficient of a fuselage (CDf) consists of the drag or the fuselage at zero angle of attack (CD0)f plus drag due to angle of attack. It can be expressed as :

CDf=(CD0)f+K( )2

For a streamlined body (CD0)f is mainly skin friction drag and depends on (a) Reynolds number, based on length of fuselage (lf),(b)surface roughness and (c) fineness ratio (Af).The fineness ratio is defined as:

Af=lf /de (3.9)

(3.8)

24

where de is the equivalent diameter given by:

( /4)de2 = Afmax

where Afmax equals the area of the maximum cross-section of fuselage.

When the fineness ratio of the fuselage is small for e.g. in case of general aviation airplanes , the fuselage is treated as a bluff body. In such cases the drag is mainly pressure drag and the drag coefficient is based on the frontal area (Afmax).

The drag coefficients of other bodies like engine nacelle, external fuel tanks, bombs can also be estimated in a similar manner.

iv) The drag coefficients of other components likelanding gear are based on areas specific to the

25

component. They should be obtain from the sources of drag data mentioned earlier.

3.2.1 Drag polar

To obtain the drag polar by adding the drag coefficients of individual components at corresponding angles of attack , needs a large amount of detailed data about the airplane geometry and drag coefficients. A typical drag polar obtained by such a procedure or by experiments on a model of the airplane appears as shown in Fig. 3.4a. When this curve is replotted as CD vs.CL

2 (Fig.3.4b), it is found that over a wide range of CL the curve is a straight line and one could write.

CD=CD0 + KCL2 (3.10)

26

Fig 3.4a Typical drag polar of a piston – engined airplane

27

CD0 is the intercept of the straight line and is called zero lift drag coefficient or parasite drag coefficient (Fig.3.4b).

The term KCL2 is called induced drag coefficient

or more appropriately lift dependent drag coefficient. K is written as:

1KA e

(3.11)

where e, called Oswald efficiency factor, includes the changes in drag due to angle of attack of the wing, the fuselage and other components (Refs.1.9, Chapter 2 & 3.3, Chapter 2).

28Fig 3.4(b) Drag polar- CD vrs.CL

2

29

It may be added that in the original definition of Oswald efficiency factor only the contribution of wing was included.

Remarks:

i) The reason why an expression like Eq.(3.10) fits the drag polar is because the lift dependent drags of wing and fuselage are proportional to the square of the angle of attack.

ii) The drag polar given by Eq.(3.10) is called parabolic drag polar.

iii) It found that CD0 is roughly equal to the sum of the minimum drag coefficients of various components plus the correction for interference .

30

iv) Parasite drag area and equivalent skin friction coefficient

The product CD0 x S is called parasite drag area. For streamlined airplanes the parasite drag is mostly skin friction drag plus a small pressure drag. The skin friction drag depends on the wetted area of the surface. The wetted area of the entire airplane is denoted by Swet and the equivalent skin friction coefficient (Cfe) is defined as :

CD0 x S = Cfe x Swet

or wetD 0 fe

SC CS

31

Reference 3.7 , Chapter 12 gives values of Cfe

for different types of airplanes.

v) The factor ‘e’ lies between 0.8 to 0.9 for airplanes with unswept wings and between 0.6 to 0.8 for those with swept wings.See Refs.3.3 & 3.4 for estimating CD0 and K.

vi) The parabolic polar is an approximation . It is inaccurate near CL =0 and CL= CLmax (Fig.3.4b).It is should not be used beyond CLmax .A quick estimate of the drag polar is carried outin example 3.1.

32

Example 3. 1

An airplane has a wing of planform area 51.22 m2

and span 20 m. It has a fuselage of frontal area 3.72 m2 and two nacelles having a total frontal area of 3.25 m2. The total planform area of horizontal and vertical tails is 18.6 m2 . Obtain a rough estimate of the drag polar in a flight at a speed of 430 kmph at sea level (s.l.). when landing gear is in retracted position.

33

Solution :

Flight speed is 430 kmph = 119.5 m/s.

Average chord of wing = S/b = 51.22/20=2.566 m.

Reynolds number (Re) based on average chord is:

66

1 1 9 .5 2 .5 6 6 2 1 1 01 4 .6 1 0

Assuming a 12% thick airfoil the (CDmin)wing at this Re would be 0.0054 (See Reference 3.4).Since the frontal area is specified, the fuselage is treated as a bluff body; (CDmin)fuselage can be taken as 0.08 (Ref.3.4).

34

The nacelle generally has a lower fineness ratio

and (CDmin)nac can be taken as 0.10.

(CDmin)tail for the tail surfaces is taken as 0.006, which is slightly higher than that for wing as the Re for tail surfaces would be smaller. The

results are presented in Table 3.1.

1.013Total0.1120.00618.6Tail surfaces 0.3250.13.25Nacelles0.3000.0803.72Fuselage0.2790.005451.22Wing

CD S (m2)CDS (m2)Part

Table 3.1 Rough estimate of CD0

35

Adding 10% for interference effects, the total parasite drag area (CD S ) is:1.013 + 0.1013 = 1.1143 m2. HenceCD0= 1.1143/51.22 = 0.0216

Wing aspect ratio is 202/51.22=7.8

Taking e = 0.83 (see reference 3.4, page A119for details) we get the drag polar as

or CD = 0.0216 + 0.049 CL2

210.02167.8 0.83D LC C

X X

36

Remarks:

i) A detailed estimation of the drag polar of Piper Cherokee airplane is presented in appendix A.

ii)Typical values of CD0 , A, e and the polar for subsonic airplanes are given in Table 3.2.

37

0.6 to 0.75

6 to 80.015 to 0.017

Highsubsonic(M around 0.8, Swept wing)

0.85to 0.9

10 to 120.018 to 0.020

Mediumspeed(M around 0.5)

0.8 to 0.9

6 to 80.022 to 0.04

Low speed (M <0.3)

Typical polareACD0Type of airplane

0.025 + 0.055CL2

0.019 + 0.04CL2

0.016 +0.06CL2

Table 3.2 Typical values of CD0, A ,e and polar

38

Note:

Table 3.2 shows that for low speed airplanes CD0 ishigher than in other cases. This is because these airplanes have exposed landing gear, bluff fuselage and struts. They also have only moderate aspect ratio (6 to 8) so that wing-span is not large and the hanger-space needed for parking the plane is not excessive.

The CD0 for high subsonic airplanes is low due to smooth surfaces, thin wings and slender fuselage. It may be added that during the design process, the values of airfoil thickness ratio, aspect ratio and angle of sweep for the wing are obtained from considerations of optimum performance.

39

3.3 Drag polar at high speeds

At this stage the reader is advised to revise background on compressible aerodynamics and gas dynamics (see Refs.1.7 & 1.8). Some important aspects are brought out in the following remarks.

(1) When the Mach number roughly exceeds a value of 0.3, the changes in the fluid density within the flow field become significant and the flow needs to be treated as compressible.

(2) In a compressible flow the changes of temperature in the flow field may be large and hence the speed of sound (a= ) may vary from point to point.

RT

40

(3) When the Mach number exceeds unity, the flow is called supersonic.

(4) When a supersonic flow decelerates, shock waves occur. The pressure, temperature, density and Mach number change discontinuously across the shock. The shocks may be normal or oblique. The Mach number behind a normal shock is subsonic; behind an oblique shock it may be subsonic or supersonic.

(5) When supersonic flow encounters a concave corner, as shown in Fig 3.5 (a), the flow changes thedirection across a shock. When such a flow encounters a convex corner, as shown in Fig 3.5.(b) the flow expands across a series of Mach waves.

41

Fig 3.5 Supersonic flow at corners a) Concave corner (b) Convex corner

(From Ref.1.7,chapter 5)

(a) (b)

42

(6) A typical flow past a diamond airfoil at supersonic Mach number is shown in Fig 3.6.

If the Mach number is low supersonic (i.e. only slightly higher than unity) and the angle , as shown in Fig 3.6, is high then instead of the attached shock waves at the leading edge, a bow shock wave may occur ahead of the airfoil. A blunt-nosed airfoil can be thought of an airfoil with large value of ‘ ’ at the leading edge and will have a bow shock at the leading edge as shown in Fig 3.7. Behind a bow shock there is a region of subsonic flow ( Fig 3.7) .

43Fig 3.6 Supersonic flow past a diamond airfoil

(From Ref.1.9, chapter1)

44Fig 3.7 Bow shock ahead of blunt-nosed airfoil

( Adapted from Ref.1.7, chapter 5 )

45

(7) Transonic flow

This type of flow occurs when the free stream Mach number is around unity. The changes in the flow and hence in the drag occurring in this range of Mach numbers can be appreciated from the following statements.

(I) In subsonic flow past an airfoil the flow has zero velocity at the stagnation point. Then the flow accelerates, it reaches a maximum value (Vmax)and later attains the free stream velocity (V ).The ratio Vmax /V is greater than unity and depends on (a) shape of airfoil (b) thickness ratio (t/c) and ( c ) angle of attack ( )

46

(II) As (Vmax/ V ) is greater than unity, the ratio of the maximum Mach number on the airfoil( M max) and free stream Mach number (M )would also be more than unity. However ( Mmax/M ) would not be equal to (Vmax /V )as the speed of sound varies from point to point on the airfoil.

(III) Critical Mach number

As M increases, Mmax also increases. The free stream Mach number for which the maximum Mach number on the airfoil equals unity is called critical Mach number (Mcrit).

47

(IV) When M exceeds Mcrit , a region of supersonic flow occurs which is terminated by a shock wave. The changes in flow pattern are shown in Fig 3.8.

(V) As free stream Mach number increases the region of supersonic flow enlarges and this region occurs on both the upper and lower surfaces of the airfoil (Figs. 3.8 c & d).

(VI) At a free stream Mach number slightly higher than unity, a bow shock is seen near the leading edge of the airfoil ( Fig. 3.8 e).

(VII) At still higher Mach numbers the bow shock approaches the leading edge and if the leading edge is sharp, then the shock waves attach to the leading edge as shown in Fig 3.6.

48Fig 3.8 Flow past airfoil near critical Mach number

(Adapted from Ref. 1.9,chapter 1)

49

(VIII) Transonic flow regime

When M is less than Mcrit the flow every where i.e. in the free stream and on the body is subsonic.

It is seen that when Mcrit < M < 1, the free stream Mach number is subsonic but there are regions of supersonic flow on the airfoil ( Figs. 3.8 c & d ) .

Further When M is slightly more than unity i.e.free stream is supersonic, there is bow shock ahead of the airfoil resulting in subsonic flow near the leading edge.

When the shock waves are attached to the leading edge ( Fig. 3.6 ) the flow is supersonic

50

every where i.e. in the free stream and on the airfoil.

The above flow features permit us to classify the flow in to three regimes.

(a) Sub-critical regime when the Mach number is subsonic in the free stream as well as on the body ( M < Mcrit ).

(b) Transonic regime when the regions of subsonic and supersonic flow are seen within the flow field.

(c) Supersonic regime when the Mach number in the free stream as well as on the airfoil is supersonic.

The extent of the transonic regime is loosely stated as between 0.8 to 1.2. However the actual extent is between Mcrit and the Mach number at

51

which the flow becomes supersonic everywhere.The extent depends on the shape of the airfoiland the angle of attack.

(8) In the transonic regime the lift coefficient and drag coefficient undergo rapid changes with Mach number ( Fig.3.9). For a chosen angle of attack the drag coefficient begins to increase near Mcrit and reaches a peak around M =1.Drag divergence Mach number

When the change in Cd with Mach number is studied experimentally, we can notice the effect of appearance of shock waves in the form of increase in drag coefficient . The beginning of the transonic region is characterized by drag divergence Mach number (MD) at which the

52

Fig 3.9 Schematic variations of Cl and Cd of an airfoil in transonic regime.

(Adapted from Ref. 1.9, chapter1)

53

increase in the drag coefficient is 0.002 over the value of Cd at sub-critical Mach numbers. It may be added that for a chosen angle of attack the value of Cd remains almost constant at sub-critical Mach numbers. As mentioned earlier the increase in the drag coefficient in the transonic region is due to appearance of shock waves and hence it is also called wave drag.

The drag divergence Mach number of an airfoil depends on its shape, thickness ratio and the angle of attack.

(9) The drag divergence Mach number of a wing depends on the drag divergence Mach number of the airfoil used, and the aspect ratio. It can be increased by incorporating sweep ( ) to the

54

wing. The geometrical parameters of the wing are shown in Fig.3.10. The beneficial effects of sweep on (a) increasing MD , (b) decreasing peak value of wave drag coefficient (CDpeak)and (c ) increasing Mach number at which CDpeak occurs are shown in Fig.3.11.

55Fig 3.10 Geometric parameters of a wing

56

Fig 3.11 Effect of wing sweep on variation of CDwith Mach number.(Adapted from Ref.1.9, chapter 1)

57

(10) Drag at supersonic speeds

At supersonic Mach numbers also the drag of a wing can be expressed as sum of the profile drag of the wing section plus the drag due to effect of finite aspect ratio . The profile drag consists of pressure drag plus the skin friction drag . The pressure drag results from the pressure distribution caused by the shock waves and expansion waves (Fig.3.6) and hence is called wave drag. At supersonic speed the skin friction drag is only a small fraction of the wave drag. The wave drag of a symmetrical aerofoil (Cdw) can be expressed as (Ref.1.7 , chapter 5 ):

58

2 2

2

4 [ ( / ) ]1

dwC t cM

The wave drag of a finite wing at supersonic speeds can also be expressed as KCL

2 ( see Ref.1.7 , chapter 5 for details). However in this case K depends on free stream Mach number (M ), aspect ratio and leading edge sweep of the wing (see Ref.1.7 for details). (11) It can be imagined that the flow past a fuselage will also show that the maximum velocity (Vmax) on the fuselage is higher than V .Consequently, a fuselage will also have a critical Mach number (Mcritf ) which depends on the fineness ratio of the fuselage. For the slender fuselage,

59

typical of high subsonic jet airplanes, Mcritf couldbe around 0.9. Above Mcritf, the drag of the fuselage will be a function of Mach number in addition to the angle of attack.

60

3.3.1 Drag polar of at high speeds

The drag polar of an airplane, which is obtained by the summing the drag coefficients of its major components, will also undergo changes as Mach number changes from subsonic to supersonic. However it is found that the approximation of parabolic polar is still valid at transonic and supersonic speeds, but CD0 and K are now functions of Mach number i.e. :

CD = CD0(M) + K(M)CL2 (3.12)

Detailed estimation of the drag polar of a subsonic jet airplane is presented in Appendix B

61

Remarks:

i) Guidelines for variations of CD0 and K for a subsonic jet transport airplane

Subsonic jet airplanes are generally designed such that there is no significant wave drag up to cruise Mach number ( Mcruise ). However to calculate the maximum speed in level flight (Vmax) or the maximum Mach number (Mmax ), we need guidelines for increase in CDo and K beyond Mcruise .Towards this end we consider the data on B727-100 airplane. Reference 3.8 gives drag polars of B727-100 at M=0.7,0.76,0.82,0.84,0.86 and 0.88. Values of CD

and CL corresponding to various Mach numbers were read and are shown in Fig. 3.12 by symbols.Following the parabolic approximation, these polars

62

were fitted with Eq.(3.12) and CD0 and K were obtained using least square technique. The fitted polars are shown as curves in Fig. 3.12. The values of CD0 and K are given in Table 3.3. and presented in Figures 3.13 (a) & (b).

Table 3.3 Variations of CD0 and K with Mach number

0.049690.052570.061010.068070.081830.103

0.016310.016340.016680.016950.017330.01792

0.70.760.820.840.860.88

KCD0M

63Fig 3.12 Drag polars at different Mach numbers

for B727-100

64Fig 3.13 (a) Parameters of drag polar -CD0 for B727-100

65Fig 3.13 (b) Parameters of drag polar- K for B727-100

66

It is seen that the drag polar and hence CD0 andK are almost constant up to M=0.76. The variations of CD0 and K between M=0.76 and 0.86, when fitted with polynomial curves give the following equations (see also Figures 3.13 a & b).

CD0=0.01634 -0.001( M-0.76)+0.11 (M-0.76)2

K= 0.05257+ (M-0.76)2 + 20.0 (M-0.76)3

Note: For M 0.76 , CD0= 0.01634 , K=0.05257

Based on these trends the variations of CD0 and K beyond Mcruise but up to Mcruise+0.1 can be expressed by Eqs. (3.13a) and (3.14a) and treated as guideline for calculation of Mmax andrange of an airplane (see also Appendix B )

(3.13)

(3.14)

67

CD=CD0cr -0.001 ( M-Mcruise)+0.11 (M-Mcruise)2 (3.13 a)K=Kcr+ (M-0.76)2 + 20.0(M-0.76)3 (3.14 a)Where CD0cr and Kcr are the values of CD0 and K atcruise Mach number. It may be pointed out that the value of 0.01634 in Eq.(3.13) has been replace by CD0cr in Eq.(3.13a). This has been done to permit use of the equation for different types of airplanes which may have their own values of CDcr (see Appendix B). Similar is the reason for using Kcr in Eq.(3.14a).

(2) Variations of CD0 and K for a fighter airplaneReference 1.8 has given drag polars of F-15 fighter airplane at M=0.8,0.95,1.2,1.4 and 2.2.These are shown in Fig 3.14. These drag polars were also fitted with Eq.(3.12) and CD0 and K were calculated. The variations of CD0 and

68

K are shown in Figs.3.15 (a) & (b). It is interesting to note that CD0 has a peak and thendecreases, whereas K increases monotonically with Mach number. It may be recalled that the Mach number, at which CD0 has the peak value, depends mainly on the sweep of the wing.

69

Fig 3.14 Drag polars at different Mach numbers for F15(Adapted from Ref.1.8, chapter2)

Please note : The origins for polars corresponding todifferent Mach numbers are shifted.

70

Fig 3.15(a) Typical variations of CD0 with Mach number for fighter airplane

71

Fig 3.15(b) Typical variations of K with Mach number for a fighter airplane

72

3.4 Drag polar at hypersonic speeds

When the free stream Mach number exceeds five, the changes in temperature and pressure behind the shock waves are large and the treatment of a flow has to be different. Hence the flows with Mach number greater than five are termed hypersonic flow. Reference 3.8 may be referred to for details. For the purpose of flightmechanics it may be mentioned that the drag polar at hypersonic speeds is given by the following modified expression (Ref. 1.1).

CD=CD0(M)+K(M)CL3/2 (3.15)

73

Note that the index of CL term is 1.5 and not 2.0

3.5 Lift to drag ratio

The ratio CL/CD is called lift to drag ratio. It is an indicator of the aerodynamic efficiency of the design of the airplane. For parabolic polar CL/CD

can be worked out as follows.

CD=CD0 +KCL2

Hence CD/CL = (CD0/CL) +KCL (3.16)

Differentiating Eq.(3.16) with CL and equating to zero gives CLmd which corresponds to minimum of (CD/CL) or maximum of (CL/CD).

74

CLmd = (CD0/K)1/2 (3.17)

CDmd = CD0 +K(CLmd)2= 2CD0 (3.18)

(L/D)max = (CLmd/CDmd) = (3.19) Note:

To show that CLmd corresponds to minimum of (CD/CL ), take second derivative of the right hand side of Eq.(3.16) and verify that it is greater than zero.

0

12 DC K

75

3.6 Other types of drag

In sections 3.1,3.2 and 3.3 we discussed the skin friction drag, pressure drag (or form drag), profile drag , interference drag , parasite drag, induced drag, lift dependent drag and wave drag. Following additional types of drags are mentioned briefly to complete the discussion on drag.

I) Cooling drag: The piston engines used in airplanes are air cooled engines. In such a situations when a part of free stream air passes over the cooling fins and accessories, some momentum is lost and this results in a drag called cooling drag.

76

II) Base drag: If the rear end of a body terminates abruptly , the area at the rear is called a base. The abrupt ending causes flow to separate and a low pressure region exists over the base. This causes a pressure drag called base drag.

III) External stores drag: Presence of external fuel tank, bombs, missiles etc. causes additional parasite drag which is called external stores drag.Antennas, lights etc. also cause parasite drag which is called protuberance drag.

IV) Leakage drag: Air leaking into and out of gaps and holes in the airplane surface causes increase in parasite called leakage drag.

77

V) Trim drag: In example 1.1 it was shown that to balance the pitching moment about c.g. (Mcg),the horizontal tail produces a lift (-Lt) in the downward direction. To compensate for this , the wing needs to produce a lift (LW) equal to the weight of the airplane plus the downward load(LW = W+Lt) . Hence the induced drag of the wing, which depends on Lw , would be more than that when the lift equals weight. This additional drag is called trim drag as the action of making Mcg equalto zero is referred to as trimming the airplane.

78

3.7 High lift devices

3.7.1 IntroductionFrom earlier discussion we know that:

Further for an airplane to take-off , the lift must at least be equal to the weight of the airplane , or

Since CL has a maximum value , we define stalling speed (Vs) as:

212 LL V SC

212

2

L

L

L W V SC

WVSC

Hence

(3.20)

(3.21)

(3.22)

79

The take-off speed (VTo) is actually higher than the stalling speed. It is easy to imagine that the take-off distance would be proportional V2

To and in turn to Vs

2. Thus to reduce the take-off distance we need to reduce Vs. Further the wing loading (W/S) is decided by other consideration like cruise. Hence CLmax should be high to reduce take-off and landing distances. The devices to increase CLmax are called high lift devices.

max

2s

L

WVSC

(3.23)

80

3.7.2 Factors limiting CLmax

Consider an airfoil at low angle of attack ( ).Figure 3.16a shows a flow visualization picture of the flow field . Boundary layers are seen on the upper and lower surfaces. As the pressure gradient is low, the boundary layers are attached. The lift coefficient is nearly zero. Now consider the same airfoil at slightly higher angle of attack (Fig.3.16b). The velocity on the upper surface is higher than that on the lower surface and consequently the pressure is lower on the upper surface as compared to that on the lower surface. The airfoil develops higher lift coefficient as compares to that in Fig.3.16a.

81

3.16a Flow past an airfoil at low angle of attack. Note: The flow is from left to right

(Adapted from Ref. 3.10 , chapter 3)

82

3.16b Flow past an airfoil at moderate angle of attack. Note: The flow is from right to left

(Adapted from Ref. 3.11 , part 3 section II B)

83

However the pressure gradient is also higher on the upper surface and the boundary layer separates ahead of the trailing edge (Fig.3.16b) . As the angle of attack approaches about 150 theseparation point approaches the leading edge of the airfoil (Fig.3.16c). Then the lift coefficient begins to decrease (Fig.3.16d) and the airfoil is said to be stalled. The value of for which Cl

equals Clmax is called stalling angle ( stall). Based on these observations , delay of stalling is an important method to increase Clmax. Since stalling is due to separation of boundary layer, many methods have been suggested for boundary layer control. In the suction method the airfoil

84

3.16c Flow past an airfoil at angle of attack near stall. Note: The flow is from left to right

(Adapted from Ref. 3.10 , chapter 3)

85

3.16d Typical Cl vrs curve

86

surface is made porous and boundary layer is sucked (Fig.3.17a) . In the blowing method, fluid is blown tangential to the surface and the low energy fluid in the boundary layer is energized (Fig.3.17b). This effect (energizing ) is achieved in a passive manner by a leading edge slot (Fig.3.17c) and a slotted flap (section 3.7.3) . See Ref.3.13, chapter 11, for other methods of boundary layer control and for further details.

87Fig. 3.17 Boundary layer control with suction and

blowing (Adapted from Ref.3.12, section 9)

88

3.7.3. Ways to increase Clmax

Beside the boundary layer control, there are two other way to increase Clmax viz. increase of camber and increase of wing area. These methods are briefly described below .

I) Increase in Clmax due to change of camber

It may be recalled that when camber of an airfoil increases, the zero lift angle ( ol) decreases and the Cl vrs curve shifts to the left (Fig.3.18) . It is observed that stall does not decrease significantly due to the increase of camber and a higher Clmax isrealized (Fig.3.18). However, the camber of the airfoil used on the wing is chosen such that minimum drag coefficient occurs near the lift coefficient

89Fig. 3.18 Increase in Clmax due to increase of camber

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corresponding to the cruise or the design lift coefficient . One of the ways to achieve the increase in camber during take-off and landing is to use flaps. In a plain flap the rear portion of the airfoil is hinged and is deflected when Clmax is required to be increased (Fig.3.19a) . In a split flap only the lower half of the airfoil is moved down (Fig.3.19b) . To observe the change in camber brought about by a flap deflection, draw a line in-between the upper and lower surfaces of the airfoil with flap deflected. This line is approximately the camber line of the flapped airfoil. The line joining the ends of the camber line is the new chord line . The difference between the ordinates of the camber line and the chord line is a measure of camber.

91Fig. 3.19 Flaps, slot and slat

(Adapted from Ref.3.7 , chapter 12)

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II) Increase in Clmax due to boundary layer control

In a slotted flap (Fig.3.19c) the effects of camber change and the boundary layer control are brought together. In this case when the flap is deflected a gap is created between the main surface and the flap (Fig.3.19c) . As the pressure on the lower side of airfoil is more than that on the upper side, the air from the lower side of theairfoil rushes to the upper side and energizes the boundary layer on the upper surface. This way the separation is delayed and Clmax increases(Fig.3.20). The slot is referred to as a passiveboundary layer control , as no blowing by external source is involved in this devise.

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Fig.3.20 Effects of camber change and boundary layer control on CLmax

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After the success of single slotted flap , the double slotted and triple slotted flaps were developed

(Figs.3.19 d and e).

III) Increase in Clmax due to change in wing area

Equation (3.20) shows that the lift can be increased when the wing area (S) is increased. An increase in wing area can be achieved if the flap, in addition to being deflected, also moves outwards and effectively increases the wing area. This is done in a Fowler flap (Fig.3.19 f) . Thus a Fowler flap incorporates three methods to increase Clmax viz change of camber, boundary layer control and increase of wing area. It may be added that while defining the Clmax in case of Fowler flap, the reference area is the original area

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of the wing and not that of the extended wing.

A zap flap is a split flap where the lower portion also moves outwards as the flap is deflected.

IV) Leading edge devices

High lift devices are also used near the leading edge of the wing. A slot near the leading edge (Fig.3.19 g) also permits passive way of energizing the boundary layer. However a permanent slot has adverse effects during cruise. Hence leading edge slat as shown in Fig.3.19h is used . When deployed it produces a slot and increase Clmax by delaying separation.

On high subsonic speed airplanes , both leading edge and trailing edge devices are used to increase Clmax(Fig.3.2.).