To Calculate the Lift and Drag Force on an Airfoil

What is Aerodynamics?

Aerodynamics is the study of motion of air, particularly as interaction with a solid object, such as an airplane wing. It is a sub-field of fluid dynamics and gas dynamics, and many aspects of aerodynamics theory are common to these fields. The term aerodynamics is often used synonymously with gas dynamics, the difference being that "gas dynamics" applies to the study of the motion of all gases, and is not limited to air.

Compressible Aerodynamics:

According to the theory of aerodynamics, a flow is considered to be compressible if the density changes along a streamline. This means that – unlike incompressible flow – changes in density are considered. In general, this is the case where the Mach number in part or all of the flow exceeds 0.3.

There are 4 types of flows:

1. Sub-Sonic Flow
2. Transient Flow
3. Super-Sonic Flow
4. Hyper-Sonic Flow

Sub-Sonic Flow: It is a type of flow where the veloity of the object is less than the velocity of air. Sice, the Mach Number is the ratio of velocity of the object to the veloctiy of the air, so the Mach Number is less than zero for the sub-sonic flows.

Transient Flow: The term Transient refers to a range of flow velocities just below and above the local speed of sound (generally taken as Mach 0.8–1.2). It is defined as the range of speeds between the critical Mach number, when some parts of the airflow over an aircraft become supersonic, and a higher speed, typically near Mach 1.2, when all of the airflow is supersonic. Between these speeds, some of the airflow is supersonic, while some of the airflow is not supersonic.

Super-Sonic Flow: It is a type of flow where the velocity of the object is greater thean the velocity of the air. Supersonic flow behaves very differently from subsonic flow. Fluids react to differences in pressure.

Thus, when the fluid finally reaches the object it strikes it and the fluid is forced to change its properties – temperature, density, pressure, and Mach number—in an extremely violent and irreversible fashion called a shock wave. The presence of shock waves, along with the compressibility effects of high-flow velocity fluids, is the central difference between the supersonic and subsonic aerodynamics regimes.

Hyper-Sonic Flow: Hypersonic speeds are speeds that are highly supersonic. This flow takes place when the speeds of object is 5 times the speed of the sound i.e. Mach 5. The hypersonic regime is a subset of the supersonic regime. Hypersonic flow is characterized by high temperature flow behind a shock wave, viscous interaction, and chemical dissociation of gas.

Parameters of Aerodynamics:

Lift: A fluid flowing past the surface of a body exerts a force on it. Lift is the component of this force that is perpendicular to the oncoming flow direction. Lift conventionally acts in an upward direction in order to counter the force of gravity, but it can act in any direction at right angles to the flow. While the common meaning of the word "lift" assumes that lift opposes weight, lift can be in any direction with respect to gravity, since it is defined with respect to the direction of flow rather than to the direction of gravity.

A fluid flowing past the surface of a body exerts a force on it. It makes no difference whether the fluid is flowing past a stationary body or the body is moving through a stationary volume of fluid. Lift is the component of this force that is perpendicular to the oncoming flow direction. Lift is always accompanied by a drag force, which is the component of the surface force parallel to the flow direction.

Drag:  In fluid dynamics, drag (sometimes called air resistance, a type of friction, or fluid resistance) is a force acting opposite to the relative motion of any object moving with respect to a surrounding fluid. This can exist between two fluid layers (or surfaces) or a fluid and a solid surface.

Drag force is proportional to the velocity for a laminar flow and the squared velocity for a turbulent flow. Even though the ultimate cause of a drag is viscous friction, the turbulent drag is independent of viscosity.

Aerodynamic Drag Components:

Drag is the force experienced by an object representing the resistance to its movement through a fluid. Sometimes called wind resistance or fluid resistance, it acts in the opposite direction to the relative motion between the object and the fluid. The example opposite shows the aerodynamic drag forces experienced by an aerofoil or aircraft wing moving through the air with constant angle of attack as the air speed is increased

The Total Aerodynamic Drag is the sum of the following components:

• Induced Drag - Due to the vortices and turbulence resulting from the turning of the air flow and the downwash associated with the generation of lift. Increases with the angle of attack. Inversely proportional to the square of the air speed. Decreases as aircraft speed increases and the angle of attack is reduced. Induced drag associated with the high angle of attack needed to maintain the lift is dominant at low air speeds.
• Form Drag or Pressure Drag - Due to the size and shape of the aerofoil. Increases with the square of air speed. Streamlined shapes designed to reduce form drag.
• Friction Drag - Arises from the friction of the air against the "skin" of the aerofoil moving through it. Increases with the surface area of the aerofoil and the square of air speed.
• Profile Drag or Viscous Drag- The sum of Friction Drag and the Form Drag.
• Parasitic Drag or Interference Drag - Incurred by the non-liftting parts of the aircraft such as the wheels, fuselage, tail fins, engines, handles and rivets. Increases with the square of air speed. Parasitic drag becomes dominant at higher air speeds.
• Wave Drag - Due to the presence of shock waves occurring on the blade tips of aircraft and projectiles. Associated with passing the sound barrier it is a sudden and dramatic increase in drag which only comes into play as the vehicle increases speed through transonic and supersonic speeds. Independent of viscous effects.

Newton's Theory of Flight 

Newton's Second Law of Motion states that:

• The force on an object is equal to its mass times its acceleration or equivalently to its rate of change of momentum

                                                             F = M a = d/dt (M v)

In other words, whenever there is a change of momentum, there must be a force causing it. 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 Law of Motion 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.

Bernoulli's Theory of Flight 

The Theory of Flight is often explained in terms Bernoulli's Equation which is a statement of the Conservation of Energy. It states that:

• For a non-viscous, incompressible fluid in steady flow, the sum of pressure, potential and kinetic energies per unit volume is constant at any point. It states that there is a direct mathematical relationship between the pressure of a fluid and the speed of that fluid

In other words, ignoring the potential energy due to altitude:

• When the velocity of a fluid increases, its pressure decreases by an equivalent amount to maintain the overall energy. This is known as Bernoulli's Principle

According to Bernoulli's Principle, the air passing over the top of an aerofoil or wing must travel further and hence faster that air the travelling the shorter distance under the wing in the same period but the energy associated with the air must remain the constant at all times. The consequence of this is that the air above the wing has a lower pressure than the air below below the wing and this pressure difference creates the lift.

Pressure Differences:

Pressure is the normal force per unit area exerted by the air on itself and on surfaces that it touches. The lift force is transmitted through the pressure, which acts perpendicular to the surface of the airfoil. Thus, the net force manifests itself as pressure differences. The direction of the net force implies that the average pressure on the upper surface of the airfoil is lower than the average pressure on the underside.

The left side of this equation represents the pressure difference perpendicular to the fluid flow. On the right hand side ρ is the density, v is the velocity, and R is the radius of curvature. This formula shows that higher velocities and tighter curvatures create larger pressure differentials and that for straight flow (R → ∞) the pressure difference is zero.

Airfoils:

A structure with curved surfaces designed to give the most favorable ratio of lift to drag in flight, used as the basic form of the wings, fins, and horizontal stabilizer of most aircraft.

Most foil shapes require a positive angle of attack to generate lift, but cambered airfoils can generate lift at zero angle of attack, whereas a symmetrical airfoil will generate zero lift at zero angle of attack. This "turning" of the air in the vicinity of the airfoil creates curved streamlines, resulting in lower pressure on one side and higher pressure on the other. This pressure difference is accompanied by a velocity difference, via Bernoulli's Principle, so the resulting flow-field about the airfoil has a higher average velocity on the upper surface than on the lower surface.

Airfoil Terminology:

The various terms related to airfoils are defined below:

• The suction surface (a.k.a. upper surface) is generally associated with higher velocity and lower static pressure.
• The pressure surface (a.k.a. lower surface) has a comparatively higher static pressure than the suction surface. The pressure gradient between these two surfaces contributes to the lift force generated for a given airfoil.

The geometry of the airfoil is described with a variety of terms:

• The leading edge is the point at the front of the airfoil that has maximum curvature (minimum radius).
• The trailing edge is defined similarly as the point of maximum curvature at the rear of the airfoil.
• The chord line is the straight line connecting leading and trailing edges. The chord length, or simply chord, is the length of the chord line. That is the reference dimension of the airfoil section.

Angle of Attack:

The angle of attack is the angle between the chord line of an airfoil and the oncoming airflow. A symmetrical airfoil will generate zero lift at zero angle of attack. But as the angle of attack increases, the air is deflected through a larger angle and the vertical component of the airstream velocity increases, resulting in more lift. For small angles a symmetrical airfoil will generate a lift force roughly proportional to the angle of attack.

As the angle of attack increases, the lift reaches a maximum at some angle; increasing the angle of attack beyond this critical angle of attack causes the upper-surface flow to separate from the wing, there is less deflection downward so the airfoil generates less lift. The airfoil is said to be stalled.

Airfoil Shape:

An airfoil with camber compared to a symmetrical airfoil

The shape of the airfoil is defined using the following geometrical parameters:

• The mean camber line or mean line is the locus of points midway between the upper and lower surfaces. Its shape depends on the thickness distribution along the chord;
• The thickness of an airfoil varies along the chord. It may be measured in either of two ways:
• Thickness measured perpendicular to the camber line.This is sometimes described as the "American convention"
• Thickness measured perpendicular to the chord line. This is sometimes described as the "British convention".

Some important parameters to describe an airfoil's shape are its camber and its thickness. For example, an airfoil of the NACA 4-digit series such as the NACA 2415 (to be read as 2 – 4 – 15) describes an airfoil with a camber of 0.02 chord located at 0.40 chord, with 0.15 chord of maximum thickness.

The lift force depends on the shape of the airfoil, especially the amount of camber (curvature such that the upper surface is more convex than the lower surface) Increasing the camber generally increases lift.

Cambered airfoils will generate lift at zero angle of attack. When the chord line is horizontal, the trailing edge has a downward direction and since the air follows the trailing edge it is deflected downward. When a cambered airfoil is upside down, the angle of attack can be adjusted so that the lift force is upwards.

Flow conditions

The ambient flow conditions which affect lift include the fluid density, viscosity and speed of flow. Density is affected by temperature, and by the medium's acoustic velocity - i.e. by compressibility effects.

Air speed and density

Lift is proportional to the density of the air and approximately proportional to the square of the flow speed. Lift also depends on the size of the wing, being generally proportional to the wing's area projected in the lift direction. In calculations it is convenient to quantify lift in terms of a lift coefficient based on these factors.

Stalling:

Airflow separating from a wing at a high angle of attack

An airfoil's maximum lift at a given airspeed is limited by boundary-layer separation. As the angle of attack is increased, a point is reached where the boundary layer can no longer remain attached to the upper surface. When the boundary layer separates, it leaves a region of recirculating flow above the upper surface. This is known as the stall, or stalling. At angles of attack above the stall, lift is significantly reduced, though it does not drop to zero. The maximum lift that can be achieved before stall, in terms of the lift coefficient, is generally less than 1.5 for single-element airfoils and can be more than 3.0 for airfoils with high-lift slotted flaps and leading-edge devices deployed.

An Experimental Test is conducted to calculate the Lift and Drag force for a NACA Airfoil for different Angle of Attack starting from (0-10) degrees.

1. Angle of Attack = 0 Degrees

Velocity Plot:

Pressure Plot:

Goal Plots for Lift and Drag Force:

The Goal Plot clearly shows that at the angle of attack of 0 degree the drag force is greater than the lift force, which is evident from the fact that the symmetrical airfoils at 0 degree produces minimal lift.

2. Angle of Attack = 2 Degrees

Velocity Plot:

Pressure Plot:

Goal Plots for Lift and Drag Force:

Now, as we have increased the Angle of Attack from 0 to 2 degrees, we observe that the lift force increases to a significant value greater than the drag force. Since, Lift is proportional to the square of the velocity of an airplane and as a plane goes faster, its lift increases and overcomes the air resistance. As a plane moves forward, its lift force increases until it equals its weight. When lift equals weight, the plane can fly.

3. Angle of Attack = 4 Degree

Velocity Plot:

Pressure Plot:

Goal Plot for Lift and Drag Force:

Now, as we have increased the Angle of Attack from 2 to 4 degrees, we observe that the lift force has nearly been doubled from the previous value and the drag force almost remains the same.

4. Angle of Attack = 6 degree                  

Velocity Plot:

Pressure Plot:

Goal Plot for Lift and Drag Force:

Now, as we have increased the Angle of Attack from 4 to 6 degrees, we observe that the lift force is gradually increasing. The lift force keeps on increasing until it balances the weight of the object.

5. Angle of Attack = 8 Degrees

Velocity Plot:

Pressure Plot:

Goal Plot for Lift and Drag Force:

6. Angle of Attack = 10 degrees

Velocity Plot:

Pressure Plot:

Goal Plot for Lift and Drag Force:

Conclusion:

1. The Angle of Attack for the Airfoil increases, the Lift force also increases.

2. The Flow Simulation of the airfoil seems to be complying with the Bernoulli's Principle i.e. the region where the pressure decreases, the veloctiy increases and the vice-versa.

3. Since the lift force is proportional to the square of velocity, the velocity also increases with every increase in AOA, thus increasing the Mach Number. The flow of the airfoil is Super-Sonic.