Unmanned Aerial Vehicles

The fluid flowing around an object exerts a force on it. You can split the force in two parts:

• one perpendicular to the flow direction, which is called lift.
• and one in the opposite direction of movement - the drag.

In air, we call these forces aerodynamic forces.

Lift and drag force of an airfoil in laminar flow. Illustration by author.

Here we only focus on the drag force. The drag force is usually an undesirable force - it slows your object down or in other words you need more energy to advance in the fluid and loose in efficiency. Therefore engineers usually try to minimize the drag - if it is a car, a boat or an airplane. The same goes for drones. But what exactly does cause drag? The cause of drag can be decomposed into two effects:

1. shear stresses acting tangential to the surface and are caused by frictional forces due to the fluids viscosity. This is the friction drag.
2. pressure stresses acting perpendicular to the surface and are caused by how the pressure is distributed around a object. This is the pressure drag or sometimes also called form drag.

The sum of these two effects in the direction of movement is the drag.

Pressure Drag

Pressure distribution of a ball moving through a fluid. Illustration by author.

Pressure drag is the strongest for blunt bodies like a ball. It is caused by a difference in pressure in front and rear of an object. While air in front of the body is compressed and thus pressure increases, the air behind the body becomes separated and turbulent, leading to a region of lower pressure. Pressure drag increases substantially in case of flow separation, which is the case when the fluid’s boundary layer detaches from the body. This causes a recirculating flow, significantly decreasing the pressure behind the body. This is called the separation region. To reduce drag forces, it is desirable to minimize flow separation at all cost. Flow separation can also lead to vortex creation, which can lead to instability and turbulence. Why does flow separation occur to begin with?

Flow separation on rear side of a ball moving through a fluid. Illustration by author.

When the fluid passes over the surface of the sphere it initially accelerates and the pressure decreases in the flow direction. Beyond a certain point the flow then decelerates and the pressure starts increasing. The increase in pressure pushes the fluid backward. But due to the oncoming fluid it cannot travel back, forcing it to detach from the surface, resulting in flow separation. Flow separation occurs at 80° for a smooth sphere in laminar flow. In case of a turbulent flow it can be delayed until 120°, which drastically reduces the drag. This is because the mixing between different flow layers transfers momentum to the fluid, allowing them to sustain a larger pressure difference. That is why for example a golf ball has dimples instead of being completely smooth. The turbulence caused them delays flow separation, reduces drag and hence allows the ball to travel further. Bodies traveling through fluid like plane wings or racing cars are usually designed in a tear-drop shape to minimize flow separation. Flow separation is delayed so much or doesn’t occur at all, that pressure drag is greatly reduced. For these type of objects it’s the shear stresses that contribute most to the total drag force.

Friction Drag

Shear stress over an airfoil caused by flow of fluid. Illustration by author.

Friction drag increases with the viscosity of the fluid and the surface of area aligned with the direction of flow. While turbulence decreases pressure drag by delaying the flow separation, it has the opposite effect on friction drag. Laminar and turbulent boundary layers show very different velocity profiles. Turbulence boundary layers have higher velocity gradients and thus produces larger shear stress. Hence to reduce friction drag, you want to maintain laminar flow for the largest possible distance around the object.

The figure illustrates typical velocity profiles for laminar and turbulent flow along a surface. Schema from Landis (2018). CFD Analysis of RAM Air Flow in an Aircraft Air Conditioning System.

If you would manage to maintain laminar flow over the wings of commercial aircraft, you could reduce the total drag by 10-15%. But this is very hard to achieve and is an open question in research. One idea that was partially successful is the so called hybrid-laminar flow control, where air is suck downwards along the surface of the wing. Another possibility is to reduce the effect of turbulent flow on friction drag. One interesting research aspect there looks at the microstructure of shark skin.

We have seen that the magnitude of friction and pressure drag depends on the surface of a body relative to the direction of flow. An obvious example is flat plat at 90° angle to the direction of flow. The flow separates easily, creating a separation region and the pressure drag is large. In this case friction drag is almost zero, since shear stresses are not aligned with the drag direction. However if you turn the plate by 90° such that the surface is aligned with the direction of flow, we have a very streamlined body and the pressure drag is small. But the friction is now much more significant.

The figure shows the influence of the angle of attack on flow separation for an airfoil. Schema from NASA.

The same logic applies to airfoils, where the angle of attack - i.e. the angle between the centerline of a wing and the direction of flow - has a large influence on the drag force. At high angles of attack (AOA) separation occurs, which significantly increases the drag force. In general it is important to remember that friction drag increases as pressure drag decreases and so these two aspects need to be carefully balanced.

Drag Force Calculation

If you would integrate the pressure stress and wall shear stresses of an object, you would obtain the exact drag force. However those information are almost never available. That is why usually the drag equation is used to represent the total drag force: $$D=C_D\frac{1}{2}\rho A v^2$$

$D$: drag force
$C_D$: drag coefficient, depending on the morphology of the object
$\rho$: fluid density
$A$: wing area
$v$: airspeed

The $C_D$ drag coefficient includes all of the hard to measure parameters such as object geometry or the effect of flow regime. It can be determined experimentally by running wind tunnel experiments or by running numerical simulations. $\rho$ is the fluids density, $v$ the relative velocity of the fluid to the object and usually assumed to be steady and uniform. Finally $A$ is a reference area, that depends on how the drag coefficient is determined. For airfoils it is usually the object’s planform area. For blunt bodies it is usually the projected frontal area.

Question 1: Pressure drag is caused by shear stresses acting along the surface.

Question 2: Does using the most streamlined shape, always result in the lowest drag?

Question 3: Which changes would likely help reduce total aerodynamic drag on a vehicle?

EXERCISE 1:

A small drone is flying at a constant speed of 15 m/s at sea level. The drone has:

• A frontal area $A=0.2 m^2$
• A drag coefficient $C_D=0.9$ ​

The air density at sea level is $ρ=1.225 kg/m^3$.

Calculate the drag force acting on the plane.

Humans have always been fascinated by the possibility to fly. Once thought to be impossible, heavier-than-air flight is only a reality because of the lift generated by aircraft wings. But lift is a complicated topic, and even to this day engineers do not entirely agree on about how it’s created. So what exactly is lift?

Forces on an Airfoil:

Lift and drag force of an airfoil in laminar flow. Illustration by author.

When fluid flows past an object, or an object like the plane wing on the image above moves through a stationary fluid, the fluid exerts a force on the object, which can be split into two components:

1. acting in the same direction as the fluid flow, called drag,
2. and a component acting perpendicular to the flow direction, called lift.

When talking about lift we’re mostly interested in streamlined bodies like an airfoil, which are designed to produce a lot of lift, but to minimize drag at the same time. Airfoils aren’t just found on airplane wings — they’re used in wind turbines, propellers, and even Formula 1 cars. They come in a huge range of shapes and sizes. One designed for an aircraft wing won’t be optimized for a propellor blade, for example. And a wing designed to fly at supersonic speeds will have a very different profile compared to one designed to fly slower than the speed of sound.

Airfoil Parameters

Chord, camber, trailing and leading edge and the angle of attack presented on an airfoil in laminar flow. Illustration by author.

Airfoil profiles can be defined using a few different parameters. The forward-most edge of the airfoil is called the leading edge, and the trailing edge is at the back of the airfoil. Drawing a straight line between the leading and trailing edges gives us the chord line. The angle between the chord line and the flow direction is called the angle of attack. Drawing a line which is midway between the upper and lower surfaces gives us the mean camber line. Camber describes how curved an airfoil is. We can have positive camber or negative camber, and a symmetrical airfoil has zero camber. Camber and the angle of attack are important parameters that will have a large influence on how much lift an airfoil can generate.

How Lift is Generated

So how does a humble teardrop shape generate enough force to lift heavy aircraft off the ground? As the fluid flows around the airfoil it creates two different types of stress which act on its surface.

1. First we have the wall shear stresses. These stresses act tangential to the object’s surface, and are caused by the frictional forces that act on the airfoil because of the fluid’s viscosity.
2. Then we have the pressure stresses. They act perpendicular to the object’s surface, and are caused by how pressure is distributed around it.

Lift is the sum of these two stresses in the direction perpendicular to the flow. The only way a fluid can create a force onto an object is through these stresses. For streamlined bodies like airfoils, the shear stresses will mostly be acting in the same direction as the flow. They make a large contribution to the drag force, but won’t contribute a significant amount to the lift force. Thus, we neglect them and say that the lift acting on an airfoil is mainly caused by the way pressure is distributed around it.

Pressure Distribution:

The figure shows the pressure distribution over a typical cambered airfoil. Schema from Chakraborty, Manash (2015). A Computational Study on two horizontally close sequential airfoils to determine conjoined pressure distribution and aerodynamic influences on each other.

You can see a typical pressure distribution on the image above. The pressure is low above the airfoil and high below it, which creates a net force with a large component in the lift direction. Also note that the low pressure on the top surface is larger in magnitude than the high pressure on the bottom surface. So the suction pressure on the top surface is what contributes most to the total lift force, in fact it’s usually around 2/3 of it. We can also see that the majority of the pressure difference is coming from the forward-most part of the airfoil.

In truth there’s nothing particularly special about the shape of an airfoil that allows it to generate lift. Any object that creates an uneven pressure distribution will generate a force in the lift direction, like a flat plate at an angle relative to the flow, for example. Airfoils are just optimized shapes that have been carefully designed to have high lift-to-drag ratios. Without a difference in pressure above and below an object there can be no lift. A symmetrical body like a bullet or a ball does not generate any lift force because there’s no pressure difference around it.

But where does the pressure distribution come from? The answer to this question is complex, and there is still much debate about the best way to explain it in a concise way. We can broadly split the different explanations into two groups - those based on Bernoulli’s Principle and those based on Newton’s third law.

Bernoulli’s Principle:

The figure shows the velocity distribution over a typical cambered airfoil. Schema from Petinrin (2017). Computational Study of Aerodynamic Flow over NACA 4412 Airfoil. British Journal of Applied Science & Technology.

Bernoulli’s Principle relates pressure and velocity of a fluid. As it can be seen on the image above, the leading edge of an airfoil creates a stagnation point where velocity drops to zero. Fluid flowing above that point on the top surface travels faster than the fluid flowing on the bottom surface. After Bernoulli’s Principle, an increase in speed occurs simultaneously with a decrease in pressure. That means the greater increase in velocity on the top surface creates a zone of lower pressure and below the bottom surface a zone of higher pressure.

But what causes the velocity difference? Two explanations are common:

1. One explanation is that the geometry of an airfoil causes the flow to be pinched together above the airfoil, but not below it. Because of the conservation of mass, this results in increased velocity above the airfoil.
2. A more complete but less intuitive explanation for the difference in velocity is based on the concept of circulation. The flow around an airfoil can be thought of as the superposition of idealized uniform irrotational flow, and circulatory flow. If we impose a condition that the flow above and below the airfoil must be parallel when leaving the trailing edge, we can calculate the exact amount of circulation that must be generated by the airfoil to do this. This is called the Kutta condition. Circulation has the effect of accelerating the flow above the airfoil and delaying the flow below it.

Newtons Third Law:

A second type of explanations are based on Newton’s third law and the momentum exchange. If we look at a wider area we can observe that the effect of an airfoil can be felt far beyond its immediate vicinity. Upstream of the airfoil the flow is being swept upwards, which is called upwash. And downstream the flow is deflected downwards, which is called downwash. A very large volume of air is being displaced by the airfoil. Newton’s third law tells us that for every action there is an equal and opposite reaction. The airfoil must be imparting a force on the air to create the downwash, and so based on Newton’s third law, there must be a corresponding reaction force acting on the airfoil. In other words an airfoil generates lift by turning more incoming air downwards than upwards.

In summary, a lift force acts on an airfoil because of the pressure distribution around it. The exact cause of this pressure distribution is complex, and can be explained in several different ways, which approach the problem from different angles. Explanations based on Bernoulli’s Principle and on Newton’s Third Law provide valuable insight into how lift is generated, although both approaches have limitations. Nevertheless, these explanations are useful and can lead to a more intuitive understanding of lift.

Stall and Angle of Attack

We can easily imagine for example that increasing the camber or the angle of attack of an airfoil will allow it to deflect a larger amount of fluid downwards, and so will increase the lift force. However beyond a critical angle of attack, we can observe a sudden decrease in the lift force and increase in the drag force. At this angle of attack the boundary layer is no longer able to remain attached to the airfoil and it detaches from the surface creating a wake behind it which affects the pressure distribution around the airfoil, significantly reducing lift and increasing drag. Flow separation is explained in more detail in the section about aerodynamical drag. The sudden reduction in lift is called stalling, and it can be very dangerous for aircraft.

Effect of camber and angle of attack on lift coefficient of an airfoil. Schema from AeroToolbox.

Different airfoil shapes can have drastically different lift characteristics. For example if an airfoil is symmetrical, and so has zero camber, the lift force will be zero for zero angle of attack - as for any symmetrical body. Aerobatic aircraft usually use symmetrical airfoils since they allow planes to fly upside down more easily. Lift is then solely generated by adjusting the angle of attack.

Control Surfaces

Illustration of a trailing edge flap for an airfoil. Original image from Wikimedia.

Airbus A310-300 with leading edge slats and trailing edge flaps. Wikimedia.

Modern aircraft wings are equipped with control surfaces - flaps and slats - which allow the shape of the airfoil to be adjusted and optimized for the different phases of flight through mechanical actuation. During take-off for example you want high lift. Extending the flaps increases the camber of the wing, which increases lift, and so flaps are extended during take-off. But the extra lift comes at the expense of increased drag, and so the flaps are retracted when cruising, since high lift is no longer needed and drag should be minimized to improve fuel consumption.

Question 1: Lift is primarily generated due to the pressure difference between the upper and lower surfaces of a body.

Question 2: Which of the following factors contribute to generating lift on a wing or airfoil? (Select all that apply)

Question 3: Flow separation on the upper surface of an airfoil usually leads to:

EXERCISE 1:

Imagine a Boeing-777 is flying at an altitude of $500 m$ with a speed of $400 km/h$. The total weight of the airplane is $250 t$ and the wing area is $427 m²$.

Remark:

The image below shows the density of air with respect to the altitude.

Change of air density with altitude. Schema from Sabrije Osmanaj.

Question 1: What is the lift coefficient of the airplane at 500m, supposed it flies straight?

Question 2: How much faster must the airplane fly at 10,000 m to maintain level flight (same lift force)?

Question 3: How much does the drag increase when flying at 10,000 m?

Question 1: Thrust using a propeller is generated because the blades create a pressure difference between the front and rear of the propeller.

Question 2: Which of the following is true about the relationship between propeller rotation speed and thrust generation? (Select all that apply)

Question 3: What is the primary purpose of a variable-pitch propeller?