# Fluid Mechanics

In order to begin to understand the complexities of designing a small UAV, it is important to first have some grasp of the fundamentals of the fluid (air) within which they operate. This page is a short guide to the most relevant principles, and is not meant to be a reference or comprehensive source for understanding fluid mechanics. It can, however, be a great starting point for understanding important concepts. These concepts will help you to have a basic idea of what is really going on in flight and to develop your own intuition for making design decisions throughout your project.

## Fluid Flow

By definition, both gases and liquids are fluids. These are classified together since priniples of fluid mechanics apply to both accurately. Both gases and liquids have weaker molecular forces than solids, creating an indefinite shape and volume. It is typically taught that liquids have an indefinite volume, but this is incorrect since they can be compressed if enough pressure is applied. As a reference, air is about 22,000 times more compressible than water. The weaker molecular forces in all fluids allow them to move and create a flow through or around a volume (such as an airplane).

When a fluid flows, a *no-slip condition* is assumed wherever it contacts a solid. This means that the flow velocity of the fluid at any point it contacts a solid is zero. The velocity increases outward from the surface until it gets to the *free stream velocity* (U_{∞}). This occurs in a quadratic fashion, as shown in Figure 1. The portion of the flow that has a velocity profile is of particular interest to us because this is where the aircraft is dealing with aerodynamic forces. It is referred to as the *boundary layer*.

### The Boundary Layer

As a fluid flows past a solid, the boundary layer “sticks” to the surface, creating a layer of slower moving fluid around the solid. This layer is typically only a few millimeteres thick. It is generally accepted that the boundary layer ends when the fluid velocity is 0.99(U_{∞}). As the fluid flows farther down the surface, the boundary layer grows in thickness. Each horizontal point along the boundary layer maintains a quadratic velocity profile, increasing in size with the thickness. The boundary layer continues to grow until either it separates from the surface or the surface ends. Much of aerodynamic design is focused on ensuring that the boundary layer stays attached to the airplane as much as possible.

Within the boundary layer, one of the main forces being dealt with is shear stress. This is essentially a force acting parallel to the surface whose magnitude depends on the rate at which the velocity is changing and properties of the fluid present. In the equation below, *τ* is the shear stress, *µ* is the dynamic viscosity of the fluid, *u* is the fluid velocity relative to the surface, and *y* is the perpendicular distance from the surface. As you can see, an increase in the rate at which the velocity changes with position would result in an increase in shear stress. This in turn causes an increase in parasitic drag on the airplane, which is comprised of skin friction and pressure drag. Further discussion on types of drag can be found in the section on drag.

### Laminar and Turbulent Flow

To delay boundary layer separation, it is desirable to maintain a laminar boundary layer for a portion of the airfoil, and then allow a transition to turbulent. Whether a fluid flow is laminar or turbulent depends upon a few different factors that are related in what is called the *Reynolds number*. The Reynolds number is a nondimensional parameter that is, on a fundamental level, the ratio of intertial to viscous forces in a fluid. In the equation below, *ρ* is the density of the fluid, *v* is the velocity, *d* is the characteristic length (chord for airfoils, mean aerodynamic chord for wings), and *µ* is again the dynamic viscosity.

A smaller Reynolds number indicates that the flow is *laminar*, meaning it is steady, smooth, and mixes very little through intermolecular forces. A larger Reynolds number indicates that the flow is *turbulent*, meaning it is unsteady, disordered, and mixes rapidly. It is also important to note that the shear stress in turbulent flow is higher because the velocity profile is steeper. For external flow, the transition from laminar to turbulent flow occurs at a value of around 500,000. A typical small UAV will operate with a Reynolds number on the order of 10,000. For flow around an airfoil, both types of flow will be present.

## Airfoils in Fluids

An essential characteristic about fixed-wing UAVs is their ability to produce lift almost entirely from their shape. This is accomplished by the thin, cambered profile of the airfoils used for the lifting surfaces. An airfoil’s shape produces lift due to some important principles of fluid flow that can be measured with reasonable accuracy by the *Bernoulli Equation*.

### The Bernoulli Equation

The fundamental governing equation of fluid mechanics is referred to as the *Navier-Stokes Equation*. This equation cannot be solved analytically, so a number of assumptions can be made under certain circumstances to produce the Bernoulli Equation. The assumptions are as follows:

- Steady flow
- Incompressible fluid
- No heat transfer
- Streamlines
- Inviscid flow
- No work is being done

These assumptions can be loosely true for a small UAV, or at least inconsequential enough that the assumption can be made. Let’s go over a few of the less obvious assumptions. We are assuming that the air around the UAV is incompressible. Although some compression is likely taking place, compressibility drag doesn’t become significant until about Mach 0.7. Streamlines are lines of fluid flow around an object, along which the Bernoulli Equation can be applied. Although it is not fully accurate to say the flow is inviscid (this would eliminate all viscous drag), we will still make this assumption to use the Bernoulli Equation. These assumptions and the others listed allow us to use the equation below for calculating properties of a fluid around an airfoil. For this equation, *V* is the local velocity, *P* is the local pressure, *ρ* is the fluid’s density, *g* is the gravitational constant, and *z* is the local vertical position.

### Lift

The Bernoulli Equation allows us to compare properties at different points along a streamline. This can be done around an airfoil, allowing us to determine a few important facts. When air flows over the top of an airfoil, we see a slight increase in vertical position and velocity, but a decrease in pressure. The exact opposite is true along the bottom of the airfoil, but to a lesser magnitude. This creates a net upward pressure gradient that produces what we refer to as lift. In the case of a symmetric airfoil (in level flight), the pressure change on each side is equal and opposite, creating a net zero pressure gradient and therefore no lift.

The lift vector is, by definition, perpendicular to the free stream (not necessarily to the airfoil itself). This puts lift in direct opposition to gravity under normal conditions. As the angle of attack is increased, the net upward pressure gradient increases, producing more lift. At higher angles of attack (typically about 12-15°) the airfoil begins to stall and lose lift. This occurs due to the boundary layer separating earlier along the airfoil due to viscous effects. Figure 3 shows the streamlines around an inclined airfoil near stall. Notice that the wake behind the airfoil also becomes quite large, increasing drag and energy lost to the air.

### Drag

In contrast to the lifting force, drag is defined as being parallel to the free stream. For small UAVs, its main source is produced by skin friction. This is a result of the shear stress acting on the surface of the airplane. Laminar flow produces significantly lower shear stress, making turbulent flow undesirable in terms of skin friction. An airfoil that can delay boundary layer separation longer will have more laminar flow, and therefore less skin friction.

Another significant source of drag is pressure drag. Since air is viscous, some energy is lost while it passes over an airfoil. What this means is that if we follow a streamline of air particles over the airfoil, it will not return to the initial vertical position past the trailing edge. This is similar to how a ball rolling down a hill can’t return to the same height on the next hill to energy being lost to friction. The result is a wake that contains energy lost to the air. This also means there is a net pressure gradient from back to front, producing pressure drag.

The rest of the drag force on small UAVs is related to how the vehicle produces lift, and is called induced drag. It is impossible to eliminate induced drag because it is coupled to lift. When a wing generates lift, it is not exactly perpendicular to the free stream. The component of the induced lift vector that is parallel to the free stream is the induced drag. This can also be thought of as energy being left behind in the air from the downwash of the UAV. It is this same principle that results in the vortices trailing behind the wingtips (see Figure 4).