The lift exerted on the wing can be explained in terms of the wing curvature, flow velocity, fluid density, and angle of attack.
This pressure difference is created by air flowing across the surfaces of the wing, where a greater velocity, denser fluid, or both will create greater lift.
As angle of attack increases, lift generally increases until flow separation occurs and the aircraft stalls.
Probably every child that has flown on an airplane has looked out the window and wondered how an aircraft can seem to float above the ground. In the simplest terms, it has to do with the motion of the craft and its wing shape. The responsible force, aerodynamic lift, is explained in terms of these factors and the behavior of the surrounding fluid during flow. However, these same factors determine the drag force on the aircraft, which must be considered alongside lift to describe the forces on an aircraft. These points can be examined in CFD simulations of fluid flow along the surface of an airfoil and determine the pressure field that provides lift.
Aerodynamic Lift Explained in Terms of Streamlines
As should be obvious from its name, lift is the force in aerodynamics that is responsible for counteracting the gravitational force and for keeping an aircraft airborne as it moves. In thinking about the behavior of lift and the other aerodynamic forces, one can likely deduce that it originates from the motion of fluid across the surface of a wing. In this way, it occurs alongside the other important aerodynamic force that depends on the velocity of the aircraft, namely drag.
What may not be obvious is how flow behavior and orientation of the flow with respect to the wing of the aircraft are determinants of lift. The shape of the wing (or airfoil), or more specifically its curvature, will also determine the lift provided by a moving fluid. There is a simple summative relation that describes the relationship between the pressure exerted by a fluid on an airfoil and the curvature of its streamlines, known as the streamline curvature theorem.
Streamline Curvature Theorem
The streamline curvature theorem is derived for steady flow conditions along a curved surface using the Euler equations. In other words, it can be derived by considering the curvature of an airfoil and how this curvature directs fluid flow. An explicit description of fluid flow across the airfoil requires a transformation into a general curvilinear coordinate system, where the curvature in the system follows the radius of curvature of the airfoil surfaces. The streamline curvature theorem is a general relationship that describes how the pressure gradient across a set of streamlines is related to the velocity of each streamline:
Streamline curvature theorem
While the above equation is normally communicated in terms of aerodynamic lift, it is a general relationship that relates the pressure gradient across streamlines to the radius of curvature of those same streamlines. This has important consequences for the construction of aircraft wings, propellers, helicopter rotors, sails, and any other curved surface that takes advantage of fluid flow for propulsion.
In general, the above equation contains a proportionality constant, called the lift coefficient, which accounts for the variation in curvature along the airfoil and the angle of attack. The latter factor is important, as it describes how lift can be varied based on the orientation of an aircraft. The lift coefficient CL is only equal to 1 for a specific angle of attack, typically about 5°, and it will vary as the angle of attack is changed (see below).
If we draw out the resulting streamlines along an airfoil, we can see how a pressure gradient arises due to steady flow along the top and bottom curved surfaces of an airfoil. The image below shows an airfoil, where the top and bottom surfaces have two different radii of curvature. The top surface has a smaller radius of curvature than the bottom surface, so the pressure gradient is negative pointing from the bottom surface to the top surface. This means the bottom surface will experience greater pressure than the top surface, which creates lift on the airfoil.
Fluid flow along the surfaces of an airfoil creates a pressure gradient due to differences in the curvature of streamlines
An important consequence of the streamline curvature theorem is that, in order for lift to occur, we must have different radii of curvature of the two airfoil surfaces. If the top and bottom radii of curvature were the same, there would be no pressure gradient between the two surfaces and there would be no lift. Three other insights can be derived from the streamline curvature theorem:
- Airflow speed is important: there is more lift when fluid flow across the airfoil is faster.
- Fluid density is important: denser fluids create more lift. In addition, low density can be compensated by higher speed to produce the same pressure gradient, and thus the same lift.
- The value of the radius of curvature at each point matters as well as the change in radius of curvature. Surfaces with smaller curvature are preferred for a given change in curvature.
Angle of Attack
The final piece that determines the direction of motion of the aircraft is its angle of attack, or the wing’s orientation with respect to fluid flow. The angle of attack defines the lift coefficient as shown above, and a lift curve can be developed from flow simulations across multiple attack angles.
Example lift curve on an airfoil [Source]
There will be some angle of attack that provides maximum lift and will determine the maximum upward trajectory of the aircraft. Eventually, when the angle of attack is too large, the lift decreases, and the aircraft will stall due to flow separation. These aspects of fluid flow and flight can be examined from fluid flow simulations with an airfoil design in a CFD package.
Solve Airfoil Design Problems With a CFD Simulator
Today’s CFD simulation applications can be used to solve aerodynamic lift problems for nearly any wing design as well as determine the other important aerodynamic forces that are present on an aircraft. These simulations involve determining fluid flow across the surface of the craft and plotting streamlines across the body of the airfoil. Finally, lift can be determined by looking at the curvature as outlined above.
Advanced CFD simulation suites can help automate these steps and can help you quickly determine lift, drag, and other forces on an aircraft during flight. The complete set of fluid dynamics analysis and simulation tools in Omnis from Cadence are ideal for defining and running CFD simulations with modern numerical approaches, including aerodynamic lift explanations in complex aircraft.