User:David Weyburne/sandbox

A new simplified explanation for aerodynamic lift is presented below. It is a hybrid of the simplified and mathematical explanations. It is based on a graphical interpretation of the mathematical equations governing fluid flow. The key to the approach is the graphical plots of the velocity profiles taken at a bunch of locations along the airfoil surface. This permits a one-to-one correspondence between the flow governing equations and the plotted profiles.

The actual aerodynamic lift argument is made in the referenced e-book and in the referenced YouTube video (a free version of the e-book is available at researchgate.com). The graphical nature of the argument means the 15 min. YouTube video, which is basically a PowerPoint slide movie, is the better presentation of the simplified explanation. To keep the Wikipedia section as short as possible, only two figures are included. The Wikipedia text can probably be further shortened by removing ~3-6 sentences that were added for the expert readers rather than the expected non-expert readers.

Explanation based on mass and momentum conservation and boundary layer shape

Recently, a new simplified aerodynamic lift explanation based on a graphical interpretation of the mathematical equations that govern fluid flow has been presented.

Velocity profile plots from a 2-D airfoil simulation showing peaks at eight locations along the top surface ($x$-direction) of an airfoil.

Pressure changes (dP/dy) plotted above the airfoil showing peaks at the same locations along the surface of the airfoil as the velocity profile plots.

The equations that govern airflow around an airfoil require the mass, momentum, and energy be conserved. An important feature of these equations is that they only become important where the velocity and pressure are changing. The mass of the free stream air approaching an airfoil surface is conserved by being redirected around the airfoil. This mass diversion causes the air velocity and direction to change as the incoming airflow approaches the airfoil surface. Changes in the air's velocity means that the momentum (equal to the mass times the velocity) is also changing. The momentum conservation equations require that the diverted incoming stream-wise momentum must be: 1) partially converted into perpendicular-to-the-flow momentum (airflow up or down) and 2) partially into pressure changes at locations where the velocity changes take place. This requirement provides the critical connection between the velocity and pressure in the near airfoil region.

To understand how this conservation requirement creates aerodynamic lift, we need to know the shape of the boundary layer around the airfoil. The boundary layer, as used here, is understood to be the region around the airfoil where the airflow velocity is different than the free stream incoming flow. For our purpose, the flow governing equations are best interpreted graphically in terms of plots of the velocity profiles. Velocity profiles are defined as the velocity values measured at a whole series of points along the perpendicular to the direction of the incoming flow from the airfoil surface to a point deep in the free stream above or below the airfoil. To understand what is going on along the airfoil, we need to look at a series of profiles along the airfoil. Care is needed because the traditional boundary layer velocity profile description found in fluid flow textbooks does not correspond to fluid flow around an aerodynamically thick object. The traditional boundary layer velocity profile view in which the velocity starts at zero on the airfoil surface and monotonically increases until it slowly plateaus to a constant value is not a description that permits conservation of mass, momentum, and energy. Actual velocity profile plots generated by solving the flow governing equations using computer simulation of airflow along an airfoil (example to the right) indicate that the velocity peaks to values higher than incoming flow value. In the velocity profile plot to the right, eight profiles along the airfoil are shown where the $y$-direction is the perpendicular to the incoming flow direction, $c$ is the chord (airfoil) length, and $u_{0}$ is the free stream velocity. The profiles all start at zero but are cut off to emphasize the peak. This peaking is a result of conserving the mass and momentum diverted by the airfoil.

What is important about this velocity peaking behavior is that it means the velocity is continuously changing in the perpendicular to the incoming flow direction for more than a chord (airfoil) length above and below the airfoil. Furthermore, the fact that the peaks are not identical along the airfoil indicates that the velocity is also changing in the incoming flow direction ($x$-direction) near the airfoil surface. The conservation of momentum equations requires that where the velocity is changing, pressure changes (dP/dy) must be present. The resulting momentum plot of the pressure gradient contributions corresponding to the above velocity profile plot is shown to the right. The equivalent below airfoil figure (not shown) also shows peaks but ones that are noticeably smaller. The pressure difference between the airfoil surface and the free stream is obtained as the area under the pressure gradient curves. The above and below pressure gradient figures reveal that although both have a low-pressure cloud-like region, the overall pressure gradient areas are larger above the airfoil which means the pressure above the airfoil is lower than the pressure below the airfoil. Aerodynamic lift is calculated as the pressure difference above and below the airfoil surfaces times the airfoil’s surface area. This pressure difference, in this case, is caused by the airfoil being slightly tilted to the incoming airflow. The overall aerodynamic lift force for the airfoil with a slight tilt to the incoming flow is positive but it is zero for the untilted symmetrical airfoil case (not unlike the effect obtained by sticking your hand out the window of a moving car).