119
edits
Changes
From CUGC Wiki
no edit summary
It can be shown that, for a glider, the aeroplane lift-to-drag ratio is identical to the glide ratio. It just needs to be noted that the aeroplane lift-to-drag ratio is not the same as the wing L/D, because the fuselage contributes only to drag but not lift.
== Downwash (advanced topic) ==
'''Warning''': readers whose working knowledge on fluid mechanics is limited are advised to Google “tip vortices” and jump to the Implications section.
=== Physical introduction ===
Circulation results in lift. For a finite span wing, at each section there must be a defined circulation value which can be determined by a closed line integral around the aerofoil. By the Stokes’s theorem (vector calculus), this implies that there is a '''distribution of vorticity''' within the integration loop. However, the loop encloses potential flow (which, by definition, can have no vorticity), and within the aerofoil contour the flow speed is zero so there cannot be vorticity either (the curl of a constant zero is zero). Therefore, this vorticity is distributed on the surface of the aerofoil. In the real flow, this corresponds to the vorticity in the boundary layer.
We have already argued that at the tip of the wing, the local wing loading is zero. If for an aerofoil section there is zero loading, this implies that the circulation for this section is zero. By the same argument, we find that the vorticity is zero for a section plane taken at the tip of the wing.
According to '''Helmholtz’s second theorem''', a vortex filament cannot end in a fluid. If there is a vortex line going spanwise when a section plane in the middle is examined, but it is not found going outwards on the section plane at the tip, the only possibility is that it has been deflected to some other direction. But in which direction? Again, according to Helmholtz’s theorems, vortex lines move with the fluid. Because the aeroplane flies forward, the flow around it is effectively going backwards. Therefore, the vortex line has been deflected into going backwards by the flow.
In a sense, we have shown that, for a wing of finite span, a '''sheet of vorticity''' is trailed behind it because vorticity is being shed off from the root to the tip of the wing. But this does not remain as a sheet forever. For the third time according to Helmholtz’s theorems, these vortex lines move with the fluid, but do not forget that the vortices can introduce swirling flow themselves. The result is that this vortex sheet rolls up into a single vortex with all the strength there is. For an aeroplane, because of the obvious symmetry, a pair of vortices are formed. These are known as tip vortices. Despite the name, it is good to understand that '''they are not produced by flow escaping around the tip''', but it is a result of a finite span wing generating lift in general. A more detailed examination of the rolling up process reveals that the distance between these tip vortices is less than the geometrical span of the wing, i.e. ''the tip vortices are closer to the fuselage than the tips are''.
[[Category:Theory]]
