I know leading edge vortices over delta wings make more lift. I understood that the core of the vortex sheet is low, so there is suction flow that makes lift. But, I can't understand how a leading edge vortex accelerates the air downward. Please explain the detailed procedure; how does the leading edge vortex make the flow faster?
It is the inclination of the wing which forces the air flowing over it downwards. This is very much like when you hold your stretched-out hand out of a car's window and into the airstream and then slowly rotate it so it has an inclination to the flow. The vortex only makes sure that the air has enough energy to follow the wing's contour instead of separating.
how does the leading edge vortex make the flow faster?
It is suction over the top surface of the wing which accelerates the air and, due to the leading edge sweep, results in a circular motion which is superimposed on the lengthwise motion. Seen this way, it is the low pressure which produces the vortex, and the accelerated, rotating air has less static pressure, in accordance with Bernoulli's law. Trying to separate cause and effect is futile - it is all happening together.
How about it doesn't! The bound vortex does not accelerate the flow down, in 3D flow around a wing, it is the wing tip vortices that are responsible for the accelerated flow downward , and that imparted energy is wasted, hence why less lift and more drag is produced by a wing relative to an aerofoil.
The bound vortex is the result viscosity and is responsible for producing the Kutta condition . See this great CFD video (not mine):
I will note that suction does not exist as a force, pressure can only act normal to a surface, and only inward. What we may describe as suction can always be explained as a higher pressure somewhere else . That is, the pressure gradient force is always positive, it is just less in some places relative to others, and hence there will be associated forces such as lift and movement generated (you don't suck water up a straw, you reduce the pressure at the top of the straw and atmospheric pressure acting on the water pushes it down in the glass and up into the straw, similarly a vacuum cleaner does not suck air in, the compressor removes air, and the higher pressure air outside is forced in, usually bringing dust and dirt with it).
So while it is hard to separate cause and effect, it is not impossible. The actual positive pressure responsible for accelerating the air above a wing is coming from the front stagnation point. There is an asymmetric pressure gradient, which is what is responsible for making the air flow over the top surface faster, relative to the bottom surface .
This is all much more complex for a delta wing, where the tip vortices may not be located at the tip, if the delta wing is responsible for pitch and roll control (no canards). In this case, the downwash inducing vortices can be located inboard of the wingtips, if the wing tips have a positive coefficient of lift.
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