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Watching this video I notice there is a big hole in the middle of the parachute, at its top when deployed. This feature seems to exist in other smaller parachutes.

For me, it seems that air can escape the parachute by this hole and thus the parachute is less efficient than a same-sized parachute without this hole. But it must serve in a way I don't imagine. What is its purpose?

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    $\begingroup$ That's a spectacular picture of the 82nd Airborne jumper you found! $\endgroup$ – FreeMan Mar 11 '16 at 15:48
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For me, it seems that air can escape the parachute by this hole

That's the point, air HAS to escape the parachute anyway, and without the hole it would do so laterally and in an uneven fashion, leading to lateral oscillations.

The hole allows the air to escape in a controlled location, avoiding undesired and possibly dangerous oscillations.

Some (most personal ones, but for example not the SR22 one) parachutes have handles that allow the user to deform the parachute: this allows some extra air to escape from the desired side, giving lateral controllability.

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    $\begingroup$ In one sense, the 'parachute with a hole' is a derivative of the wind sock. The hole and the tapering help stabilise the sock / parachute. $\endgroup$ – Philip Oakley Mar 11 '16 at 19:42
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In order for a parachute to move through the air, all of the air underneath it needs to move out of its way. If a concave round parachute didn't have any kind of hole in it, it would fill with a convex mass of high-pressure air which would push air that was underneath it outward. If it had an overly large hole, the air pressure would be accelerated inward but wouldn't face much resistance going there. If a parachute has a properly-sized hole, the pressures that would push air inward and those which would push air outward will roughly balance, with the effect that as much air as possible will remain underneath the parachute as long as possible thus blocking its downward motion.

To be sure, it's not possible for a parachute to prevent all of the air below it (or even any of it) from moving horizontally at all, but reducing the outward pressure will greatly increase the time required for the air below the parachute to get out of its way, and thus improve the effectiveness of the parachute.

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