

Problem 5.11: Air Cushion Vehicle 

The sketch (a) shows the crosssection of an air cushion vehicle of the "peripheral jet" type, first developed by Christopher Cockerell in the mid1950s. A fan draws air from the ambient atmosphere at pressure p_{a} and compresses it to a stagnation pressure p_{o} in the reservoir inside the vehicle. The air then exhausts downward through a narrow slot of width w at the periphery of the vehicle (w<<R), creating a condition that maintains a positive gage pressure (an "air cushion") underneath the vehicle. This allows the vehicle to float at a height h above the ground. This problem is concerned with the mechanics of how the "air cushion" is maintained the vehicle holds itself off the ground. In what follows we assume that the vehicle has total mass M and is circular with outer radius R, the slot has a width w that is very small compared with R, the jet issues vertically downward from the slot as shown, and the vehicle's elevation h is always small compared with R. A fan or turbine maintains the pressure in the reservoir at pressure D p_{0} = p _{a} + D p. In what follows, we proceed with the assumption that the flow from the reservoir to the slot is steady, incompressible and inviscid. To simplify matters, we shall also assume (tentatively) that the pressure p_{b} under the vehicle is just slightly greater than p_{a}, such that p_{b}  p _{0} << D p. and justify this à posteriori by showing that our solution is consistent with is assumption. Given: r, M, g, R, w, D p, where w<<R, h<<R 
General Case: (a) Derive an expression for the outflow velocity V_{j} in the slot jet. 

(d) Show that the assumption (1) is satisfied in Case 1 because h>>w. 
(e) Obtain an expression for the vehicle's ground clearance h for Case 1 in terms of the given quantities. 
