THE CRITICAL CROSS-SECTION OF A VORTEX

This paper discusses the problem of critical-flow cross-sections in vo rtex flows. It is shown that there are two different types of vortex f lows, A-type and B-type vortices (say). An A-type vortex approaches it s critical flow state as its cross-sectional area increases and depart s from the critical state as the cross-sectional area is decreased. Th is property is associated with the particular dependence of total pres sure and circulation on the stream function, and it holds for both sub critical and supercritical A-type vortices. On the other hand, both su bcritical and supercritical B-type vortices approach their critical fl ow states as their cross-sectional areas are decreased and depart from their critical states for increasing cross-sectional area. As was sho wn by Benjamin, setting the first variation of the flow force with res pect to the stream function equal to zero leads to Euler's equation of motion. The second variation also vanishes if the corresponding flow state is critical. In this case the sign of the third variation decide s whether the flow is an A-type or a B-type vortex. Within the framewo rk of inviscid-fluid flow theory the type of a vortex is preserved unl ess vortex breakdown occurs. Making use of the knowledge that vortex f lows are controlled by two different types of critical-flow cross-sect ions a variety of vortex flow phenomena are investigated, including th e two types of inlet vortices that are observed upstream of jet engine s, the behavior of vortex valves, the flow characteristics of liquid-f uel atomizers and the bath tub vortex.