A GENERALIZATION OF ARNOLD'S STRANGENESS INVARIANT

The purpose of this paper is to introduce an infinite sequence {Stk}k of mutually independent topological invariants of smooth closed plane curves, which is proved to be a natural extension of the rotation number and the strangeness invariant defined by Arnold in [Ar]. We prove a formula to express Stk by using the invariants [Formula: see text] which are defined in [Oz]. The jumps of Stk at perestroikas of three types (namely cusp point, triple point, and self tangent point perestroika) are investigated, and as a consequence we find that Stk have the order in the sense of Vassiliev equal to 1.

Balance of turbulence energy in the region of jet-flow establishment

To complete a previous investigation reported earlier and carried out in an air jet issuing from a 1·0ft. diameter nozzle into still air, additional measurements were made of one-point triple-velocity and two-point double-velocity correlations and the spatial mean gradients of the axial turbulent velocities. The one-point pressure/axial-velocity correlation term was evaluated by using piezo-electric ceramic elements in conjunction with total- and static-pressure probes. For the measurement of spatial mean gradients of the axial-velocity fluctuation, special probes with two parallel wires were constructed. The axial scale of turbulence was found to be almost twice as large as the other two scales. The tangential scale was the smallest. All terms of the integral and differential forms of the turbulence energy equation were evaluated throughout the region and the results are given in dimensionless form.