This paper deals with tip leakage flow structure in subsonic axial compressor rotor blades row under different operating conditions. Analyses are based on flow simulation utilizing computational fluid dynamic technique. Three different circumstances at near stall condition are considered in this respect. Tip leakage flow frequency spectrum was studied through surveying instantaneous static pressure signals imposed on blades surfaces. Results at the highest flow rate, close to the stall condition, showed that the tip vortex flow fluctuates with a frequency close to the blade passing frequency. In addition, pressure signals remained unchanged with time. Moreover, equal pressure fluctuations at different passages guaranteed no peripheral disturbances. Tip leakage flow frequency decreased with reduction of the mass flow rate and its structure was changing with time. Spillage of the tip leakage flow from the blade leading edge occurred without any backflow in the trailing edge region. Consequently, various flow structures were observed within every passage between two adjacent blades. Further decrease in the mass flow rate provided conditions where the spilled flow ahead of the blade leading edge together with trailing edge backflow caused spike stall to occur. This latter phenomenon was accompanied by lower frequencies and higher amplitudes of the pressure signals. Further revolution of the rotor blade row caused the spike stall to eventuate to larger stall cells, which may be led to fully developed rotating stall.

A random mapping partitions the set {1, 2, ···, m} into components, where i and j are in the same component if some functional iterate of i equals some functional iterate of j. We consider various functionals of these partitions and of samples from it, including the number of components of ‘small' size and of size O(m) as m → ∞the size of the largest component, the number of components, and various symmetric functionals of the normalized component sizes. In many cases exact results, while available, are uniformative, and we consider various approximations. Numerical and simulation results are also presented. A central tool for many calculations is the ‘frequency spectrum', both exact and asymptotic.

Individuals in a population which grows according to the rules defining the simple branching process can mutate to novel allelic forms. We obtain limit theorems for the number of alleles present in any generation, the total number of alleles ever seen and the number of the generation containing the last mutation event.

In addition we define a notion of frequency spectrum for each generation as the expected number of alleles having a given number of representatives. As the generation number increases we prove the existence of a limiting notion of the frequency spectrum and discuss its upper tail behaviour. Our results here are incomplete and we make some conjectures which are supported by informal argument and specific examples.