The effect of phase-difference on the spreading rate of a jet

The role of initial phase-difference between fundamental and subharmonic instability waves on the spreading rate of a circular jet under bimodal excitation is investigated theoretically. For all Strouhal numbers considered, the initial growth rate of the subharmonic was found to be a maximum when the two waves are initially in-phase. The effect of phase-differe nce on the subharmonic's peak is dependent on Strouhal number. Development of the momentum thickness along the jet is characterized by regions of stepwise growth. The momentum thickness first increases as the fundamental amplifies. This growth region is followed by a region where the decay of the fun- damental and the growth of the subharmonic result in a constant momentum thickness. Once the fundamental has fully decayed, the momentum thickness grows again as a result of the subharmonic's amplification. The growth rate is dependent on the initial phase-difference, and is maximum when the subharmonic suffers max- imum amplification. UMERICAL simulation of vortex-pairin g interactions in two-dimensional mixing layers1'2 has shown that the pair- ing process is dependent on the phase-difference between the fundamental and subharmonic instability waves. The smaller the initial phase-difference between the two modes, the faster the coalescence of the two vortices. In the limiting case where the two components are antiphase, the numerical results of Patnaik et al. 1 showed that the vortex-pairing interaction is actually suppressed and replaced by shredding interactions. In the experiment of Zhang et al. 3 of a two-dimensional shear layer under bimodal excitation, significantly different merging patterns were observed as a result of changing the in- itial phase-difference between the fundamental and sub- harmonic. Monkewitz4 modified Kelly's5 temporal stability analysis of a spatially periodic mixing layer to include an ar- bitrary phase-difference between the fundamental and sub- harmonic instability waves. The initial growth rate was found to vary continuously from maximum when both waves are in- phase, to a minimum when they are out of phase. These in- vestigations indicate the importance of the phase-difference on the fundamental-subharmonic interaction. In the plane shear-layer case, Ho and Huang6 have shown that the spreading rates can be greatly manipulated by con- trolling the vortex-pairing process. Since this pairing process is dependent on the phase-difference between the fundamen- tal and subharmonic instabilities,1"5 one would expect the spreading rate to be dependent on this phase-difference. The present work is concerned with the technologically important problem of a round jet where the spreading rate can be manipulated via bimodal excitation at fundamental and sub- harmonic frequencies. In the present analysis, the energy equations for each flow component are used to study the ef- fect of initial phase-differencing between the two excitation components on their spatial developments and on the spreading rate. In the jet mixing region, the vortex-pairing process results in the growth of the subharmonic. The loca- tion of pairing and, thus, of the subharmonic's amplification depends on the random initial fluctuations at the jet exit and, hence, on the initial phase-difference. The question arises as to how much energy is carried by the subharmonic