Nonlinear Instability and Amplitude Equation

Terminology used here is as given in Fig. 5.1, showing the variation of computed lift coefficient C , with time for an impulsive start of the flow past a cylinder. It is noted that the amplitude of the lift variation always attains the same value (not shown here) and the corresponding time period is given by T = 27r/o e, irrespective of the method used for the computation, provided the same grid is used for these calculations. 

Below Rccri, the real part ar is negative i.e. the small disturbances damp, while above Rccri the flow becomes temporally unstable in the linear sense that would amplify the velocity and vorticity field, that can be traced from the lift variation itself. Presence of the nonlinear term does not allow uninhibited growth of such disturbances. Passage of from negative to 

Landau did not address the issue of phase angle (see Landau (1944)), it was later derived by treating the landau coefficient /, as a complex quantity-as given in the last section. Despite the nonlinearity of Eqn. (5.1.7), it is readily integrable to provide 

While lots of attention have been paid on the real part of Landau s equation, the imaginary part has not been analyzed in great detail. As A approaches its asymptotic value Ag, the circular frequency ( ) also reaches its as miptotic value, LOg. Thus, the Strouhal number is found to be amplitude dependent and is given by. 

Instead of using this equation, in the literature, there are few models proposed by which the frequency or Strouhal number of the shedding is fixed. Koch (1985) proposed a resonance model that fixes it for a particular location in the wake by a local linear stability analysis. Upstream of this location, flow is absolutely unstable and downstream, the flow displays convective instability. Nishioka Sato (1973) proposed that the frequency selection is based on maximum spatial growth rate in the wake. The vortex shedding phenomenon starts via a linear instability and the limit cycle-like oscillations result from nonlinear super critical stability of the flow, describ-able by Eqn. (5.3.1). 

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