Instability of Flow Past a Cylinder

Vortex shedding behind a circular cylinder is explained theoretically as a Hopf bifurcation which is a consequence of linear temporal instability of the flow. In this point of view, the above temporal instability is moderated by nonlinearity of the system, that is quite adequately explained by Landau equation, as given in Landau (1944) and Drazin Reid (1981). Earlier numerical investigations by Zebib (1987), Jackson (1987) and Morzynski Thiele (1993) have identified the onset of vortex shedding to be at a critical Reynolds number (Rccr) between 45 and 46. 

However, different values of critical Reynolds number Rccr for flow past a circular cylinder reported by various investigators show lack of unanimity in defining an universal number. For example, Batchelor (1988) conjectured it to be between 30 and 40 Landau Lifshitz (1959) quoted it as 34. Experimentally, different investigations obtained different Rccr - Kovasznay (1949) obtained it as 40 Strykowski Sreenivasan (1990) found it to be between 45 and 46 Roshko (1954) reported a value of 50 Kiya et al. (1982) have reported it to be 52 Tordella Cancelli (1991) have reported it as 53 and Homann (1936) has reported it to be 65.2, a maximum value so far. 

Theoretically, flow criticality is related to the onset of global linear instability, performed numerically by Jackson (1987), Zebib (1987), Morzynski Thiele (1993), who all have reported 45 Rccr 46. For steady flows, we will identify this critical Reynolds number as Rccn, for the ease of future discussion. Similarly, we will identify the critical Reynolds number value indicated in Homann s experiment as Rccr Hopf bifurcation describes the passage of a dynamical system from a steady state to a periodic state as a typical bifurcation parameter is varied, that in this case is the Reynolds number (Golubitsky Schaefer (1984)). The results of the numerical investigations mentioned above, relate to study of the flow system unimpeded by noise or perturbations- barring numerical errors. 

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