ABSTRACT
In Chap. 3, we familiarized ourselves with different aspects of spatial instability theory. We specifically noted that flow instability depends intimately upon the way the flow is excited. A distinction was made between wall and free stream modes for the solution of the Orr-Sommerfeld equation in Secs. 3.6.5 and 3.7.1. Unfortunately, classical stability theory approaches are incapable of making the connection between instability and receptivity of flows. To circumvent this, in many recent studies, fluid flow transition from the laminar to the turbulent state has been viewed as a consequence of phase transitions and/or bifurcation(s) of a dynamical system [353, 360]. The response of a dynamical system is a convolution of transfer function with the input to the system, in the spectral plane. In stability theories, one is only interested in interpreting system dynamics in terms of the transfer function of the dynamical system. Furthermore, focus on instability implicitly considers only small perturbations. This aids in linearizing the system dynamics and one can superpose solutions. This is the basis of normal mode analysis, and flow instability is often interpreted in terms of the least stable mode only. However, this approach does not produce information about the response field in the neighborhood of the exciter. This has been rectified in receptivity approaches in Sec. 3.6, where one correlates the input with the transfer function for disturbances which evolve in space. The role of disturbance amplitude was also earlier highlighted in the pioneering pipe flow experiments of [297].
