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One of the principal tenets in developing a dynamical system theory is to study the relationship between cause and effects. This is true for a fluid dynamical system characterized by a large number of degrees of freedom, as compared to other dissipative dynamical systems in many fields of physics. Experimental verification of any theory is imperative, and in this respect, theories of instabilities are difficult propositions. This is because instability theories rely on omnipresent imperceptible ambient disturbances as input to produce response, specifically in the limit of vanishingly small input that is needed in the dynamical system approach. Mathematically, the instability problem involves seeking the output of a system governed by a homogeneous differential equation, subject to a homogeneous boundary and initial conditions. Implicit in this is the requirement of an equilibrium state whose instability is studied, and for which imperceptible omnipresent disturbance resides and draws energy for its growth. For example, flow past a circular cylinder displays unsteadiness above a critical Reynolds number (based on oncoming flow speed and diameter of the cylinder), even when one is considering uniform flow over a perfectly smooth cylinder. Whereas this can be rationalized for experimental investigation where the presence of background disturbances cannot be ruled out, the situation is far from straightforward for computational efforts. Roles of various numerical sources of error triggering instability for uniform flow past a smooth circular cylinder is complicated. This issue has been dealt with in [469]. Inability to compute the equilibrium flow past a circular cylinder at relatively high Reynolds numbers is due to the presence of adverse pressure gradient experienced by the flow on the lee side of the cylinder. The situation is equally difficult for the flow over a very long flat plate. As the equilibrium flow is obtained with significant precision, it is possible to study the flow past a flat plate as a receptivity problem, as has been done experimentally to study the existence of TS waves by Schubauer and Skramstad [405], where the disturbances were created by a vibrating ribbon inside the boundary layer.
We have already identified a few drawbacks of the linear instability theory formulated by a homogeneous governing equation with homogeneous boundary conditions, in search of eigenvalues to explain growth of disturbances.
In Chapter 1, we have discussed the historical development of the field of instability and receptivity. Helmholtz [177] first provided some theoretical ideas regarding hydrodynamic instability. About a decade later, the works of Reynolds [365], Rayleigh [350, 351] and Kelvin [224] produced experimental and theoretical results that laid the foundation of stability theory. According to Betchov and Criminale [28], stability is defined as the property of the flow describing its resistance to grow due to small imposed disturbances. We note that the background disturbances do not have to be small (as noted experimentally by Reynolds [365]); we will also see in this chapter that the growth noted experimentally in [405] for the zero pressure gradient boundary layer occurs over a short streamwise distance. The original question of transition to turbulence was not addressed directly in theoretical studies, as most of these were related to finding conditions for growth of background disturbances by developing the linear stability theory. This theory investigated the ability of an equilibrium state to retain its undisturbed laminar state for stability.
Instability studies began by a linear theory resulting in Rayleigh's stability equation and a corresponding theorem, [351, 353, 356], with focus on inviscid temporal instability. This theorem was based on an incorrect assumption that viscous action in fluid flow is dissipative and can be neglected to obtain a more critical instability limit. It was strange for fluid dynamicists to accept this, as researchers in other disciplines of mechanics and electrical sciences, geophysics and engineering were aware of the role of resistive instability, which can arise in fluid flow only by viscous action. Viscous action can give rise to phase shift or time delay. A basic oscillator is governed by an equation with time delay as. This is equivalent to providing anti-diffusion as noted in, with the second term destabilizing the oscillator via the time delay, τ. Despite this rudimentary observation, only when scientists failed to explain disturbance growth for zero pressure gradient boundary layer, were alternatives sought [321, 495] via the Orr–Sommerfeld equation, which has viscous diffusion included for disturbance equations.
Although we understand the importance of viscous diffusion, we begin by describing inviscid instability, as it demonstrates the logic behind Rayleigh's early works and his theorem to explain the concept of flow instability.
In Chapter 1, we have stated that in this book, the study of flow instability will be performed using dynamical system theory. For flow instability, we will follow the schematic shown in Figure 2.1 for flows undergoing transition to turbulence from a laminar state. The idea behind this path dates back to the famous pipe flow experiment of Osborne Reynolds, who understood that the phenomenon of transition depends upon the prevalent background disturbances. For this reason, Reynolds designed the experimental setup with utmost care to minimize sources of disturbances. The time of performing experiments were also so chosen that the disturbances were further minimized. Thus, the transition phenomenon significantly depends on the input to the system, referred to as the receptivity of the system.
The concept of receptivity is reflected in Figure 2.1, where the dynamical system is identified by the box with thick borders and input to this system is marked on the top. There are alternative processes which are marked inside the box indicating various mechanisms responsible for transition. The output of the system is the turbulent flow, shown at the bottom of the schematic. It has been noted in [405] that for experimentally generating Tollmien–Schlichting waves, a vibrating ribbon excited time-harmonically at a single frequency was successful, while acoustic excitation of a free stream was not effective. Readers should note that the eigenvalue analysis (as we will describe in Chapter 4) is not only incapable of distinguishing between wall and free stream excitation, but also incapable of distinguishing between vortical and acoustic excitations. This prompted researchers to initiate studies about the propensity of equilibrium flows to be more receptive to one type of input excitation over the other. This is the essence of ‘receptivity’, a term coined by Morkovin [298], whose study not only discusses amplitude of input excitation, but also quality, that is, different types of physical input excitations.
The initial state of the dynamical system is represented by an equilibrium flow. In this chapter, we will look at a few representative equilibrium flows with the help of which certain transition mechanisms will be explained in the book. The equilibrium flows are obtained from different levels of hierarchy of conservation equations for fluid flows. Readers are encouraged to peruse the books [412] and [551] for a range of equilibrium states for internal and external flows.