Introduction

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.

Introduction

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.

The subject of this book has kept fluid dynamicists occupied for nearly two centuries, ever since the governing equation for fluid motion was developed by C. L. Navier and G. G. Stokes. While mathematicians are concerned with exploring the existence and uniqueness of a solution to the Navier–Stokes equation, physicists are fascinated by the fact that some “exact” laminar solutions are not observable. Early pioneers attributed this facet of the solution to the lack of stability of such exact or equilibrium solutions. This is how instability of fluid flows became a major subject of investigation, with near-unanimity among scientists that base flow is capable of feeding omnipresent background disturbances leading to transition.

As there are many equilibrium flows, the routes by which disturbances grow can also be vastly different. For example, the classical pipe flow experiment of Osborne Reynolds demonstrated that the transition of laminar flow to seemingly chaotic turbulent flow depends on the flow velocity and size of the pipe. Thus, the experiment highlighted the relationship of the instability with physical parameters. Now we know that the transition to turbulence in pipe flow depends upon the Reynolds numbers and background disturbances. The fact that quantitative description of transition to turbulent pipe flow still eludes us will encourage potential readers to embark upon research in this challenging field.

One of the early forays in flow instability studies has been the development of the eigenvalue analysis. It was adopted by Kelvin and Helmholtz to qualitatively explain interfacial instabilities such as those arising during the creation of surface gravity waves in lakes and oceans. The eigenvalue analysis remains the pedagogical tool to explain the phenomenon of instability and introduce the dispersion relation between spatial and temporal scales. In a similar vein of studying disturbance growth as an inviscid phenomenon, with equal ingenuity, Rayleigh developed the governing stability equation and a theorem to explain instabilities for jets afflicted by temporally growing disturbances. The failure of this inviscid theory in explaining flow over a flat plate prompted the development of viscous linear instability theory, known as the Orr–Sommerfeld equation. From the wave-like solutions obtained from this equation grow, in space, the well-known Tollmien–Schlichting (TS) waves.

Introduction

Turbulence continues to be a largely unsolved problem of physics, despite accurate numerical results available for some canonical problems. One of the dominant approaches in studying turbulence is nonlinear dynamics, sharing certain universal properties of fully developed turbulence. The other approaches include studies where turbulence is traced as a receptivity problem starting from the excitation of an equilibrium flow by input disturbances and the disturbances propagate via multiple instabilities accounting for the overall growth processes. This latter approach has been the one followed in this book so far. In Figure 2.1, a schematic of flow transition indicated the dynamical system approach as a possible route. Two other such roadmaps are now presented in Figure 14.1, and these are from [90] and [385], both of which classify transition routes based on the amplitude of excitation only. According to Saric et al. [385] the amplitude of input excitation increases for routes followed along A to E in Figure 14.1. In the other road-map, Cherubini et al. [90] also cites the primary instability associated with TS waves as due to low amplitude excitation, as in the path A due to [385] with routes are somewhat similar in these maps. In explaining the relation between instability experiments and receptivity analysis in Chapter 5, it is now clear that TS wave or wave-packet is strictly an artifact of experiments created to validate spatial instability theory. Discussion in Chapter 6 also establishes that transition can be initiated in many ways, with harmonic wall excitation (as in [405]) as just one of the many routes described in Chapter 6. The classification of a route as bypass transition is therefore an anachronism, as the original connotation of it in [298, 364], was absence of TS wave or wave-packet in any route being the rule (and not exceptional cases) for the canonical flow past zero pressure gradient boundary layer. The same can be said about the transient growth processes, which are marked as routes B, C and D in Figure 14.1(a) due to spanwise modulation, mean flow distortion or due to some bypass route - as one of the many possibilities whose generic route happens to be the spatio-temporal route espoused correctly since the necessary approaches developed in [418] and demonstrated in [34, 451, 452, 508, 509] for both two- and three-dimensional transition routes for wall excitation.

Introduction

In Chapter 1, we have discussed chronologically the theory of instability, starting with the works of Helmholtz [177], Reynolds [365], Rayleigh [351], and Kelvin [224], purely as a discourse about the different facets of the phenomena that we identify as the transition to turbulence. Although laminar flow can be stationary, background disturbances grow in space and time to create turbulent flow. Thus, the main issue in transition research is in identifying how disturbances display spatio-temporal growth. The research built upon the idea of imperceptible disturbances feeding upon the equilibrium flow and the resultant growth being so overwhelming that it takes the initial equilibrium state to another state that will be space–time dependent. Although it is imperative to explain how turbulence comes into being, there remain a few unexplored steps following which the laminar flow becomes transitional and turbulent. In the previous chapter, we emphasized the role of instability studies in this search, but one of the central issues of establishing any instability theory posed as an eigenvalue problem, lies in the difficulty of physically verifying such a theory. There was a major roadblock in experimentally verifying instability theories that continued till very recently. The subject started with the erroneous concept that viscous action is dissipative, and hence, an inviscid theory was considered appropriate and the theorems due to Rayleigh and FjØrtoft came into existence to explain temporal growth of inviscid disturbances. Prior to this investigation, Rayleigh [350] was successful in explaining the motion of jets by this temporal theory. However, this could not explain the instability of flow over a flat plate, and two new concepts came to the forefront. First, the significance and importance of viscous actions was seized upon, as diffusive actions were known to create instabilities for various mechanical and electro-mechanical systems. Second, as a reaction to the failure of temporal growth of disturbances for the zero pressure gradient boundary layer, researchers started on the path of spatial instability theory. In this theory, one has to fix a time scale and look for complex wavenumbers which satisfies the basic requirements of eigenvalue analysis. We have noted the discovery of Tollmien–Schlichting (TS) waves from this theory, proved to be difficult to verify experimentally.

Computing Space–Time Dependent Flows

It has been highlighted that the equilibrium flow described in the previous chapter requires high accuracy computations so that the subsequent investigation of receptivity and instability is not affected by numerical artifacts. It is imperative that all spatial and temporal scales are resolved accurately. For example, for the same equilibrium flow, if the imposed excitation level is increased, one may observe different types of transition with different wavenumbers and frequency spectra in the disturbance field. Thus, the behaviors perceived for different perturbed flows are not due to difference in the governing principles – rather these are due to altered boundary and initial conditions. These auxiliary conditions, in general, are given as either Dirichlet or Neumann boundary conditions. In convective heat transfer, one may have to deal with Robin or mixed boundary conditions. There are multiple aspects in computing governing equations for transitional and turbulent flows accurately. For example, one has to resolve all the space–time scales. Specifically, in numerically treating the space–time dependence of the problem, the discretization or integration process must be handled simultaneously. This last aspect is often overlooked, and spatial and temporal discretizations are treated separately. In Chapter 1, we have noted that in studies of instability and receptivity, the dispersion relation plays a central role. A poorly constructed numerical method will not follow this relation; in certain cases, the choice of numerical parameters are such that one incurs large dispersion errors. This aspect has to be studied carefully before one initiates numerical activity in studying instability and transition, which has been highlighted also in [412, 413].

Speaking about spectra and resolution of space and time variation of variables, one would require a good understanding of waves. Although waves were introduced in the context of hyperbolic partial differential equations [540], dispersive waves are present for any flows governed by other types of partial differential equations too [412, 553]. We have noted in Chapter 1, that waves are created during the Kelvin– Helmholtz instability, for which the equilibrium flow is given by a uniform velocity profile and the disturbance field is governed by the Laplace equation (elliptic partial differential equation). In developing high accuracy computing methods in [413], it has been noted that numerical treatment of parabolic and elliptic partial differential equations requires that the discrete equations have the same formalism used for hyperbolic partial differential equations.