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.

Introduction

By now what has been established is: (a) The onset of instability is due to the growth of imperceptible background disturbances for the canonical zero pressure gradient boundary layer or the boundary layer formed over a semi-infinite flat plate. Here, the connotation of zero pressure gradient has to be understood in the context of the discussions in Section 7.3; that the streamwise pressure gradient on the semi-infinite flat plate is variable in nature, as shown in Figure 7.8. The transition that the semi-infinite flat plate undergoes will be a strong function of the flow receptivity depending on the location of the exciter with respect to the leading edge of the plate. For the same reason, results reported in literature or books that employ the Blasius boundary layer should be viewed as pedagogic exercises primarily [133, 388] with poor reliability. (b) For experimental verification of instability theories, one has to follow a receptivity approach, with respect to deterministic excitation, for this allows replicability of the experiment. (c) Such experiments have to be for the conditions of the theory, just as it has been for the spatial instability theory, which requires that time periodic excitation be inside the boundary layer, even though the real excitation is completely different from such monochromatic excitation. This has led to the incorrect identification of TS waves as the precursors of natural transition to turbulence, and misdirecting researchers right from the beginning of the twentieth century. (d) Even when conducting experiments based on time aperiodic excitation [96, 144, 290], researchers and experimenters always tried to locate TS waves to explain their experiments. (e) However, when researchers investigated further for more generic spatio-temporal route of transition using linear theory, the concept of spatio-temporal wave front (STWF) emerged in [418, 451], which was later identified by solving the Navier–Stokes equation without linearization [419, 422]. Variety of researchers have tried to explain such events as due to transient growth and/or bypass transition [299, 364], or by nonmodal growth [229, 398, 532] and so on.

The generic nature of STWF for seemingly different routes of transition via alternate routes for two-dimensional transition has been explained in the previous two chapters and is based upon research by Sundaram et al. [509]. The researchers no longer have to go looking for TS waves for every episode of transition.

Introduction

In Chapter 9, two nonlinear theories have been described to predict the onset and growth of disturbance fields. For incompressible flows, these are derived from the Navier–Stokes equation without resorting to any assumptions, based on the disturbance mechanical energy (DME) and the disturbance enstrophy transport equation (DETE), and are given in Eqs. (9.7) to (9.11). In studying transitional and turbulent flows, one begins with the details of the receptivity of the equilibrium flow to imposed perturbations. These are ingested in the flow and evolve via various instability stages. The onset or the receptivity stage strongly depends on the way the equilibrium flow is excited. It has been clearly described in Chapters 5, 6, 8 and 9 that there are two prototypical routes of causing transition for the flow over the canonical semi-infinite flat plate [298]: (i) where the boundary layer is excited at the wall [32, 34, 134, 509], experimentally investigated in [236, 405, 565], and (ii) where the flow transition is triggered by free stream excitations, studied theoretically and experimentally in [225, 267, 431, 559]. The second route of excitation has been originally conceptualized by Taylor [510] in trying to quantify the effects of free stream turbulence. This has been endorsed by Monin and Yaglom [295] subsequently. In this chapter, the methods which trace disturbances from the onset to fully developed turbulent stages are discussed. Thus, it is essential that one understands the genesis and growth of disturbances in the first place. That there is multiplicity of point of views about the receptivity or onset stage itself, is well known.

Even for the canonical flow over a semi-infinite flat plate, various aspects of flow transition have been emphasized by different researchers. For example, Saric et al. [385] provided a roadmap of transition, by highlighting the role of amplitude of the imposed perturbation, noting that weak disturbances inside the boundary layer can cause instabilities that can be described by the Orr–Sommerfeld equation. With higher amplitude of imposed perturbation, nonlinear interactions can directly occur in the form of secondary instabilities, bypassing the primary linear instability [299] with turbulent spots appearing directly. Such a bypass route is also noted for high free stream turbulence in [364].

Introduction to Free Stream Excitation

The receptivity to free stream disturbances was introduced in Chapter 5 from the perspective of its role in creating TS waves. This problem is addressed by few researchers due to the failure reported in the experiments of [405]. Its authors could not create TS waves by acoustic excitation from the free stream, as was theoretically predicted earlier by Tollmien and Schlichting using linear spatial instability theory.

Reasons for these are many-fold: first, an acoustic wave is three-dimensional and thus does not excite two-dimensional TS waves. Free stream excitation is not monochromatic, and the linear spatial theory demands monochromatic excitation. It is shown in Sub-section 5.3.2 that receptivity of the laminar boundary layer to free stream disturbance convecting with free stream speed shows very weak coupling, unless one follows the bypass route (convection speed is significantly lower than the free stream speed). Experimental work also started with that reported in [510]; to estimate the dependence of critical Reynolds number on free stream turbulence (FST). While discussing Taylor's work in [295], the authors (of this treatise) conjectured that FST consisting of convected vortices is responsible for creating adverse pressure gradients locally, which gives rise to unsteady separation. Such separations trigger the rapid vortex-induced instability leading to transition. The assumption in [295] is that the “effect is connected with the generation of fluctuations of longitudinal pressure gradient by these disturbances, leading to the random formation of individual spots of unstable S-shaped velocity profile.” Exciting a shear layer by sources outside it has been experimentally investigated later in [109, 225, 226, 267].

Dietz [109] created disturbances inside the boundary layer by vibrating a ribbon in the free stream at a single frequency. This is supposed to supplement the experiment in [405] to create TS waves by free stream excitation at a monochromatic frequency. Dietz [109] actually did not demonstrate progressive TS wave, and instead plotted velocity profiles which looked like the TS wave eigenfunctions, shown in [412]. It is pertinent to note that even the STWF displays a velocity profile in [29], which has the same wall-normal distribution of streamwise disturbance velocity. So the demonstration in [109] is qualitative only. In Sub-section 5.3.2, the coupling mechanism between wall and free stream excitation is explained.

Introduction

It has already been explained how linear theories display unconditional stability for flow in a pipe and plane Couette flows. Even for plane Poiseuille flow, the computed critical Reynolds number obtained by linear theory is about almost six times higher than that has been observed experimentally by Davies and White [106]. Also, of interest is the classical experiment of Reynolds [203] for the pipe flow, in which the critical Reynolds number was raised to a value of 12,830 after careful control of the experimental conditions. Monin and Yaglom [295] noted that “in the case of a tube with sharp entrance, pushed through the plane wall of the reservoir, the end of the tube will create considerable disturbances, and Recr will equal approximately 2800. Conversely, if the degree of disturbance at the intake into the tube is decreased strongly by some means or other, we can delay the transition from laminar to turbulent flow until very high Reynolds numbers.” It has been noted that Pfenninger [332] could delay transition in pipe flow for up to Re = 100, 000, by arranging twelve special screens to damp disturbances in the flow approaching the inlet of the tube. Interestingly, Reynolds [203] noted the events in his experiments “at once suggested the idea that the condition might be one of instability for disturbance of a certain magnitude and stable for smaller disturbances.” While the relation between response with amplitude of input is an attribute of nonlinear instability, authors in [295] interpret the situation somewhat differently by concluding that “Reynolds number in itself is not a unique criterion for transition to turbulence; for a flow in a tube it is apparently impossible to find a universal critical value of Recr such that for Re ≥ Recr the flow regime is bound to be turbulent. To establish the upper value of Re for laminar flows in tubes it is necessary to have some knowledge of the level of inlet turbulence of the laminar flows considered.” To support this conjecture, a figure showing the variation of Reynolds number at transition is plotted against turbulence intensity in Figure 10.1 for flow over a zero pressure gradient boundary layer.

Historical Introduction

In many natural or engineering fluid flows, turbulence is the natural state. However, in the eighteenth century, when fluid dynamicists were divided into two distinct and separate schools of thought belonging to hydrodynamics and hydraulics, they did not have the benefit of the Navier–Stokes equation (NSE, not developed at that time) that governs all incompressible fluid flow behavior. The hydrodynamics practitioners did not envisage the importance of viscous flow, as it was thought to be confined to a very narrow region near the body placed in a flowing fluid, while most of the flow region was considered to be inviscid. This was the justification for the use of Euler's equation. Hydraulics practitioners approached their problems with charts and tables obtained empirically from actual observations. This segregation of thought continued for another century until the advent of the boundary layer theory, proposed by Prandtl [338], which we will visit later in the book.

After the derivation of the viscous flow equations [311, 501] by introducing the constitutive relation between the stress and rate of strain to obtain the Navier–Stokes equation [20, 412], Stokes tested the equations using pipe flow experiments. There was absolutely no match between the “exact” solution of the Navier–Stokes equation and the experimental observations. There could have been various reasons for this: for example, the analytical solution of the Navier–Stokes equation is obtained after many simplifying assumptions; moreover, the correctness of the constitutive relation and no-slip condition has not been rigorously established even today, and so what is an “exact” solution? We will revisit the constitutive relation between the stress and rate of strain while discussing the Rayleigh–Taylor instability problem.

It is pertinent to note that, in the absence of rigorous proof, even today, the no-slip condition is considered as a modeling approximation. Although Batchelor [20] noted that for Newtonian fluid flow, the absence of slip at a rigid wall is now amply confirmed by direct observation and by the correctness of its many consequences under normal conditions. In microfluidics, continuum equations are often solved with slip boundary conditions, while solving the Navier–Stokes equation [217].

Introduction to ow instability

In retrospective, one can observe now that the analytical solution obtained by Stokes for pipe flow was acutally for a laminar flow, while the real flow was turbulent for the operating conditions.

Baroclinic Instability

Baroclinic instability of flow is of interest in multiple branches of engineering, geophysics and astrophysics. It is considered very important in explaining motion in earth's atmosphere and in the ocean. Rayleigh–Taylor instability is an example of a baroclinic instability introduced in Chapter 1, as a special case of the inviscid Kelvin– Helmholtz instability. Rayleigh–Taylor instability, in its simple connotation, arises when a heavier fluid is made to rest on top of a lighter one, as the initial equilibrium state. If the heavier and lighter fluids are perfectly aligned with the direction of gravity, then such an equilibrium state would remain stable. If an instability misaligns this configuration, then the ensuing turning moment will unabatedly destabilize the initial state. However, if the opposite initial configuration of a lighter fluid resting on top of heavier fluid is perturbed, then the turning moment will restore the equilibrium state.

This qualitative explanation of Rayleigh–Taylor instability can be given more meaning, physically and mathematically, by considering the Euler equation for a compressible flow. Taking the curl of the momentum conservation equation in its primitive variable form, one can depict the corresponding inviscid vorticity transport equation as

where and are the velocity and vorticity vectors, and the last term on the right-hand side is the baroclinic contribution to vorticity generation by the misalignment of the pressure gradient from the density gradient. Thus, for baroclinic flows, pressure is not dependent on density alone, but also depends on the temperature. In contrast, an atmospheric flow is called barotropic, for which the pressure depends only on density, and thus, the cross product is identically zero. It should also be noted that the term, is due to compressibility and prediction of such instabilities should justifiably depend upon bulk viscosity for the constitutive relation between stress and rates of strain tensors.

In atmospheric motion, baroclinic instability is the main source for the formation of cyclones and anti-cyclones, occurring mainly in temperate latitudes. At the tropics, the atmospheric flow is barotropic. The baroclinic instability is observed to contribute to the formation of eddies in the mesoscale. However, such instabilities are for a rapidly rotating fluid, which also displays strong density stratification. The nondimensional parameter, Richardson number, is an indicator of the strength of stratification, with large values indicative of stable stratification.

Instability of Mixed Convection Flows

At this point it would be relevant to examine heat transfer effects for incompressible fluid flows governed by the Navier–Stokes equation, along with the mass conservation equation. Any reckoning of heat transfer, demands consideration of energy conservation, taking into account possible heat transfer in the flow, originating from the boundary and initial conditions. In forced convection, the flow is caused by an external agency, such as by a fan, a pump or by atmospheric wind. In contrast, free convection is caused by buoyancy force that exists only in the fluid. Mixed convection flow is a combination of free or natural convection (without any background flow of the medium) and forced convection [496], and in the present context, the examination about very low speed flow, with heat transfer taking place because of a small temperature gradient. Such heat transfer can be modeled by Boussinesq approximation. This approximation is limited however to small temperature gradients.

For the small temperature gradients that are responsible for heat transfer effects from the boundaries, it is to be investigated how the associated buoyancy force comes into play. This can be attempted through the Boussinesq approximation, on the premise that the buoyancy force is the result of change in density. Otherwise, the density is treated as constant, as representative of a reference temperature (call it T0), and the temperature differential as δT = (T − T0). The buoyancy force is then inserted in the momentum equation, in an appropriate direction, given by δρg in the direction of gravity (say, along the y- axis). Introducing the volumetric expansion coefficient as, the buoyancy force is given by. As further explained in the next section, the reason is that there is an added variable to deal with for the mixed convection flows. And, energy equation is intrinsic to all study of flows that involve heat transfer. Thus, study of corresponding perturbation fields would require consideration of entropic disturbances.

Mixed convection flows are found in many natural and engineering devices, as in geophysical fluid dynamics and various engineering applications that are affected by mixed convection flow instabilities. Studies of these flows in the presence of different disturbance sources are important. It has been shown [412, 420] that vortical excitation of small amplitude can couple to create thermal fluctuations, implying possible flow control in many engineering devices.

Introduction

Experimental study of flow transition began with the famous pipe flow experiment [365], in which Reynolds took pipes of different diameters and fitted them carefully with bell-mouth shaped entry sections. In Chapter 4, one noted that favorable pressure gradient delays transition by attenuating disturbances. The bell-mouth accelerates the flow, and the resultant favorable pressure gradient stabilizes the flow. Apart from this, Reynolds took other precautions to note that the transition in the pipe flow depends upon the non-dimensional parameter, now known as the Reynolds number, Re = Vd/ν, where V is the centreline velocity and d is the diameter of the pipe. He found that with all extra precautions taken against disturbance growth, the flow can be kept laminar up to Re = 12, 830. It was also noted by Reynolds that this critical value is very sensitive to disturbances in the oncoming flow, before it enters the pipe. Although this might also indicate receptivity of the flow, Reynolds remarked that “this at once suggested the idea that the condition might be one of instability for disturbance of certain magnitude and stable for smaller disturbances.” The relation between input and output amplitudes during disturbance growth is a typical attribute of nonlinear instability. There are other flows, e.g. the Couette flow, which are found to be linearly stable for all Reynolds numbers. This prompted researchers [282, 298] to suggest nonlinear routes of instabilities for such flows. However, it is interesting to note that some authors [399] have stated without proof that “it is easy to verify that the nonlinear terms of the incompressible Navier–Stokes equations are energy preserving: the role of the nonlinear terms is the distribution, scattering and transfer of energy, but this reorganization is accomplished in a conservative manner. Energy growth or decay can only come from linear processes.” The presented explanation in this book is contrary to this point of view, with enough evidences provided to show the central and important role of nonlinearity in causing transition [471].

One of the constraints of classical linear instability theories performing local analysis is the adoption of the parallel flow assumption of the equilibrium flow. This has been addressed in review articles by Chomaz [93] and Theofilis [519], with the main emphasis on giving up on local analysis in favor of a global analysis.