The subject of hydrodynamic stability or stability of fluid flow is one that is most important in the fields of aerodynamics, hydromechanics, combustion, oceanography, atmospheric sciences, astrophysics, and biology. Laminar or organized flow is the exception rather than the rule to fluid motion. As a result, exactly what may be the reasons or causes for the breakdown of laminar flow has been a central issue in fluid mechanics for well over a hundred years. And, even with progress, it remains a salient question for there is yet to be a definitive means for prediction. The needs for such understanding are sought in a wide and diverse list of fluid motions because the stability or instability mechanisms determine, to a great extent, the performance of a system. For example, the under prediction of the laminar to turbulent transitional region on aircraft – that is due to hydrodynamic instabilities – would lead to an underestimation of a vehicle's propulsion system and ultimately result in an infeasible engineering design. There are numerous such examples.

The seeds for the writing of this book were sown when one of us (WOC) was contacted by two friends, namely Philip Drazin and David Crighton with the suggestion that it was perhaps time for a new treatise devoted to the subject of stability of fluid motion. A subsequent review was taken by asking many colleagues as to their assessment of this thought and, if this was positive, what should a new writing of this subject entail?

Discussion

When a parallel or nearly parallel mean flow does not have an inflection point, viscous effects are important and the Orr-Sommerfeld equation (2.31) must be considered in order to determine the stability of the flow. This is in contrast to solving the much simpler Rayleigh's equation (2.32) where it is generally believed that, for flows with an inflection point, the most unstable mode is inviscid in nature.

In this chapter we will examine the temporal stability characteristics of various well known profiles. These profiles include bounded flows, such as plane Poiseuille and Couette flows, semi-bounded flows, such as the Blasius boundary layer and the more general Falkner-Skan family, and unbounded flows, such as jets, wakes and mixing layers. Other well-known profiles are given in the exercise section. We restrict our attention to two-dimensional disturbances, since according to Squire's theorem, if a three-dimensional disturbance is unstable, there corresponds a more unstable two-dimensional disturbance.

For bounded flows, DiPrima & Habetler (1969) proved that the spectrum of the Orr-Sommerfeld equation consists of an infinite number of discrete eigenvalues, and that the spectrum is complete. Contrary to this, if perturbations for this flow are considered inviscidly, then there are no discrete modes. As a result, only a continuous spectrum is possible. Thus, any arbitrary initial disturbance can be decomposed and expressed as a linear combination of the eigenfunctions. For unbounded flows, stability calculations for various flows have uncovered only a finite number of eigenvalues.

Introduction

In this chapter, we discuss the breakdown of hydrodynamic instability, a theory that is initially characterized by a system of linear equations, as discussed in great detail in Chapters 2–8. Breakdown thus implies that the linear assumption is becoming invalid and the flow now has several modes interacting and amplifying. This interaction can then transfer energy to modes not yet dominant in the flow. The culmination of this breakdown process is a turbulent flow. One might suppose that the characteristics of the breakdown stage depends on the initial conditions – as receptivity – as well as freestream conditions such as vorticity and freestream turbulence. Today, we understand much about this initial stage and the linear amplification stage but have only limited knowledge for the nonlinear processes of many flows (cf. Chapter 9) because the complete Navier-Stokes equations must be solved and tracing measurements in this stage back to their origin to ascertain the cause and effect is challenging.

The major goal of this text has been to present the subject of hydrodynamic instability processes for many different engineering problems. The initial chapters demonstrated that this understanding can most often be achieved with linear systems. However, as was somewhat evident in Chapters 8 and 9, the transition from a laminar to turbulent flow is extremely complicated. This Chapter and the next will expose the reader to issues effecting hydrodynamic instabilities, the nonlinear breakdown of modes after linear growth, and we will summarize a condensed history of methods that have been used to predict loss of laminar flow and onset of transition to turbulence.

Introduction

The previous chapters have outlined and validated various theoretical and computational methodologies to characterize hydrodynamic instabilities. This chapter serves to cursorily summarize techniques to control flows of interest. In some situations, the instabilities may require suppressive techniques while, in other situations, enhancing the amplification of the disturbance field is desirable. Similarly, enhanced mixing is an application where disturbance amplification may be required to obtain the goal. Small improvements in system performance often lead to beneficial results. For example, Cousteix (1992) noted that 45 percent of the drag for a commercial transport transonic aircraft is due to skin friction drag on the wings, fuselage, fin, etc., and that a 10–15 percent reduction of the total drag can be expected by maintaining laminar flow over the wings and the fin. Hence, flow control methods that can prevent the onset of turbulence could lead to significant performance benefits to the aircraft industry. For aircraft, as well as many other applications, the flow starts from a smooth laminar state that is inherently unstable and develops instability waves. These instability waves grow exponentially, interact nonlinearly, and lead ultimately to fully developed turbulence or flow separation. Therefore, one goal of a good control system is to inhibit, if not eliminate, instabilities that lead to the deviation from laminar to turbulent flow state. Because it is beyond the scope of this text to cover all possible flow control methodologies, this chapter will primarily highlight passive control techniques, wave-induced forcing, feed forward and feedback flow control, and the optimal flow control approach applied to suppression of boundary layer instabilities that maintains laminar flow.

The initial-value problem

The fundamental needs for specifying an initial-value problem for stability investigations are not in any way different from those that have long since been established in the theory of partial differential equations. This is especially true in view of the fact that the governing equations are linear. Thus, by knowing the boundary conditions as well as the particular initial specification, the problem is, in principle, complete. Unfortunately, in this respect, classical theory deals almost exclusively with second order systems and, as such, few problems in this area can be cast in terms of well known orthogonal functions. For the equations that are the bases of shear flow instability, however, it is only the inviscid problem that is second order (Rayleigh equation) and even this limiting equation does not have a detailed set of known functional solutions. The more serious case where viscous effects are retained, then the minimum requirement is an equation that is fourth order (Orr-Sommerfeld equation) and even this, as previously noted, is fortuitous. An a priori inspection would have led one to believe that the full three-dimensional system should be sixth order, such as that discussed for the case of the Ekman boundary layer, for example. The net result is one where there are neither known closed solutions nor mutual orthogonality. It is only the accompanying Squire equation, where the solutions are coupled to those of the Orr-Sommerfeld equation, that eventually makes for sixth order.

For purely temporal disturbances k is real, and if it is given together with the complete set of the flow parameters, then we can solve the complex wave frequency ω from (10.26) as the eigenvalue. The IMSL library routine GVLCG has been used to obtain ω. We characterize the spatial-temporal disturbances of a given wave number kr for a given set of flowparameters with the spatial amplification curves ωr = 0. There are at least two such curves for a given set of flow parameters in the case of convective instability. One corresponds to the sinuous mode and the other to the varicose mode. For each mode, we start with an initial guess of ki for a given kr. Then solve for ωr and ωi using the IMSL routine GVCCG. If ωr = 0 the guess was perfect, if not we find ki by using the Newton integration method with a reduced value of ∣ωr∣. With the new ki and the original kr we update (ωr, ωi) by means of the IMSL routine GVCCG. We repeat this procedure until the IMSL routine gives ωr = 0.

Annular jets are encountered in many industrial processes. Their stability has been studied in the contexts of ink-jet printing (Hertz and Hermanrud, 1983; Sanz and Meseguer, 1985), encapsulation (Lee and Wang, 1989; Kendall, 1986), gas absorption (Baird and Davidson, 1962), and atomization (Crapper, Dombrowski, and Pyott, 1975; Lee and Chen, 1991; Shen and Li, 1996; Villermaux, 1998). Shen and Li analyzed the spatial-temporal instability of an annular liquid jet surrounded by an inviscid gas. Hu and Joseph (1989) investigated the temporal instability of a three-layered liquid core-annular flow. The instability of annular layeres has been used to model the formation of liquid bridges in microairways in lungs (Newhouse and Pozrikidis, 1992). The related problems of liquid bridge instability are reviewed by Alexander (1998). Annular jet instability is also of considerable theoretical interest because it includes many other flow instabilities as special cases (Meyer and Weihs, 1987). Moreover, it serves to establish knowledge of the fluid physics of flows with two distinctive curved fluid-fluid interfaces subjected to different shear forces, capillary forces, and inertial forces under variable gravitational conditions.

An Annular Jet

Consider the flow of a fluid in an annulus enclosing another fluid, which is surrounded by yet another fluid inside a circular pipe of radius Rω as shown in Figure 10.1. The axis of the pipe aligns with the direction of the acceleration due to gravity g. All three fluids are incompressible.

The phenomena of the breakup of liquid sheets and jets are encountered in nature as well as in various industrial applications. A good understanding of these phenomena requires a sound basic scientific knowledge of the dynamics of flows involving interfaces between different fluids. This book is the outcome of the author's inquiry into this fundamental knowledge. My understanding of the subject matter has been consolidated gradually through direct and indirect collaborations with my students and colleagues. The objective and scope of this book in the context of related existing works are explained in Chapter 1. Chapters 2 to 5 are devoted to exposition of the onset of sheet breakup. Chapters 6 to 10 discuss jet breakup. A perspective of the challenging aspects of the subject, including the nonlinear evolution subsequent to the onset of instability and nanojets, is sketched in Chapter 11. Some additional topics related to the breakup of a liquid body into smaller parts are discussed in the epilogue. Readers are expected to have the equivalent of at least an undergraduate background in science or engineering. In the theoretical development I have strived for mathematical rigor, numerical accuracy, and rational approximation. However, mathematics has not been used just for the sake of mathematics. I have depended on comparisons between different theories and experiments to establish physical concepts. Practical applications of the concepts are pointed out in appropriate places. The references relevant to each chapter are listed at the end of the chapter.

Nonuniform liquid sheets are encountered in various industrial applications including radiation cooling in space (Chubb et al., 1994), and paper making (Soderberg and Alfredson, 1998). Many of the works cited below were motivated by applications in surface coating, fuel spray formation, nuclear safety, and other industrial processes. The spatial variation of sheet thicknesses in these applications is necessitated by the conservation of mass flow across the cross section perpendicular to the flow direction. For example, the thickness of a planar sheet of constant width must decrease in the flow direction due to the gravitational acceleration. Consequently the local Weber number, based on the local thickness and velocity, changes spatially. In particular We may be greater than one in part of the sheet and smaller than one in the rest. If one locally applies the concept of absolute and convective instability in a uniform sheet, then part of the sheet may experience convective instability, while the remaining part may experience absolute instability. Depending on the relative location of the regions of We > 1 and We < 1, one would expect different physical consequences to the entire flow. The objective of this chapter is to elucidate the effect of the spatial variation of We on the dynamics of sheet breakup by properly applying the concept developed in the previous chapter for a sheet of uniform thickness.