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Chapter 6 an Introduction to Hydrodynamic Stability Chapter 6 An Introduction to Hydrodynamic Stability Anubhab Roy and Rama Govindarajan Abstract In this chapter, our objective is twofold: (1) to describe common physical mechanisms which cause flows to become unstable, and (2) to introduce recent viewpoints on the subject. In the former, we present some well-known instabili- ties, and also discuss how surface tension and viscosity can act as both stabilisers and destabilisers. The field has gone through a somewhat large upheaval over the last two decades, with the understanding of algebraic growth of disturbances, and of absolute instability. In the latter part we touch upon these aspects. 6.1 Introduction “... not every solution of the equations of motion, even if it is exact, can actually occur in Nature. The flows that occur in Nature must not only obey the equations of fluid dynamics, but also be stable.” – Landau & Lifshitz Flow stability has preoccupied fluid dynamicists for centuries. The ubiquitous nature of turbulence, and the inability to offer a universal theory of it, led many to try tackling what was perceived as a simpler problem: of transition from a laminar state to a turbulent one. In the last two centuries, such efforts many-a-times have offered remarkable insight and helped make successful predictions. The description has undergone major reforms from time to time, and the complete process is not fully understood. The transition to turbulence begins usually with an instability of the laminar state, which is the subject of this brief review. An effort is made to address the basic tenets of hydrodynamic stability with a focus on a few recent viewpoints on the subject. R. Govindarajan () Engineering Mechanics Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bangalore 560064, India e-mail: [email protected] J.M. Krishnan et al. (eds.), Rheology of Complex Fluids, 131 DOI 10.1007/978-1-4419-6494-6 6, © Springer Science+Business Media, LLC 2010 132 A. Roy and R. Govindarajan Fig. 6.1 Normal modes. (a) A double pendulum exhibiting 2 normal modes – symmetric and anti-symmetric oscillations (b) A vibrating string with the first seven of its infinite normal modes Early studies in hydrodynamic stability arose in the theory of water waves by Newton followed by seminal work by Laplace, Cauchy, Poisson and later, by Stokes. Incidentally, most of the early work was done before the world got acquainted with Fourier series, although Cauchy employed defacto Fourier transforms [5, 6]. The advent of harmonic analysis facilitated the study of stability of mechanical sys- tems by introducing the notion of normal modes. A normal mode is a pattern of oscillations in which the entire system oscillates in space/time with the same wave- length/frequency. A coupled pendulum with two degrees of freedom has two normal modes, whereas a vibrating string has infinitely many normal modes, often labelled as harmonics in musical instruments (Fig. 6.1). The beauty of normal modes lies in their ability to offer a complete description for the evolution of any arbitrary dis- turbance in most cases. Helmholtz, Kelvin and Rayleigh were among the earliest to use normal modes to describe problems in hydrodynamic stability. We will discuss some instabilities named after them. 6.2 Waves in Fluids The variety of available restoring mechanisms allow a plethora of waves in flu- ids [20]. To name a few – compressibility gives rise to sound waves, variation in buoyancy leads to gravity waves, inertial waves owe their origin to rotation, mag- netic fields lead to Alfven waves, whereas a latitudinal variation of Coriolis force leads to Rossby waves. A typical example to understand the generation of oscillations in fluids is internal gravity waves. Consider a fluid at rest, whose density .z/ increases with the depth z, i.e., the fluid is stably stratified, with heavier fluid at the bottom and lighter fluid above. A fluid parcel of density 0 at z0, when displaced upward by a distance ız,is now surrounded by lighter fluid and thus sinks. In the absence of retarding forces due to viscosity, it would then overshoot its mean position and reach a neighbourhood of heavier fluid, where it experiences an upward thrust (Fig. 6.2). From Newton’s second law we have d2ız 0 Dı g; (6.1) dt 2 6 An Introduction to Hydrodynamic Stability 133 Fig. 6.2 Oscillations in a continuously and stably stratified fluid where g is the acceleration due to gravity and ı is the density difference between the parcel and the ambient fluid at z0 C ız.Sinceı is infinitesimal, it can be ex- pressed as ı Dız d=dz. We may rewrite (6.1)as 2 d ız C 2 D 2 N ız 0; (6.2) dt s g d where N D dz is the oscillation frequency, called the Brunt–V¨ais¨al¨a frequency. Thus, buoyancy provides a restoring force in this oscillatory motion. The reader can easily predict the effect of viscous forces on this oscillatory motion. Such wave-like motion in fluids is abundant in nature. Ripples generated on the surface of a pond to giant waves in oceans – all can be appreciated as oscillations over a quiescent flow state. 6.3 Instabilities We have given an example of a stable system in the above section, where the os- cillations are at best neutral, i.e., continue for ever with the same amplitude, and in most real situations, are damped out by a retarding force. What of the opposite situation, i.e., one in which small oscillations grow in amplitude? This is a common way in which a flow becomes unstable. In the above example, if the sign of d=dz were to be flipped, i.e., if the fluid were to be unstably stratified, one can see that the parcel of fluid would get accelerated away from its starting position rather than oscillate. In many flows, one may have growing oscillations instead, which could either be followed quite abruptly by a transition to turbulence, or be the first step in a long and eventful march towards turbulence. Flow instabilities occur all around 134 A. Roy and R. Govindarajan us in both desirable and undesirable situations. Thanks to the efforts of numerous remarkable mathematicians, engineers and physicists, a lot is known about them, but more continues to bewilder us. We consider a few model flow situations, concentrating on the physical mech- anisms responsible for destabilising simple orderly flows and taking them to a completely different state. We first discuss linear instability, i.e., when the orderly state is given a very small perturbation, and examine whether the resulting oscil- lations grow or decay. For extensions to more complicated flows, the interested reader may consult standard textbooks, some of which are listed at the end of this chapter. 6.3.1 Effect of Shear: Kelvin–Helmholtz Instability The classical problem serving as an excellent introduction to hydrodynamic instabil- ity is that of Kelvin–Helmholtz. Imagine two uniform streams of different velocities flowing past each other. Such a scenario is very commonly observed in nature, and is known as a mixing layer. The jump in the velocity across a layer of infinitesi- mal thickness is described as a vortex sheet (Fig. 6.3b). The vortex sheet consists of a constant density of point vortices, with their vorticities pointing out of the pa- per. The physical mechanism for the instability as described by Batchelor [2] relies purely on vortex dynamics. Let the vortex sheet be perturbed by a sinusoidal distur- bance so that the perturbed interface is located at D sin kx. In the neighbourhood of the nodes (A and B), the positive vorticity induces a clockwise circulating veloc- ity field. If @=@x > 0, the crest and trough move away from each other, leading to vorticity being swept off from nodes like A, whereas if @=@x < 0, the crest and trough come closer to each other, leading to vorticity being swept into nodes like B. Thus, accumulation of vorticity at points like B takes place unboundedly in the linear, non-dissipative scenario, giving exponential growth. This mechanism translates into the celebrated Rayleigh–Fjørtoft criterion in inviscid parallel shear flows – that there should exist a vorticity maximum somewhere in the base flow for instability to occur. To see the linear instability mathematically, we begin with the Euler equations in two dimensions (which are equivalent to the Navier–Stokes equa- tions without viscosity). Here, u is the component of the velocity in the streamwise Fig. 6.3 Kelvin–Helmholtz instability. (a) represents the unperturbed state and (b)the corresponding vortex sheet base state. A sinusoidal perturbation is imposed on the vortex sheet (c)which amplifies through vorticity induction (d) 6 An Introduction to Hydrodynamic Stability 135 direction x,andv the component in the normal direction z. We then split all flow quantities into their mean and a perturbed value, such as u D U.z/COu.x; z;t/.Intwo dimensions for an incompressible flow, the continuity equation r:u enables us to write both components of the velocity vector u in terms of the stream function , as u D @ =@y,andv D @ =@x. The perturbed quantities are considered in their normal mode form, such as O .x; z;t/ D .z/ expŒik.x ct/,wherek and c are the wavenumber and wave speed, respectively, with c D !=k, ! being the fre- quency of the disturbance. In the discussion here, we allow !, and therefore c,tobe a complex quantity, whose imaginary part tells us whether the disturbance grows or decays. We take the disturbance to be much smaller than the undisturbed flow, and so compared to the perturbation, we may neglect products of perturbation quantities, and the resulting equations are linear in the perturbations.
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