Stability and Transition in Shear Flows

Hydrodynamic stability theory is concerned with the response of a laminar flow to a disturbance of small or moderate amplitude. If the flow returns to its original laminar state one defines the flow as stable, whereas if the disturbance grows and causes the laminar flow to change into a different state, one defines the flow as unstable. Instabilities often result in turbulent fluid motion, but they may also take the flow into a different laminar, usually more complicated state. Stability theory deals with the mathematical analysis of the evolution of disturbances superposed on a laminar base flow. In many cases one assumes the disturbances to be small so that further simplifications can be justified. In particular, a linear equation governing the evolution of disturbances is desirable. As the disturbance velocities grow above a few percent of the base flow, nonlinear effects become important and the linear equations no longer accurately predict the disturbance evolution. Although the linear equations have a limited region of validity they are important in detecting physical growth mechanisms and identifying dominant disturbance types.

We begin this section by deriving the stability equations for infinitesimal disturbances when effects due to viscosity are negligible. Stability calculations of this sort were among the first in the field of hydrodynamic stability theory. We will assume parallel flow. Let U _i = U(y)δ_1i be the base flow, i.e., a flow in the x-direction that varies with y (see Figure 2.1). If this flow is substituted into the disturbance equations (1.6) and the nonlinear and viscous terms are omitted, the resulting equations can be written as and the continuity equation is

We will consider the governing equations for infinitesimal disturbances in parallel flows. Let U _i = U(y)δ_1i be the parallel base flow, i.e., a flow in the x-direction that only depends on the wall-normal direction y (see Figure 2.1 defining the coordinate system and base flow). If this mean velocity profile is introduced into the disturbance equations (1.6) and the nonlinear terms are neglected, the resulting equations can be written:

This chapter will reexamine the equations governing the evolution of small disturbances in a viscous fluid. The emphasis will be different, though. Rather than concentrating on the eigenvalue problem, we will investigate the equations in the form of an initial value problem. What may seem to be merely a question of formalism or notation will turn out to have significant implications on the way we characterize the behavior of infinitesimal disturbances and the tools we use to study them.

The previous chapters have been devoted to the linear theory of hydrodynamic instabilities. This means that only the development of disturbances with infinitesimal amplitude can be described reliably. As soon as larger amplitudes are obtained (through an instability, for example), the linearized equations are rendered invalid, and nonlinear effects become important and have to be taken into account. For wavelike disturbances Fourier components no longer evolve independently but are all coupled together through wave-triad interactions. Typically this implies that waves with larger wave numbers than those included in the initial conditions are needed to describe the nonlinearly developing solution. In physical space smaller scales are introduced and the evolution of the disturbance becomes more complicated.

In previous chapters we developed a mathematical framework to analyze the stability characteristics of shear flows. We addressed instabilities of inviscid flows, and the effects of viscosity, transient behavior, and various effects of nonlinearities. The examples chosen have concentrated on the mathematical tools rather than an accurate modeling of realistic flow behavior. However, few applications of hydrodynamic stability theory deal with these idealized flow situations, and additional effects have to be taken into account. Varying pressure gradients, three-dimensionality of the mean flow, rotation and curvature, surface tension for free-surface flows and compressibility of the fluid medium are but a few of the effects that arise in realistic situations. Although the mathematical techniques introduced in previous chapters carry over to more complex flows, we will devote this chapter to the study of selected complications of the basic flow and their effect on the temporal growth of infinitesimal perturbations.

Up to now we have only been concerned with the temporal evolution of disturbances. However, it is conceivable that the physical situation requires the modeling of the disturbance amplitude/energy as a spatially growing quantity. Vibrating ribbons or harmonic point sources are only two of many situations where a spatial framework is more appropriate than a temporal one. In this section we develop the mathematical basis for a spatial linear stability analysis and discuss the relation between the temporal and spatial settings.

Secondary instability theory deals with the stability analysis of finite-amplitude steady or quasi-steady states that resulted from an earlier primary instability. In many cases a secondary instability is a precursor of transition to turbulent flow. Consequently, secondary instability analysis is sometimes used for transition prediction. In this chapter we will mainly cover the theoretical framework of secondary instability analysis. The role of secondary instabilities in the transition process of various shear flows will be discussed in Chapter 9.

We have spent several chapters discussing various type of instabilities in a number of flow situations. In this final chapter we will consider how these instabilities may trigger laminar-turbulent transition. The transition process is often complicated and can follow many possible routes. We shall try to classify these routes based on the mechanisms responsible for the disturbance growth. In this introductory section we will discuss three typical scenarios for the simplified flow situation of a parallel boundary layer growing in time; in the subsequent sections examples of more complicated flow situations will be given. First, however, we will briefly summarize some earlier attempts to classify transition.