This text has covered some historical and more advanced theoretical and computational techniques to predict the onset of transitional flows with linear methods, the amplification and interaction of these linear modes in the nonlinear regime, and the matching of these predictions with empirical models. Furthermore, some methods of control have been developed and discussed in the chapter on flow control. Here, we address issues associated with investigating hydrodynamic instabilities using experimental techniques. These issues include the experimental facility, model configuration, and instrumentation, all of which impact the understanding of hydrodynamic instabilities.

Because the authors have primary expertise in theory and computation, we readily acknowledge the topics in this chapter are based on literature from leading scientists and engineers in the field of transitional flows. This chapter serves as an introduction to the experimental process. The content of this chapter is primarily based on the review by Saric (1994b) and a text by Smol'yakov & Tkachenko (1983).

Experimental Facility

Because the theoretical and computational modeling of a hydrodynamic instability process is the goal, two key aspects of the flow must be carefully documented in the experiment before studying the instabilities. First, the physical properties of the flow environment must be understood within the experimental facility. The makeup of the facility dictates the background (or freestream) disturbances and the spatial-temporal characteristics of the flow environment.

The incoming freestream environment should be understood and characterized before commencing with a discussion of the use of artificial disturbances, that are typically the manner hydrodynamic instabilities are investigated.

Polar coordinates

As has been mentioned earlier, there are many flows that require the formulation of the stability problem to be cast in coordinate systems other than Cartesian. For example, pipe flow is perhaps the most notable example. Then, there is the case that is now referred to as stability of Couette flow and has been examined both theoretically and experimentally by Taylor (1921, 1923). In this case, concentric cylinders rotate relative to each other to produce the flow. Free flows, such as the jet and wake, can be thought of as round rather than plane. If the boundary layer occurs on a curved wall, then Görtler vortices result. All of these examples can be described in terms of polar coordinates. And, at the outset, it should be recognized that, not only will the governing mathematics be different from that that has been used up to this point but the resulting physics may have novel characteristics as well.

The prevailing basis for the flows that have been examined heretofore has been that the flows are parallel or almost parallel. Then, the solutions for the disturbances were all of the form of plane waves that propagate in the direction of the mean flow or, more generally, obliquely to the mean flow. And, as we have learned, if solid boundaries are present in the flow, viscosity is a cause for instability. Now there are flows that will have curved streamlines and this leads to the possibility of a centrifugal force and its influence must be considered.

History, background, and rationale

In examining the dynamics of any physical system the concept of stability becomes relevant only after first establishing the possibility of equilibrium. Once this step has been taken, the concept becomes ubiquitous, regardless of the actual system being probed. As expressed by Betchov & Criminale (1967), stability can be defined as the ability of a dynamical system to be immune to small disturbances. It is clear that the disturbances need not necessarily be small in magnitude but the fact that the disturbances become amplified as a result and then there is a departure from any state of equilibrium the system had is implicit. Should no equilibrium be possible, then it can already be concluded that that particular system in question is statically unstable and the dynamics is a moot point.

Such tests for stability can be and are made in any field, such as mechanics, astronomy, electronics and biology, for example. In each case from this list, there is a common thread in that only a finite number of discrete degrees of freedom are required to describe the motion and there is only one independent variable. Like tests can be made for problems in continuous media but the number of degrees of freedom becomes infinite and the governing equations are now partial differential equations instead of the ordinary variety. Thus, conclusions are harder to obtain in any general manner but it is not impossible.

Introduction

The consideration of flows when the fluid is compressible presents a great many difficulties. The basic mathematics requires far more detail in order to make a rational investigation. The number of dependent variables is increased. And, regardless of the specific mean flow that is under scrutiny, the boundary conditions can be quite involved. Such observations will become more than obvious as the bases for examining the stability of such flows are established.

By their nature, the physics of compressible flows implies that there are now fluctuations in the density as well as the velocity and pressure. And, the density can be altered by pressure forces and the temperature. As a result, the laws of thermodynamics must be considered along with the equations for the conservation of mass and momentum. Consequently, a new set of governing equations must be derived. Moreover, this set of equations must be valid for flows than range from slightly supersonic to those that are hypersonic; i.e., M, the Mach number defined for the flow is of order one or larger. Flows that are characterized by a Mach number that is small compared to 1 and simply have a mean density that is inhomogeneous are those flows that satisfy what is known as the Boussinesq approximation and will be examined in Chapter 7. Suffice it for now to say that it is the force of gravity that plays a key role in such cases.

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.