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Introduction

1.1 What is Turbulent Flow?

Turbulent flow is ubiquitous in nature and technology. From breaking waves on a beach, to vortical eddies in the atmosphere that shake an airplane in flight, to flow across the hull of a submarine and separating into the ocean, to the flow disturbing the landing of a helicopter on the flight deck of a ship, fluid flows in a turbulent state. Why turbulent flow is so common is equivalent to the question of why flows often depart from a laminar state to become turbulent. The answer to this question can explain the ubiquitousness of turbulent flow and simultaneously provide a useful definition of what is meant by turbulent flow.

It is a fact that in many fluid flows there are sources of perturbation that are persistent and inevitable. These can arise from minute imperfections on boundary surfaces or slight variability in the incoming flow field. It is also conceivable [1, 2] that perturbations of a molecular origin can occur, as in the kinds of organized molecular behavior that leads to Brownian motion. How a fluid flow reacts to the presence of perturbations lies at the heart of whether or not the fluid motion is turbulent or will become turbulent. When perturbations to the velocity field appear in an otherwise laminar flow, the action of viscous forces is to diffuse the local momentum excess associated with the disturbance. Depending on the strength of the perturbation and the effectiveness of the viscous smoothing, the flow disturbances may grow, leading to the appearance of three-dimensional (3D), non-steady, disorganized motion that is referred to as turbulence or else be damped leading to the maintenance of laminar flow. The balance in this case is between the inertia of fluid particles in motion and the viscous forces acting on them to regularize the local flow field.

As will be considered more formally in the next chapter, the Reynolds number where and are, respectively, characteristic velocity and length scales, and is the kinematic viscosity, is a measure of the ratio of inertia to viscous forces and so has a large role to play in characterizing whether fluid flow is turbulent or not. For small values of the Reynolds number, internal viscous forces dominate and the flow tends to be laminar, or laminarize if it is not initially so. For high Reynolds numbers viscous smoothing is insufficient to prevent the growth of instabilities, with turbulent flow being the result.

The transition of laminar flow to turbulence can follow a number of different routes depending on the magnitude and nature of the perturbations that are present [3]. For slight perturbations a linear instability is triggered that in a boundary layer, for example, would be manifest as a pattern of streamwise disturbances known as Tollmein-Schlichting waves. As they develop downstream vortical structures appear in the flow whose breakdown and interactions signal the appearance of turbulent flow. If the initial perturbation to a laminar flow is sufficiently large, then bypass transition may occur for which there is a rapid development of the vortical structures leading to turbulence. Some aspects of the vortical structures occurring in transition and turbulent flow are discussed in Chapter 8.

Once initiated, turbulent flow persists unless there is a change in external conditions that could remove or reduce the mechanisms leading to instability. For example, flow acceleration in some circumstances has been observed to relaminarize the flow in boundary layers [4]. In many situations, such as pipe and channel flows, and boundary and mixing layers that will be discussed in this book, the flows may be seen to evolve from an upstream laminar state, through transition to the fully turbulent state downstream.

The subject of turbulence is primarily concerned with describing and predicting the quantitative and qualitative properties of fluid flow specifically in turbulent flow regions. The study of the conditions that might lead to the appearance of turbulent flow is the main interest of stability theory. Some overlap between these fields can be expected since the conditions leading to flow breakdown in transition may have some role in the fully turbulent region as well [5]. Moreover, turbulent flow may contain structural features in the form of vortices that also populate the transitional flow region [6]. Studying such aspects of transition can have benefit for the study of turbulent flow as well. For those flows where laminar, transitional, and turbulent flow exist simultaneously, as in a developing boundary layer, it can be important to predict the extent and properties of each separate region.

It will be seen subsequently that there are numerous ways that turbulent fluid motion differs from laminar motion, and this has important consequences for such aspects of flow analysis as predicting the forces on bodies or the diffusion of contaminants within the flow field. Moreover, the special properties of turbulent motion make it much more difficult than laminar flow to either solve for the fluid velocity field via numerical simulations or measure it in physical experiments. It will be seen that limitations of a variety of sorts shadow the analysis of turbulent flow so that in the study of any particular flow decisions often have to be made as to what methods ought to be brought to bear in studying the flow and how and to what extent they should be deployed. The combination of an important need to predict turbulent flow behavior and the fact that analysis and measurement techniques are not without limitations leads to the need for fluids engineers to acquire some degree of proficiency in understanding the advantages and disadvantages of available techniques and how best to interpret what they say about the flow field.

1.2 Examples of Turbulent Flow

Considering some specific examples where turbulent flow is present and important can help make clear the wide range of situations that are encompassed in the study of turbulence. Among the important categories of phenomena involving turbulent motion are those associated with the flow adjacent to solid surfaces. In the near-wall field lies the origin of the viscous and pressure forces that affect the motion of bodies. Turbulence produced in boundary layers is a common occurrence that has significant consequences for the strength and nature of the boundary forces. Turbulent flow is an integral part of the movement of automobiles and trucks, as seen in Figure 1.1 where knowledge from wind tunnel testing and simulations has been used to recreate the kind of smoke pattern to be expected in the flow around semi-tractor trailers. Turbulence develops over the trailer as boundary layers that separate into a turbulent wake that has a large influence on the overall drag force. Complicated turbulent eddying appears in the gap between the cab and trailer, and in the underbody with considerable consequences for drag and stability.

Turbulence produced on the wings and fuselage of an aircraft must be taken into account in the design process. For a wing at high angle of attack, as seen in Figure 1.2, turbulent vortices filling the wake and shedding from the edges of the wing are inextricably tied to the dynamics of the airplane by their effect on the pressure field. Similarly, turbulent flow is an important part of producing drag and lift forces affecting the hulls and keels of boats and such objects as runners, sky divers, skiers, ice skaters, and projectiles including baseballs, footballs, and golf balls.

Figure 1.1 Artist's rendering of smoke visualized flow past a tractor trailer. The main areas affected by turbulence include boundary layers on the trailer surface, the tractor-trailer gap, under the chassis, and the large area of turbulence behind the vehicle. Courtesy of Don-Bur (Bodies and Trailers) Ltd.

Figure 1.2 Fog wind tunnel visualization of a NACA 4412 airfoil at a low-speed flow (Re = 20,000). Turbulence fills the massively separated flow on the back of the airfoil and vortices roll up in the trailing edge wake. Image by Georgepehli, Smart Blade GmbH.

Turbulent flow next to walls is also an essential part of the forces in internal flows such as channels and pipes. For low Reynolds numbers, the flow in a pipe will be laminar, that is Poiseulle flow, but beyond a transition Reynolds number depending on such factors as the smoothness of the boundaries and incoming flow, the motion in the pipe will be turbulent. In a smooth-walled pipe transition occurs at a Reynolds number based on mean flow velocity and diameter of approximately 2000 [7], a condition that is often exceeded in engineering applications. Under some circumstances, such as the presence of blockages, turbulent flow occurs in the human lung and heart. Turbulence is an essential aspect of the combustion process in engines, and the flow through heat exchangers, turbines, and numerous other devices.

Oftentimes, turbulent flows occur away from the immediate effects of solid boundaries in what are known as free shear flows. In the case of mixing layers two streams of differing velocity come together, leading to the development, amplification, and merging of vortical structures that have arisen via Kelvin-Helmholtz instability. An example of this is illustrated in Figure 1.3 where clouds mark the presence of a mixing layer formed from the presence of high wind shear aloft in the atmosphere.

Figure 1.3 Clouds forming over Mount...