Introduction to Chemical Engineering Fluid Mechanics

INTRODUCTION

As discussed in Chapters 2 and 3, laminar flows tend to be unstable at large Reynolds number. A transition to turbulence occurs when Re exceeds a critical value, which depends on the system but is often in the thousands to hundreds of thousands. Turbulence is ubiquitous in everyday life, the natural environment, and large-scale, continuous processes. One of its hallmarks is a marked increase in wall shear stresses. That is indicated by much higher friction factors for flow in pipes (Fig. 2.2) or past flat plates (Fig. 3.5), and is a reflection of the more rapid cross-stream transfer of momentum in turbulent than in laminar flow. In laminar flow, momentum transfer across streamlines is due only to the molecular-level friction that underlies viscosity. In turbulent flow, transient eddies or vortices of varying size are superimposed on the average motion. The eddies, which vary randomly from instant to instant, transport momentum at rates that greatly exceed those from the molecular mechanism.

The random velocity fluctuations associated with the eddies ensure that no two turbulent flows are ever exactly the same. Deterministic modeling of flow details must be abandoned and statistical descriptions used instead. In engineering, the momentary fluctuations are of secondary interest and it is desired mainly to predict time-smoothed or time-averaged quantities. Even so, all available tools must be brought to bear: dimensional analysis, experimentation, analytical theory, and computation. With the present state of knowledge, experimental findings are particularly crucial. Whereas the details of a laminar flow often can be predicted from first principles, turbulence calculations nearly always contain an element of empiricism.

This chapter is organized as follows. After several aspects of turbulence are described qualitatively, there is a discussion of insights into velocity and length scales that can be gained by combining key observations with dimensional analysis. How to reformulate the continuity and Navier–Stokes equations in terms of time-smoothed quantities, a procedure called Reynolds averaging, is then described. The concept of an eddy diffusivity is introduced, which is the classical approach for making turbulence calculations practical. Several examples then illustrate the application of this concept to shear flows in conduits and boundary layers.