INTRODUCTION THE NAVIER-STOKES EQUATION FLOW PAST A SPHERE AT LOW REYNOLDS NUMBERS INVISCID FLOW THE BERNOULLI EQUATION TURBULENCE Introduction What Is Turbulence? Describing Turbulence Laminar and Turbulent Flow Origin of Turbulence BOUNDARY LAYERS Introduction Laminar Boundary Layers and Turbulent Boundary Layers Wakes How Thick are Boundary Layers? Some Flows Are “All Boundary Layer” Internal Boundary Layers FLOW SEPARATION FLOW PAST A SPHERE AT HIGH REYNOLDS NUMBERS SETTLING OF SPHERES Introduction Towing vs. Settling Dimensional Analysis Settling at Low Reynolds NumbersCHAPTER 3 FLOW PAST A SPHERE II: STOKES’ LAW, THE BERNOULLI EQUATION, TURBULENCE, BOUNDARY LAYERS, FLOW SEPARATION INTRODUCTION 1 So far we have been able to cover a lot of ground with a minimum of material on fluid flow. At this point I need to present to you some more topics in fluid dynamics—inviscid fluid flow, the Bernoulli equation, turbulence, boundary layers, and flow separation—before returning to flow past spheres. This material also provides much of the necessary background for discussion of many of the topics on sediment movement to be covered in Part II. But first we will make a start on the nature of flow of a viscous fluid past a sphere. THE NAVIER-STOKES EQUATION 2 The idea of an equation of motion for a viscous fluid was introduced in the Chapter 2. It is worthwhile to pursue the nature of this equation a little further at this point. Such an equation, when the forces acting in or on the fluid are those of viscosity, gravity, and pressure, is called the Navier–Stokes equation, after two of the great applied mathematicians of the nineteenth century who independently derived it. 3 It does not serve our purposes to write out the Navier–Stokes equation in full detail. Suffice it to say that it is a vector partial differential equation. (By that I mean that the force and acceleration terms are vectors, not scalars, and the various terms involve partial derivatives, which are easy to understand if you already know about differentiation.) The single vector equation can just as well be written as three scalar equations, one for each of the three coordinate directions; this just corresponds to the fact that a force, like any vector, can be described by its scalar components in the three coordinate directions. 4 The Navier–Stokes equation is notoriously difficult to solve in a given flow problem to obtain spatial distributions of velocities and pressures and shear stresses. Basically the reasons are that the acceleration term is nonlinear, meaning that it involves products of partial derivatives, and the viscous-force term contains second derivatives, that is, derivatives of derivatives. Only in certain special situations, in which one or both of these terms can be simplified or neglected, can the Navier–Stokes equation be solved analytically. But numerical solutions of the full Navier–Stokes equation are feasible for a much wider range of flow problems, now that computers are so powerful. 35FLOW PAST A SPHERE AT LOW REYNOLDS NUMBERS 5 We will make a start on the flow patterns and fluid forces associated with flow of a viscous fluid past a sphere by restricting consideration to low Reynolds numbers ρUD/μ (where, as before, U is the uniform approach velocity and D is the diameter of the sphere). Figure 3-1. Steady flow of a viscous fluid at very low Reynolds numbers (“creeping flow”) past a sphere. The flow lines are shown in a planar section parallel to the flow direction and passing through the center of the sphere. 6 At very low Reynolds numbers, Re << 1, the flow lines relative to the sphere are about as shown in Figure 3-1. The first thing to note is that for these very small Reynolds numbers the flow pattern is symmetrical front to back. The flow lines are straight and uniform in the free stream far in front of the sphere, but they are deflected as they pass around the sphere. For a large distance away from the sphere the flow lines become somewhat more widely spaced, indicating that the fluid velocity is less than the free-stream velocity. Does that do damage to your intuition? One might have guessed that the flow lines would be more crowded together around the midsection of the sphere, reflecting a greater velocity instead—and as will be shown later in this chapter, that is indeed the case at much higher Reynolds numbers. (See a later section for more on what I have casually called flow lines here.) For very low Reynolds numbers, however, the effect of “crowding”, which acts to increase the velocity, is more than offset by the effect of viscous retardation, which acts to decrease the velocity. 7 The velocity of the fluid is everywhere zero at the sphere surface (remember the no-slip condition) and increases only slowly away from the sphere, even in the vicinity of the midsection: at low Reynolds numbers, the retarding effect of the sphere is felt for great distances out into the fluid. You will see later in this chapter that the zone of retardation shrinks greatly as the Reynolds number 36increases, and the “crowding” effect causes the velocity around the midsection of the sphere to be greater than the free-stream velocity except very near the surface of the sphere; more on that later. Figure 3-2. Coordinates for description of the theoretical distribution of velocity in flow past a sphere at very low Reynolds numbers (creeping flow). 8 If you would like to see for yourself how the velocity varies in the vicinity of the sphere, Equations 3.1 give the theoretical distribution of velocity v, as a function of distance r from the center of the sphere and the angle θ measured around the sphere from 0° at the front point to 180° at the rear point (Figure 3-2): ur=−U cosθ1 −3R2r⎛ ⎝ ⎜ +R32r3⎞ ⎠ ⎟ (3.1) uθ= U sinθ1 −3R4r⎛ ⎝ ⎜ −R34r3⎞ ⎠ ⎟ This result was obtained by Stokes (1851) by specializing the Navier–Stokes equations for an approaching flow that is so slow that accelerations of the fluid as it passes around the sphere can be ignored, resulting in an equation that can be solved analytically. I said in Chapter 2 that fluid density ρ is needed as a variable to describe the drag force on a sphere because accelerations are produced in the fluid as the sphere moves through it. If these accelerations are small enough, however, it is reasonable