fluidsCh1 - Chapter 1 Motivation This chapter adapted from...

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Chapter 1 Motivation This chapter, adapted from Tritton’s book, presents the need for more ad- vanced tools. For internal flows, the canonical case is the flow in a circular pipe. The student should be familiar with the laminar (Poiseuille) parabolic velocity profile, with the Moody diagram for pressure loss in a pipe, and with the modified Bernoulli equation including friction and minor losses. So we understand laminar friction and pressure drop, but how about entrance losses? Secondary flows at the entrance and exit? Flows in curved pipes? Pulsatile flow? At the undergraduate level, external flows are studied in relation with drag in uniform flows (usually involving empirical dimensionless curves of drag coefficients, and a description of flow separation and the drag crisis), flat plate boundary layers, and possibly some simple potential flows such as the flow around a cylinder. In this chapter, we revisit these topics to motivate the introduction of the more advanced concepts required to understand e.g. flow separation. We will end up with many questions marks, many unresolved difficulties to be addressed later in the course. 1.1 Internal Flow 1.1.1 A simple problem There are a number of practical design issues related to the pipe flow issuing from a head tank (Fig. 1.1). There are also some added features (e.g. about entrance pressure drop) that can be explained on the basis of undergraduate 17
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18 CHAPTER 1. MOTIVATION Figure 1.1: A simple pipe flow set-up tools. And an engineer should not overlook these approaches: simple first! But there are also a number of facts that show the need for more advanced concepts, more sophisticated tools (and more advanced mathematics). On the practical side, it would be important to maintain constant flow rate. This can be achieved in several ways: use a constant-displacement pump, insulated from the pipe by a baffled plenum; or use an overflow pipe, so the supply rate into the head tank needs no monitoring; etc. You might want to discuss alternative designs. By running this experiment over a wide range of Reynolds numbers, and making sure the pipe is long enough so entrance and exit losses can be neglected (how long is that?), one can collect the data represented on the Moody diagram (Fig. 1.2), here shown for a smooth pipe. The analysis was carried out in the previous chapter. Refer to the previous chapter for the basic analysis, using Bernoulli’s equation and control volume in the pipe. For automated calculations, one
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1.1. INTERNAL FLOW 19 10 2 10 3 10 4 10 5 10 6 10 7 10 8 10 -3 10 -2 10 -1 10 0 Reynolds number Friction factor Figure 1.2: Sketch of Moody diagram for pipe friction can make use e.g. of Colebrook’s formula for the friction factor f 1 f = - 2 .log 10 ( e 0 3 . 7 + 2 . 51 Re D f ) , (1.1) where e 0 is the pipe’s relative roughness (Fig. 1.2). We now turn to some of the details.
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