# fluidsCh9 - Chapter 9 Flow Separation and Secondary Flow...

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Unformatted text preview: Chapter 9 Flow Separation and Secondary Flow Secondary flows imply a primary flow. There are many variants of this idea. In this chapter, we leave aside secondary flows that can be attributed to instabilities of the primary flow, which rely on linearization (see Ch. 11) of the equations for disturbance to the baseline solution and may be irrotational. Here, vorticity always plays an essential role, for example when its direction is modified by the flow (flow in a bend) or when vorticity is added at the boundaries. In some instances, the secondary flow is a region separated from the primary flow by a streamline that attaches to smooth surfaces or sharp edges. Examples (Fig. 9.2) are encountered in pipe entrance flows, aerodynamics, and wake formation: in all instances the performance is affected, and it is imperative to include the correct physics in any model of such flows. The entire range of Reynolds numbers is affected. In this chapter, we only peel the outermost layers of these difficult topics. See Fig. 9.1) 9.1 Curved channel First we examine the secondary flow in a curved channel (possibly a model for a bend in a river) (Fig. 9.3). The primary flow would be irrotational and governed by Euler’s normal equation: pressure must be larger on the outside in order to redirect the flow inward; therefore the speed must increase inward. 189 190 CHAPTER 9. FLOW SEPARATION AND SECONDARY FLOW E.S. Taylor’s movie: Secondary flows • All dependent on vorticity (potential flow have unique solutions). • Flow separation – vorticity from boundary, requires pressure gradients (see lec- tures) • No flow separation – Flow in a bend: material lines and vortex lines – Is the bathtub vortex really 2-dimensional? – Necklace vortex – Ekman layers Figure 9.1: Secondary flows: the role of vorticity, movie by E.S. Taylor 9.1. CURVED CHANNEL 191 Figure 9.2: Secondary flow: creeping flow in a wedge, forward- and backward- facing steps, diffuser, wake of a sphere, necklace vortex around an obstacle in a boundary layer. 192 CHAPTER 9. FLOW SEPARATION AND SECONDARY FLOW Figure 9.3: Irrotational flow in a curved channel The simple version, sometimes found in undergraduate texts, is possible if we ignore friction at the walls....
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fluidsCh9 - Chapter 9 Flow Separation and Secondary Flow...

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