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Unformatted text preview: Chapter 8 Narrow Flows A distinct type of approximation is associated with different scaling in differ ent directions: narrow flows encompass boundary layers, but also jets, mixing layers, shear layers, wakes and similar configurations where the streamwise length scale is much larger than transverse scales. The simplifications to the NS equations are less drastic than in the limits of small or large Re, but still sufficient to allow solutions. More importantly, they include distinct flow physics and phenomena (first recognized by Prandtl) that may need to be taken in consideration more generally. Again we focus on broad phenomena and useful techniques: indepth study is left for courses on viscous flows and turbulent flows. The movie by Abernathy (Fig. 8.1) serves as a useful introduction. A first viewing will provide useful background, on which the analysis below can build; a second viewing, as time allows, is recommended. 8.1 Flat plate boundary layers Undergraduate classes (fluid mechanics, convective heat transfer) should mention some basics about boundary layers (BL). Beyond a basic description, a controlvolume analysis is repeated below; further details can be obtained by using approximate velocity profiles in K` arm` ans integral method, with which a graduate student in fluid mechanics should be familiar (see, possi bly, convective heat transfer course; more comments below). Flat plate, of course, is code for something other than the geometry: what it really means is that the freestream speed is constant, which in turn 165 166 CHAPTER 8. NARROW FLOWS F. Abernathys movie: Boundary layers narrow flow, solid boundary flat plate means zero pressure gradient (ZPG) growth in streamwise direction wall stress larger near leading edge effect of pressure gradient vorticity, circulation Flat plate: vorticity from leading edge, not from wall surface Note: separation ahead of bluff body! see chapter on flow separation. transition to turbulence Figure 8.1: Basics of boundary layers, in F. Abernathys movie. 8.1. FLAT PLATE BOUNDARY LAYERS 167 Figure 8.2: ZPGBL definition sketch implies, through Bernoullis equation, that the streamwise pressure gradi ent vanishes. Thus, a more technically correct terminology is zeropresure gradient boundary layer (ZPGBL). 2D steady flow is assumed. Then, the cartesian directions are very distinct. The streamwise direction includes the primary motion, and will be our xdirection. Normal to the plate is the transverse direction, labeled as y. Normal to both is the spanwise direction, parallel to the leading edge of the plate, as the zdirection. In 2D flows, there are no variations or velocity component in the zdirection. The distance along the plate will be denoted indifferently as x or as a generic distance L , freestream velocity is U ....
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 Spring '10
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