With sufficient increases the flow can separate in this case as well even when

# With sufficient increases the flow can separate in

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With sufficient increases the flow can separate in this case as well, even when the surface is quite smooth—even flat. This can occur for flow over airfoils at high angles of attack, resulting in “stalling” of the wing and loss of lift. Thus, an understanding of this mechanism for separation is also very important. 2.5 Flow Visualization In the context of laboratory experiments flow visualization is an absolute necessity, and it was within this context that the techniques we will treat here were first introduced. It is worth noting that even for theoretical analyses much can often be learned from a well-constructed sketch that emphasizes the key features of the flow physics in a situation of interest. But even more important is the fact that now CFD can produce details of 3-D, time-dependent fluid flows that simply could not previously have been obtained, even in laboratory experiments. So visualization is all the more important in this context. The visualization techniques we will describe in this section are very standard (and there are others); they are: streamlines, pathlines and streaklines. We will devote a subsection to a brief discussion of each of these, introducing their mathematical representations and physical interpretations. Before beginning this we note one especially important connection amongst these three representations of a flow field: they are all equivalent for steady flows . 2.5.1 Streamlines We have already seen some examples of streamlines; the curves appearing in Figs. 2.16 and 2.17, which we called trajectories of fluid elements, were actually streamlines, and in the preceding sub- section we introduced the notion of a dividing streamline—without saying exactly what a streamline is. We remedy these omissions with the following definition. Definition 2.13 A streamline is a continuous line within a fluid such that the tangent at each point is the direction of the velocity vector at that point. One can check that Figs. 2.16 and 2.23 are drawn in this manner; Fig. 2.24 provides more detail. In this figure we consider a 2-D case because it is more easily visualized, but all of the ideas we present work equally well in three space dimensions. The figure shows an isolated portion
42 CHAPTER 2. SOME BACKGROUND: BASIC PHYSICS OF FLUIDS of a velocity field obtained either via laboratory experiments ( e.g. , particle image velocimetry mentioned in Chap. 1) or from a CFD calculation. We first recall from basic physics that the u v y x Figure 2.24: Geometry of streamlines. velocity components are given by u = dx dt , and v = dy dt . (2.17) It is then clear from the figure that the local slope of the velocity vector is simply v/u . Thus, v u = dy/dt dx/dt , or dx u = dy v parenleftbigg = dz w in 3D parenrightbigg . (2.18) Equations (2.18) are often viewed as the defining relations for a streamline. The fact (given in the definition) that a streamline is everywhere tangent to the velocity field is especially clear if we write these as (in 2D) dy dx = v u . (2.19) Furthermore, if the velocity field ( u, v ) T is known (as would be the case with PIV data or CFD simulations), streamlines can be constructed by solving the differential equation (2.19). We note,

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