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

With sufficient increases the flow can separate in

This preview shows page 47 - 49 out of 164 pages.

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
Image of page 47
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,
Image of page 48
Image of page 49

You've reached the end of your free preview.

Want to read all 164 pages?

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern

Stuck? We have tutors online 24/7 who can help you get unstuck.
A+ icon
Ask Expert Tutors You can ask You can ask You can ask (will expire )
Answers in as fast as 15 minutes
A+ icon
Ask Expert Tutors