But their solutions consist merely the fluid particle spatial location at each

But their solutions consist merely the fluid particle

This preview shows page 54 - 56 out of 164 pages.

they result from direct application of Newton’s second law. But their solutions consist merely of the fluid particle spatial location at each instant of time, as depicted in Fig. 3.1. This figure shows two different fluid particles and their particle paths for a short period of time. Notice that it is the location of the fluid parcel at each time that is given, and this can be obtained directly by X z y 2 (1) 2 X x particle path #1 (2) particle path #2 X (0) 1 X 1 1 (2) X (1) X 2 (0) Figure 3.1: Fluid particles and trajectories in Lagrangian view of fluid motion. solving the corresponding equations. The notation X (0) 1 represents particle #1 at time t = 0, with X denoting the position vector ( x, y, z ) T . There are several important features of this representation requiring some explanation. First, it can be seen that the fluid parcel does not necessarily retain its size and shape during its motion. Later, when we derive the equation for mass conservation we will require that the mass of the fluid element remain fixed; hence, if the density is changing, which might well be the case, the volume must also change. Second, we can think of changes in shape as having arisen due to interactions with neighboring fluid elements (not shown, but recall Fig. 2.9); we will treat this in more detail when we derive the momentum equations. We next observe that although the velocity is not directly calculated, it is easily deduced since, e.g. , dx dt = u . Thus, if a sequence of locations of the fluid parcel is known for a period of time, it is easy to calculate its velocity (and acceleration) during this same period. Furthermore, we can think about obtaining values of any other fluid property ( e.g. , temperature or pressure) at this sequence of locations by simply “measuring” them as we ride through the fluid on the fluid parcel. Finally, it must be emphasized that in order to obtain a complete description of the flow field using this approach it is necessary to track a very large number of fluid parcels. From a practical standpoint,
Image of page 54
3.1. LAGRANGIAN & EULERIAN SYSTEMS; THE SUBSTANTIAL DERIVATIVE 49 either experimentally or computationally, this can present a significant burden. Furthermore, it is rather typical in engineering applications to need to know fluid properties and behavior at specific points in a flow field. In the context of a Lagrangian description it is difficult to specify, a priori , which fluid parcel to follow in order to obtain the desired information at some later time. 3.1.2 The Eulerian viewpoint An alternative to the Lagrangian representation is the Eulerian view of a flowing fluid. As noted above, this corresponds to a coordinate system fixed in space, and within which fluid properties are monitored as functions of time as the flow passes fixed spatial locations. Figure 3.2 is a simple representation of this situation. It is evident that in this case we need not be explicitly typical measurement locations x z y Figure 3.2: Eulerian view of fluid motion.
Image of page 55
Image of page 56

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