Despite the emphasis we have placed on finding dimensionless parameters and

Despite the emphasis we have placed on finding

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lectures we will not be providing any further treatment of these. Despite the emphasis we have placed on finding dimensionless parameters and writing governing equations in dimensionless form, we feel it is essential to remark that, especially in constructing modern CFD codes, such an approach is seldom, if ever, taken. There is a very fundamental reason for this stemming from the extreme generality of the Navier–Stokes equations. In particular, these equations represent essentially all possible fluid motions, and the physical details may differ drastically from one situation to the next. In turn, this implies that the specific dimensionless parameters needed for a complete description of the flow also will vary widely. Thus, it is basically not possible to account for this in a general CFD code in any way other than by writing the code for the original unscaled physical equations. 3.7 Summary We conclude this chapter with a brief recap of the topics we have treated. We began with a discussion in which Lagrangian and Eulerian reference frames were compared, and we noted that the former is more consistent with application of Newton’s second law of motion while the latter provided formulations in terms of variables more useful in engineering practice. We then introduced
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100 CHAPTER 3. THE EQUATIONS OF FLUID MOTION the substantial derivative that permits expressing Lagrangian motions in terms of an Eulerian reference frame. We next provided a brief review of the parts of vector calculus that are crucial to derivation of the equations of motion, namely Gauss’s theorem and the transport theorems; we then proceeded to derive the equations of motion. The first step was to obtain the so-called “continuity” equation which represents mass conservation. The differential form of this was derived, and then by basically working backwards we produced a control-volume formulation that is valuable for “back-of-the-envelope” engineering calculations. We next began derivation of the differential form of the equations of momentum balance—the Navier–Stokes equations. This began by stating a more general form of Newton’s second law— one more appropriate for application to fluid elements, as needed for describing fluid flows. We expressed this in terms of acceleration (times mass per unit volume) and a sum of body and surface forces acting on an isolated fluid element. Then we treated the surface terms in detail, ultimately employing Newton’s law of viscosity to obtain formulas for the viscous stresses that generate most (except for pressure) of the surface forces. This led to the final form of the equations of motion, and we then discussed some of the basic mathematics and physics of these equations on a term-by-term basis. The final section of the chapter was devoted to treatment of scaling and dimensional analysis of the governing equations in order to determine the important dimensionless parameters for any given flow situation. One of the prime uses of such information is to allow application of data
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