We can observe two distinct flow regimes indicated by the data and a possible

We can observe two distinct flow regimes indicated by

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We can observe two distinct flow regimes indicated by the data, and a possible transitional regime between the two—not unlike our earlier intuitive description of the transition to turbulence (recall Fig. 2.22). In the first of these ( Re less than approximately 20) the dimensionless force varies linearly with Reynolds number (a fact that can be derived analytically). In the last regime ( Re greater than about 200) the force depends on the square of Re . In this case we would expect that ρF μ 2 slashBig Re 2
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3.6. SCALING AND DIMENSIONAL ANALYSIS 97 F/ μ 2 0.16 2 Re 10 9 10 7 10 5 10 3 10 1 10 1 10 2 10 3 10 4 10 5 3 π Re laminar turbulent transitional ρ 1 turbulent laminar transitional Re Figure 3.16: Dimensionless force on a sphere as function of Re ; plotted points are experimental data, lines are theory (laminar) and curve fit (turbulent). should be approximately constant. That is, ρF μ 2 slashbiggparenleftbigg ρ UD μ parenrightbigg 2 = F ρU 2 D 2 const. This suggests a different, but related, parameter by means of which to analyze the data. Namely, recall that pressure is force per unit area; so we have F/D 2 p , and it follows that we might also correlate data using the quantity p ρU 2 as the dimensionless parameter. It is of interest to recall that this is precisely the scaled pressure arising in our earlier analysis of the Navier–Stokes equations. In practice, it is more common to utilize a reference pressure, often denoted p , and define the dimensionless pressure coefficient as C p p p 1 2 ρU 2 . (3.59) Then the above plot can also be presented as C p vs. Re , a common practice in fluid dynamics. We remark, however, that while a single value of the force on the sphere will fairly well characterize physics of this experiment, this is not true of pressure. Typically, C p will need to be determined at various spatial locations.
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98 CHAPTER 3. THE EQUATIONS OF FLUID MOTION 3.6.4 Physical description of important dimensionless parameters In the preceding sections we have encountered the dimensionless parameters Re , Fr and C p . The first two of these were shown to completely characterize the nature of solutions to the N.–S. equa- tions in a wide range of physical circumstances involving incompressible flow. The last appeared as a quantity useful for data correlations associated with force on an object due to fluid flow around it, as a function of Re . In the present section we will treat these in somewhat more detail, especially with regard to their physical interpretations, and we will introduce a few other widely-encountered dimensionless parameters. Reynolds Number The Reynolds number, Re , can be described as a ratio of inertial to viscous forces. A simple way to see this is to recall Eqs. (3.50) and note that if Re is large the diffusive terms (viscous force terms) will be small; hence, flow behavior will be dominated by the inertial forces (and possibly also pressure and body forces). A more precise way to obtain this characterization is to note that the inertial forces are associated with accelerations and thus come from F inertial = ma . Now observe that m = ρL 3 , and a
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