Thus one of the two dimensionless parameters also sometimes termed dimensionle

# Thus one of the two dimensionless parameters also

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as required. Thus, one of the two dimensionless parameters (also sometimes termed “dimensionless groups”) is P 0 F F 0 = ρF μ 2 . The second dimensionless parameter is, of course, P 1 which we immediately recognize as the Reynolds number (because ρ/μ = 1 ) from our earlier scaling analysis of the equations of motion. Indeed, we should have expected from the start that this parameter would have to occur. Application to Data Analysis At this point it is worthwhile to consider how the preceding results can actually be used. As we have indicated earlier, one of the main applications of this approach is correlation of data from laboratory (and, now, computer) experiments. In particular, it is important to “collapse” data from a range of related experiments as much as possible before attempting a correlation, and this is precisely what can be accomplished with properly-chosen dimensionless parameters. (We comment that this can be foreseen from the scaling analysis of the governing equations in light of the fact that these must always produce the same solution for a given set dimensionless parameters, independent of the physical parameter values that led to these.) It is useful to first consider collecting data from the experiment treated in the preceding section in the absence of dimensional analysis. Based on physical arguments we can deduce that there are at least four dimensional quantities each of which might be varied to produce the force F that is the subject of the experimental investigations. Probably, the most easily changed would be the velocity, U ; but without performing dimensional analysis it would be difficult to argue that none of the remaining quantities would need to be varied. This would result in a large number of experiments producing data that would be difficult to interpret. For example, it is easily argued that the force on the sphere of Fig. 3.15 would change as its diameter is varied, so one set of experiments might involve measuring forces over a range of velocities for each of several different sphere diameters. But viscosity and density are also important physical parameters, so at least in priniciple, one would expect to have to vary these quantities as well. We now demonstrate that all of this is unnecessary when dimensional analysis is employed. In the present case we have found that only two dimensionless variables are needed to completely characterize data associated with the forces acting on a sphere immersed in a fluid flow, viz. , a dimensionless force defined as ρF/μ 2 and the Reynolds number, Re = ρUD/μ . This implies that we can in advance choose a fluid and the temperature at which the experiments are to be run (thereby setting ρ and μ ), select a diameter D of the sphere that will fit into the wind tunnel (or tow tank) being used, and then run the experiments over a range of Re by simply varying the flow speed U . For each such run of the experiment we measure the force F and nondimensionalize it with the scaling μ 2 (which is fixed once the temperature is set). Figure 3.16 provides a plot of such data.

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