Figure P.10.3 provides an instantaneous view of a submerged, circular jet of water. The water coming from the nozzle at the top was visualized using laserinduced fluorescence. In the lower part of the image the Kolmogorov scales are resolved. As will be outlined, length ratios in the photograph can be used to estimate the jet velocity. In particular, the objective is to infer the Reynolds number based on the nozzle diameter and mean velocity.

visualized using laser-induced fluorescence as described in Dimotakis et al. (1983).

Image reproduced from Van Dyke (1982, p. 97) with permission from P. E. Dimotakis.

At an axial distance z from the nozzle, let vm be the maximum smoothed velocity (that at the center of the jet), let u₁ be the large-eddy velocity scale (fluctuation in v_z), and let L₁ and L₂ be the length scales for the largest and smallest eddies, respectively. It is useful to know that the turbulence intensity in a circular jet is u_1⁄v_m = 0.24 (Panchapakesan and Lumley, 1993). The correlation for v_m(z) in Eq. (10.6-24) is applicable far from the nozzle.

(a) Show that Re can be expressed as a function of just two ratios, z⁄L₁ and L_1⁄L₂.

(b) Focusing near the bottom of the image, assume that the large-eddy size is that of the largest “vortex” that can be seen and that the small-eddy size (Kolmogorov length) is the thickness of the thinnest “threads.”

Measurements on an enlargement yield z = 200, L₁ = 20, and L₂ = 0.3, all in arbitrary units. Show that these lengths imply that Re = 1900. This agrees with the reported value of 2300 to one digit, which is perhaps better than might have been expected.

(c) There is some subjectivity in choosing the largest vortex and difficulty in precisely measuring thread widths. A value of L₁ as large as 35 (the approximate jet radius) would be reasonable, as would any L₂ between 0.2 and 0.4. Which of these dimensions is the more important source of uncertainty in estimating Re?

Transcribed Image Text:

Figure P10.3 An axisymmetric water jet directed downward into water,