Solution to Problem 7.1

1) Profiles of v_rms are sketched below, with x = 20D shown as a solid line and x = 40D shown as a dashed line. The strongest turbulent fluctuations occur at the points of maximum mean shear. Because the maximum shear is weaker at the downstream station, the fluctuations are weaker here as well.

2) First, let B(x) be the width of the jet at position x. While many eddy scales coexist in a turbulent jet, it is the biggest scales that dominate mixing. The biggest eddies have the scale of the flow, which for a jet is B. We find the sign of rotation of these eddies by imaging the motion of a rigid stick placed in the flow. Because the sign of the shear (and thus vortex rotation) reverses at mid-jet, a single eddy cannot grow larger than the half-width of the jet. Thus, we can say more specifically that the largest eddies in the jet will be the scale of the half-width, B/2.

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