Left: Experimental setup showing the reservoirs (RT, RB), pump (P), valve (V), comb (C), blades (WL, WR), and weight (W). Middle: Fluctuations in film thickness from turbulent velocity fluctuations with smooth walls and a comb. Right: Thickness fluctuations with smooth and rough wall. Adapted from Ref. 28 . Reprinted with permission from T. Tran, P. Chakraborty, N. Guttenberg, A. Prescott, H. Kellay, W. I. Goldburg, N. Goldenfeld, and G. Gioia, Nat. Phys. 6, 438 (2010). Copyright 2010, Nature Publishing Group.
Energy spectra E(k) in cm3/s2 measured at the centerline. The upper curve (□) is the enstrophy cascade and the lower curve (◯) is the energy cascade data. While they are only guides to the eye, the straight lines are the expected slopes for the energy and enstrophy cascades (Kr67 or dimensional reasoning). The spectra are normalized such that .
Probability density functions (pdfs) of longitudinal velocity differences δu(r) measured for the energy cascade (a) at R λ = 50, and for the enstrophy cascade (b) at R λ = 610. The first three values of r here are in the inertial range as determined by the power law scaling region of the structure functions. The dashed line is a gaussian function with zero mean and a standard deviation of unity. The mean and variance of the velocity data have been normalized so that if they are gaussian, they will lie on top of this curve. None of these pdfs can be truly gaussian as their third moments cannot vanish, 34 but the energy cascade data deviate much more than the enstrophy cascade data. The pdfs at different r do not have the same shape, indicating a lack of self-similarity.
Flatness of the velocity derivative F η vs. R λ with ∂ x u estimated using the central difference method. The energy cascade data (□) and the enstrophy cascade data (△) fall into two sections with distinct R λ. A curved dashed line is shown (f(R λ) = 8(log R λ)−1/2) which suggests a (sub-) logarithmic approach to zero.
Flatness F(r) of P(δu(r)) depending on the scale r (normalized by channel width w). The two upper curves are for energy cascade data (◯: R λ = 60, □: R λ = 90) and the two bottom curves are for enstrophy cascade data (△: R λ = 490, ⋄: R λ = 690). The arrows denote the beginning and end of the inertial range for each curve, as determined by the power law region of the structure functions. All of the curves start out above the gaussian value of 3, but the energy cascade data remain above and seem to asymptote to 3 as r increases. The enstrophy cascade curves cross guassianity in the inertial range and then asymptote to a value smaller than 3.
Three measured structure functions whose scaling exponents are determined using extended self-similarity (ESS). That is, S n (r) is plotted vs. S 3(r). From the bottom to the top, the lines are S 2(r), S 4(r), and S 6(r). Here R λ = 490. The data exhibit the enstrophy cascade.
Normalized scaling exponents of the nth-order structure functions out to n = 10 for the enstrophy cascade with R λ = 490. The data set denoted by squares (□) is extracted from measurements spanning a decade in r. The open circles (◯) denote slopes deduced using extended self-similarity. 39 The triangles (△) denote measurements obtained using the method from JW. 18 All methods agree very well with each other.
Intermittency exponent μ vs. R λ for energy cascade (◯) and the enstrophy cascade (◯). The value of μ for the energy cascade is roughly constant while μ for enstrophy cascade appears to be a decreasing function of R λ.
The third order structure function S 3(r) vs. r for several R λ. The sign of S 3(r) is positive for r in the inertial range of each case. This indicates that energy is being transferred to large scales, in agreement with the prediction for the inverse energy cascade of 2D turbulence.
Rank-ordered log-log plot of velocity differences for two sets of experimental data. The curves initially behave as power laws. The power law exponent may be used to estimate the highest order of structure functions that can be accurately measured.
The structure function integrand of order n = 1, 2, and 6 plotted versus the velocity difference. The area under the curves is finite, indicating that this order structure function can be accurately measured.
Average (⟨·⟩) and standard deviation values (σ) of intermittency measures for energy and enstrophy data. All mean values show significant deviations from the non-intermittent standard. The enstrophy cascade standard deviations are large due to the R λ-dependence. Recall that for 3D, μ ≃ 0.2. 1
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