Unlikely existence of kx−1 spectral law in wall turbulence: An observation of the atmospheric surface layer

Abstract

For wall turbulence, there has been predicted a range of streamwise wavenumbers k x such that the spectral density of streamwise velocity fluctuations is proportional to k −1 x . The existence or nonexistence of this k −1x law is examined here. We observe the atmospheric surface layer over several months, select suitable data, and use them to synthesize the energy spectrum that would represent wall turbulence at a very high Reynolds number. The result is not consistent with the k −1 x law. It is rather consistent with a recent correction to the prediction of a model of energy-containing eddies that are attached to the wall. The reason for these findings is discussed mathematically.

Wall-attached and wall-detached eddies in wall-bounded turbulent flows

“…the y ∼ λ x scaling of the pre-multiplied velocity spectrum, the k −1 scaling is much more difficult to find (Mouri, Morinaga & Haginoya 2019). Hence, although Hwang (2015), Nickels et al 2005and we are looking for the parts of the velocity spectrum that represent contributions from wall-attached eddies, the claimed regions in Hwang (2015) and Nickels et al (2005) and by us will be different.…”

“…A. Yang and X. Zheng 10 3 10 5 10 7 10 2 10 4 10 6 10 3 10 5 10 7 10 2 10 4 10 6 10 3 10 5 10 7 10 2 10 4 10 6 inter-scale interaction of amplitude modulation, where the (loosely defined) small-scale eddies are modulated by the (loosely defined) large-scale eddies. This form of inter-scale interaction has received considerable attention in the recent literature (Hutchins & Marusic 2007;Mathis et al 2009;Chung & McKeon 2010;Guala, Metzger & McKeon 2011;Ganapathisubramani et al 2012;Agostini & Leschziner 2014;Anderson 2016;Dogan, Hanson & Ganapathisubramani 2016;Pathikonda & Christensen 2017Howland & Yang 2018;Liu, Wang & Zheng 2019). In anticipation of the results, we briefly review one of the previously used quantifications of amplitude modulation in Mathis et al (2009): where u S and u L are the (loosely defined) small-and large-scale components of the streamwise velocity, E(•) is the envelope of the bracketed signal and STD(•) is the standard deviation of the bracketed quantity.…”

Logarithmic and nonlogarithmic scaling laws of two-point statistics in wall turbulence

“…Thus far, we have studied two-point cumulants, which are useful if wall turbulence is modeled as a superposition of finite-size motions like those of the attached eddies. Although it is usual to study energy spectra [18], their wavelengths do not correspond to any particular size of the individual motions [19].…”

“…(3a) implies that it is always affected by the turbulence thickness δ. Such an effect is through largest eddies of h e = δ, which contribute to all of nonzero cumulants [19].…”

“…Although previous studies added assumptions [15][16][17][18], they are not consistent with that hypothesis if their results do not satisfy Eq. (3a), aside from whether they appear reasonable or not [11,19].…”