On the linkage between the k −5/3 spectral and k −7/3 cospectral scaling in high-Reynolds number turbulent boundary layers

• Published in: Physics of fluids (1994) Vol. 29; no. 6
• Main Authors: Li, Dan, Katul, Gabriel G.
• Format: Journal Article
• Language: English
• Published: Melville American Institute of Physics 01.06.2017; American Institute of Physics (AIP)
• Subjects: Atmospheric boundary layer; Crossovers; Decay rate; Energy spectra; Fluid dynamics; Fluid flow; Flux; GEOSCIENCES; High Reynolds number; Kinetic energy; Momentum; Physics; Relaxation time; Reynolds number; Scaling; Time dependence; Turbulent boundary layer; Wavelengths
• ISSN: 1070-6631; 1089-7666
• DOI: 10.1063/1.4986068

Connections between the “−5/3” spectral and “−7/3” cospectral scaling exponents characterizing the inertial subranges of the wall-normal energy spectrum and the turbulent momentum flux cospectrum are explored in the equilibrium layer of high-Reynolds number turbulent boundary layers. Previous laboratory experiments and field measurements featured here in the atmospheric boundary layer show that the “−7/3” scaling in the momentum flux cospectrum F u w ( k ) commences at lower wavenumbers (around kz = 3) than the “−5/3” scaling in the wall-normal energy spectrum E w w ( k ) (around kz = 6), where k is the streamwise wavenumber and z is the distance from the surface. A satisfactory explanation as to why F u w ( k ) attains its “−7/3” inertial subrange scaling earlier than E w w ( k ) in wavenumber space remains elusive. A cospectral budget (CSB) model subject to several simplifications and closure schemes offers one viewpoint. In its simplest form, the CSB model assumes a balance at all k between the production term and a Rotta-like pressure decorrelation term with a prescribed wavenumber-dependent relaxation time scale. It predicts the “−7/3” scaling for F u w ( k ) from the “−5/3” scaling in E w w ( k ) , thereby recovering earlier results derived from dimensional considerations. A finite flux transfer term was previously proposed to explain anomalous deviations from the “−7/3” cospectral scaling in the inertial subrange using a simplified spectral diffusion closure. However, this explanation is not compatible with an earlier commencement of the “−7/3” scaling in F u w ( k ) . An alternative explanation that does not require a finite flux transfer is explored here. By linking the relaxation time scale in the slow-component of the Rotta model to the turbulent kinetic energy (TKE) spectrum, the earlier onset of the “−7/3” scaling in F u w ( k ) is recovered without attainment of a “−5/3” scaling in E w w ( k ) . The early onset of the “−7/3” scaling at smaller k is related to a slower than k −5/3 decay in the TKE spectrum at the crossover from production to inertial scales.