Scale separation effects on mechanisms of boundary layer turbulence

Correlations between specific velocity and vorticity components dictate how the distributions of mean momentum and turbulence kinetic energy are realized in the turbulent boundary layer. For turbulent inertia to remain dynamically significant at arbitrarily high Reynolds number, differences of these correlations must remain non-zero. This motivates the study of velocity vorticity products under the influence of increasing scale separation. Through the use of both laboratory and field data, scale separation between relevant velocity and vorticity components is shown to increase with distance from the wall and Reynolds number. Time-delayed correlations between the vertical velocity and spanwise vorticity fluctuations reveal that only very slight variations in their average phase relation would cause significant variations in the mean transport of momentum. Spectral analyses are used to explore previous observations of scale selection between the participating velocity and vorticity cornponents. The wavelengths corresponding to the peaks in the relevant velocity and vorticity component spectra are used to describe scale separation effects. The variations in the wavelength ratios are shown to correlate with the scaling properties that follow from the magnitude ordering of terms in the mean momentum equation. Scale separation is observed to arise via spatial confinement, and spatial dispersion. In the region where the mean viscous force is of leading order, the mechanism of vortex stretching generates motions bearing concentrated vorticity that, with increasing Reynolds number, are confined to a smaller fraction of the viscous region flow volume. In the region where the mean dynamics are inertially dominated, the characteristic vortical motions are sparsely dispersed over a domain whose thickness asymptotically grows like the boundary layer thickness. In the region y + ≲ 40, the streamwise lengths of the correlations affiliated with turbulent inertia are seen to scale with the square root of the Reynolds number, while those affiliated with the gradient of turbulence kinetic energy are seen to scale with the Reynolds number itself

An invariant representation of mean inertia: theoretical basis for a log law in turbulent boundary layers

A refined scaling analysis of the two-dimensional mean momentum balance (MMB) for the zero-pressure-gradient turbulent boundary layer (TBL) is presented and experimentally investigated up to high friction Reynolds numbers, \unicode[STIX]{x1D6FF}^{+}. For canonical boundary layers, the mean inertia, which is a function of the wall-normal distance, appears instead of the constant mean pressure gradient force in the MMB for pipes and channels. The constancy of the pressure gradient has led to theoretical treatments for pipes/channels, that are more precise than for the TBL. Elements of these analyses include the logarithmic behaviour of the mean velocity, specification of the Reynolds shear stress peak location, the square-root Reynolds number scaling for the log layer onset and a well-defined layer structure based on the balance of terms in the MMB. The present analyses evidence that similarly well-founded results also hold for turbulent boundary layers. This follows from transforming the mean inertia term in the MMB into a form that resembles that in pipes/channels, and is constant across the outer inertial region of the TBL. The physical reasoning is that the mean inertia is primarily a large-scale outer layer contribution, the ‘shape’ of which becomes invariant of \unicode[STIX]{x1D6FF}^{+} with increasing \unicode[STIX]{x1D6FF}^{+}, and with a ‘magnitude’ that is inversely proportional to \unicode[STIX]{x1D6FF}^{+}. The present analyses are enabled and corroborated using recent high resolution, large Reynolds number hot-wire measurements of all the terms in the TBL MMB

Statistical evidence of an asymptotic geometric structure to the momentum transporting motions in turbulent boundary layers

The turbulence contribution to the mean flow is reflected by the motions producing the Reynolds shear stress (〈-uv〉) and its gradient. Recent analyses of the mean dynamical equation, along with data, evidence that these motions asymptotically exhibit self-similar geometric properties. This study discerns additional properties associated with the uv signal, with an emphasis on the magnitudes and length scales of its negative contributions. The signals analysed derive from high-resolution multi-wire hot-wire sensor data acquired in flat-plate turbulent boundary layers. Space-filling properties of the present signals are shown to reinforce previous observations, while the skewness of uv suggests a connection between the size and magnitude of the negative excursions on the inertial domain. Here, the size and length scales of the negative uv motions are shown to increase with distance from the wall, whereas their occurrences decrease. A joint analysis of the signal magnitudes and their corresponding lengths reveals that the length scales that contribute most to 〈-uv〉 are distinctly larger than the average geometric size of the negative uv motions. Co-spectra of the streamwise and wall-normal velocities, however, are shown to exhibit invariance across the inertial region when their wavelengths are normalized by the width distribution, W(y), of the scaling layer hierarchy, which renders the mean momentum equation invariant on the inertial domain.This article is part of the themed issue 'Toward the development of high-fidelity models of wall turbulence at large Reynolds number'

Self-similarity in the inertial region of wall turbulence

The inverse of the von Kármán constant κ is the leading coefficient in the equation describing the logarithmic mean velocity profile in wall bounded turbulent flows. Klewicki [J. Fluid Mech. 718, 596 (2013)] connects the asymptotic value of κ with an emerging condition of dynamic self-similarity on an interior inertial domain that contains a geometrically self-similar hierarchy of scaling layers. A number of properties associated with the asymptotic value of κ are revealed. This is accomplished using a framework that retains connection to invariance properties admitted by the mean statement of dynamics. The development leads toward, but terminates short of, analytically determining a value for κ. It is shown that if adjacent layers on the hierarchy (or their adjacent positions) adhere to the same self-similarity that is analytically shown to exist between any given layer and its position, then κ≡Φ(-2)=0.381966..., where Φ=(1+√5)/2 is the golden ratio. A number of measures, derived specifically from an analysis of the mean momentum equation, are subsequently used to empirically explore the veracity and implications of κ=Φ(-2). Consistent with the differential transformations underlying an invariant form admitted by the governing mean equation, it is demonstrated that the value of κ arises from two geometric features associated with the inertial turbulent motions responsible for momentum transport. One nominally pertains to the shape of the relevant motions as quantified by their area coverage in any given wall-parallel plane, and the other pertains to the changing size of these motions in the wall-normal direction. In accord with self-similar mean dynamics, these two features remain invariant across the inertial domain. Data from direct numerical simulations and higher Reynolds number experiments are presented and discussed relative to the self-similar geometric structure indicated by the analysis, and in particular the special form of self-similarity shown to correspond to κ=Φ(-2)

Streamwise velocity statistics in turbulent boundary layers that spatially develop to high Reynolds number

Well-resolved measurements of the streamwise velocity in zero pressure gradient turbulent boundary layers are presented for friction Reynolds numbers up to 19,670. Distinct from most studies, the present boundary layers undergo nearly a decade increase in Reynolds number solely owing to streamwise development. The profiles of the mean and variance of the streamwise velocity exhibit logarithmic behavior in accord with other recently reported findings at high Reynolds number. The inner and mid-layer peaks of the variance profile are evidenced to increase at different rates with increasing Reynolds number. A number of statistical features are shown to correlate with the position where the viscous force in the mean momentum equation loses leading order importance, or similarly, where the mean effect of turbulent inertia changes sign from positive to negative. The near-wall peak region in the 2-D spectrogram of the fluctuations is captured down to wall-normal positions near the edge of the viscous sublayer at all Reynolds numbers. The spatial extent of this near-wall peak region is approximately invariant under inner normalization, while its large wavelength portion is seen to increase in scale in accord with the position of the mid-layer peak, which resides at a streamwise wavelength that scales with the boundary layer thickness