A collaborative experimental effort employing the minimally perturbed atmospheric surface-layer flow over the salt playa of western Utah has enabled us to map coherence in turbulent boundary layers at very high Reynolds numbers, Re τ ∼ O(10 6 ). It is found that the large-scale coherence noted in the logarithmic region of laboratory-scale boundary layers are also present in the very high Reynolds number atmospheric surface layer (ASL). In the ASL these features tend to scale on outer variables (approaching the kilometre scale in the streamwise direction for the present study). The mean statistics and two-point correlation map show that the surface layer under neutrally buoyant conditions behaves similarly to the canonical boundary layer. Linear stochastic estimation of the three-dimensional correlation map indicates that the low momentum fluid in the streamwise direction is accompanied by counter-rotating roll modes across the span of the flow. Instantaneous flow fields confirm the inferences made from the linear stochastic estimations. It is further shown that vortical structures aligned in the streamwise direction are present in the surface layer, and bear attributes that resemble the hairpin vortex features found in laboratory flows. Ramp-like high shear zones that contribute significantly to the Reynolds shear-stress are also present in the ASL in a form nearly identical to that found in laboratory flows. Overall, the present findings serve to draw useful connections between the vast number of observations made in the laboratory and in the atmosphere.

The properties of the mean momentum balance in turbulent boundary layer, pipe and channel flows are explored both experimentally and theoretically. Available highquality data reveal a dynamically relevant four-layer description that is a departure from the mean profile four-layer description traditionally and nearly universally ascribed to turbulent wall flows. Each of the four layers is characterized by a predominance of two of the three terms in the governing equations, and thus the mean dynamics of these four layers are unambiguously defined. The inner normalized physical extent of three of the layers exhibits significant Reynolds-number dependence. The scaling properties of these layer thicknesses are determined. Particular significance is attached to the viscous/Reynolds-stress-gradient balance layer since its thickness defines a required length scale. Multiscale analysis (necessarily incomplete) substantiates the four-layer structure in developed turbulent channel flow. In particular, the analysis verifies the existence of at least one intermediate layer, with its own characteristic scaling, between the traditional inner and outer layers. Other information is obtained, such as (i) the widths (in order of magnitude) of the four layers, (ii) a flattening of the Reynolds stress profile near its maximum, and (iii) the asymptotic increase rate of the peak value of the Reynolds stress as the Reynolds number approaches infinity. Finally, on the basis of the experimental observation that the velocity increments over two of the four layers are unbounded with increasing Reynolds number and have the same order of magnitude, there is additional theoretical evidence (outside traditional arguments) for the asymptotically logarithmic character of the mean velocity profile in two of the layers; and (in order of magnitude) the mean velocity increments across each of the four layers are determined. All of these results follow from a systematic train of reasoning, using the averaged momentum balance equation together with other minimal assumptions, such as that the mean velocity increases monotonically from the wall.

Turbulent boundary layer measurements above a smooth wall and sandpaper roughness are presented across a wide range of friction Reynolds numbers, δ + 99 , and equivalent sand grain roughness Reynolds numbers, k + s (smooth wall: 2020 δ + 99 21 430, rough wall: 2890 δ + 99 29 900; 22 k + s 155; and 28 δ + 99 /k + s 199). For the rough-wall measurements, the mean wall shear stress is determined using a floating element drag balance. All smooth-and rough-wall data exhibit, over an inertial sublayer, regions of logarithmic dependence in the mean velocity and streamwise velocity variance. These logarithmic slopes are apparently the same between smooth and rough walls, indicating similar dynamics are present in this region. The streamwise mean velocity defect and skewness profiles each show convincing collapse in the outer region of the flow, suggesting that Townsend's (The Structure of Turbulent Shear Flow, vol. 1, 1956, Cambridge University Press.) wall-similarity hypothesis is a good approximation for these statistics even at these finite friction Reynolds numbers. Outer-layer collapse is also observed in the rough-wall streamwise velocity variance, but only for flows with δ + 99 14 000. At Reynolds numbers lower than this, profile invariance is only apparent when the flow is fully rough. In transitionally rough flows at low δ + 99 , the outer region of the inner-normalised streamwise velocity variance indicates a dependence on k + s for the present rough surface.

The present study explores the effects of Reynolds number, over three orders of magnitude, in the viscous wall region of a turbulent boundary layer. Complementary experiments were conducted both in the boundary layer wind tunnel at the University of Utah and in the atmospheric surface layer which flows over the salt flats of the Great Salt Lake Desert in western Utah. The Reynolds numbers, based on momentum deficit thickness, of the two flows were Rθ=2×103 and Rθ≈5×106, respectively. High-resolution velocity measurements were obtained from a five-element vertical rake of hot-wires spanning the buffer region. In both the low and high Rθ flows, the length of the hot-wires measured less than 6 viscous units. To facilitate reliable comparisons, both the laboratory and field experiments employed the same instrumentation and procedures. Data indicate that, even in the immediate vicinity of the surface, strong influences from low-frequency motions at high Rθ produce noticeable Reynolds number differences in the streamwise velocity and velocity gradient statistics. In particular, the peak value in the root mean square streamwise velocity profile, when normalized by viscous scales, was found to exhibit a logarithmic dependence on Reynolds number. The mean streamwise velocity profile, on the other hand, appears to be essentially independent of Reynolds number. Spectra and spatial correlation data suggest that low-frequency motions at high Reynolds number engender intensified local convection velocities which affect the structure of both the velocity and velocity gradient fields. Implications for turbulent production mechanisms and coherent motions in the buffer layer are discussed.

Elements of the first-principles-based theory of Wei et al. (J. Fluid Mech., vol. 522, 2005, p. 303), Fife et al. (Multiscale Model. Simul., vol. 4, 2005a, p. 936; J. Fluid Mech., vol. 532, 2005b, p. 165) and Fife, Klewicki & Wei (J. Discrete Continuous Dyn. Syst., vol. 24, 2009, p. 781) are clarified and their veracity tested relative to the properties of the logarithmic mean velocity profile. While the approach employed broadly reveals the mathematical structure admitted by the time averaged Navier–Stokes equations, results are primarily provided for fully developed pressure driven flow in a two-dimensional channel. The theory demonstrates that the appropriately simplified mean differential statement of Newton's second law formally admits a hierarchy of scaling layers, each having a distinct characteristic length. The theory also specifies that these characteristic lengths asymptotically scale with distance from the wall over a well-defined range of wall-normal positions, y. Numerical simulation data are shown to support these analytical findings in every measure explored. The mean velocity profile is shown to exhibit logarithmic dependence (exact or approximate) when the solution to the mean equation of motion exhibits (exact or approximate) self-similarity from layer to layer within the hierarchy. The condition of pure self-similarity corresponds to a constant leading coefficient in the logarithmic mean velocity equation. The theory predicts and clarifies why logarithmic behaviour is better approximated as the Reynolds number gets large. An exact equation for the leading coefficient (von Kármán coefficient κ) is tested against direct numerical simulation (DNS) data. Two methods for precisely estimating the leading coefficient over any selected range of y are presented. These methods reveal that the differences between the theory and simulation are essentially within the uncertainty level of the simulation. The von Kármán coefficient physically exists owing to an approximate self-similarity in the flux of turbulent force across an internal layer hierarchy. Mathematically, this self-similarity relates to the slope and curvature of the Reynolds stress profile, or equivalently the slope and curvature of the mean vorticity profile. The theory addresses how, why and under what conditions logarithmic dependence is approximated relative to the specific mechanisms contained within the mean statement of dynamics.

Spanwise vorticity measurements have been performed in zero-pressure-gradient boundary layers over the range 1010 < Rθ < 4850 (Rθ ≡ U∞ θ/ν, where U∞ is the free-stream velocity and θ is the momentum deficit thickness) using a four-wire probe. In addition, experiments quantifying the spatial and temporal resolution required to obtain an accurate statistical representation of the small-scale structure of wall-bounded turbulence were performed. Furthermore, a thorough investigation of statistical convergence for a variety of fluctuating quantities was performed. Comparisons with earlier high-resolution studies indicate that the maximum value of u′/uτ increases with increasing Reynolds number over the given Rθ range (u′ ≡ r.m.s. u, and uτ is the friction velocity). It is suggested that detecting this dependence provides a good measure of probe resolution. In general it was found that statistics of velocity gradients were distinctly more sensitive to finite probe size than velocity statistics. Wire spacing experiments suggest that Wyngaard's (1969) criterion is to a good approximation valid even under anisotropic conditions. Furthermore, it was found that instantaneously spatial averaging of ∂u/∂t caused significant attenuation in the resulting r.m.s., and that this averaging procedure is sensitive to the level of mean shear. A simple method of estimating how noise in the u-velocity signals enters into the ∂u/∂y signals is presented. The convergence study shows that statistical convergence criteria developed from free-shear flows severely underestimates the averaging times required in boundary layers. A table of general convergence criteria is provided.

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This study investigates how and why dynamical self-similarities emerge with increasing Reynolds number within the canonical wall flows beyond the transitional regime. An overarching aim is to advance a mechanistically coherent description of turbulent wall-flow dynamics that is mathematically tractable and grounded in the mean dynamical equations. As revealed by the analysis of Fife, Klewicki & Wei (J. Discrete Continuous Dyn. Syst. A, vol. 24, 2009, pp. 781-807), the equations that respectively describe the mean dynamics of turbulent channel, pipe and boundary layer flows formally admit invariant forms. These expose an underlying self-similar structure. In all cases, two kinds of dynamical self-similarity are shown to exist on an internal domain that, for all Reynolds numbers, extends from O(ν/u τ ) to O(δ), where ν is the kinematic viscosity, u τ is the friction velocity and δ is the half-channel height, pipe radius, or boundary layer thickness. The simpler of the two self-similarities is operative on a large outer portion of the relevant domain. This self-similarity leads to an explicit analytical closure of the mean momentum equation. This self-similarity also underlies the emergence of a logarithmic mean velocity profile. A more complicated kind a self-similarity emerges asymptotically over a smaller domain closer to the wall. The simpler self-similarity allows the mean dynamical equation to be written as a closed system of nonlinear ordinary differential equations that, like the similarity solution for the laminar flat-plate boundary layer, can be numerically integrated. The resulting similarity solutions are demonstrated to exhibit nearly exact agreement with direct numerical simulations over the solution domain specified by the theory. At the Reynolds numbers investigated, the outer similarity solution is shown to be operative over a domain that encompasses ∼40 % of the overall width of the flow. Other properties predicted by the theory are also shown to be well supported by existing data.