B. J. McKeon

Scaling of the streamwise velocity component in turbulent pipe flow

Statistics of the streamwise velocity component in fully developed pipe flow are examined for Reynolds numbers in the range 5.5 x 10^4 ≤ ReD ≤ 5.7 x 10^6. Probability density functions and their moments (up to sixth order) are presented and their scaling with Reynolds number is assessed. The second moment exhibits two maxima: the one in the viscous sublayer is Reynolds-number dependent while the other, near the lower edge of the log region, follows approximately the peak in Reynolds shear stress. Its locus has an approximate (R^+)^{0.5} dependence. This peak shows no sign of ‘saturation’, increasing indefinitely with Reynolds number. Scalings of the moments with wall friction velocity and

A critical-layer framework for turbulent pipe flow

(U_{cl}-overline{U})

Large-eddy simulation of large-scale structures in long channel flow

are examined and the latter is shown to be a better velocity scale for the outer region, y/R > 0.35, but in two distinct Reynolds-number ranges, one when ReD 7 x 10^4. Probability density functions do not show any universal behaviour, their higher moments showing small variations with distance from the wall outside the viscous sublayer. They are most nearly Gaussian in the overlap region. Their departures from Gaussian are assessed by examining the behaviour of the higher moments as functions of the lower ones. Spectra and the second moment are compared with empirical and theoretical scaling laws and some anomalies are apparent. In particular, even at the highest Reynolds number, the spectrum does not show a self-similar range of wavenumbers in which the spectral density is proportional to the inverse streamwise wavenumber. Thus such a range does not attract any special significance and does not involve a universal constant.

A new friction factor relationship for fully developed pipe flow

A model-based description of the scaling and radial location of turbulent fluctuations in turbulent pipe flow is presented and used to illuminate the scaling behaviour of the very large scale motions. The model is derived by treating the nonlinearity in the perturbation equation (involving the Reynolds stress) as an unknown forcing, yielding a linear relationship between the velocity field response and this nonlinearity. We do not assume small perturbations. We examine propagating helical velocity response modes that are harmonic in the wall-parallel directions and in time, permitting comparison of our results to experimental data. The steady component of the velocity field that varies only in the wall-normal direction is identified as the turbulent mean profile. A singular value decomposition of the resolvent identifies the forcing shape that will lead to the largest velocity response at a given wavenumber–frequency combination. The hypothesis that these forcing shapes lead to response modes that will be dominant in turbulent pipe flow is tested by using physical arguments to constrain the range of wavenumbers and frequencies to those actually observed in experiments. An investigation of the most amplified velocity response at a given wavenumber–frequency combination reveals critical-layer-like behaviour reminiscent of the neutrally stable solutions of the Orr–Sommerfeld equation in linearly unstable flow. Two distinct regions in the flow where the influence of viscosity becomes important can be identified, namely wall layers that scale with R^(+1/2) and critical layers where the propagation velocity is equal to the local mean velocity, one of which scales with R^(+2/3) in pipe flow. This framework appears to be consistent with several scaling results in wall turbulence and reveals a mechanism by which the effects of viscosity can extend well beyond the immediate vicinity of the wall. The model reproduces inner scaling of the small scales near the wall and an approach to outer scaling in the flow interior. We use our analysis to make a first prediction that the appropriate scaling velocity for the very large scale motions is the centreline velocity, and show that this is in agreement with experimental results. Lastly, we interpret the wall modes as the motion required to meet the wall boundary condition, identifying the interaction between the critical and wall modes as a potential origin for an interaction between the large and small scales that has been observed in recent literature as an amplitude modulation of the near-wall turbulence by the very large scales.

Friction factors for smooth pipe flow

We investigate statistics of large-scale structures from large-eddy simulation (LES) of turbulent channel flow at friction Reynolds numbers Re_τ = 2K and 200K (where K denotes 1000). In order to capture the behaviour of large-scale structures properly, the channel length is chosen to be 96 times the channel half-height. In agreement with experiments, these large-scale structures are found to give rise to an apparent amplitude modulation of the underlying small-scale fluctuations. This effect is explained in terms of the phase relationship between the large- and small-scale activity. The shape of the dominant large-scale structure is investigated by conditional averages based on the large-scale velocity, determined using a filter width equal to the channel half-height. The conditioned field demonstrates coherence on a scale of several times the filter width, and the small-scale–large-scale relative phase difference increases away from the wall, passing through π/2 in the overlap region of the mean velocity before approaching π further from the wall. We also found that, near the wall, the convection velocity of the large scales departs slightly, but unequivocally, from the mean velocity.

The near-neutral atmospheric surface layer: turbulence and non-stationarity.

The friction factor relationship for high-Reynolds-number fully developed turbulent pipe flow is investigated using two sets of data from the Princeton Superpipe in the range 31×10^3 ≤ ReD ≤ 35×10^6. The constants of Prandtl’s ‘universal’ friction factor relationship are shown to be accurate over only a limited Reynolds-number range and unsuitable for extrapolation to high Reynolds numbers. New constants, based on a logarithmic overlap in the mean velocity, are found to represent the high-Reynolds-number data to within 0.5%, and yield a value for the von Karman constant that is consistent with the mean velocity profiles themselves. The use of a generalized logarithmic law in the mean velocity is also examined. A general friction factor relationship is proposed that predicts all the data to within 1.4% and agrees with the Blasius relationship for low Reynolds numbers to within 2.0%.

On coherent structure in wall turbulence

Friction factor data from two recent pipe flow experiments are combined to provide a comprehensive picture of the friction factor variation for Reynolds numbers from 10 to 36,000,000.

Interactions within the turbulent boundary layer at high Reynolds number

The neutrally stable atmospheric surface layer is used as a physical model of a very high Reynolds number, canonical turbulent boundary layer. Challenges and limitations with this model are addressed in detail, including the inherent thermal stratification, surface roughness and non-stationarity of the atmosphere. Concurrent hot-wire and sonic anemometry data acquired in Utahs western desert provide insight to Reynolds number trends in the axial velocity statistics and spectra.

Asymptotic scaling in turbulent pipe flow

A new theory of coherent structure in wall turbulence is presented. The theory is nthe first to predict packets of hairpin vortices and other structure in turbulence, nand their dynamics, based on an analysis of the Navier–Stokes equations, under an nassumption of a turbulent mean profile. The assumption of the turbulent mean acts nas a restriction on the class of possible structures. It is shown that the coherent nstructure is a manifestation of essentially low-dimensional flow dynamics, arising from na critical-layer mechanism. Using the decomposition presented in McKeon & Sharma n(J. Fluid Mech., vol. 658, 2010, pp. 336–382), complex coherent structure is recreated nfrom minimal superpositions of response modes predicted by the analysis, which take nthe form of radially varying travelling waves. The leading modes effectively constitute na low-dimensional description of the turbulent flow, which is optimal in the sense of ndescribing the resonant effects around the critical layer and which minimally predicts nall types of structure. The approach is general for the full range of scales. By way nof example, simple combinations of these modes are offered that predict hairpins nand modulated hairpin packets. The example combinations are chosen to represent nobserved structure, consistent with the nonlinear triadic interaction for wavenumbers nthat is required for self-interaction of structures. The combination of the three leading nresponse modes at streamwise wavenumbers 6; 1; 7 and spanwise wavenumbers n±6; ±6; ±12, respectively, with phase velocity 2/3, is understood to represent a nturbulence ‘kernel’, which, it is proposed, constitutes a self-exciting process analogous nto the near-wall cycle. Together, these interactions explain how the mode combinations nmay self-organize and self-sustain to produce experimentally observed structure. The nphase interaction also leads to insight into skewness and correlation results known in nthe literature. It is also shown that the very large-scale motions act to organize hairpin-like nstructures such that they co-locate with areas of low streamwise momentum, nby a mechanism of locally altering the shear profile. These energetic streamwise nstructures arise naturally from the resolvent analysis, rather than by a summation of nhairpin packets. In addition, these packets are modulated through a ‘beat’ effect. The nrelationship between Taylor’s hypothesis and coherence is discussed, and both are nshown to be the consequence of the localization of the response modes around the ncritical layer. A pleasing link is made to the classical laminar inviscid theory, whereby nthe essential mechanism underlying the hairpin vortex is captured by two obliquely ninteracting Kelvin–Stuart (cat’s eye) vortices. Evidence for the theory is presented nbased on comparison with observations of structure in turbulent flow reported in the experimental and numerical simulation literature and with exact solutions reported in nthe transitional literature.

Pitot probe corrections in fully developed turbulent pipe flow

Simultaneous streamwise velocity measurements across the vertical direction obtained in the atmospheric surface layer (Re_τ ≃ 5 × 10^5) under near thermally neutral conditions are used to outline and quantify interactions between the scales of turbulence, from the very-large-scale motions to the dissipative scales. Results from conditioned spectra, joint probability density functions and conditional averages show that the signature of very-large-scale oscillations can be found across the whole wall region and that these scales interact with the near-wall turbulence from the energy-containing eddies to the dissipative scales, most strongly in a layer close to the wall, z^+ ≲ 10^3. The scale separation achievable in the atmospheric surface layer appears to be a key difference from the low-Reynolds-number picture, in which structures attached to the wall are known to extend through the full wall-normal extent of the boundary layer. A phenomenological picture of very-large-scale motions coexisting and interacting with structures from the hairpin paradigm is provided here for the high-Reynolds-number case. In particular, it is inferred that the hairpin-packet conceptual model may not be exhaustively representative of the whole wall region, but only of a near-wall layer of z^+ = O(10^3), where scale interactions are mostly confined.