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 withR^+1/2and critical layers where the propagation velocity is equal to the local mean velocity, one of which scales withR^+2/3in 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.

Known structures and self-sustaining mechanisms of wall turbulence are reviewed and explored in the context of the scale interactions implied by the nonlinear advective term in the Navier–Stokes equations. The viewpoint is shaped by the systems approach provided by the resolvent framework for wall turbulence proposed by McKeon & Sharma (J. Fluid Mech., vol. 658, 2010, pp. 336–382), in which the nonlinearity is interpreted as providing the forcing to the linear Navier–Stokes operator (the resolvent). Elements of the structure of wall turbulence that can be uncovered as the treatment of the nonlinearity ranges from data-informed approximation to analysis of exact solutions of the Navier–Stokes equations (so-called exact coherent states) are discussed. The article concludes with an outline of the feasibility of extending this kind of approach to high-Reynolds-number wall turbulence in canonical flows and beyond.

The measurements by Zagarola & Smits (1998) of mean velocity profiles in fully developed turbulent pipe flow are repeated using a smaller Pitot probe to reduce the uncertainties due to velocity gradient corrections. A new static pressure correction (McKeon & Smits 2002) is used in analysing all data and leads to significant differences from the Zagarola & Smits conclusions. The results confirm the presence of a power-law region near the wall and, for Reynolds numbers greater than $230\,{\times}\,10^3$ ($R^+\,{>}\,5\,{\times}\,10^3$), a logarithmic region further out, but the limits of these regions and some of the constants differ from those reported by Zagarola & Smits. In particular, the log law is found for $600\,{<}\, y^+\,{<}\,0.12R^+$ (instead of $600\,{<}\,y^+\,{<}\,0.07R^+$), and the von Kármán constant $\kappa$, the additive constant $B$ for the log law using inner flow scaling, and the additive constant $B^*$ for the log law using outer scaling are found to be $0.421 \pm 0.002$, $5.60 \pm 0.08 $ and $1.20 \pm 0.10$, respectively, with 95% confidence level (compared with $0.436 \pm 0.002$, $6.15 \pm 0.08$, and $1.51 \pm 0.03$ found by Zagarola & Smits). The data also confirm that the pipe flow data for Re$_D\,{\le}\,13.6\,{\times}\,10^6$ (as a minimum) are not affected by surface roughness.

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

In this work, we demonstrate how physical principles—such as symmetries, invariances and conservation laws—can be integrated into the dynamic mode decomposition (DMD). DMD is a widely used data analysis technique that extracts low-rank modal structures and dynamics from high-dimensional measurements. However, DMD can produce models that are sensitive to noise, fail to generalize outside the training data and violate basic physical laws. Our physics-informed DMD (piDMD) optimization, which may be formulated as a Procrustes problem, restricts the family of admissible models to a matrix manifold that respects the physical structure of the system. We focus on five fundamental physical principles—conservation, self-adjointness, localization, causality and shift-equivariance—and derive several closed-form solutions and efficient algorithms for the corresponding piDMD optimizations. With fewer degrees of freedom, piDMD models are less prone to overfitting, require less training data, and are often less computationally expensive to build than standard DMD models. We demonstrate piDMD on a range of problems, including energy-preserving fluid flow, the Schrödinger equation, solute advection-diffusion and three-dimensional transitional channel flow. In each case, piDMD outperforms standard DMD algorithms in metrics such as spectral identification, state prediction and estimation of optimal forcings and responses.