We study turbulent flows in a smooth straight pipe of circular cross--section up to $Re_τ \approx 6000$ using direct--numerical-simulation (DNS) of the Navier--Stokes equations. The DNS results highlight systematic deviations from Prandtl friction law, amounting to about $2\%$, which would extrapolate to about $4\%$ at extreme Reynolds numbers. Data fitting of the DNS friction coefficient yields an estimated von Kármán constant $k \approx 0.387$, which nicely fits the mean velocity profile, and which supports universality of canonical wall-bounded flows. The same constant also applies to the pipe centerline velocity, thus providing support for the claim that the asymptotic state of pipe flow at extreme Reynolds numbers should be plug flow. At the Reynolds numbers under scrutiny, no evidence for saturation of the logarithmic growth of the inner peak of the axial velocity variance is found. Although no outer peak of the velocity variance directly emerges in our DNS, we provide strong evidence that it should appear at $Re_τ \gtrsim 10^4$, as a result of turbulence production exceeding dissipation over a large part of the outer wall layer, thus invalidating the classical equilibrium hypothesis.
This paper presents a method for calculating the wall shear rate in pipe turbulent flow. It collapses adequately the data measured in laminar flow and turbulent flow into a single flow curve and gives the basis for the design of turbulent flow viscometers. Key words: non-Newtonian, wall shear rate, turbulent, rheometer
A Lie-group based similarity theory is developed for both momentum and energy distributions in a turbulent pipe flow, leading to asymptotic logarithmic profiles of mean velocity and turbulent kinetic energy. Both channel and pipe data over a wide range of Re yield 0.45 to be the universal Karman constant. A new spatial invariant characterizing outer dynamics is discovered and validated by reliable experimental data. The theory predicts the mean velocity profile (MVP) with 99% accuracy for high Re experimental data (up to 40 millions), and offers a quantitative explanation for recent observatio…
This paper uses the estimates of phase-locked parameters at the onset of bursting presented in a companion paper to derive logarithmic correlations for turbulent friction factor losses in time-independent power law fluids. Two different techniques for analysis were used. They gave logarithmic correlations with different coefficients. But both correlations predicted 264 data points of friction factors published in the literature with a standard error of 1.3%. The derivations show how modern understanding of coherent structures embedded in turbulent flow fields can be used to derive engineering …
We report a theory deriving bulk flow scaling for canonical wall-bounded flows. The theory accounts for the symmetries of boundary geometry (flat plate channel versus circular pipe) by a variational calculation for a large-scale energy length, which characterizes its bulk flow scaling by a simple exponent, i.e. $m=4$ for channel and 5 for pipe. The predicted mean velocity shows excellent agreement with several dozen sets of quality empirical data for a wide range of the Reynolds number (Re), with a universal bulk flow constant $κ\approx0.45$. Predictions for dissipation and turbulent transport…
Fully 3-dimensional computations of flow through a long pipe demand a huge number of degrees of freedom, making it very expensive to explore parameter space and difficult to isolate the structure of the underlying dynamics. We therefore introduce a `2+epsilon' dimensional model of pipe flow which is a minimal 3-dimensionalisation of the axisymmetric case: only sinusoidal variation in azimuth plus azimuthal shifts are retained, yet the same dynamics familiar from experiments are found. In particular the model retains the subcritical dynamics of fully resolved pipe flow, capturing realistic loca…
Using extensive direct numerical simulations, the dynamics of laminar-turbulent fronts in pipe flow is investigated for Reynolds numbers between $Re=2000$ and $5500$. We here investigate the physical distinction between the fronts of weak and strong slugs both by analysing the turbulent kinetic energy budget and by comparing the downstream front motion to the advection speed of bulk turbulent structures. Our study shows that weak downstream fronts travel slower than turbulent structures in the bulk and correspond to decaying turbulence at the front. At $Re\simeq 2900$ the downstream front spee…
A multi-layer model of an energy length function is developed by employing recent results of the authors. The theory predicts the complete, mean streamwise turbulent kinetic-energy profile (MKP), in good agreement with empirical data for a wide range of Reynolds numbers (Re). In particular, a critical $Re_τ$ is predicted, beyond which a scaling anomaly appears and MKP develops a second peak.
The dynamical behavior of propagating structures, determined from a Karhunen-Lo`eve decomposition, in turbulent pipe flow undergoing reverse transition to laminar flow is investigated. The turbulent flow data is generated by a direct numerical simulation started at a fully turbulent Reynolds number of Re_τ=150, which is slowly decreased until Re_τ=95. At this low Reynolds number the high frequency modes decay first, leaving only the decaying streamwise vortices. The flow undergoes a chugging phenomena, where it begins to relaminarize and the mean velocity increases. The remaining propagating m…
This study performs a large-eddy simulation of turbulent separated and reattached flow in an enlarged annular pipe. A vortex ring is periodically shed from the sudden expansion part. A longitudinal vortex occurs around the vortex ring, making the flow three-dimensional. As a result, the vortex ring becomes unstable downstream and splits into small vortices. A tubular longitudinal vortex structure occurs downstream of the reattachment point near the wall surface on the inner pipe side. A low-frequency fluctuation occurs at each pipe diameter ratio. The smaller the pipe diameter ratio is, the mo…
Symmetry reduction by the method of slices is applied to pipe flow in order to quotient the stream-wise translation and azimuthal rotation symmetries of turbulent flow states. Within the symmetry-reduced state space, all travelling wave solutions reduce to equilibria, and all relative periodic orbits reduce to periodic orbits. Projections of these solutions and their unstable manifolds from their $\infty$-dimensional symmetry-reduced state space onto suitably chosen 2- or 3-dimensional subspaces reveal their interrelations and the role they play in organising turbulence in wall-bounded shear f…
Local dissipation scales are a manifestation of the intermittent small-scale nature of turbulence. We report the first experimental evaluation of the distribution of local dissipation scales in turbulent pipe flows for a range of Reynolds numbers, 2.4x10^4<=Re_D<=7.0x10^4. Our measurements at the nearly isotropic pipe centerline and within the anisotropic logarithmic layer show excellent agreement with distributions that were previously calculated from numerical simulations of homogeneous isotropic box turbulence and with those predicted by theory. The reported results suggest a universality o…
We present convincing evidence of a direct connection between the pressure drop, viscous dissipation, and the turbulent energy spectrum. We use this finding to explain Nikuradse's experimental results of pressure drop for turbulent flows in rough pipes, in terms of a modified Kolmogorov length scale that varies with the surface roughness. Furthermore, we use Laufer's measurements of turbulent energy spectra in pipe flow to calculate -- to a good approximation -- the turbulent component of the pressure drop directly from an averaged turbulent energy spectrum. We also show that the incompressibi…
The results of a comparative analysis based upon a Karhunen-Loève expansion of turbulent pipe flow and drag reduced turbulent pipe flow by spanwise wall oscillation are presented. The turbulent flow is generated by a direct numerical simulation at a Reynolds number $Re_τ= 150$. The spanwise wall oscillation is imposed as a velocity boundary condition with an amplitude of $A^+ = 20$ and a period of $T^+ = 50$. The wall oscillation results in a 27% mean velocity increase when the flow is driven by a constant pressure gradient. The peaks of the Reynolds stress and root-mean-squared velocities shi…
The transition to turbulence in pipe flow does not follow the scenario familiar from Rayleigh-Benard or Taylor-Couette flow since the laminar profile is stable against infinitesimal perturbations for all Reynolds numbers. Moreover, even when the flow speed is high enough and the perturbation sufficiently strong such that turbulent flow is established, it can return to the laminar state without any indication of the imminent decay. In this parameter range, the lifetimes of perturbations show a sensitive dependence on initial conditions and an exponential distribution. The turbulence seems to be…
We investigate calcium carbonate scale formation at high Reynolds numbers in a large pipe rig facility. The calcium carbonate solution is produced from the injection, at a T-joint inlet, of pH-stabilized sodium carbonate and calcium chloride aqueous solutions. A scanning electron microscopy analysis of the deposited mass along the pipe indicates that after an initial transient regime of ion-by-ion crystal growth, calcium carbonate scale is dominated by particulate deposition. While limescale formation in regions that are closer to the pipe's entrance can be described as the heterogeneous surfa…
The drag reduction induced by superhydrophobic surfaces is investigated in turbulent pipe flow. Wetted superhydrophobic surfaces are shown to trap gas bubbles in their asperities. This stops the liquid from coming in direct contact with the wall in that location, allowing the flow to slip over the air bubbles. We consider a well defined texture with streamwise grooves at the walls in which the gas is expected to be entrapped. This configuration is modelled with alternating no-slip and shear-free boundary conditions at the wall. With respect to classical turbulent pipe flow, a substantial drag …
We propose a generalization of Hopf fibrations to quotient the streamwise translation symmetry of turbulent pipe flows viewed as dynamical systems. In particular, we exploit the geometric structure of the associate high dimensional state space, which is that of a principal fiber bundle. The relation between the comoving frame velocity $U_{d}$ associated with the dynamical phase of an orbit in the bundle and the Taylor's hypothesis is investigated. As an application, Laser-Induced-Fluorescence techniques are exploited to capture planar fluorescent dye concentration fields tracing a turbulent pi…
The Karman constant κ- widely used in atmospheric science and engineering turbulence modelling, and proposed by Prandtl in 1925 and von Karman in 1930 to describe the mean velocity of a turbulent wall-bounded flow - leads to a logarithmic profile in an overlap region near the wall. For over eighty years, its value was believed to be ~0.41. But more recently, many argue that it is not a constant, because of measured variations in different flows and at different Reynolds numbers (Re). Here, a multi-layer analytic theory is shown to lead to a re-interpretation of κas a global constant for both t…
A three-layer asymptotic structure for turbulent pipe flow is proposed, revealing in terms of intermediate variables, the existence of a Reynolds-number invariant logarithmic region. It provides a theoretical foundation for addressing important questions in the scaling of the streamwise mean velocity and variance. The key insight emerging from the analysis is that the scale separation between two adjacent layers is proportional to $\sqrt{Re_τ}$, rather than $Re_τ$. This suggests that, in order to realise Reynolds-number asymptotic invariance, much higher Reynolds numbers may be necessary to ac…
A study of the the main features of low- and high amplitude steady streamwise wall transpiration applied to pipe flow is presented. The effect of the two transpiration parameters, amplitude and wavenumber, on the flow have been investigated by means of direct numerical simulation at a moderate turbulent Reynolds number. The behaviour of the three identified mechanisms that act in the flow: modification of Reynolds shear stress, steady streaming and generation of non-zero mean streamwise gradients, have been linked to the transpiration parameters. The observed trends have permitted the identifi…
Turbulent pipe flows exhibit organizational states (OSs) that are labelled by discrete azimuthal wavenumber modes and are reminiscent of the traveling wave solutions of low Reynolds number regimes. The discretized time evolution of the OSs, obtained through stereoscopic particle image velocimetry, is shown to be non-Markovian for data acquisition carried out at a structure-resolved sampling rate. In particular, properly defined time-correlation functions for the OS transitions are observed to decay as intriguing power laws, up to a large-eddy time horizon, beyond which they decorrelate at much…
Chaotic dynamics of low-dimensional systems, such as Lorenz or Rössler flows, is guided by the infinity of periodic orbits embedded in their strange attractors. Whether this also be the case for the infinite-dimensional dynamics of Navier--Stokes equations has long been speculated, and is a topic of ongoing study. Periodic and relative periodic solutions have been shown to be involved in transitions to turbulence. Their relevance to turbulent dynamics---specifically, whether periodic orbits play the same role in high-dimensional nonlinear systems like the Navier--Stokes equations as they do in…
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 modes, permitting comparison of our results to experimental data, and identify the steady component of the velocit…
Isolated patches of turbulence in transitional straight pipes are sustained by a strong instability at their upstream front, where the production of turbulent kinetic energy (TKE) is up to five times higher than in the core. Direct numerical simulations presented in this paper show no evidence of such strong fronts if the pipe is bent. We examine the temporal and spatial evolution of puffs and slugs in a toroidal pipe with pipe-to-torus diameter ratio $δ= D/d = 0.01$ at several subcritical Reynolds numbers. Results show that the upstream overshoot of TKE production is at most one-and-a-half ti…
Experiments and numerical simulations were conducted to investigate the dispersion of turbulent jets issuing from realistic pipe geometries. The effect of jet densities and Reynolds numbers on vertical buoyant jets were investigated, as they emerged from the side wall of a circular pipe, through a round orifice. Particle image velocimetry (PIV) and planar laser-induced fluorescence (PLIF) techniques were employed simultaneously to provide time-averaged flow velocity and concentrations fields. Large eddy simulation (LES) was applied to provide further detail with regards to the three-dimensiona…
The recent discovery of unstable travelling waves (TWs) in pipe flow has been hailed as a significant breakthrough with the hope that they populate the turbulent attractor. We confirm the existence of coherent states with internal fast and slow streaks commensurate in both structure and energy with known TWs using numerical simulations in a long pipe. These only occur, however, within less energetic regions of (localized) `puff' turbulence at low Reynolds numbers (Re=2000-2400), and not at all in (homogeneous) `slug' turbulence at Re=2800. This strongly suggests that all currently known TWs si…
In shear flows turbulence first occurs in the form of localized structures (puffs/spots) surrounded by laminar fluid. We here investigate such spatially intermittent flows in a pipe experiment showing that turbulent puffs have a well defined interaction distance, which sets the minimum spacing of puffs as well as the maximum observable turbulent fraction. Two methodologies are employed here. Starting from a laminar flow puffs can be created by locally injecting a jet of fluid through the pipe wall. When the perturbation is applied periodically at low frequencies, as expected, a regular sequenc…
Equilibrium solutions are believed to structure the pathways for ergodic trajectories in a dynamical system. However, equilibria are atypical for systems with continuous symmetries, i.e. for systems with homogeneous spatial dimensions, whereas relative equilibria (traveling waves) are generic. In order to visualize the unstable manifolds of such solutions, a practical symmetry reduction method is required that converts relative equilibria into equilibria, and relative periodic orbits into periodic orbits. In this article we extend the fixed Fourier mode slice approach, previously applied 1-dim…
Direct numerical simulations (DNS) of particle-laden turbulent flow in straight, mildly curved and strongly bent pipes are performed in which the solid phase is modelled as small heavy spherical particles. A total of seven populations of dilute particles with different Stokes numbers, one-way coupled with their carrier phase, are simulated. The objective is to examine the effect of the curvature on micro-particle transport and accumulation. It is shown that even a slight non-zero curvature in the flow configuration strongly impact the particle concentration map such that the concentration of i…
We propose and apply a Fourier-based symmetry reduction scheme to remove, or quotient, the streamwise translation symmetry of Laser-Induced-Fluorescence measurements of turbulent pipe flows that are viewed as dynamical systems in a high-dimensional state space. We also explain the relation between Taylor's hypothesis and the comoving frame velocity $U_{d}$ of the turbulent orbit in state space. In particular, in physical space we observe flow structures that deform as they advect downstream at a speed that differs significantly from $U_{d}$. Indeed, the symmetry reduction analysis of planar dy…
Almost 50 years ago Bob Brodkey and his student, Corino, conceived of and carried out a visualization experiment for the very near wall region of a turbulent pipe flow that, together with the turbulent boundary layer visualization of Kline et al., excited the turbulence research community. Using a high speed movie camera mounted on a lathe bed that recorded magnified images in a frame of reference moving with the flow, they observed the motions of sub-micron particles in the sub-, buffer- and log-layers. Surprisingly, these motions were not nearly so locally random as was the general view of t…
The swirling secondary flow in curved pipes is studied in three-space dimensions using a weakly compressible Smoothed Particle Hydrodynamics (WCSPH) formulation coupled to new non-reflecting outflow boundary conditions. A large-eddy simulation (LES) model for turbulence is benchmarked with existing experimental data. After validation of the present model against experimental results for a 90 grades pipe bend, a detailed numerical study aimed at reproducing experimental flow measurements for a wide range of Reynolds numbers has been performed for different pipe geometries, including U pipe bend…
We experimentally assess the capabilities of an active, open-loop technique for drag reduction in turbulent wall flows recently introduced by Quadrio et al. [J. Fluid Mech., v.627, 161, (2009)]. The technique consists in generating streamwise-modulated waves of spanwise velocity at the wall, that travel in the streamwise direction. A proof-of-principle experiment has been devised to measure the reduction of turbulent friction in a pipe flow, in which the wall is subdivided into thin slabs that rotate independently in the azimuthal direction. Different speeds of nearby slabs provide, although i…
We show that the main results of the analysis of the friction factor for turbulent pipe flow reported in G. Gioia and P. Chakraborty (GC), Phys. Rev. Lett. 96, 044502 (1996) can be recovered by assuming the Heisenberg closure hypothesis for the turbulent spectrum. This highlights the structural features of the turbulent spectrum underlying GC's analysis.
As everyone knows who has opened a kitchen faucet, pipe flow is laminar at low flow velocities and turbulent at high flow velocities. At intermediate velocities there is a transition wherein plugs of laminar flow alternate along the pipe with "flashes" of a type of fluctuating, non-laminar flow which remains poorly known. We show experimentally that the fluid friction of flash flow is diagnostic of turbulence. We also show that the statistics of flash flow are in keeping with Kolmogorov's phenomenological theory of turbulence (so that, e.g., the energy spectra of both flash flow and turbulent …
We investigated the propagation of turbulent fronts in pipe flow at high Reynolds numbers by direct numerical simulation. We used a technique combining a moving frame of reference and an artificial damping to isolate the fronts in short periodic pipes, which enables us to explore the bulk Reynolds number up to Re = $10^5$ with affordable computation power. We measured the propagation speed of the downstream front and observed that a fit of $1.971-(Re/1925)^{-0.825}$ (in unit of bulk speed) well captures this speed above $Re\simeq 5000$. The speed increases monotonically as Re increases, in sta…
In this visualisation, the transition from laminar to turbulent flow is characterised by the intermittent ejection of wall fluid into the outer stream. The normalised thickness of the viscous flow layer reaches an asymptotic value but the physical thickness drops exponentially after transition. The critical transition pipe Reynolds number can be obtained simply by equating it with the asymptotic value of the normalised thickness of viscous flow layer. Key words: Transition, critical stability Reynolds number, critical transition Reynolds number, non-Newtonian pipe flow
The recent theoretical discovery of families of travelling wave solutions in pipe flow at Reynolds numbers lower than the transitional range naturally raises the question of their relevance to the turbulent transition process. Here a series of numerical experiments are conducted in which we look for the spatial signature of these travelling waves in transitionary flows. Working within a periodic pipe of 5D (diameters) length, we find that travelling waves with low wall shear stresses (lower branch solutions) are on a surface which separates initial conditions which uneventfully relaminarise an…
A streamwise-constant model is presented to investigate the basic mechanisms responsible for the change in mean flow occuring during pipe flow transition. Using a single forced momentum balance equation, we show that the shape of the velocity profile is robust to changes in the forcing profile and that both linear non-normal and nonlinear effects are required to capture the change in mean flow associated with transition to turbulence. The particularly simple form of the model allows for the study of the momentum transfer directly by inspection of the equations. The distribution of the high- an…
Drag reduction and subcritical turbulence in controlled pipe flows
Politecnico di Milano（米兰理工大学）；Univ. Lille, CNRS, ONERA, Arts et Metiers Institute of Technology, Centrale Lille（里尔大学、法国国家科学研究中心、法国航空发展局、巴黎国立高等工艺学院、里尔中央理工学院）；Massachusetts Institute of Technology（麻省理工学院）
A volume-filtered Euler-Lagrange large eddy simulation methodology is used to predict the physics of turbulent liquid-solid slurry flow through a horizontal pipe. A dynamic Smagorinsky model based on Lagrangian averaging is employed to account for the sub-filter scale effects in the liquid phase. A fully conservative immersed boundary method is used to account for the pipe geometry on a uniform cartesian grid. The liquid and solid phases are coupled through volume fraction and momentum exchange terms. Particle-particle and particle-wall collisions are modeled using a soft-sphere approach. A se…
The friction f is the property of wall-bounded flows that sets the pumping cost of a pipeline, the draining capacity of a river, and other variables of practical relevance. For highly turbulent rough-walled pipe flows, f depends solely on the roughness length scale r, and the f -- r relation may be expressed by the Strickler empirical scaling f $\propto$ r$^{\frac{1}{3}}$. Here, we show experimentally that for soap film flows that are the two-dimensional (2D) equivalent of highly turbulent rough-walled pipe flows, f $\propto$ r and the f -- r relation is not the same in 2D as in 3D. Our findin…
