One of the key observations in the Princeton Superpipe was the late start of the logarithmic mean velocity overlap layer at a wall distance of the order of $10^3$ inner units. Between $y^+\approx 150$, the start of the overlap layer in zero pressure gradient turbulent boundary layers, and $y^+\approx 10^3$, the Superpipe profile is modeled equally well by a power law or a log-law with a larger slope than in the overlap layer. In this paper it is shown that the mean velocity profile in turbulent plane channel flow exhibits analogous characteristics, namely a sudden decrease of logarithmic slope (increase of $κ$) at a $y^+_{\mathrm{break}}\approx 600$, which marks the start of the actual overlap layer. This demonstration results from the first construction of the complete inner and outer asymptotic expansions up to order $\mathcal{O}(\mathrm{Re}_τ)^{-1}$ from mean velocity profiles of direct numerical simulations (DNS) at moderate Reynolds numbers. A preliminary analysis of a Couette flow DNS, on the other hand, yields an increase of logarithmic slope (decrease of $κ$) at a $y^+_{\mathrm{break}}\approx 400$. The correlation between the sign of the slope change and the flow symmetry motivates the hypothesis that the breakpoint between the possibly universal short inner logarithmic region and the actual overlap log-law corresponds to the penetration depth of large-scale turbulent structures originating from the opposite wall.
The Variable Density and Speed of Sound Vessel (VDSSV) produces subsonic turbulent flows that are both compressible and observable at all scales with existing instrumentation including hot wires and particle tracking. We realize this objective by looking at the flow of a heavy gas (sulfur hexafluoride $SF_6$), with a speed of sound almost three times lower than for air. By switching between air and $SF_6$, we isolate the influence of the turbulent Mach number (up to $M_t$ = 0.17) on turbulence statistics from the influences of changes in the Reynolds number (up to $R_λ$ = 1600), and boundary c…
Studies in the literature on plane turbulent wall jets on flat surfaces, have invariably considered either the nozzle initial conditions or the asymptotic conditions far downstream, as scaling parameters for the streamwise variations of length and velocity scales. These choices, however, do not square with the notion of self similarity which is essentially a "local" concept. We first demonstrate that the streamwise variations of velocity and length scales in wall jets show remarkable scaling with local parameters i.e. there appear to be no imposed length and velocity scales. Next, it is shown …
The power laws in the overlap region of two layer theory of turbulent boundary layer have been obtained by Izakson-Millikan-Kolmogorov hypothesis. The solution of the open functional equation is not unique. The power laws for velocity \& skin friction have equivalence with log laws for velocity \& skin friction, for large Reynolds numbers.
A major challenge in computation of engineering flows is to derive and improve turbulence models built on turbulence physics. Here, we present a physics-based modified $k-ω$ equation for canonical wall bounded turbulent flows (boundary layer, channel and pipe), predicting both mean velocity profile (MVP) and streamwise mean kinetic energy profile (SMKP) with high accuracy over a wide range of Reynolds number ($Re$). The result builds on a multi-layer quantification of wall flows, which allows a significant modification of the $k-ω$ equation. Three innovations are introduced: First, an adjustme…
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…
Direct Numerical Simulations of turbulent channel flows at friction Reynolds number 550, 1000, 1500, are used to analyse the turbulent production, transfer and dissipation mechanisms in the compound space of scales and wall-distances by means of the Kolmogorov equation generalized to inhomogeneous anisotropic flows. Two distinct peaks of scale-energy source are identified. The first stronger one belongs to the near-wall cycle. Its location in the space of scales and physical space is found to scale in viscous units while its intensity grows slowly with $Re$, indicating a near-wall modulation. …
An experiment was performed using Dual-plane-SPIV in the LMFL boundary layer facility to determine all of the derivative moments needed to estimate the average dissipation rate of the turbulent kinetic energy, $\varepsilon$, and its Reynolds stress counterpart the dissipation tensor, $\varepsilon_{ij}$. For this experiment, the Reynolds number was $Re_θ= 7500$ or $Re_τ= 2300$. Part I of this contribution \cite{stanislas20} presented in short the experiment and discussed in detail the dissipation profile and all twelve derivative moments required to compute it. The data were compared to a chann…
We track the secondary bifurcations of coherent states in plane Couette flow and show that they undergo an incomplete periodic doubling cascade that ends with a crisis bifurcation. We introduce a symbolic dynamics for the orbits and show that the ones that exist fall into the universal sequence described by Metropolis, Stein and Stein for unimodal maps. The periodic orbits cover much of the turbulent dynamics in that their temporal evolution overlaps with turbulent motions when projected onto a plane spanned by energy production and dissipation.
We argue that important elements of the dynamics of wall-bounded flows reside at the wall-normal position $y_p^+$ corresponding to the peak of the Reynolds shear stress. Specializing to pipe and channel flows, we show that the mean momentum balance in the neighborhood of $y_p^+$ is distinct in character from those in the classical inner and outer layers. We revisit empirical data to confirm that $y_p^+ = O(R^{1/2})$ and show that, in a neighborhood of order $R^{1/2}$ around $y_p^+$, only the viscous effects balance pressure-gradient terms. Here, R is the Reynolds number based on friction veloc…
The symmetries and similarities of the zero-pressure-gradient turbulent boundary layer (ZPGTBL) are investigated to derive the full set of similarity variables, to derive the similarity equations, and to obtain a higher-order approximate solution of the mean velocity profile. Previous analyses have not resulted in all the similarity variables. We perform a symmetry analysis of the equations for ZPGTBL using Lie dilation groups, and obtain local, leading-order symmetries of the equations. The full set of similarity variables were obtained in terms of the boundary layer parameters. The friction …
We study turbulent channel flows of monodisperse and polydisperse suspensions of finite-size spheres by means of Direct Numerical Simulations using an immersed boundary method to account for the dispersed phase. Suspensions with 3 different Gaussian distributions of particle radii are considered (i.e. 3 different standard deviations). The distributions are centered on the reference particle radius of the monodisperse suspension. In the most extreme case, the radius of the largest particles is 4 times that of the smaller particles. We consider two different solid volume fractions, 2% and 10%. W…
An experiment was performed using SPIV in the LMFL boundary layer facility to determine all the derivative moments needed to estimate the average dissipation rate of the turbulence kinetic energy, $\varepsilon = 2 ν\langle s_{ij}s_{ij} \rangle$ where $s_{ij}$ is the fluctuating strain-rate and $\langle~\rangle$ denotes ensemble averages. Also measured were all the moments of the full average deformation rate tensor, as well as all of the first, second and third fluctuating velocity moments except those involving pressure. The Reynolds number was $Re_θ= 7500$ or $Re_τ= 2300$. The results are pr…
The convection velocity ($U_C$) of turbulent structures has been studied in adverse-pressure-gradient (APG) turbulent boundary layers (TBLs) for a wide range of Reynolds numbers $Re_τ= 1400 - 4000$. The study is based on estimation of the convection velocity using decomposed streamwise skewness factor introduced in (Drozdz A., Elsner W., Int. J. of Heat and Fluid Flow. 63 (2017) 67\-74) and verified by means of two-point correlation method. It was shown that in the overlapping region of APG flows, the convection velocity profiles (when scaled in viscous units) reassemble the universal logarith…
Simultaneous measurements, such as the combination of particle image velocimetry (PIV) for velocity fields with planar laser induced fluorescence (PLIF) for species fields, are widely used in experimental turbulent combustion applications for the analysis of a plethora of complex physical processes. Such physical analyses are driven by the interpretation of spatial correlations between these fields by the experimenter. However, these correlations also imply some amount of intrinsic redundancy; the simultaneous fields contain overlapping information content. The goal of this work lies in the qu…
The decomposition of hydrodynamic and acoustic components of cavity flows can aid the understanding of the flow-acoustic interaction, which produces a variety of adverse effects in applications. We apply the approach of combining the resolvent analysis and Doak's momentum potential theory to examine the input-output flow-acoustic characteristics of compressible flow over an open cavity at Re=10,000. The methodology can decompose hydrodynamic and acoustic components from an input-output framework. We further localize the forcing at the leading-edge and front wall of the cavity and filter with a…
There are several approaches to describe flows with particles e.g. Lattice-Gas Automata (LGA), Lattice-Boltzmann method (LBM) or smoothed particle hydrodynamics (SPH). These approaches do not use fixed grids on which the Navier-Stokes equations are solved via e.g. finite volume method. The flow is simulated using a multitude of particles or particle density distributions, which interacts and due to statistical laws and an even more fundamental approach than the Navier-Stokes equation, the averaged flow variables can be derived. After a short summary of the most popular particle methods the new…
The original turbulent pipe flow experiments in the Princeton "Superpipe" by Zagarola & Smits (1997, 1998) at unprecedented laboratory Reynolds numbers have started an ongoing vigorous debate on the logarithmic law in the mean velocity profile $U^+(y^+)$ and the intimately related question of Pitot probe corrections for mean shear, viscous effects and turbulence level. Considering that the Pitot probe diameter $d^+$ exceeded 7000 wall units at the highest Reynolds number, the various \textcolor{black}{traditional Pitot} corrections had to be extended into uncharted territory \textcolor{black}{…
With the availability of new high-Reynolds-number ($Re$) databases of turbulent boundary layers (TBLs) it has been possible to identify in detail certain regions of the boundary layer with more complex behavior. In this study we consider a unique database at moderately-high $Re$, with a near-constant adverse pressure gradient (APG) (Pozuelo {\it et al.}, {\it J. Fluid Mech.}, {\bf 939}, A34, 2022), and perform spectral analysis of the Reynolds stresses, focusing on the streamwise component. We assess different regions of the APG TBL, comparing this case with the zero-pressure-gradient (ZPG) TB…
We present the first general metric for attractor overlap (MAO) facilitating an unsupervised comparison of flow data sets. The starting point is two or more attractors, i.e., ensembles of states representing different operating conditions. The proposed metric generalizes the standard Hilbert-space distance between two snapshots to snapshot ensembles of two attractors. A reduced-order analysis for big data and many attractors is enabled by coarse-graining the snapshots into representative clusters with corresponding centroids and population probabilities. For a large number of attractors, MAO i…
We perform direct numerical simulation (DNS) to study the clustering of small, heavy, monodisperse particles subject to collision-coagulation in turbulent flow (i.e., colliding particles always coagulate (coalesce) into large ones). We find that collision-coagulation causes the radial distribution function (RDF) of the particles to decrease strongly at particle separation distances $r$ close to the particle diameter $d$. However, the RDF do not decrease indefinitely but approach a finite value in the limit of $r\to d$. We study how the characteristics of this "depletion zone" relate to the par…
The performances of a new data processing technique, namely the Empirical Mode Decomposition, are evaluated on a fully developed turbulent velocity signal perturbed by a numerical forcing which mimics a long-period flapping. First, we introduce a "resemblance" criterion to discriminate between the polluted and the unpolluted modes extracted from the perturbed velocity signal by means of the Empirical Mode Decomposition algorithm. A rejection procedure, playing, somehow, the role of a high-pass filter, is then designed in order to infer the original velocity signal from the perturbed one. The q…
The sources of wall-pressure fluctuations in turbulent channel flow are studied using a novel framework. The wall-pressure power spectral density (PSD) $(ϕ_{pp}(ω))$ is expressed as an integrated contribution from all wall-parallel plane pairs, $ϕ_{pp}(ω)=\int_{-δ}^{+δ}\int_{-δ}^{+δ}Γ(r,s,ω)\,\mathrm{dr}\,\mathrm{ds}$, using the Green's function. Here, $Γ(r,s,ω)$ is termed the net source cross spectral density (CSD) between two wall-parallel planes, $y=r$ and $y=s$ and $δ$ is the half channel height. Direct Numerical Simulation (DNS) data at friction Reynolds number of $180$ and $400$ are used…
Flow between concentric spheres of radius ratio $η= r_\mathrm{i}/r_\mathrm{o} = 0.35$ is studied in a 3 m outer diameter experiment. We have measured the torques required to maintain constant boundary speeds as well as localized wall shear stress, velocity, and pressure. At low Ekman number $E = 2.1\times10^{-7}$ and modest Rossby number $0.07 < Ro < 3.4$, the resulting flow is highly turbulent, with a Reynolds number ($Re=Ro/E$) exceeding fifteen million. Several turbulent flow regimes are evident as $Ro$ is varied for fixed $E$. We focus our attention on one flow transition in particular, be…
Direct numerical simulations of the Taylor-Couette (TC) problem, the flow between two coaxial and independently rotating cylinders, have been performed. The study focuses on TC flow with mild curvature (small gap) with a radius ratio of $η=r_i/r_o=0.909$, an aspect ratio of $Γ=L/d=2π/3$, and a stationary outer cylinder. Three inner cylinder Reynolds of $1\cdot10^5$, $2\cdot10^5$ and $3\cdot 10^5$ were simulated, corresponding to frictional Reynolds numbers between $Re_τ\approx 1400$ and $Re_τ\approx 4000$. An additional case with a large gap, $η=0.5$ and driving of $Re=2\cdot10^5$ was also per…
The transport equations for velocity variances are investigated using data from DNS of incompressible channel flows at $Re_τ$ up to 5200. Each term in the transport equation has been spectrally decomposed to expose the contribution of turbulence at different length scales to the processes governing the flow of energy in the wall-normal direction, in scale and among components. The outer-layer turbulence is dominated by very large-scale streamwise elongated modes. Away from the wall, production occurs primarily in these large-scale streamwise-elongated modes in the streamwise velocity, but diss…
The problem of discerning key features of steady turbulent flow adjacent to a wall has drawn the attention of some of the most noted fluid dynamicists of all time. Standard examples of such features are found in the mean velocity profiles of turbulent flow in channels, pipes or boundary layers. The aim of this review article is to expound the essence of some elementary theoretical efforts in this regard. Possibly the best known of them, and certainly the simplest, is the argument (obtained independently) by Izakson (1937) and Millikan (1939). They showed that if an inner scaling and an outer s…
We investigate statistics of large-scale structures from large-eddy simulation (LES) of turbulent channel flow at friction Reynolds numbers $Re_τ= 2 {\rm k}$ and $200 {\rm k}$. To properly capture the behaviour of large-scale structures, 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…
The current theories on the mechanism for polymer drag-reduction (DR) are generally applicable for long-chain flexible polymers that form viscoelastic solutions. Rigid polymer solutions that generate DR seemingly lack prevalent viscoelastic characteristics. They do, however, demonstrate larger viscosities and a noticeable shear-thinning trend, well approximated by generalized Newtonian models. The following experimental investigation scrutinized the flow statistics of an aqueous xanthan gum solution in a turbulent channel flow, with friction Reynolds numbers ($Re_τ$) between 160 and 680. The a…
This paper is devoted to the investigation of the origin and mechanism of randomization in late boundary layer transition over a flat plate without pressure gradient. The flow randomization is a crucial phase before flow transition to the turbulent state. According to existing literatures, the randomization was caused by the big background noises and non-periodic spanwise boundary conditions. It was assumed that the large ring structure is affected by background noises first, and then the change of large ring structure affects the small length scales quickly, which directly leads to randomizat…
We propose a novel cluster-based reduced-order modelling (CROM) strategy of unsteady flows. CROM combines the cluster analysis pioneered in Gunzburger's group (Burkardt et al. 2006) and and transition matrix models introduced in fluid dynamics in Eckhardt's group (Schneider et al. 2007). CROM constitutes a potential alternative to POD models and generalises the Ulam-Galerkin method classically used in dynamical systems to determine a finite-rank approximation of the Perron-Frobenius operator. The proposed strategy processes a time-resolved sequence of flow snapshots in two steps. First, the sn…
