# Journal of Turbulence

Publications
Shear flow turbulence in oscillatory fluid motions is of theoretical interest and practical relevance, since the onset of turbulence can drastically change the transport properties and mixing efficiency. To supplement former theoretical and experimental investigations on the transition to turbulence in Sexl-Womersley (SW) flows, we perform three-dimensional direct numerical simulations (DNS) of oscillatory pipe flows at three Womersley numbers (Wo ∈ {26, 13, 5}) and one constant Reynolds number (Reτ = 1440) based on the friction velocity and the pipe diameter. For this, the incompressible Navier-Stokes equations are solved in cylindrical coordinates using a fourth-order-accurate finite-volume method on staggered grids, motivated by Schumann’s volume balance procedure. We generate a well-correlated high-Reynolds-number initial flow field for the oscillatory flows by means of a DNS of a statistically steady pipe flow at Reτ = 1440 . To underline the reliability of the DNS results for the oscillatory pipe flows, we validate the finite-volume method, the spatial resolution of the computational grid and the length of the computational domain by comparing the results for the statistically steady pipe flow with experimental data obtained by laser Doppler anemometry (LDA). Comparing the statistical moments of the velocity components up to the fourth order shows good agreement with the corresponding LDA data. When started from the turbulent initial velocity field, the oscillatory flows relaminarise or reach a conditionally or fully turbulent state, depending on Wo. The peak flow rates decrease with increasing Wo, while the relaxation phase for the initially steady flow converging to a purely oscillating flow increases with increasing Wo. For the highest Wo considered, the flow completely relaminarises and we do not find any instabilities close to the wall, as expected from former stability analyses. On the other hand, we confirm the existence of turbulent bursts and increasing turbulence intensity during the deceleration phase of the flow and relaminarisation in the acceleration phase for Wo = 13 in agreement with experimental results in the literature. By analysing selected terms of the transport equations for the mean and turbulent kinetic energy, we demonstrate the transport of turbulent kinetic energy from the axial to the radial and azimuthal velocity components during flow deceleration.

In this paper, we investigate the use of compactly supported divergence-free wavelets for the representation of the Navier-Stokes solution. After reminding the theoretical construction of divergence-free wavelet vectors, we present in detail the bases and corresponding fast algorithms for 2D and 3D incompressible flows. In order to compute the nonlinear term, we propose a new method which provides in practice with the Hodge decomposition of any flow: this decomposition enables us to separate the incompressible part of the flow from its orthogonal complement, which corresponds to the gradient component of the flow. Finally we show numerical tests to validate our approach.

We investigate the statistics of orientation of small, neutrally buoyant, spherical tracers whose center of mass is displaced from the geometrical center. If appropriate-sized particles are considered, a linear relation can be derived between the horizontal components of the orientation vector and the same components of acceleration. Direct numerical simulations are carried out, showing that such relation can be used to reconstruct the statistics of acceleration fluctuations up to the order of the gravitational acceleration. Based on such results, we suggest a novel method for the local experimental measurement of accelerations in turbulent flows.

We report recent results from a high resolution numerical study of fluid particles transported by a fully developed turbulent flow. Single particle trajectories were followed for a time range spanning more than three decades, from less than a tenth of the Kolmogorov time-scale up to one large-eddy turnover time. We present some results concerning acceleration statistics and the statistics of trapping by vortex filaments.

We report results from a high resolution numerical study of fluid particles transported by a fully developed turbulent flow. Single particle trajectories were followed for a time range spanning more than three decades, from less than a tenth of the Kolmogorov time-scale up to one large-eddy turnover time. We present results concerning acceleration statistics and the statistics of trapping by vortex filaments conditioned to the local values of vorticity and enstrophy. We distinguish two different behaviors between the joint statistics of vorticity and centripetal acceleration or vorticity and longitudinal acceleration. Comment: 8 pages, 6 figures

We compute the joint distribution of relative velocities and separations of identical inertial particles suspended in randomly mixing and turbulent flows. Our results are obtained by matching asymptotic forms of the distribution. The method takes into account spatial clustering of the suspended particles as well as singularities in their motion (so-called 'caustics'). It thus takes proper account of the fractal properties of phase space and the distribution is characterised in terms of the corresponding phase-space fractal dimension D_2. The method clearly exhibits universal aspects of the distribution (independent of the statistical properties of the flow): at small particle separations R and not too large radial relative speeds |V_R|, the distribution of radial relative velocities exhibits a universal power-law form \rho(V_R,R) \sim |V_R|^{D_2-d-1} provided that D_2 < d+1 (d is the spatial dimension) and that the Stokes number St is large enough for caustics to form. The range in V_R over which this power law is valid depends on R, on the Stokes number, and upon the nature of the flow. Our results are in good agreement with results of computer simulations of the dynamics of particles suspended in random velocity fields with finite correlation times. In the white-noise limit the results are consistent with those of [Gustavsson and Mehlig, Phys. Rev. E84 (2011) 045304].

In a previous communication (W.J.T. Bos and J.-P. Bertoglio 2006, Phys. Fluids, 18, 031706), a self-consistent Markovian triadic closure was presented. The detailed derivation of this closure is given here, relating it to the Direct Interaction Approximation and Quasi-Normal types of closure. The time-scale needed to obtain a self-consistent closure for both the energy spectrum and the scalar variance spectrum is determined by evaluating the correlation between the velocity and an advected displacement vector-field. The relation between this latter correlation and the velocity-scalar correlation is stressed, suggesting a simplified model of the latter. The resulting closed equations are numerically integrated and results for the energy spectrum, scalar fluctuation spectrum and velocity-displacement correlation spectrum are presented for low, unity and high values of the Schmidt number.

A novel mapping closure approximation (MCA) technique is developed to construct a model for the conditional dissipation rate (CDR) of a scalar in homogeneous turbulence. It is shown that the CDR model from amplitude mapping closure is incorrect in asymptotic behavior for unsymmetric binary mixing. The correct asymptotic behavior can be described by the CDR model formulated by the MCA technique. The MCA approach is outlined for constructing successive approximation to probability density function (PDF) and conditional moment.

The behavior of the second-order Lagrangian structure functions on state-of-the-art numerical data both in two and three dimensions is studied. On the basis of a phenomenological connection between Eulerian space-fluctuations and the Lagrangian time-fluctuations, it is possible to rephrase the Kolmogorov $4/5$-law into a relation predicting the linear (in time) scaling for the second order Lagrangian structure function. When such a function is directly observed on current experimental or numerical data, it does not clearly display a scaling regime. A parameterization of the Lagrangian structure functions based on Batchelor model is introduced and tested on data for $3d$ turbulence, and for $2d$ turbulence in the inverse cascade regime. Such parameterization supports the idea, previously suggested, that both Eulerian and Lagrangian data are consistent with a linear scaling plus finite-Reynolds number effects affecting the small- and large-time scales. When large-time saturation effects are properly accounted for, compensated plots show a detectable plateau already at the available Reynolds number. Furthermore, this parameterization allows us to make quantitative predictions on the Reynolds number value for which Lagrangian structure functions are expected to display a scaling region. Finally, we show that this is also sufficient to predict the anomalous dependency of the normalized root mean squared acceleration as a function of the Reynolds number, without fitting parameters.

Wind turbines operate in the atmospheric boundary layer, where they are exposed to the turbulent atmospheric flows. As the response time of wind turbine is typically in the range of seconds, they are affected by the small scale intermittent properties of the turbulent wind. Consequently, basic features which are known for small-scale homogeneous isotropic turbulence, and in particular the well-known intermittency problem, have an important impact on the wind energy conversion process. We report on basic research results concerning the small-scale intermittent properties of atmospheric flows and their impact on the wind energy conversion process. The analysis of wind data shows strongly intermittent statistics of wind fluctuations. To achieve numerical modeling a data-driven superposition model is proposed. For the experimental reproduction and adjustment of intermittent flows a so-called active grid setup is presented. Its ability is shown to generate reproducible properties of atmospheric flows on the smaller scales of the laboratory conditions of a wind tunnel. As an application example the response dynamics of different anemometer types are tested. To achieve a proper understanding of the impact of intermittent turbulent inflow properties on wind turbines we present methods of numerical and stochastic modeling, and compare the results to measurement data. As a summarizing result we find that atmospheric turbulence imposes its intermittent features on the complete wind energy conversion process. Intermittent turbulence features are not only present in atmospheric wind, but are also dominant in the loads on the turbine, i.e. rotor torque and thrust, and in the electrical power output signal. We conclude that profound knowledge of turbulent statistics and the application of suitable numerical as well as experimental methods are necessary to grasp these unique features (...)

The bottleneck phenomenon in three-dimensional turbulence is generally associated with the dissipation range of the energy spectrum. In the present work, it is shown by using a two-point closure theory, that in two-dimensional turbulence it is possible to observe a bottleneck at the large scales, due to the effect of friction on the inverse energy cascade. This large-scale bottleneck is directly related to the process of energy condensation, the pile-up of energy at wavenumbers corresponding to the domain size. The link between the use of friction and the creation of space-filling structures is discussed and it is concluded that the careless use of hypofriction might reduce the inertial range of the energy spectrum.

Large scale coherent streaks are artificially forced in a well developed turbulent boundary layer at $\Redstar \approx 1000$ using an array of cylindrical roughness elements. Measures of the velocity field with particle image velocimetry reveal the presence of well reproducible, streamwise oriented, steady coherent streaks. We show that the amplitude of these coherent streaks transiently grows in space. The position of the maximum amplitude is proportional to the spanwise wavelength of the streaks and the most amplified spanwise wavelength is of very large scale $\lz \approx 6 \dblo$. These results are in good agreement with the recent predictions based on the optimal transient growth analysis of turbulent mean flows.

A new variational problem for upper bounds on the rate of energy dissipation in body-forced shear flows is formulated by including a balance parameter in the derivation from the Navier-Stokes equations. The resulting min-max problem is investigated computationally, producing new estimates that quantitatively improve previously obtained rigorous bounds. The results are compared with data from direct numerical simulations. Comment: 15 pages, 7 figures

We discuss continuous cascade models and their potential for modelling the energy dissipation in a turbulent flow. Continuous cascade processes, expressed in terms of stochastic integrals with respect to L\'evy bases, are examples of ambit processes. These models are known to reproduce experimentally observed properties of turbulence: The scaling and self-scaling of the correlators of the energy dissipation and of the moments of the coarse-grained energy dissipation. We compare three models: a normal model, a normal inverse Gaussian model and a stable model. We show that the normal inverse Gaussian model is superior to both, the normal and the stable model, in terms of reproducing the distribution of the energy dissipation; and that the normal inverse Gaussian model is superior to the normal model and competitive with the stable model in terms of reproducing the self-scaling exponents. Furthermore, we show that the presented analysis is parsimonious in the sense that the self-scaling exponents are predicted from the one-point distribution of the energy dissipation, and that the shape of these distributions is independent of the Reynolds number.

A direct numerical simulation (DNS) of a channel flow with one curved surface was performed at moderate Reynolds number (Re_tau = 395 at the inlet). The adverse pressure gradient was obtained by a wall curvature through a mathematical mapping from physical coordinates to Cartesian ones. The code, using spectral spanwise and normal discretization, combines the advantage of a good accuracy with a fast integration procedure compared to standard numerical procedures for complex geometries. The turbulent flow slightly separates on the profile at the lower curved wall and is at the onset of separation at the opposite flat wall. The thin separation bubble is characterized with a reversal flow fraction. Intense vortices are generated near the separation line on the lower wall but also at the upper wall. Turbulent normal stresses and kinetic energy budget are investigated along the channel.

In this paper we study a well-known three--dimensional turbulence model, the filtered Clark model, or Clark-alpha model. This is Large Eddy Simulation (LES) tensor-diffusivity model of turbulent flows with an additional spatial filter of width alpha ($\alpha$). We show the global well-posedness of this model with constant Navier-Stokes (eddy) viscosity. Moreover, we establish the existence of a finite dimensional global attractor for this dissipative evolution system, and we provide an anaytical estimate for its fractal and Hausdorff dimensions. Our estimate is proportional to $(L/l_d)^3$, where $L$ is the integral spatial scale and $l_d$ is the viscous dissipation length scale. This explicit bound is consistent with the physical estimate for the number of degrees of freedom based on heuristic arguments. Using semi-rigorous physical arguments we show that the inertial range of the energy spectrum for the Clark-$\aa$ model has the usual $k^{-5/3}$ Kolmogorov power law for wave numbers $k\aa \ll 1$ and $k^{-3}$ decay power law for $k\aa \gg 1.$ This is evidence that the Clark$-\alpha$ model parameterizes efficiently the large wave numbers within the inertial range, $k\aa \gg 1$, so that they contain much less translational kinetic energy than their counterparts in the Navier-Stokes equations.

The use of simplified models of turbulent flows provides an appealing possibility to study the collision rate of turbulent suspensions, especially in conditions relevant to astrophysics, which require large time scale separations. To check the validity of such approaches, we used a direct numerical simulation (DNS) velocity field, which satisfies the Navier-Stokes equations (although it neglects the effect of the suspended particles on the flow field), and a kinematic simulation (KS) velocity field, which is a random field designed so that its statistics are in accord with the Kolmogorov theory for fully-developed turbulence. In the limit where the effects of particle inertia (characterised by the Stokes number) are negligible, the collision rates from the two approaches agree. As the Stokes number St increases, however, we show that the DNS collision rate exceeds the KS collision rate by orders of magnitude. We propose an explanation for this phenomenon and explore its consequences. We discuss the collision rate $R$ for particles in high Reynolds number flows at large Stokes number, and present evidence that $R\propto \sqrt{{\rm St}}$.

Nearly homogeneous and isotropic, highly turbulent flow, generated by an original multi-scale injector is experimentally studied. This multi-scale injector is made of three perforated plates shifted in space such that the diameter of their holes and their blockage ratio increase with the downstream distance. The Multi-Scale Turbulence Injector (hereafter, MuSTI) is compared with a Mono-Scale Turbulence Injector (MoSTI), the latter being constituted by only the last plate of MuSTI. This comparison is done for both cold and reactive flows. For the cold flow, it is shown that, in comparison with the classical mono-scale injector, for the MuSTI injector: (i) the turbulent kinetic energy is roughly twice larger, and the kinetic energy supply is distributed over the whole range of scales. This is emphasized by second and third order structure functions. (ii) the transverse fluxes of momentum and energy are enhanced, (iii) the homogeneity and isotropy are reached earlier ($\approx 50$%), (iv) the jet merging distance is the relevant scaling length-scale of the turbulent flow, (v) high turbulence intensity ($\approx 15$%) is achieved in the homogeneous and isotropic region, although the Reynolds number based on the Taylor microscale remains moderate ($Re_\lambda \approx 80$). In a second part, the interaction between the multi-scale generated turbulence and the premixed flame front is investigated by laser tomography. A lean V-shaped methane/air flame is stabilised on a heated rod in the homogeneous and isotropic region of the turbulent flow. The main observation is that the flame wrinkling is hugely amplified with the multi-scale generated injector, as testified by the increase of the flame brush thickness. Comment: 29 pages, 21 figures, submitted to Journal of Turbulence

Bacteria and plankton populations living in oceans and lakes reproduce and die under the in- fluence of turbulent currents. Turbulent transport interacts in a complex way with the dynamics of populations because the typical reproduction time of microorganism is within the inertial range of turbulent time scales. In the present manuscript we quantitatively investigate the effect of flow compressibility on the dynamics of populations. While a small compressibility can be induced by several physical mechanisms, like density mismatch or the finite size of microorganisms with respect to the fluid turbulence, its effect on the the carrying capacity of the ecosystem can be dramatic. We report, for the first time, how a small compressibility can produce a sizeable reduction in the carrying capacity, due to an integrated effect made possible by the long replication times of the organisms with respect to turbulent time scales. A full statistical quantification of the fluctuations of population concentration field leads to data collapse over a broad range in parameter space.

Rich recirculation patterns have been recently discovered in the electrically conducting flow subject to a local external magnetic termed "the magnetic obstacle" [Phys. Rev. Lett. 98 (2007), 144504]. This paper continues the study of magnetic obstacles and sheds new light on the core of the magnetic obstacle that develops between magnetic poles when the intensity of the external field is very large. A series of both 3D and 2D numerical simulations have been carried out, through which it is shown that the core of the magnetic obstacle is streamlined both by the upstream flow and by the induced cross stream electric currents, like a foreign insulated insertion placed inside the ordinary hydrodynamic flow. The closed streamlines of the mass flow resemble contour lines of electric potential, while closed streamlines of the electric current resemble contour lines of pressure. New recirculation patterns not reported before are found in the series of 2D simulations. These are composed of many (even number) vortices aligned along the spanwise line crossing the magnetic gap. The intensities of these vortices are shown to vanish toward to the center of the magnetic gap, confirming the general conclusion of 3D simulations that the core of the magnetic obstacle is frozen. The implications of these findings for the case of turbulent flow are discussed briefly. Comment: 14 pages, 9 figures, submitted to Journal of Turbulence

The results of an analysis of turbulent pipe flow based on a Karhunen-Loeve decomposition are presented. The turbulent flow is generated by a direct numerical simulation of the Navier-Stokes equations using a spectral element algorithm at a Reynolds number Re_\tau=150. This simulation yields a set of basis functions that captures 90% of the energy after 2,453 modes. The eigenfunctions are categorised into two classes and six subclasses based on their wavenumber and coherent vorticity structure. Of the total energy, 81% is in the propagating class, characterised by constant phase speeds; the remaining energy is found in the non propagating subclasses, the shear and roll modes. The four subclasses of the propagating modes are the wall, lift, asymmetric, and ring modes. The wall modes display coherent vorticity structures near the wall, the lift modes display coherent vorticity structures that lift away from the wall, the asymmetric modes break the symmetry about the axis, and the ring modes display rings of coherent vorticity. Together, the propagating modes form a wave packet, as found from a circular normal speed locus. The energy transfer mechanism in the flow is a four step process. The process begins with energy being transferred from mean flow to the shear modes, then to the roll modes. Energy is then transfer ed from the roll modes to the wall modes, and then eventually to the lift modes. The ring and asymmetric modes act as catalysts that aid in this four step energy transfer. Physically, this mechanism shows how the energy in the flow starts at the wall and then propagates into the outer layer. Comment: 28 pages, 20 figures. Updated with reviewer's comments / suggestions

The classical structure-function (SF) method in fully developed turbulence or for scaling processes in general is influenced by large-scale energetic structures, known as infrared effect. Therefore, the extracted scaling exponents $\zeta(n)$ might be biased due to this effect. In this paper, a detrended structure-function (DSF) method is proposed to extract scaling exponents by constraining the influence of large-scale structures. This is accomplished by removing a $1$st-order polynomial fitting within a window size $\ell$ before calculating the velocity increment. By doing so, the scales larger than $\ell$, i.e., $r\ge \ell$, are expected to be removed or constrained. The detrending process is equivalent to be a high-pass filter in physical domain. Meanwhile the intermittency nature is retained. We first validate the DSF method by using a synthesized fractional Brownian motion for mono-fractal processes and a lognormal process for multifractal random walk processes. The numerical results show comparable scaling exponents $\zeta(n)$ and singularity spectra $D(h)$ for the original SFs and DSFs. When applying the DSF to a turbulent velocity obtained from a high Reynolds number wind tunnel experiment with $Re_{\lambda}\simeq 720$, the 3rd-order DSF demonstrates a clear inertial range with $\mathcal{B}_3(\ell)\simeq 4/5\epsilon \ell$ on the range $10<\ell/\eta<1000$, corresponding to a wavenumber range $0.001<k\eta<0.1$. This inertial range is consistent with the one predicted by the Fourier power spectrum. The directly measured scaling exponents $\zeta(n)$ (resp. singularity spectrum $D(h)$) agree very well with a lognormal model with an intermittent parameter $\mu=0.33$. Due to large-scale effects, the results provided by the SFs are biased.

Statistics of kinetic energy and transfer localized in space and scale are examined by using the datasets of high-resolution direct numerical simulations of three-dimensional incompressible homogeneous isotropic turbulence with the Taylor micro-scale Reynolds number up to 732. We find that the standard deviation spectra of the local energy σe and that of energy transfer σt decrease with a representative wavenumber kα approximately as σe ∝ k − 5/3 α and σt ∝ k −1/2 α in the inertial subrange, respectively.

We extend the generalized Langevin model, originally developed for the Lagrangian fluid particle velocity in constant-density shear-driven turbulence, to variable-density (VD) pressure-gradient-driven flows. VD effects due to non-uniform mass concentrations (e.g. mixing of different species) are considered. In the extended model large density fluctuations leading to large differential fluid accelerations are accounted for. This is an essential ingredient to represent the strong coupling between the density and velocity fields in VD hydrodynamics driven by active scalar mixing. The small scale anisotropy, a fundamentally "non-Kolmogorovian" feature of pressure-gradient-driven flows, is captured by a tensorial stochastic diffusion term. The extension is so constructed that it reduces to the original Langevin model in the limit of constant density. We show that coupling a Lagrangian mass-density particle model to the proposed extended velocity equation results in a statistical representation of VD turbulence that has important benefits. Namely, the effects of the mass flux and the specific volume, both essential in the prediction of VD flows, are retained in closed form and require no explicit closure assumptions. The paper seeks to describe a theoretical framework necessary for subsequent applications. We derive the rigorous mathematical consequences of assuming a particular functional form of the stochastic momentum equation coupled to the stochastic density field in VD flows. A previous article discussed VD mixing and developed a stochastic Lagrangian model equation for the mass-density. Second in the series, this article develops the momentum equation for VD hydrodynamics. A third, forthcoming paper will combine these ideas on mixing and hydrodynamics into a comprehensive framework: it will specify a model for the coupled problem and validate it by computing a Rayleigh-Taylor flow.

We present the results of Direct Numerical Simulations (DNS) of turbulent flows seeded with millions of passive inertial particles. The maximum Taylor's Reynolds number is around 200. We consider particles much heavier than the carrier flow in the limit when the Stokes drag force dominates their dynamical evolution. We discuss both the transient and the stationary regimes. In the transient regime, we study the growt of inhomogeneities in the particle spatial distribution driven by the preferential concentration out of intense vortex filaments. In the stationary regime, we study the acceleration fluctuations as a function of the Stokes number in the range [0.16:3.3]. We also compare our results with those of pure fluid tracers (St=0) and we find a critical behavior of inertia for small Stokes values. Starting from the pure monodisperse statistics we also characterize polydisperse suspensions with a given mean Stokes. Comment: 13 pages, 10 figures, 2 tables

We investigate the dynamo problem in the limit of small magnetic Prandtl number ($\Pm$) using a shell model of magnetohydrodynamic turbulence. The model is designed to satisfy conservation laws of total energy, cross helicity and magnetic helicity in the limit of inviscid fluid and null magnetic diffusivity. The forcing is chosen to have a constant injection rate of energy and no injection of kinetic helicity nor cross helicity. We find that the value of the critical magnetic Reynolds number ($\Rm$) saturates in the limit of small $\Pm$. Above the dynamo threshold we study the saturated regime versus $\Rm$ and $\Pm$. In the case of equipartition, we find Kolmogorov spectra for both kinetic and magnetic energy except for wave numbers just below the resistive scale. Finally the ratio of both dissipation scales (viscous to resistive) evolves as $\Pm^{-3/4}$ for $\Pm < 1$

Exact structure function equations are an efficient means of obtaining asymptotic laws such as inertial range laws, as well as all measurable effects of inhomogeneity and anisotropy that cause deviations from such laws. "Exact" means that the equations are obtained from the Navier-Stokes equation or other hydrodynamic equations without any approximation. A pragmatic definition of local homogeneity lies within the exact equations because terms that explicitly depend on the rate of change of measurement location appear within the exact equations; an analogous statement is true for local stationarity. An exact definition of averaging operations is required for the exact equations. Careful derivations of several inertial range laws have appeared in the literature recently in the form of theorems. These theorems give the relationships of the energy dissipation rate to the structure function of acceleration increment multiplied by velocity increment and to both the trace of and the components of the third-order velocity structure functions. These laws are efficiently derived from the exact velocity structure function equations. In some respects, the results obtained herein differ from the previous theorems. The acceleration-velocity structure function is useful for obtaining the energy dissipation rate in particle tracking experiments provided that the effects of inhomogeneity are estimated by means of displacing the measurement location. Comment: accepted by Journal of Turbulence

The statistical properties of fluid particles transported by a fully developed turbulent flow are investigated by means of high resolution direct numerical simulations. Single trajectory statistics is investigated in a time range spanning more than three decades, from less than a tenth of the Kolmogorov timescale up to one large-eddy turnover time. Acceleration and velocity statistics show a neat quantitative agreement with recent experimental results. Trapping effects in vortex filaments give rise to enhanced small-scale intermittency on Lagrangian observables.

We report results on the geometrical statistics of the vorticity vector obtained from experiments in electromagnetically forced rotating turbulence. A range of rotation rates $\Omega$ is considered, from non-rotating to rapidly rotating turbulence with a maximum background rotation rate of $\Omega=5$ rad/s (with Rossby number much smaller than unity). Typically, in our experiments ${\rm{Re}}_{\lambda}\approx 100$. The measurement volume is located in the centre of the fluid container above the bottom boundary layer, where the turbulent flow can be considered locally statistically isotropic and horizontally homogeneous for the non-rotating case, see van Bokhoven et al., Phys. Fluids 21, 096601 (2009). Based on the full set of velocity derivatives, measured in a Lagrangian way by 3D Particle Tracking Velocimetry, we have been able to quantify statistically the effect of system rotation on several flow properties. The experimental results show how the turbulence evolves from almost isotropic 3D turbulence ($\Omega\lesssim 0.2$ rad/s) to quasi-2D turbulence ($\Omega\approx 5.0$ rad/s) and how this is reflected by several statistical quantities. In particular, we have studied the orientation of the vorticity vector with respect to the three eigenvectors of the local strain rate tensor and with respect to the vortex stretching vector. Additionally, we have quantified the role of system rotation on the self-amplification terms of the enstrophy and strain rate equations and the direct contribution of the background rotation on these evolution equations. The main effect is the strong reduction of extreme events and related (strong) reduction of the skewness of PDFs of several quantities such as, for example, the intermediate eigenvalue of the strain rate tensor and the enstrophy self-amplification term.

Among existing subgrid scale models for large-eddy simulation (LES) some are time-reversible in the sense that the dynamics evolve backwards in time after a transformation $\bm u \rightarrow -\bm u$ at every point in space. In practice, reversible subgrid models reduce the numerical stability of the simulations since the effect of the subgrid scales is no longer strictly dissipative. This lack of stability constitutes often a criterion to reject this kind of models. The aim of this paper is to examine whether time-reversibility can constitute a criterion that a subgrid model has to fulfill, or has not to. Thereto we investigate by direct numerical simulation the time-dependence of the kinetic energy of the resolved scales when the velocity is reversed in all or part of the lengthscales of the turbulent flow. These results are compared with results from existing LES subgrid models. It is argued that the criterion of time-reversibility to assess subgrid models is incompatible with the main underlying assumption of LES.

We demonstrate that the spacetime statistics of the birth of turbulent spots in boundary layers can be reconstructed qualitatively from the average behaviour of macroscopic measures in the transition zone. The conclusion in [11. Vinod , N. and Govindarajan , R. 2004. Pattern of breakdown of laminar flow into turbulent spots. Physical Review Letters, 93: 114501 [CrossRef], [PubMed], [Web of Science ®]View all references] that there exists a connection between the patterns in laminar instability and the birth of turbulent spots is strengthened. We examine why the relationship between instability and transition to turbulence is manifest in some cases and appears to be totally absent in others. Novel cellular automaton-type simulations of the transition zone are conducted, and the pattern of spot birth is obtained from secondary instability analysis. The validity of the hypothesis of concentrated breakdown, according to which most turbulent spots originate at a particular streamwise location, is assessed. The predictions made lend themselves to straightforward experimental verification.

A new computational framework for the simulation of turbulent flow through complex objects and along irregular boundaries is presented. This is motivated by the application of metal foams in compact heat-transfer devices, or as catalyst substrates in process-engineering. The flow-consequences of such complicated objects are incorporated by adding explicit multiscale forcing to the Navier-Stokes equations. The forcing represents the simultaneous agitation of a wide spectrum of length-scales when flow passes through the complex object. It is found that a considerable modulation of the traditional energy cascading can be introduced with a specific forcing strategy. In spectral space, forcing yields strongly localized deviations from the common Kolmogorov scaling law, directly associated with the explicitly forced scales. In addition, the accumulated effect of forcing induces a significant non-local alteration of the kinetic energy including the spectrum for the large scales. Consequently, a manipulation of turbulent flow can be achieved over an extended range, well beyond the directly forced scales. The turbulent mixing of a passive scalar field is also investigated, in order to quantify the physical-space modifications of transport processes in multiscale forced turbulence. The surface-area and wrinkling of level-sets of the scalar field are monitored as measures of the influence of explicit forcing on the local and global mixing efficiency.

This contribution covers the topics presented by the authors at the {\it `Fundamental Problems of Turbulence, 50 Years after the Marseille Conference 1961"} meeting that took place in Marseille in 2011. It focuses on some of the mathematical approaches to fluid dynamics and turbulence. This contribution does not pretend to cover or answer, as the reader may discover, the fundamental questions in turbulence, however, it aims toward presenting some of the most recent advances in attacking these questions using rigorous mathematical tools. Moreover, we consider that the proofs of the mathematical statements (concerning, for instance, finite time regularity, weak solutions and vanishing viscosity) may contain information as relevant, to the understanding of the underlying problem, as the statements themselves.

The paper presents a two-dimensional immersed interface technique for the vortex-in-cell (VIC) method for the simulation of flows past complex geometries. The particle-mesh VIC algorithm is augmented by a local particle-particle (PP) correction term in a particle-particle particle-mesh (PPPM) context to resolve sub-grid scales incurred by the presence of the immersed interface. The PP correction furthermore allows mesh and particle resolution to be disjoined by explicitly resolving sub-grid scales on the particles. This PPPM algorithm uses an influence matrix technique to annihilate the anisotropic subgrid scales and an exact PP correction term.

The surge of turbulence research in the first half of the twentieth century is reviewed with the focus on the institutional environment that provided an umbrella for the activities in this field. Before the Second World War, the International Congresses for Applied Mechanics served as the main stage for presenting research results on turbulence. After the War, the International Union of Theoretical and Applied Mechanics (IUTAM), organizations such as the Division of Fluid Dynamics of the American Physical Society and new journals (the Journal of Fluid Mechanics and The Physics of Fluids) added to the institutional framework for turbulence activities and publications. The development of these activities is illustrated with examples from the correspondence of some of the involved actors (such as George K. Batchelor and Francois N. Frenkiel, the editors of the new journals; both were also involved with the organization of the Marseille events in 1961).

Although much progress has been made in pipe and channel flow turbulence over the past 50 years since the Turbulence Colloquium Marseille 1961 (TCM 1961), these simple wall-bounded turbulent flows have continuously baffled us, refusing to reveal their true and complete nature. In a broader sense, almost everything we thought we knew about wall-bounded turbulent flows, including the scaling laws that appear in textbooks, has been the subject of further scrutiny. An overview of the progress that has been made since TCM 1961 is presented, followed by discussions on unresolved issues that would require further investigations.

We have performed large-eddy simulation with subgrid scale (LES-SGS) stretched-vortex model of an inclined sonic jet into a supersonic crossflow at Mach 3.6. The main flow features generated by the gas-dynamic interactions of the jet with the supersonic crossflow, such as barrel shock, shear layer, and counter-rotating vortex pair, are numerically captured by the employed LES-SGS. The transition and spatial development of the jet into a supersonic crossflow have been shown to be strongly dependent on the inflow conditions of the crossflow. This result indicates that correct turbulent inflow conditions are necessary to predict the main flow characteristics, dispersion and mixing of a gaseous jet in a supersonic, turbulent crossflow using LES-SGS. This work presents a methodology for the generation of realistic synthetic turbulent inflow conditions for LES of spatially developing, supersonic, turbulent, wall-bounded flows. The methodology is applied to the study of a supersonic turbulent flow over a flat wall interacting with an inclined jet. The effects of inflow conditions on the spatial development of the inclined jet are discussed, and then the results are compared with the available experimental data. Also, the dominant vortical structures generated by the jet/turbulent boundary layer interaction are identified as sheets, tilted tubes and discontinuous rings, and a visualization of their spatiotemporal development is provided. The identified vortical structures are shown to be enveloped by the helium mass-fraction isosurface, thus showing the important role of those structures in the dispersion of a gaseous jet in a supersonic crossflow.

The high-Reynolds-number channel flow is simulated by numerical approach at coarse resolution, in which the instantaneous acceleration is decomposed into filtered and subgrid parts, and then both components are modeled. The filtered acceleration is modeled in the framework of the large-eddy simulation approach. The model for the subgrid acceleration is based on two stochastic processes. The first is for its norm and is based on statistical universalities in fragmentation under scaling symmetry, providing correlation of subgrid forcing across the channel. The second is for its orientation and is based on the Brownian motion on a unit sphere in order to represent a stochastic relaxation toward full isotropy away from the wall. Two main parameters of the stochastic process include the Reynolds number based on the friction velocity and the channel half-width. In order to assess the capability of the model proposed, the paper illustrates its application versus recent high-Reynolds-number direct numerical simulations, including direct numerical simulations performed in this paper.

A new formulation of scale-adaptive simulation (SAS) approach for complex wall-bounded shear flows is presented. This approach makes use of a unique modelling representation and requires moderate computational costs. Based on the Rotta original transport equation for turbulence integral length scale, the suggested model is able to resolve unsteady turbulent structures with sufficient spatial resolution. Similar to the classical SAS-turbulence model (SST–SAS) proposed by Menter et al. [55. F.R. Menter and Y. Egorov, The scale adaptive simulation method for unsteady turbulent flow prediction, Part 1: Theory and Model Description, Flow Turbul. Combust 85 (2010), pp. 113–138. DOI: 10.1007/s10494-010-9264-5.[CrossRef], [Web of Science ®]View all references] that represents an improvement of the proven turbulence model for unsteady calculations, the new model appears advantageous where classical LES (large eddy simulation) or hybrid LES–RANS models are too expensive. However, in its zonal formulation, the new model provides accurate predictions in (1) so-called stable flows (e.g. channel flow) in which the classical SAS () model will not be able to switch from Reynolds-averaged Navier–Stokes (RANS) to scale-resolving simulations without an explicit introduction of synthetic turbulence, (2) flows with relatively weak instabilities also challenging for the classical SAS-type models (e.g. swirling flow with sudden expansion). Such improvements result from the appropriately treating additional source terms in modified transport equation of dissipation rate. For demonstration, comparisons with experimental data and direct numerical simulation are provided.

Direct numerical simulation (DNS) of turbulent isothermal-wall bounded flow subjected to favourable and adverse pressure gradient (FPG, APG) at low Mach number is investigated. The FPG/APG is obtained by imposing a concave/convex curvature on the top wall of a plane channel. The flows on the bottom and top walls are a tripped turbulent and laminar boundary layers, respectively. It is observed that the flow reaches equilibrium in the FPG region and the first and second order statistics are strongly influenced by the pressure gradients. For FPG/APG regions very near the bottom plane wall, the correlations between the streamwise pressure gradient and the spanwise vorticity flux and between the spanwise pressure gradient and the streamwise vorticity flux in the wall-normal direction are high on the wall and quickly drops to a negligible value within the viscous sublayer. Related flow physics and linkage to energy balance transport is discussed.

A direct numerical simulation (DNS) of a turbulent channel flow with a lower curved wall is performed at Reynolds number Reτ ≃ 617 at inlet. This adverse-pressure gradient turbulent flow is characterized by strong peaks of turbulent kinetic energy at both walls, as a consequence of the breakdown of more organized flow structures. To elucidate the underlying instability scenario, low-speed streak structures are extracted from the turbulent flow field and base flows formed with conditional streak averages, superimposing the mean streamwise velocity profile, are used for linear stability analyses. The size and shape of the counter-rotating streamwise vortices associated with the instability modes are shown to be reminiscent of the coherent vortices emerging from the streak skeletons in the direct numerical simulation. The distance of the streak's centre from the wall is used as a criterion for the conditional averages and the corresponding streak base flows are characterised by more or less pronounced contours of inflection points in the averaging windows normal to the wall. It is shown that the strength of instability of the streak base flows can be inferred from a simplified 1D stability analysis, using local inflectional profiles at different spanwise locations.

Our study is focused on a phenomenon often encountered in flow carrying pipes, since flow instabilities caused by geometric features may generate acoustic signals and, thereafter, interact with these signals in such a way that powerful pure tones are produced. A modern example is found in the so-called ‘singing risers’, or the gas pipes connecting gas production platforms to the transport network. But the flow generated resonance in a fully corrugated circular pipe may be silenced by the addition of relatively low frequency flow oscillations induced by an acoustic generator. Experiments reported here, aimed at investigating in more detail the coupling between the flow in the pipe, the acoustically generated flow oscillations and the emitted resulting noise, are performed in a specifically designed facility. A rectangular transparent channel using glass walls enables us to use optical techniques to describe in detail the flow field in the corrugation vicinity, in addition to more standard hot-wire anemometry and acoustic pressure measurements with microphones, with and without the acoustically generated low-frequency oscillations.

Three-dimensional direct numerical simulation (DNS) of hydrogen-air turbulent plane jet premixed flames, which are composed of jet with unburnt mixture gas and surrounding burnt gas for flame holding, has been conducted for two cases of mean streamwise velocity of the jet, 100 m/s and 350 m/s. Fully-developed homogeneous isotropic turbulence is superimposed on the mean flow. A detailed kinetic mechanism including 12 reactive species and 27 elementary reactions is considered. Eddy structures which have large-scale in space are produced for both cases, whereas the mechanism of the eddy formation depends on the inlet velocity. Although combustion condition of the present DNS with inlet velocity 100 m/s is classified into corrugated flamelets regime, unburnt mixture islands frequently appear behind the main flame body. The creation of these islands is closely related to the fine-scale eddies in the unburnt gas and the separated unburnt mixture contributes to increase of heat release rate and turbulent burning velocity. Effects of shear and turbulent intensity on characteristics of heat release rate and tangential strain rate of the jet flames are investigated statistically.

Direct numerical simulations (DNSs) of open-channel flows in the presence of an air-water interface were performed to examine the effects of interface deformation on the turbulence structures. In the water-driven turbulence, flows characterized by either of two Froude numbers (Fr=0.2 and 0.8) were examined and compared. A coupled level-set and volume-of-fluid (CLSVOF) method was employed to track the interface. The mean and root-mean-square of the air-water interface elevation varied rapidly with the spanwise distance as Fr increased. The air that contacted the water was entrained into the turbulent flow. At high Fr, all turbulent normal stresses on the air side of the interface were high near the sidewall. Moreover, all terms of the Reynolds shear stress were intensified at the mixed-boundary corner on the air side of the interface. At high Fr, all terms except for the pressure-strain term in the budget of the turbulent kinetic energy increased at the mixed-boundary corner on the air side of the interface. Two-point correlations between the streamwise vortex and the velocity fluctuations provided structural information about the near-wall streaky structures and the inner secondary flows in the cross-stream plane. Linear stochastic estimates of the conditionally averaged flow field showed that the inner secondary flow consisted of not only in-plane velocity components but also streamwise velocity components.

We analyse the performance of the explicit algebraic subgrid-scale (SGS) stress model (EASSM) in large eddy simulation (LES) of plane channel flow and the flow in a channel with streamwise periodic hill-shaped constrictions (periodic hill flow) which induce separation. The LESs are performed with the Code_Saturne which is an unstructured collocated finite volume solver with a second-order spatial discretisation suitable for LES of incompressible flow in complex geometries. At first, performance of the EASSM in LES of plane channel flow at two different resolutions using the Code_Saturne and a pseudo-spectral method is analysed. It is observed that the EASSM predictions of the mean velocity and Reynolds stresses are more accurate than the conventional dynamic Smagorinsky model (DSM). The results with the pseudo-spectral method were, in general, more accurate. In the second step, LES with the EASSM of flow separation in the periodic hill flow is compared to LES with the DSM, no SGS model and a highly resolved LES data using the DSM. Results show that the mean velocity profiles, the friction and pressure coefficients, the length and shape of the recirculation bubble, as well as the Reynolds stresses are considerably better predicted by the EASSM than the DSM and the no SGS model simulations. It was also observed that in some parts of the domain, the resolved strain-rate and SGS shear stress have the same sign. The DSM cannot produce a correct SGS stress in this case, in contrast to the EASSM.

Wind farm-atmosphere interaction is complicated by the effect of turbine array configuration on momentum, scalar and kinetic energy fluxes. Wind turbine arrays are often arranged in rectilinear grids and, depending on the prevailing wind direction, may be perfectly aligned or perfectly staggered. The two extreme configurations are end members with a spectrum of infinite possible layouts. A wind farm of finite length may be modeled as an added roughness or as a canopy in large-scale weather and climate models. However, it is not clear which analogy is physically more appropriate. Also, surface scalar flux, including heat, moisture and trace gas (e.g. CO2), are affected by wind farms, and need to be properly parameterized in large-scale models. Experiments involving model wind farms, in aligned and staggered configurations, were conducted in a thermally controlled boundary-layer wind tunnel. Measurements of the turbulent flow were made using a custom x-wire/cold-wire probe. Particular focus was placed on studying the effect of wind farm layout on flow adjustment, momentum and scalar fluxes, and turbulent kinetic energy distribution. The flow statistics exhibit similar turbulent transport properties to those of canopy flows, but retain some characteristic surface-layer properties in a limited region above the wind farms as well. The initial wake growth over columns of turbines is faster in the aligned wind farm. However, the overall wake adjusts within and grows more rapidly over the staggered farm. The flow equilibrates faster and the overall momentum absorption is higher for the staggered compared to the aligned farm, which is consistent with canopy scaling and leads to a larger effective roughness. Surface heat flux is found to be altered by the wind farms compared to the boundary-layer flow without turbines, with lower flux measured for the staggered wind farm.

Strong anisotropy in turbulent flows may be induced by body forces, Coriolis, buoyancy, Lorentz, and/or by large-scale gradients. These effects combined to the redistribution pressure terms are first identified by an angle dependence of the wave vector k in Fourier space, the directionality. The resulting anisotropic structure is not taken into account in classical phenomenological theory, using essentially 'isotropised' dimensional analysis. Besides, it is generally hidden in practical engineering models by means of tuned constants, which may vary if the flow changes of nature. In this paper, different examples of anisotropic turbulence are revisited and compared to each other in order to shed light on fundamental aspects of this specific turbulence. To begin with, flows without energy production like rotating turbulence are considered. In this case, isotropy is broken by mean of third-order correlations in the equations. These correlations quantify the interscale energy transfer, and must be investigated at three-point, or triad by triad in Fourier space. This allows to account for the role of typical anisotropic frequency 2 Omega cos theta(k), with theta(k) the angle of the wave vector to the axis of rotation, and to simultaneously restore the role of phase coherence. We pursue the discussion with a second flow case, with production, quasi-static magnetohydrodynamics. This illustrates turbulence forced towards two-dimensional structure by an explicit Ohmic dissipation term. Linear dynamics displays an angle (called Moreau, or Shebalin) capable of reflecting the basic anisotropy in models as simple as K - epsilon. In the final phase of transition towards 2D structure, however, dynamics are essentially driven by third-order velocity correlations, and both successive linear and nonlinear phases yield counter-intuitive anisotropic results. The last case considered here is the turbulent mixing induced by a Rayleigh-Taylor instability. It is shown that anisotropy plays a central role in the dynamics of the mixing zone by means of an angular dimensionality parameter similar to the Moreau angle but for the density field, and appearing in a global model of buoyancy-drag equation.

A necessary condition for the accurate prediction of turbulent flows using large-eddy simulation (LES) is the correct representation of energy transfer between the different scales of turbulence in the LES. For scalar turbulence, transfer of energy between turbulent length scales is described by a transport equation for the second moment of the scalar increment. For homogeneous isotropic turbulence, the underlying equation is the well-known Yaglom equation. In the present work, we study the turbulent mixing of a passive scalar with an imposed mean gradient by homogeneous isotropic turbulence. Both direct numerical simulations (DNS) and LES are performed for this configuration at various Schmidt numbers, ranging from 0.11 to 5.56. As the assumptions made in the derivation of the Yaglom equation are violated for the case considered here, a generalised Yaglom equation accounting for anisotropic effects, induced by the mean gradient, is derived in this work. This equation can be interpreted as a scale-by-scale energy-budget equation, as it relates at a certain scale r terms representing the production, turbulent transport, diffusive transport and dissipation of scalar energy. The equation is evaluated for the conducted DNS, followed by a discussion of physical effects present at different scales for various Schmidt numbers. For an analysis of the energy transfer in LES, a generalised Yaglom equation for the second moment of the filtered scalar increment is derived. In this equation, new terms appear due to the interaction between resolved and unresolved scales. In an a-priori test, this filtered energy-budget equation is evaluated by means of explicitly filtered DNS data. In addition, LES calculations of the same configuration are performed, and the energy budget as well as the different terms are thereby analysed in an a-posteriori test. It is shown that LES using an eddy viscosity model is able to fulfil the generalised filtered Yaglom equation for the present configuration. Further, the dependence of the terms appearing in the filtered energy-budget equation on varying Schmidt numbers is discussed.

The statistics of the velocity and passive scalar gradients in rotating turbulence are studied using Lagrangian stochastic models. Models for the velocity gradients are derived generalizing the approach proposed by L. Chevillard and C. Meneveau [Phys. Rev. Lett. 97, No. 17, Article ID 174501 (2006)], whereas the scalar gradients are described using the model proposed by M. Gonzalez [Phys. Fluids 21, No. 5, Paper No. 055104, 9 p. (2009; Zbl 1183.76219)]. The non-Gaussian and anisotropic statistics of the gradients are analyzed, and compared with available results in the literature. It is found that the models reproduce the observation that rotation tends to reduce small-scale intermittency for both velocity and scalar gradients. The models predict the skewness of the transverse velocity gradient components in the perpendicular plane and its non-monotonic dependence on the rotation rate. The models also reproduce the anisotropy in the scalar gradient at intermediate Rossby numbers. Furthermore, we show that the anisotropy is reached at an intermediate rotation rate, and the maximum coincides with a transition in the relative importance of the self- and cross-production terms for the scalar gradient.

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