Velocity Correlation Tensor Research Articles

Superstatistical Wind Fields from Pointwise Atmospheric Turbulence Measurements

Accurate models of turbulent wind fields have become increasingly important in the atmospheric sciences, e.g., for the determination of spatiotemporal correlations in wind parks, the estimation of individual loads on turbine rotor and blades, or the modeling of particle-turbulence interaction in atmospheric clouds or pollutant distributions in urban settings. Because of the difficult task of resolving the fields across a broad range of scales, one oftentimes has to invoke stochastic wind field models that fulfill specific, empirically observed, properties. Whereas commonly used Gaussian random field models solely control second-order statistics (i.e., velocity correlation tensors or kinetic energy spectra), we explicitly show that our extended model emulates the effects of higher-order statistics as well. Most importantly, the empirically observed phenomenon of small-scale intermittency, which can be regarded as one of the key features of atmospheric turbulent flows, is reproduced with a very high level of accuracy and at considerably low computational cost. Our method is based on a multipoint statistical description of turbulent velocity fields that consists of a superposition of multivariate Gaussian statistics with fluctuating covariances. We propose a new and efficient sampling algorithm for this Gaussian scale mixture and demonstrate how such “superstatistical” wind fields can be constrained on a certain number of real-world measurement data points from a meteorological mast array.Received 12 April 2022Revised 26 July 2022Accepted 22 August 2022DOI:https://doi.org/10.1103/PRXEnergy.1.023006Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasAtmospheric scienceBoundary layersShear flowsStochastic processesStructure & turbulence of boundary layersSustainabilityTurbulenceWind energyEnergy Science & TechnologyNonlinear DynamicsFluid DynamicsStatistical Physics

Surrogate modelling of wind fields from point-wise atmospheric turbulence measurements

We present an advanced model for the generation of synthetic wind fields that can be understood as an extension of the well-known Mann model. In contrast to such Gaussian random field models which control second-order statistics (i.e., velocity correlation tensors or spectra), we demonstrate that our extended model incorporates the effects of higherorder statistics as well. In particular, the empirically observed phenomenon of small-scale intermittency, a key feature of atmospheric turbulent flows, can be reproduced with high accuracy and at considerably low computational cost. Our method is based on a recently developed multipoint statistical description of a turbulent velocity field [J. Friedrich et al., J. Phys. Complex. 2 045006 (2021)] and consists of a superposition of multivariate Gaussian statistics with fluctuating covariances. Furthermore, we explicitly show how such superstatistical Mann fields can be constraint on a certain number of point-wise measurement data. We give an outlook on the relevance of such surrogate wind fields in the context of fatigue loads on wind turbines.

Geometry of two-point velocity correlation tensor of homogeneous isotropic turbulence with helicity in sol-manifold coordinates

In this paper, a geometric model of homogeneous isotropic turbulence with helicity in three dimensions using a family of the quadratic forms of Riemannian metrics generated by the two-point correlation tensor field is investigated. We show that this metric induces a sol-metric on infinite cylinder R×T2 as a sol-manifold. Using the corresponded Hamiltonian equation of the geodesic flow of our sol metric on the infinite cylinder R×T2, we find three functionally independent first integrals and then the geodesic flow is integrable. Moreover, we consider how the terms of this family of quadratic forms influence the length scales of turbulent motion. The conformal invariance of the corresponding manifolds on our Reimanian metric is proved using the symmetry group of the system of von Kármán–Howarth and Chkhetiani–Kurien equations in the two limits of finite and infinite Reynolds numbers. This property corresponds to the conservation of angles between the intersecting curves located on the manifold under the action of these scaling groups.

Re τ scaling of POD modes in plane channel flow

This article studies how the proper orthogonal decomposition eigenvectors and eigenvalues of the two-point velocity correlation tensor scale with Reτ in turbulent channel flows. To this effect, the two-point correlation tensor is computed from velocity fields extracted from the Direct Numerical Simulations (DNS) of plane channel flows at Reτ = 547 and Reτ = 934. The analysis reveals that the eigenvalues exhibit a high degree of scaling with Reτ, across a very wide range of streamwise and spanwise wavenumbers. The eigenvectors also show near complete independence from Reτ, as long as the wall-normal lengthscales larger than the channel height are removed. The poor Reτ scaling of turbulent structures larger than the channel height is well documented in the literature, and thus one would not expect eigenvectors corresponding to these scales to exhibit favorable Reτ scaling. Two-point velocity correlations and their eigenvectors are also computed using Large Eddy Simulations (LES) at Reτ = 1000 and compared to the results of the DNS at Reτ = 934. Both the correlations and eigenvectors matched very well between LES and DNS.

Generation of anisotropy in turbulent flows subjected to rapid distortion.

A computational tool for the anisotropic time-evolution of the spectral velocity correlation tensor is presented. We operate in the linear, rapid distortion limit of the mean-field-coupled equations. Each term of the equations is written in the form of an expansion to arbitrary order in the basis of irreducible representations of the SO(3) symmetry group. The computational algorithm for this calculation solves a system of coupled equations for the scalar weights of each generated anisotropic mode. The analysis demonstrates that rapid distortion rapidly but systematically generates higher-order anisotropic modes. To maintain a tractable computation, the maximum number of rotational modes to be used in a given calculation is specified a priori. The computed Reynolds stress converges to the theoretical result derived by Batchelor and Proudman [Quart. J. Mech. Appl. Math. 7, 83 (1954)QJMMAV0033-561410.1093/qjmam/7.1.83] if a sufficiently large maximum number of rotational modes is utilized; more modes are required to recover the solution at later times. The emergence and evolution of the underlying multidimensional space of functions is presented here using a 64-mode calculation. Alternative implications for modeling strategies are discussed.

Fine-Scale Turbulent Noise Predictions from Non-Axisymmetric Jets

In this work a model for fine-scale turbulent noise predictions from non-axisymmetric, heated and unheated jets is developed. This model, called GENO, improves upon the previous noise prediction methods JeNo and MGBK by including the effects of mean flow refraction from non-circular geometry, as well as a more complete model for the self, shear, and enthalpy noise sources. The Green's function for non-axisymmetric flows is determined by solving the Lilley-Goldstein equation in terms of coupled azimuthal modes. Additionally, the noise source model evaluates the fourth order velocity correlation tensor I ijkl for the general case and the relationship between non-zero components is found. This is combined with an empirical noise source model for heated jets. Acoustic results using GENO are shown for aspect ratio 2:1 and 4:1 rectangular jets, under unheated (SP7) and heated (SP46) conditions, and the comparisons agree favorably with experimental data.

Model-free simulation approach to molecular diffusion tensors

In the present work, we propose a simple model-free approach for the computation of molecular diffusion tensors from molecular dynamics trajectories. The method uses a rigid body trajectory of the molecule under consideration, which is constructed a posteriori by an accumulation of quaternion-based superposition fits of consecutive conformations. From the rigid body trajectory, we compute the translational and angular velocities of the molecule and by integration of the latter also the corresponding angular trajectory. All quantities can be referred to the laboratory frame and a molecule-fixed frame. The 6 × 6 diffusion tensor is computed from the asymptotic slope of the tensorial mean square displacement and, for comparison, also from the Kubo integral of the velocity correlation tensor. The method is illustrated for two simple model systems - a water molecule and a lysozyme molecule in bulk water. We give estimations of the statistical accuracy of the calculations.

Eddy Shape, Orientation, Propagation, and Mean Flow Feedback in Western Boundary Current Jets

AbstractThis manuscript revisits a study of eddy–mean flow interactions in an idealized model of a western boundary current extension jet using properties of the horizontal velocity correlation tensor to diagnose characteristics of average eddy shape, orientation, propagation, and mean flow feedback. These eddy characteristics are then used to provide a new description of the eddy–mean flow interactions observed in terms of different ingredients of the eddy motion. The diagnostics show patterns in average eddy shape, orientation, and propagation that are consistent with the signatures of jet instability in the upstream region and wave radiation in the downstream region. Together they give a feedback onto the mean flow that gives the downstream character of the jet and drives the jet's recirculation gyres. A breakdown of the eddy forcing into contributions from individual terms confirms the expected role of cross-jet gradients in meridional eddy tilt in stabilizing the jet to its barotropic instability; however, it also reveals important roles played by the along-jet evolution of eddy zonal–meridional elongation. It is the mean flow forcing derived from these patterns that acts to strengthen and extend the jet downstream and forces the time-mean recirculation gyres. This understanding of the dependence of mean flow forcing on eddy structural properties suggests that failure to adequately resolve eddy elongation could underlie the weakened jet strength, extent, and changed recirculation structure seen in this idealized model for reduced spatial resolutions. Further, it may suggest new ideas for the parameterization of this forcing.

Perturbative evaluation of universal numbers in homogeneous shear turbulence

We employ Leslie's perturbation scheme coupled with renormalization group calculations to obtain the anisotropic velocity correlation tensor in the inertial range of homogeneous shear turbulence. Hence we evaluate two universal numbers associated with the anisotropic part of the equal-time correlation tensor in the leading order of the perturbative scheme. Our theoretical results for these universal numbers are found to be in fairly good agreement with experimental values as well as estimates coming from direct numerical simulations.

The Extended Symmetry Lie Algebra and the Asymptotic Expansion of the Transversal Correlation Function for the Isotropic Turbulence

The extended symmetry of the functional of length determined in an affine spaceK3of the correlation vectors for homogeneous isotropic turbulence is studied. The two-point velocity-correlation tensor field (parametrized by the time variablet) of the velocity fluctuations is used to equip this space by a family of the pseudo-Riemannian metricsdl2(t)(Grebenev and Oberlack (2011)). First, we observe the results obtained by Grebenev and Oberlack (2011) and Grebenev et al. (2012) about a geometry of the correlation spaceK3and expose the Lie algebra associated with the equivalence transformation of the above-mentioned functional for the quadratic formdlD22(t)generated bydl2(t)which is similar to the Lie algebra constructed by Grebenev et al. (2012). Then, using the properties of this Lie algebra, we show that there exists a nontrivial central extension wherein the central charge is defined by the same bilinear skew-symmetric formcas for the Witt algebra which measures the number of internal degrees of freedom of the system. For the applications in turbulence, as the main result, we establish the asymptotic expansion of the transversal correlation function for large correlation distances in the frame ofdlD22(t).

Infinite dimensional Lie algebra associated with conformal transformations of the two-point velocity correlation tensor from isotropic turbulence

We deal with homogeneous isotropic turbulence and use the two-point velocity correlation tensor field (parametrized by the time variable t) of the velocity fluctuations to equip an affine space K3 of the correlation vectors by a family of metrics. It was shown in Grebenev and Oberlack (J Nonlinear Math Phys 18:109–120, 2011) that a special form of this tensor field generates the so-called semi-reducible pseudo-Riemannian metrics ds2(t) in K3. This construction presents the template for embedding the couple (K3, ds2(t)) into the Euclidean space \({\mathbb{R}^3}\) with the standard metric. This allows to introduce into the consideration the function of length between the fluid particles, and the accompanying important problem to address is to find out which transformations leave the statistic of length to be invariant that presents a basic interest of the paper. Also we classify the geometry of the particles configuration at least locally for a positive Gaussian curvature of this configuration and comment the case of a negative Gaussian curvature.

A second-order moment method applied to gas-solid risers

Second-order moment method of particles is proposed on the basis of the kinetic theory of granular flow. Closure equations for the third-order velocity moments are presented to account for the increase of the probability of collisions of particles on the basis of the elementary kinetic theory and order of magnitude analysis. The boundary conditions for the set of equations describing flow of particles are proposed with the consideration of the momentum exchange by collisions between the wall and the particles. The distributions of velocity, concentration and moments of particles are predicted. Simulated results are compared with experimental data measured by Tartan and Gidaspow and Bhusarapu et al. in risers, and Tsuji et al. in a vertical pipe. The effects of the closure equations for the third-order velocity moments and the fluid-particle velocity correlation tensor on flow behavior of particles are analyzed. © 2012 American Institute of Chemical Engineers AIChE J, 2012

A geometry of the correlation space and a nonlocal degenerate parabolic equation from isotropic turbulence

AbstractConsidering the metric tensor ds2(t), associated with the two‐point velocity correlation tensor field (parametrized by the time variable t) in the space 𝒦3of correlation vectors, at the first part of the paper we construct the Lagrangian system (Mt,ds2(t)) in the extended space 𝒦3 × R+ for homogeneous isotropic turbulence. This allows to introduce into the consideration common concept and technics of Lagrangian mechanics for the application in turbulence. Dynamics in time of (Mt,ds2(t)) (a singled out fluid volume equipped with a family of pseudo‐Riemannian metrics) is described in the frame of the geometry generated by the 1‐parameter family of metrics ds2(t) whose components are the correlation functions that evolve according to the von Kármán‐Howarth equation. This is the first step one needs to get in a future analysis the physical realization of the evolution of this volume. It means that we have to construct isometric embedding of the manifold Mt equipped with metric ds2(t) into R3 with the Euclidean metric. In order to specify the correlation functions, at the second part of this paper we study in details an initial‐boundary value problem to the closure model [19,26] for the von Kármán‐Howarth equation in the case of large Reynolds numbers limit.

GEOMETRIC REALIZATION OF THE TWO-POINT VELOCITY CORRELATION TENSOR FOR ISOTROPIC TURBULENCE

A new geometric view of homogeneous isotropic turbulence is contemplated employing the two-point velocity correlation tensor of the velocity fluctuations. We show that this correlation tensor generates a family of pseudo-Riemannian metrics. This enables us to specify the geometry of a singled out Eulerian fluid volume in a statistical sense. We expose the relationship of some geometric constructions with statistical quantities arising in turbulence.

Determination of two-dimensional space–time correlations in jet flows using simultaneous PIV and LDV measurements

The two-dimensional space–time turbulence statistics of free shear jet flows in the form of the two-point velocity correlation tensor are important for aeroacoustic noise source modelling based on the acoustic analogy approach. This paper presents a direct application of the Point-Referenced Global Correlation (PRGC) technique to measure the components of this correlation tensor for a sector of the flow field in two jet configurations. The PRGC approach combines single point and global measurement techniques and enables two-point space–time correlations over a region of the flow to be obtained. The technique is applied to a single stream jet and a co-axial coplanar jet at a Mach number of 0.24 using commercial Laser Doppler velocimetry (LDV) and low-speed particle image velocimetry (PIV) systems. Results for the one-dimensional correlations are shown to compare well with two-point measurements. The results for the two-dimensional space–time correlations are presented and the characteristics for both configurations discussed.

Amplification of magnetic fields by dynamo action in Gaussian-correlated helical turbulence

We investigate the growth and structure of magnetic fields amplified by kinematic dynamo action in turbulence with nonzero kinetic helicity. We assume a simple Gaussian velocity correlation tensor, which allows us to consider very large magnetic Reynolds numbers, up to 1012. We use the kinematic Kazantsev–Kraichnan model of dynamo and find a complete numerical solution for the correlation functions of growing magnetic fields.

Representing anisotropy of two-point second-order turbulence velocity correlations using structure tensors

A locally homogeneous representation for the two-point, second-order turbulent velocity fluctuation Rij(x,r)=⟨ui′(x)uj′(x+r)⟩ is formulated in terms of three linearly independent structure tensors [Kassinos et al., J. Fluid Mech. 428, 213 (2001)]: Reynolds stress Bij, dimensionality Dij, and stropholysis Qijk∗. These structure tensors are single-point moments of the derivatives of vector stream functions that contain information about the directional and componential anisotropies of the correlation. The representation is a sum of several rotationally invariant component tensors. Each component tensor scales like a power law in r, while its variation in r/r depends linearly on the structure tensors. Continuity and self-consistency constraints reduce the number of degrees of freedom in the model to 17. A finite Re correction is introduced to the representation for separations of the order of Kolmogorov’s length scale. To evaluate our representation, we construct a model correlation by fitting the representation to correlations calculated from direct numerical simulation (DNS) of homogeneous turbulence and channel flow. Comparison of the model correlation to the DNS data shows that the representation can capture the character of the anisotropy of two-point second-order velocity correlation tensors.

From single-scale turbulence models to multiple-scale and subgrid-scale models by Fourier transform

A theoretical method based on mathematical physics formalism that allows transposition of turbulence modeling methods from URANS (unsteady Reynolds averaged Navier–Stokes) models, to multiple-scale models and large eddy simulations (LES) is presented. The method is based on the spectral Fourier transform of the dynamic equation of the two-point fluctuating velocity correlations with an extension to the case of non-homogenous turbulence. The resulting equation describes the evolution of the spectral velocity correlation tensor in wave vector space. Then, we show that the full wave number integration of the spectral equation allows one to recover usual one-point statistical closure whereas the partial integration based on spectrum splitting gives rise to partial integrated transport models (PITM). This latter approach, depending on the type of spectral partitioning used, can yield either a statistical multiple-scale model or subfilter transport models used in LES or hybrid methods, providing some appropriate approximations are made. Closure hypotheses underlying these models are then discussed by reference to physical considerations with emphasis on identification of tensorial fluxes that represent turbulent energy transfer or dissipation. Some experiments such as the homogeneous axisymmetric contraction, the decay of isotropic turbulence, the pulsed turbulent channel flow and a wall injection induced flow are then considered as typical possible applications for illustrating the potentials of these models.

Broadband slat noise prediction based on CAA and stochastic sound sources from a fast random particle-mesh (RPM) method

The application of a low-cost computational aeroacoustics (CAA) approach to a slat noise problem is studied. A fast and efficient stochastic method is introduced to model the unsteady turbulent sound sources in the slat-cove of a high-lift airfoil. It is based on the spatial convolution of spatiotemporal white-noise and can reproduce target distributions of turbulence kinetic energy and length scales, such as that provided by a RANS computation of the time-averaged turbulent flow problem. The computational method yields a perfectly solenoidal velocity field. For homogeneous isotropic turbulence, the complete second-order two-point velocity correlation tensor is realized exactly. Two RANS turbulence models are applied to the slat noise problem to study how sensitive the aeroacoustics predictions depend on turbulence kinetic energy predictions. Results for the sound generation at the slat are given for a Menter SST turbulence model with and without Kato–Launder modification. The aeroacoustic simulations yield a characteristic narrow band spectrum that compares very well with the experimental data. The directivities found point toward an edge noise mechanism at the slat as the main cause for slat noise sound generation.

A rigorous framework for Leslie's model of homogeneous anisotropic turbulence

Leslie, in the book Developments in the Theory of Turbulence (1973 Oxford: Clarendon), offered a very simple and intuitively founded model for the case of homogeneous anisotropic turbulence. Here, we offer a rigorous reformulation of Leslie's model leading to a general form for the velocity correlation tensor that satisfies realizability conditions like symmetry in its tensor indices and the condition of solenoidality arising from the incompressibility condition. The anisotropic part of the correlation tensor involves two non-dimensional constants—one arising from the pure strain term and the other from the term that induces distortion in the wave vector space; the rapid pressure term does not contribute. The estimates for these non-dimensional constants yield encouraging results within this framework.