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Coherence of velocity fluctuations in turbulent flows
Abstract
We investigate the spatio-temporal quantity of coherence for turbulent velocity fluctuations at spatial distances of the order or larger than the integral length scale . Using controlled laboratory experiments, an exponential decay as a function of distance is observed with a decay rate which depends on the flow properties. The same law is observed in two different flows indicating that it can be a generic property of turbulent flows.
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Statistical descriptions of coherent flow motions in the atmospheric boundary layer have many applications in the wind engineering community. For instance, the dynamical characteristics of large-scale motions in wall turbulence play an important role in predicting the dynamical loads on buildings, or the fluctuations in the power distribution across wind farms. Davenport (Quarterly Journal of the Royal Meteorological Society, 1961, Vol. 372, 194-211) performed a seminal study on the subject and proposed a hypothesis that is still widely used to date. Here, we demonstrate how the empirical formulation of Davenport is consistent with the analysis of Baars et al. (Journal of Fluid Mechanics, 2017, Vol. 823, R2) in the spirit of Townsend’s attached-eddy hypothesis in wall turbulence. We further study stratification effects based on two-point measurements of atmospheric boundary-layer flow over the Utah salt flats. No self-similar scaling is observed in stable conditions, putting the application of Davenport’s framework, as well as the attached-eddy hypothesis, in question for this case. Data obtained under unstable conditions exhibit clear self-similar scaling and our analysis reveals a strong sensitivity of the statistical aspect ratio of coherent structures (defined as the ratio of streamwise and wall-normal extent) to the magnitude of the stability parameter.
The buffeting response of a suspension bridge in complex terrain is studied based on full-scale data from sonic anemometers and accelerometers installed on the Lysefjord Bridge in Norway. The influence of the topography on the wind field and the associated effects on the displacement response are investigated. The two main wind conditions observed display different characteristics, which requires a case by case approach, and a refined description of the wind co-coherence. The identification of the mode shapes and modal damping ratios is carried out using ambient vibration data as a verification procedure. The buffeting response of the bridge is computed in the frequency domain, and compared to the recorded response in terms of standard deviation and power spectral density of the displacement response. The overall comparison is found satisfactory, although some discrepancies are observed. These are attributed to large yaw angles and non-stationary wind fluctuations caused by the influence of a complex topography on the flow properties.
Digital particle image velocimetry (DPIV) is a non-intrusive analysis technique that is very popular for mapping flows quantitatively. To get accurate results, in particular in complex flow fields, a number of challenges have to be faced and solved: The quality of the flow measurements is affected by computational details such as image pre-conditioning, sub-pixel peak estimators, data validation procedures, interpolation algorithms and smoothing methods. The accuracy of several algorithms was determined and the best performing methods were implemented in a user-friendly open-source tool for performing DPIV flow analysis in Matlab.
This article reviews evidence concerning the cornerstone dissipation scaling of turbulence theory: epsilon = C epsilon U3/L, with C-epsilon = const., epsilon the dissipation rate of turbulent kinetic energy U-2, and L an integral length scale characterizing the energy-containing turbulent eddies. This scaling is intimately linked to the Richardson-Kolmogorov equilibrium cascade. Accumulating evidence shows that a significant nonequilibrium region exists in various turbulent flows in which the energy spectrum has Kolmogorov's -5/3 wave-number scaling over a wide wave-number range, yet C-epsilon similar to Re-I(m)/Re-L(n), with m approximate to 1 approximate to n, Re-I a global/inlet Reynolds number, and Re-L a local turbulence Reynolds number.
The aerodynamics of horizontal axis wind turbine wakes is studied. The contents is directed towards the physics of power extraction by wind turbines and reviews both the near and the far wake region. For the near wake, the survey is restricted to uniform, steady and parallel flow conditions, thereby excluding wind shear, wind speed and rotor setting changes and yawed conditions. The emphasis is put on measurements in controlled conditions. For the far wake, the survey focusses on both single turbines and wind farm effects, and the experimental and numerical work are reviewed; the main interest is to study how the far wake decays downstream, in order to estimate the effect produced in downstream turbines. The article is further restricted to horizontal axis wind turbines and excludes all other types of turbines.
Methods are developed for modeling the dynamic response of horizontal axis wind turbines to atmospheric turbulence. The dynamic system is made as simple as possible, while still retaining essential physical characteristics. The report describes how to model a turbine system using the 5 degrees-of-freedom to represent the lower frequency motions of the turbine system. The rotor is assumed to be rigid and is three-bladed for simplicity. Both the structural dynamic properties for this system and the aerodynamic influence coefficients which determine how the system will respond to atmospheric turbulence are developed. Two different wake models are used to compute the induced velocities at the rotor disk. The reader is introduced to the basic concepts involved in modeling the turbulent fluctuations of the wind. Using an approximation concept, the spatial variations of the turbulence are modeled with the first few terms of a series expansion. In addition, this report covers the use of filtered noise as a modeling approximation for the turbulence input and provides the necessary data for modeling the turbulence inputs for a wide range of turbine sizes. Finally, the governing equations for the turbine system and the turbulence inputs are combined to yield a state space description of the entire system. It is shown how decoupling of these equations allows power spectral densities of the various turbine responses to be computed. The computer program used to perform these computations is also discussed and listed.
A general mathematical theory of magnetic field generation by inductive fluid motion (the dynamo theory) is developed with reference to the generation of the magnetic fields of the earth and sun. Attention is given to the more fundamental aspects of the problem with a treatment of those basic results of magnetohydrodynamics which are the foundation of the theory. An effort is made to find a common framework for the diverse dynamo theories which have evolved in recent years. The key concept in the theory developed here is that of the lack of reflectional symmetry of a fluid flow, the simplest measure of which is helicity. The invariance and topological interpretation of helicity are examined and its influence in turbulent flows with and without magnetic fields is discussed. Consideration is given to laminar dynamo theory, Braginskii's theory (1964) for weakly asymmetric systems, the structure and solution of the dynamo equations, and dynamically consistent dynamos. Bibtex entry for this abstract Preferred format for this abstract (see Preferences) Find Similar Abstracts: Use: Authors Title Keywords (in text query field) Abstract Text Return: Query Results Return items starting with number Query Form Database: Astronomy Physics arXiv e-prints
Using a physics-based approach, we infer the impact of the coherence of atmospheric turbulence on the power fluctuations of wind farms. Application of the random-sweeping hypothesis reveals correlations characterized by advection and turbulent diffusion of coherent motions. Those contribute to local peaks and troughs in the power spectrum of the combined units at frequencies corresponding to the advection time between turbines, which diminish in magnitude at high frequencies. Experimental inspection supports the results from the random-sweeping hypothesis in predicting spectral characteristics, although the magnitude of the coherence spectrum appears to be over-predicted. This deviation is attributed to the presence of turbine wakes, and appears to be a function of the turbulence approaching the first turbine in a pair.
Matching-index-of-refraction (MIR) has been used for obtaining high-quality flow visualization data for the fundamental nuclear thermal-hydraulic researches. By this method, distortions of the optical measurements such as PIV and LDV have been successfully minimized using various combinations of the model materials and the working fluids. This study investigated a novel 3D printing technology for manufacturing models and an oil-based working fluid for matching the refractive indices. Transparent test samples were fabricated by various rapid prototyping methods including selective layer sintering (SLS), stereolithography (SLA), and vacuum casting. As a result, the SLA direct 3D printing was evaluated to be the most suitable for flow visualization considering manufacturability, transparency, and refractive index. In order to match the refractive indices of the 3D printing models, a working fluid was developed based on the mixture of herb essential oils, which exhibit high refractive index, high transparency, high density, low viscosity, low toxicity, and low price. The refractive index and viscosity of the working fluid range 1.453–1.555 and 2.37–6.94 cP, respectively. In order to validate the MIR method, a simple test using a twisted prism made by the SLA technique and the oil mixture (anise and light mineral oil) was conducted. The experimental results show that the MIR can be successfully achieved at the refractive index of 1.51, and the proposed MIR method is expected to be widely used for flow visualization studies and CFD validation for the nuclear thermal-hydraulic researches.
§1. We shall denote by u α ( P ) = u α ( x 1 , x 2 , x 3 , t ), α = 1, 2, 3, the components of velocity at the moment t at the point with rectangular cartesian coordinates x 1 , x 2 , x 3 . In considering the turbulence it is natural to assume the components of the velocity u α ( P ) at every point P = ( x 1 , x 2 , x 3 , t ) of the considered domain G of the four-dimensional space ( x 1 , x 2 , x 3 , t ) are random variables in the sense of the theory of probabilities (cf. for this approach to the problem Millionshtchikov (1939) Denoting by Ᾱ the mathematical expectation of the random variable A we suppose that ῡ ² α and (d u α /d x β ) ² ― are finite and bounded in every bounded subdomain of the domain G .
In a previous paper it has been established that almost all spatially periodic motions of an infinite homogenous conducting fluid give magnetohydrodynamic dynamo action for almost all values of the magnetic diffusivity or resistivity lambda . It was shown that there is a dynamo action if and only if for some real vector j there is a growing magnetic field solution of the form B(x, t) = H(x) exp (pt + ij\cdot x), where the complex vector function H(x) has the same periodicity as the motion. The complex growth rate p was studied in a first-order limit of small j to obtain the above result. In this sequel the special case of spatially periodic motions with their three components functions of the two Cartesian coordinates y and z only is considered. The first-order method establishes dynamo action for only half, in a certain sense, of the motion-resistivity combinations. It is shown that the two-dimensional spatially periodic motion u = (cos y - cos z, sin z, sin y) is a first-order dynamo, at least for almost all resistivities. Three similar motions, which are not first-order dynamos for any resistivity, are also studied. It is proved that multiple-scale versions of all three can give growing magnetic fields for certain resistivities when terms of higher order in j are included. Heuristic descriptions of all four dynamo mechanisms are given. A numerical method is described for determining the most rapidly growing magnetic field solution of the above form, and results for all four motions are presented, giving the growth rate Re p as a function of j, for a range of resistivities lambda down to about 10-2. The first motion gives growing solutions for all resistivities in this range, the others give dynamo action only for resistivities below critical values near unity. The numerical and analytic results agree.
Tennekes [J. Fluid Mech. 67, 561 (1975)] estimated the time decorrelation of inertial-range excitation in isotropic turbulence by assuming effective statistical independence of the one-time distributions of inertial-range and energy-range excitation. This picture has been challenged by Yakhot, Orszag, and She [Phys. Fluids A 1, 184 (1989)], who studied forced turbulence by renormalization-group (RNG) methods. The analysis given in the present paper leads to the conclusion that (a) precise coherence between energy-range and inertial-range excitation is needed to inhibit sweeping effects; (b) in the case of randomly forced turbulence, this coherence is impossible and Tennekes’ picture is unavoidable; and (c) the RNG analysis does not demonstrate inhibition of sweeping; instead, it discards sweeping effects at the outset. To augment the present study, an advected passive scalar is examined by computer simulation. Sweeping effects on small scales survive even in the case of long-time advection by a frozen velocity field. The observed probability distributions resemble those for the alignment of vorticity and velocity observed in flow simulations.
Spatial statistics in the measured inflow of the long-term inflow and structural test (LIST) turbine was investigated by estimating the coherence for the three turbulence components. Several standard coherence models were also compared for inflow turbulence with estimates from the field measurements. It was observed that the Davenport exponential models fails to account for reductions in coherence at low frequencies and over large separations. It was also observed that the Mann uniform shear turbulence model predicted coherence spectra for all turbulence components.
In isotropic ‘box’ turbulence without a mean flow, the Lagrangian frequency spectrum extends to frequencies of order ([varepsilon] is the rate of dissipation of kinetic energy per unit mass and ν is the kinematic viscosity of the fluid). This leads to an estimate that makes the r.m.s. value of du/dt of order . The Eulerian frequency spectrum, however, extends to higher frequencies than its Lagrangian counterpart; this is caused by spectral broadening associated with large-scale advection of dissipative eddies. As a consequence, the r.m.s. value of [partial partial differential]u/[partial partial differential]t at a fixed observation point is (apart from a numerical factor) times as large as the r.m.s. value of du/dt (RΛ is the turbulence Reynolds number based on the Taylor microscale). The results of a theoretical analysis based on these premises agree with data obtained by Comte-Bellot, Shlien and Corrsin. The analysis also suggests that the Eulerian frequency spectrum has a behaviour in the inertial subrange, and that it is not governed by Kolmogorov similarity.
A short review of experimental findings is given, followed by a theoretical derivation, based on Taylor's hypothesis, of formulas for lateral coherences. It is assumed that the flow is stationary and homogeneous. Explicit formulas are derived assuming an energy spectrum pertaining to the inertial subrange. Even when the last assumption is not fulfilled, there are only four different types of non-zero velocity coherences. These four coherences correspond to the combinations uu, vv, ww, and uv, where u, v, and w are the longitudinal, the transversal, and the vertical component of the turbulent velocity with respect to the direction of the horizontal mean wind velocity U. In the case of small displacements relative to the scale of turbulence, the coherences are shown to be universal functions of the non-dimensional frequency nD/U, where n is the frequency and D the lateral displacement. It is shown that these theoretical formulas for spectral coherences are in good agreement with atmospheric data. Finally, the role of the scale of the turbulence is discussed.
Three-dimensional incompressible flows lacking parity-invariance (reversal of coordinates) can display a large-scale instability analogous to the α-effect of magnetohydrodynamics, but essentially anisotropic. At low Reynolds number R, the unstable modes have wavenumbers (R2). A specific example is given which leads to the growth of a very strong large-scale Beltrami flow. Nonlinear saturation of the instability is obtained by feed-back on the small-scale flow. All the results are illustrated by full simulations of the three-dimensional Navier-Stokes equations.
This paper firstly considers the history of wind engineering in five rather arbitrary time periods—the “traditional” period (up to 1750), the “empirical” period (1750–1900), the “establishment” period (1900–1960), the period of growth (1960–1980), and the modern period (1980 onwards). In particular it considers the development of the discipline in terms of the socio-economic and intellectual contexts of the time. This leads to a description of the current state of the discipline and a forward look at possible developments, again taking into consideration the likely socio-economic and intellectual changes in the next few decades.
- S Fauve
- J Herault
- G Michel
- F Pétrélis
S. Fauve, J. Herault, G. Michel, and F. Pétrélis, Instabilities on a turbulent background, Journal of Statistical Mechanics: Theory and Experiment 2017, 064001
(2017).