Correlations Of Velocity Fluctuations Research Articles

Estimating Velocity Correlations Using Single Pixel PIV

In this work, we analyze the possibility of determining correlations of turbulent velocity fluctuations from PIV data with improved spatial resolution. The analysis is based on the method presented by Oldenziel et al. (2023), where a joint probability density function of the velocity at two locations is calculated from the single-pixel ensemble correlation for these locations. Synthetic PIV images are used to discuss the occurrence of random and systematic errors and to derive the number of image pairs required for a determination of the correlation coefficient with desired uncertainty, depending on the particle image density. Compared to classical window correlation, the presented method can improve the spatial resolution by about one order of magnitude if a very large number of PIV image pairs are available and/or many statistically independent points with identical flow properties can be used. As an example of application, we show the spatial distribution of a two-point correlation in a compressible turbulent boundary layer flow for a reference point near the wall.

Estimating Velocity Correlations Using Single Pixel PIV

In this work, we analyze the possibility of determining correlations of turbulent velocity fluctuations from PIV data with improved spatial resolution. The analysis is based on the method presented by Oldenziel et al. (2023), where a joint probability density function of the velocity at two locations is calculated from the single-pixel ensemble correlation for these locations. Synthetic PIV images are used to discuss the occurrence of random and systematic errors and to derive the number of image pairs required for a determination of the correlation coefficient with desired uncertainty, depending on the particle image density. Compared to classical window correlation, the presented method can improve the spatial resolution by about one order of magnitude if a very large number of PIV image pairs are available and/or many statistically independent points with identical flow properties can be used. As an example of application, we show the spatial distribution of a two-point correlation in a compressible turbulent boundary layer flow for a reference point near the wall.

Dominant heat transfer mechanism with conical roughness in a cubical box in turbulent Rayleigh–Bénard convection

In the present study, we numerically investigate the effect of Prandtl number on the heat transfer mechanism in turbulent Rayleigh–Bénard convection inside a cubical box endowed with conical roughness elements. The Rayleigh number is kept fixed at Ra=108, while the Prandtl number (Pr) varies from 1 to 50. In contrast to the monotonic increasing trend of the Nusselt number (Nu∼Pr0.27) in the two-dimensional (2D) roughness explored previously [Sharma et al., “Influence of Prandtl number in turbulent Rayleigh-Bénard convection over rough surfaces,” Phys. Rev. Fluids 7, 104609 (2022)], it assumes an invariant behavior (∼Pr0.012) for three dimensional (3D), though it is approximately 50% higher than its smooth counterpart. Flow intensity, measured in terms of Reynolds number (Re), drops with increasing Pr showing consistently lower magnitude for the 3D configuration. The addition of roughness elements is observed to disrupt the preferred orientation of large-scale circulation (LSC). The effect is predominant for lower Pr, where the roughness interferes most with the natural bias of LSC toward the diagonal planes of the cubical box. The analysis of plume statistics reveals that both coverage and intensity of plumes are augmented for the roughened cell. Increased homogeneity in the flow at higher Pr is reflected by the emergence of a more pronounced and distinguishable peak in probability density functions of temperature and velocity. Temporal spectra and variance data substantiate augmented intensity of fluctuations in the rough cell, while behavioral differences in the flow at different Pr are elucidated by using cross correlation of vertical velocity and temperature fluctuations.

Pressure and velocity fluctuations characteristics of the tip clearance flow in an axial compressor stage at the near-stall condition

Pressure and velocity fluctuations characteristics of tip clearance flow in the rotor and stator of the axial compressor are studied by large eddy simulation (LES) at the near stall condition. In this paper, our work mainly emphasizes that the spatiotemporal evolution of complex flow structure, the pressure fluctuation, the velocity fluctuation of tip clearance flows, probability density function (PDF) of velocity and pressure fluctuations in the upstream and downstream of the rotor. Pressure fluctuations of the downstream rotor blade gradually decrease from the tip leading edge to the trailing edge. The motion law of leakage vortex may be related to the passage of the blade channel and the leakage vortexes mainly affect the region from the leading edge to the middle of the blade tip, but has little effect near the trailing edge. The frequency amplitude of suction surface is significantly higher than that of the pressure surface, and the frequency amplitude decreases along the flow direction. The passage shock has a major impact on the space-time correlation for pressure fluctuation. The intensity of space-time correlation of the downstream clearance flow of the rotor tip is similar to that of the upstream scraping vortex flow of the rotor tip. The strong space-time correlation for pressure fluctuation mainly concentrates in the separated flow on both sides of the rotor. Downstream flow has a better local temporal correlation than that of the upstream flow. The space-time correlation decreases more rapidly in velocity fluctuation. The space-time correlations of pressure and velocity fluctuations are vital for understanding the behaviors of unsteady flow characteristics of tip clearances at the near-stall conditions.

Simulation of Pressure–Velocity Correlations by Green’s Function Based on Reynolds Stress Model

Cost-effective wind energy harvesting by wind turbines in urban areas needs to strengthen the required flow field properties, such as mean velocity, turbulence, and its distribution. This paper conducts a series of CFD simulations to investigate the characteristics and related mechanisms of flow within the cavity, considering the force–turbulence interactions at the RANS scales. The pressure–velocity correlation term is formulated and solved by the elliptic relaxation equation to compensate for the Reynolds stress overestimation. Numerical simulations of flow over an open cavity with the proposed model are compared with corresponding PIV data. The results show that the mean velocity and the fluctuation velocity along the streamwise direction exist a slightly favorable pressure gradient. While the fluctuation velocity and fluctuation pressure show different correlation characteristics along the streamwise direction. Moreover, the pressure–velocity fluctuation correlation becomes obvious near the upper corner of the cavity due to the favorable pressure gradient. Hence, the leading and trailing locations of the cavity are both obvious favorable regions and further emphasis should be put on both high-accurate simulation methods and practical applications.

A statistical analysis of velocity and acceleration fluctuations of inertial particles in particle-laden turbulent Couette flow

Dynamics of the particle phase in a particle-laden turbulent flow is strongly influenced by the fluctuating velocity and vorticity field of the fluid phase. The present work mainly focuses on exploring the statistics of velocity and acceleration of the particle phase in a particle-laden turbulent Couette flow. Direct numerical simulations have been performed for particle-laden turbulent Couette flow with two different Reynolds numbers, Reδ=750 and 1300, in the presence of sub-Kolmogorov sized inertial particles for multiple Stokes numbers (Stokes number ≫1). The inter-particle and wall-particle interactions have also been considered to be elastic. We report the distribution functions for the linear and rotational velocities and accelerations in the presence of particle roughness. From the particle equation of rotational motion, we arrive at the expression where the fluctuating angular acceleration αi′ of the particle is expressed as the ratio of a linear combination of fluctuating rotational velocities of particle (ωi′) and fluid angular velocity (Ωi′) to the particle rotational relaxation time τr. The analysis is done using probability density function plots and Jensen–Shannon divergence-based method to assess the similarity between the particle net rotational acceleration distributions f(αi′), with (i) the distributions of particle acceleration component arises from fluctuating fluid angular velocity computed in the particle-Lagrangian frame f((Ω′i/τr)pl), (ii) fluctuating particle angular velocity f(ω′i/τr), and (iii) the fluid angular velocity f((Ω′i/τr)e) computed in the fluid Eulerian grids. The analysis leads to the conclusion that for a wide range of Reynolds and Stokes numbers, f(αi′) can be represented with a Gaussian white noise with a pre-estimated strength that can be calculated from the temporal decorrelation correlation of fluid-phase angular velocity fluctuations at Eulerian grid (Ω′i/τr)e.

Reynolds number effect on statistics of turbulent flows over periodic hills

The wall-resolved large-eddy simulations of turbulent flows over periodic hills are carried out to study the Reynolds number effect on flow statistics. Five different Reynolds numbers ranging from 2800 to 37 000 are considered. The present simulations are validated by comparing the time-averaged flow statistics with those from the literature. The Reynolds number effect is first examined on the skin friction and pressure coefficients, the isosurfaces of p′ and Q criteria, and the vertical profiles of flow statistics. The results show that (1) at most locations the magnitude of friction coefficient decreases with the increase in Reynolds number, while the pressure coefficient varies in the opposite direction; (2) smaller turbulence structures arise at higher Reynolds numbers; and (3) the mean velocities and Reynolds stresses in general exhibit asymptotic behaviors with the increase in Reynolds number. The statistical properties of turbulence structures are further examined via the probability density function and time correlation of velocity fluctuations. At last, the dynamics in the separation bubble is investigated by examining the flow statistics and the budget equation of mean kinetic energy (MKE) on the coordinate with its origin fixed at the recirculation center, and the power spectral density of the velocity fluctuations. Similarities are in general observed for the mean velocities, Reynolds stresses, and the MKE budget in the rear part of the separation bubble. The mean convection term and turbulence convection term are observed playing a key role on the decrease in bubble size with the increase in Reynolds number.

Nonequilibrium fractional correlation functions and fluctuation–dissipation in linear viscoelasticity

A simple model for time retarded fractional density and velocity hydrodynamic fluctuations correlations of a linear viscoelastic fluid under an external gradient is proposed. A fractional generalized Langevin equation with a Caputo’s fractional time derivative is formulated for a power-law memory kernel with long correlation noise. The fractional density and velocity fluctuations correlation functions, as well as the fractional light scattering spectra, and the fractional intermediate scattering function (FISF) are calculated analytically for power-law viscoelasticity. Their relation to the longitudinal modulus of the viscoelastic fluid is also obtained. We find that for different frequency intervals appropriate to perform isothermal light scattering experiments using Fabry–Perot interferometry and photon correlation, the non-equilibrium fractional structure factor may be smaller or larger than its fractional equilibrium values. We also discuss the validity of a fractional fluctuation–dissipation theorem for this model in terms of the asymptotic behavior of the equilibrium fractional correlation functions of the model.

On the similarity of shear deformation of a granular massif and a fragmented medium in the seismically active area

In recent research, the dynamics of the medium located in the seismic region at the boundary of tectonic plates is considered as the behavior of a complex open system that is in a state of self-organized criticality. Such an approach results from the very laws of earthquake generation and the complex structure of these areas. The network of faults and cracks makes seismic zones significantly heterogeneous and fragmented. Therefore, discrete models are increasingly used to model the dynamics of these media. The basis for comparing the model and the full-scale object serves the statistical regularities of their dynamic deformation. Relying on this concept, in the paper it is modeled the shear dynamics of a granular massif composed of identical cubic granules and is compared system’s statistical characteristics with the similar characteristics obtained for the earthquake generation zone. Shear deformation is carried out by means of the box consisting of two parts — movable and immovable ones. The movable part possesses the cover which receives kinetic energy from the granular massif in the process of shear deformation. For numerical simulations of the shear dynamics, the discrete element method is applied. The numerical calculations result in the distribution of cover’s kinetic energy jumps simulating the perturbations transmitted from the granular system to an external medium. It turned out that the distribution for these perturbations is the power dependence with an exponent that is inherent in earthquakes (Gutenberg-Richter law). Before and after large perturbations it is observed the swarms of smaller perturbations which are the analogues of foreshocks and aftershocks. The distributions of element’s velocity fluctuations and the correlation of velocity fluctuations are calculated as well. It is revealed the similarity of distributions for velocity fluctuations in the model massif and in the seismically active region of California, which includes the San Andreas fault. Moreover, the similarity of corresponding correlation functions is shown. They both are the functions of the stretched exponent. The obtained result indicates that shear processes in granular massifs and natural seismic processes in the San Andreas Fault are statistically similar.

Coherent structure characteristics of the swirling flow during turbulent mixing in a multi-inlet vortex reactor

Improving the efficiency of the production of uniformly sized functional nanoparticles for pharmaceutical and agricultural applications has been a problem of great interest. The macroscale multi-inlet vortex Reactor (macro-MIVR) could potentially be used for this purpose due to its ability to achieve the rapid mixing necessary for the flash nanoprecipitation nanoparticle fabrication technique. In the presented work, the coherent structures, a key contributor to the turbulent mixing, were investigated for the turbulent swirling flow within the MIVR. The two-point spatial correlations of velocity and concentration fluctuations at various basepoints were measured from instantaneous velocity and concentration fields obtained using simultaneous stereoscopic particle image velocimetry and planar laser-induced fluorescence. The basepoint locations were chosen as the middle and at the edge of the partially mixed concentration spiral arms. The correlations were found to be elliptical in shape, inclined, and peaked at the basepoints. A region near the basepoint was positively correlated and was surrounded by negatively correlated regions. Autocorrelations of concentration were also elliptical and curved toward the center of the reactor. The linear stochastic estimation was used to interpret the coherent structure features that would result in the observed spatial correlations. The linear stochastic estimates of the velocity fields were computed directly from the cross correlations of the tangential velocity fluctuations with the concentration fluctuations. The estimated conditional velocity fields revealed obliquely oriented counter-rotating vortical structures that stir the fluid from high-concentration regions to low-concentration regions, and the orientation of these vortical structures depended on the local concentration gradient.

Space–time characteristics of turbulence in minimal flow units

Turbulent flows in minimal flow units (MFUs) provide intrinsic information on near-wall physics of turbulence and help to develop useful models for numerical simulations. In the present study, MFUs at friction Reynolds numbers ranging from 1000 to 4000 are simulated. The space–time spectra and correlations of velocity and pressure fluctuations are computed. Two models, i.e., the elliptic approximation and the local amplitude modulated wave method are tested against the MFU data, and their performances in representing and reconstructing space–time spectra and correlations are evaluated. The models are then utilized to analyze the space–time characteristics of MFUs. An important feature of MFUs – the Reynolds number independence – holds for space–time statistics as well as the energy spectra. Comparisons are further carried out among the space–time statistics of velocity components and pressure, at different heights and scales. It is shown that despite the anisotropy in space, the space–time distributions of the three components of velocity fluctuations are quite similar. The scale dependence of convection speed is weak even near the wall, but that of sweeping velocity is always strong. The spanwise-temporal spectra and correlations are also presented, of which the proper modeling requires future work.

Monte Carlo study of noise velocity fluctuations and microscopic carrier transport in monolayer transition metal dichalcogenides

The microscopic transport properties of electrons and holes in MoS2, MoSe2, WS2 and WSe2 are studied by means of an ensemble Monte Carlo simulator. Moreover, instantaneous carrier velocity fluctuations are analyzed in depth from a microscopic point of view, including their power spectral density, which is directly related to thermal noise in the GHz range. Results show important differences in the mean free path and momentum relaxation time for electrons and holes: valence band transport has, in general, a more ballistic character, particularly for tungsten based transition metal dichalcogenides. Electrons (especially in MoSe2 and WSe2) have faster momentum relaxation times in the whole electric field range under study. Transport in subsidiary valleys is relevant for electrons, while in the case of holes it only has a minor influence, being noteworthy only for WS2 at high electric fields. Holes present larger decay times of the velocity fluctuations correlation function: the faster decorrelation in the conduction band is associated to a larger scattering activity, a general trend observed in all the materials under study. The results indicate that it is to be expected a much larger thermal noise for p-type devices in the GHz range as compared to n-type devices: materials containing selenium have reduced levels of noise generated by velocity fluctuations, especially for electrons.

Investigation of large-scale coherent structures and momentum interference of twin rectangular jets

An experimental study was performed to investigate the effects of nozzle spacing between twin rectangular jets on the non-linear momentum interference and spatial evolution of large-scale flow structures. Five test cases of different spacing ratios (S/de = 1.8, 2.8, 3.7, 5.5 and 7.3) were examined. Two-point correlation, joint probability density function and swirling strength were used to characterize the large-scale structures. Turbulent/non-turbulent interface along the inner and outer shear layers were investigated. The results showed that due to the interactions in the merging region, the spatial correlation of streamwise velocity fluctuations was considerably lower along the inner shear layer than along the outer shear layer. As a result of the mixing process, the swirling strength of retrograde vortices generated along the inner shear layer was also lower than that of the prograde vortices. Along the symmetry line, longitudinal and transverse integral length scales increased linearly as the downstream distance from the jets increases. For the longitudinal length scale, the slope of the linear region was independent of the spacing ratio, while for the transverse length scale, the slope increases as the spacing ratio increases. For cases of S/de < 3.7, due to the suppression effects of the adjacent jet, the jump in the conditionally-averaged mean streamwise velocity profile around the turbulent/non-turbulent interface was lower for the inner shear layer compared to the outer shear layer.

Aerodynamic noise reduction of circular cylinder by longitudinal grooves

Aerodynamic noise reduction of a circular cylinder using longitudinal grooves was experimentally investigated by measuring noise and velocity fields with a microphone and particle image velocimetry, respectively. Experiments were performed in an acoustic wind tunnel operating at subcritical Reynolds numbers, Re = (2–6) ​× ​104. The results indicate that the aerodynamic noise from a circular cylinder decreased by 8 and 5 ​dB compared with a smooth circular cylinder when the ratio of groove-depth to cylinder-diameter is set to 0.010 and 0.017, respectively. Velocity field measurements suggested a downstream shift in flow separation on the circular cylinder and a corresponding decrease in turbulence intensities near the wake of the grooved circular cylinder. Furthermore, the spanwise correlation of streamwise velocity fluctuations decreased near the wake of the grooved circular cylinder. These results demonstrated that aerodynamic noise reduction and variation in turbulence structure occurred in circular cylinders with longitudinal grooves at subcritical Reynolds numbers.

Mechanisms of broadband noise generation on metal foam edges

The turbulent flow over a porous trailing edge of a NACA 0018 airfoil is experimentally investigated to study the link between the hydrodynamic flow field and the acoustic scattering. Four porous trailing edges, obtained from open-cell metal foams, are tested to analyze the effects on far-field noise of the permeability of the material and of the hydrodynamic communication between the two sides of the airfoil. The latter is assessed by filling the symmetry plane of two of the porous trailing edges with a thin layer of adhesive that acts as a solid membrane. Experiments are performed at a zero degree angle of attack. Far-field noise measurements show that the most permeable metal foam reduces noise (up to 10 dB) with respect to the solid trailing edge for Strouhal numbers based on the chord below 16. At higher nondimensional frequencies, a noise increase is measured. The porous inserts with an adhesive layer show no noise abatement in the low frequency range, but only a noise increase at higher frequency. The latter is, therefore, attributed to surface-roughness noise. Flow field measurements, carried out with time-resolved planar particle image velocimetry, reveal correlation of near-wall velocity fluctuations between the two sides of the permeable trailing edges only within the frequency range where noise abatement is reported. This flow communication suggests that permeable treatments abate noise by distributing the impedance jump across the foam in the streamwise direction, promoting noise scattering from different chordwise locations along the inserts. This is further confirmed by noise source maps obtained from acoustic beamforming. For the frequency range where noise reduction is measured, the streamwise position of the main noise emission depends on the permeability of the insert. At higher frequencies, noise is scattered from upstream the trailing edge independently of the test case, in agreement with the roughness-generated noise assumption.

Control of the coherent structure dynamics of a film cooling flow by plasma aerodynamic actuation

A numerical study of film cooling flow from a cylindrical hole with and without plasma aerodynamic actuation is conducted using large eddy simulation. A phenomenological plasma model is employed to solve the electrohydrodynamic body force generated by DBD plasma actuator. The main attention is focused on the unsteady behaviors of coherent structures, which determine the film cooling efficiency. The time-averaged flow field parameters are analyzed, and results show that the plasma aerodynamic actuation suppresses the formation of downstream separation spiral node vortices by injecting energy into the boundary layer so as to improve the film cooling efficiency. Moreover, the spatial-temporal evolution processes of coherent structures are presented and discussed, which show that the hairpin vortices decrease in size and number, and stay closer to the wall because of the downward force and energy injection effect generated by the plasma aerodynamic actuation, then break into small-scale streaks in a short downstream distance. Subsequently, discussions about the space correlations of velocity fluctuations coupled with kinetic-energy spectra are made using the discrete Fourier transform, and results show that the space correlation coefficients become larger and periodically oscillate in near filed region, indicating that hairpin vortices are distributed more orderly, and they occurs irregular oscillations around zero in a shorter downstream distance confirming that hairpin vortices happen to breakup earlier. Furthermore, owing to the plasma aerodynamic actuation, the energetic wavenumbers are increased and their corresponding values of the kinetic-energy spectra are decreased, signifying that the hairpin vortices reduce in size and lose their strength in spectral space. Besides, the plasma aerodynamic actuation significantly decreases the surface area of the mixing interface inferring that the entrainment of the crossflow by hairpin vortices is suppressed.

HYDRODYNAMIC CHARACTERISTICS OF VORTEX MOTION INSIDE THE HEMISPHERICAL DIMPLE

Experimental researches of the features of the generation and evolution mechanism of coherent vortex structures, which are formed inside the hemispherical cavity, and space-time characteristics of the fields of velocity and pressure fluctuations that they create, both within the cavity, and in its wake, were conducted. Experiments were carried out in hydrodynamic flume on the hydraulically smooth plate with hemispherical dimple. The symmetric and asymmetric large-scale vortical systems inside a dimple are found out depending on the flow regime, and location and periodicity of their ejections are shown. The evolution of tornado-like vortices subjected to a switch mechanism that is result in appearance of low-frequency modulating transversal oscillations of vortex motion inside the hemispherical dimple. The fields of velocity, dynamic and wall pressure fluctuations by a group of hot-films, miniature piezoelectric sensors of pressure fluctuations are studied. Space-time correlations of velocity and pressure fluctuations, which generate the coherent large-scale vortex structures, the circulation flow and the vortex structure of the shear layer, which are formed inside the hemispherical dimple and ejected out into the boundary layer, are obtained. It was determined that the largest values of the coefficient of the space-time correlations of the wall pressure fluctuations observed on the aft and side walls of the hemispherical dimple, and the lowest – in its forward bottom part. It was established that the cross-correlations of the longitudinal velocity fluctuations are higher than the cross-correlations of the velocity and wall pressure fluctuations, and the cross-correlations of the dynamic and wall pressure fluctuations. On the basis of short-time spectrum and correlation analyses, the non-stationary in tame and non-uniform in space characteristics of the coherent vortical systems inside and past the open dimple have been found. Temporal periodicity of cross-correlation characteristics of the wall pressure fluctuations between two sensors, located on the fixed distance corresponds to frequencies of the shear and wake modes of resonance oscillations in the dimple. Registration of the time lag, at which the cross-correlation or anticorrelation has maximum, enables to determine the group convective velocity of the coherent vortical structures during their formation and evolution, as well as the phase of development and direction of vortex motion inside the dimple.

Settling of finite-size particles in turbulence at different volume fractions

We study the settling of finite-size rigid spheres in quiescent fluid and in sustained homogeneous isotropic turbulence (HIT) by direct numerical simulations using an immersed boundary method to account for the dispersed solid phase. We consider semi-dilute and dense suspensions of rigid spheres with solid volume fractions $$\phi =0.5{-}10\%$$ , solid-to-fluid density ratio $$R=1.02$$ , and Galileo number (i.e., the ratio between buoyancy and viscous forces) $$Ga=145$$ . In HIT, the nominal Reynolds number based on the Taylor microscale is $$Re_{\lambda } \simeq 90$$ , and the ratio between the particle diameter and the nominal Kolmogorov scale is $$(2a)/\eta \simeq 12$$ (being a the particle radius). We find that in HIT the mean settling speed is less than that in quiescent fluid for all $$\phi $$ . For $$\phi =0.5\%$$ , the mean settling speed in HIT is $$8\%$$ less than in quiescent fluid. However, by increasing the volume fraction the difference in the mean settling speed between quiescent fluid and HIT cases reduces, being only $$1.7\%$$ for $$\phi =10\%$$ . Indeed, while at low $$\phi $$ the settling speed is strongly altered by the interaction with turbulence, at large $$\phi $$ this is mainly determined by the (strong) hindering effect. This is similar in quiescent fluid and in HIT, leading to similar mean settling speeds. On the contrary, particle angular velocities are always found to increase with $$\phi $$ . These are enhanced by the interaction with turbulence, especially at low $$\phi $$ . In HIT, the correlations of particle lateral velocity fluctuations oscillate around zero before decorrelating completely. The time period of the oscillation seems proportional to the ratio between the integral lengthscale of turbulence and the particle characteristic terminal velocity. Regarding the mean square particle displacement, we find that it is strongly enhanced by turbulence in the direction perpendicular to gravity, even at the largest $$\phi $$ . Finally, we investigate the collision statistics for all cases and find the interesting result that the collision frequency is larger in quiescent fluid than in HIT for $$\phi =0.5{-}1\%$$ . This is due to frequent drafting–kissing–tumbling events in quiescent fluid. The collision frequency becomes instead larger in HIT than in still fluid for $$\phi =5{-}10\%$$ , due to the larger relative approaching velocities in HIT, and to the less intense drafting–kissing–tumbling events in quiescent fluid. The collision frequency also appears to be almost proportional to the estimate for small inertial particles uniformly distributed in space, though much smaller. Concerning the turbulence modulation, we find that the mean energy dissipation increases almost linearly with $$\phi $$ , leading to a large reduction of $$Re_{\lambda }$$ .

Modeling space‐time correlations of velocity fluctuations in wind farms

AbstractAn analytical model for the streamwise velocity space‐time correlations in turbulent flows is derived and applied to the special case of velocity fluctuations in large wind farms. The model is based on the Kraichnan‐Tennekes random sweeping hypothesis, capturing the decorrelation in time while including a mean wind velocity in the streamwise direction. In the resulting model, the streamwise velocity space‐time correlation is expressed as a convolution of the pure space correlation with an analytical temporal decorrelation kernel. Hence, the spatiotemporal structure of velocity fluctuations in wind farms can be derived from the spatial correlations only. We then explore the applicability of the model to predict spatiotemporal correlations in turbulent flows in wind farms. Comparisons of the model with data from a large eddy simulation of flow in a large, spatially periodic wind farm are performed, where needed model parameters such as spatial and temporal integral scales and spatial correlations are determined from the large eddy simulation. Good agreement is obtained between the model and large eddy simulation data showing that spatial data may be used to model the full spatiotemporal structure of fluctuations in wind farms.

Fractional Time Fluctuations in Viscoelasticity: A Comparative Study of Correlations and Elastic Moduli.

We calculate the transverse velocity fluctuations correlation function of a linear and homogeneous viscoelastic liquid by using a generalized Langevin equation (GLE) approach. We consider a long-ranged (power-law) viscoelastic memory and a noise with a long-range (power-law) auto-correlation. We first evaluate the transverse velocity fluctuations correlation function for conventional time derivatives and then introduce time fractional derivatives in their equations of motion and calculate the corresponding fractional correlation function. We find that the magnitude of the fractional correlation is always lower than the non-fractional one and decays more rapidly. The relationship between the fractional loss modulus and is also calculated analytically. The difference between the values of for two specific viscoelastic fluids is quantified. Our model calculation shows that the fractional effects on this measurable quantity may be three times as large as compared with its non-fractional value. The fact that the dynamic shear modulus is related to the light scattering spectrum suggests that the measurement of this property might be used as a suitable test to assess the effects of temporal fractional derivatives on a measurable property. Finally, we summarize the main results of our approach and emphasize that the eventual validity of our model calculations can only come from experimentation.