Structure Functions In Turbulence Research Articles

Universal relative scaling of longitudinal structure functions in shear-dominated turbulence

Shear significantly influences turbulence in the energy-containing range of shear-dominated flows, and the longitudinal structure functions do not have a universal form as they do in homogeneous isotropic turbulence. Despite this, the relative scaling of structure functions exhibits universal sub-Gaussian behaviour in shear-dominated flows, in particular for turbulent boundary layers, channels and Taylor–Couette flows. Our investigation of a turbulent vertical buoyancy layer at $Pr = 0.71$ using direct numerical simulation shows this universality even in moderate-Reynolds-number buoyancy-driven but shear-dominated boundary layers. It is demonstrated that the universality is related to the energy density of the eddies, which attains a hierarchical equilibrium in the energy-containing range of shear-dominated turbulence. We conjecture that the universal sub-Gaussian behaviour of the energy density of the energy-containing range, which was considered to be non-trivial in prior studies, is related to the universal anomalous scaling exponents of the inertial subrange turbulence. Based on this conjecture, we propose a hypothesis that relates large-scale eddies and the intermittent dissipation field in shear-dominated turbulence, highlighting a relationship between large and small scales. A phenomenological model is also developed to predict the scaling, which is verified using data from a turbulent boundary layer, half-channel and vertical buoyancy layer at friction Reynolds numbers spanning four orders of magnitude. Excellent agreement is observed.

Spatial-temporal structure functions in Burgers turbulence driven by an Ornstein-Uhlenbeck process

In Burgers turbulence driven by an Ornstein-Uhlenbeck process, for which the characteristic time scale is much larger than the characteristic time scale of energy transfer, we find the spatial-temporal third-order structure function follows a separation-of-variables form: the spatial dependence follows the classic results calculated from the Karman-Howarth-Monin equation, and the temporal dependence is determined by the temporal correlation of the Ornstein-Uhlenbeck process. This result confirms the forcing dependence of the spatial-temporal structure function and provides an intuitive way of analyzing sparsely measured spatial-temporal turbulence data.

Structure functions of turbulence through a degraded channel bed

This investigation deals with the turbulence characteristics for flow through degraded channel beds consisting of uniform and bimodal sediment particles. Three-dimensional velocity was measured throughout the depth at three different streamwise locations over a degraded bed. The velocity data were then processed to observe the spatial distributions of the longitudinal and vertical velocity components in vector form, turbulence indicator and length scales at different locations. The main intent of the present study was to observe the effect of bed degradation on the intermittency and anisotropy in higher order scaling exponents using the structure functions methodology. The extended self-similarity (ESS) technique was employed to estimate the scaling exponents from higher order structure functions of streamwise velocity component. The results show that below the bed-level (level of the bed surface measured before the degradation started), the scaling exponents deviate more from the theoretical values than that above the bed-level indicating higher intermittency of scaling exponents in the vicinity of the bed. Furthermore, data were analyzed to observe the scaling exponents from higher order mixed structure functions. To this end, the SO(3) symmetry decomposition was utilized to differentiate the isotropic and anisotropic parts of the scaling exponents. The anisotropic scaling exponents obtained from higher order mixed structure functions possess higher values than that of isotropic part. The experimental findings also specify that as one moves close to the degraded bed, the scaling exponents obtained from the higher order mixed structure functions deviate more significantly. However, according to the present experimental results, no significant differences were observed in the values of the scaling exponents and intermittency when they were compared between uniform and bimodal sedimentary beds. It is expected that the results of the present study will essentially lead to new contributions to the existing knowledge in the field of turbulence through degraded and scoured channel beds.

Global expressions for high-order structure functions in Burgers turbulence

Since the famous work by Kolmogorov on incompressible turbulence, the structure-function theory has been a key foundation of modern turbulence study. Due to the simplicity of Burgers turbulence, structure functions are calculated to arbitrary orders, which provides numerous implications for other compressible turbulent systems. We present the derivation of exact forcing-scale resolving expressions for high-order structure functions of the burgers turbulence. Compared with the previous theories, where the structure functions are calculated in the inertial range based on the statistics of shocks, our expressions link high-order structure functions in different orders without extra information on the flow structure and are valid beyond the inertial range, therefore they are easily checked by numerical simulations.

Parameterization of Outer-scale on DECam Point-spread Function

Abstract A method to better parameterize the Outer-Scale (L 0) of the Point-Spread Function (PSF) in Dark Energy Camera is described. The PSF characterizes the atmospheric turbulence and diffraction in astronomical images. In particular, the Outer-Scale describes the maximum coherence length in the turbulent atmosphere and leads to a maximum value of the turbulence structure function. We model the PSF by computing a pattern of star features, which are used to fit our PSF model. For each star, we apply a weight function which maximizes the signal-to-noise ratio. We find difficulty in parameterizing L 0, thus, we propose to change the current Gaussian weight function to the von Kaŕmań weight function which extends to larger radii. We find that by changing to the von Kaŕmań, we double our total sensitivity but lose 1st order sensitivity to L 0. Therefore, ulterior methods need to be explored to parameterize L 0.

Structure Functions of Turbulence Through a Degraded Channel Bed

Structure Functions of Turbulence Through a Degraded Channel Bed

The “Maggie” filament: Physical properties of a giant atomic cloud

Context. The atomic phase of the interstellar medium plays a key role in the formation process of molecular clouds. Due to the line-of-sight confusion in the Galactic plane that is associated with its ubiquity, atomic hydrogen emission has been challenging to study. Aims. We investigate the physical properties of the “Maggie” filament, a large-scale filament identified in H I emission at line-of-sight velocities, vLSR ~−54 km s−1. Methods. Employing the high-angular resolution data from The H I/OH Recombination line survey of the inner Milky Way (THOR), we have been able to study H I emission features at negative vLSR velocities without any line-of-sight confusion due to the kinematic distance ambiguity in the first Galactic quadrant. In order to investigate the kinematic structure, we decomposed the emission spectra using the automated Gaussian fitting algorithm GAUSSPY+. Results. We identify one of the largest, coherent, mostly atomic H I filaments in the Milky Way. The giant atomic filament Maggie, with a total length of 1.2 ± 0.1 kpc, is not detected in most other tracers, and it does not show signs of active star formation. At a kinematic distance of 17 kpc, Maggie is situated below (by ≈500 pc), but parallel to, the Galactic H I disk and is trailing the predicted location of the Outer Arm by 5−10 km s−1 in longitude-velocity space. The centroid velocity exhibits a smooth gradient of less than ±3 km s−1 (10 pc)−1 and a coherent structure to within ±6 km s−1. The line widths of ~10 km s−1 along the spine of the filament are dominated by nonthermal effects. After correcting for optical depth effects, the mass of Maggie’s dense spine is estimated to be 7.2−1.9+2.5 × 105 M⊙. The mean number density of the filament is ~4 cm−3, which is best explained by the filament being a mix of cold and warm neutral gas. In contrast to molecular filaments, the turbulent Mach number and velocity structure function suggest that Maggie is driven by transonic to moderately supersonic velocities that are likely associated with the Galactic potential rather than being subject to the effects of self-gravity or stellar feedback. The probability density function of the column density displays a log-normal shape around a mean of ⟨NH I⟩ = 4.8 × 1020 cm−2, thus reflecting the absence of dominating effects of gravitational contraction. Conclusions. While Maggie’s origin remains unclear, we hypothesize that Maggie could be the first in a class of atomic clouds that are the precursors of giant molecular filaments.

Phase-factor spectra of turbulent phase screens.

The optical phase ϕ is a key quantity in the physics of light propagating through a turbulent medium. In certain respects, however, the statistics of the phase factor, ψ=exp⁡(iϕ), are more relevant than the statistics of the phase itself. Here, we present a theoretical analysis of the 2D phase-factor spectrum Fψ(κ) of a random phase screen. We apply the theory to four types of phase screens, each characterized by a power-law phase structure function, Dϕ(r)=(r/rc)γ (where rc is the phase coherence length defined by Dϕ(rc)=1rad2), and a probability density function pα(α) of the phase increments for a given spatial lag. We analyze phase screens with turbulent (γ=5/3) and quadratic (γ=2) phase structure functions and with normally distributed (i.e., Gaussian) versus Laplacian phase increments. We find that there is a pronounced bump in each of the four phase-factor spectra Fψ(κ). The precise location and shape of the bump are different for the four phase-screen types, but in each case it occurs at κ∼1/rc. The bump is unrelated to the well-known "Hill bump" and is not caused by diffraction effects. It is solely a characteristic of the refractive-index statistics represented by the respective phase screen. We show that the second-order ψ statistics (covariance function, structure function, and spectrum) characterize a random phase screen more completely than the second-order ϕ counterparts.

Advective structure functions in anisotropic two-dimensional turbulence

Abstract

FastSF: A parallel code for computing the structure functions of turbulence

Turbulence is a complex phenomenon in fluid dynamics involving nonlinear interactions between multiple scales. Structure functions are popular diagnostics in the study of statistical properties properties of turbulent flows (Frisch, 1995; Kolmogorov, 1941a, 1941b). Some of the earlier works comprising of such analysis are those of Gotoh et al. (2002), Ishihara et al. (2003), and Ishihara & Gotoh (2009) for three-dimensional (3D) hydrodynamic turbulence; Yeung et al. (2005) and Ray et al. (2008) for passive scalar turbulence; Biferale et al. (2004) for two-dimensional (2D) hydrodynamic turbulence; and Kunnen et al. (2008), Kaczorowski & Xia (2013), and Bhattacharya et al. (2019) for turbulent thermal convection. Structure functions are two-point statistical quantities; thus, an accurate computation of these quantities requires averaging over many points. However, incorporation of a large number of points makes the computations very expensive and challenging. Therefore, we require an efficient parallel code for accurate computation of structure functions. In this paper, we describe the design and validation of the results of fastSF, a parallel code to compute the structure functions for a given velocity or scalar field.

Stirring anisotropic turbulence with an active grid

We study spectra and high-order structure functions in anisotropic wind tunnel turbulence, which is generated using an active grid. In the first experiment, we impose homogeneous shear turbulence with a constant gradient of the mean flow and (approximately) homogeneous turbulent fluctuations. We measure mixed structure functions of order 2, 3, 4, 6, and 10 using an array of two-component hotwires. These structure functions, which vanish for isotropic turbulence, display scaling with scaling exponents that highlight intermittency: the return to isotropy at small scales of large fluctuations is much slower than expected on the basis of a simple Kolmogorov-like scaling argument [J. L. Lumley, “Similarity and the turbulent energy spectrum,” Phys. Fluids 10, 855 (1967)]. In the second experiment, we impose anisotropy in otherwise homogeneous turbulence through the time modulation of the active grid. This is done by driving the grid using signals from a turbulence (shell) model, which acts as a convenient turbulent random signal generator. In this way, different statistical properties of different velocity components could be imposed. Similar to the first experiment, our interest is in the return to isotropy of the small-scale turbulent fluctuations, which is quantified using second-order quantities such as spectra and correlation functions. Also, in this case, the strongly anisotropic correlations induced by the forcing at large scales tend to return to isotropy at small, inertial-range scales, but with the imprint of large-scale anisotropy retained.

Reynolds Number Dependence of the Structure Functions in Homogeneous Turbulence

We compare the predictions of stochastic closure theory (SCT) (Birnir in J Nonlinear Sci 23:657–688, 2013a. https://doi.org/10.1007/s00332-012-9164-z) with experimental measurements of homogeneous turbulence made in the variable density turbulence tunnel (Bodenschatz et al. in Rev Sci Instrum 85(9):093908, 2014) at the Max Planck Institute for Dynamics and Self-Organization in Göttingen. While the general form of SCT contains infinitely many free parameters, the data permit us to reduce the number to seven, only three of which are active over the entire range of Taylor–Reynolds numbers. Of these three, one parameter characterizes the variance of the mean-field noise in SCT and another characterizes the rate in the large deviations of the mean. The third parameter is the decay exponent of the Fourier variables in the Fourier expansion of the noise, which characterizes the smoothness of the turbulent velocity. SCT compares favorably with velocity structure functions measured in the experiment. We considered even-order structure functions ranging in order from two to eight as well as the third-order structure functions at five Taylor–Reynolds numbers (\(R_\lambda \)) between 110 and 1450. The comparisons highlight several advantages of the SCT, which include explicit predictions for the structure functions at any scale and for any Reynolds number. We observed that finite-\(R_\lambda \) corrections, for instance, are important even at the highest Reynolds numbers produced in the experiments. SCT gives us the correct basis function to express all the moments of the velocity differences in turbulence in Fourier space. These turn out to be powers of the sine function indexed by the wavenumbers. Here, the power of the sine function is the same as the order of the moment of the velocity differences (structure functions). The SCT produces the coefficients of the series and so determines the statistical quantities that characterize the small scales in turbulence. It also characterizes the random force acting on the fluid in the stochastic Navier–Stokes equation, as described in this paper.

Similarities between the structure functions of thermal convection and hydrodynamic turbulence

In this paper, we analyze the scaling of velocity structure functions of turbulent thermal convection. Using high-resolution numerical simulations, we show that the structure functions scale similar to those of hydrodynamic turbulence, with the scaling exponents in agreement with the predictions of She and Leveque [“Universal scaling laws in fully developed turbulence,” Phys. Rev. Lett. 72, 336–339 (1994)]. The probability distribution functions of velocity increments are non-Gaussian with wide tails in the dissipative scales and become close to Gaussian in the inertial range. The tails of the probability distribution follow a stretched exponential. We also show that in thermal convection, the energy flux in the inertial range is less than the viscous dissipation rate. This is unlike in hydrodynamic turbulence where the energy flux and the dissipation rate are equal.

Towards mapping turbulence in the intra-cluster medium

X-ray observations of the hot gas filling the intra-cluster medium (ICM) provide a wealth of information on the dynamics of clusters of galaxies. The global equilibrium of the ICM is believed to be ensured by non-thermal and thermal pressure support sources, among which gas movements and the dissipation of energy through turbulent motions. Accurate mapping of turbulence using X-ray emission lines is challenging due to the lack of spatially resolved spectroscopy. Only future instruments such as the X-ray Integral Field Unit (X-IFU) on Athena will have the spatial and spectral resolution to quantitatively investigate the ICM turbulence over a broad range of spatial scales. Powerful diagnostics for these studies are line shift and the line broadening maps, and the second-order structure function. When estimating these quantities, instruments will be limited by uncertainties of their measurements, and by the sampling variance (also known as cosmic variance) of the observation. Here, we extend the formalism started in our companion Paper I to include the effect of statistical uncertainties of measurements in the estimation of these line diagnostics, in particular for structure functions. We demonstrate that statistics contribute to the total variance through different terms, which depend on the geometry of the detector, the spatial binning and the nature of the turbulent field. These terms are particularly important when probing the small scales of the turbulence. An application of these equations is performed for the X-IFU, using synthetic turbulent velocity maps of a Coma-like cluster. Results are in excellent agreement with the formulas both for the structure function estimation (≤3%) and its variance (≤10%). The expressions derived here and in Paper I are generic, and ensure an estimation of the total errors in any X-ray measurement of turbulent structure functions. They also open the way for optimisations in the upcoming instrumentation and in observational strategies.

Cascade of turbulent energy and scalar variance in DC electrokinetic turbulence

Abstract Turbulent flow can be generated by DC electrokinetic (EK) mechanism resulted from the electric conductivity and permittivity variation in fluids, as has been demonstrated by Varshney et al (2016), where a −1.4 slope of velocity power spectrum is observed. Here, we theoretically investigate the scaling exponents of both velocity and scalar structure functions in DC EK turbulence. The scaling exponents of the second order structure functions of velocity and scalar in the so-called electric body force (EBF) dominant subrange are 2/5 (equivalent to the −7/5 slope of velocity power spectrum) and 4/5 respectively. The theory directly explains the experimental results of Varshney et al. (2016). Based on Karman–Howarth equation with forcing terms, the third order structure function of velocity which is the representative of energy transport process is investigated in the DC EK turbulence. Compared with the conventional hydrodynamic turbulence, the energy transport in DC EK turbulence has an additional scaling subrange and is governed by additional small length scales.

Exact third-order structure functions for two-dimensional turbulence

We derive and investigate exact expressions for third-order structure functions in stationary isotropic two-dimensional turbulence, assuming a statistical balance between random forcing and dissipation both at small and large scales. Our results extend previously derived asymptotic expressions in the enstrophy and energy inertial ranges by providing uniformly valid expressions that apply across the entire non-dissipative range, which, importantly, includes the forcing scales. In the special case of white noise in time forcing this leads to explicit predictions for the third-order structure functions, which are successfully tested against previously published high-resolution numerical simulations. We also consider spectral energy transfer rates and suggest and test a simple robust diagnostic formula that is useful when forcing is applied at more than one scale.

Turbulence Measurements from Five-Beam Acoustic Doppler Current Profilers

AbstractTwo new five-beam acoustic Doppler current profilers—the Nortek Signature1000 AD2CP and the Teledyne RDI Sentinel V50—are demonstrated to measure turbulence at two energetic tidal channels within Puget Sound, Washington. The quality of the raw data is tested by analyzing the turbulent kinetic energy frequency spectra, the turbulence spatial structure function, the shear in the profiles, and the covariance Reynolds stresses. The five-beam configuration allows for a direct estimation of the Reynolds stresses from along-beam velocity fluctuations. The Nortek’s low Doppler noise and high sampling frequency allow for the observation of the turbulent inertial subrange in both the frequency spectra and the turbulence structure function. The turbulence parameters obtained from the five-beam acoustic Doppler current profilers are validated with turbulence data from simultaneous measurements with acoustic Doppler velocimeters. These combined results are then used to assess a turbulent kinetic energy budget in which depth profiles of the turbulent kinetic energy dissipation and production rates are compared. The associated codes are publicly available on the MATLAB File Exchange website.

Outer region scaling using the freestream velocity for nonuniform open channel flow over gravel

The theoretical basis for outer region scaling using the freestream velocity for nonuniform open channel flows over gravel is derived and tested for the first time. Owing to the gradual expansion of the flow within the nonuniform case presented, it is hypothesized that the flow can be defined as an equilibrium turbulent boundary layer using the asymptotic invariance principle. The hypothesis is supported using similarity analysis to derive a solution, followed by further testing with experimental datasets. For the latter, 38 newly collected experimental velocity profiles across three nonuniform flows over gravel in a hydraulic flume are tested as are 43 velocity profiles previously published in seven peer-reviewed journal papers that focused on fluid mechanics of nonuniform open channel over gravel. The findings support the nonuniform flows as equilibrium defined by the asymptotic invariance principle, which is reflective of the consistency of the turbulent structure's form and function within the expanding flow. However, roughness impacts the flow structure when comparing across the published experimental datasets. As a secondary objective, we show how previously published mixed scales can be used to assist with freestream velocity scaling of the velocity deficit and thus empirically account for the roughness effects that extend into the outer region of the flow. One broader finding of this study is providing the theoretical context to relax the use of the elusive friction velocity when scaling nonuniform flows in gravel bed rivers; and instead to apply the freestream velocity. A second broader finding highlighted by our results is that scaling of nonuniform flow in gravel bed rivers is still not fully resolved theoretically since mixed scaling relies to some degree on empiricism. As researchers resolve the form and function of macroturbulence in the outer region, we hope to see the closing of this research gap.

Scaling of velocity and scalar structure functions in ac electrokinetic turbulence.

Electrokinetic (EK) turbulence or electrohydrodynamic (EHD) turbulence has been recently achieved in different fluids under both ac [G. Wang et al., Lab Chip 14, 1452 (2014)10.1039/C3LC51403J; Phys. Rev. E 93, 013106 (2016)10.1103/PhysRevE.93.013106] and dc electric fields [A. Varshney et al., Soft Matter 12, 1759 (2016)10.1039/C5SM02316E]. Here, through dimensional analysis, scaling laws of both velocity and electric conductivity structure functions in the forced cascade region of ac EK turbulence can be predicated (similar to Bolgiano-Obukhov scaling law in turbulent Rayleigh-Bénard convection), in either macroscale or microscale flows. In the forced cascade region, EK force, which relies on the direct cascade of conductivity structures, injects energy directly into a wide spectral region to sustain the flow disturbance. The scaling exponents of the second-order velocity and conductivity structures are 2/5 and 4/5, respectively. In addition to the scaling regions, two characteristic small length scales are derived for both weak and strong electric body forces, respectively. This theoretical investigation can significantly enhance our understanding of EK or EHD turbulence while forced by an ac electric field. It can further broaden our understanding of the forced cascade region of forced turbulence and make the manipulation of the turbulent cascade process more flexible and controllable.

Longitudinal and transverse structure functions in high Reynolds-number magneto-hydrodynamic turbulence

We investigate the scaling behavior of longitudinal and transverse structure functions in homogeneous and isotropic magneto-hydrodynamic (MHD) turbulence by means of an exact hierarchy of structure function equations as well as by direct numerical simulations of two- and three-dimensional MHD turbulence. In particular, rescaling relations between longitudinal and transverse structure functions are derived and utilized in order to compare different scaling behavior in the inertial range. It is found that there are no substantial differences between longitudinal and transverse structure functions in MHD turbulence. This finding stands in contrast to the case of hydrodynamic turbulence which shows persistent differences even at high Reynolds numbers. We propose a physical picture that is based on an effective reduction of pressure contributions due to local regions of same magnitude and alignment of velocity and magnetic field fluctuations. Finally, our findings underline the importance of the pressure term for the actually observed scaling differences in hydrodynamic turbulence.