Third-order Correlation Research Articles

Validation of a closing procedure for fourth-order RANS turbulence models with DNS data in an incompressible zero-pressure-gradient turbulent boundary layer

Among factors affecting the accuracy of flow simulations with Reynolds-Averaged Navier–Stokes turbulence models is modeling turbulent diffusion processes. With the use of the Gram–Charlier series expansions, the turbulent diffusion in fourth-order one-point statistical closures of the Reynolds-Averaged Navier–Stokes equations can be modeled without introducing unknown model coefficients and without assuming turbulence being Gaussian. Terms representing turbulent diffusion processes in transport equations for second- and third-order velocity correlations do not require any modeling in such closures. In this regard, fourth-order closures are a more accurate alternative to lower-order closures where turbulent diffusion is modeled on semi-empirical or Gaussian turbulence assumptions. In the current paper, the accuracy of the closing procedure based on the Gram–Charlier series expansions is evaluated using data of direct numerical simulations in an incompressible zero-pressure-gradient turbulent boundary layer over a flat plate. One-point third-, fourth-, and fifth-order velocity moments were extracted for this purpose from the dataset collected by the Fluid Dynamics Group at the Universidad Politécnica de Madrid at two streamwise locations Reθ=4101 and 5200 that correspond to channels and pipes at δ+=1331 and 1626. Results demonstrate that the truncated Gram–Charlier series expansions are an accurate approximation of the fifth-order velocity moments in the considered flow.

A-priori validation of a second-order moment combustion model via DNS database

The newly proposed combustion model, the second-order moment (SOM) turbulent combustion model is validated a-priori using the DNS database of combustion. It is found that the assumption of neglecting the density fluctuation and the third-order fluctuation correlation terms is acceptable. The fluctuation averaged closure method for the mean reaction-rate coefficient can be a promising method. The comparison of the terms in the correlation equations indicates that the chemical reaction term plays a dominant role, followed by the production term and the diffusion term. The predicted production term is in good agreement with the exact production term. The validation of the dissipation term shows that the performance of the closure is not promising but can be acceptable due to its less contribution. The predicted reaction terms are validated and discussed. Two modified reaction-term closure methods are proposed for different fluctuation correlation equations. It is concluded that the SOM combustion model is a promising one for the future development of turbulent combustion models.

CFHTLenS: a Gaussian likelihood is a sufficient approximation for a cosmological analysis of third-order cosmic shear statistics

We study the correlations of the shear signal between triplets of sources in the Canada-France-Hawaii Lensing Survey (CFHTLenS) to probe cosmological parameters via the matter bispectrum. In contrast to previous studies, we adopted a non-Gaussian model of the data likelihood which is supported by our simulations of the survey. We find that for state-of-the-art surveys, similar to CFHTLenS, a Gaussian likelihood analysis is a reasonable approximation, albeit small differences in the parameter constraints are already visible. For future surveys we expect that a Gaussian model becomes inaccurate. Our algorithm for a refined non-Gaussian analysis and data compression is then of great utility especially because it is not much more elaborate if simulated data are available. Applying this algorithm to the third-order correlations of shear alone in a blind analysis, we find a good agreement with the standard cosmological model: $\Sigma_8$=$\sigma_8$ $(\Omega_{\rm m}/0.27)^{0.64}$=$0.79^{+0.08}_{-0.11}$ for a flat $\Lambda\rm CDM$ cosmology with $h=0.7\pm0.04$ ($68\%$ credible interval). Nevertheless our models provide only moderately good fits as indicated by $\chi^2/{\rm dof}=2.9$, including a $20\%$ r.m.s. uncertainty in the predicted signal amplitude. The models cannot explain a signal drop on scales around 15 arcmin, which may be caused by systematics. It is unclear whether the discrepancy can be fully explained by residual PSF systematics of which we find evidence at least on scales of a few arcmin. Therefore we need a better understanding of higher-order correlations of cosmic shear and their systematics to confidently apply them as cosmological probes.

Relaxation dynamics of the Lieb-Liniger gas following an interaction quench: A coordinate Bethe-ansatz analysis

We investigate the relaxation dynamics of the integrable Lieb-Liniger model of contact-interacting bosons in one dimension following a sudden quench of the collisional interaction strength. The system is initially prepared in its noninteracting ground state and the interaction strength is then abruptly switched to a positive value, corresponding to repulsive interactions between the bosons. We calculate equal-time correlation functions of the nonequilibrium Bose field for small systems of up to five particles via symbolic evaluation of coordinate Bethe-ansatz expressions for operator matrix elements between Lieb-Liniger eigenstates. We characterize the relaxation of the system by comparing the time-evolving correlation functions following the quench to the equilibrium correlations predicted by the diagonal ensemble and relate the behavior of these correlations to that of the quantum fidelity between the many-body wave function and the initial state of the system. Our results for the asymptotic scaling of local second-order correlations with increasing interaction strength agree with the predictions of recent generalized thermodynamic Bethe-ansatz calculations. By contrast, third-order correlations obtained within our approach exhibit a markedly different power-law dependence on the interaction strength as the Tonks-Girardeau limit of infinitely strong interactions is approached.

Measuring the growth of matter fluctuations with third-order galaxy correlations

Measurements of the linear growth factor $D$ at different redshifts $z$ are key to distinguish among cosmological models. One can estimate the derivative $dD(z)/d\ln(1+z)$ from redshift space measurements of the 3D anisotropic galaxy two-point correlation $\xi(z)$, but the degeneracy of its transverse (or projected) component with galaxy bias $b$, i.e. $\xi_{\perp}(z) \propto\ D^2(z) b^2(z)$, introduces large errors in the growth measurement. Here we present a comparison between two methods which break this degeneracy by combining second- and third-order statistics. One uses the shape of the reduced three-point correlation and the other a combination of third-order one- and two-point cumulants. These methods use the fact that, for Gaussian initial conditions and scales larger than $20$ $h^{-1}$Mpc, the reduced third-order matter correlations are independent of redshift (and therefore of the growth factor) while the third-order galaxy correlations depend on $b$. We use matter and halo catalogs from the MICE-GC simulation to test how well we can recover $b(z)$ and therefore $D(z)$ with these methods in 3D real space. We also present a new approach, which enables us to measure $D$ directly from the redshift evolution of second- and third-order galaxy correlations without the need of modelling matter correlations. For haloes with masses lower than $10^{14}$ $h^{-1}$M$_\odot$, we find $10%$ deviations between the different estimates of $D$, which are comparable to current observational errors. At higher masses we find larger differences that can probably be attributed to the breakdown of the bias model and non-Poissonian shot noise.

Second-order adjoint sensitivity analysis methodology (2nd-ASAM) for computing exactly and efficiently first- and second-order sensitivities in large-scale linear systems: II. Illustrative application to a paradigm particle diffusion problem

This work presents an illustrative application of the second-order adjoint sensitivity analysis methodology (2nd-ASAM) to a paradigm neutron diffusion problem, which is sufficiently simple to admit an exact solution, thereby making transparent the underlying mathematical derivations. The general theory underlying 2nd-ASAM indicates that, for a physical system comprising Nα parameters, the computation of all of the first- and second-order response sensitivities requires (per response) at most (2Nα+1) “large-scale” computations using the first-level and, respectively, second-level adjoint sensitivity systems (1st-LASS and 2nd-LASS). Very importantly, however, the illustrative application presented in this work shows that the actual number of adjoint computations needed for computing all of the first- and second-order response sensitivities may be significantly less than (2Nα+1) per response. For this illustrative problem, four “large-scale” adjoint computations sufficed for the complete and exact computations of all 4 first- and 10 distinct second-order derivatives. Furthermore, the construction and solution of the 2nd-LASS requires very little additional effort beyond the construction of the adjoint sensitivity system needed for computing the first-order sensitivities. Very significantly, only the sources on the right-sides of the diffusion (differential) operator needed to be modified; the left-side of the differential equations (and hence the “solver” in large-scale practical applications) remained unchanged.All of the first-order relative response sensitivities to the model parameters have significantly large values, of order unity. Also importantly, most of the second-order relative sensitivities are just as large, and some even up to twice as large as the first-order sensitivities. In the illustrative example presented in this work, the second-order sensitivities contribute little to the response variances and covariances. However, they have the following major impacts on the computed moments of the response distribution: (a) they cause the “expected value of the response” to differ from the “computed nominal value of the response”; and (b) they contribute decisively to causing asymmetries in the response distribution. Indeed, neglecting the second-order sensitivities would nullify the third-order response correlations, and hence would nullify the skewness of the response. Consequently, any events occurring in a response's long and/or short tails, which are characteristic of rare but decisive events (e.g., major accidents, catastrophes), would likely be missed. The 2nd-ASAM is expected to affect significantly other fields that need efficiently computed second-order response sensitivities, e.g., optimization, data assimilation/adjustment, model calibration, and predictive modeling.

Four-Photon Imaging with Thermal Light

In a near-field four-photon correlation measurement, ghost imaging with classical incoherent light is investigated. By applying the Klyshko advanced-wave picture, we consider the properties of four-photon spatial correlation and find that the fourth-order spatial correlation function can be decomposed into multiple lower-order correlation functions. On the basis of the spatial correlation properties, a proof-of-principle four-photon ghost imaging is proposed, and the effect of each part in a fourth-order correlation function on imaging is also analyzed. In addition, the similarities and differences among ghost imaging by fourth-, second-, and third-order correlations are also discussed. It is shown that the contrast and visibility of fourth-order correlated imaging are improved significantly, while the resolution is unchanged. Such studies can be very useful in better understanding multi photon interference and multi-channel correlation imaging.

Third-order optical intensity correlation measurements of pseudo-thermal light

Third-order Hanbrury Brown—Twiss and double-slit interference experiments with a pseudo-thermal light are performed by recording intensities in single, double and triple optical paths, respectively. The experimental results verifies the theoretical prediction that the indispensable condition for achieving a interference pattern or ghost image in Nth-order intensity correlation measurements is the synchronous detection of the same light field by each reference detector, no matter the intensities recorded in one, or two, or N optical paths. It is shown that, when the reference detectors are scanned in the opposite directions, the visibility and resolution of the third-order spatial correlation function of thermal light is much better than that scanned in the same direction, but it is no use for obtaining the Nth-order interference pattern or ghost image in the thermal Nth-order interference or ghost imaging.

Effect of turbulence on NO formation in swirling combustion

Turbulence affects both combustion and NO formation. Fluctuation correlations are ideally used for quantitative analysis. From the instantaneous chemical reaction rate expression, ignoring the third-order correlation terms, the averaged reaction rate will have four terms, including the term of averaged-variable product, a concentration fluctuation correlation term, and temperature-concentration fluctuation correlation term. If the reaction-rate coefficient is denoted as K, the temperature fluctuation would be included in the K fluctuation. In order to quantitatively study the effect of turbulence on NO formation in methane-air swirling combustion, various turbulence-chemistry models are tested. The magnitudes of various correlations and their effects on the time-averaged reaction rate are calculated and analyzed, and the simulation results are compared with the experimental measurement data. The results show that among various correlation moments, the correlation between the reaction-rate coefficient K fluctuation with the concentration fluctuation is most important and is a strong nonlinear term.

Ultrabroadband direct detection of nonclassical photon statistics at telecom wavelength

Broadband light sources play essential roles in diverse fields, such as high-capacity optical communications, optical coherence tomography, optical spectroscopy, and spectrograph calibration. Although a nonclassical state from spontaneous parametric down-conversion may serve as a quantum counterpart, its detection and characterization have been a challenging task. Here we demonstrate the direct detection of photon numbers of an ultrabroadband (110 nm FWHM) squeezed state in the telecom band centred at 1535 nm wavelength, using a superconducting transition-edge sensor. The observed photon-number distributions violate Klyshko's criterion for the nonclassicality. From the observed photon-number distribution, we evaluate the second- and third-order correlation functions, and characterize a multimode structure, which implies that several tens of orthonormal modes of squeezing exist in the single optical pulse. Our results and techniques open up a new possibility to generate and characterize frequency-multiplexed nonclassical light sources for quantum info-communications technology.

Mapping sediment-dispersal characteristics using seismic geomorphology: Late Paleogene to Neogene, Qinan Sag, Huanghua Depression, China

The main objective of this article has been to demonstrate the utility of stratal slice images for exploring the sequence stratigraphy and sedimentology of complex depositional systems. The seismic-geomorphology study and sedimentary interpretation were performed to map sediment-dispersal characteristics from late Paleogene to Neogene in Qinan Sag, located in Huanghua Depression of China. The Qinan Sag is underlain by a non-marine Mesozoic and Cenozoic-age stratigraphic section. Main data types in this study area are lithology, wire-line logs and 3D seismic. The main study strata, Member one of Shahejie Formation to Minghuazhen Formation, divided into two second-order sequences and four third-order sequences. The older of the two second-order sequences, SE, corresponds to the formations from Kongdian to Dongying. The younger of the two second-order sequences, SN, correspond to Guantao and Minghuazhen formations. Guided by third-order sequence-stratigraphic correlations from seismic and wireline-log data, we prepared stratal slices from a three-dimensional seismic volume to reveal high-resolution (10-m) sediment dispersal characteristics in a relative geologic-time domain. Using techniques of 3-D seismic geomorphology, we defined four types of depositional systems: braided deltas, deltas, braided rivers and meandering rivers. Stratal slices indicated that the depositional evolution of these interest sequences was from braided-delta front to delta front, braided river and finally meandering river system. Many factors related to the structural evolution history controlled the sediment-dispersal characteristics, such as the basin type, dynamic mechanism, faulting activity, subsidence rate and paleo-high.

Decay of Temperature Fluctuations in Dusty Fluid Homogeneous Turbulence Prior to the Ultimate Period in Presence of Coriolis Force

Using deissler’s method we have studied the decay of temperature fluctuations in dusty fluid homogeneous turbulence before the final period in presence of coriolis force and have considered correlations between fluctuating quantities at two- and three-point. The equations for two- and three-point correlation is obtained and the set of equations is made to determinate by neglecting the forth-order correlation in comparison to the second- and third-order correlations. For solving the correlation equations are converted to spectral form by taking their Fourier transform. Finally, integrating the energy spectrum over all wave numbers, the energy decay law of temperature fluctuations in homogeneous dusty fluid turbulence before the final period in presence of coriolis force is obtained.

Scale-dependent cyclone-anticyclone asymmetry in a forced rotating turbulence experiment

We characterize the statistical and geometrical properties of the cyclone-anticyclone asymmetry in a statistically steady forced rotating turbulence experiment. Turbulence is generated by a set of vertical flaps which continuously inject velocity fluctuations towards the center of a tank mounted on a rotating platform. We first characterize the cyclone-anticyclone asymmetry from conventional single-point vorticity statistics. We propose a phenomenological model to explain the emergence of the asymmetry in the experiment, from which we predict scaling laws for the root-mean-square velocity in good agreement with the experimental data. We further quantify the cyclone-anticyclone asymmetry using a set of third-order two-point velocity correlations. We focus on the correlations which are nonzero only if the cyclone-anticyclone symmetry is broken. They offer two advantages over single-point vorticity statistics: first, they are defined from velocity measurements only, so an accurate resolution of the Kolmogorov scale is not required; second, they provide information on the scale-dependence of the cyclone-anticyclone asymmetry. We compute these correlation functions analytically for a random distribution of independent identical vortices. These model correlations describe well the experimental ones, indicating that the cyclone-anticyclone asymmetry is dominated by the large-scale long-lived cyclones.

Testing locality and noncontextuality with the lowest moments

The quest for fundamental tests of quantum mechanics is an ongoing effort. We here address the question of what are the lowest possible moments needed to prove quantum nonlocality and noncontextuality without any further assumptions---in particular, without the often assumed dichotomy. We first show that second-order correlations can always be explained by a classical noncontextual local-hidden-variable theory. Similar third-order correlations also cannot violate classical inequalities in general, except for a special state-dependent noncontextuality. However, we show that fourth-order correlations can violate locality and state-independent noncontextuality. Finally we obtain a fourth-order continuous-variable Bell inequality for position and momentum, which can be violated and might be useful in Bell tests, closing all loopholes simultaneously.

Size effect in through-thickness conductivity of heterogeneous plates

In this work, the size effect on the effective through-thickness conductivity of heterogeneous plates expressed in second-order Hashin–Shtrikman bounds and a third-order correlation approximation is studied. By taking into account the homogeneous temperature boundary conditions, the exact Green operator for the plate is first established. Then, the respective bounds and correlation approximation are constructed. With the help of the method based on the fast Fourier transform (FFT), the bounds and correlation approximation for the effective through-thickness conductivity are computed for the plates reinforced or weaken randomly either by spherical particles or unidirectional fibers. The numerical results show that the size effect of the effective through-thickness conductivity is more significant than the one of the effective in-plane conductivity.

Effect of Jet Injection on Flow Structure in Local Pier Scour

This study investigates experimentally the effect of jet injection on the flow structure around a pier-scour hole. Experiments are conducted with 1-jet, 3-jets and without jet injection under clear-water scour conditions. The results show that the jet(s) alter(s) the turbulent characteristics and vertical component of the velocity, causing a reduction of the strength of the down flow. A quadrant analysis for the Reynolds shear stress values reveals that the sweeps and ejections have the main contributions on the Reynolds shear stress production in the near-bed flow. The third-order correlations of velocity fluctuations show an obvious distinction for the jet injection experiments in comparison to that without injections, especially inside the scour hole.

Application of Millionschikov's hypothesis to homogeneous and isotropic turbulence possessing helicity

In this manuscript we construct a deductive theory for the stationary homogeneous turbulence, which is invariant under translations and rotations but lacks reflexional symmetry, considering the appropriate forms for the second- and third-order correlation functions. Appropriate invariant forms are considered for some second and third order correlation tensors, formed from the fluctuating components of velocity, acceleration, components of external force and pressures. In the light of the formulation of Millionschikov’s hypothesis we would point out further the treatment of first author [4] on the final period decay of isotropic turbulence using the points-merging technique employed here. We include also some enlightments of the process of future work that could be undertaken in this field of research.

Deciphering the Catalysis-Associated Conformational Changes of Human Adenylate Kinase 1 with Single-Molecule Spectroscopy

Human adenylate kinase isoenzyme 1 (AK1) is the key enzyme in maintaining the cellular energy homeostasis. The catalysis-associated conformational changes of AK1 involve large-amplitude rearrangements. To decipher the conformational changes of AK1 at the single-molecule level, we tagged AK1 with two identical fluorophores, one near the substrate-binding site and the other at the boundary of the core domain. We found that magnesium ion binding to AK1 increases the structural heterogeneity of AK1, whereas ADP binding reduces the structural heterogeneity. We exploited the hidden Markov model to extract the underlying catalysis-associated conformational dynamics and determined thermodynamic parameters of the multiple catalytic pathways. The third-order correlation difference calculated from photon fluctuation traces reveals the irreversible nature of the conformational motions, suggesting that single-molecule AK1 is in a nonequilibrium steady state. This discovery offers a fresh viewpoint to look into the molecular mechanisms of cellular biochemistry.

Enhanced photon correlations due to strong laser-atom-cavity coupling

We investigate the resonant quantum dynamics of a laser-pumped real or artificial two-level single-atom system embedded in a leaking microcavity. We found that for stronger laser-atom-cavity couplings the generated microcavity photons exhibit larger steady-state correlations. In particular, the second- and third-order photon correlation functions are greater than the corresponding ones obtained for an incoherent light source, respectively. Furthermore, the emitted microcavity photon flux is enhanced in comparison to weaker coupling cases.

An Eulerian model for particles nonisothermally carried by a compressible fluid

An Eulerian model describing the transport of particles that nonisothermally interact with their carrying flow is developed in the general time-dependent, three-dimensional frame. The Eulerian model derives from the Lagrangian kinematic, momentum and energy particle equations based on a combination of the filtered Liouville equation and the method of moments. The filtered Liouville equation governs the filtered density function of the particle velocities and temperature in phase space. Moments of the Liouville equation result in a coupled system of Eulerian governing equations for the particle number density, velocity and temperature and their second moments. The equations are closed by expressing third order moments in terms of lower order moments based on the assumption that third-order correlations are negligible. The particle transport equations are two-way coupled with the inviscid, Euler equations governing compressible flows. Consistent with the hyperbolic conservation form of the Eulerian–Eulerian model, a flux-vector splitting and characteristics-based Roe averaging method discretizes the equations. Computations of a cloud of particles initially at rest in a uniform, subsonic flow and in an accelerated flow behind a normal shock in one dimension, show very good comparison with a recent higher order WENO based Eulerian–Lagrangian model.