Lagrangian Acceleration Research Articles

Cosmological Constraints from the Redshift Dependence of the Alcock–Paczynski Effect: Possibility of Estimating the Nonlinear Systematics Using Fast Simulations

Abstract The tomographic Alcock–Paczynski (AP) method is so far the best method in separating the AP signal from the redshift space distortion (RSD) effects and deriving powerful constraints on cosmological parameters using the ≲ 40 h − 1 Mpc clustering region. To guarantee that the method can be easily applied to the future large-scale structure surveys, we study the possibility of estimating the systematics of the method using the fast simulation method. The major contribution of the systematics comes from the nonzero redshift evolution of the RSD effects, which is quantified by ξ ˆ Δ s ( μ , z ) in our analysis, and estimated using the BigMultidark exact N-body simulation and approximate COmoving Lagrangian Acceleration (COLA) simulation samples. We find about 5%/10% evolution when comparing the ξ ˆ Δ s ( μ , z ) measured as z = 0.5/z = 1 to the measurements at z = 0. We checked the inaccuracy in the 2pCFs computed using COLA, and find it 5–10 times smaller than the intrinsic systematics of the tomographic AP method, indicating that using COLA to estimate the systematics is good enough. Finally, we test the effect of halo bias, and find ≲1.5% change in ξ ˆ Δ s when varying the halo mass within the range of 2 × 1012–1014 M ⊙. We will perform more studies to achieve an accurate and efficient estimation of the systematics in the redshift range of z = 0–1.5.

On the Unsteady Wake of a Rigid Plate Under Constant Acceleration and Deceleration

AbstractLaboratory experiments are performed to explore distinct features of the unsteady wake induced by a rigid plate undergoing a series of accelerated and decelerated phases in a quiescent water tank. Each plate motion pattern is defined by a constant acceleration from the rest followed by the same magnitude deceleration until rest; i.e., both of them lasting the same time interval. Flow statistics are obtained for various plate acceleration/deceleration motions in a plane normal to the plate span using particle image velocimetry. Results revealed that the relative near-wake bulk flow underwent acceleration during 75% of the plate cycle; the associated maximum velocity was reached approximately at the middle of the deceleration phase in all the cases. Transversal velocity profiles in the near wake were significantly modulated by the magnitude and sign of the plate acceleration. The dynamics of the large-scale coherent motions shed by the plate was also dependent on plate acceleration pattern. However, the near-wake nondimensional circulation strength reached a plateau roughly at the end of the acceleration phase and was approximately conserved despite the modulation of the deceleration phase. In the early stage of the plate motion, the Lagrangian flow acceleration was dominant; however, the local acceleration dominated at the end of the cycle.

Beyond two-point statistics: using the minimum spanning tree as a tool for cosmology

ABSTRACT Cosmological studies of large-scale structure have relied on two-point statistics, not fully exploiting the rich structure of the cosmic web. In this paper we show how to capture some of this cosmic web information by using the minimum spanning tree (MST), for the first time using it to estimate cosmological parameters in simulations. Discrete tracers of dark matter such as galaxies, N-body particles or haloes are used as nodes to construct a unique graph, the MST, that traces skeletal structure. We study the dependence of the MST on cosmological parameters using haloes from a suite of COmoving Lagrangian Acceleration (COLA) simulations with a box size of $250\ h^{-1}\, {\rm Mpc}$, varying the amplitude of scalar fluctuations (As), matter density (Ωm), and neutrino mass (∑mν). The power spectrum P and bispectrum B are measured for wavenumbers between 0.125 and 0.5 $h\, {\rm Mpc}^{-1}$, while a corresponding lower cut of ∼12.6 $h^{-1}\, {\rm Mpc}$ is applied to the MST. The constraints from the individual methods are fairly similar but when combined we see improved 1σ constraints of $\sim 17{{\ \rm per\ cent}}$ ($\sim 12{{\ \rm per\ cent}}$) on Ωm and $\sim 12{{\ \rm per\ cent}}$ ($\sim 10{{\ \rm per\ cent}}$) on As with respect to P (P + B) thus showing the MST is providing additional information. The MST can be applied to current and future spectroscopic surveys (BOSS, DESI, Euclid, PSF, WFIRST, and 4MOST) in 3D and photometric surveys (DES and LSST) in tomographic shells to constrain parameters and/or test systematics.

General Basset–Boussinesq–Oseen equation: existence, uniqueness, approximation and regularity of solutions

The Basset–Boussinesq–Oseen equation (BBO equation) is the mathematical formulation of the Lagrangian acceleration of a spherical particle, moving with Lagragian velocity, in an unsteady flow as the sum of the viscous, gravitational, buoyancy, virtual mass, and Basset forces acting respectively on the particle. Despite the widespread use of the equation in applications, the basic properties of its solutions have remained unexplored. Here we fill this gap by proving global existence and uniqueness of solutions in an appropriate partially ordered Banach space. The method also gives an approximate solution to the problem with accurate result which is easy to implement. Meanwhile, regularity properties of the solutions are proved under some conditions which gives a representation of the solutions. Illustrative examples exhibit the efficiency of our method.

Persistent accelerations disentangle Lagrangian turbulence

Particles in turbulence frequently encounter extreme accelerations between extended periods of quiescence. The occurrence of extreme events is closely related to the intermittent spatial distribution of intense flow structures such as vorticity filaments. This mixed history of flow conditions leads to very complex particle statistics with a pronounced scale dependence, which presents one of the major challenges on the way to a non-equilibrium statistical mechanics of turbulence. Here, we introduce the notion of persistent Lagrangian acceleration, quantified by the squared particle acceleration coarse-grained over a viscous time scale. Conditioning Lagrangian particle data from simulations on this coarse-grained acceleration, we find remarkably simple, close-to-Gaussian statistics for a range of Reynolds numbers. This opens the possibility to decompose the complex particle statistics into much simpler sub-ensembles. Based on this observation, we develop a comprehensive theoretical framework for Lagrangian single-particle statistics that captures the acceleration, velocity increments as well as single-particle dispersion.

A scanning particle tracking velocimetry technique for high-Reynolds number turbulent flows

We propose a novel robust 3D particle tracking technique based on a scanning laser setup. The method yields Lagrangian statistics in densely seeded turbulent flows with good spatial and temporal resolution, overcoming some of the inherent difficulty with line-of-sight-based volumetric methods. To do this, we have developed an effective triangulation method greatly reducing ghost particle reconstruction using images from two cameras. A laser sheet is rapidly traversed (‘scanned’) across a measurement volume illuminating only a thin slice of the flow at a time. Particle images are taken at closely spaced, overlapping nominal laser sheet locations giving multiple intensity recordings for each individual particle. The laser-sheet intensity varies as a Gaussian across its thickness, which is here exploited to deduce the particle’s probable location along the scan direction to sub-sheet number resolution by fitting a similar Gaussian profile to the particle’s multiple intensity recordings. The method is presently verified via numerical experiment using a DNS database. Following successful reconstruction of a time series of 3D particle fields, particle tracks are formed from which all components of Lagrangian velocity and acceleration are calculated.

Sharp transitions in rotating turbulent convection: Lagrangian acceleration statistics reveal a second critical Rossby number

In RB convection for fluids with Prandtl number $Pr\gtrsim 1$, rotation beyond a critical (small) rotation rate is known to cause a sudden enhancement of heat transfer which can be explained by a change in the character of the BL dynamics near the top and bottom plates of the convection cell. Namely, with increasing rotation rate, the BL signature suddenly changes from Prandtl--Blasius type to Ekman type. The transition from a constant heat transfer to an almost linearly increasing heat transfer with increasing rotation rate is known to be sharp and the critical Rossby number $Ro_{c}$ occurs typically in the range $2.3\lesssim Ro_{c}\lesssim 2.9$ (for Rayleigh number $Ra=1.3\times 10^9$, $Pr=6.7$, and a convection cell with aspect ratio $\Gamma=\frac{D}{H}=1$, with $D$ the diameter and $H$ the height of the cell). The explanation of the sharp transition in the heat transfer points to the change in the dominant flow structure. At $1/Ro\lesssim 1/Ro_c$ (slow rotation), the well-known LSC is found: a single domain-filling convection roll made up of many individual thermal plumes. At $1/Ro\gtrsim 1/Ro_c$ (rapid rotation), the LSC vanishes and is replaced with a collection of swirling plumes that align with the rotation axis. In this paper, by numerically studying Lagrangian acceleration statistics, related to the small-scale properties of the flow structures, we reveal that this transition between these different dominant flow structures happens at a second critical Rossby number, $Ro_{c_2}\approx 2.25$ (different from $Ro_{c_1}\approx 2.7$ for the sharp transition in the Nusselt number $Nu$; both values for the parameter settings of our present numerical study). When statistical data of Lagrangian tracers near the top plate are collected, it is found that the root-mean-square (rms) values and the kurtosis of the horizontal acceleration of these tracers show a sudden increase at $Ro_{c_2}$.

Real-Time Identification of Vortex Shedding in the Wake of a Circular Cylinder

Lagrangian saddles, identified as nonparallel intersections of finite-time Lyapunov exponent (FTLE) ridges, have previously been shown to indicate the timing of von Kármán vortex shedding in the wake of a circular cylinder using two-dimensional simulation data. To expand the application of this method, two-dimensional, two-component velocity data were collected in the wake of a circular cylinder using particle image velocimetry at a Reynolds number of 19,000. This experimental data, as well as three-dimensional simulation data at a Reynolds number of 400, were used to calculate FTLE fields. The Lagrangian saddle found upstream of a forming vortex was shown to accelerate away from the cylinder surface as the vortex begins to shed for both cases, in agreement with the previous work. These saddles are impossible to track in real time, however, because future flowfield data are needed for the computation of the FTLE fields. To detect the Lagrangian saddle acceleration without using FTLE, the saddle dynamics are connected to physical quantities measurable in real time. The acceleration of the Lagrangian saddle occurs simultaneously with a maximum in lift in the numerical case and with a minimum in the static pressure at a location slightly upstream of the mean separation location in both cases, albeit with a lag time in the experimental case. This allows the time at which the von Kármán vortex sheds, determined objectively by the acceleration of the Lagrangian saddle away from the circular cylinder, to be identified by distinct signatures in the spatiotemporal evolution of the pressure distribution on the cylinder surface.

Lagrangian acceleration in Rayleigh-Bénard convection at various aspect ratios

Lagrangian dynamics of turbulent Rayleigh-B\'enard convection is studied experimentally with 3D particle tracking velocimetry at various aspect ratios $G$. Insight on the pair dispersion, conditional acceleration probability density functions, and roll dynamics induced by $G$ was found from nearly $3.5\ifmmode\times\else\texttimes\fi{}{10}^{7}$ events.

Bias in particle tracking acceleration measurement

We investigate sources of systematic error (bias) in acceleration statistics derived from Lagrangian particle tracking data and demonstrate techniques to eliminate or minimise these bias errors introduced during processing. Numerical simulations of particle tracking experiments in isotropic turbulence show that the main sources of bias error arise from noise due to random position errors and selection biases introduced during numerical differentiation. We outline the use of independent measurements and filtering schemes to eliminate these biases. Moreover, we test the validity of our approach in estimating the statistical moments and probability densities of the Lagrangian acceleration. Finally, we apply these techniques to experimental particle tracking data and demonstrate their validity in practice with comparisons to available data from the literature. The general approach, which is not limited to acceleration statistics, can be applied with as few as two cameras and permits a substantial reduction in the position accuracy and sampling rate required to adequately measure the statistics of Lagrangian acceleration.Graphical abstractSources of bias error in Lagrangian Particle Tracking measurements are explored. Methods are presented and validated to correct acceleration statistics for the main sources of systematic errors introduced by random position error and filtering, allowing for a substantial improvement in the effective temporal resolution of particle tracking measurements.

Velocity and acceleration statistics in rapidly rotating Rayleigh-Bénard convection.

Background rotation causes different flow structures and heat transfer efficiencies in Rayleigh-Bénard convection (RBC). Three main regimes are known: rotation-unaffected, rotation-affected and rotation-dominated. It has been shown that the transition between rotation-unaffected and rotation-affected regimes is driven by the boundary layers. However, the physics behind the transition between rotation-affected and rotation-dominated regimes are still unresolved. In this study, we employ the experimentally obtained Lagrangian velocity and acceleration statistics of neutrally buoyant immersed particles to study the rotation-affected and rotation-dominated regimes and the transition between them. We have found that the transition to the rotation-dominated regime coincides with three phenomena; suppressed vertical motions, strong penetration of vortical plumes deep into the bulk and reduced interaction of vortical plumes with their surroundings. The first two phenomena are used as confirmations for the available hypotheses on the transition to the rotation-dominated regime while the last phenomenon is a new argument to describe the regime transition. These findings allow us to better understand the rotation-dominated regime and the transition to this regime.

Acceleration statistics of tracer particles in filtered turbulent fields

We present results from direct numerical simulations of tracer particles advected in filtered velocity fields to quantify the impact of the scales of turbulence on Lagrangian acceleration statistics. Systematically removing spatial scales reduces the frequency of extreme acceleration events, consistent with the notion that they are rooted in the small-scale structure of turbulence. We also find that acceleration variance and flatness as a function of filter scale closely resemble experimental results of neutrally buoyant, finite-sized particles, corroborating the picture that particle size determines the scale on which turbulent fluctuations are sampled.

Lattice Boltzmann simulations of high-order statistics in isotropic turbulent flows

The lattice Boltzmann method (LBM) is coupled with the multiple-relaxation-time (MRT) collision model and the three-dimensional 19-discrete-velocity (D3Q19) model to resolve intermittent behaviors on small scales in isotropic turbulent flows. The high-order scaling exponents of the velocity structure functions, the probability distribution functions of Lagrangian accelerations, and the local energy dissipation rates are investigated. The self-similarity of the space-time velocity structure functions is explored using the extended self-similarity (ESS) method, which was originally developed for velocity spatial structure functions. The scaling exponents of spatial structure functions at up to ten orders are consistent with the experimental measurements and theoretical results, implying that the LBM can accurately resolve the intermittent behaviors. This validation provides a solid basis for using the LBM to study more complex processes that are sensitive to small scales in turbulent flows, such as the relative dispersion of pollutants and mesoscale structures of preferential concentration of heavy particles suspended in turbulent flows.

Imaginary particle tracking accelerometry based on time-resolved velocity fields

An accurate calculation of material acceleration is important for particle image velocimetry-based pressure reconstruction. Therefore, an imaginary particle tracking accelerometry (IPTA) approach based on time-resolved velocity fields is described in this paper for a better determination of acceleration. Multi-velocity fields and a least squares polynomial fitting of the velocity along imaginary particle trajectories are introduced to improve the acceleration accuracy. The process of imaginary particle tracking is operated iteratively until a convergence condition is satisfied. Then the Lagrangian acceleration (or the material acceleration in the Eulerian coordinates) is acquired by the first-order time derivation of the fitting polynomial. In addition, the sensitivity of the IPTA approach to different levels of noise and parameters that affect its performance is investigated. A criterion is proposed to determine these parameters when using IPTA to calculate the acceleration. Performance of the IPTA method is compared with other velocity-based accelerometry methods, including both Eulerian and Lagrangian methods. Assessments are conducted in a synthetic solid body rotation flow, a synthetic flow of a vortex ring, and an experimental jet flow. The results show that IPTA is a robust method for experimental acceleration determination that can both improve the accuracy of acceleration and provide better physical characteristics of the flow field.

COLA with scale-dependent growth: applications to screened modified gravity models

We present a general parallelized and easy-to-use code to perform numerical simulations of structure formation using the COLA (COmoving Lagrangian Acceleration) method for cosmological models that exhibit scale-dependent growth at the level of first and second order Lagrangian perturbation theory. For modified gravity theories we also include screening using a fast approximate method that covers all the main examples of screening mechanisms in the literature. We test the code by comparing it to full simulations of two popular modified gravity models, namely f(R) gravity and nDGP, and find good agreement in the modified gravity boost-factors relative to ΛCDM even when using a fairly small number of COLA time steps.

Theoretical accuracy in cosmological growth estimation

We elucidate the importance of the consistent treatment of gravity-model specific non-linearities when estimating the growth of cosmological structures from redshift space distortions (RSD). Within the context of standard perturbation theory (SPT), we compare the predictions of two theoretical templates with redshift space data from COLA (COmoving Lagrangian Acceleration) simulations in the normal branch of DGP gravity (nDGP) and General Relativity (GR). Using COLA for these comparisons is validated using a suite of full N-body simulations for the same theories. The two theoretical templates correspond to the standard general relativistic perturbation equations and those same equations modelled within nDGP. Gravitational clustering non-linear effects are accounted for by modelling the power spectrum up to one loop order and redshift space clustering anisotropy is modelled using the Taruya, Nishimichi and Saito (TNS) RSD model. Using this approach, we attempt to recover the simulation's fiducial logarithmic growth parameter $f$. By assigning the simulation data with errors representing an idealised survey with a volume of $10\mbox{Gpc}^3/h^3$, we find the GR template is unable to recover fiducial $f$ to within 1$\sigma$ at $z=1$ when we match the data up to $k_{\rm max}=0.195h$/Mpc. On the other hand, the DGP template recovers the fiducial value within $1\sigma$. Further, we conduct the same analysis for sets of mock data generated for generalised models of modified gravity using SPT, where again we analyse the GR template's ability to recover the fiducial value. We find that for models with enhanced gravitational non-linearity, the theoretical bias of the GR template becomes significant for stage IV surveys. Thus, we show that for the future large data volume galaxy surveys, the self-consistent modelling of non-GR gravity scenarios will be crucial in constraining theory parameters.

Efficient simulations of large-scale structure in modified gravity cosmologies with comoving Lagrangian acceleration

We implement an adaptation of the COLA approach, a hybrid scheme that combines Lagrangian perturbation theory with an N-body approach, to model non-linear collapse in chameleon and symmetron modified gravity models. Gravitational screening is modeled effectively through the attachment of a suppression factor to the linearized Klein-Gordon equations. The adapted COLA approach is benchmarked, with respect to an N-body code both for the $\Lambda$CDM scenario and for the modified gravity theories. It is found to perform well in the estimation of the dark matter power spectra, with consistency of 1 % to $k\sim2.5$ h/Mpc. Redshift space distortions are shown to be effectively modeled through a Lorentzian parameterization with a velocity dispersion fit to the data. We find that COLA performs less well in predicting the halo mass functions, but has consistency, within $1\sigma$ uncertainties of our simulations, in the relative changes to the mass function induced by the modified gravity models relative to $\Lambda$CDM. The results demonstrate that COLA, proposed to enable accurate and efficient, non-linear predictions for $\Lambda$CDM, can be effectively applied to a wider set of cosmological scenarios, with intriguing properties, for which clustering behavior needs to be understood for upcoming surveys such as LSST, DESI, Euclid and WFIRST.

Lagrangian acceleration statistics in a turbulent channel flow

Lagrangian acceleration statistics in a fully developed turbulent channel flow at $Re_\tau = 1440$ are investigated, based on tracer particle tracking in experiments and direct numerical simulations. The evolution with wall distance of the Lagrangian velocity and acceleration time scales is analyzed. Dependency between acceleration components in the near-wall region is described using cross-correlations and joint probability density functions. The strong streamwise coherent vortices typical of wall-bounded turbulent flows are shown to have a significant impact on the dynamics. This results in a strong anisotropy at small scales in the near-wall region that remains present in most of the channel. Such statistical properties may be used as constraints in building advanced Lagrangian stochastic models to predict the dispersion and mixing of chemical components for combustion or environmental studies.

Resolving vorticity and dissipation in a turbulent boundary layer by tomographic PTV and VIC+

The existing time-resolved tomographic particle image velocimetry (PIV) measurements by Jodai and Elsinga (J Fluid Mech 795:611–633; Jodai, Elsinga, J Fluid Mech 795:611–633, 2016) in a turbulent boundary layer (Reθ = 2038) are reprocessed using tomographic particle tracking velocimetry (PTV) and vortex-in-cell-plus (VIC+). The resulting small-scale flow properties, i.e. vorticity and turbulence dissipation, are compared. The VIC+ technique was recently proposed and uses the concept of pouring time into space to increase reconstruction quality of instantaneous velocity. The tomographic PTV particle track measurements are interpolated using VIC+ to a dense grid, making use of both particle velocity and Lagrangian acceleration. Comparison of the vortical structures by visualization of isosurfaces of vorticity magnitude shows that the two methods return similar coherent vortical structures, but their strength in terms of vorticity magnitude is increased when using VIC+, which suggests an improvement in spatial resolution. Further statistical evaluation shows that the root mean square (rms) of vorticity fluctuations from tomographic PIV is approximately 40% lower in comparison to a reference profile available from a DNS simulation, while the VIC+ technique returns rms vorticity fluctuations to within 10% of the reference. The dissipation rate is heavily underestimated by tomographic PIV with approximately 50% damping, whereas the VIC+ analysis yields a dissipation rate to within approximately 5% for y+ > 25. The fact that dissipation can be directly measured by a volumetric experiment is novel. It differs from existing approaches that involve 2d measurements combined with isotropic turbulence assumptions or apply corrections based on sub-grid scale turbulence modelling. Finally, the study quantifies the spatial response of VIC+ with a sine-wave lattice analysis. The results indicate a twofold increase of spatial resolution with respect to cross-correlation interrogation.

Characterisation of the Eulerian and Lagrangian accelerations in the intermediate field of turbulent circular jets

ABSTRACTA rich data-set of Lagrangian trajectories from 3D particle tracking velocimetry is used to study the structure of various acceleration components, vorticity, and strain in the intermediate field of a circular jet at Reynolds number Re = 6000. The total acceleration is decomposed into three distinctive sets: (1) streamwise–radial; (2) tangential–normal; and (3) local–convective components. Probability density function (PDF) and joint distributions of each set are characterised at various radial locations from the jet core within a streamwise band 16 ≤ x/dh ≤ 17, where dh is the diameter of the pipe. The PDF of the relative angle between the acceleration components and the velocity vector is also included to aid the characterisation. Results show that the acceleration components are described by two distinctive distributions: one of them exhibits symmetry and heavy tails, while the other is best fitted by a power-law type. The tails of acceleration PDFs are heavier with larger radial distance from the core. The increased departure from the Gaussian distribution with the distance from the core is a result of the increasing turbulence levels promoted by the mean shear. The variation of the third and fourth moments between the streamwise–tangential and the radial–normal accelerations indicate the anisotropy of the jet. Joint PDF of each acceleration decomposition exhibits distinctive distribution that appears to depend from the distance from the jet core. However, the vorticity and strain show similar PDF across radial distances. Finally, complementary analysis of a jet from a semicircular pipe shows the footprint of the nozzle geometry in the acceleration structure of jets.