Computation Of Particle Trajectories Research Articles

A time-step-robust algorithm to compute particle trajectories in 3-D unstructured meshes for Lagrangian stochastic methods

Abstract The purpose of this paper is to propose a time-step-robust cell-to-cell integration of particle trajectories in 3-D unstructured meshes in particle/mesh Lagrangian stochastic methods. The main idea is to dynamically update the mean fields used in the time integration by splitting, for each particle, the time step into sub-steps such that each of these sub-steps corresponds to particle cell residence times. This reduces the spatial discretization error. Given the stochastic nature of the models, a key aspect is to derive estimations of the residence times that do not anticipate the future of the Wiener process. To that effect, the new algorithm relies on a virtual particle, attached to each stochastic one, whose mean conditional behavior provides free-of-statistical-bias predictions of residence times. After consistency checks, this new algorithm is validated on two representative test cases: particle dispersion in a statistically uniform flow and particle dynamics in a non-uniform flow.

Under What Conditions Does Transverse Macrodispersion Exist in Groundwater Flow?

AbstractIn recent years there has been vigorous debate whether asymptotic transverse macrodispersion exists in steady three‐dimensional (3D) groundwater flows in the purely advective limit. This question is tied to the topology of 3D flow paths (termed the Lagrangian kinematics), specifically whether streamlines can undergo braiding motions or can wander freely in the transverse direction. In this study we determine which Darcy flows do admit asymptotic transverse macrodispersion for purely advective transport on the basis of the conductivity structure. We prove that porous media with smooth, locally isotropic hydraulic conductivity exhibit zero transverse macrodispersion under pure advection due to constraints on the Lagrangian kinematics of these flows, whereas either non‐smooth or locally anisotropic conductivity fields can generate transverse macrodispersion. This has implications for upscaling locally isotropic porous media to the block scale as this can result in a locally anisotropic conductivity, leading to non‐zero macrodispersion at the block scale that is spurious in that it does not arise for the fully resolved Darcy scale flow. We also show that conventional numerical methods for computation of particle trajectories do not explicitly preserve the kinematic constraints associated with locally isotropic Darcy flow, and propose a novel psuedo‐symplectic method that preserves these constraints. These results provide insights into the mechanisms that govern transverse macrodispersion in groundwater flow, and unify seemingly contradictory results in the literature in a consistent framework. These insights call into question the ability of smooth, locally isotropic conductivity fields to represent flow and transport in real heterogeneous porous media.

Finite-volume flux reconstruction and semi-analytical particle tracking on triangular prisms for finite-element-type models of variably-saturated flow

Consistent particle tracking relies on conforming velocity fields that ensure local mass conservation on elements. Cell-centered finite-volume and mixed finite-element methods result in conforming velocity fields but this is not the case for continuous Galerkin methods, such as the standard finite element method (FEM). Nonetheless standard FEM is often used for subsurface flow modeling because it yields a continuous approximation of hydraulic heads, and it naturally handles unstructured grids and full material tensors. Acknowledging these advantages and the wide-spread use of finite-element-type simulations, we present a postprocessing method that reconstructs a cell-centered finite-volume solution from a finite-element-type solution of the variably-saturated subsurface flow equation to obtain conforming, mass-conservative fluxes. Using the linear average velocity field derived from these fluid fluxes, we employ element-wise analytical solutions for triangular prisms to compute particle trajectories and associated travel times. As a result, we can compute consistent particle trajectories for variably-saturated flow solutions generated by node-centered methods, such as finite element or finite difference methods, that do not yield conforming velocity fields. Our flux reconstruction solves a linear elliptic problem whose size is on the order of the number of elements, which is computationally much faster than solving the initial, non-linear transient variably-saturated flow equation. Compared to other postprocessing schemes, our flux reconstruction is numerically stable, fast to compute, and does not induce severe numerical artifacts when applied to heterogeneous domains with strongly varying velocities. However, these advantages come with a comparably high coding effort and the necessity of solving a global system of equations. We show that the results of our flux reconstruction are close to the node-centered primal solution for variably saturated three-dimensional flow with heterogeneous material properties.

On the origin of jet-like features in bow shock pulsar wind nebulae

ABSTRACT Bow shock pulsar wind nebulae are a large class of non-thermal synchrotron sources associated to old pulsars that have emerged from their parent supernova remnant and are directly interacting with the interstellar medium. Within this class a few objects show extended X-ray features, generally referred as ‘jets’, that defies all the expectations from the canonical MHD models, being strongly misaligned respect to the pulsar direction of motion. It has been suggested that these jets might originate from high energy particles that escape from the system. Here we investigate this possibility, computing particle trajectories on top of a 3D relativistic MHD model of the flow and magnetic field structure, and we show not only that beamed escape is possible, but that it can easily be asymmetric and charge separated, which as we will discuss are important aspects to explain known objects.

Quantifying Discretization Errors in Electrophoretically-Guided Micro Additive Manufacturing.

This paper presents process models for a new micro additive manufacturing process termed Electrophoretically-guided Micro Additive Manufacturing (EPμAM). In EPμAM, a planar microelectrode array generates the electric potential distributions which cause colloidal particles to agglomerate and deposit in desired regions. The discrete microelectrode array nature and the used pulse width modulation (PWM) technique for microelectrode actuation create unavoidable process errors—space and time discretization errors—that distort particle trajectories. To combat this, we developed finite element method (FEM) models to study trajectory deviations due to these errors. Mean square displacement (MSD) analysis of the computed particle trajectories is used to compare these deviations for several electrode geometries. The two top-performing electrode geometries evaluated by MSD were additionally investigated through separate case studies via geometry variation and MSD recomputation. Furthermore, separate time-discretization error simulations are also studied where electrode actuating waveforms were simulated. The mechanical impulse of the electromechanical force, generated from these waveforms is used as the basis for comparison. The obtained results show a moderate MSDs variability and significant differences in the computed mechanical impulses for the actuating waveforms. The observed limitations of the developed process model and of the error comparison technique are briefly discussed and future steps are recommended.

An Approximate Time-Parallel Method for the Fast and Accurate Computation of Particle Trajectories in a Magnetic Field*

A temporal multiscale hybridization method is presented that carefully couples coarse scale gyrokinetic models with exact charged particle solution trajectories (that is, with full phase information) in a magnetic field. The approach is based on the careful approximation of a sum, generally employed for time-parallel (TP) computing applications. While the hybridization method presented is highly parallelizable, a computational efficiency gain is seen from considering serial computations only. A complete numerical method is only presented for the aforementioned charged particle application, however, the general approach depicted likely has relevance to a wide swath of challenging multiscale/multiphysics problems. Additionally, the approach has obvious relevance to TP computing applications (such as variable selection on which to perform TP calculations and fine scale sampling strategies).

GPU acceleration of Eulerian–Lagrangian particle-laden turbulent flow simulations

Acceleration of an existing MPI-based, particle-laden turbulent flow simulation code is achieved using up to four NVIDIA GPU devices. The overall design is to transfer the entire flow velocity, temperature, and humidity fields to each device, and compute particle trajectories entirely on the GPU hardware. For one-way coupled turbulent flow simulations, accurate simulations can be achieved for less computational cost than the original CPU implementation for particle numbers above 106. Above 107 particles, the GPU version is roughly 14 times faster than the original CPU implementation. The effects of interpolation order, precision of the transferred Eulerian fields, and sub-time-stepping for fast particle dynamics are discussed.

Numerical investigation of the interaction of a finite-size particle with a tangentially moving boundary

The motion of a finite-size particle in steady two-dimensional lid- and shear-driven square-cavity flows is investigated. The coupled equations are solved numerically using a discontinuous Galerkin-finite-element method (DG-FEM) combined with the so-called smoothed-profile method (SPM). The spectral convergence enables an accurate and efficient computation of particle trajectories without moving grids. Particle trajectories are obtained by solving the Navier–Stokes and Newtons’s equations for the particle motion using simulations without additional model assumptions fully resolving all scales down to the flow in the lubrication gap. Particle trajectories are compared with streamlines and trajectories from one-way coupling. In the shear-driven square cavity with convex streamlines finite-size particles suffer a significant displacement effect when passing the moving boundary closely. While inertia displaces the particle towards outer streamlines, the finite-size effect alone displaces the particle towards inner streamlines. For weakly inertial particles the latter displacement effect is shown to be qualitatively similar to the displacement modelled by inelastic collision in a one-way-coupling approach.

Parallel dilute particulate flow simulations in the human nasal cavity

When simulating time-dependent particulate flows, one faces the dilemma that the domain decomposition used for fluid simulation is not optimal for parallel computation of particle trajectories. Therefore, the article proposes and compares two parallelization strategies for the particle phase based on the fixed domain decomposition approach used in the open source lattice Boltzmann framework OpenLB (http://www.openlb.net). The communication optimal strategy is found to be more efficient in the case of homogeneously distributed particles. Convergence studies and performance tests are conducted using a simplified geometry of the human lungs and show excellent parallel speedup. The implemented strategy is used to simulate time-dependent particulate flows of micro-particles in a patient-specific geometry of a human nasal cavity including paranasal sinuses. Dilute, uniformly distributed particles are released once at the start of inspiration, as well as repeatedly during the entire inspiratory cycle, which leads to a more homogeneous distribution. It is found that the deposition rates vary for the different injection methods.

On the use of the finite-time Lyapunov exponent to reveal complex flow physics in the wake of a mechanical valve

The finite-time Lyapunov exponent (FTLE) is a popular tool to extract characteristic features of flows that cannot be revealed by other criteria. However, even if the computational cost of computing particle trajectories in space and time has been reduced and optimized in a considerable number of works, the main challenge probably consists in increasing the spatial resolution of Lagrangian coherent structures locally, i.e., where the FTLE field reaches its maximum values. On the other hand, most of experimental data are obtained in planes so that the FTLE field is computed without the out-of-plane particle movements. To investigate which physics the FTLE can capture in flows that are highly three dimensional, the criterion is computed from high-speed stereoscopic particle image velocimetry measurements of the pulsatile flow that develops behind a bi-leaflet mechanical heart valve. The similitude is based on the Womersley number, and experiments are performed for a lower Reynolds number than the physiologic value to obtain sufficiently resolved data in space and time. It is found that the vortex shedding is well captured and that its development can be decomposed into four successive phases. The longest phase occurs near the peak flow rate and exhibits a break of symmetry similar to the one appearing in the wakes of two side-by-side cylinders in the regime when the separation between the cylinders is of the order of their diameter. Specifically, a vortex street with alternating vortex sheddings is observed in a narrow wake behind one of the leaflets, whereas single large vortices develop inside a wide wake downstream of the other leaflet. It appears that these patterns are difficult or even impossible to discern with classical Eulerian vortex identification techniques. The Strouhal numbers of vortex-shedding frequencies, obtained from continuous wavelet transforms and based on the apparent height of the leaflets, are also close to those found in the flow behind two cylinders. By invoking the Taylor hypothesis, an approximate three-dimensional reconstruction of the flow can be obtained and a three-dimensional FTLE field is deduced, which provides a very detailed view of the vortex structures that form and develop in the wakes of the leaflets.

Material flow visualization in Friction Stir Welding via particle tracing

This work deals with the modeling of the material flow in Friction Stir Welding (FSW) processes using particle tracing method. For the computation of particle trajectories, three accurate and computationally efficient integration methods are implemented within a FE model for FSW process: the Backward Euler with Sub-stepping (BES), the 4-th order Runge–Kutta (RK4) and the Back and Forth Error Compensation and Correction (BFECC) methods. Firstly, their performance is compared by solving the Zalesak’s disk benchmark. Later, the developed methodology is applied to some FSW problems providing a quantitative 2D and 3D view of the material transport in the process area. The material flow pattern is compared to the experimental evidence.

Lagrangian transport in a microtidal coastal area: the Bay of Palma, island of Mallorca, Spain

Abstract. Coastal transport in the Bay of Palma, a small region in the island of Mallorca, Spain, is characterized in terms of Lagrangian descriptors. The data sets used for this study are the output for two months (one in autumn and one in summer) of a high resolution numerical model, ROMS (Regional Ocean Model System), forced atmospherically and with a spatial resolution of 300 m. The two months were selected because of their different wind regime, which is the main driver of the sea dynamics in this area. Finite-size Lyapunov exponents (FSLEs) were used to locate semi-persistent Lagrangian coherent structures (LCS) and to understand the different flow regimes in the bay. The different wind directions and regularity in the two months have a clear impact on the surface bay dynamics, whereas only topographic features appear clearly in the bottom structures. The fluid interchange between the bay and the open ocean was studied by computing particle trajectories and residence time (RT) maps. The escape rate of particles out of the bay is qualitatively different, with a 32% greater escape rate of particles to the ocean in October than in July, owing to the different geometric characteristics of the flow. We show that LCSs separate regions with different transport properties by displaying spatial distributions of residence times on synoptic Lagrangian maps together with the location of the LCSs. Correlations between the time-dependent behavior of FSLE and RT are also investigated, showing a negative dependence when the stirring characterized by FSLE values moves particles in the direction of escape.

CFD methodology for sedimentation tanks: The effect of secondary phase on fluid phase using DPM coupled calculations

Computational Fluid Dynamics (CFD) methods are employed in order to simulate the 3D hydrodynamics and flow behaviour in a sedimentation tank. Unlike most of the previous numerical investigations, in the present paper the momentum exchange between the primary and the secondary phase is taken into account, using a Lagrangian method (discrete phase model) with two-way coupled calculations. By computing particle trajectories the proposed numerical model can track the momentum gained or lost by the particle stream that follows that trajectory and these quantities can be incorporated in the subsequent continuous phase calculations. Thus, while the continuous phase always impacts the discrete phase, the effect of the discrete phase trajectories on the continuum can be incorporated. This interchange affects fluid velocity, especially in the case of large particles sizes, which have a greater relaxation time in relation to the characteristic time of the tank. The present investigation compares a series of numerical simulations for a sedimentation tank with varying particle diameters and volume fractions, in order to identify the influence of the secondary phase to the primary phase and vice-versa and the way that this influence affects the efficiency of the tank.

Planar positron channeling radiation generated in an acoustic superlattice

Planar positron channeling in an acoustic superlattice and stimulation of the emission of channeling radiation at resonance of the particle motion with the acoustic field excited in a PbTiO3 single crystal is considered in the framework of classical mechanics and electrodynamics. Based on computed particle trajectories, spectral-angular distributions of channeling radiation influenced by resonant ultrasound have been simulated. The presented method explains the influence of ultrasonic waves on the intensity of channeling radiation in a rather simple and descriptive manner.

Lattice Boltzmann simulations of pinched flow fractionation

Direct numerical simulations of microsphere motion through a microfluidic separation device (pinched flow fractionation, PFF, device) were performed using the lattice Boltzmann method. The results were compared with the original experimental work on PFF by Yamada et al. (2004) . The effects of the pinched segment width and the ratio between the particle solution and diluent flow rates were studied. Both analyses showed agreement with the experimental trends. Previous modelling of PFF has relied on the assumption that particles follow streamlines, and this assumption was evaluated. The simulations indicated that large particles experience a lift force due to a region of low pressure between the particle and the wall of the pinch. The lift force attracts the particles to the wall as they exit the pinched segment. Smaller particles experience a much weaker force. This force may provide an explanation for the experimental observation that a sharp expansion provides better separation performance than a gradual expansion, an effect that cannot be explained by streamline analyses. ► Microfluidic pinched flow fractionation was simulated. ► Computed particle trajectories were consistent with experimental work in the literature. ► Particles do not follow streamlines when they interact with walls. ► Deviation was caused by low pressure region between particle and wall.

Ballistics Model for Particles on a Horizontal Plane in a Vacuum Propelled by a Vertically Impinging Gas Jet

In previous work, a time-stepped numerical algorithm to compute particle trajectories consisting of lunar soil blown by the engine exhaust of a lunar lander was developed. The time-stepped integration method relies on the gas density, velocity, and temperature fields, calculated by computational fluid dynamics simulations, to compute the forces and accelerations acting on single noninteracting particles. In this paper, a computationally efficient particle ballistics model is presented where the trajectory is estimated by computing the vertical position (axial coordinate x) as a function of the horizontal position (radial coordinate y) using a constant horizontal velocity and a vertical acceleration approximated as a power-law. The unknown parameters of the model are determined by fitting the ballistics trajectory path to a matrix of trajectories generated by the time-step integration method using the rocket exhaust gas properties predicted by computational fluid dynamics software. Also in previous work, a strictly empirical trajectory model was developed to satisfy the need for a computationally efficient method of computing particle trajectories. This new model (like the previous model) is expressed as a time-independent trajectory path function. However, the method of this current work is based on physical laws of motion, unlike the previous empirical model.

A Novel Anisotropic Fast Marching Method and its Application to Blood Flow Computation in Phase-contrast MRI

Phase-contrast MRI (PC MRI) can be used to assess blood flow dynamics noninvasively inside the human body. The acquired images can be reconstructed into flow vector fields. Traditionally, streamlines can be computed based on the vector fields to visualize flow patterns and particle trajectories. The traditional methods may give a false impression of precision, as they do not consider the measurement uncertainty in the PC MRI images. In our prior work, we incorporated the uncertainty of the measurement into the computation of particle trajectories. As a major part of the contribution, a novel numerical scheme for solving the anisotropic Fast Marching problem is presented. A computing time comparison to state-of-the-art methods is conducted on artificial tensor fields. A visual comparison of healthy to pathological blood flow patterns is given. The comparison shows that the novel anisotropic Fast Marching solver outperforms previous schemes in terms of computing time. The visual comparison of flow patterns directly visualizes large deviations of pathological flow from healthy flow. The novel anisotropic Fast Marching solver efficiently resolves even strongly anisotropic path costs. The visualization method enables the user to assess the uncertainty of particle trajectories derived from PC MRI images.

Numerical Simulation of Roughness Effects Inside a Brick-lined Cyclone Separator

An industrial cyclone used for separating dust from a hot gas mixture was studied by means of numerical simulation. Because of the high gas temperatures (about 1075 K), the inner surfaces of the cyclone separator and the inlet pipe must be lined with refractory bricks, causing high wall roughness. The virtual wall model was used to simulate this roughness, and a new model extension accounting for joints between the bricks is proposed and validated experimentally. As the cyclone inlet mass load was relatively low (0.01 kg/kg), the Eulerian-Lagrangian approach was used for the simulations. The continuous phase conservation equations were solved in the Eulerian reference frame on a fixed grid, whereas the discrete phase properties were determined by computing particle trajectories through the computational domain in a Lagrangian reference frame. Besides inlet particle diameters ranging from 1 μm to 100 μm, different particle materials with varying particle densities had to be considered. The measurements and the Muschelknautz analytical model were compared to the numerical simulations regarding pressure loss, particle spectra at the outlet, and the fractional separation efficiency. The Muschelknautz model exhibited some difficulties with the kind of wall roughness considered here (especially joints between bricks). Moreover, this method was not designed to handle multiple different particle densities simultaneously. Nevertheless, comparisons show good agreement between numerical simulations, measurements and theoretical predictions.

Numerical simulations of granular dynamics: I. Hard-sphere discrete element method and tests

We present a new particle-based (discrete element) numerical method for the simulation of granular dynamics, with application to motions of particles on small solar system body and planetary surfaces. The method employs the parallel N-body tree code pkdgrav to search for collisions and compute particle trajectories. Collisions are treated as instantaneous point-contact events between rigid spheres. Particle confinement is achieved by combining arbitrary combinations of four provided wall primitives, namely infinite plane, finite disk, infinite cylinder, and finite cylinder, and degenerate cases of these. Various wall movements, including translation, oscillation, and rotation, are supported. We provide full derivations of collision prediction and resolution equations for all geometries and motions. Several tests of the method are described, including a model granular “atmosphere” that achieves correct energy equipartition, and a series of tumbler simulations that show the expected transition from tumbling to centrifuging as a function of rotation rate.

Mixing processes in the vitreous chamber induced by eye rotations

In this paper, we study a model of flow in the vitreous humour in the posterior chamber of the human eye, induced by saccadic eye rotations. We concentrate on the effect of the shape of the chamber upon the mixing properties of the induced flows. We make particle image velocimetry measurements of the fluid velocity in a transparent plastic (Perspex) model of the posterior chamber during sinusoidal torsional oscillations about a vertical axis. We use a Newtonian fluid to model the vitreous humour, which is most realistic when either the vitreous humour is liquefied or has been replaced by purely viscous tamponade fluids. The model of the posterior chamber is a sphere with an indentation, representing the effect of the lens. In spite of the purely periodic forcing, a steady streaming flow is generated, which plays a fundamental role in the mixing processes in the domain. The streaming flow differs markedly from that in a perfect sphere, and its topological characteristics change substantially as the frequency of oscillation varies. We discuss the flow characteristics in detail and show that, for physiological parameter values, the Péclet number (based on a suitable measure of the steady streaming velocity) is large, suggesting that advection strongly dominates over diffusion for mass transport phenomena. We also compute particle trajectories based on the streaming velocity and use these to investigate the stirring properties of the flow.