Time-dependent Velocity Field Research Articles

Control of neural transport for normalising flows

Inspired by normalising flows, we analyse the bilinear control of neural transport equations by means of time-dependent velocity fields restricted to fulfil, at any time instance, a simple neural network ansatz. The L1 approximate controllability property is proved, showing that any probability density can be driven arbitrarily close to any other one in any time horizon. The control vector fields are built explicitly and inductively and this provides quantitative estimates on their complexity and amplitude. This also leads to statistical error bounds when only random samples of the target probability density are available.

Migrating Carrollian particles on magnetized black hole horizons

By considering a misaligned (asymptotically uniform) magnetic field in the background of a rotating black hole, we uncover a possibility for a highly nontrivial motion of Carrollian particles on the black hole horizon that is characterized by a time-dependent velocity field and reminds us (because of its latitudinal oscillations) of a ``monarch butterfly migration.''

Vortex-induced sound prediction of slat noise from time-resolved particle image velocimetry data

A data-driven method for predicting the vortex-induced sound from time-resolved velocimetry data is presented and applied to the sound generated by flow passing through the slat in a multi-element high-lift airfoil. The time-dependent velocity fields in the slat-cove region of the 30P30N multi-element airfoil are obtained from time-resolved particle image velocimetry measurements, and a low-order reconstruction is achieved by using the rank-one vbnm modes from spectral proper orthogonal decomposition. The pressure force and associated dipole sound are then computed via the application of the force and acoustic partitioning methods (Seo et al. in Phys Fluids 34(5):053607, 2002) which involve volume integrals of the product of the second invariant of the velocity gradient tensor and geometry-dependent influence fields. The method enables estimation of the dipole sound generated by local flow structures, and the results are shown to be consistent with theory of vortex sound. Comparison with the measured sound data suggests that while the shear layer modes are responsible for the tonal noise, the interactions between the shear layer modes and other parts of the wing also generate a substantial level of flow noise.

Computational study of three-dimensional Lagrangian transport and mixing in a stirred tank reactor

The detection of compartments and dead zones as well as the estimation of the mixing efficiency in stirred tanks are of vital interest for a variety of biochemical and chemical processes. Here, numerically derived time-dependent 3D fluid velocity fields of a stirred tank reactor are computed using the Lattice Boltzmann Method. Mixing in the stirred tank reactor is analysed by means of Lagrangian Coherent Structures which allow to unravel the mixing states of complex flows. This Lagrangian analysis is achieved by computing Finite Time Lyapunov Exponents and applying recent trajectory-based network methods on the three-dimensional flow. The results reveal a zone of low interaction in the upper region of the stirred tank reactor and five additional transient compartments. The trajectory-based network analysis detects low cross-mixing of fluid parcels between different compartments but a very high mixing of fluid parcels inside of each compartment. This high interaction is also found in an analysis of the Finite Time Lyapunov Mixing Intensity. Time-averaging of the fluid velocity field prior to the Lagrangian analysis is considered to extract the most influential Lagrangian Coherent Structures.

Self-Similar Solutions of a Gravitating Dark Fluid

In this paper, a fluid model is presented which contains the general linear equation of state including the gravitation term. The obtained spherical symmetric Euler equation and the continuity equations were investigated with the Sedov-type time-dependent self-similar ansatz which is capable of describing physically relevant diffusive and disperse solutions. The result of the space and time-dependent fluid density and radial velocity fields are presented and analyzed. Additionally, the role of the initial velocity on the kinetic and total energy densities of the fluid is discussed. This leads to a model, which can be considered as a simple model for a dark-fluid.

Planktos: An Agent-Based Modeling Framework for Small Organism Movement and Dispersal in a Fluid Environment with Immersed Structures.

Multiscale modeling of marine and aerial plankton has traditionally been difficult to address holistically due to the challenge of resolving individual locomotion dynamics while being carried with larger-scale flows. However, such problems are of paramount importance, e.g., dispersal of marine larval plankton is critical for the health of coral reefs, and aerial plankton (tiny arthropods) can be used as effective agricultural biocontrol agents. Here we introduce the open-source, agent-based modeling software Planktos targeted at 2D and 3D fluid environments in Python. Agents in this modeling framework are relatively tiny organisms in sufficiently low densities that their effect on the surrounding fluid motion can be considered negligible. This library can be used for scientific exploration and quantification of collective and emergent behavior, including interaction with immersed structures. In this paper, we detail the implementation and functionality of the library along with some illustrative examples. Functionality includes arbitrary agent behavior obeying either ordinary differential equations, stochastic differential equations, or coded movement algorithms, all under the influence of time-dependent fluid velocity fields generated by computational fluid dynamics, experiments, or analytical models in domains with static immersed mesh structures with sliding or sticky collisions. In addition, data visualization tools provide images or animations with kernel density estimation and velocity field analysis with respect to deterministic agent behavior via the finite-time Lyapunov exponent.

Dispersion of a fluid plume during radial injection in an aquifer

This study outlines a model for radial injected fluid flow with mechanical dispersion in a vertically confined porous aquifer. Existing studies have investigated fully segregated fluid flows in this setting, where the injected fluid and resident fluid form a propagating sharp interface. The present study uses the geometry of these sharp interfaces as a basis for the velocity field to take into account dispersion and buoyancy/viscosity effects. By differentiating the radial position of the sharp interface with respect to time, a time dependent radial velocity field governing the flow is obtained. Evaluating this radial velocity at the moment the original interface were to intersect a given position gives a velocity field that is a function of the position coordinates inside the aquifer. Using this velocity field, the fluids saturation profile resulting from mechanical dispersion can be found analytically. It is shown that the concentration of the injected fluid smoothly decays around the position of the corresponding sharp interface, allowing for the injected fluid to be present in detectable quantities beyond the extent of these solutions. This concentration spread should be considered in defining outer boundaries on fluids in injection well projects such as carbon sequestration or groundwater applications.

Experimental Observation of Lagrangian Coherent Structures in Perturbed Rayleigh-Benard Convection

Abstract Rayleigh-Benard convection is natural convection that appears in a fluid layer with heated bottom and cooled top planes. The velocity fields of the Rayleigh-Benard convection may oscillate slightly by even oscillatory instability by controlling the temperature difference of the two planes. In such flow, some fluid particles may be transported chaotically in Lagrangian description, even when the velocity fields seem to be stable in Eulerian description. In order to clarify the chaotic structures of the flow, it is essential to identify the invariant structures associated with the velocity fields. Therefore, in this study, we measure the two-dimensional velocity fields of the perturbed Rayleigh-Benard convection by Particle Image Velocimetry (PIV) and numerically detect the Lagrangian coherent structures (LCSs), which correspond to invariant manifolds in autonomous systems, by integrating the obtained time-dependent velocity fields.

ResNet-LDDMM: Advancing the LDDMM Framework Using Deep Residual Networks.

In deformable registration, the Riemannian framework - Large Deformation Diffeomorphic Metric Mapping, or LDDMM for short - has inspired numerous techniques for comparing, deforming, averaging and analyzing shapes or images. Grounded in flows of vector fields, akin to the equations of motion used in fluid dynamics, LDDMM algorithms solve the flow equation in the space of plausible deformations, i.e., diffeomorphisms. In this work, we make use of deep residual neural networks to solve the non-stationary ODE (flow equation) based on an Euler's discretization scheme. The central idea is to represent time-dependent velocity fields as fully connected ReLU neural networks (building blocks) and derive optimal weights by minimizing a regularized loss function. Computing minimizing paths between deformations, thus between shapes, turns to find optimal network parameters by back-propagating over the intermediate building blocks. Geometrically, at each time step, our algorithm searches for an optimal partition of the space into multiple polytopes, and then computes optimal velocity vectors as affine transformations on each of these polytopes. As a result, different parts of the shape, even if they are close (such as two fingers of a hand), can be made to belong to different polytopes, and therefore be moved in different directions without costing too much energy. Importantly, we show how diffeomorphic transformations, or more precisely bilipshitz transformations, are predicted by our registration algorithm. We illustrate these ideas on diverse registration problems of 3D shapes under complex topology-preserving transformations. We thus provide essential foundations for more advanced shape variability analysis under a novel joint geometric-neural networks Riemannian-like framework, i.e., ResNet-LDDMM.

Time-resolved velocity and pressure field quantification in a flow-focusing device for ultrafast microbubble production

We present an experimental study of the time-dependent velocity and pressure fields during high-speed microbubble production in a flow-focusing device. Particle tracking velocimetry is used to resolve velocities up to 30 m/s at timescales down to 100 ns, while numerical simulations give access to both the gas and liquid velocity and pressure oscillations.

Large eddy fire simulation applications from nuclear industry

Abstract A computational study has been carried out for predicting the behaviour of a pool fire source using the field-model based code Fire Dynamics Simulator (FDS). Time dependent velocity and temperature fields are predicted along with the resulting changes in the plume structure and its width. Firstly, a grid study was performed to find out the best grid size for this purpose. Then calculations were done which showed a very good agreement with earlier reported experimental based correlations for the temperature of the plume region. These studies have been extended to use this field-model based tools for modelling particular separate effect phenomena like puffing frequency and to validate against experimental data. There are several applications in nuclear industry like room fires, wildland fires, smoke or ash disposal, hydrogen transport in nuclear reactor containment, natural convection in building flows etc. In this paper the use of FDS with the advanced Large Eddy Simulation (LES) based CFD turbulence model is described for various applications: Fire simulation for Alpha storage, Bhabhatran teletherapy, pool fire for transport casks, fire PSA of a representative NPP, exhaust air fan buildings of a process plant and smoke dispersion in large fires around NPPs.

Experimental quantification of oscillating flow in finite-length straight elastic vessels for Newtonian and non-Newtonian fluids

To better understand the complex flow in arteries it is necessary to analyze the fluid–structure interaction between the time dependent velocity field, the non-Newtonian fluid, and the elastic blood vessels and to investigate the wall-shear stress distribution that develops in the elastic vessel during an oscillation cycle. The scope of this study is to quantify the influence of the viscoelastic fluid properties on the fluid–structure interaction and the wall-shear stress of a straight elastic vessel by analyzing oscillatory flow of both a Newtonian reference fluid and a non-Newtonian fluid. Unlike in Womersley’s analyses, an elastic vessel of finite length with a non-Newtonian fluid is investigated in this study. Furthermore, quantitative benchmark data for numerical fluid–structure interaction methods are provided. Time-resolved particle-image velocimetry, static pressure measurements, and wall detection are used to measure the velocity field, the pressure distribution, and the vessel dilatation with high temporal and spatial resolution. The mechanical properties of the vessel material are determined by a dynamic mechanical analysis and the viscosities of the Newtonian and the non-Newtonian fluid are measured by a rheometer. The fluid–structure interaction is measured for a sinusoidal oscillating flow in a Reynolds number range 472≤ReNF≤971 and 503≤ReNNF≤1065 for the Newtonian reference fluid and the non-Newtonian fluid. The Womersley number range is 5.97≤WoNF≤8.53 for the Newtonian reference fluid and 6.14≤WoNNF≤8.79 for the non-Newtonian fluid. The results show that the wall-shear stress maximum increases up to 32%–38% for the non-Newtonian fluid despite a 10%–45% higher wall-shear rate for the Newtonian reference fluid. Furthermore, the amplitudes of the dilatation and the pressure in the vessel are more pronounced for the non-Newtonian fluid.

The dynamics of parallel-plate and cone–plate flows

Rotational rheometers are the most commonly used devices to investigate the rheological behavior of liquids in shear flows. These devices are used to measure rheological properties of both Newtonian and non-Newtonian, or complex, fluids. Two of the most widely used geometries are flow between parallel plates and flow between a cone and a plate. A time-dependent rotation of the plate or cone is often used to study the time-dependent response of the fluid. In practice, the time dependence of the flow field is ignored, that is, a steady-state velocity field is assumed to exist throughout the measurement. In this study, we examine the dynamics of the velocity field for parallel-plate and cone–plate flows of Newtonian fluids by finding analytical solutions of the Navier–Stokes equation in the creeping flow limit. The time-dependent solution for parallel-plate flow is relatively simple as it requires the velocity to have a linear dependence on radial position. Interestingly, the time-dependent solution for cone–plate flow does not allow the velocity to have a linear dependence on radial position, which it must have at the steady state. Here, we examine the time-dependent velocity fields for these two flows, and we present results showing the time dependence of the torque exerted on both the stationary and rotating fixtures. We also examine the time dependence of spatial non-homogeneities of the strain rate. Finally, we speculate on the possible implications of our results in the context of shear banding, which is often observed in parallel-plate and cone–plate flows of complex fluids.

Observing traveling waves in glaciers with remote sensing: new flexible time series methods and application to Sermeq Kujalleq (Jakobshavn Isbræ), Greenland

Abstract. The recent influx of remote sensing data provides new opportunities for quantifying spatiotemporal variations in glacier surface velocity and elevation fields. Here, we introduce a flexible time series reconstruction and decomposition technique for forming continuous, time-dependent surface velocity and elevation fields from discontinuous data and partitioning these time series into short- and long-term variations. The time series reconstruction consists of a sparsity-regularized least-squares regression for modeling time series as a linear combination of generic basis functions of multiple temporal scales, allowing us to capture complex variations in the data using simple functions. We apply this method to the multitemporal evolution of Sermeq Kujalleq (Jakobshavn Isbræ), Greenland. Using 555 ice velocity maps generated by the Greenland Ice Mapping Project and covering the period 2009–2019, we show that the amplification in seasonal velocity variations in 2012–2016 was coincident with a longer-term speedup initiating in 2012. Similarly, the reduction in post-2017 seasonal velocity variations was coincident with a longer-term slowdown initiating around 2017. To understand how these perturbations propagate through the glacier, we introduce an approach for quantifying the spatially varying and frequency-dependent phase velocities and attenuation length scales of the resulting traveling waves. We hypothesize that these traveling waves are predominantly kinematic waves based on their long periods, coincident changes in surface velocity and elevation, and connection with variations in the terminus position. This ability to quantify wave propagation enables an entirely new framework for studying glacier dynamics using remote sensing data.

PIV mapping of pressure and velocity fields in the plane magnetohydrodynamic Couette flow

We present the first simultaneous mapping of two-dimensional, time-dependent velocity and pressure fields in a plane Couette flow pervaded by a transverse magnetic field. While electromagnetic forces are strongest in fluids of high electric conductivity such as liquid metals, their opacity excludes optical optical measurement methods. We circumvent this difficulty using a transparent electrolyte (Sulfuric acid), whose weaker conductivity is offset by higher magnetic fields. We describe an experimental rig based on this idea, where the Couette flow is entrained by a tape immersed in sulfuric acid and positioned flush onto the bore of large superconducting magnet, so that most of the flow is pervaded by a sufficiently homogeneous transverse magnetic field. Velocity and pressure fields are obtained by means of a bespoke PIV system, capable of recording the fluid's acceleration as well as its velocity. Both fields are then fed into a finite difference solver that extracts the pressure field from the magnetohydrodynamic governing equations. This method constitutes the first implementation of the pressure PIV technique to an MHD flow. Thanks to it, we obtain the first experimental velocity and pressure profiles in an MHD Couette flows and show that the transitional regime between laminar and turbulent states is dominated by near-wall, isolated, anisotropic perturbations.

On the convergence of augmented Lagrangian method for optimal transport between nonnegative densities

The dynamical formulation of the optimal transport problem, introduced by J.D. Benamou and Y. Brenier [4], amounts to find a time dependent space density and velocity field minimizing a transport energy between two densities. In order to solve this problem, an algorithm has been proposed to estimate the saddle point of a Lagrangian. We study the convergence of this algorithm in the most general case where initial and final densities may vanish on regions of the transportation domain. Under these assumptions, the main difficulty of our study is the proof of existence of a saddle point and of uniqueness of the density-momentum component, as it leads to deal with non-regular optimal transportation maps. For these reasons, a detailed study of the regularity properties of the velocity field associated to an optimal transportation map is required.

Unsteady Secondary Motion of Pulsatile Turbulent Flow through a Double 90°-Bend Duct

We investigate turbulent flow with highly pulsating axial velocity passing through a duct with both first and second bends. The time-dependent velocity fields downstream of the bends were measured using time-resolved stereo particle image velocimetry for the steady case (Reynolds number Re = 36,700) and the pulsatile case (Re = 37,800 and Womersley number α = 59.1). Proper orthogonal decomposition (POD) of the in-plane velocity data isolates the energetic structures of the secondary flow. The modes downstream of the first bend have a Dean motion (mode 0), single swirl (mode 1), and double swirl (mode 2), which agree with those of previous studies on steady turbulent flow. Downstream of the second bend, additional vortices appear in the modes owing to the secondary flow originating in the first bend. The modal structure of the pulsatile case is virtually the same as that of the steady case. To our knowledge, we are the first to find swirl switching in pulsatile flow, whereas the switching has been reported only for steady cases. We further conduct a time-frequency analysis via wavelet transformation onto the POD time coefficient, showing intermittency in energy of the mode associated with swirl switching.

Analytical solutions of incompressible laminar channel and pipe flows driven by in-plane wall oscillations

Emerging flow control strategies have been proposed to tackle long-lasting problems, for instance, precise mixing of chemicals and turbulent drag reduction. Employing actuators imposing in-plane wall oscillations are particularly popular. This paper investigates incompressible laminar rectangular channel and circular pipe flows driven by uniform and traveling wave in-plane wall oscillations. A comprehensive set of exact analytical solutions are presented describing parallel and concentric flows. Dimensionless groups are identified, and it is described how they characterize the one- and two-dimensional time-dependent velocity and pressure fields. The solutions enable to compute the oscillating boundary layer thickness. It is demonstrated that the dimensionless groups and the boundary layer thickness narrows the region of interest within the parameter space. In particular, the oscillating boundary layer thickness obtained from these laminar flows estimates a “radius of action” within which flow features can be altered to boost mixing or reduce turbulent friction drag. The results are suitable for software validation and verification, may open the way to promising complex wall oscillations, and ease the optimization task that delays the industrial application of flow controls.

Axial conduction and dissipation in oscillatory laminar pipe flow at low and high frequencies

The problem of fully developed laminar fluid flow in pipes, driven by an oscillatory pressure gradient, can be solved exactly for the time-dependent velocity field and related quantities such as flow rate and tidal displacement. When dissipation is neglected and the momentary axial variation of temperature is assumed to be linear, the corresponding thermal energy equation describing heat transfer along a pipe connecting two reservoirs at different temperatures can also be solved to yield exact solutions for the time-dependent temperature field, axial heat flux, and effective axial conductivity. In this paper, it is shown that these exact solutions for the unsteady temperature field are invalid at low Womersley numbers because the momentary axial variation of temperature is not linear. When the thermal energy equation is written in quasisteady form, approximate quasisteady analytical solutions can be found for the temperature field, which yield effective axial conductivities several orders of magnitude greater than those given by the low-Womersley-number, unsteady-flow solution. It is also shown that the conditions under which effects of dissipation on axial heat transfer become significant, at high Womersley numbers, can be determined by a simple criterion. When dissipation is significant, exact solutions for the unsteady temperature field are invalid at high Womersley numbers because the momentary axial variation of temperature is also nonlinear.

Superconvergence of Arbitrary Lagrangian--Eulerian Discontinuous Galerkin Methods for Linear Hyperbolic Equations

In this paper, we study the superconvergence properties of an arbitrary Lagrangian--Eulerian discontinuous Galerkin (ALE-DG) method with approximations to one-dimensional linear hyperbolic equations. The ALE-DG method is a mesh moving method; we need to deal with the new difficulties brought by the time-dependent test function space and grid velocity field. Since the time derivative cannot commute with the space projections for the ALE-DG method, we will introduce the material derivative in our analysis. With the help of the scaling argument and material derivative, we build a special interpolation function by constructing the correction functions and prove that the numerical solution is superclose to the interpolation function in the $L^2$-norm. The order of the superconvergence is 2k+1 when piecewise polynomials of degree at most $k$ are used. We also rigorously prove a (2k+1)th-order superconvergence rate for the domain and cell average and at the downwind points in the maximal and average norm. Furthermore, we prove that the function value approximation is superconvergent with a rate $k+2$ at the right Radau points and a superconvergence rate $k+1$ for the derivative approximation at all interior left Radau points. All theoretical findings are confirmed by numerical experiments.