Lagrangian Velocity Research Articles

3D-PTV flow measurements of Newtonian and non-Newtonian fluid blending in a batch reactor in the transitional regime

Lagrangian trajectories obtained through 3D Particle Tracking Velocimetry (3D-PTV) measurements have been used to visualize the flow field of Newtonian and non-Newtonian fluids in a flat-bottomed vessel. The vessel, of diameter T=180 mm, was equipped with a 6-blade Rushton turbine of diameter D=T/3 and four baffles of width b=T/10. The experiments were carried out in the transitional flow regime (73≤Re≤1,257). The velocities and Lagrangian accelerations in the flows have been calculated from the time-resolved tracer coordinates. Non-Newtonian fluids exhibited a smaller impeller flow number compared to Newtonian fluids. The distributions of shear rate have been obtained via interpolation of the Lagrangian velocity data in a 3D Eulerian grid. In the impeller region, the mean shear rate was, to a first approximation, proportional to the impeller rotational speed, although a more detailed analysis revealed influences of both rheology and Reynolds number. The mean Lagrangian acceleration scaled with the mean shear rate raised to the power of 1.8.

Lagrangian velocity and acceleration measurements in plume-rich regions of turbulent Rayleigh-Bénard convection

An experimental particle tracking velocimetry is used to investigate the Lagrangian velocity and acceleration in plume-rich regions of turbulent Rayleigh-B\'enard convection. The measured acceleration variances show regions with and without abounded number of plumes have different Ra-dependent power laws. Further examination reveals that turbulent flow in the plume-rich regions is governed by a mixed dynamics with contribution from both thermal plumes and turbulent background fluctuations.

Lagrangian dynamics and heat transfer in porous-media convection

Abstract

Correction to: Initial time dependence of wind- and density-driven Lagrangian residual velocity in a tide-dominated bay

Correction to: Initial time dependence of wind- and density-driven Lagrangian residual velocity in a tide-dominated bay

Lagrangian turbulence in the woods

Abstract

Initial time dependence of wind- and density-driven Lagrangian residual velocity in a tide-dominated bay

The nonlinear effect of the summer southeast wind and density on the 3D structures of the full Lagrangian residual velocity (LRV) was quantified for a generally nonlinear system, using Jiaozhou Bay (JZB), China as the test site. In the tidally energetic JZB, the basic patterns of the wind- and density-driven full LRVs were found to be consistent with semi-analytical solutions but highly dependent on initial times. The wind-driven full LRVs at different tidal phases flowed similarly downwind over shoals and upwind in the deep region; however, the main branches could migrate across nearly half of the bay. A density-driven, clockwise flow was dominant in the western inner bay at low tide, but it almost disappeared at high tide. The effect of density generally enhanced the outward flow in the surface layer and inward flow in the bottom layer. Along-trajectory integrated momentum balances indicated that viscosity was the main factor responsible for the time dependence of the wind-driven full LRVs, while viscosity, barotropic and baroclinic pressure gradients were the main drivers of the intra-tidal variations in the density-driven full LRVs. Generally, the summer wind and density had opposing effects, although their influence was weaker than that of the tide and could not change the patterns of the tide-driven full LRVs. When analysing the effects of wind and density on the coastal circulation in JZB, both the 3D structures and the possibility of a high initial tidal phase dependency should be considered.

On small-scale and large-scale intermittency of Lagrangian statistics in canopy flow

Abstract

Visualization of submerged turbulent jets using particle tracking velocimetry

Over the past few decades, advances have been made in using particle image velocimetry (PIV) and particle tracking velocimetry (PTV) for mapping of Lagrangian velocity and acceleration flow fields. With PIV, Lagrangian trajectories are not measured directly; rather, hypothetical trajectories must be constructed from sequences of Eulerian velocity snapshots. Because PTV directly measures actual trajectories, it provides distinct advantages over PIV, especially for trajectories with abrupt changes in direction. In this work, a novel particle tracking algorithm is described, then applied to track trajectories of tracer particles in submerged turbulent jets. The Reynolds numbers ranged from 1000 to 25,000, thereby covering laminar, transitioning-to-turbulence, and fully turbulent flow regimes. The novel particle tracking algorithm is designed to handle flows with very high particle concentrations, thereby resolving small-scale flow structures. Trajectories are tracked with high velocity gradients, sharp curvatures, cycloids, abrupt changes in direction, and strong recirculation—all of which are inaccessible via construction from PIV sequences. Most trajectories measured in this work are at least 500 camera frames (time steps) long, with many being more than 3000 frames long.Graphic abstract

Roughness, inertia, and diffusion effects on anomalous transport in rough channel flows

We study how the complex interplay between roughness, inertia, and diffusion controls the tracer transport in rough channels. We first simulate the flow and tracer transport over wide ranges of channel roughness, Reynolds number ($Re$), and P\'eclet number ($Pe$) observable in nature. $Pe$ exerts a first-order control on first-passage time distributions, and the effect of roughness on the tracer transport becomes evident with the increase in $Re$. The interplay between the roughness and $Re$ causes eddy flows, which intensify or suppress anomalous transport depending on $Pe$. At infinite $Pe$, the late-time scaling follows a universal power-law scaling, which is explained by conducting a scaling analysis. With extensive numerical simulations and stochastic modeling, we show that the roughness, inertia, and diffusion effects are encoded in Lagrangian velocity statistics represented by velocity distribution and correlation. We finally predict the anomalous transport using a stochastic model that considers the Lagrangian velocity statistics.

The diffusing-velocity random walk: a spatial-Markov formulation of heterogeneous advection and diffusion

Abstract

An experimental method of simultaneously studying refractive index and dynamic properties of transparent materials

Based on the electromagnetic loading device CQ-4, an experimental method of simultaneously measuring the refractive index and high pressure sound velocity of transparent material is established. The ramp wave compression experiment of PMMA is carried out under a pressure of 14 GPa. The velocity history curves of PMMA sample rear surface are obtained by dual laser heterodyne velocimetry (DLHV). The velocity curve shows obvious double wave structure, which indicates the elastic-plastic transition. The refractive index particle velocity optical characteristics and Lagrangian sound velocity particle velocity dynamic characteristics of PMMA are obtained simultaneously with the experimental data processing.

Smoothed particle hydrodynamics method from a large eddy simulation perspective. Generalization to a quasi-Lagrangian model

The present work deals with some recent developments regarding the inclusion of the Large-Eddy Simulation (LES) in the weakly compressible Smoothed Particle Hydrodynamics (SPH) framework. Previously {see the work of Di Mascio et al. [Phys. Fluids 29, 4 (2017)]}, this goal was achieved by applying a Lagrangian filter to the Navier–Stokes equations for compressible fluids and, then, approximating the differential operators in a SPH fashion. Since the Lagrangian nature of the derived scheme turned out to be an obstacle for accurate simulations of high Reynolds number problems, the above approach is here modified to obtain a quasi-Lagrangian LES-SPH model. This relies on the addition of a small velocity deviation to the actual Lagrangian velocity based on the particle shifting technique and on the inclusion of the tensile instability control technique for eliminating the onset of the tensile instability in the fluid regions characterized by large vorticity and negative pressure. The proposed model is successfully tested in both two-dimensional and three-dimensional frameworks by simulating the evolution of freely decaying turbulence problems and comparing the outputs with the available theoretical results and solutions from other numerical models.

Stochastic Variational Formulations of Fluid Wave\u2013Current Interaction

We are modelling multiscale, multi-physics uncertainty in wave–current interaction (WCI). To model uncertainty in WCI, we introduce stochasticity into the wave dynamics of two classic models of WCI, namely the generalised Lagrangian mean (GLM) model and the Craik–Leibovich (CL) model. The key idea for the GLM approach is the separation of the Lagrangian (fluid) and Eulerian (wave) degrees of freedom in Hamilton’s principle. This is done by coupling an Euler–Poincaré reduced Lagrangian for the current flow and a phase-space Lagrangian for the wave field. WCI in the GLM model involves the nonlinear Doppler shift in frequency of the Hamiltonian wave subsystem, which arises because the waves propagate in the frame of motion of the Lagrangian-mean velocity of the current. In contrast, WCI in the CL model arises because the fluid velocity is defined relative to the frame of motion of the Stokes mean drift velocity, which is usually taken to be prescribed, time independent and driven externally. We compare the GLM and CL theories by placing them both into the general framework of a stochastic Hamilton’s principle for a 3D Euler–Boussinesq (EB) fluid in a rotating frame. In other examples, we also apply the GLM and CL methods to add wave physics and stochasticity to the familiar 1D and 2D shallow water flow models. The differences in the types of stochasticity which arise for GLM and CL models can be seen by comparing the Kelvin circulation theorems for the two models. The GLM model acquires stochasticity in its Lagrangian transport velocity for the currents and also in its group velocity for the waves. However, the CL model is based on defining the Eulerian velocity in the integrand of the Kelvin circulation relative to the Stokes drift velocity induced by waves driven externally. Thus, the Kelvin theorem for the stochastic CL model can accept stochasticity in its both its integrand and in the Lagrangian transport velocity of its circulation loop. In an “Appendix”, we also discuss dynamical systems analogues of WCI.

Spatial distribution of particles in turbulent channel flow of dilute suspensions

We experimentally investigated the kinematics and suspension of particles in the near-wall region of a horizontal turbulent channel flow of water. The particles were glass beads with a diameter of 0.014H, where H is the full channel height. The experiments involved dilute particle suspensions at bulk volumetric concentrations (ϕ) of 0.05, 0.12, and 0.27%, and at Reynolds numbers (Re) of 20,200, 40,400, and 60,500. Measurement of Lagrangian position and velocity of the particles was carried out using 3D particle tracking velocimetry. At the lowest Re, the results showed a large near-wall accumulation of particles due to gravitational settling at all three bulk concentrations. The analysis of turbulence statistics suggested negligible effect due to inter-particle collisions even when the local volumetric particle concentration reached 2%. With increasing Re to 40,400 and 60,500, different concentration profiles were observed for ϕ = 0.05% with respect to the profiles of ϕ = 0.12 and 0.27%. A bi-model particle concentration distribution, with an inner and outer peak, was present for ϕ = 0.05% at Re = 40,400 and 60,500. In contrast, the particles at ϕ = 0.12 and 0.27% had a wall-peaking profile. The autocorrelation function of the streamwise Lagrangian velocity and wall-normal dispersion of the particles showed greater stability of particle motions in the near-wall region for ϕ = 0.12 and 0.27% with respect to ϕ = 0.05% at the two higher Re. The observations are associated with a stronger particle-wall lubrication at the higher volumetric concentrations of ϕ = 0.12 and 0.27%.

The [formula omitted]-ALE-SPH model: An arbitrary Lagrangian-Eulerian framework for the [formula omitted]-SPH model with particle shifting technique

The behaviour of a weakly-compressible SPH scheme obtained by rewriting the Navier-Stokes equations in an arbitrary Lagrangian-Eulerian (ALE) format is studied. Differently from previous works on ALE, which generally adopt conservative variables (i.e. mass and momentum) and rely on the use of Riemann solvers inside the spatial operators, the proposed model is expressed in terms of primitive variables (i.e. density and velocity) and is written by using the standard differential formulations of the weakly-compressible SPH schemes. Similarly to ALE-SPH models, the arbitrary velocity field is obtained by modifying the pure Lagrangian velocity of the material point through a velocity δu→ given by a Particle Shifting Technique (PST). We show that the above-mentioned ALE-SPH equations are, however, unstable when they are integrated in time. The instability appears in the form of large volume variations in those fluid regions characterised by high velocity strain rates. Nonetheless, the scheme can be stabilised if appropriate diffusion terms are included in both the equations of density and mass. This latter scheme, hereinafter called δ-ALE-SPH scheme, is validated against reference benchmark test-cases: the viscous flow around an inclined elliptical cylinder, the lid-driven cavity and a dam-break flow impacting a vertical wall.

Surface-only ferrofluids

We devise a novel surface-only approach for simulating the three dimensional free-surface flow of incompressible, inviscid, and linearly magnetizable ferrofluids. A Lagrangian velocity field is stored on a triangle mesh capturing the fluid's surface. The two key problems associated with the dynamic simulation of the fluid's interesting geometry are the magnetization process transitioning the fluid from a non-magnetic into a magnetic material, and the evaluation of magnetic forces. In this regard, our key observation is that for linearly incompressible ferrofluids, their magnetization and application of magnetic forces only require knowledge about the position of the fluids' boundary. Consequently, our approach employs a boundary element method solving the magnetization problem and evaluating the so-called magnetic pressure required for the force evaluation. The magnetic pressure is added to the Dirichlet boundary condition of a surface-only liquids solver carrying out the dynamical simulation. By only considering the fluid's surface in contrast to its whole volume, we end up with an efficient approach enabling more complex and realistic ferrofluids to be explored in the digital domain without compromising efficiency. Our approach allows for the use of physical parameters leading to accurate simulations as demonstrated in qualitative and quantitative evaluations.

Anomalous Transport in Three‐Dimensional Discrete Fracture Networks: Interplay Between Aperture Heterogeneity and Injection Modes

AbstractWe study how the interplay between fracture aperture heterogeneity and tracer injection mode controls fluid flow and tracer transport in three‐dimensional (3D) discrete fracture networks (DFNs). The direct 3‐D DFN simulations show that tracer injection mode has substantial effects on tracer spreading across all levels of aperture heterogeneity. The key controlling factor for effective transport is the initial Lagrangian velocity distribution, which is determined by the interplay between injection mode and aperture heterogeneity. The fundamental difference between initial Lagrangian velocity distribution and domain‐scale Eulerian velocity distribution plays a vital role in determining anomalous transport. We effectively capture the observed anomalous transport using an upscaled transport model that incorporates initial velocity distribution, stationary velocity distribution, velocity correlation length, and average advective tortuosity. With the upscaled transport model, we accurately capture the evolution of Lagrangian velocity distribution and predict longitudinal spreading in 3‐D DFN.

Unsteady Ekman‐Stokes Dynamics: Implications for Surface Wave‐Induced Drift of Floating Marine Litter

AbstractWe examine Stokes drift and wave‐induced transport of floating marine litter on the surface of a rotating ocean with a turbulent mixed layer. Due to Coriolis‐Stokes forcing and surface wave stress, a second‐order Eulerian‐mean flow forms, which must be added to the Stokes drift to obtain the correct wave‐induced Lagrangian velocity. We show that this wave‐driven Eulerian‐mean flow can be expressed as a convolution between the unsteady Stokes drift and an “Ekman‐Stokes kernel.” Using this convolution, we calculate the unsteady wave‐driven contribution to particle transport. We report significant differences in both direction and magnitude of transport when the Eulerian‐mean Ekman‐Stokes velocity is included.

Near-Surface Transport Properties and Lagrangian Statistics during Two Contrasting Years in the Adriatic Sea

This paper describes the near-surface transport properties and Lagrangian statistics in the Adriatic semi-enclosed basin using synthetic drifters. Lagrangian transport models were used to simulate synthetic trajectories from the mean flow fields obtained by the Massachusetts Institute of Technology general circulation model (MITgcm), implemented in the Adriatic from October 2006 until December 2008. In particular, the surface circulation properties in two contrasting years (2007 had a mild winter and cold fall, while 2008 had a normal winter and hot summer) are compared here. In addition, the Lagrangian statistics for the entire Adriatic Basin after removing the Eulerian mean circulation for numerical particles were calculated. The results indicate that the numerical particles were slower in this simulation when compared with the real drifters. This is because of the reduced energetic flow field generated by the MIT general circulation model during the selected years. The numerical results showed that the balanced effects of the wind-driven recirculation in the northernmost area(which would be a sea response to the Bora wind field) and the Po River discharge cause the residence times to be similar during the two selected years (182 and 185 days in 2007 and 2008, respectively). Furthermore, the mean angular momentum, diffusivity, and Lagrangian velocity covariance values are smaller than in the real drifter observations, while the maximum Lagrangian integral time scale is the same.

Turbulence-obstacle interactions in the Lagrangian framework: Applications for stochastic modeling in canopy flows

Lagrangian stochastic models are widely used to predict and analyze turbulent dispersion in complex environments, such as in various terrestrial and marine canopy flows. However, due to a lack of empirical data, it is still not understood how particular features of highly inhomogeneous canopy flows affect the Lagrangian statistics. In this work, we study Lagrangian short time statistics by analyzing empirical Lagrangian trajectories in sub-volumes of space that are small in comparison with the canopy height. For the analysis we used 3D Lagrangian trajectories measured in a dense canopy flow model in a wind-tunnel, using an extended version of real-time 3D particle tracking velocimetry (3D-PTV). One of our key results is that the random turbulent fluctuations due to the intense dissipation were more dominant than the flow's inhomogeneity in affecting the short-time Lagrangian statistics. This amounts to a so-called quasi-homogeneous regime of Lagrangian statistics at small scales. Using the Lagrangian dataset we calculate the Lagrangian autocorrelation function and the second-order Lagrangian structure-function, and extract associated parameters, namely a Lagrangian velocity decorrelation timescale, $T_i$, and the Kolmogorov constant, $C_0$. We demonstrate that in the quasi-homogeneous regime, both these functions are well represented using a second-order Lagrangian stochastic model that was designed for homogeneous flows. Furthermore, we show that the spatial variations of the Lagrangian separation of scales, $T_i/\tau_\eta$, and the Kolmogorov constant, $C_0$, cannot be explained by the variation of the Reynolds number, $Re_\lambda$, in space, and that $T_i/\tau_\eta$ was small as compared with homogeneous turbulence predictions at similar $Re_\lambda$. We thus hypothesize that this occurred due to the so-called "wake production", and show empirical results supporting our hypothesis.