Lagrangian Mean Research Articles

Momentum, energy and vorticity balances in deep-water surface gravity waves

The particle trajectories in irrotational, incompressible and inviscid deep-water surface gravity waves are open, leading to a net drift in the direction of wave propagation commonly referred to as the Stokes drift, which is responsible for catalysing surface wave-induced mixing in the ocean and transporting marine debris. A balance between phase-averaged momentum density, kinetic energy density and vorticity for irrotational, monochromatic and spatially periodic two-dimensional water waves is derived by working directly within the Lagrangian reference frame, which tracks particle trajectories as a function of their labels and time. This balance should be expected as all three of these quantities are conserved following particles in this system. Vorticity in particular is always conserved along particles in two-dimensional inviscid flow, and as such even in its absence it is the value of the vorticity that fundamentally sets the drift, which in the Lagrangian frame is identified as the phase-averaged momentum density of the system. A relationship between the drift and the geometric mean water level of particles is found at the surface, which highlights connections between the geometry and dynamics. Finally, an example of an initially quiescent fluid driven by a wavelike pressure disturbance is considered, showing how the net momentum and energy from the surface pressure disturbance transfer to the wave field, and recognizing the source of the mean Lagrangian drift as the net momentum required to generate an irrotational surface wave by any conservative force.

The Dirichlet problem for the Lagrangian mean curvature equation

The Dirichlet problem for the Lagrangian mean curvature equation

Upscaling transport in heterogeneous media featuring local-scale dispersion: Flow channeling, macro-retardation and parameter prediction

Many theoretical treatments of transport in heterogeneous Darcy flows consider advection only. When local-scale dispersion is neglected, flux weighting persists over time; mean Lagrangian and Eulerian flow velocity distributions relate simply to each other and to the variance of the underlying hydraulic conductivity field. Local-scale dispersion complicates this relationship, potentially causing initially flux-weighted solute to experience lower-velocity regions as well as Taylor-type macrodispersion due to transverse solute movement between adjacent streamlines. To investigate the interplay of local-scale dispersion with conductivity log-variance, correlation length, and anisotropy, we perform a Monte Carlo study of flow and advective-dispersive transport in spatially-periodic 2D Darcy flows in large-scale, high-resolution multivariate Gaussian random conductivity fields. We observe flow channeling at all heterogeneity levels and quantify its extent. We find evidence for substantial effective retardation in the upscaled system, associated with increased flow channeling, and observe limited Taylor-type macrodispersion, which we physically explain. A quasi-constant Lagrangian velocity is achieved within a short distance of release, allowing usage of a simplified continuous-time random walk (CTRW) model we previously proposed in which the transition time distribution is understood as a temporal mapping of unit time in an equivalent system with no flow heterogeneity. The numerical data set is modeled with such a CTRW; we show how dimensionless parameters defining the CTRW transition time distribution are predicted by dimensionless heterogeneity statistics and provide empirical equations for this purpose.

Optimal Regularity for Lagrangian Mean Curvature Type Equations

Optimal Regularity for Lagrangian Mean Curvature Type Equations

Stability of the Generalized Lagrangian Mean Curvature Flow in Cotangent Bundle

Stability of the Generalized Lagrangian Mean Curvature Flow in Cotangent Bundle

An analytical scaling law model framework for short-time dust emission from belt conveyor

Aiming at explaining the short-time dust emission from belt conveyors, an analytical scaling law model framework is introduced, employing the same assumptions and wall models as traditional numerical simulations. Further, assuming a piecewise mean velocity profile, the Lagrangian mean velocity of dust particles can be written as a general expression. This expression is validated in various cases, including the 2D Belt, 3D Belt, 2D Tire, and 3D Tire cases. Numerical results show that the theoretical prediction, with key parameters fixed at us=5,κ=0.4187, and E=9.793, fits well with the simulated Lagrangian mean velocity profiles. For example, in the 2D Belt case, short-time agreement is achieved within t⩽1 s, indicating the model’s effectiveness in predicting dust dispersion within this time frame.

Hessian estimates for the Lagrangian mean curvature flow

Hessian estimates for the Lagrangian mean curvature flow

Effects of buoyancy on the dispersion of drugs released intrathecally in the spinal canal.

This paper investigates the transport of drugs delivered by direct injection into the cerebrospinal fluid (CSF) that fills the intrathecal space surrounding the spinal cord. Because of the small drug diffusivity, the dispersion of neutrally buoyant drugs has been shown in previous work to rely mainly on the mean Lagrangian flow associated with the CSF oscillatory motion. Attention is given here to effects of buoyancy, arising when the drug density differs from the CSF density. For the typical density differences found in applications, the associated Richardson number is shown to be of order unity, so that the Lagrangian drift includes a buoyancy-induced component that depends on the spatial distribution of the drug, resulting in a slowly evolving cycle-averaged flow problem that can be analysed with two-time scale methods. The asymptotic analysis leads to a nonlinear integro-differential equation for the spatiotemporal solute evolution that describes accurately drug dispersion at a fraction of the cost involved in direct numerical simulations of the oscillatory flow. The model equation is used to predict drug dispersion of positively and negatively buoyant drugs in an anatomically correct spinal canal, with separate attention given to drug delivery via bolus injection and constant infusion.

Existence of Entire Solutions to the Lagrangian Mean Curvature Equations in Supercritical Phase

Existence of Entire Solutions to the Lagrangian Mean Curvature Equations in Supercritical Phase

Ancient solutions and translators of Lagrangian mean curvature flow

AbstractSuppose that ℳ is an almost calibrated, exact, ancient solution of Lagrangian mean curvature flow in $\mathbf {C} ^{n}$ C n . We show that if ℳ has a blow-down given by the static union of two Lagrangian subspaces with distinct Lagrangian angles that intersect along a line, then ℳ is a translator. In particular in $\mathbf {C} ^{2}$ C 2 , all almost calibrated, exact, ancient solutions of Lagrangian mean curvature flow with entropy less than 3 are special Lagrangian, a union of planes, or translators.

Numerical simulation and coupling mechanism study of acoustic-inertial micromixer

Acoustic-inertial micromixers exhibit a wider operating range and higher mixing efficiency than conventional mixers. We have carried out experimental studies and numerical simulations of acoustic-inertial micromixers. It is found that the acoustic streaming will be changed in morphology and intensity by the background flow, and the coupling mechanism between them is obtained. Specifically, we developed the numerical model of acoustic-inertial micromixer by using perturbation theory and the Generalized Lagrangian Mean (GLM) theory. By dividing the flow variables into zero-order, first-order and second-order variables, we obtained the control equations representing background flow, acoustic response and time-averaged streaming flow. Then realized the decoupling analysis of the acoustic-inertial coupling phenomenon. It is found that in terms of acoustic streaming flow, the intensity extreme of it increases as the background flow rate increases. The flow velocity at 5Vpp can reach 0.46 m/s under 500 μL/min, which is 30 times for the same condition under 10 μL/min. However, the streaming morphology no longer behaves as a symmetric vortex distributed on both sides of the sharp edge, which is stretched and twisted by the background flow and structure at a high flow rate. The percentage of acoustic contribution at different working conditions is also investigated quantitatively, and the acoustic contribution index (ACI) will generally decrease to less than 0.5 after the background flow dominates the flow field. The results of the study can provide a reference for the design of acoustic-inertial micromixers with larger flow ranges and higher efficiencies.

Observations of transient wave-induced mean drift profiles caused by virtual wave stresses in a two-layer system

An experimental study of long interfacial gravity waves was conducted in a closed wave tank containing two layers of viscous immiscible fluids. The study focuses on the development in time of the mean particle drift that occurs close to the interface where the two fluids meet. From a theoretical analysis by Weber & Christensen (Eur. J. Mech.-B/Fluids, vol. 77, 2019, pp. 162–170) it is predicted that the growing drift in this region is associated with the action of the virtual wave stress. This effect has not been explored experimentally before. Interfacial waves of different amplitudes and frequencies were produced by a D-shaped paddle. Particle tracking velocimetry was used to track the time development of the Lagrangian mean drift. The finite geometry of the wave tank causes a mean return flow that is resolved by mass transport considerations. The measurements clearly demonstrate the importance of the virtual wave stress mechanism in generating wave drift currents near the interface.

Eternal solutions to almost calibrated Lagrangian and symplectic mean curvature flows

In this paper, we prove some nonexistence results for the Type II singularity to the almost calibrated Lagrangian mean curvature flow and the symplectic mean curvature flow.

Regularity for convex viscosity solutions of Lagrangian mean curvature equation

Abstract We show that convex viscosity solutions of the Lagrangian mean curvature equation are regular if the Lagrangian phase has Hölder continuous second derivatives.

Interannual Variability of the Mass-Weighted Isentropic Zonal Mean Meridional Circulation in the Northern Hemisphere Winter

Abstract The atmospheric general circulation diagnosed by the mass-weighted isentropic zonal mean exhibits an extratropical direct (ETD) circulation driven by eddy momentum transport. We investigated regional atmospheric modulations associated with the year-to-year variability of ETD circulation in boreal winter using a reanalysis dataset. Composite analyses showed that the interannual variability of ETD circulation accompanies a seesaw-like variation between the Aleutian low (AL) and Icelandic low (IL), which coincides with the Pacific–North American teleconnection pattern and North Atlantic dipole anomaly. A diagnosis using isentropic airmass fluxes representing the geographical Lagrangian mean motion indicated that the significant intensification of upper-tropospheric poleward warm airflow over the eastern North Pacific is responsible for the enhanced ETD circulation, together with the intensified equatorward lower cold airflow over East Asia and eastern North America. The resulting enhanced upward propagation of the stationary planetary waves corresponds to the intensification of the divergence of the Eliassen–Palm flux, which explains the modulated ETD circulation in view of the balance between the Coriolis force and eddy momentum flux convergence. These results suggest that the AL–IL seesaw promotes variations in the global general circulation, and the anomalous ETD circulation interacts with several teleconnection patterns involving the modulation of planetary waves through anomalous meridional heat transport by the basinwide-scale eddies.

Study of hybrid passive-active micromixers under an acoustic field

This paper presents a numerical investigation of the mixing performance of two hybrid passive active micromixers subjected to an acoustic field. The study employs the Generalized Lagrangian Mean (GLM) theory and the convection-diffusion equation to analyze concentration profiles within the micromixers, providing a comprehensive assessment of their performance. Perturbation theory is employed to solve the equations of zeroth-, first-, and second-order, resulting in the determination of the final flow velocity from the zeroth- and second-order solutions. Notably, the study incorporates a non-zero background laminar flow to explore the interaction between acoustic waves and a moving laminar flow. The findings reveal that while the presence of sharp-edge structures on the channel walls significantly enhances mixing quality, this improvement is not observed that much when rectangular structures.

Lagrangian reduction and wave mean flow interaction

How does one derive models of dynamical feedback effects in multiscale, multiphysics systems such as wave mean flow interaction (WMFI)? We shall address this question for hybrid dynamical systems, defined as systems whose motion can be expressed as the composition of two or more Lie-group actions. Hybrid systems abound in fluid dynamics. Examples include: the dynamics of complex fluids such as liquid crystals; wind-driven waves propagating with the currents moving on the sea surface; turbulence modelling in fluids and plasmas; and classical–quantum hydrodynamic models in molecular chemistry. From among these examples, the motivating question here is: How do wind-driven waves produce ocean surface currents?The paper first summarises the geometric mechanics approach for deriving hybrid models of multiscale, multiphysics motions in ideal fluid dynamics. It then illustrates this approach for WMFI in the examples of 3D WKB waves and 2D wave amplitudes governed by the nonlinear Schrödinger (NLS) equation propagating in the frame of motion of an ideal incompressible inhomogeneous Euler fluid flow. The results for these examples tell us that the mean flow in WMFI does not create waves, although it does transport the waves. However, feedback in the opposite direction is possible, since the 3D WKB and 2D NLS wave dynamics discussed here do in fact create circulatory mean flow.

Symmetries of the one-dimensional hyperbolic Lagrangian mean curvature flow

Symmetries of the one-dimensional hyperbolic Lagrangian mean curvature flow

Computing Lagrangian means

Lagrangian averaging plays an important role in the analysis of wave–mean-flow interactions and other multiscale fluid phenomena. The numerical computation of Lagrangian means, e.g. from simulation data, is, however, challenging. Typical implementations require tracking a large number of particles to construct Lagrangian time series, which are then averaged using a low-pass filter. This has drawbacks that include large memory demands, particle clustering and complications of parallelisation. We develop a novel approach in which the Lagrangian means of various fields (including particle positions) are computed by solving partial differential equations (PDEs) that are integrated over successive averaging time intervals. We propose two strategies, distinguished by their spatial independent variables. The first, which generalises the algorithm of Kafiabad (J. Fluid Mech., vol. 940, 2022, A2), uses end-of-interval particle positions; the second uses directly the Lagrangian mean positions. The PDEs can be discretised in a variety of ways, e.g. using the same discretisation as that employed for the governing dynamical equations, and solved on-the-fly to minimise the memory footprint. We illustrate the new approach with a pseudo-spectral implementation for the rotating shallow-water model. Two applications to flows that combine vortical turbulence and Poincaré waves demonstrate the superiority of Lagrangian averaging over Eulerian averaging for wave–vortex separation.

Perturbation analysis of nonlinear partial reflected wave on a sloping bottom

In this paper, we use the perturbation method to develop a new mathematical derivation to describe nonlinear partial standing wave over uniformly sloping bottoms. In the Lagrangian coordinate system, the particle trajectories are obtained as a function of the nonlinear order parameter ɛ and the bottom slope α to the second order of the perturbation. The setups and mean return flow for an arbitrary bottom in the Lagrangian framework are also found as part of the solutions. The direct influences of reflection coefficient, wave steepness and bottom slope on the surface profiles of partially reflected waves are also derived. This nonlinear analytical solution is verified by reduction to the classical Stokes solution of progressive waves in both the deep water and constant depth limit. This solution allows to describe the successive deformation of partial reflected wave profiles and water particle trajectories prior to wave breaking to be described. The dynamic properties including mass transport, and Lagrangian mean level and radiation stress for nonlinear partially reflected waves on various sloping bottoms are investigated.