Lagrangian Transformation Research Articles

The coupled Hudson River estuarine-plume response to variable wind and river forcings

Observations of the Hudson River plume were taken in the spring of 2006 in conjunction with the Lagrangian Transport and Transformation Experiment using mooring arrays, shipboard observations, and satellite data. During this time period, the plume was subjected to a variety of wind, buoyant, and shelf forcings, which yield vastly different responses in plume structure including a downstream recirculating eddy. During weak and downwelling winds, the plume formed a narrow buoyant coastal current that propagated downstream near the internal wave speed. Freshwater transport during periods when the downwelling wind was closely aligned with the coast was near the river discharge values. During periods with a cross-shore component to the wind, freshwater transport in the coastal current estimated by the mooring array is less than the river discharge due to a widening of the plume that leads to the internal Rossby radius scaling for the plume width to be invalid. The offshore detachment of plume and formation of a downstream eddy that is observed surprisingly persisted for 2 weeks under a variety of wind forcing conditions. Comparison between mooring, shipboard, and satellite data reveal the downstream eddy is steady in time. Shipboard transects yield a freshwater content equal to the previous 3 days of river discharge. The feature itself was formed due to a large discharge following a strong onshore wind. The plume was then further modified by a brief upwelling wind and currents influenced by the Hudson Shelf Valley. The duration of the detachment and downstream eddy can be explained using a Wedderburn number which is largely consistent with the wind strength index described by Whitney and Garvine (J Geophys Res 110:C03014 1997).

Study on the chaotic behavior of mining rock seepage system

One dimensional non-steady, non-Darcy flow of water in a rock stratum was reduced into a system described by six ordinary differential equations involving five controlling parameters. Through response computations and time series analysis, chaotic behavior in the reduced system was discussed in details. Firstly, the dynamical response of the reduced system under a set of parameters was calculated, and the power spectrum of the attractor was obtained through fast Lagrangian transformation; then the phase space was reconstructed by fixing embedding dimension to be 6 and delay time to range from 1 to 20, and the correlation dimension of the attractor was calculated based on the curves under the coordinates of logarithm of correlation integral vs. logarithm of covering radius; and lastly, the Lyapunov indices of the attractor were calculated by using Gram-Schmit’s orthogonalization method. The results show that the power spectrum of the attractor is continuous; the correlation dimension of the attractor is equal to 2.36; among the Lyapunov indices, LE1, LE2, LE3 are positive, LE5, LE6 are negative, and LE4 fluctuates near zero. All the analysis indicates that there may exist chaos in the system of non-steady, non-Darcy flow.

On the vacua of mass-deformed Gaiotto-Tomasiello theories

We write explicit Lagrangian and supersymmetry transformation rules using the component fields in the ${\cal N}=2,3$ GT theories. In the component field expansion, the manifestation of an additional ${\cal N}=1$ supersymmetry is verified in the ${\cal N}=3$ GT theory. We find maximal supersymmetry preserving mass-deformation of the GT theories and their classical supersymmetric discrete vacua. Some interesting aspects of the set of discrete vacua are discussed in comparison with the ABJM case.

Towards a three-dimensional moving body incompressible flow solver with a linear deformable model

In this study, a three-dimensional fluid–structured parallelized solver is extended from the previous work (Niu et al., 2009 [1]) for moving body simulations. Based on the unified Eulerian and Lagrangian coordinate transformations, the unsteady three-dimensional incompressible Navier–Stokes equations with artificial compressibility (Chorin, 1967 [2]) in a dual-time stepping approach are first derived. To implement unsteady flow calculations, the dual-time stepping strategy including the LU decomposition method is used in the pseudo-time iteration and the second-order accurate backward difference is adopted to discretize the unsteady flow terms. Also, a third-order Roe type flux limited splitting is derived to evaluate the spatial difference of the convective fluxes. The original FORTRAN code is converted to the MPI code and tested on a 64-CPU IBM SP2. The parallel strategy here is based on the partitions of all do-loops in the original FORTRAN code and transferring the calculations inside the do-loop into different CPUs. The partition of the do-loop can be applied on the innermost loop, only or the last two inner loops depending on two-dimensional or three-dimensional problems. This kind of the parallel data partition of the loops is independent of what kind of the explicit or implicit type numerical algorithm used. Therefore, the current parallel approach can take advantage of the MPI language fully to transfer data efficiently among CPUs even for solving the governing equation implicitly. The test results show that a significant reduction of computing time in running the model and a near-linear speed up rate is achieved up to 32 CPUs at IBM SP2. The speed up rate is as high as 31 for using 64 IBM SP2 processors The test shows efficient parallel processing to provide prompt simulation of 3D cavity, unsteady dropping airfoil and blood flows in an aortic tube with a linear elastic modeling of wall motion is included here.

High‐frequency forcing and subtidal response of the Hudson River plume

The Lagrangian Transport and Transformation Experiments (LaTTE) document the physical, biological, and chemical evolution of the Hudson River plume during the spring seasons of 2004, 2005, and 2006. While plume variability due to river discharge, subinertial frequency variability in winds, and ambient shelf circulation are important, the observations show that the plume reacts directly to higher‐frequency forcing as well. Mooring records during 2005 and 2006 show that fortnightly variability in tidal mixing is manifested as fortnightly changes in plume stratification. Diurnal variability related to forcing by the sea‐land breeze system (SLBS) is apparent in the Hudson River plume during the 2005 experiment. The SLBS, while episodic, accounts for ∼15% of the kinetic energy in surface currents in the New York Bight apex during the summer months with individual SLBS events providing up to 50% of the total kinetic energy. Simulations of the plume, using the Regional Ocean Modeling System (ROMS), demonstrate there is a subtidal response to high‐frequency forcing. Spring‐neap variability in tidal mixing modifies the estuary outflow Rossby and Froude numbers, resulting in increased transport (80% of river discharge) in the New Jersey coastal current during spring tides with lower transport (60% of river discharge) during neap tides. SLBS variability results in greater storage of river discharge water in the recirculating bulge region and increases the net transport of freshwater along the Long Island coast while significantly reducing freshwater supplied to the New Jersey coastal current to as little as 30% of the total river discharge.

Development of a moving artificial compressibility solver on unified coordinates

AbstractBased on the unified Eulerian and Lagrangian coordinate transformations, the unsteady incompressible Navier–Stokes equations with artificial compressibility effects are developed. As we know, the Eulerian coordinates cause excessive numerical diffusion across flow discontinuities, slip lines in particular. The Lagrangian coordinates, on the other hand, can resolve slip lines sharply but cause severe grid deformation, resulting in large errors and even breakdown of the computation. Recently, Hui et al. (J. Comput. Phys. 1999; 153:596–637) have introduced a unified coordinate system that moves with velocity hq, q being the velocity of the fluid particle. It includes the Eulerian system as a special case when h=0, and the Lagrangian when h=1, and was shown for the two‐dimensional unsteady Euler equations of compressible flow to be superior than both Eulerian and Lagrangian systems. In the framework of unified coordinates, our work will derive the unsteady incompressible flow equations and moving geometry equations, when hq equals grid velocity in conservation form and is updated simultaneously during each time step. Thus, the accurate estimation of geometry conservation and controlling the grid velocity or the h value based on the unified approach can keep numerical stability and avoid computation breakdown caused by moving body or boundary layers (considered as slip lines in Lagrangian coordinates). Also, the existing high‐resolution Riemann solver can be extended to discretize the current unified incompressible flow equations. Our benchmark tests including the lid‐driven cavity flow and backward step flow, oscillating flat plat and pulsating stenotic tube are used to validate the computations. The results verify the accuracy and robustness of the unified artificial compressibility solver on the moving body simulation. Copyright © 2009 John Wiley & Sons, Ltd.

Towards frame-like gauge invariant formulation for massive mixed symmetry bosonic fields

In this paper, as a first step towards frame-like gauge invariant formulation for massive mixed symmetry bosonic fields, we consider mixed tensors, corresponding to Young tableau with two rows with k ⩾ 2 boxes in the first row and only one box in the second row. We construct complete Lagrangian and gauge transformations describing massive particles in (anti) de Sitter space–time with arbitrary dimension d ⩾ 4 and investigate all possible massless and partially massless limits.

Global solvability of the free-boundary problem for one-dimensional motion of a self-gravitating viscous radiative and reactive gas

In this paper we consider a system of equations describing the one-dimensional motion of a self-gravitating, radiative and chemically reactive gas having the free-boundary. For arbitrary large, smooth initial data we prove the unique existence, global in time, of a classical solution of the corresponding problem with fixed domain, obtained by the Lagrangian mass transformation.

Bulge Formation of a Buoyant River Outflow

Observations taken during the Lagrangian Transport and Transformation Experiment (LaTTE) in 2005 indicated that the Hudson's river outflow formed a bulge of recirculating fluid that limits the volume of fresh water that is advected away in a coastal current. Focusing on an event that began with downwelling winds we made estimates of the fresh‐water flux in the coastal current and the fresh water inventory of the bulge. The coastal current was characterized by a surface advected plume in thermal wind balance. However, the freshwater transport in the coastal current was less than 1/2 of the total freshwater outflow. The bulge extended 30 km from the coast and 40 km in the along‐shore direction and was evident in ocean color imagery. Recirculation in the bulge region was also apparent in daily averaged surface current radar data, but this flow pattern was obscured in the hourly data by tidal and wind‐forcing even in the diurnal band. Nevertheless, many aspects of the Hudson's outflow are consistent with recent laboratory experiments and numerical simulations of buoyant discharges. The growing bulge transports the river's outflow to the head of the Hudson shelf valley where it crosses the 50 m isobath. Previous work in this region indicates that frontal features reside along this isobath. We observed fresh water being transported along this isobath and is suggestive of a rapid cross‐shelf transport pathway for fresh water. Both the bulge formation and cross‐shelf transport have significant biogeochemical implications.

On Transonic Shocks in Two‐Dimensional Variable‐Area Ducts for Steady Euler System

This paper concerns transonic shocks in compressible inviscid flow passing a two- dimensional variable-area duct for the complete steady Euler system. The flow is supersonic at the entrance of the duct, whose boundaries are slightly curved. The condition of impenetrability is posed on the boundaries. After crossing a nearly flat shock front, which passes through a fixed point on the boundary of the duct, the flow becomes subsonic. We show that to ensure the stability of such shocks, pressure should not be completely given at the exit: it only should be given with freedom one, that is, containing an unknown constant to be determined by the upstream flow and the profile of the duct. Careful analysis shows that this is due to the requirement of conservation of mass in the duct. We used Lagrangian transformation and characteristic decomposition to write the Euler system as a 2 × 2 system, which is valid for general smooth flows. Due to such a simplification, we can employ the theory of boundary value problems for elliptic equations to discuss well-posedness or ill-posedness of transonic shock problems in variable-area duct for various conditions giving at the exit.

Numerical methods for modeling transient flow in distribution systems

The authors compared two numerical methods, one Eulerian‐based, the other Lagrangian‐based, for modeling hydraulic transients in water distribution systems. The method of characteristics (MOC), considered to be the most accurate of the Eulerian methods in its representation of the governing equations, requires a number of steps or calculations to solve a typical transient pipe flow problem. The wave characteristic method (WCM), a Lagrangian approach, tracks movement and transformation of pressure waves and computes new conditions either at fixed intervals or only at times when a change actually occurs. The WCM requires orders of magnitude fewer pressure and flow calculations, allowing large systems to be solved in an expeditious manner. Furthermore, because the method is continuous in both time and space, it is less sensitive to the structure of the network and to the length of the simulation process, resulting in improved computational efficiency. Results indicated that both the MOC and WCM provide fast and accurate results for small pipe systems of short duration transients. However, WCM was superior for transient analysis of large distribution systems because it yields accurate results in a fraction of the computational time required for MOC. Pressure transients can adversely affect the quality of treated water because of potential intrusion by pathogens associated with transient events. Accurate modeling studies can help utility managers determine adequate surge protection, strengthen the integrity of their systems, and minimize costly disruptions to service.

Full Lagrangian methods for calculating particle concentration fields in dilute gas-particle flows

The paper discusses two Full Lagrangian methods for calculating the particle concentration and velocity fields in dilute gas-particle flows. The methods are mathematically similar but crucially different in application. By examining the analytical solution for two-phase stagnation-point flow, it is shown that Osiptsov's method is more general than that of Fernandez de la Mora and Rosner. In Osiptsov's method, the Jacobian of the Eulerian–Lagrangian transformation is computed by integration along particle pathlines. The particle concentration is then obtained algebraically from the Lagrangian form of the particle continuity equation. It is shown that the correct specification of the initial conditions is non-trivial and of vital importance. A technique to alleviate problems of mathematical ‘stiffness’ at small Stokes numbers is also described. Full Lagrangian methods require knowledge of the fluid velocity gradient field and, if the carrier flowfield is calculated numerically, differentiation of a ‘noisy’ field can result in serious errors. The paper describes a method for reducing these errors. The incompressible, inviscid flow over a cylinder provides a useful test case for validation and the Osiptsov method proves its worth by revealing a region, hitherto unknown, of crossing particle pathlines in the mathematical solution. Crossing pathlines and their relationship to Robinson's integral are then discussed, and calculations of particle flow through a turbine cascade at high Mach numbers are presented to illustrate the engineering potential of the method.

Steady non-Newtonian flow of Vitreloy-1 in continuous extrusion

Abstract The steady thermal flow evolution of Vitreloy-1 (Zr 41.25 Ti 13.75 Cu 12.5 Ni 10 Be 22.5 ) during continuous extrusion is simulated by means of finite element modeling. The conjugate coupling between fluid dynamics and heat transfer via convection, viscous dissipation, and temperature-dependent properties is implicitly accounted for in the finite element solver. The flow model incorporates the viscosity dependence on temperature and strain-rate, hence it accounts for flow deceleration near the walls as a consequence of cooling, as well as for non-Newtonian effects such as shear thinning. At the melt-mold interface, shear-rate dependent slip and imperfect thermal contact were accommodated. In order to assess the extent to which crystallization occurs during the flow a Lagrangian transformation of coordinates is implemented to track the temporal response of flow elements from a moving frame of reference. For a 5 mm channel within a 100 mm thick copper mold, where the melt is injected above its melting point (~1000 K) at 1 mm/s and the mold walls are actively cooled at 373 K, the simulation predicts that the melt is quenched to near glass transition (~600 K) at a rate fast enough such that crystallization is entirely evaded, as the nose of the TTT curve (~900 K) is bypassed by virtually all flow elements, and consequently 5 mm purely amorphous plate is extruded.

N= 2 supersymmetric scalar-tensor couplings

We determine the general coupling of a system of scalars and antisymmetric tensors, with at most two derivatives and undeformed gauge transformations, for both rigid and local N=2 supersymmetry in four-dimensional spacetime. Our results cover interactions of hyper, tensor and double-tensor multiplets and apply among others to Calabi-Yau threefold compactifications of Type II supergravities. As an example, we give the complete Lagrangian and supersymmetry transformation rules of the double-tensor multiplet dual to the universal hypermultiplet.

Transient non-isothermal behavior during the growth and collapse of spherical fluid shells

This theoretical study explores the influence of inertia, surface tension, and viscous dissipation on the transient heat transfer during the growth and collapse of fluid shells. The shell is spherical, the fluid is Newtonian, and the flow is induced by a constant driving pressure. The coupled heat and flow equations are solved numerically using the Cobody (Lagrangian) transformation and a central difference discretization in space. The results are described in terms of four dimensionless numbers, namely, the Reynolds number, the capillary number, the Peclet number and the Brinkman number. In particular, it is found that viscous dissipation and surface tension have a significant effect on the temperature evolution within the fluid, namely on the temperature buildup within the fluid shell.

Influence of non‐isothermal fluid properties on the growth of spherical fluid shells

This numerical study explores the influence of the time dependence of material fluid parameters on the transient temperature evolution during the growth of fluid shells. The shell is spherical, the fluid is Newtonian, and the flow is induced by a constant driving pressure. The coupled heat and flow equations are solved numerically using the cobody (Lagrangian) transformation and a central difference discretization in space. The range of material values is adjusted from existing experiments. It is generally found that the variation in viscosity, surface tension and specific heat can have a significant influence on both the growth rate and temperature evolution. Thermal conductivity is found to be of little influence.

Dynamics of a multimodule variable geometry manipulator

As robotic manipulators gain more importance in space operations, it is becoming imperative to understand their distinctive dynamics and control characteristics. With the increasing complexity of space robot, efficient algorithms are required for their simulation. The present study uses an order- N algorithm, based on the Lagrangian approach and velocity transformations, to simulate the planar dynamics of a novel orbiting manipulator with arbitrary number of slewing and deployable flexible links. The relatively general formulation accounts for the interactions between orbital, librational, slewing, deployment, and vibrational degrees of freedom, and thus is applicable to a large class of manipulator systems of contemporary interest. A parametric analysis of the system dynamics is carried out to investigate effects of initial disturbances, system parameters and maneuver profiles. The study suggests considerable coupling between the rigid body motion and structural vibrations. Obviously, this would significantly affect the manipulator's performance in tracking a prescribed trajectory.

A Lagrangian analysis of advection-diffusion equation for a three dimensional chaotic flow

The advection-diffusion equation is studied via a global Lagrangian coordinate transformation. The metric tensor of the Lagrangian coordinates couples the dynamical system theory rigorously into the solution of this class of partial differential equations. If the flow has chaotic streamlines, the diffusion will dominate the solution at a critical time, which scales logarithmically with the diffusivity. The subsequent rapid diffusive relaxation is completed on the order of a few Lyapunov times, and it becomes more anisotropic the smaller the diffusivity. The local Lyapunov time of the flow is the inverse of the finite time Lyapunov exponent. A finite time Lyapunov exponent can be expressed in terms of two convergence functions which are responsible for the spatio-temporal complexity of both the advective and diffusive transports. This complexity gives a new class of diffusion barrier in the chaotic region and a fractal-like behavior in both space and time. In an integrable flow with shear, there also exist fast and slow diffusion. But unlike that in a chaotic flow, a large gradient of the scalar field across the KAM surfaces can be maintained since the fast diffusion in an integrable flow is strictly confined within the KAM surfaces.

Planar Dynamics of Flexible Manipulators with Slewing Deployable Links

Space manipulators present several features uncommon to ground-based robots: They are highly flexible, are often mobile, and have a degree of redundancy. As space robots become more complex, efficient algorithms are required for their simulation and control. The present study uses an order N algorithm, based on the Lagrangian approach and velocity transformations, to simulate the planar dynamics of an orbiting manipulator with arbitrary number of slewing and deployable flexible links. The relatively general formulation accounts for interactions between orbital, librational, slewing, deployment, and vibrational degrees of freedom and, thus, is applicable to a large class of manipulator systems of contemporary interest. A parametric analysis of the system dynamics suggests significant coupling between the rigid-body motion and structural vibrations. Obviously, this would affect the manipulator's performance. A nonlinear controller based on the feedback linearization technique is developed to regulate the rigid degrees of freedom. Nomenclature />/ = vector describing the position of F/ relative to F0 d = translation of the frame F/ from the tip of the (/th-l)body dm/ = mass of the infinitesimal element located on the /th body E AI , EIj = structural stiffness of the /th body in the

Order-N Formulation and Dynamics of Multi-Unit Flexible Space Manipulators

As robotic manipulators gain more importance in space operations, it is becoming imperative to understand their distinctive dynamics and control characteristics. With the increasing complexity of space robots, efficient algorithms are now required for their simulation. The present study uses an order-N algorithm, based on the Lagrangian approach and velocity transformations, to simulate the planar dynamics of a novel orbiting manipulator with arbitrary number of slewing and deployable flexible links. The relatively general formulation accounts for interactions between orbital, librational, slewing, deployment, and vibrational degrees of freedom, and thus is applicable to a large class of manipulator systems of contemporary interest. A parametric analysis of the system dynamics is carried out to investigate the effects of initial disturbances, variation of system parameters and maneuver profiles. The study suggests significant coupling between the rigid body motion and structural vibrations. As a result, the system's flexibility can significantly affect the manipulator's performance.