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NASA NTRS1981

A simple model of the Lagrangian-mean flow produced by dissipating planetary waves

A simple equation for the Lagrangian-mean flow induced by damped planetary waves is derived. The flow computed for stationary planetary waves of a beta-plane is found to be generally poleward and downward during winter and appears to be about twice as strong as the diabatic circulation in the lower stratosphere. An important factor in determining the high-latitude Lagrangian-mean flow field is the subtropical jet stream which blocks planetary wave propagation toward the equatorial regions. Computations using two types of Lagrangian-mean boundary conditions at the surface show that incorrect orographic forcing distorts the Lagrangian-mean flow up to three scale heights or more above ground.

Schoeberl, M. R.↗

NASA NTRS2004

Lagrangian Assimilation of Satellite Data for Climate Studies in the Arctic

Under this grant we have developed and tested a new Lagrangian model of sea ice. A Lagrangian model keeps track of material parcels as they drift in the model domain. Besides providing a natural framework for the assimilation of Lagrangian data, it has other advantages: 1) a model that follows material elements is well suited for a medium such as sea ice in which an element retains its identity for a long period of time; 2) model cells can be added or dropped as needed, allowing the spatial resolution to be increased in areas of high variability or dense observations; 3) ice from particular regions, such as the marginal seas, can be marked and traced for a long time; and 4) slip lines in the ice motion are accommodated more naturally because there is no internal grid. Our work makes use of these strengths of the Lagrangian formulation.

Lindsay, Ronald W.↗

DOE OSTI2025

Developing a Lagrangian Frame Transformation on Satellite Data to Study Cloud Microphysical Transitions in Arctic Marine Cold Air Outbreaks

Abstract Arctic marine cold air outbreaks (CAOs) generate distinct and dynamic cloud regimes due to intense air‐sea interactions. To understand the temporal evolution of CAO cloud properties and compare different CAO events, a Lagrangian perspective is particularly useful. We developed a novel technique that enables the conversion of inherently Eulerian satellite data into a Lagrangian framework, combining the broad spatiotemporal coverage of satellite observations with the advantages of Lagrangian tracking. This technique was applied to eight CAO cases associated with a recent field campaign. Our results reveal a striking contrast among the cases in terms of cloud‐top phase transitions, providing new insights into the evolution of CAO cloud properties.

Lagrangian analysis↗

DOE OSTI2024

A High-Order Eulerian–Lagrangian Runge–Kutta Finite Volume (EL–RK–FV) Method for Scalar Nonlinear Conservation Laws

Abstract We present a class of high-order Eulerian–Lagrangian Runge–Kutta finite volume methods that can numerically solve Burgers’ equation with shock formations, which could be extended to general scalar conservation laws. Eulerian–Lagrangian (EL) and semi-Lagrangian (SL) methods have recently seen increased development and have become a staple for allowing large time-stepping sizes. Yet, maintaining relatively large time-stepping sizes post shock formation remains quite challenging. Our proposed scheme integrates the partial differential equation on a space-time region partitioned by linear approximations to the characteristics determined by the Rankine–Hugoniot jump condition. We trace the characteristics forward in time and present a merging procedure for the mesh cells to handle intersecting characteristics due to shocks. Following this partitioning, we write the equation in a time-differential form and evolve with Runge–Kutta methods in a method-of-lines fashion. High-resolution methods such as ENO and WENO-AO schemes are used for spatial reconstruction. Extension to higher dimensions is done via dimensional splitting. Numerical experiments demonstrate our scheme’s high-order accuracy and ability to sharply capture post-shock solutions with large time-stepping sizes.

Chen, Jiajie↗

DOE OSTI2025

One-Dimensional Multi-Velocity Capabilities for Arbitrary Lagrangian-Eulerian Normal Contact Mechanics

Lagrangian and Arbitrary Lagrangian Eulerian (ALE) hydrodynamics codes such as FLAG form the backbone of many mission-critical multi physics simulations at Los Alamos National Laboratory. Critical to pre forming high fidelity simulations with these codes are Lagrangian and ALE contact algorithms, which allow materials to collide, slide, and sep arate throughout a simulation.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1972

Lagrangian methods in the analysis of nonlinear wave interactions in plasma

An averaged-Lagrangian method is developed for obtaining the equations which describe the nonlinear interactions of the wave (oscillatory) and background (nonoscillatory) components which comprise a continuous medium. The method applies to monochromatic waves in any continuous medium that can be described by a Lagrangian density, but is demonstrated in the context of plasma physics. The theory is presented in a more general and unified form by way of a new averaged-Lagrangian formalism which simplifies the perturbation ordering procedure. Earlier theory is extended to deal with a medium distributed in velocity space and to account for the interaction of the background with the waves. The analytic steps are systematized, so as to maximize calculational efficiency. An assessment of the applicability and limitations of the method shows that it has some definite advantages over other approaches in efficiency and versatility.

Galloway, J. J.↗

NASA NTRS1977

Microscopic Lagrangian description of warm plasmas. I - Linear wave propagation. II - Nonlinear wave interactions

It is pointed out that the conventional iterative analysis of nonlinear plasma wave phenomena, which involves a direct use of Maxwell's equations and the equations describing the particle dynamics, leads to formidable theoretical and algebraic complexities, especially for warm plasmas. As an effective alternative, the Lagrangian method may be applied. It is shown how this method may be used in the microscopic description of small-signal wave propagation and in the study of nonlinear wave interactions. The linear theory is developed for an infinite, homogeneous, collisionless, warm magnetoplasma. A summary is presented of a perturbation expansion scheme described by Galloway and Kim (1971), and Lagrangians to third order in perturbation are considered. Attention is given to the averaged-Lagrangian density, the action-transfer and coupled-mode equations, and the general solution of the coupled-mode equations.

Kim, H.↗

NASA NTRS1978

A relation between the Lagrangian and Eulerian turbulent velocity autocorrelations

Direct computation of the Lagrangian autocorrelation is not feasible since generally one cannot measure the turbulent velocity of each fluid particle. A model whereby the Lagrangian autocorrelation is determined in terms of a domain integral of a set of regular Eulerian autocorrelations is advanced. The Eulerian autocorrelations are to be acquired concurrently at all positions in the flow field. Three novel averaging procedures are utilized for obtaining the relationship between the Lagrangian and Eulerian autocorrelations. This relationship is not constrained to either homogeneous or isotropic turbulence.

Koper, C. A., Jr.↗

NASA NTRS1980

A Lagrangian mean theory of wave, mean-flow interaction with applications to nonacceleration and its breakdown

A review is given of new Lagrangian mean theory of wave transport. Attention is focused on the so-called 'nonacceleration' theorem, and it is shown that such a theorem arises naturally in the Lagrangian mean framework. Also discussed is a simple example of the Stokes drift, a concept which is central to nonacceleration. The Lagrangian mean theory substantially simplifies and unifies the understanding of wave driving in cases where nonacceleration is violated because of wave transience and dissipation. Moreover, the theory has given new insights in one particular case, that of Rossby gravity wave, mean-flow interaction. These insights have successfully explained some hitherto unresolved paradoxes in the theory of the quasi-biennial oscillation of zonal wind in the equatorial stratosphere. Some brief remarks are also made concerning some of the outstanding difficulties of the theory in need of future investigation.

Dunkerton, T.↗

NASA NTRS1981

The Lagrangian-mean motions forced by steady, dissipating equatorial waves. I

Waves are treated with a normal mode structure in order to determine the steady mean motion of the atmosphere that can be induced by dissipating equatorial waves. A model is developed which comprises a continuously stratified atmosphere at rest on the equatorial beta-plane. It is assumed that waves are excited by the corrugated bottom and are in a steady state, that dissipation is due to Newtonian cooling and Rayleigh friction, steadiness in wave magnitude is up to the second order, the waves have a long wave length, wave induced mean flows do not affect the waves, mean flows are steady, and dissipation mechanisms for the mean flows are the same as for the waves. Disturbance equations are formulated, along with Eulerian- and Lagrangian-mean flows, and the nonexistence of cross equatorial mean flows is demonstrated. Kelvin waves are shown to possess a Lagrangian-mean meridional circulation which is the same as the Eulerian-mean circulation. In the Boussinesq limit, however, neither the Eulerian- nor the Lagrangian-mean meridional circulations are caused by Kelvin waves. Further examination is made of Rossby-gravity waves and n = 1 westward propagating inertio-gravity waves.

Takahashi, M.↗

NASA NTRS1983

Microscopic Lagrangian description of warm plasmas. IV - Macroscopic approximation

The averaged-Lagrangian method is applied to linear wave propagation and nonlinear three-wave interaction in a warm magnetoplasma, in the macroscopic approximation. The microscopic Lagrangian treated by Kim and Crawford (1977) and by Galloway and Crawford (1977) is first expanded to third order in perturbation. Velocity integration is then carried out, before applying Hamilton's principle to obtain a general description of wave propagation and coupling. The results are specialized to the case of interaction between two electron plasma waves and an Alfven wave. The method is shown to be more powerful than the alternative possibility of working from the beginning with a macroscopic Lagrangian density.

Kim, H.↗

NASA NTRS1984

A Lagrangian method for a two layer shallow water model

One of the biggest drawbacks of Eulerian methods is the presence of the nonlinear convective terms which leads to inaccurate representations of advection and fronts. An alternative and conceptually simpler approach is the Lagrangian approach in which the fluid particles themselves are tracked and equations are derived based on local spatial interaction. In this formulation the nonlinear convective terms do not appear. The main drawback with the Lagrangian formulation, however, is that the local spatial interactions are time dependent. That is, at each time level, one must know the current neighbors of a given particle to accurately compute the forces acting on the particle. The Voroni Mesh which is continuously deforming and always linking nearest neighbors is needed. The effectiveness of a simple Lagrangian method for simulating the onset of a Kelvin-Helmholtz instability in a 2 layer fluid is demonstrated.

Augenbaum, J. M.↗

NASA NTRS1987

A combined Eulerian-Lagrangian analysis for computation of two-phase flows

A combined Eulerian-Lagrangian analysis, which combines a linearized block implicit Navier-Stokes analysis for the continuous phase with a Lagrangian analysis for the motion of the droplet phase to simulate evaporating two-phase flows, has been developed. A unique aspect of this analysis is that the Lagrangian equations for the droplet motion have been transformed into the Eulerian computational space using coordinate transformation resulting in better computational efficiency. Use of the present implicit procedure for the continuous phase makes it possible to efficiently use a locally highly refined mesh. The coupling of the continuous phase and the droplet phase analyses is such that it allows the use of convergence acceleration techniques for steady-state problems, as well as making it possible to simulate transient flows. A rapidly evaporating two-phase flow caused by spray of liquid nitrogen in gaseous nitrogen has been used as a test problem to demonstrate the computer code using this analysis.

Sabnis, J. S.↗

NASA NTRS1990

Integration of the shallow water equations on the sphere using a vector semi-Lagrangian scheme with a multigrid solver

A vector semi-Lagrangian semi-implicit two-time-level finite-difference integration scheme for the shallow water equations on the sphere is presented. A C-grid is used for the spatial differencing. The trajectory-centered discretization of the momentum equation in vector form eliminates pole problems and, at comparable cost, gives greater accuracy than a previous semi-Lagrangian finite-difference scheme which used a rotated spherical coordinate system. In terms of the insensitivity of the results to increasing timestep, the new scheme is as successful as recent spectral semi-Lagrangian schemes. In addition, the use of a multigrid method for solving the elliptic equation for the geopotential allows efficient integration with an operation count which, at high resolution, is of lower order than in the case of the spectral models. The properties of the new scheme should allow finite-difference models to compete with spectral models more effectively than has previously been possible.

Bates, J. R.↗

NASA NTRS1991

Parallel computing using a Lagrangian formulation

A new Lagrangian formulation of the Euler equation is adopted for the calculation of 2-D supersonic steady flow. The Lagrangian formulation represents the inherent parallelism of the flow field better than the common Eulerian formulation and offers a competitive alternative on parallel computers. The implementation of the Lagrangian formulation on the Thinking Machines Corporation CM-2 Computer is described. The program uses a finite volume, first-order Godunov scheme and exhibits high accuracy in dealing with multidimensional discontinuities (slip-line and shock). By using this formulation, a better than six times speed-up was achieved on a 8192-processor CM-2 over a single processor of a CRAY-2.

Liou, May-Fun↗

NASA NTRS1991

Lagrangian and Eulerian statistics obtained from direct numerical simulations of homogeneous turbulence

Direct numerical simulation is used to study dispersion in decaying isotropic turbulence and homogeneous shear flow. Both Lagrangian and Eulerian data are presented allowing direct comparison, but at fairly low Reynolds number. The quantities presented include properties of the dispersion tensor, isoprobability contours of particle displacement, Lagrangian and Eulerian velocity autocorrelations and time scale ratios, and the eddy diffusivity tensor. The Lagrangian time microscale is found to be consistently larger than the Eulerian microscale, presumably due to the advection of the small scales by the large scales in the Eulerian reference frame.

Squires, Kyle D.↗

NASA NTRS1992

An extended Lagrangian method

A unique formulation of describing fluid motion is presented. The method, referred to as 'extended Lagrangian method', is interesting from both theoretical and numerical points of view. The formulation offers accuracy in numerical solution by avoiding numerical diffusion resulting from mixing of fluxes in the Eulerian description. Meanwhile, it also avoids the inaccuracy incurred due to geometry and variable interpolations used by the previous Lagrangian methods. Unlike the Lagrangian method previously imposed which is valid only for supersonic flows, the present method is general and capable of treating subsonic flows as well as supersonic flows. The method proposed in this paper is robust and stable. It automatically adapts to flow features without resorting to clustering, thereby maintaining rather uniform grid spacing throughout and large time step. Moreover, the method is shown to resolve multi-dimensional discontinuities with a high level of accuracy, similar to that found in one-dimensional problems.

Liou, Meng-Sing↗

NASA NTRS1993

A Generalized Eulerian-Lagrangian Analysis, with Application to Liquid Flows with Vapor Bubbles

Under a NASA MSFC SBIR Phase 2 effort an analysis has been developed for liquid flows with vapor bubbles such as those in liquid rocket engine components. The analysis is based on a combined Eulerian-Lagrangian technique, in which Eulerian conservation equations are solved for the liquid phase, while Lagrangian equations of motion are integrated in computational coordinates for the vapor phase. The novel aspect of the Lagrangian analysis developed under this effort is that it combines features of the so-called particle distribution approach with those of the so-called particle trajectory approach and can, in fact, be considered as a generalization of both of those traditional methods. The result of this generalization is a reduction in CPU time and memory requirements. Particle time step (stability) limitations have been eliminated by semi-implicit integration of the particle equations of motion (and, for certain applications, the particle temperature equation), although practical limitations remain in effect for reasons of accuracy. The analysis has been applied to the simulation of cavitating flow through a single-bladed section of a labyrinth seal. Models for the simulation of bubble formation and growth have been included, as well as models for bubble drag and heat transfer. The results indicate that bubble formation is more or less 'explosive'. for a given flow field, the number density of bubble nucleation sites is very sensitive to the vapor properties and the surface tension. The bubble motion, on the other hand, is much less sensitive to the properties, but is affected strongly by the local pressure gradients in the flow field. In situations where either the material properties or the flow field are not known with sufficient accuracy, parametric studies can be carried out rapidly to assess the effect of the important variables. Future work will include application of the analysis to cavitation in inducer flow fields.

Dejong, Frederik J.↗