Lagrangian Markers Research Articles

Numerical simulation of the instability of an inviscid liquid jet in a coflowing compressible airstream

The nonlinear interfacial instability of a liquid jet in a coflowing compressible airstream is studied numerically. A high-resolution scheme which has second-order accuracy in space and time is coupled with a Lagrangian marker particle algorithm to visualize the large-scale motion of the interfaces in compressible flow. A numerical algorithm based on an approximate equation of state of a compressible liquid is developed to allow this two-fluid system to be governed by the nonlinear unsteady Euler equations in conservative form. The initial growth of small disturbances given by the simulations agrees well with linear theory. The process of jet disruption in compressible flow is demonstrated to consist of the formation of liquid spikes, interweaving of the gas and liquid and stretching and detachment of the liquid main center core.

Eigenvalue problems in three-dimensional Euler flows

A simple analysis shows that there are two eigenvalue problems associated with vorticity. Dynamically, the vorticity tends to be a simultaneous eigenvector of the rate-of-strain tensor S and the pressure hessian P at the point of maximum enstrophy, as shown in the numerical simulations. This suggests that intense vortex stretching occurs at particular fluid particles. Indeed, high vorticity regions are more localized in Lagrangian marker space than in physical space. It is also shown that the P alignment holds valid even kinematically for random fields.

Do Singular Structures Move with an Inviscid Fluid?

Starting with analysis of a simple case of the inviscid Burgers equation, we raise the question of the title as a fundamental characterization of general inviscid fluid motion. In particular, formation of high vorticity-gradient regions in two-dimensional Euler flow is studied numerically both in physical space and in the Lagrangian marker space. Each singular structure is shown to be fixed in the marker space for some time, even though the strain and the vorticity gradient are not clearly separated in scale.

Meteorological Modeling of Arrival and Deposition of Fallout at Intermediate Distances Downwind of the Nevada Test Site

A three-dimensional atmospheric transport and diffusion model is used to calculate the arrival and deposition of fallout from 13 selected nuclear tests at the Nevada Test Site (NTS) in the 1950s. Results are used to extend NTS fallout patterns to intermediate downwind distances (300 to 1200 km). The radioactive cloud is represented in the model by a population of Lagrangian marker particles, with concentrations calculated on an Eulerian grid. Use of marker particles, with fall velocities dependent on particle size, provides a realistic simulation of fallout as the debris cloud travels downwind. The three-dimensional wind field is derived from observed data, adjusted for mass consistency. Terrain is represented in the grid, which extends up to 1200 km downwind of NTS and has 32-km horizontal resolution and 1-km vertical resolution. Ground deposition is calculated by a deposition-velocity approach. Source terms and relationships between deposition and exposure rate are based on work by Hicks. Uncertainty in particle size and vertical distributions within the debris cloud (and stem) allow for some model "tuning" to better match measured ground-deposition values. Particle trajectories representing different sizes and starting heights above ground zero are used to guide source specification. An hourly time history of the modeled fallout pattern as the debris cloud moves downwind provides estimates of fallout arrival times. Results for event HARRY illustrate the methodology. The composite deposition pattern for all 13 tests is characterized by two lobes extending out to the north-northeast and east-northeast, respectively, at intermediate distances from NTS. Arrival estimates, along with modeled deposition values, augment measured deposition data in the development of data bases at the county level; these data bases are used for estimating radiation exposure at intermediate distances downwind of NTS. Results from a study of event TRINITY are also presented.

Nonlinear free-surface flow at a two-dimensional bow

Unsteady free-surface flow at the bow of a steadily moving, two-dimensional body is solved using a modified Eulerian-Lagrangian technique. Lagrangian marker particles are distributed on both the free surface and the far-field boundary. The flow field corresponding to an inviscid, double-body solution is used for the initial condition. Solutions are obtained over a range of Froude numbers for bodies of three different shapes: a vertical step, a faired profile, and a bulbous bow. A transition Froude number exists at which the bow wave begins to overturn and break. The value of the transition Froude number depends on the bow shape. A stagnation point is observed to be present below the free surface during the initial stage of the wave formation. For flows occurring above the transition Froude number, the stagnation point remains trapped below the free surface as the wave overturns. Below the transition Froude number, the stagnation point rises to the surface as the crest of the transient bow wave moves upstream and away from the body.

PIC and its progeny

The Particle-in-Cell (PIC) method was developed to solve numerically for the strongly contorting dynamics of several materials in two or three space dimensions. Related numerical techniques with Lagrangian marker particles are also discussed in this brief survey. An extensive bibliography lists many of the earlier publications describing these techniques.

A three-dimensional numerical model of particulate transport for coastal waters

A Lagrangian marker particle in Eulerian finite difference cell solution to the three-dimensional incompressible mass transport equation was developed for predicting particulate transport in coastal and estuarine waters. Special features of the solution procedure include a finite difference grid network which translates horizontally and vertically with the mean particle motion and expands with the dispersive growth of the marker particle cloud. The cartesian vertical coordinate of the three-dimensional mass transport equation has been transformed, using instantaneous water column depth to allow adaptation to flow situations with a temporally and spatially varying bottom topography and free surface. Results from this model for turbulent diffusion and advection of a uniform plug flow of sediment in an unbounded uniform flow field with various sediment settling velocities were in excellent agreement with the corresponding analytic solutions. Using current information from a two-dimensional vertically averaged hydrodynamic's model, the model was utilized to predict the long term diffusion and advection of dilute neutrally and negatively buoyant suspended sediment clouds resulting from a hypothetical instantaneous release of dredge spoil waste at Brown's Ledge in Rhode Island Sound.

Random-Walk Simulation of Gradient-Transfer Processes Applied to Dispersion of Stack Emission from Coal-Fired Power Plants

Abstract A numerical solution to the three-dimensional advection-diffusion equation is developed and applied to the dispersion of power plant stack contaminants throughout the boundary layer. The method employs Lagrangian marker particles undergoing variable random-walk displacements to simulate a gradient-transfer process. The size of the particle displacements is directly related to the magnitude of the vertical and horizontal diffusivities which can be any functions of space and time. Using recent atmospheric turbulence data, empirical expressions for the eddy diffusivities are derived for the entire boundary layer in terms of common meteorologic parameters. Reasonable agreement is found between the numerical predictions and actual fly-ash data collected in the vicinity of a 500 MW coal-fired power station. The random-walk technique has a number of distinct advantages over both finite-difference and particle-in-a-cell methods. It is mathematically simple, computationally fast, and requires only modest ...

MODIFIED PARTIAL-CELL METHOD FOR FREE-SURFACE INCOMPRESSIBLE FLOW SIMULATIONS

A modified partial-cell method, SAMPAC, is presented to numerically calculate the time-dependent, viscous, incompressible flow of a fluid with a free surface. Lagrangian marker particles and partial cells are used to delineate the free surface, refine the surface velocity calculation, and enforce rigorous mass conservation in fluid. The axisymmetric Navier-Stokes equations are solved by a time-average, central-space differencing scheme that gives second-order accuracy to the computation, while the pressure field is found through direct matrix inversion. The versatility and accuracy of the method are illustrated in three sample calculations, which demonstrate satisfactory agreement with experimental and analytical results.

A method of time-centering the lagrangian Marker particle computation

A method of time-centering the lagrangian Marker particle computation

Numerical modeling of buoyancy dominated dispersal using a Lagrangian approach

A three-dimensional Lagrangian pollutant dispersal model for passive contaminants has been modified to allow simulation of dispersal of heavy gases and moist, buoyant plumes, e.g. cooling tower plumes. The model uses output from a two-dimensional Navier-Stokes solver for description of the atmospheric boundary layer. For heavy gas dispersion modeling, Lagrangian markers representing the emission are released into the flow field and are displaced by mean winds, turbulent velocities derived from the Langevin equation, and buoyancy forces resulting from density differences between the pollutant and the atmosphere. In the case of moist, buoyant plume dispersal, the Lagrangian markers represent water in both liquid and vapor phases and carry both dry energy (sensible heat) and latent heat of condensation/evaporation, and accelerations due to buoyancy are related to temperature differences. Model evaluations were made using experimental tests of passive and heavy gas dispersion in simulated atmospheric surface layers. Predicted concentration fields are in accord with measurements. Model simulations of cooling tower plume dispersal in the planetary boundary layer were made using field data from the single natural draft cooling tower at the Philippsburg site in West Germany and the French energy central at Bugey. Predicted plume rise and visible plume outline are in good agreement with observations for various meteorological and source conditions.