Donald L. Ermak

Brownian dynamics with hydrodynamic interactions

A method for simulating the Brownian dynamics of N particles with the inclusion of hydrodynamic interactions is described. The particles may also be subject to the usual interparticle or external forces (e.g., electrostatic) which have been included in previous methods for simulating Brownian dynamics of particles in the absence of hydrodynamic interactions. The present method is derived from the Langevin equations for the N particle assembly, and the results are shown to be consistent with the corresponding Fokker–Planck results. Sample calculations on small systems illustrate the importance of including hydrodynamic interactions in Brownian dynamics simulations. The method should be useful for simulation studies of diffusion limited reactions, polymer dynamics, protein folding, particle coagulation, and other phenomena in solution.

A computer simulation of charged particles in solution. I. Technique and equilibrium properties

A computer technique is presented for simulating the translational motion of ions in a liquid solution. In the model the diffusive motion of each ion is perturbed by the electrostatic force of the surrounding ions. Several polyelectrolyte systems of spherical polyions (10–50 A in radius) and small ions (∼1 A in radius) have been studied. For each system the polyion electrostatic shielding length and the average potential energy of each ion species was calculated. When the shielding length was sufficiently short, the computer results and the predictions of the zero polyion concentration Debye–Huckel theory were in good agreement.

Numerical integration of the Langevin equation: Monte Carlo simulation

Monte Carlo simulation techniques are derived for solving the ordinary Langevin equation of motion for a Brownian particle in the presence of an external force. These methods allow considerable freedom in selecting the size of the time step, which is restricted only by the rate of change in the external force. This approach is extended to the generalized Langevin equation which uses a memory function in the friction force term. General simulation techniques are derived which are independent of the form of the memory function. A special method requiring less storage space is presented for the case of the exponential memory function.

AN ANALYTICAL MODEL FOR AIR POLLUTANT TRANSPORT AND DEPOSITION FROM A POINT SOURCE

An atmospheric transport and deposition model is presented for pollutants emitted from an elevated point source over flat terrain. The model is obtained from the analytic solution of the atmospheric diffusion equation with the coefficients of eddy diffusion taken to be functions of downwind distance and the average wind velocity taken to be constant. Ground deposition of the pollutants is accounted for by (1) including a gravitational settling term in the atmospheric diffusion equation and by (2) applying an absorptive boundary condition at the ground surface. In order to facilitate application of the model, the results for the general situation that includes settling and deposition (and where the distribution is no longer Gaussian) are expressed in terms of the Grassian plume parameters, and specifically the crosswind and vertical standard deviation functions U,,(X) and g=,(x)_ Graphs of the pollutant air concentration and ground deposition flux are presented for a number of deposition conditions.

Analysis of Burro series 40-m3 lng spill experiments

The U.S. Department of Energy sponsored a series of nine field experiments (Burro series) conducted jointly in 1980 by the Naval Weapons Center, China Lake, California, and the Lawrence Livermore National Laboratory to determine the transport and dispersion of vapor from spills of liquefied natural gas (LNG) on water. The spill volume ranged from 24 to 39 m3, the spill rate from 11.3 to 18.4 m3/min, the wind speed from 1.8 to 9.1 m/s, and the atmospheric stability from unstable to slightly stable. An extensive array of instrumentation was deployed both upwind and downwind of the spill pond. Wind speed and direction, gas concentration, temperature, humidity, and heat flux from the ground were measured at different distances from the spill point and at different elevations relative to ground level. The wind and gas-concentration data were analyzed to further define the fluid dynamic and thermodynamic processes associated with the dispersion of the gas cloud. Data pertaining to differential boiling of LNG and observed rapid phase-transition explosions were also analyzed. The principal conclusions are summarized as follows: The turbulent processes in the lower atmospheric boundary layer dominated the transport and dispersion of gas for all experiments except Burro 8. Burro 8 was conducted under very low wind-speed conditions, and the gravity flow of the cold gas displaced the atmospheric flow, causing the wind speed within the cloud to drop essentially to zero. This has profound implications for hazard prediction from large accidental spills. High-frequency (3–5 Hz) gas-concentration measurements indicate that peak concentrations within the flammable limits are common with 10-s-average concentrations above 1%. This implies a larger flammable extent than averaged data or calculations would indicate. Differential boiloff of LNG was observed with resultant enrichment of ethane and propane in the cloud at later times. This ethane-enriched region propagates downwind and represents an additional hazard since it is more easily detonated than the methane-rich region. Energetic rapid phase transition (RPT) explosions, though not expected, did occur under at least two different circumstances during the Burro 6 and 9 tests. These explosions were large enough to damage the facility and raise questions about the coupling of the RPT-produced shock wave into the ethane-rich region of the cloud.

A comparison of dense gas dispersion model simulations with burro series LNG spill test results

Abstract The predictions from three vapor dispersion models for cold dense gas releases are compared with the results from several 40 m 3 LNG spill experiments conducted at China Lake, California, in 1980. The models vary considerably in the degree to which they approximate important physical phenomena and include restricting assumptions. The simplest model (GD), a modified Gaussian plume model, predicted a vapor cloud that was always too high and too narrow by a factor of 1.5 to 3. The second model (SLAB), a layer-averaged conservation equation model with one independent spatial variable (downwind distance), generally predicted the maximum distance to the lower flammability limit (LFL) and cloud width quite well. SLAB assumes the vertical concentration distribution is nearly uniform so that the vertical concentration gradient (∂ c /∂ z ) is essentially zero from the ground up through most of the cloud and then very steep at the top of the cloud. This was generally not the case in these experiments, especially in the high wind speed tests, where the vertical concentration gradient was found to be more gradual throughout the cloud. The final model (FEM3) is a fully three-dimensional conservation equation model that generally predicted the concentration distribution in time and space rather well. A particular achievement of this model was the prediction of a bifurcated cloud structure observed in one experiment conducted with a low ambient wind speed. Both the SLAB and the FEM3 models accurately predicted the length of time required for the cloud to disperse to a level below the LFL, even in the low wind speed test where the vapor cloud lingered over the source region for a considerable length of time after the LNG spill was terminated.

A computer simulation of charged particles in solution. II. Polyion diffusion coefficient

A computer simulation study was conducted on the effects of the polyion–small ion electrostatic interaction upon the translational diffusion of polyions in polyelectrolyte solutions at thermodynamic equilibrium. This interaction increased the polyion diffusion coefficient and the increase was strongly dependent upon the polyion charge and size, and the small ion charge and mobility, while only slightly affected by changes in the polyion concentration. The increase was largest when there were only enough counterions to balance the charge of the polyions. Increasing the number of counterion–byion pairs per polyion decreased the polyion diffusion coefficient toward its uncharged value. The computer results were markedly different from those predicted by the self‐consistent field theory of Stephen and a comparison of the two is presented.

A Lagrangian stochastic diffusion method for inhomogeneous turbulence

Abstract A Lagrangian stochastic method of solving the diffusion equation for inhomogeneous turbulence is presented in this paper. This numerical method uses (1) a first-order approximation for the spatial variation of the eddy diffusivity and (2) the first three particle position moments to define a skewed, non-Gaussian particle position probability density function. Two cases of the variation of the eddy diffusivity near a boundary are considered. In the first case, the eddy diffusivity varies linearly with height and is zero at the boundary. The method handles this case efficiently and accurately by using the first few terms of the series representation of the analytic solution to construct components of the non-Gaussian position probability density function that are important near the boundary. In the second case, the eddy diffusivity is constant near the boundary, and the well-known solution for this case – a Gaussian function reflected at the boundary – is used. Comparison of numerical simulation results to analytic solutions of the diffusion equation show that this method is accurate. For cases where the eddy diffusivity varies linearly with height to zero at the boundary, we demonstrate that this method can be significantly more efficient than the commonly used method that assumes a Gaussian particle position probability density function.

A review of recent field tests and mathematical modelling of atmospheric dispersion of large spills of Denser-than-air gases

Large-scale spills of hazardous materials often produce gas clouds which are denser than air. The dominant physical processes which occur during dense-gas dispersion are very different from those recognized for trace gas releases in the atmosphere. Most important among these processes are stable stratification and gravity flow. Dense-gas flows displace the ambient atmospheric flow and modify ambient turbulent mixing. Thermodynamic and chemical reactions can also contribute to dense-gas effects. Some materials flash to aerosol and vapor when released and the aerosol can remain airborne, evaporating as it moves downwind, causing the cloud to remain cold and dense for long distances downwind. Dense-gas dispersion models, which include phase change and terrain effects have been developed and are capable of simulating many possible accidental releases. A number of large-scale field tests with hazardous materials such as liquefied natural gas (LNG), ammonia (NH3), hydrofluoric acid(HF) and nitrogen tetroxide(N2O4) have been performed and used to evaluate models. The tests have shown that gas concentrations up to ten times higher than those predicted by trace gas models can occur due to aerosols and other dense-gas effects. A methodology for model evaluation has been developed which is based on the important physical characteristics of dense-gas releases.

FEM3 model simulations of selected Thorney Island Phase I trials

Abstract FEM3 is a three-dimensional computer model that was designed for simulating the atmospheric dispersion of heavy gas releases. The model, which has been