Howard C. Rodean

The universal constant for the Lagrangian structure function

The Langevin equation has been used for many years to model the dispersion of passive scalars in turbulent flow. It is a stochastic differential equation for the incremental change of Lagrangian particle velocity as a function of the sum of a deterministic term and a stochastic term. The stochastic term is the product of a coefficient and an incremental Wiener process. The coefficient can be written as (C0e)1/2, where C0 is a universal constant associated with the Lagrangian structure function and e is the mean rate of turbulent kinetic energy dissipation. There is considerable uncertainty about the value of C0. The values obtained by different investigators are reviewed. A value of C0=5.7 is calculated for the constant‐stress region in the neutral boundary layer.

A comparison between two stochastic diffusion models in a complex three-dimensional flow

A Random Displacement Model (RDM) and a Langevin Equation Model (LEM) are used to simulate point releases in a complex flow around a building. The flow field is generated by a three-dimensional finite element model that uses the standardk-ε model to parameterize the turbulence. The RDM- and LEM-calculated concentration fields are compared, with particular emphasis on the structure in regions with high turbulence and/or recirculation. RDM and LEM results are similar qualitatively, but RDM tends to predict lower concentration levels. In part this is due to the higher early-time diffusion. However, the expected convergence at later times is prevented by the interaction of the diffusion with the strongly inhomogeneous mean flow.

Relationship for Condensed Materials among Heat of Sublimation, Shock‐Wave Velocity, and Particle Velocity

The shock‐wave velocity U and the particle velocity u in many condensed materials are linearly related (in the absence of phase changes) according to the equation U = a + bu, where a and b are empirical constants. If the shock compression does not produce phase changes, a is approximately equal to a0, the “adiabatic,” “bulk,” or “hydrodynamic” sound speed at the initial condition. On the basis of a theoretical analysis in which it is assumed that a = a0, it is proposed that a0 and b are related to the initial cohesive energy Eχ0 by the equation Eχ0 = − a02 / 2b2, and that this relation is exact (neglecting the residual zero‐point energy) at zero pressure and temperature. This equation is consistent with experimental data for 32 metals and 11 alkali‐metal halides if Eχ0 is identified as a heat of sublimation Hs. The definition of Hs is a function of the material and is the energy required to transform the material from the solid state to an un‐ionized gas (diatomic in the case of the alkali‐metal halides, perhaps a diatomic–monatomic mixture for the alkali metals, and monatomic in the case of the other metals). This suggests that the molecular bonds of gases may be preserved in the condensed state, or perhaps they become effective in the shock‐compression process.The shock‐wave velocity U and the particle velocity u in many condensed materials are linearly related (in the absence of phase changes) according to the equation U = a + bu, where a and b are empirical constants. If the shock compression does not produce phase changes, a is approximately equal to a0, the “adiabatic,” “bulk,” or “hydrodynamic” sound speed at the initial condition. On the basis of a theoretical analysis in which it is assumed that a = a0, it is proposed that a0 and b are related to the initial cohesive energy Eχ0 by the equation Eχ0 = − a02 / 2b2, and that this relation is exact (neglecting the residual zero‐point energy) at zero pressure and temperature. This equation is consistent with experimental data for 32 metals and 11 alkali‐metal halides if Eχ0 is identified as a heat of sublimation Hs. The definition of Hs is a function of the material and is the energy required to transform the material from the solid state to an un‐ionized gas (diatomic in the case of the alkali‐metal halides, ...

Evaluation of relations among stress‐wave parameters and cohesive energy of condensed materials

The shock‐wave velocity U and the particle velocity u for many condensed materials are linearly related by the equation U =a + b u along one or more sections of the Hugoniot. Departures from linearity can usually be attributed to porosity, elastic‐wave precursors, or phase changes. If there are no such effects to cause nonlinearity, a is approximately equal to the adiabatic, bulk, or hydrodynamic sound velocity ak. Two equations involving the cohesive energy Ec are compared for 56 metals and 13 simple compounds (12 alkali metal halides and MgO): Ec = −(1/2)(a/b)2 and |Ec|=aμ2, where aμ is the shear wave velocity. It is shown that the experimental data are such that the energy of sublimation Es ≈ (1/2)(a/b)2 for the metals and compounds considered, Es≈ aμ2 for the metals, but Es≈ 0.4 aμ2 for the compounds. It is concluded that the shock‐wave parameter equation, |Ec| = Es = (1/2)(a/b)2 is preferred because it applies without coefficient adjustments to both metals and simple compounds, and it may be applied ...

Thermodynamic relations for shock waves in materials with a linear relation between shock‐wave and particle velocities

The Rankine‐Hugoniot relations for shock waves and the empirical linear relation between the shock‐wave and particle velocities define an incomplete thermodynamic description of the states along the Hugoniot curve. This incomplete description defines the following along the Hugoniot: (1) internal energy and pressure as functions of specific volume, (2) the ratio of enthalpy to internal energy, (3) the ratio of the changes in enthalpy and internal energy across a shock wave, and (4) the relation between the Gruneisen coefficient and the effective isentropic exponent. We use the Dugdale‐MacDonald relation for the Gruneisen coefficient at low pressure, an assumed constant value for the specific heat at constant volume, and reasonable physical assumptions for extremely strong shock waves together with the incomplete thermodynamic state description to define the following along the Hugoniot: (5) the Gruneisen coefficient, (6) the effective isentropic exponent, (7) the ratio of specific heats, and (8) thermal a...

A structure for models of hazardous materials with complex behavior

Abstract Most atmospheric dispersion models used to assess the environmental consequences of accidental releases of hazardous chemicals do not have the capability to simulate the pertinent chemical and physical processes associated with the release of the material and its mixing with the atmosphere. The purpose of this paper is to present a materials sub-model with the flexibility to simulate the chemical and physical behaviour of a variety of materials released into the atmosphere. The model, which is based on thermodynamic equilibrium, incorporates the ideal gas law, temperature-dependent vapor pressure equations, temperature-dependent dissociation reactions, and reactions with atmospheric water vapor. The model equations, written in terms of pressure ratios and dimensionless parameters, are used to construct equilibrium diagrams with temperature and the mass fraction of the material in the mixture as coordinates. The models versatility is demonstrated by its application to the release of UF 6 and N 2 O 4 , two materials with very different physical and chemical properties.

Effects of a Spill of LNG on Mean Flow and Turbulence under Low Wind Speed, Slightly Stable Atmospheric Conditions

Of the many liquefied natural gas (LNG) spill experiments in the 1980 Burro and 1981 Coyote series at the Naval Weapons Center (NWC), China Lake, California, only one was observed to affect the mean flow and turbulence in the near-surface atmospheric boundary layer. This experiment, Burro 8, was conducted under atmospheric conditions that permitted the gravity flow of the cold, dense gas to be almost independent of the atmospheric boundary layer. The mean flow kinetic energy was damped proportionately more than the turbulent kinetic energy. These effects of the Burro 8 LNG spill were observed at only one instrument station.

The Boundary Condition Problem

We have not discussed one problem that must be faced in using the stochastic Lagrangian models of turbulent diffusion that we studied in the preceding chapters: How should we treat the trajectories of “marked particles” near the boundaries of the turbulent atmospheric boundary layer? The following discussion is in the context of two-dimensional simulations with vertical (z) and longitudinal (x) coordinates. We are concerned with the boundaries at the bottom and the top. The lower boundary is the bottom of the computational domain that generally corresponds to the land or water surface. This boundary can act as a perfect reflector, a sink (deposition), or a source (evaporation, resuspension, etc.). There can be two upper boundaries: that of the computational domain and, at a lower level, that of the turbulent atmospheric boundary layer. In the latter case, there can be entrainment into or out of the boundary layer.

Criteria for Stochastic Models of Turbulent Diffusion

We have mentioned two criteria for stochastic Lagrangian models of turbulent diffusion: Durbin’s (1984) requirement that a Langevin model reduce to the diffusion equation and Thomson’s (1987) well-mixed criterion. It is appropriate at this point to review the several criteria that have been proposed. These were the subject of Thomson’s classic paper in which he considered the following five: 1) The well-mixed condition: If the particles of a tracer are initially well-mixed (in both position and velocity space) in a turbulent flow, will they remain so? 2) Is the small-time behavior of the velocity distribution of the particles from a point source correct? 3) Are the Eulerian equations derived from the Lagrangian model compatible with the true Eulerian equations? 4) Are the forward and reverse formulations of the dispersion consistent? 5) Does the model reduce to a diffusion equation as the Lagrangian timescale tends toward zero? Thomson showed in section 3 of his paper that a generalized version of criterion 1 is sufficient to ensure that criteria 2–5 are satisfied. Specifically, he proved mathematically that criteria 2–4 are equivalent to criterion 1 and that criterion 5 is a weaker condition than criterion 1. [We stated in section 6.3 that a model that satisfies criterion 5 does not necessarily satisfy criterion 1.] In the following chapter, we use material from Thomson’s sections 2–4 to expand upon his section 5 on the “simplest” solution for a Langevin model of nonstationary, inhomogeneous three-dimensional diffusion in Gaussian turbulence.

Stochastic Differential Equations for Turbulent Diffusion

We begin with the model proposed by Wilson et al. (1983) and rigorously derived by Thomson (1984, 1987): (6.1a)