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- Author or Editor: Lars P. Prahm x
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Abstract
The gradient transfer theory for turbulent diffusion is reformulated in order to obtain an improved method for applied dispersion studies. The basic innovation is that diffusivity of single Fourier components of the concentration field is treated separately, i.e., spectral turbulent diffusivity coefficients are introduced. The value of the diffusivity decreases with increasing wave vector k of the concentration spectrum. The rate of growth of an expanding cloud of material thus becomes dependent on the stage of growth. This is in qualitative agreement with the statistical dispersion theory. It is shown that the assumption of k-dependent diffusivity leads to a nonlocal flux-gradient relation. A new function, the turbulent diffusivity transfer function, is introduced. The turbulent diffusive flux depends on concentration gradients at all points in the space. The diffusion equation is written in terms of the turbulent diffusivity transfer function. The width of the turbulent diffusivity transfer function is shown to determine the validity of the traditional gradient transfer theory formulation. The turbulent dispersion can be considered as Gaussian when the size of the cloud is considerably larger than the size of the most energetic turbulent eddies. These eddies determine the width of the turbulent diffusivity transfer function.
In general, it is shown that the shape of the cloud is non-Gaussian and the width, computed in terms of spectral turbulent diffusivity coefficients, is smaller than in a Gaussian distribution. This deviation decreases with increasing size of the cloud.
The present theory reveals properties in agreement with experiments and Lagrangian statistical dispersion theory and has the advantage of being an Eulerian method which can be used for air pollution dispersion models treating multiple, interacting sources.
Abstract
The gradient transfer theory for turbulent diffusion is reformulated in order to obtain an improved method for applied dispersion studies. The basic innovation is that diffusivity of single Fourier components of the concentration field is treated separately, i.e., spectral turbulent diffusivity coefficients are introduced. The value of the diffusivity decreases with increasing wave vector k of the concentration spectrum. The rate of growth of an expanding cloud of material thus becomes dependent on the stage of growth. This is in qualitative agreement with the statistical dispersion theory. It is shown that the assumption of k-dependent diffusivity leads to a nonlocal flux-gradient relation. A new function, the turbulent diffusivity transfer function, is introduced. The turbulent diffusive flux depends on concentration gradients at all points in the space. The diffusion equation is written in terms of the turbulent diffusivity transfer function. The width of the turbulent diffusivity transfer function is shown to determine the validity of the traditional gradient transfer theory formulation. The turbulent dispersion can be considered as Gaussian when the size of the cloud is considerably larger than the size of the most energetic turbulent eddies. These eddies determine the width of the turbulent diffusivity transfer function.
In general, it is shown that the shape of the cloud is non-Gaussian and the width, computed in terms of spectral turbulent diffusivity coefficients, is smaller than in a Gaussian distribution. This deviation decreases with increasing size of the cloud.
The present theory reveals properties in agreement with experiments and Lagrangian statistical dispersion theory and has the advantage of being an Eulerian method which can be used for air pollution dispersion models treating multiple, interacting sources.
Abstract
The moment method for accurate Eulerian advection computation, originally suggested by Egan and Mahoney (1972), was developed and tested for problems in two dimensions by Pedersen and Prahm (1973, 1974). A two-dimensional version was discussed by Egan and Mahoney (1972) but their mathematical formulation was incomplete. The necessary corrections and additions are presented here.
Abstract
The moment method for accurate Eulerian advection computation, originally suggested by Egan and Mahoney (1972), was developed and tested for problems in two dimensions by Pedersen and Prahm (1973, 1974). A two-dimensional version was discussed by Egan and Mahoney (1972) but their mathematical formulation was incomplete. The necessary corrections and additions are presented here.
Abstract
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Abstract
The pseudospectral dispersion model (Christensen and Prahm, 1976) is adapted for simulation of the long-range transmission of sulphur pollutants in the European region, covering an area of about 4000 km × 4000 km. Regional “background” concentrations of sulphur oxides are found to be highly dependent on distant sources and to correlate poorly with local source strength during the considered three- and four-day episodes. The simulation is based on emission data, given in squares of about 50 km × 50 km and on synoptic wind fields derived from observed wind velocities of the 850 mb level and the surface level. The two-dimensional model includes a constant vertical mixing depth. Appropriate values for the deposition and the transformation rates of SO2 and
Abstract
The pseudospectral dispersion model (Christensen and Prahm, 1976) is adapted for simulation of the long-range transmission of sulphur pollutants in the European region, covering an area of about 4000 km × 4000 km. Regional “background” concentrations of sulphur oxides are found to be highly dependent on distant sources and to correlate poorly with local source strength during the considered three- and four-day episodes. The simulation is based on emission data, given in squares of about 50 km × 50 km and on synoptic wind fields derived from observed wind velocities of the 850 mb level and the surface level. The two-dimensional model includes a constant vertical mixing depth. Appropriate values for the deposition and the transformation rates of SO2 and
Abstract
An Eulerian model, describing the dispersion of pollutants in gases and fluids, is developed. The model is based on numerical integration of the dispersion equation, including the effect of advection, diffusion, sinks and of multiple sources. The pseudospectral method is employed for numerical integration of the dispersion equation. The model is not limited to physical problems with periodic boundary conditions, as imposed by the spectral technique. A filtering procedure prevents instabilities caused by aliasing interactions. The emphasis is placed on numerical tests relevant to air pollution studies. Pseudodiffusion is not present in this model. The error in numerical integration is brought down to a few percent of the concentration level of the pollutant. To our knowledge, the model is the most accurate Eulerian model presently available for dispersion calculations. Applications to air pollution studies are discussed. Accuracy of 19 different numerical methods is compared.
Abstract
An Eulerian model, describing the dispersion of pollutants in gases and fluids, is developed. The model is based on numerical integration of the dispersion equation, including the effect of advection, diffusion, sinks and of multiple sources. The pseudospectral method is employed for numerical integration of the dispersion equation. The model is not limited to physical problems with periodic boundary conditions, as imposed by the spectral technique. A filtering procedure prevents instabilities caused by aliasing interactions. The emphasis is placed on numerical tests relevant to air pollution studies. Pseudodiffusion is not present in this model. The error in numerical integration is brought down to a few percent of the concentration level of the pollutant. To our knowledge, the model is the most accurate Eulerian model presently available for dispersion calculations. Applications to air pollution studies are discussed. Accuracy of 19 different numerical methods is compared.
Abstract
Further development of the spectral turbulent diffusivity concept is presented with the aim of obtaining an Eulerian dispersion model applicable for multiple interacting sources. The theory is applied for studies of plume dispersion in a field of a homogeneous and stationary turbulence. A continuous plume is considered as consisting of an infinite number of expanding puffs. The puffs' center of mass fluctuates following the long-wave range of the turbulent velocity fluctuation spectrum. The center-of-mass fluctuations are assigned to phases of the Fourier coefficients of the concentration distribution. The standard deviation of the velocity of the phase fluctuations is dependent on the wave vector of the Fourier coefficient. Time-averaging results in a spectral phase diffusivity coefficient.
It is shown that the rate of growth and the center-line concentration obtained by the spectral diffusivity model are in agreement with results predicted by the Lagrangian statistical theory. For a narrow plume, it is shown that the plume width is proportional to the time of travel, while for a narrow puff, the 3/2-power dependence is found. For a narrow distribution, the concentration shape deviates, however, from a Gaussian shape, in contradiction to results of the statistical theory.
It is shown that only two external parameters are required in the spectral turbulent diffusivity model. These are the long-wave range diffusivity coefficient K 0 and the wave vector k m of the most energetic turbulent eddies. An Eulerian integro-differential transport equation is the final result of the model. This equation can also be used for dispersion in case of space- and time-dependent parameters. We suggest a procedure for a direct experimental test of the spectral turbulent diffusivity concept.
Abstract
Further development of the spectral turbulent diffusivity concept is presented with the aim of obtaining an Eulerian dispersion model applicable for multiple interacting sources. The theory is applied for studies of plume dispersion in a field of a homogeneous and stationary turbulence. A continuous plume is considered as consisting of an infinite number of expanding puffs. The puffs' center of mass fluctuates following the long-wave range of the turbulent velocity fluctuation spectrum. The center-of-mass fluctuations are assigned to phases of the Fourier coefficients of the concentration distribution. The standard deviation of the velocity of the phase fluctuations is dependent on the wave vector of the Fourier coefficient. Time-averaging results in a spectral phase diffusivity coefficient.
It is shown that the rate of growth and the center-line concentration obtained by the spectral diffusivity model are in agreement with results predicted by the Lagrangian statistical theory. For a narrow plume, it is shown that the plume width is proportional to the time of travel, while for a narrow puff, the 3/2-power dependence is found. For a narrow distribution, the concentration shape deviates, however, from a Gaussian shape, in contradiction to results of the statistical theory.
It is shown that only two external parameters are required in the spectral turbulent diffusivity model. These are the long-wave range diffusivity coefficient K 0 and the wave vector k m of the most energetic turbulent eddies. An Eulerian integro-differential transport equation is the final result of the model. This equation can also be used for dispersion in case of space- and time-dependent parameters. We suggest a procedure for a direct experimental test of the spectral turbulent diffusivity concept.
