Advection velocity turbulent

Explain the difference between the dispersion coefficient, dis, in a river and in the atmosphere. How is dis related to the mean advection velocity and to lateral turbulent diffusivity in each case ... 

On a large scale, particles (as well as gases) are moved through the atmosphere by advection and turbulence, i.e., horizontal and vertical winds (Wexler et al. 1994 Seinfeld and Pandis 1998). Simultaneous with these large-scale motions are the smaller-scale processes that can transport particles across surface boundary layers (e.g., at the Earth s surface) and thus remove them. As discussed earlier, diffusion is the dominant removal mechanism for small particles because of their high diffusion coefficients and low gravitational settling velocities. Because of their very small sizes, nanoparticles can slip... 

The smallest spatial scale at which outdoor air pollution is of concern corresponds to the air volume affected by pollutant chemical emissions from a single point source, such as a smokestack (Fig. 4-24). Chemicals are carried downwind by advection, while turbulent transport (typically modeled as Fick-ian transport) causes the chemical concentrations to become more diluted. Typically, smokestacks produce continuous pollutant emissions, instead of single pulses of pollutants thus, steady-state analysis is often appropriate. At some distance downwind, the plume of chemical pollutants disperses sufficiently to reach the ground the point at which this occurs, and the concentrations of the chemicals at this point and elsewhere, can be estimated from solutions to the advection-dispersion-reaction equation (Section 1.5), given a knowledge of the air (wind) velocity and the magnitude of Fickian transport. 

In the case of turbulent advection velocity, the transported quantity in the PBE (i.e. the NDF) fluctuates around its mean value. These fluctuations are due to the nonlinear convection term in the momentum equation of the continuous phase. In turbulent flows usually the Reynolds average is introduced (Pope, 2000). It consists of calculating ensemble-averaged quantities of interest (usually lower-order moments). Given a fluctuating property of a turbulent flow f>(t,x), its Reynolds average at a fixed point in time and space can be written as... 

The quantity k is related to the intensity of the turbulent fluctuations in the three directions, k = 0.5 u u. Equation 41 is derived from the Navier-Stokes equations and relates the rate of change of k to the advective transport by the mean motion, turbulent transport by diffusion, generation by interaction of turbulent stresses and mean velocity gradients, and destmction by the dissipation S. One-equation models retain an algebraic length scale, which is dependent only on local parameters. The Kohnogorov-Prandtl model (21) is a one-dimensional model in which the eddy viscosity is given by... 

The physical transport of particles in a river occurs by two primary modes bedload and suspended load. Bedload consists of material moved along the bed of the river by the tractive force exerted by flowing water. Bedload may roll or hop along the bottom, and individual particles may remain stationary for long periods of time between episodes of movement. Suspended load consists of material suspended within the flow and that is consequently advected by flowing water. Rivers and streams are naturally turbulent, and if the upward component of turbulence is sufficient to overcome the settling velocity of a particle, then it will tend to remain in suspension because the particles become resuspended before they can settle to the bottom of the flow. Suspended load consists of the finest particles transported by a river, and in general is composed of clay- and silt-sized... 

In the Lagrangian approach, individual parcels or blobs of (miscible) fluid added via some feed pipe or otherwise are tracked, while they may exhibit properties (density, viscosity, concentrations, color, temperature, but also vorti-city) that distinguish them from the ambient fluid. Their path through the turbulent-flow field in response to the local advection and further local forces if applicable) is calculated by means of Newton s law, usually under the assumption of one-way coupling that these parcels do not affect the flow field. On their way through the tank, these parcels or blobs may mix or exchange mass and/or temperature with the ambient fluid or may adapt shape or internal velocity distributions in response to events in the surrounding fluid. 

In order to make the transport model adaptable to measurement results some simplifications are used. Vertical and lateral components of wind are neglible, the mean transport velocity U in x-direction is steady the pollutant transfer by advection in the drift direction is greater than by turbulent diffusion at the ground total reflection is assumed. For the case that the concentration at any point in space is independent of t and that the diffusivities are independent of x, y and z the simplified diffusion equation of the K-therory /8/ becomes... 

Recall our short discussion in Section 18.5 where we learned that turbulence is kind of an analytical trick introduced into the theory of fluid flow to separate the large-scale motion called advection from the small-scale fluctuations called turbulence. Since the turbulent velocities are deviations from the mean, their average size is zero, but not their kinetic energy. The kinetic energy is proportional to the mean value of the squared turbulent velocities, Mt2urb, that is, of the variance of the turbulent velocity (see Box 18.2). The square root of this quantity (the standard deviation of the turbulent velocities) has the dimension of a velocity. Thus, we can express the turbulent kinetic energy content of a fluid by a quantity with the dimension of a velocity. In the boundary layer theory, which is used to describe wind-induced turbulence, this quantity is called friction velocity and denoted by u. In contrast, in river hydraulics turbulence is mainly caused by the friction at the... 

Although dispersion can be described by the same law as diffusion, its nature is different. Dispersion is the result of the velocity shear, that is, of the velocity difference between adjacent streamlines in an advective flow. Due to turbulent exchange perpendicular to the direction of flow, water parcels continuously change the streamline along which they move. Since these streamlines move at different speeds, each water parcel has its own individual history of speed and thus its individual mean velocity. 

The velocities of air and water frequently vary with time, as is evident to anyone who has stood in a gusty wind or swum in a turbulent river. Consequently, any estimate of flux density due to advection by a turbulent fluid flow must involve a time period over which flow variations and corresponding fluctuations of chemical concentration are averaged. Often the fluctuations in time are faster than the instruments for determining velocity and chemical concentration can follow, and the instruments inherently provide averaged values. In other situations, instruments can easily detect and measure the... 

FIGURE 2-4 Transport of a chemical in a river. At time zero, a pulse injection is made at a location defined as distance zero in the river. As shown in the upper panel, at successive times C, t2, and t3, the chemical has moved farther downstream by advection, and also has spread out lengthwise in the river by mixing processes, which include turbulent diffusion and the dispersion associated with nonuniform velocity across the river cross section. Travel time between two points in the river is defined as the time required for the center of mass of chemical to move from one point to the other. Chemical concentration at any time and distance may be calculated according to Eq. [2-10]. As shown in the lower panel, Cmax, the peak concentration in the river at any time t, is the maximum value of Eq. [2-10] anywhere in the river at that time. The longitudinal dispersion coefficient may be calculated from the standard deviation of the concentration versus distance plot, Eq. [2-7]. 

The key physical mechanisms of dispersion on this scale are the differences in mean wind speed and velocity (Uc), Uh within and above the canopy. In porous canopies, the mean wind speed normalised on Uh, i.e. (Uc/Uh) is greater than the turbulence intensities ctv/Uh, o-w/Uh 1/10, so that the cloud/plume is advected by the main wind within as well as above the canopy. In addition the topological and wake dispersion processes (described in Section 2.4.2) are as significant as turbulent eddying for dispersing matter both horizontally and vertically within the canopy. 

The vertical salt flow between the bottom and the surface water equals 5b as well and any variation in 5b causes variations in the depth of the halocline, which changes until the net vertical turbulent salt flow through the halocline is Sb or the vertical advection of the halocline equals the entrainment velocity. 

The physical interpretation of the terms in the equation is not necessary obvious. The first term on the LHS denotes the rate of accumulation of the kinematic turbulent momentum flux within the control volume. The second term on the LHS denotes the advection of the kinematic turbulent momentum flux by the mean velocity. In other words, the left hand side of the equation constitutes the substantial time derivative of the Re3molds stress tensor The first and second terms on the RHS denote the production... 

The first term on the LHS denotes the rate of accumulation of the velocity variance v - within the control volume. The second term on the LHS denotes the advection of the velocity variance by the mean velocity. The first term on the RHS denotes the production of velocity variance by the mean velocity shears. The momentum flux v vl is usually negative, thus it results in a positive contribution to variance when multiplied by a negative sign. The second term on the RHS denotes a turbulent transport term. It describes how variance is moved around by the turbulent eddies v. The third term on the RHS describes how variance is redistributed by pressure perturbations. This term is often associated with oscillations in the fluid (e.g., like buoyancy or gravity waves.) The fourth term on the RHS is called the pressure redistribution term. The factor in square brackets consists of the sum of three terms (i.e.,... 

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