Turbulent flux

The mean turbulent flux of concentration in the vertical direction z is w x. Assuming that this turbulent flux is proportional to the gradient of concentration, and in the direction from higher to lower concentrations, an overall diffusivity K can be defined ... 

The first two terms on the right side of Equafion (40) describe the contributions from transport by advection and by turbulent flux, respectively. The separation of the motion flux into advection and turbulent flux is somewhat arbitrary depending upon the circumstances the averaging time can be anything from a few minutes to a year or even more. 

Since in most situations the perturbation quantities (V and c() are not explicitly resolved, it is not possible to evaluate the turbulent flux term directly. Instead, it must be related to the distribution of averaged quantities - a process referred to as parameterization. A common assumption is to relate the turbulent flux vector to the gradient of the averaged tracer distribution, which is analogous with the molecular diffusion expression. Equation (35). 

Development in recent years of fast-response instruments able to measure rapid fluctuations of the wind velocity (V ) and of fhe tracer concentration (c ), has made it possible to calculate the turbulent flux directly from the correlation expression in Equation (41), without having to resort to uncertain assumptions about eddy diffusivities. For example, Grelle and Lindroth (1996) used this eddy-correlation technique to calculate the vertical flux of CO2 above a foresf canopy in Sweden. Since the mean vertical velocity w) has to vanish above such a flat surface, the only contribution to the vertical flux of CO2 comes from the eddy-correlation term c w ). In order to capture the contributions from all important eddies, both the anemometer and the CO2 instrument must be able to resolve fluctuations on time scales down to about 0.1 s. 

The application of DQMOM to the closed composition PDF transport equation is described in detail by Fox (2003). If the IEM model is used to describe micromixing and a gradient-diffusivity model is used to describe the turbulent fluxes, the CFD model will have the form... 

An additional equation is required to describe the turbulent flux. The usual approach is to define an eddy diffusivity Kj (with units of area/time) such that... 

The mean profiles of velocity, temperature and solute concentration are relatively flat over most of a turbulent flow field. As an example, in Figure 1.24 the velocity profile for turbulent flow in a pipe is compared with the profile for laminar flow with the same volumetric flow rate. As the turbulent fluxes are very high but the velocity, temperature and concentration gradients are relatively small, it follows that the effective diffusivities (iH-e), (a+eH) and (2+ed) must be extremely large. In the main part of the turbulent flow, ie away from the walls, the eddy diffusivities are much larger than the corresponding molecular diffusivities ... 

In analogy to (4.55), the turbulent flux terms in (4.78) are usually modeled by invoking a gradient-diffusion hypothesis 21... 

Authors efforts in this part of the work have been concentrated on developing turbulence closures for the statistical description of two-phase turbulent flows. The primary emphasis is on development of models which are more rigorous, but can be more easily employed. The main subjects of the modeling are the Reynolds stresses (in both phases), the cross-correlation between the velocities of the two phases, and the turbulent fluxes of the void fraction. Transport of an incompressible fluid (the carrier gas) laden with monosize particles (the dispersed phase) is considered. The Stokes drag relation is used for phase interactions and there is no mass transfer between the two phases. The particle-particle interactions are neglected the dispersed phase viscosity and pressure do not appear in the particle momentum equation. 

Using the standard k-e model for compressible flows [5], the turbulent fluxes are modeled in the following way ... 

The quantities pvy vt are the Reynolds stresses and have been extensively studied by hot-wire anemometry techniques (81, G7). The other turbulent fluxes have not yet received much attention. 

Miinnich, K. O., W. B. Clarke, K. H. Fisscher, D. Flothmann, B. Kromer, W. Roether, U. Siegenthaler, Z. Top, and W. Weiss, Gas exchange and evaporation studies in a circular wind tunnel, continuous radon-222 measurements at sea, and tritium/helium-3 measurements in a lake . In Turbulent Fluxes Through the Sea Surface, Wave Dynamics and Predictions, H. Favre and K. Hasselmann, Eds., Plenum, New York, 1978, pp. 151-165. 

The turbulent fluxes of carbon at the lower boundary of the UQL (HUPL) and the upper boundary of the thermocline (HT) are described by the functions ... 

The turbulent fluxes of carbon on the thermocline-deep ocean border is considered to be proportional to the coefficient kT of the difference in carbon concentrations in the bordering layers ... 

Dollard, G.J. Unsworth, M.H. (1983) Field measurements of turbulent fluxes of wind driven fog drops to a grass surface. Atmospheric Environment, 17, 775-80. 

The Eulerian (bottom-up) approach is to start with the convective-diffusion equation and through Reynolds averaging, obtain time-smoothed transport equations that describe micromixing effectively. Several schemes have been proposed to close the two terms in the time-smoothed equations, namely, scalar turbulent flux in reactive mixing, and the mean reaction rate (Bourne and Toor, 1977 Brodkey and Lewalle, 1985 Dutta and Tarbell, 1989 Fox, 1992 Li and Toor, 1986). However, numerical solution of the three-dimensional transport equations for reacting flows using CFD codes are prohibitive in terms of the numerical effort required, especially for the case of multiple reactions with... 

Fig. 9.6 Concentrations of O3 (a) and NO2 (b) computed using RAMS (dotted line with triangles) and SURFPRO (solid line with diamonds) turbulent fluxes and scaling parameters vs. observations (grey squares)...

We now complete the derivation by introducing two final assumptions, that the turbulent flux C u jc i is linearly related to the gradients of ,... 

Direct measurement of turbulent fluctuations in canopies was not possible until relatively recently so their first recourse was to simple, first-order eddy diffusivity or X -theory models of turbulent fluxes. Typically the vertical eddy flux of a scalar, C, or of mean momentum, that is windspeed, U, was written as3... 

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