Turbulent diffusion

Turbulent diffusion is the transport of mass, heat, or momentum within a system due to random and chaotic time dependent motions. It occurs when turbulent fluid systems reach critical conditions in response to shear flow, which results from a combination of steep concentration gradients, density gradients, and high velocities. It occurs much more rapidly than molecular diffusion and is therefore extremely important for problems concerning mixing and transport in systems dealing with combustion, contaminants, dissolved oxygen, and solutions in industry. In these fields, turbulent diffusion acts as an excellent process for quickly reducing the concentrations of a species in a fluid or environment, in cases where this is needed for rapid mixing during processing, or rapid pollutant or contaminant reduction for safety. Turbulent diffusion is the transport of mass, heat, or momentum within a system due to random and chaotic time dependent motions. It occurs when turbulent fluid systems reach critical conditions in response to shear flow, which results from a combination of steep concentration gradients, density gradients, and high velocities. It occurs much more rapidly than molecular diffusion and is therefore extremely important for problems concerning mixing and transport in systems dealing with combustion, contaminants, dissolved oxygen, and solutions in industry. In these fields, turbulent diffusion acts as an excellent process for quickly reducing the concentrations of a species in a fluid or environment, in cases where this is needed for rapid mixing during processing, or rapid pollutant or contaminant reduction for safety. However, it has been extremely difficult to develop a concrete and fully functional model that can be applied to the diffusion of a species in all turbulent systems due to the inability to characterize both an instantaneous and predicted fluid velocity simultaneously. In turbulent flow, this is a result of several characteristics such as unpredictability, rapid diffusivity, high levels of fluctuating vorticity, and dissipation of kinetic energy. Atmospheric dispersion, or diffusion, studies how pollutants are mixed in the environment. There are many factors included in this modeling process, such as which level of atmosphere(s) the mixing is taking place, the stability of the environment and what type of contaminant and source is being mixed. The Eulerian and Lagrangian (discussed below) models have both been used to simulate atmospheric diffusion, and are important for a proper understanding of how pollutants react and mix in different environments. Both of these models take into account both vertical and horizontal wind, but additionally integrate Fickian diffusion theory to account for turbulence. While these methods have to use ideal conditions and make numerous assumptions, at this point in time, it is difficult to better calculate the effects of turbulent diffusion on pollutants. Fickian diffusion theory and further advancements in research on atmospheric diffusion can be applied to model the effects that current emission rates of pollutants from various sources have on the atmosphere. Using planar laser-induced fluorescence (PLIF) and particle image velocimetry (PIV) processes, there has been on-going research on the effects of turbulent diffusion in flames. Main areas of study include combustion systems in gas burners used for power generation and chemical reactions in jet diffusion flames involving methane (CH4), hydrogen (H2) and nitrogen (N2). Additionally, double-pulse Rayleigh temperature imaging has been used to correlate extinction and ignition sites with changes in temperature and the mixing of chemicals in flames. The Eulerian approach to turbulent diffusion focuses on an infinitesimal volume at a specific space and time in a fixed frame of reference, at which physical properties such as mass, momentum, and temperature are measured. The model is useful because Eulerian statistics are consistently measurable and offer great application to chemical reactions. Similarly to molecular models, it must satisfy the same principles as the continuity equation below, where the advection of an element or species is balanced by its diffusion, generation by reaction, and addition from other sources or points, and the Navier–Stokes equations. ∂ c i ∂ t + ∂ ∂ x j ( u j , c i ) = D i ∂ 2 c i ∂ x j ∂ x j + R i ( c 1 , . . . , c N , T ) + S i ( x , t ) {displaystyle {partial c_{i} over partial t}+{partial over partial x_{j}}(u_{j},c_{i})=D_{i}{partial ^{2}c_{i} over partial x_{j}partial x_{j}}+R_{i}(c_{1},...,c_{N},T)+S_{i}(x,t)} i = 1 , 2 , . . . , N {displaystyle {i=1,2,...,N}} Where c i {displaystyle c_{i}} = species concentration of interest, u j {displaystyle u_{j}} = velocityt= time, x j {displaystyle x_{j}} = direction, D i {displaystyle D_{i}} = molecular diffusion constant, R i {displaystyle R_{i}} = rate of c i {displaystyle c_{i}} generated reaction, S i {displaystyle S_{i}} = rate of c i {displaystyle c_{i}} generated by source. Note that c i {displaystyle c_{i}} is concentration per unit volume, and is not mixing ratio ( k g / k g {displaystyle kg/kg} ) in a background fluid. If we consider an inert species (no reaction) with no sources and assume molecular diffusion to be negligible, only the advection terms on the left hand side of the equation survive. The solution to this model seems trivial at first, however we have ignored the random component of the velocity plus the average velocity in uj= ū + uj’ that is typically associated with turbulent behavior. In turn, the concentration solution for the Eulerian model must also have a random component cj= c+ cj’. This results in a closure problem of infinite variables and equations and makes it impossible to solve for a definite ci on the assumptions stated.