Turbulent diffusion tensor

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 Reynolds stress tensor v v. The first and second terms on the RHS denote the production of the kinematic turbulent momentum flux by the mean velocity shears. The third term on the RHS denotes the transport of the kinematic momentum flux by turbulent motions (turbulent diffusion). This latter term is unknown and constitutes the well known moment closure problem in turbulence modeling. The fourth and fifth terms on the RHS denote the turbulent transport by the velocity-pressure-gradient correlation terms (pressure diffusion). The sixth term on the RHS denotes the redistribution by the return to isotropy term. In the engineering literature this term is called the pressure-strain correlation, but is nevertheless characterized by its redistributive nature (e.g., [132]). The seventh term on the RHS denotes the molecular diffusion of the turbulent momentum flux. The eighth term on the RHS denotes the viscous dissipation term. This term is often abbreviated by the symbol... 

Other authors (e.g., Dhotre et al, 2008, 2009 HiU et al, 1995 Lain et al, 2000 Masood and Delgado, 2014 Moraga et al, 2003 Mudde and Simonin, 1999 Niceno et al, 2008) have incorporated such a term into the phase interaction force in the momentum balances, its basis being in the above concept of a drifting velocity (Dehbi, 2008 VioUet and Simonin, 1994). In aU cases, assumptions have to be made about the (isotropic) turbulent dif-fusivity or the turbulent diffusivity tensor in this term. Also this topic is covered by the recent review by Pourtousi et al (2014). 

Note that 7Zu = 0 due to the continuity equation. Thus, the pressure-rate-of-strain tensor s role in a turbulent flow is to redistribute turbulent kinetic energy among the various components of the Reynolds stress tensor. The pressure-diffusion term T is defined... 

Vedula, P., P. K. Yeung, and R. O. Fox (2001). Dynamics of scalar dissipation in isotropic turbulence A numerical and modeling study. Journal of Fluid Mechanics 433, 29-60. Verman, B., B. Geurts, and H. Kuertan (1994). Realizability conditions for the turbulent stress tensor in large-eddy simulations. Journal of Fluid Mechanics 278, 351-362. Vervisch, L. (1991). Prise en compte d effets de cinetique chimique dans lesflammes de diffusion turbulente par Tapproche fonction densite de probabilite. Ph. D. thesis, Universite de Rouen, France. 

As indicated, the flux may be expressed either in units of molecules/m2 s or in units of kg/m2 s. Here, p and n are the density and number density of air, respectively, and K is called the eddy diffusion coefficient. This quantity must be treated as a tensor because atmospheric diffusion is highly anisotropic due to gravitational constraints on the vertical motion and large-scale variations in the turbulence field. Eddy diffusivity is a property of the flowing medium and not specific to the tracer. Contrary to molecular diffusion, the gradient is applied to the mixing ratio and not to number density, and the eddy diffusion coefficient is independent of the type of trace substance considered. In fact, aerosol particles and trace gases are expected to disperse with similar velocities. 

The additional introduction of turbulent viscosity to the stress tensor in Eq. (17.60) is called the Reynold s stress tensor, while in the energy equation the fluctuating component increases the thermal diffusivity by a turbulent contribution ( turb)- The same correction appears in the conservation equations of chemical species, where the presence of a diflusion associated to fluctuations is introduced. 

The first term on the right-hand side of eq. (5-19) represents heat transfer due to conduction, or the diffusion of heat, where the effective conductivity, keff, contains a correction for turbulent simulations. The second term represents heat transfer due to the diffusion of species, where Jj, i is the diffusion flux defined in Section 5-2.1.4. The third term involves the stress tensor, (tij)eff, a collection of velocity gradients, and represents heat loss through viscous dissipation. The... 

On the right-hand side of the constitutive equation, Eq. (1.3), a diffusion term has been added, as proposed by Sureshkumar and Beris [81], so that in turbulent simulations the high wavenumber contributions of the conformation tensor do not diverge during the numerical integration of this equation in time. This parallels the introduction of a numerical diffusion term in any scalar advection equation (e.g., a concentration equation with negligible molecular diffusion) that is solved along with the flow equations under turbulent conditions [82]. In Eq. (1.3), Dq is the dimensionless numerical diffusivity [54-56]. The issue of the numerical diffusivity is further discussed in Sections 1.3.2 and 1.4.3. 

These tensors represent the average viscous stress, turbulent stress and diffusion stress due to the phase slip, respectively. 

---