Turbulence scalar flux

Returning to Eq. (166), the third term on the left-hand side involves the turbulent scalar fluxes, defined by... 

The last term on the right-hand side is unclosed and represents scalar transport due to velocity fluctuations. The turbulent scalar flux ( , varies on length scales on the order of the turbulence integral scales Lu, and hence is independent of molecular properties (i.e., v and T).17 In a CFD calculation, this implies that the grid size needed to resolve (4.70) must be proportional to the integral scale, and not the Batchelor scale as required in DNS. In this section, we look at two types of models for the scalar flux. The first is an extension of turbulent-viscosity-based models to describe the scalar field, while the second is a second-order model that is used in conjunction with Reynolds-stress models. 

Fox (1996b) has investigated the properties of the VCIEM model, and has shown that correlation between the velocity and scalar fields due to the turbulent scalar flux can have a significant effect on the molecular mixing rate predicted by (6.86). 

A somewhat less computationally demanding approach for calculating the composition field is based on the one-point joint composition PDF, f ), instead of the one-point joint velocity-composition PDF, f v,yf). With this approach, information on the turbulent flow / velocity field must be provided by appropriate flow, turbulence, scalar-flux and micro-mixing models. The reaction rate can still be exactly dealt with. A one-point joint composition PDF transport equation similar to the one-point joint velocity-composition PDF transport equation, (12.4.2-2), can be derived. For statistically stationary flow ... 

Thus, the closure problem reduces to finding an appropriate expression for the scalar flux (Ujtp). In high-Reynolds-number turbulent flows, the molecular transport term is again negligible. Thus, the scalar-flux term is responsible for the rapid mixing observed in turbulent flows. 

In locally isotropic turbulence, the fluctuating velocity gradient and scalar gradient will be uncorrelated, and sf will be null. Thus, at sufficiently high Reynolds number, the scalar-flux dissipation is negligible. 

Furthermore, in stationary isotropic turbulence the scalar flux is related to the mean scalar gradient by... 

In order to understand the physical basis for turbulent-diffusivity-based models for the scalar flux, we first consider a homogeneous turbulent flow with zero mean velocity gradient18 and a uniform mean scalar gradient (Taylor 1921). In this flow, velocity fluctuations of characteristic size... 

The scalar flux is then proportional to a turbulent-diffusion coefficient ... 

The turbulent diffusivity defined by (4.74) is proportional to the turbulent viscosity defined by (4.46). Turbulent-diffusivity-based models for the scalar flux extend this idea to arbitrary mean scalar gradients. The standard gradient-diffusion model has the form... 

Similar techniques have been employed to derive ASM models for turbulent reacting flows (Adumitroaie et al. 1997). It can be noted from (3.102) on p. 84 that the chemical source term will affect the scalar flux. For example, for a scalar involved in a first-order... 

In Section 3.3, the general transport equations for the means, (3.88), and covariances, (3.136), of 0 are derived. These equations contain a number of unclosed terms that must be modeled. For high-Reynolds-number flows, we have seen that simple models are available for the turbulent transport terms (e.g., the gradient-diffusion model for the scalar fluxes). Invoking these models,134 the transport equations become... 

There is no direct information on scalar transport due to velocity fluctuations. A PDF scalar-flux model is required to describe turbulent scalar transport.2... 

Turbulent mixing (i.e., the scalar flux) transports fluid elements in real space, but leaves the scalars unchanged in composition space. This implies that in the absence of molecular diffusion and chemistry the one-point composition PDF in homogeneous turbulence will remain unchanged for all time. Contrast this to the velocity field which quickly approaches a multi-variate Gaussian PDF due, mainly, to the fluctuating pressure term in (6.47). 

As noted in Chapter 1, the composition PDF description utilizes the concept of turbulent diffusivity (Tt) to model the scalar flux. Thus, it corresponds to closure at the level of the k-e and gradient-diffusion models, and should be used with caution for flows that require closure at the level of the RSM and scalar-flux equation. In general, the velocity, composition PDF codes described in Section 7.4 should be used for flows that require second-order closures. On the other hand, Lagrangian composition codes are well suited for use with an LES description of turbulence. 

Unlike Lagrangian composition codes that use two-equation turbulence models, closure at the level of second-order RANS turbulence models is achieved. In particular, the scalar fluxes are treated in a consistent manner with respect to the turbulence model, and the effect of chemical reactions on the scalar fluxes is treated exactly. 

Adumitroaie, V., D. B. Taulbee, and P. Givi (1997). Explicit algebraic scalar-flux models for turbulent reacting flows. AIChE Journal 43, 1935-1946. 

The vanishing effect of molecular diffusivity on turbulent dispersion Implications for turbulent mixing and the scalar flux. Journal of Fluid Mechanics 359, 299-312. 

The terms in this equation have physical interpretations analogous to those in the momentum flux equation (1.394), except for the additional term (i.e., the second term on the RHS), which is a production/loss term related to the mean scalar quantity gradient. Physically, this term suggests production of the scalar quantity flux when there is a momentum flux in a mean scalar quantity gradient. The turbulent momentum flux implies a turbulent movement of the fluid. If that movement occurs across a mean scalar quantity gradient, then the scalar quantity fluctuation would be expected. 

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