Scalar flux derivation

Furthermore, space and time derivatives of mean quantities can be easily related to space and time derivatives of /u>(V, t/e x, l). For example, starting from (3.84), the time derivative of the scalar flux is given by... 

Like the Reynolds stresses, the scalar flux obeys a transport equation that can be derived from the Navier-Stokes and scalar transport equations. We will first derive the transport equation for the scalar flux of an inert scalar from (2.99), p. 48, and the governing equation for inert-scalar fluctuations. The latter is found by subtracting (3.89) from (1.28) (p. 16), and is given by... 

The derivation of the scalar-flux transport equation proceeds in exactly the same manner as with the Reynolds stresses. We first multiply (2.99), p. 48, by

[Pg.101]

The transport equation for the scalar flux of a reacting scalar [Pg.103]

This type of model is usually referred to as an algebraic scalar-flux model. Similarmodels for the Reynolds-stress tensor are referred to as algebraic second-moment (ASM) closures. They can be derived from the scalar-flux transport equation by ignoring time-dependent and spatial-transport terms. 

Like the Reynolds stresses, the scalar flux obeys a transport equation that was derived in Section 3.3 ... 

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... 

A joint statistic of particular interest is the scalar flux uitransport equation, and to compare the result to (3.102) on p. 84. 

In transported PDF methods (Pope 2000), the closure model for A, V, ip) will be a known function26 ofV. Thus, (U,Aj) will be closed and will depend on the moments of U and their spatial derivatives.27 Moreover, Reynolds-stress models derived from the PDF transport equation are guaranteed to be realizable (Pope 1994b), and the corresponding consistent scalar flux model can easily be found. We shall return to this subject after looking at typical conditional acceleration and conditional diffusion models. 

We shall see that a conditional acceleration model in the form of (6.48) is equivalent to a stochastic Lagrangian model for the velocity fluctuations whose characteristic correlation time is proportional to e/k. As discussed below, this implies that the scalar flux (u,

joint velocity, composition PDF level, and thus that a consistent scalar-flux transport equation can be derived from the PDF transport equation. 

Let us now derive phenomenological equations of the kind (5.193) corresponding to the expression (5.205). As has been mentioned before, each flux is a linear function of all thermodynamic forces. However the fluxes and thermodynamic forces that are included in the expression (5.205) for the dissipative function, have different tensor properties. Some fluxes are scalars, others are vectors, and the third one represents a second rank tensor. This means that their components transform in different ways under the coordinate transformations. As a result, it can be proven that if a given material possesses some symmetry, the flux components cannot depend on all components of thermodynamic forces. This fact is known as Curie s symmetry principle. The most widespread and simple medium is isotropic medium, that is, a medium, whose properties in the equilibrium conditions are identical for all directions. For such a medium the fluxes and thermodynamic forces represented by tensors of different ranks, cannot be linearly related to each other. Rather, a vector flux should be linearly expressed only through vectors of thermodynamic forces, a tensor flux can be a liner function only of tensor forces, and a scalar flux - only a scalar function of thermodynamic forces. The said allows us to write phenomenological equations in general form... 

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 ... 

Finally, before leaving our exploration of the dusty gas model, we must compare the large pore (or high pressure) limiting form of its flux relations with the corresponding results derived in Chapter 4 by detailed solution of the continuum equations in a long capillary. The relevant equations are (4,23) and (4,25), to be compared with the corresponding scalar forms of equations (5.23) and (5.24). Equations (4.25) and (5.24).are seen to be identical, while (4,23) and (5.23) differ only in the pressure diffusion term, which takes the form... 

The divergence operator is the three-dimensional analogue of the differential du of the scalar function u x) of one variable. The analogue of the derivative is the net outflow integral that describes the flux of a vector field across a surface S... 

In a very general sense, the flux of a quantity G is proportional at a given location to the gradient of the scalar field produced by the flux, a(x, y, z). Mathematically, one obtains the contributions of the three components with the gradient of a, grad a, from the partial derivative of a at the coordinates x, y, z which for the flux G results in ... 

During a diffusion process, e.g. the migration of an additive from a plastic into the atmosphere, a change in the concentration of the diffusing substance takes place at every location throughout the plastic. The mass flux caused by diffusion is represented by a vector quantity whereas the concentration c and its derivative of time t is a scalar quantity and is connected by the flux with help of the divergence operator. The following example serves to emphasize this relationship. 

---