Scalar-variance transport equation

Starting with the scalar transport equation, a transport equation for the inert-scalar variance was derived in Section 3.3 ((3.105), p. 85)  

Assuming that the scalar flux (ut(p) has been closed,25 this equation has two unclosed terms. The first is the scalar-variance flux ut(p 2), which is usually closed by invoking a 

23 The standard example is A + B — P, where the reaction time scale is much shorter than the Kolmogorov time scale. 

24 In other words, reaction can only occur once A and B have diffused together so that they coexist at the same spatial location. 

25 It is imperative that the same closure for the scalar flux be used in (4.70) to find the scalar mean and the scalar-variance-production term P f,. 

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The inert-scalar-variance transport equation can then be written as... 

Thus the SR model yields the standard scalar-variance transport equation for homogeneous flow ... 

The failure of first-order moment closures for the treatment of mixing-sensitive reactions has led to the exploration of higher-order moment closures (Dutta and Tarbell 1989 Heeb and Brodkey 1990 Shenoy and Toor 1990). The simplest closures in this category attempt to relate the covariances of reactive scalars to the variance of the mixture fraction (I 2). The latter can be found by solving the inert-scalar-variance transport equation ((3.105), p. 85) along with the transport equation for (f). For example, for the one-step reaction in (5.54) the unknown scalar covariance can be approximated by... 

On the other hand, the form of the scalar-variance transport equation will depend on Gs and For example, applying the model to the mixture fraction 6 in the absence of... 

Comparing (5.377) with (3.105) on p. 85 in the high-Reynolds-number limit (and with e = 0), it can be seen that (5.378) is a spurious dissipation term.149 This model artifact results from the presumed form of the joint composition PDF. Indeed, in a transported PDF description of inhomogeneous scalar mixing, the scalar PDF relaxes to a continuous (Gaussian) form. Although this relaxation process cannot be represented exactly by a finite number of delta functions, Gs and M1 1 can be chosen to eliminate the spurious dissipation term in the mixture-fraction-variance transport equation.150... 

This is the approach taken in transported PDF methods, as discussed in detail in Chapter 6. Most of the other closure methods discussed in Chapter 5 require knowledge of the scalar variance, which can be found from a transport equation as shown next. 

The transport equation for the variance of an inert scalar (

and Reynolds averaging the resultant expression. This process leads to an unclosed term of the form 2[Pg.103]

For a homogeneous scalar field with an isotropic filter, the conditional expected value of the scalar will have the property (+U,

transport equation can be derived for the residual scalar variance defined by11... 

In the equilibrium-chemistry limit, the turbulent-reacting-flow problem thus reduces to solving the Reynolds-averaged transport equations for the mixture-fraction mean and variance. Furthermore, if the mixture-fraction field is found from LES, the same chemical lookup tables can be employed to find the SGS reacting-scalar means and covariances simply by setting x equal to the resolved-scale mixture fraction and x2 equal to the SGS mixture-fraction variance.88... 

For each test case, a non-reacting scalar (e.g., mixture fraction) should be used to determine the spatial distribution of its mean and variance (i.e., (f) and (f/2 . These results can then be compared with those found by solving the RANS transport equations (i.e., (4.70), p. 120 and (4.90), p. 125) with identical values for (U) and Tt. Fike-wise, the particle-weight distribution should be compared with the theoretical value (i.e., (7.74)). While small fluctuations about the theoretical value are to be expected, a systematic deviation almost always is the result of inconsistencies in the particle-convection algorithm. 

Generalized Transport Equations for Specific Turbulent Fluxes and Variances of the scalar variables... 

The general transport equation for the scalar quantity variance can be formulated, in analogy to the procedure applied for momentum, by subtracting (1.456) from (1.455) to obtain an equation for the turbulent fluctuations (e.g., [153] [167] [154]) ... 

Thus, the transport equation for the general scalar quantity -0 variance is... 

In scalar mixing studies and for infinite-rate reacting flows controlled by mixing, the variance of inert scalars is of interest since it is a measure of the local instantaneous departure of concentration from its local instantaneous mean value. For non-reactive flows the variance can be interpreted as a departure from locally perfect mixing. In this case the dissipation of concentration variance can be interpreted as mixing on the molecular scale. The scalar variance equation (1.462) can be derived from the scalar transport equation... 

CFD providers treat gas-phase combustion by using a mixture fraction model (Wang et al., 2006). The model is based on the solution of the transport equations for the fuel and oxidant mixture fractions as scalars and their variances. The combustion chemistry of the mixture fractions is modeled by using the equilibrium model through the minimization of the Gibbs free energy, which assumes that the chemistry is rapid enough to assure chemical equilibrium at the molecular level. Therefore, individual component concentrations for the species of interest are derived from the predicted mixture fraction distribution. 

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