Scalar variance

The expression x (J)P(j - l)x(j) in eq. (41.4) represents the variance of the predictions, y(j), at the value x(j) of the independent variable, given the uncertainty in the regression parameters P(/). This expression is equivalent to eq. (10.9) for ordinary least squares regression. The term r(j) is the variance of the experimental error in the response y(J). How to select the value of r(j) and its influence on the final result are discussed later. The expression between parentheses is a scalar. Therefore, the recursive least squares method does not require the inversion of a matrix. When inspecting eqs. (41.3) and (41.4), we can see that the variance-covariance matrix only depends on the design of the experiments given by x and on the variance of the experimental error given by r, which is in accordance with the ordinary least-squares procedure. 

The degree of local mixing in a RANS simulation is measured by the scalar variance (complete mixing (i.e., (j> — (j>) is uniform at the SGS) up to (4>max — (4>))((4>) — 4>min) where () is the mean concentration and max and r/>min are the maximum and minimum values, respectively. The rate of local mixing is controlled by the scalar dissipation rate (Fox, 2003). The scalar time scale analogous to the turbulence integral time scale is (Fox, 2003) as follows ... 

In the statistical theory of fluid mixing presented in Chapter 3, well macromixed corresponds to the condition that the scalar means () are independent of position, and well micromixed corresponds to the condition that the scalar variances are null. An equivalent definition can be developed from the residence time distribution discussed below. 

The next step is to derive an expression for the scalar variance defined by... 

In Section 3.2, we show that under the same conditions the right-hand side of (1.24) is equal to the negative scalar dissipation rate ((3.45), p. 70). Thus, the micromixing time is related to the scalar dissipation rate e and the scalar variance by... 

Heuristically, the SGS distribution of a scalar field 0(x, t) can be used to estimate the composition PDF by constructing a histogram from all SGS points within a particular CFD grid cell.30 Moreover, because the important statistics needed to describe a scalar field (e.g., its expected value (0) or its variance (e//2)) are nearly constant on sub-grid... 

For homogeneous binary mixing of an inert scalar, the scalar mean will remain constant so that (e/>(x, t)) = (c/Tx, ())) = p. Thus, the rate of scalar mixing can be quantified in terms of the scalar variance ([Pg.84]

At intermediate times, the scalar variance will be a decreasing function of time. We can thus define the intensity of segregation - a measure of the departure of the scalar field... 

As mentioned above, the one-point PDF description does not provide the length-scale information needed to predict the decay rate of the scalar variance. For this purpose, a... 

Moreover, like the relationship between the turbulence Taylor microscale and the dissipation rate e, X t) is related to the scalar variance decay rate by17... 

Note that from its definition, the scalar spatial correlation function is related to the scalar variance by... 

Thus, 4> (k, t) d roughly corresponds to the amount of scalar variance located at point k in wavenumber space at time t. Similar statements can be made concerning the relationship between and the scalar flux (ut(p), and between [Pg.90]

By definition, the scalar variance can be found directly from the scalar energy spectrum by integrating over wavenumber space ... 

Thus, E k, t) Ak represents the amount of scalar variance located at wavenumber k. For isotropic turbulence, the scalar integral length scale is related to the scalar energy spectrum by... 

Note that as Re/, goes to infinity with Sc constant, both the turbulent energy spectrum and the scalar energy spectrum will be dominated by the energy-containing and inertial/inertial-convective sub-ranges. Thus, in this limit, the characteristic time scale for scalar variance dissipation defined by (3.55) becomes... 

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]

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

The molecular-transport term rV2( 2) will be negligible at high Reynolds number. The scalar-variance-production term V4, is defined by... 

Thus, (3.105) has three unclosed terms the scalar flux Uj), the scalar variance flux (Uj(p/2), and the scalar dissipation rate e, defined by... 

The scalar-dissipation wavenumber /cd is defined in terms of /cdi by /cd = Sc1/2kdi-Like the fraction of the turbulent kinetic energy in the dissipation range kn ((2.139), p. 54), for a fully developed scalar spectrum the fraction of scalar variance in the scalar dissipation range scales with Reynolds number as... 

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... 

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[Pg.144]

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 Vj,. 

The terms involving y in the SR model equations correspond to the fraction of the scalar-variance production that falls into a particular wavenumber band. In principle, yn could be found from the scalar-flux spectrum (Fox 1999). Instead, it is convenient to use a self-similarity hypothesis that states that for Sc = 1 at spectral equilibrium the fraction of scalar variance that lies in a particular wavenumber band will be independent of V. Applying this hypothesis to (4.103)-(4.106) yields 31... 

The scalar variance is found by summation of the spectral energies ... 

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

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