**2.8 The central moments of scalar quantities using random square waves**

It was shown that equations (3) involve central moments like <sup>2</sup> *f* , <sup>3</sup> *f* , <sup>4</sup> *f* , which, as mentioned, must be converted to the square waves representation. The general form of the central moments is defined as

$$\overline{f^{\theta}} = \overline{\left[F - \overline{F}\right]^{\theta}} \qquad \qquad \theta = 1, 2, 3, \dots \tag{21}$$

For any statistical phenomenon, the first order central moment (ߠ=1 (is always zero. Using equations (14) and (15), Schulz & Janzen (2009) showed that the second order central moment ( <sup>2</sup> *<sup>f</sup>* for ߠ=2 (is given by

$$\overline{f^{\cdot^2}} = f\_1^{\cdot^2} n + f\_2^{\cdot^2} (1 - n) = n(1 - n) \left(1 - \alpha\_f\right)^2 \left(F\_p - F\_n\right)^2 \tag{22}$$

or, normalizing the RMS value (*f* '2)

One Dimensional Turbulent Transfer

Equations (13) through (20) and (28) lead to

flux 

**2.9.2 The correlation coefficient functions** *<sup>θ</sup> f ω* Equations (3) involve turbulent fluxes like *fω* , 2 *f*

positive and negative fluctuations, as shown in figure 3

*f fn f n*

 

Rearranging, the turbulent scalar flux is expressed as

, 2 2

and ߱, obtained for ߚ = 1.0(. Is this case, , 1 *<sup>f</sup> r*

, represented by *W*, was used to normalize

 

The correlation coefficient function is now given by

 

*f*

*f*

 *r*

*r*

of <sup>2</sup> 

for ݂ఏ, we have

Equations (23), (27) and (30) lead to the correlation coefficient function

*f n n*

*f n n*

 

variables that must be expressed as functions of *n*, ߙ, ߚ and <sup>2</sup>

Using Random Square Waves – Scalar/Velocity and Velocity/Velocity Interactions 15

any power of *f* and ߱, the superposition coefficient ߚ must be used to account for an "imperfect" superposition between the scalar and the velocity fluctuations. Therefore the

*f* is calculated as shown in equation (28), with ߚ being equally applied for the

1 2 1 2 11 1 1 *d u*

 <sup>2</sup> <sup>2</sup> 1 2 1 (1 ) 2 1 12 2 *pn f n n n n f FF n n*

<sup>2</sup>

*n n f pF Fn <sup>f</sup>*

 

*n n*

 

1 <sup>1</sup> <sup>1</sup>

Schulz el al. (2010) used this equation together with data measured by Janzen (2006). The "ideal" turbulent mass flux at gas-liquid interfaces was presented (perfect superposition of *f*

Considering *r* as defined by equation (27), it is now a function of *n* and ߚ only. Generalizing

1 2 1 2 11 1 1 *d u f fn f n fn f n*

2 1

 , 2 2 21 2 21 2

*n n f n n*

*f n n n n*

 

 

2 1

(1 ) (1 ) (2 1)

2

 

 

1 1

 , 3 *f* , 4 *f* 

 

 

2

 , and 2 2 

1 () 1

<sup>1</sup> <sup>1</sup> 1 1

(32)

 

*fn f n* (28)

, which are unknown

. For products between

 

(30)

(31)

 

(29)

*nn nn*

with , 0 1 *<sup>f</sup> r*

 

*f* , as shown in Figure 5.

 

   *f f* . The measured peak

 

(33)

 

$$f \cdot f\_2 = \frac{\sqrt{f^2}}{\left(F\_p - F\_n\right)} = \sqrt{n(1-n)}\left(1 - \alpha\_f\right) \qquad\qquad\qquad\alpha\_f = 1 - \frac{\sqrt{f^2}}{\left(F\_p - F\_n\right)\sqrt{n(1-n)}}\tag{23}$$

This form is useful to obtain the reduction function ߙ from experimental data, using the normalized mean profile and the RMS profile, as shown by Schulz & Janzen (2009). Equation (23) shows that diffusion, or other causes that inhibit the fluctuations and imply in 0 *<sup>f</sup>* , imposes a peak of *f'*2 lower than 0.5.

The general central moments (ߠ=1, 2, 3 (...for the scalar fluctuation *f* are given by

$$\overline{f^{\,\,\theta}} = f\_1^{\,\theta}n + f\_2^{\,\,\theta}(1-n) = n(1-n)\Big[\left(1-n\right)^{\theta-1} + \left(-1\right)^{\theta}\left(n\right)^{\theta-1}\Big]\left(F\_p - F\_n\right)^{\theta}\left(1-a\_f\right)^{\theta}\tag{24}$$

or, normalizing the ߠth root (݂ఏ)

$$f^{\prime}|\_{\theta} = \frac{\sqrt[\theta]{f^{\prime}}}{\left(F\_{p} - F\_{n}\right)} = \sqrt[\theta]{n\left(1 - n\right)\left[\left(1 - n\right)^{\theta - 1} + \left(-1\right)^{\theta}\left(n\right)^{\theta - 1}\right]} \left(1 - a\_{f}\right) \tag{25}$$

The functional form of the statistical quantities shown here must be obtained solving the transformed turbulent transport equations (that is, the equations involving these quantities). Equations (21) through (25) show that, given *n* and ߙ, it is possible to calculate all the central moments ( *f* statistical profiles) needed in the one-dimensional equations for scalar transfer.
