**2.9.2 The correlation coefficient functions** *<sup>θ</sup> f ω*

14 Hydrodynamics – Advanced Topics

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

1 2 1 11 1 1 *pn f f fn f n n n n n F F*

1 1 ' 1 1 11 *<sup>f</sup>*

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

**2.9 The covariances and correlation coefficient functions using random square waves** 

*F f* 

*<sup>f</sup> <sup>r</sup>*

*r* is a function of *z*, and 0 1 *r* . As it is clear from equations (26) and (27), *r* is also the normalized turbulent flux of *F* and reaches a peak amplitude less than or equal to 1.0, a range convenient for the present method, coinciding with the defined functions *n*, ߙ, ߚ, also bounded by 0.0 and 1.0 (as shown in table 1). The present method allows to express *r* as

between ߱ and *f* is given by the correlation coefficient function, *r*, defined as

*f* in equation (2) is the turbulent flux of *F* along *z*. The statistical correlation

2 2

*f*

1 1

(24)

 

> 

(25)

statistical profiles) needed in the one-dimensional equations for scalar

1

<sup>1</sup> *<sup>f</sup> p n*

2

(23)

, is defined as the mean product between scalar

(27)

(26)

*f FF n n*

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

*<sup>f</sup> f n nn n*

' 1 1 *<sup>f</sup>*

2

*p n <sup>f</sup> f n <sup>n</sup> F F*

*<sup>f</sup>* , imposes a peak of *f'*2 lower than 0.5.

 

*p n*

*F F*

 

**2.9.1 The turbulent flux of the scalar** *F* The turbulent scalar flux, denoted by *F*

fluctuations (*f*) and velocity fluctuations (߱)

dependent on *n*, the normalized mean profile of *F*.

2

or, normalizing the ߠth root (݂ఏ)

central moments ( *f*

transfer.

Thus 

0 

Equations (3) involve turbulent fluxes like *fω* , 2 *f* , 3 *f* , 4 *f* , which are unknown variables that must be expressed as functions of *n*, ߙ, ߚ and <sup>2</sup> . For products between 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 flux *f* is calculated as shown in equation (28), with ߚ being equally applied for the positive and negative fluctuations, as shown in figure 3

$$
\overline{\alpha \, f} = \alpha\_d \left[ f\_1 n \beta + f\_2 \left( 1 - n \right) (1 - \beta) \right] + \alpha\_u \left[ f\_1 n (1 - \beta) + f\_2 \left( 1 - n \right) \beta \right] \tag{28}
$$

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

$$\overline{\alpha o f} = \sqrt{\alpha^2} \left( F\_p - F\_n \right) \left( 1 - \alpha\_f \right) (1 - n) n \left( 2\beta - 1 \right) \left| \sqrt{\frac{\beta + n - 2\beta n}{1 - \left( \beta + n - 2\beta n \right)}} + \sqrt{\frac{1 - \left( \beta + n - 2\beta n \right)}{\beta + n - 2\beta n}} \right| \tag{29}$$

Rearranging, the turbulent scalar flux is expressed as

$$\overline{\alpha \, f} = \frac{n(1-n)\left(1-\alpha\_f\right)\sqrt{\alpha^2}\left(F\_p - F\_n\right)}{\sqrt{n(1-n) + \frac{\beta(1-\beta)}{\left(2\beta - 1\right)^2}}}\tag{30}$$

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

$$\left|r\right|\_{o,f} = \frac{\overline{o\,f}}{\sqrt{\overline{o\,^2}\sqrt{f^2}}} = \sqrt{\frac{n\left(1-n\right)}{n\left(1-n\right) + \frac{\beta\left(1-\beta\right)}{\left(2\beta-1\right)^2}}} \qquad \text{with} \quad 0 \le \left|r\right|\_{o,f}\Big|\le 1\tag{31}$$

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* and ߱, obtained for ߚ = 1.0(. Is this case, , 1 *<sup>f</sup> r* , and 2 2 *f f* . The measured peak

of <sup>2</sup> , represented by *W*, was used to normalize *f* , as shown in Figure 5.

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

$$
\overline{\rho a f^{\theta}} = \alpha\_d \left[ f\_1^{\theta} n \beta + f\_2^{\theta} (1 - n)(1 - \beta) \right] + \alpha\_u \left[ f\_1^{\theta} n (1 - \beta) + f\_2^{\theta} (1 - n) \beta \right] \tag{32}
$$

The correlation coefficient function is now given by

$$\left.r\right|\_{\alpha,f^{\theta}} = \frac{\overline{\alpha f^{\theta}}}{\sqrt{f^{\frac{2\theta}{2\theta}}\sqrt{\alpha^2}}} = \sqrt{\frac{n(1-n)}{n(1-n) + \frac{\beta(1-\beta)}{\left(2\beta-1\right)^2}}} \left| \frac{\overline{\left[\left(1-n\right)^{\theta} - \left(-n\right)^{\theta}\right]}}{\sqrt{\left[\left(1-n\right)^{2\theta-1} + \left(-1\right)^{2\theta}\left(n\right)^{2\theta-1}\right]}} \right|\tag{33}$$

One Dimensional Turbulent Transfer

at *n*=0.5. For 0<*n*<0.5 the flux <sup>2</sup> *f*

.

**2.10.1 Simple derivatives** 

The *pth*-order space derivative

behavior of <sup>2</sup> *f*

of *n*, ߚ, <sup>ߙ</sup> and <sup>2</sup>

Using Random Square Waves – Scalar/Velocity and Velocity/Velocity Interactions 17

 <sup>3</sup>

, 4 7 7 1

Equations (34a) and (36a) can be used to analyze the general behavior of the flux <sup>2</sup> *f*

These equations involve the factor 1 2 *n* , which shows that this flux changes its direction

mentioned example of gas-liquid mass transfer, the positive sign indicates a flux entering into the bulk liquid, while the negative sign indicates a flux leaving the bulk liquid. This

from numerical simulations. A similar change of direction is observed for the flux <sup>4</sup> *f*

The equations of items 2.9.1 and 2.9.2 confirm that the normalized turbulent fluxes are expressed as functions of *n* and ߚ only, while the covariances may be expressed as functions

The governing differential equations (2) and (3) involve the derivatives of several mean quantities. The different physical situations may involve different physical principles and boundary conditions, so that "particular" solutions may be found. For the example of interfacial mass transfer reported in the cited literature (e.g. Wilhelm & Gulliver, 1991; Jähne & Monahan, 1995; Donelan, et al., 2002; Janzen et al., 2010, 2011), *Fp* is taken as the constant saturation concentration of gas at the gas-liquid interface, and *Fn* is the homogeneous bulk liquid gas concentration. In this chapter this mass transfer problem is considered as

> *p p p p p n F n F F z z*

eventual previous knowledge about the time evolution of *Fp* and *Fn*. For interfacial mass transfer the time evolution of the mass concentration in the bulk liquid follows equation (38) (Wilhelm &

*t* 

example, because it involves an interesting definition of the time derivative of *Fn*.

Gulliver, 1991; Jähne & Monahan, 1995; Donelan, et al., 2002; Janzen et al., 2010, 2011)

, 5 5 1

<sup>1</sup> *<sup>f</sup>*

<sup>1</sup> *<sup>f</sup>*

*r*

*r*

easily analyzed through the polynomial <sup>4</sup> <sup>4</sup> 1 *n n* .

**2.10 Transforming the derivatives of the statistical equations** 

*p p F z* 

The time derivative of the mean concentration, *<sup>F</sup>*

3 3

(36b)

(36c)

is positive, while for 0.5<*n*<1.0, it is negative. In the

.

,

*n n*

*n n*

4 4

was described by Magnaudet & Calmet (2006) based on results obtained

is obtained directly from equation (8), and is given by

(37)

, is also obtained from equation (8) and

*n n*

*n n*

Fig. 5. Normalized "ideal" turbulent fluxes for ߚ=1 using measured data. *W* is the measured peak of <sup>2</sup> . *z* is the vertical distance from the interface. Adapted from Schulz et al. (2011a). Equation (32) is used to calculate covariances like <sup>2</sup> *f* , <sup>3</sup> *f* , <sup>4</sup> *f* , present in equations (3). For example, for ߠ=2, 3 and 4 the normalized fluxes are given, respectively, by:

$$\left.r\right|\_{o,f^{2}} = \frac{\overline{\alpha \, f^{2}}}{\sqrt{f^{4}}\sqrt{\alpha \, ^{2}}} = \sqrt{\frac{n(1-n)}{n(1-n) + \frac{\beta\left(1-\beta\right)}{\left(2\beta-1\right)^{2}}}} \left|\frac{(1-2n)}{\sqrt{\left[\left(1-n\right)^{3}+\left(n\right)^{3}\right]}}\right|\tag{34a}$$

$$\left.r\right|\_{o,f}\,^3 = \frac{\overline{o\,f^3}}{\sqrt{f^6}\sqrt{o\,\overline{o}^2}} = \sqrt{\frac{n\left(1-n\right)}{n\left(1-n\right) + \frac{\beta\left(1-\beta\right)}{\left(2\beta-1\right)^2}}} \left|\frac{\left[\left(1-n\right)^3 + n^3\right]}{\sqrt{\left[\left(1-n\right)^5 + \left(n\right)^5\right]}}\right|\right.\tag{34b}$$

$$\left.r\right|\_{o,f,4} = \frac{\overline{o\,f^4}}{\sqrt{f^8}\sqrt{\overline{o\,\phi^2}}} = \sqrt{\frac{n(1-n)}{n(1-n) + \frac{\beta\left(1-\beta\right)}{\left(2\beta-1\right)^2}}} \left|\frac{\left[\left(1-n\right)^4 - n^4\right]}{\sqrt{\left[\left(1-n\right)^7 + \left(n\right)^7\right]}}\right|\tag{34c}$$

As an ideal case, for =1 (perfect superposition) equation 33 furnishes

$$\left| r \right|\_{o, f^{\theta}} = \frac{\overline{o \, f^{\theta}}}{\sqrt{f^{\frac{2\theta}{2\theta}}} \sqrt{o^{\theta}}^{2}} = \left| \frac{\left[ \left( 1 - n \right)^{\theta} - \left( -n \right)^{\theta} \right]}{\sqrt{\left[ \left( 1 - n \right)^{2\theta - 1} + \left( -1 \right)^{2\theta} \left( n \right)^{2\theta - 1} \right]}} \right| \tag{35}$$

and the normalized covariances <sup>2</sup> *f* , <sup>3</sup> *f* , <sup>4</sup> *f* , for ߠ=2, 3 and 4, are then given, respectively, by:

$$\left.r\right|\_{oo,f} ^2 = \left\{ \frac{\left(1 - 2n\right)}{\sqrt{\left[\left(1 - n\right)^3 + \left(n\right)^3\right]}} \right\} \tag{36a}$$

Fig. 5. Normalized "ideal" turbulent fluxes for ߚ=1 using measured data. *W* is the measured

 

 

 

, 4 8 2 7 7

=1 (perfect superposition) equation 33 furnishes

, 2 2 21 2 21

*f n n*

1 1 *<sup>f</sup> n n <sup>f</sup> <sup>r</sup>*

> , <sup>3</sup> *f* , <sup>4</sup> *f*

<sup>1</sup> *<sup>f</sup>*

<sup>3</sup>

, 6 2 5 5

*f n n n*

*f n n n n*

<sup>2</sup>

, 4 2 3 3

<sup>3</sup> <sup>3</sup> <sup>3</sup>

*n n f n n*

*f n n n n*

<sup>4</sup> <sup>4</sup> <sup>4</sup>

*n n f n n*

*f n n n n*

 

*n*

*n n*

1 2

<sup>2</sup> , 3 3

<sup>1</sup> <sup>1</sup> <sup>1</sup> 2 1

<sup>1</sup> <sup>1</sup> <sup>1</sup> 2 1

<sup>1</sup> <sup>1</sup> <sup>1</sup> 2 1

(3). For example, for ߠ=2, 3 and 4 the normalized fluxes are given, respectively, by:

Equation (32) is used to calculate covariances like <sup>2</sup> *f*

*f*

*f*

*f*

and the normalized covariances <sup>2</sup> *f*

*r*

*r*

*r*

As an ideal case, for

respectively, by:

2

*r*

. *z* is the vertical distance from the interface. Adapted from Schulz et al. (2011a).

 , <sup>3</sup> *f* , <sup>4</sup> *f* 

2

2

2

1 ()

1 1

 

1 1

 

 

1 1 2

 

, for ߠ=2, 3 and 4, are then given,

(36a)

, present in equations

(34a)

(34b)

(34c)

(35)

peak of <sup>2</sup> 

$$\left.r\right|\_{o\circ,f^{3}} = \left\{ \frac{\left[\left(1-n\right)^{3} + n^{3}\right]}{\sqrt{\left[\left(1-n\right)^{5} + \left(n\right)^{5}\right]}} \right\} \tag{36b}$$

$$\left.r\right|\_{o,f,4} = \left\{ \frac{\left[\left(1-n\right)^4 - n^4\right]}{\sqrt{\left[\left(1-n\right)^7 + \left(n\right)^7\right]}} \right\} \tag{36c}$$

Equations (34a) and (36a) can be used to analyze the general behavior of the flux <sup>2</sup> *f* . These equations involve the factor 1 2 *n* , which shows that this flux changes its direction at *n*=0.5. For 0<*n*<0.5 the flux <sup>2</sup> *f* is positive, while for 0.5<*n*<1.0, it is negative. In the mentioned example of gas-liquid mass transfer, the positive sign indicates a flux entering into the bulk liquid, while the negative sign indicates a flux leaving the bulk liquid. This behavior of <sup>2</sup> *f* was described by Magnaudet & Calmet (2006) based on results obtained from numerical simulations. A similar change of direction is observed for the flux <sup>4</sup> *f* , easily analyzed through the polynomial <sup>4</sup> <sup>4</sup> 1 *n n* .

The equations of items 2.9.1 and 2.9.2 confirm that the normalized turbulent fluxes are expressed as functions of *n* and ߚ only, while the covariances may be expressed as functions of *n*, ߚ, <sup>ߙ</sup> and <sup>2</sup> .
