**2.7 Velocity fluctuations and the RMS velocity**

In figure 1 the scalar variable is represented oscillating between two homogeneous values. But nothing was said about the velocity field that interacts with the scalar field. It may also be bounded by homogeneous velocity values, but may as well have zero mean velocities in the entire physical domain, without any evident reference velocity. This is the case, for example, of the problem of interfacial mass transfer across gas-liquid interfaces, the application shown by Schulz et al. (2011a). In such situations, it is more useful to use the rms velocity <sup>2</sup> as reference, as commonly adopted in turbulence. For the one-dimensional case, with null mean motion, all equations must be derived using only the vertical velocity fluctuations ߱. It is necessary, thus, to obtain equations for <sup>2</sup> and for the velocity fluctuations (like equations 14 and 15 for f) considering the random square waves approximation. An auxiliary velocity scale *U* is firstly defined, shown in figure 4, considering "downwards" (߱ௗ) and "upwards" (߱௨) fluctuations, which amplitudes are functions of *z*.

One Dimensional Turbulent Transfer

The basic functions *n*, ߙ, ߚ, <sup>2</sup>

Physical

Maximum

Minimum

central moments is defined as

moment ( <sup>2</sup> *<sup>f</sup>* for ߠ=2 (is given by

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

equations. Some of their general characteristics are described in table 1.

dimensional form and having an undetermined maximum value.

Using Random Square Waves – Scalar/Velocity and Velocity/Velocity Interactions 13

to calculate the statistical quantities of the one-dimensional equations for scalar-velocity interactions. Further, incorporating them into equations (2) and (3), a closed set of equations for these functions is generated. In other words, the one dimensional turbulent transport problem reduces to the calculation of these functions, defined *a priori* to their inclusion in the

The RMS velocity may be normalized to be also bounded by the (absolute) values of 0.0 and 1.0. Because the position of the maximum value depends on the situation under study, needing more detailed explanations, the table is presented with the RMS velocity in

Function *n* ߙ ߚ <sup>2</sup>

Dimension Nondimensional Nondimensional Nondimensional Velocity

value 0 0 0 0

A further conclusion is that, because four functions need to be calculated, it implies that only four equations must be transformed to the random square waves representation in this one-dimensional situation. As a consequence, only lower order statistical quantities present in these equations need to be transformed, which is a positive consequence of this approximation, because the simplifications (and associated deviations) will not be propagated to the much higher order terms (they will not be present in the set of equations).

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

*f FF* 1,2,3,...

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

> 2 2 22 2 1 2 1 11 *f pn f f n f nn n FF*

(21)

 

Table 1. Characteristics of the functions defined for one dimensional scalar transport.

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

ground Partition Reduction Superposition Ref. velocity

value 1 1 1 Undetermined

, defined in items 2.3 through 2.7, are used in the sequence

(22)

Fig. 4. The definition of the partition function *m* and the velocity scale *U*. Upwards (-) and downwards (+) velocities are shown. The dark and light gray areas are equal, so that the mean velocity is zero.

Using *m* for the partition function of the velocity, the scale *U* shown in figure 4 is defined as the integration of the upper or the lower parts of the graph in Figure 4, as

$$
\mathcal{U} = \alpha\_d m \qquad \text{and} \qquad \mathcal{U} = -\alpha\_u \left(1 - m\right) \tag{16}
$$

Equation (17) describes the zero mean velocity (remembering that ߱௨ is negative)

$$
\rho \alpha\_d \, m + \alpha \iota\_u \left(1 - m\right) = 0 \qquad \text{or} \qquad \mathsf{U} - \mathsf{U} = 0 \tag{17}
$$

*U* is a function of *z*. Let us now consider the RMS velocity <sup>2</sup> , which is calculated as

$$
\overline{\alpha^2} = m\alpha\_d^2 + (1 - m)(-\alpha\_u)^2 \qquad \text{and} \qquad \overline{\sqrt{\alpha^2}} = \sqrt{m\alpha\_d^2 + (1 - m)(-\alpha\_u)^2} \tag{18}
$$

*U* and <sup>2</sup> may be easily related. From equations (13), (16), and (18) it follows that

$$\mathcal{U} = \sqrt{\alpha^2} \sqrt{\left[1 - \left(\beta + n - 2\beta \, n\right)\right] \left(\beta + n - 2\beta \, n\right)}\tag{19}$$

Finally, the velocity fluctuations may be related to <sup>2</sup> , *n* and using equations (16) and (19)

$$\alpha\_{\rm id} = \sqrt{\alpha^2} \sqrt{\frac{\beta + n - 2\beta n}{1 - \left(\beta + n - 2\beta n\right)}} \qquad \text{and} \qquad \alpha\_{\rm u} = -\sqrt{\alpha^2} \sqrt{\frac{1 - \left(\beta + n - 2\beta n\right)}{\beta + n - 2\beta n}} \tag{20}$$

2 is a function of *z* and is used as basic parameter for situations in which no evident reference velocities are present. For the example of interfacial mass transfer, <sup>2</sup> is zero at the water surface (*z*=0) and constant ( 0 ) in the bulk liquid ( *z* ).

Fig. 4. The definition of the partition function *m* and the velocity scale *U*. Upwards (-) and downwards (+) velocities are shown. The dark and light gray areas are equal, so that the

Using *m* for the partition function of the velocity, the scale *U* shown in figure 4 is defined as

*<sup>d</sup>* and *U m*

*m m d u* and 2 2 <sup>2</sup> <sup>1</sup>

may be easily related. From equations (13), (16), and (18) it follows that

 

is a function of *z* and is used as basic parameter for situations in which no evident

<sup>2</sup> *U*

 

reference velocities are present. For the example of interfacial mass transfer, <sup>2</sup>

*d u m m* 1 0 or *U U* 0 (17)

 12 2 *nn nn* (19)

 and <sup>2</sup> 1 2 2 *<sup>u</sup>*

  , *n* and

*<sup>u</sup>* 1 (16)

, which is calculated as

 *m m d u* (18)

using equations (16) and

is zero at

*n n n n*

 

(20)

 

the integration of the upper or the lower parts of the graph in Figure 4, as

Equation (17) describes the zero mean velocity (remembering that ߱௨ is negative)

*U m* 

> 

> >

*n n n n*

 

> 

the water surface (*z*=0) and constant ( 0 ) in the bulk liquid ( *z* ).

Finally, the velocity fluctuations may be related to <sup>2</sup>

<sup>2</sup> 2 1 2 *<sup>d</sup>*

*U* is a function of *z*. Let us now consider the RMS velocity <sup>2</sup>

2 2 <sup>2</sup> 1

 

*U* and <sup>2</sup> 

(19)

2 

mean velocity is zero.

The basic functions *n*, ߙ, ߚ, <sup>2</sup> , defined in items 2.3 through 2.7, are used in the sequence to calculate the statistical quantities of the one-dimensional equations for scalar-velocity interactions. Further, incorporating them into equations (2) and (3), a closed set of equations for these functions is generated. In other words, the one dimensional turbulent transport problem reduces to the calculation of these functions, defined *a priori* to their inclusion in the equations. Some of their general characteristics are described in table 1.

The RMS velocity may be normalized to be also bounded by the (absolute) values of 0.0 and 1.0. Because the position of the maximum value depends on the situation under study, needing more detailed explanations, the table is presented with the RMS velocity in dimensional form and having an undetermined maximum value.


Table 1. Characteristics of the functions defined for one dimensional scalar transport.

A further conclusion is that, because four functions need to be calculated, it implies that only four equations must be transformed to the random square waves representation in this one-dimensional situation. As a consequence, only lower order statistical quantities present in these equations need to be transformed, which is a positive consequence of this approximation, because the simplifications (and associated deviations) will not be propagated to the much higher order terms (they will not be present in the set of equations).
