**2.1 Governing equations for transport of scalars: Unclosed statistical set**

The turbulent transfer equations for a scalar *F* are usually expressed as

equation furnished an upper limit for the normalized RMS value, which is not reached

The random square waves were also used by Schulz et al. (1991) to quantify the so called "intensity of segregation" in the superficial boundary layer formed during mass transport, for which the explanations of segregation scales found in Brodkey (1967) were used. The time constant of the intensity of segregation, as defined in the classical studies of Corrsin (1957, 1964), was used to correlate the mass transfer coefficient across the water surface with more usual parameters, like the Schmidt number and the energy dissipation rate. Random square waves were also applied by Janzen (2006), who used the techniques of Particle Image Velocimetry (PIV) and Laser Induced Fluorescence (LIF) to study the mass transfer at the air-water interface, and compared his measurements with the predictions of Schulz & Schulz (1991) employing *ad hoc* concentration profiles. Further, Schulz & Janzen (2009) confirmed the upper limit for the normalized RMS of the concentration fluctuations by taking into account the effect of diffusion, also evaluating the thickness of diffusive layers and the role of diffusive and turbulent transports in boundary layers. A more detailed theoretical relationship for the RMS of the concentration fluctuation showed that several

Intending to present the methodology in a more organized manner, Schulz et al. (2011a) showed a way to "model" the records of velocity and mass concentration (that is, to represent them in an *a priori* simplified form) for a problem of mass transport at gas-liquid interfaces. The fluctuations of these variables were expressed through the so called "partition, reduction, and superposition functions", which were defined to simplify the oscillating records. As a consequence, a finite number of basic parameters was used to express all the statistical quantities of the equations of the problem in question. The extension of this approximation to different Transport Phenomena equations is demonstrated in the present study, in which the mentioned statistical functions are derived for general scalar transport (called here "scalar-velocity interactions"). A first application for velocity fields is also shown (called here "velocity-velocity interactions"). A useful consequence of this methodology is that it allows to "close" the turbulence equations, because the number of equations is bounded by the number of basic parameters used. In this chapter we show 1) the *a priori* modeling (simplified representation) of the records of turbulent variables, presenting the basic definitions used in the random square wave approximation (following Schulz et al., 2011a); 2) the generation of the usual statistical quantities considering the random square wave approximation (scalar-velocity interactions); 3) the application of the methodology to a one-dimensional scalar transport problem, generating a closed set of equations easy to be solved with simple numerical resources; and 4) the extension of the study of Schulz & Johannes (2009) to velocity fields (velocity-velocity

Because the method considers primarily the oscillatory records itself (*a priori* analysis), and not phenomenological aspects related to physical peculiarities (*a posteriori* analysis, like the definition of a turbulent viscosity and the use of turbulent kinetic energy and its dissipation

rate), it is applicable to any phenomenon with oscillatory characteristics.

**2.1 Governing equations for transport of scalars: Unclosed statistical set**  The turbulent transfer equations for a scalar *F* are usually expressed as

different statistical profiles of turbulent mass transfer may be interrelated.

when diffusion is taken into account.

interactions).

**2. Scalar-velocity interactions** 

$$\frac{\partial \overline{\mathbf{F}}}{\partial \mathbf{\hat{t}}} + \overline{\mathbf{V}}\_{\mathbf{i}} \frac{\partial \overline{\mathbf{F}}}{\partial \mathbf{x}\_{\mathbf{i}}} = \frac{\partial}{\partial \mathbf{x}\_{\mathbf{i}}} \left( \mathbf{D}\_{\mathbf{F}} \frac{\partial \overline{\mathbf{F}}}{\partial \mathbf{x}\_{\mathbf{i}}} - \overline{\mathbf{v}\_{\mathbf{i}} \mathbf{f}} \right) + \overline{\mathbf{g}} \ \mathbf{y} \ \mathbf{i} = \mathbf{1} \ \mathbf{2} \ \mathbf{3} \tag{1}$$

where *F* and *f* are the mean scalar function and the scalar fluctuation, respectively. *Vi* (*i* = 1, 2, 3) are mean velocities and *vi* are velocity fluctuations, *t* is the time, *xi* are the Cartesian coordinates, *g* represents the scalar sources and sinks and *DF* is the diffusivity coefficient of *F*. For one-dimensional transfer, without mean movements and generation/consumption of *F*, equation (1) with *x3*=*z* and *v3*=߱ is simplified to

$$\frac{\partial \overline{\mathbf{F}}}{\partial \mathbf{f}} = \frac{\partial}{\partial \mathbf{z}} \left( \mathbf{D}\_{\mathbf{F}} \frac{\partial \overline{\mathbf{F}}}{\partial \mathbf{z}} - \overline{\mathbf{a} \, \mathbf{f}} \right) \tag{2}$$

As can be seen, a second variable, given by the mean product ωf , is added to the equation of *F* , so that a second equation involving ωf and *F* is needed to obtain solutions for both variables. Additional statistical equations may be generated averaging the product between equation (1) and the instantaneous fluctuations elevated to some power ( *f* ). As any new equation adds new unknown statistical products to the problem, the resulting system is never closed, so that no complete solution is obtained following strictly statistical procedures (closure problem). Studies on turbulence consider a low number of statistical equations (involving only the first statistical moments), together with additional equations based on *ad hoc* models that close the systems. This procedure seems to be the most natural choice, because having already obtained equation (2), it remains to model the new parcel *ω f a posteriori* (that is, introducing hypotheses and definitions to solve it). An example is the combined use of the Boussinesq hypothesis (in which the turbulent viscosity/diffusivity is defined) with the Komogoroff reasoning about the relevance of the turbulent kinetic energy and its dissipation rate. The ߢെߝ model for statistical turbulence is then obtained, for which two new statistical equations are generated, one of them for *k* and the other for ߝ. Of course, new unknown parameters appear, but also additional *ad hoc* considerations are made, relating them to already defined variables.

In the present chapter, as done by Schulz et al. (2011a), we do not limit the number of statistical equations based on *a posteriori* definitions for *ω f* . Convenient *a priori* definitions are used on the oscillatory records, obtaining transformed equations for equation (1) and additional equations. The central moments of the scalar fluctuations, *f F F* , ߠ =1, 2, 3,… are considered here. For example, the one-dimensional equations for ߠ=2, 3 and 4, are given by

$$\frac{1}{2}\frac{\partial \overline{f^2}}{\partial t} + \overline{f}\overline{o}\frac{\partial \overline{F}}{\partial z} + \frac{1}{2}\frac{\partial \overline{f^2}}{\partial z} = D\_F \left( \overline{f \frac{\partial^2 f}{\partial z^2}} \right) \tag{3a}$$

$$\frac{1}{3}\frac{\overline{\partial}\overline{f^3}}{\overline{\partial}t} + \overline{f^2}\frac{\partial\overline{F}}{\overline{\partial}t} + \overline{f^2}o\frac{\partial\overline{F}}{\overline{\partial}z} + \frac{1}{3}\frac{\overline{\partial}o\overline{f^3}}{\overline{\partial}z} = D\_\mathbb{F}\left(\overline{f^2}\frac{\partial^2\overline{F}}{\overline{\partial}z^2} + \overline{f^2}\frac{\overline{\partial^2f}}{\overline{\partial}z^2}\right) \tag{3b}$$

One Dimensional Turbulent Transfer

**field -** *n*

depend on *z*.

for example, the central moments *f F F*

oscillations justifies these corrective parcels.

This definition also implies that

Using Random Square Waves – Scalar/Velocity and Velocity/Velocity Interactions 7

2 1 *tt t* is the time interval for the average operation. Equation (4) generates a mean value *F z*( ) for 1 2 *z zz* and 1 2 *t tt* . Any statistical quantity present in equations 3, like,

The method described in the next sections allows to obtain the relevant statistical quantities

**2.3 Bimodal square wave: Mean values using a time-partition function for the scalar** 

The basic assumptions made to "model" the original oscillatory records may be followed considering Figure 2. In this sense, figure 2a is a sketch of the original record of the scalar variable *F* at a position 1 2 *z zz* , as shown in the gray vertical plane of Figure 1. The objective of this analysis is to obtain an equation for the mean function *F z*( ) for 1 2 *t tt* , which is also shown in figure 2a. The values of the scalar variable during the turbulent transfer are affected by both the advective turbulent movements and diffusion. Discarding diffusion, the value of *F* would ideally alternate between the limits *Fp* and *Fn* (the bimodal square wave), as shown in Figure 2b (the fluid particles would transport only the two mentioned *F* values). This condition was assumed as a first simplification, but maintaining the correct mean, in which *F z*( ) is unchanged. It is known that diffusion induces fluxes governed by *F* differences between two regions of the fluid (like the Fourier law for heat transfer and the Fick law for mass transfer). These fluxes may significantly lower the amplitude of the oscillations in small patches of fluid, and are taken into account using *Fp*-*P* and *Fn*+*N* for the two new limiting *F* values, as shown in Figure 2c. The parcels *P* and *N*

In other words, the amplitude of the square oscillations is "adjusted" (modeled), in order to approximate it to the mean amplitude of the original record. As can be seen, the aim of the method is not only to evaluate *F* adequately, but also the lower order statistical quantities that depend on the fluctuations, which are relevant to close the statistical equations. The parcels *P* and *N* were introduced based on diffusion effects, but any cause that inhibits

The first statistical parameter is represented by *n*, and is defined as the fraction of the time for which the system is at each of the two *F* values (equations 5 and 6), being thus named as

> ( ) of the observation *<sup>p</sup> t at F P*

> > of the observation *<sup>n</sup> t at F N*

(5)

(6)

"partition function". This function n depends on *z* and is mathematically defined as

( ) <sup>1</sup>

*F* remains the same in figures 2a, b and c. The constancy between figures 2b and c is obtained using mass conservation, implying that *P* and *N* are related through equation (7):

*t*

*n*

*n*

*t*

simplify notation, both coordinates (*z*, *t*) are dropped off in the rest of the text.

of the governing equations, like the mean function *F* , using simplified records of *F*.

, is defined according to equation (4). To

$$\frac{1}{4}\frac{\partial\overline{f^4}}{\partial t} + \overline{f^3}\frac{\partial\overline{F}}{\partial t} + \overline{f^3}o\frac{\partial\overline{F}}{\partial z} + \frac{1}{4}\frac{\partial\overline{o\,f^4}}{\partial z} = D\_{\overline{F}}\left(\overline{f^3}\frac{\partial^2\overline{F}}{\partial z^2} + \overline{f^3}\frac{\partial^2\overline{f}}{\partial z^2}\right) \tag{3c}$$

In this example, equation (3a) involves *F* and *f* of equation (2), but adds three new unknowns. The first four equations (2) and (3 a, b, c) already involve eleven different statistical quantities: *F* , 2 *f* , 3 *f* , 4 *f* , *f* , 2 *f* , 3 *f* , 4 *f* , <sup>2</sup> 2 *<sup>f</sup> <sup>f</sup> z* , <sup>2</sup> 2 2 *<sup>f</sup> <sup>f</sup> z* , and

2 3 2 *<sup>f</sup> <sup>f</sup> z* , and the "closure" is not possible. The general equation for central moments, for

any ߠ, is given by [20]

$$\frac{1}{\partial} \frac{\overline{\partial \, f^{\theta}}}{\partial t} + \overline{f^{\theta - 1}} \frac{\partial \overline{F}}{\partial t} + \overline{f^{\theta - 1}} \alpha \frac{\partial \overline{F}}{\partial z} + \frac{1}{\theta} \frac{\partial \overline{\alpha \, f^{\theta}}}{\partial z} = D\_{\overline{F}} \left( \overline{f^{\theta - 1}} \frac{\partial^{2} \overline{F}}{\partial z^{2}} + \overline{f^{\theta - 1}} \frac{\overline{\partial^{2} f}}{\partial z^{2}} \right) \tag{3d}$$

(using ߠ=1 reproduces equation (2)).

As mentioned, the method models the records of the oscillatory variables, using random square waves. The number of equations is limited by the number of the basic parameters defined "*a priori*".

### **2.2 "Modeling" the records of the oscillatory variables**

As mentioned in the introduction, the term "modeling" is used here as "representing in a simplified way". Following Schulz et al. (2011a), consider the function *F*(*z*, *t*) shown in Figure 1. It represents a region of a turbulent fluid in which the scalar quantity *F* oscillates between two functions *Fp* (*p*=previous) and *Fn* (*n*=next) in the interval *z*1<*z*<*z*2. Turbulence is assumed stationary.

Fig. 1a. A two-dimensional random scalar field F oscillating between the boundary functions *Fp*(t) and *Fn*(t).

Fig. 1b. Sketch of the region shown in figure (1a). Turbulence is stationary. Adapted from Schulz et al. (2011a)

The time average of *F(z, t)* for 1 2 *z zz* , indicated by *Fzt* (,) is defined as usual

$$\overline{F}(z,t) = \frac{1}{\Delta t} \int\_{t\_1}^{t\_2} F(z,t)dt \qquad \text{for} \qquad z\_1 < z < z\_2 \tag{4}$$

4 4 2 2 3 3 3 3

 

*<sup>f</sup> F F <sup>f</sup> <sup>F</sup> <sup>f</sup> f f Df f t t zz z z* 

unknowns. The first four equations (2) and (3 a, b, c) already involve eleven different

 , 2 *f* , 3 *f* , 4 *f* , <sup>2</sup>

, and the "closure" is not possible. The general equation for central moments, for

1 1 1 1

 

*<sup>f</sup> F F <sup>f</sup> <sup>F</sup> <sup>f</sup> f f Df f t t zz z z*

As mentioned, the method models the records of the oscillatory variables, using random square waves. The number of equations is limited by the number of the basic parameters

As mentioned in the introduction, the term "modeling" is used here as "representing in a simplified way". Following Schulz et al. (2011a), consider the function *F*(*z*, *t*) shown in Figure 1. It represents a region of a turbulent fluid in which the scalar quantity *F* oscillates between two functions *Fp* (*p*=previous) and *Fn* (*n*=next) in the interval *z*1<*z*<*z*2. Turbulence is

   

*F*

1 1

1 1

**2.2 "Modeling" the records of the oscillatory variables** 

 

In this example, equation (3a) involves *F* and *f*

statistical quantities: *F* , 2 *f* , 3 *f* , 4 *f* , *f*

(using ߠ=1 reproduces equation (2)).

Fig. 1a. A two-dimensional random scalar field F oscillating between the boundary

functions *Fp*(t) and *Fn*(t).

2 3 2 *<sup>f</sup> <sup>f</sup> z* 

any ߠ, is given by [20]

defined "*a priori*".

assumed stationary.

4 4 *<sup>F</sup>*

2 2

2 2

Fig. 1b. Sketch of the region shown in figure (1a). Turbulence is stationary. Adapted from

1 2

(4)

Schulz et al. (2011a)

The time average of *F(z, t)* for 1 2 *z zz* , indicated by *Fzt* (,) is defined as usual

*F z t F z t dt for z z z <sup>t</sup>*

2

<sup>1</sup> (, ) (,) *t*

1

*t*

 

2 2

of equation (2), but adds three new

2 *<sup>f</sup> <sup>f</sup> z* 

 , <sup>2</sup> 2 2 *<sup>f</sup> <sup>f</sup> z* 

(3c)

, and

(3d)

2 1 *tt t* is the time interval for the average operation. Equation (4) generates a mean value *F z*( ) for 1 2 *z zz* and 1 2 *t tt* . Any statistical quantity present in equations 3, like, for example, the central moments *f F F* , is defined according to equation (4). To

simplify notation, both coordinates (*z*, *t*) are dropped off in the rest of the text.

The method described in the next sections allows to obtain the relevant statistical quantities of the governing equations, like the mean function *F* , using simplified records of *F*.
