**1. Introduction**

The mathematical treatment of phenomena that oscillate randomly in space and time, generating the so called "statistical governing equations", is still a difficult task for scientists and engineers. Turbulence in fluids is an example of such phenomena, which has great influence on the transport of physical proprieties by the fluids, but which statistical quantification is still strongly based on *ad hoc* models. In turbulent flows, parameters like velocity, temperature and mass concentration oscillate continuously in turbulent fluids, but their detailed behavior, considering all the possible time and space scales, has been considered difficult to be reproduced mathematically since the very beginning of the studies on turbulence. So, statistical equations were proposed and refined by several authors, aiming to describe the evolution of the "mean values" of the different parameters (see a description, for example, in Monin & Yaglom, 1979, 1981).

The governing equations of fluid motion are nonlinear. This characteristic imposes that the classical statistical description of turbulence, in which the oscillating parameters are separated into mean functions and fluctuations, produces new unknown parameters when applied on the original equations. The generation of new variables is known as the "closure problem of statistical turbulence" and, in fact, appears in any phenomena of physical nature that oscillates randomly and whose representation is expressed by nonlinear conservation equations. The closure problem is described in many texts, like Hinze (1959), Monin & Yaglom (1979, 1981), and Pope (2000), and a general form to overcome this difficulty is matter of many studies.

As reported by Schulz et al. (2011a), considering scalar transport in turbulent fluids, an early attempt to theoretically predict RMS profiles of the concentration fluctuations using "ideal random signals" was proposed by Schulz (1985) and Schulz & Schulz (1991). The authors used random square waves to represent concentration oscillations during mass transfer across the air-water interface, and showed that the RMS profile of the concentration fluctuations may be expressed as a function of the mean concentration profile. In other words, the mean concentration profile helps to know the RMS profile. In these studies, the authors did not consider the effect of diffusion, but argued that their

One Dimensional Turbulent Transfer

*F*, equation (1) with *x3*=*z* and *v3*=߱ is simplified to

made, relating them to already defined variables.

given by

Using Random Square Waves – Scalar/Velocity and Velocity/Velocity Interactions 5

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 F F <sup>D</sup> <sup>ω</sup><sup>f</sup> tz z

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 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

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

3,… are considered here. For example, the one-dimensional equations for ߠ=2, 3 and 4, are

2 22

 

*<sup>f</sup> Ff f f D <sup>f</sup> t zz <sup>z</sup>* 

3 3 2 2 2 2 2 2

 

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

additional equations. The central moments of the scalar fluctuations, *f F F*

2 2 *<sup>F</sup>*

1 1

3 3 *<sup>F</sup>*

1 1

equation (1) and the instantaneous fluctuations elevated to some power ( *f*

, *i* = 1, 2, 3. (1)

(2)

). As any new

, ߠ =1, 2,

(3b)

(3a)

2

2 2

i Fi ii i FF F V D vf <sup>g</sup> t xx x 

equation furnished an upper limit for the normalized RMS value, which is not reached when diffusion is taken into account.

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 different statistical profiles of turbulent mass transfer may be interrelated.

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 interactions).

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.
