**5. Conclusions**

It was shown that the methodology of random square waves allows to obtain a closed set of equations for one-dimensional turbulent transfer problems. The methodology adopts *a priori* models for the records of the oscillatory variables, defining convenient functions that allow to "adjust" the records and to obtain predictions of the mean profiles. This is an alternative procedure in relation to the *a posteriori* "closures" generally based on *ad hoc* models, like the

One Dimensional Turbulent Transfer

Using the definitions

For �� constant and defining *A*=(����):

2

Using equations (AI1) and (AI4)

Solving equation (AI5) for *IJ*:

Rearranging equation (AI6):

1

Using Random Square Waves – Scalar/Velocity and Velocity/Velocity Interactions 29

2

and

*c c*

*c c*

(1 2 ) <sup>1</sup> 2 1 2 *dn dn n d d IJ <sup>n</sup> n n A IJ IJ A A S n nA dz dz d z d z* 

2 2

*d z*

2 (1 2 ) 2 11 2 1 <sup>1</sup>

 (AI-7)

*d z*

*d n n dn n n S n nA S A n nA <sup>A</sup>*

2 2

(1 2 ) 1 (1 2 ) 2 1 <sup>1</sup>

*n dn n A S An n <sup>n</sup>*

1

<sup>2</sup>

(1 2 ) 2 2 2 1

*dn dn dn d n n dn SA n S An n dz dz dz d z d z dn d z*

1 2 2

(1 2 ) 2 1

2 2

2 3 2 3

2

 

2 *<sup>D</sup> <sup>S</sup> Ke* :

1 1

2

2 2 2 2 2

*d n n dn S n nA S A d z d z*

2

2

 

*c*

(AI-3)

(AI-4)

(AI-5)

(AI-6)

2

 

2 2

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

2

*n n dn n nA A IJ A d z*

2 2

*d z d z*

*A d z IJ <sup>A</sup> dn*

*IJ dn <sup>A</sup>*

Differentiating equation (AI7) and using equation (AI1):

1

*A d z*

2 2

*<sup>A</sup> d n S n*

 

2 2

1 (1 2 ) 1 1 2

2 2

1 1

*n n*

*n n <sup>f</sup> Ke IJ*

<sup>1</sup> 1 1 (1 2 ) 1 <sup>2</sup>

*n n n IJ n IJ z z*

(1 ) 1 1

*n n S n n z z*

2 1

use of turbulent diffusivities/viscosities, together with physical/phenomenological reasoning about relevant parameters to be considered in these diffusivities/viscosities. The basic functions are: the partition functions, the reduction coefficients and the superposition coefficients. The obtained transformed equations for the one-dimensional turbulent transport allow to obtain predictions of these functions.

In addition, the RMS of the velocity was also used as a basic function. The equations are nonlinear. An improved analysis of the one-dimensional scalar transfer through air-water interfaces was presented, leading to mean curves that superpose well with measured mean concentration curves for gas transfer. In this analysis, different constant values were used for ߙ, ߢ and the second derivative at the interface, allowing to obtain well behaved and realistic mean profiles. Using the constant ߙ values, the system of equations for onedimensional scalar turbulent transport could be reduced to only one equation for *n*; in this case, a third order differential equation. In the sequence, a first application of the methodology to velocity fields was made, following the same procedures already presented in the literature for mass concentration fields. The form of the reduction coefficient function for the velocity fluctuations was calculated from measured data found in the literature, and plotted as a function of *n*, generating a cloud of points. As for the case of mass transfer, ߙ௨ presents a minimum peak in the region of the boundary layer (maximum peak for 1-ߙ௨). Because this methodology considers *a priori* definitions, applied to the records of the random

parameters, it may be used for different phenomena in which random behaviors are observed.

### **6. Acknowledgements**

The first author thanks: 1) Profs. Rivadavia Wollstein and Beate Frank (Universidade Regional de Blumenau), and Prof. Nicanor Poffo, (Conjunto Educacional Pedro II, Blumenau), for relevant advises and 2) "Associação dos Amigos da FURB", for financial support.
