**7. Appendix I: Obtaining equation (51)**

The starting point is the set of equations (45), (46), and the definition (47). The "\*" was dropped from *z\** and *IJ\** in order to simplify the representation of the equations. The main equation (45) (or 50a) then is written as

$$\mathbf{S}\left(\mathbf{1} - n\right) = \mathbf{S}\frac{d^2 n}{dz^2} - \frac{d\left(II\right)}{dz} \tag{A1-1}$$

Equation (46), for ߠ=2, is presented as:

$$\begin{aligned} &-K\_f \ln(1-n)(1-a\_f)^2 + \sqrt{\frac{\ln(1-n)}{n(1-n)} + \frac{\beta(1-\beta)}{\left(2\beta-1\right)^2}} \Big[n(1-n)\Big]^{1/2} \sqrt{\alpha^2} \left(1-a\_f\right) \frac{\partial n}{\partial z} + \\ &+ \frac{1}{2} \frac{\partial}{\partial z} \Bigg[\sqrt{\frac{1}{n(1-n) + \frac{\beta(1-\beta)}{\left(2\beta-1\right)^2}}} \Big[(1-2n)\Big] \Big[n(1-n)(1-a\_f)^2 \Big] \sqrt{\alpha^2} \Bigg] = \\ &= D\_f \left\{ \frac{\partial^2 \left[(1-n)(1-a\_c)\right]}{\partial z^2} + \frac{\partial^2 \left[-n(1-a\_c)\right]}{\partial z^2} \right\} n(1-n)(1-a\_f) \end{aligned} \tag{A1-2}$$

Using the definitions 2 2 1 1 <sup>1</sup> <sup>1</sup> 2 1 *n n <sup>f</sup> Ke IJ n n* and 2 *<sup>D</sup> <sup>S</sup> Ke* :

$$\begin{aligned} & -n\left(1 - n\right)\left(1 - \alpha\_c\right)^2 + l\frac{\partial}{\partial z} + \frac{1}{2}\frac{\partial}{\partial z}\left\{ (1 - 2n)\left(1 - \alpha\_c\right)lI \right\} = \\ & = S\left\{ \frac{\partial^2 \left[ (1 - n)\left(1 - \alpha\_c\right) \right]}{\partial z^2} + \frac{\partial^2 \left[ -n\left(1 - \alpha\_c\right) \right]}{\partial z^2} \right\} n\left(1 - n\right)\left(1 - \alpha\_c\right) \end{aligned} \tag{A1-3}$$

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

$$-n\left(1-n\right)A^2 + \text{I)}\frac{dn}{dz} - \text{I)}\,A\frac{dn}{dz} + \frac{\left\{1-2n\right\}}{2}A\frac{d\,\text{I}}{dz} = -2S\left|\frac{d^2n}{dz^2}\right|n\left(1-n\right)A^2\tag{A1-4}$$

Using equations (AI1) and (AI4)

28 Hydrodynamics – Advanced Topics

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

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.

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

The "\*" was dropped from *z\** and *IJ\** in order to simplify the representation of the equations.

 <sup>2</sup> <sup>2</sup> <sup>1</sup> *d n d IJ n S*

 

 

*f f f*

 

*n n <sup>n</sup> Kn n n n <sup>z</sup> n n*

 

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

 

1 1

2

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

2 1

*d z d z* (AI-1)

  (AI-2)

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

 

*f*

relevant advises and 2) "Associação dos Amigos da FURB", for financial support.

The starting point is the set of equations (45), (46), and the definition (47).

 

 

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

2 2

2

*c c f f*

2 2

*n n <sup>D</sup> n n z z*

1 1 12 1 1

*nn n <sup>z</sup> n n*

2 1 (1 ) 1 1

 

transport allow to obtain predictions of these functions.

**6. Acknowledgements** 

**7. Appendix I: Obtaining equation (51)** 

The main equation (45) (or 50a) then is written as

Equation (46), for ߠ=2, is presented as:

$$\begin{aligned} -n(1-n)A^2 - \frac{(1-n)(1-2n)}{2}A + I(1-A)\frac{dn}{dz} &= \\ -2S\left\{\frac{d^2n}{dz^2}\right\}n(1-n)A^2 - S\frac{(1-2n)}{2}A\left\{\frac{d^2n}{dz^2}\right\} \end{aligned} \tag{A1-5}$$

Solving equation (AI5) for *IJ*:

$$\text{(iI)} = \frac{-2S\left\{\frac{d^2n}{dz^2}\right\}n(1-n)A^2 - S\frac{(1-2n)}{2}A\left\{\frac{d^2n}{dz^2}\right\} + n(1-n)A^2 + \frac{(1-n)(1-2n)}{2}A}{(1-A)\frac{dn}{dz}}\tag{A1-6}$$

Rearranging equation (AI6):

$$\frac{\left(1 - A\right)}{A} \mathbf{I} = \frac{-S\left[2An(1 - n) + \frac{(1 - 2n)}{2}\right]\left\{\frac{d^2 n}{dz^2}\right\} + (1 - n)\left[\frac{2n(A - 1) + 1}{2}\right]}{\frac{dn}{dz}}\tag{A1-7}$$

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

$$\begin{aligned} &\frac{\left(1-A\right)\left[S\frac{d^2n}{dz^2}-\left(1-n\right)\right]}{A\left\{2A\left[\frac{dn}{dz}-2n\frac{dn}{dz}\right]-\frac{dn}{dz}\right\}\left\{\frac{d^2n}{dz^2}\right\}-S\left\{2A\,n(1-n)+\frac{(1-2n)}{2}\right\}\left\{\frac{d^3n}{dz^3}\right\}+\dots} \\ &=\frac{\frac{dn}{dz}}{dz} \end{aligned}$$

One Dimensional Turbulent Transfer

applicability of equation (51).

experimental data.

in which

Using Random Square Waves – Scalar/Velocity and Velocity/Velocity Interactions 31

A first attempt was made using the second order Finite Differences Method and the solver device from the Microsoft Excel® table, intending to solve the problem with simple and practical tools, but the results were not satisfactory. It does not imply that the Finite Differences Method does not apply, but only that we wanted more direct ways to check the

The second attempt was made using Runge-Kutta methods, also furnished in mathematical tables like Excel ®, maintaining the objective of solving the one-dimensional problem with simple tools. In this case, the results were adequate, superposing well the

The Runge-Kutta methods were developed for ordinary differential equations (ODEs) or systems of ODEs. Equation (AI-10) is a nonlinear differential equation, so that it was

> 2 2 *d n*

12 3 ( )/ *dw ff f d z*

*<sup>n</sup> n A An n w n f A w*

2 1 1

*<sup>j</sup> <sup>A</sup>*

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

<sup>3</sup>

*w*

(1 2 ) 2 11

*A n AA n n <sup>j</sup>* (AII-5)

*<sup>n</sup> <sup>f</sup> A An n <sup>j</sup>* (AII-6)

2

2 2

2

<sup>2</sup>

<sup>3</sup> 11 12 2

(1 2 ) 2 1

Figure 6 shows that 3th, 4th and 5th orders Runge-Kutta methods were applied to obtain numerical results for the profile of *n*. This Appendix shows a summary of the use of the 5th order method. Of course, similar procedures were followed for the lower orders. As usual in this chapter, equations (AII-1) up to (AII-3) use the nondimensional variable *z* without the star "\*" (that is, it corresponds to z\*). Considering "*y*" the dependent variable in a given ODE, the of 5th order method, presented by Butcher (1964) appud Chapra and Canale (2006), is written as follows

1 1345 <sup>6</sup> 7 32 12 32 7

 

 

(AII-7)

*dn <sup>j</sup> dz* (AII-1)

*dz* (AII-2)

(AII-3)

(AII-4)

necessary to first rewrite it as a system of ODEs, as follows

<sup>1</sup>

2

90 *k k*

*y y*

*x*

12 12 1

*AA n*

 <sup>2</sup> 2 2 2 2 2 11 <sup>1</sup> <sup>1</sup> 2 (1 2 ) 2 11 2 1 <sup>1</sup> 2 2 *dn n A A dn n dz dz dn d z n dn n A S An n <sup>n</sup> d z d n d z dn d z* (AI-8) Multiplying by 2 *dn d z* and simplifying *dn d z* : <sup>2</sup> <sup>2</sup> 2 <sup>2</sup> <sup>2</sup> 2 3 3 2 2 2 1 1 2 12 1 (1 2 ) 2 1 2 2 11 1 1 2 (1 2 ) 2 1 2 *<sup>A</sup> d n dn S n A d z d z dn d n SA n d z d z dn n dn S An n d z d z dn n A n A d z n dn S An n d z* <sup>2</sup> 2 2 11 1 2 *n A d n n d z* (AI-9)

Rearranging (after multiplying the equation by A and using *S*=1/ߢ(:

$$\begin{aligned} &A\left[2\operatorname{An}(1-n)+\frac{(1-2n)}{2}\right]\frac{d^3n}{dz^3}\frac{dn}{dz}+\\ &+A\left[-\left[2\operatorname{An}(1-n)+\frac{(1-2n)}{2}\right]\frac{d^2n}{dz^2}+\kappa(1-n)\left[\frac{2n(A-1)+1}{2}\right]+\epsilon\right]\frac{d^2n}{dz^2}+\\ &+\frac{\left\{1+2A\left[A\left(1-2n\right)-1\right]\right\}}{A}\left(\frac{dn}{dz}\right)^2\end{aligned}\tag{A1-10}$$
  $\begin{aligned} &+\kappa\left\{(A-1)(1-n)-A\left[A\left(1-2n\right)-\left(\frac{3}{2}-2n\right)\right]\right\}\left|\left(\frac{dn}{dz}\right)^2\right|\\ &=0\end{aligned}$ 

Equation (AI10) is the equation (51) presented in the text.
