**2.11.1 Obtaining the transformed equations for the one-dimensional transport of F**

Equation (2) may be transformed to its random square waves correspondent using equations (2), (8), (30), (37), and (40), leading to

$$K\_f \left( 1 - n \right) = D\_f \frac{d^2 n}{dz^2} - \frac{d}{dz} \left| \frac{n \left( 1 - n \right) \left( 1 - \alpha\_f \right) \sqrt{\overline{\alpha^2}}}{\sqrt{n \left( 1 - n \right) + \frac{\beta \left( 1 - \beta \right)}{\left( 2\beta - 1 \right)^2}}} \right| \tag{45}$$

In the same way, equation (3d) is transformed to its random square waves correspondent using equations (3d), (8), (24), (32), (37), (41), and (44), leading to

$$\begin{split} &-K\_{f}n(1-n)\left[\left(1-n\right)^{\theta-1}+\left(-1\right)^{\theta}\left(n\right)^{\theta-1}\right]\left(1-\alpha\_{f}\right)^{\theta}+\\ &+K\_{f}n(1-n)^{2}\left[\left(1-n\right)^{\theta-2}+\left(-1\right)^{\theta-1}\left(n\right)^{\theta-2}\right]\left(1-\alpha\_{f}\right)^{\theta-1}+\\ &+\sqrt{\frac{\left[n\left(1-n\right)\right]^{3-\theta}}{n\left(1-n\right)+\frac{\beta\left(1-\beta\right)}{\left(2\beta-1\right)^{2}}}}\left[\left(1-n\right)^{\theta-1}-\left(-n\right)^{\theta-1}\right]\left[n\left(1-n\right)\right]^{\left(\theta-1\right)/2}\sqrt{\alpha^{2}}\left(1-\alpha\_{f}\right)^{\theta-1}\frac{\left\|\begin{matrix} \alpha\\ \alpha^{2} \end{matrix}\right\|^{2}}{\left(2\beta-1\right)^{2}}+\\ &+\frac{1}{\theta}\frac{\left\|\begin{matrix} \alpha\\ \beta\end{matrix}\right\|^{2}}{\left(\alpha(1-n)+\frac{\beta\left(1-\beta\right)}{\left(2\beta-1\right)^{2}}\right)}\left[\sqrt{\left(1-n\right)^{\theta}-\left(-n\right)^{\theta}}\left[\left(n\left(1-n\right)\left(1-\alpha\_{f}\right)^{2}\right]^{\theta/2}\sqrt{\alpha^{2}}\right]\right]=\\ \end{split}$$

One Dimensional Turbulent Transfer

2

*( n) d n dn A An n*

1 2 2 1

to the following governing equation for *n* (see appendix 1)

<sup>3</sup> 11 12 2 <sup>0</sup> 2

trivial analytical solution for the extreme case *A*=0 (or 1

1 2

<sup>3</sup> 11 12 2 , <sup>2</sup>

 <sup>2</sup> <sup>2</sup> <sup>1</sup> \* *d n <sup>n</sup> d z* 

3 3

*dn <sup>κ</sup> A n AA n n*

*f A n AA n n j*

*dn d j dw f f j w where*

2

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

(1 2 ) 21 .

\* \* <sup>3</sup>

*dz dz dy f*

, ,

1

 

  2

*and*

3

 

*dz\* dz\**

Using Random Square Waves – Scalar/Velocity and Velocity/Velocity Interactions 21

Equation (50a) is used to obtain *dIJ*/*dz*\*, which leads, when substituted into equation (50b),

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

1

*<sup>f</sup>* ), in the form

 sin \*

(52)

2

(53)

sin *<sup>z</sup> <sup>n</sup>* (51)

*dz\**

of equations derived from (51) and solved with Runge-Kutta methods is given by:

*( n) d n n A AA n dn d n A An n <sup>κ</sup> <sup>n</sup> dz\* A dz\* dz\**

2 11 12 12 1 1 2 2 1 <sup>1</sup>

2 2

2

Thus, the one-dimensional problem is reduced to solve equation (51) alone. It admits non-

But this effect of diffusion for all 0<*z\**<1 is considered overestimated. Equation (51) was presented by Schulz et al. (2011a), but with different coefficients in the last parcel of the first member (the parcel involving 3/2-2*n* in equation (51) involved 1-*n* in the mentioned study). Appendix 1 shows the steps followed to obtain this equation. Numerical solutions were obtained using Runge-Kutta schemes of third, fourth and fifth orders. Schulz et al. (2011a) presented a first evaluation of the *n* profile using a fourth order Runge-Kutta method and comparing the predictions with the measured data of Janzen (2006). An improved solution was proposed by Schulz et al. (2011b) using a third order Runge-Kutta method, in which a good superposition between predictions and measurements was obtained. In the present chapter, results of the third, fourth and fifth orders approximations are shown. The system

or

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

(1 2 ) 2 11 12 12 1 2 1 <sup>1</sup> , 2 2

*n n A AA n*

2

Equations (53) were solved as an initial value problem, that is, with the boundary conditions expressed at *z*\*=0. In this case, *n*(0)=1 and *j*(0)=~-3 (considering the experimental data of Janzen, 2006). The value of *w*(0) was calculated iteratively, obeying the boundary condition 0<*n*(1)<0.01. The Runge-Kutta method is explicit, but iterative procedures were used to

$$\begin{split} &= D\_{f}n(1-n)\Big[\left(1-n\right)^{\theta-2}+\left(-1\right)^{\theta-1}\left(n\right)^{\theta-2}\Big]\left(1-a\_{f}\right)^{\theta-1}\frac{\partial^{2}}{\partial z^{2}}^{n}+\\ &+D\_{f}\left[\left(1-n\right)^{\theta-2}\frac{\partial^{2}\left[\left(1-n\right)\left(1-a\_{f}\right)\right]}{\partial z^{2}}+\left(-n\right)^{\theta-2}\frac{\partial^{2}\left[-n\left(1-a\_{f}\right)\right]}{\partial z^{2}}\right]n\left(1-n\right)\Big(1-a\_{f}\right)^{\theta-1} \end{split} \tag{46}$$

#### **2.11.2 Simplified case of interfacial heat/mass transfer**

Although involving few equations for the present case, the set of the coupled nonlinear equations (45) and (46) may have no simple solution. As mentioned, the original onedimensional problem needs four equations. But as the simplified solution of interfacial transfer using a mean constant *<sup>f</sup> <sup>f</sup>* is considered here, only three equations would be needed. Further, recognizing in equations (45) and (46) that ߚ and 2 appear always together in the form

$$II = \frac{n(1-n)\left(1-\alpha\_f\right)\sqrt{\alpha^2}}{\sqrt{n(1-n) + \frac{\beta\left(1-\beta\right)}{\left(2\beta-1\right)^2}}}\tag{47}$$

It is possible to reduce the problem to a set of only two coupled equations, for *n* and the function *IJ*. Thus, only equations (45) and (46) for ߠ=2 are necessary to close the problem when using *<sup>f</sup> <sup>f</sup>* . Defining (1 ) *A <sup>f</sup>* the set of the two equations is given by

$$K\_f \left(1 - n\right) = D\_f \frac{d^2 n}{dz^2} - \frac{d\left(I\right)}{dz} \tag{48a}$$

$$-K\_f \, n \, (1 - n)A^2 + (I) \frac{dn}{dz} + \frac{A}{2} \frac{d}{dz} \Big[ (I) (1 - 2n) \Big] = -2D\_f \, n \, (1 - n)A^2 \frac{d^2 n}{dz^2} \tag{48b}$$

Equations (48) may be presented in nondimensional form, using *z*\*=*z*/*E*, with *E*=*z*2-*z*1, and *S*=1/ߢ=*Df*/*KfE*<sup>2</sup>

$$II^\* = \frac{n(1-n)(1-\alpha\_f)\left(\sqrt{\alpha^2} \;/\, KE\right)}{\sqrt{n(1-n) + \frac{\beta(1-\beta)}{\left(2\beta-1\right)^2}}}\tag{49}$$

$$S(1-n) = S\frac{d^2n}{dz^{\star^2}} - \frac{d\left(II^{\star}\right)}{dz^{\star}}\tag{50a}$$

$$-n(1-n)A^2 + (l|l^\*)\frac{dn}{dz^\*} + \frac{A}{2}\frac{d}{dz^\*}[(l|l^\*)(1-2n)] = -2Sn(1-n)A^2\frac{d^2n}{dz^{\*2}}\tag{50b}$$

<sup>2</sup> <sup>1</sup> 2 12

(1 ) 1 1 1 1 1

*n n D n n n n z z*

 

2 2

Although involving few equations for the present case, the set of the coupled nonlinear equations (45) and (46) may have no simple solution. As mentioned, the original onedimensional problem needs four equations. But as the simplified solution of interfacial

1 1

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

*n n*

2 *f f*

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

It is possible to reduce the problem to a set of only two coupled equations, for *n* and the function *IJ*. Thus, only equations (45) and (46) for ߠ=2 are necessary to close the problem

> <sup>2</sup> <sup>2</sup> 1 *f f d n d IJ K nD*

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

(50b)

 

2 1

2

 

2

*d z d z* (50a)

(48b)

*dn <sup>A</sup> d d <sup>n</sup> K n n A IJ IJ n D n n A dz dz d z*

Equations (48) may be presented in nondimensional form, using *z*\*=*z*/*E*, with *E*=*z*2-*z*1, and

11 /

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

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

*dn <sup>A</sup> d d <sup>n</sup> n n A IJ \* IJ \* n Sn n A dz\* dz\* dz\**

*n n <sup>f</sup> KE*

\* <sup>1</sup> <sup>1</sup>

2

*n n*

2 1

2

2

*<sup>f</sup>* the set of the two equations is given by

*d z d z* (48a)

*f f*

1 2 2

*f f*

2

 

*<sup>f</sup>* is considered here, only three equations would be

(47)

2

(49)

appear always

(46)

*z*

*<sup>n</sup> Dn n n <sup>n</sup>*

 

11 1 1

**2.11.2 Simplified case of interfacial heat/mass transfer** 

 *<sup>f</sup>* 

*<sup>f</sup>* . Defining (1 ) *A*

*IJ*

needed. Further, recognizing in equations (45) and (46) that ߚ and 2

transfer using a mean constant

together in the form

when using

*S*=1/ߢ=*Df*/*KfE*<sup>2</sup>

 *<sup>f</sup>* 

*f f*

2 2

Equation (50a) is used to obtain *dIJ*/*dz*\*, which leads, when substituted into equation (50b), to the following governing equation for *n* (see appendix 1)

$$\begin{split} &A\left[2\operatorname{An}(1-n)+\frac{(1-2n)}{2}\right]\frac{d^3n}{dz^3}\frac{dn}{dz^4}+\\ &+A\left\{-\left[2\operatorname{An}(1-n)+\frac{(1-2n)}{2}\right]\frac{d^2n}{dz^2}+\kappa(1-n)\left[\frac{2\operatorname{n}(A-1)+1}{2}\right]+\frac{\left\{1+2A\left[A(1-2n)-1\right]\right\}}{A}\left(\frac{dn}{dz^4}\right)^2\right]\frac{d^2n}{dz^4}+\left(\mathfrak{S1}\right)^2\\ &+\kappa\left\{(A-1)(1-n)-A\left[A(1-2n)-\left(\frac{3}{2}-2n\right)\right]\right\}\left(\frac{dn}{dz^4}\right)^2=0\end{split}$$

Thus, the one-dimensional problem is reduced to solve equation (51) alone. It admits nontrivial analytical solution for the extreme case *A*=0 (or 1 *<sup>f</sup>* ), in the form

$$\frac{d^2n}{dz^{\star^2}} = \kappa(1-n) \qquad \text{or} \qquad n = 1 - \frac{\sin\left(\sqrt{\kappa}z^{\star}\right)}{\sin\left(\sqrt{\kappa}\right)}\tag{52}$$

But this effect of diffusion for all 0<*z\**<1 is considered overestimated. Equation (51) was presented by Schulz et al. (2011a), but with different coefficients in the last parcel of the first member (the parcel involving 3/2-2*n* in equation (51) involved 1-*n* in the mentioned study). Appendix 1 shows the steps followed to obtain this equation. Numerical solutions were obtained using Runge-Kutta schemes of third, fourth and fifth orders. Schulz et al. (2011a) presented a first evaluation of the *n* profile using a fourth order Runge-Kutta method and comparing the predictions with the measured data of Janzen (2006). An improved solution was proposed by Schulz et al. (2011b) using a third order Runge-Kutta method, in which a good superposition between predictions and measurements was obtained. In the present chapter, results of the third, fourth and fifth orders approximations are shown. The system of equations derived from (51) and solved with Runge-Kutta methods is given by:

$$\begin{aligned} \left| \frac{dn}{dz} = j\_- \cdot \frac{dj}{dz} = w\_- \cdot \frac{dw}{dy} = \frac{f\_1 + f\_2}{f\_3} \qquad \text{where} \\ \left| f\_1 = -A \left\{ -\left[ 2A \, n(1-n) + \frac{(1-2n)}{2} \right] w + \kappa (1-n) \left[ \frac{2n(A-1)}{2} + 1 \right] + \frac{\left\{ 1 + 2A \left[ A(1-2n) - 1 \right] \right\}}{A} \right] w \right\} \\ \left| f\_2 = -\kappa \left\{ (A-1)(1-n) - A \left[ A(1-2n) - \left( \frac{3}{2} - 2n \right) \right] \right\} j^2, \\ \quad \text{and} \\ f\_3 = A \left[ 2A \, n(1-n) + \frac{(1-2n)}{2} \right] j. \end{aligned} \tag{53}$$

Equations (53) were solved as an initial value problem, that is, with the boundary conditions expressed at *z*\*=0. In this case, *n*(0)=1 and *j*(0)=~-3 (considering the experimental data of Janzen, 2006). The value of *w*(0) was calculated iteratively, obeying the boundary condition 0<*n*(1)<0.01. The Runge-Kutta method is explicit, but iterative procedures were used to

One Dimensional Turbulent Transfer

Fig. 7a. Predictions of *n* for *n*"(0) = 3.056, and

�~0.40. Fourth order Runge-Kutta.

**3. Velocity-velocity interactions** 

case, the mean product

Using Random Square Waves – Scalar/Velocity and Velocity/Velocity Interactions 23

Fig. 7b. Predictions of *n* for �=0.003 and 2.99812 ≤ n''(0) ≤ 3.2111. Third order

, i, j = 1, 2, 3. (54)

(55)

Runge-Kutta

The aim of this section is to present some first correlations for a simple velocity field. In this case, the flow between two parallel plates is considered. We follow a procedure similar to that presented by Schulz & Janzen (2009), in which the measured functional form of the reduction function is shown. As a basis for the analogy, some governing equations are first presented. The Navier-Stokes equations describe the movement of fluids and, when used to

 

*u* appears as a new variable, in addition to the mean velocity *U* .

*p* is the mean pressure, � is the kinematic viscosity of the fluid and *Bi* is the body force per unit mass (acceleration of the gravity). For stationary one-dimensional horizontal flows between two parallel plates, equation (1), with *x1*=*x*, *x3*=*z*, *v1*=*u* and *v3*=�, is simplified to:

> <sup>1</sup> *p U <sup>u</sup> xz z*

This equation is similar to equation (2) for one dimensional scalar fields. As for the scalar

In this chapter, no additional governing equation is presented, because the main objective is to expose the analogy. The observed similarity between the equations suggests also to use

quantify turbulent movements, they are usually rewritten as the Reynolds equations:

*i i j i ii i j VV V <sup>p</sup> <sup>V</sup> v v <sup>B</sup> t xx x x* 

*jj j* 1

the partition, reduction and superposition functions for this velocity field.

 

evaluate the parameters at *z\**=0 applying the quasi-Newton method and the Solver device of the Excel® table. Appendix 2 explains the procedures followed in the table. The curves of figure 6a were obtained for 0.001 0.005 , a range based on the � experimental values of Janzen (2006), for which ~0.003<�<~0.004. The values *A*=0.5 and *n*"(0)=3.056 were used to calculate *n* in this figure. As can be seen, even using a constant *A*, the calculated curve *n*(*z*\*) closely follows the form of the measured curve. Because it is known that *<sup>f</sup>* is a function of *z*\*, more complete solutions must consider this dependence. The curve of Schulz et al. (2011a) in figure 6a was obtained following different procedures as those described here. The curves obtained in the present study show better agreement than the former one.

Fig. 6a. Predictions of *n* for *n*"(0) = 3.056. Fourth order Runge-Kutta.

Fig. 6b. Predictions of *n* for � = 0.0025, and - 0.0449 ≤n"(0) ≤ 3.055. Fifth order Runge-Kutta

Fig. 6b. was obtained with following conditions for the pairs [*A*, *n*"(0)]: [0.2, 0.00596], [0.25, - 0.0145], [0.29, -0.04495], [0.35, 1.508], [0.4, 1.8996], [0.45, 1.849], [0.5, 2.509], [0.55, 3.0547], [0.62, 2.9915], [0.90, 0.00125]. Further, *n*'(0) = -3 for *A* between 0.20 and 0.62, and *n*'(0) = -1 for *A*=0.90.

Figure 7a shows results for �~0.4, that is, having a value around 100 times higher than those of the experimental range of Janzen (2006), showing that the method allows to study phenomena subjected to different turbulence levels. � � (*Kf E*2/*Df*) is dependent on the turbulence level, through the parameters *E* and *Kf*, and different values of these variables allow to test the effect of different turbulence conditions on *n*. Figure 7b presents results similar to those of figure 6a, but using a third order Runge-Kutta method, showing that simpler schemes can be used to obtain adequate results.

As the definitions of item 3 are independent of the nature of the governing differential equations, it is expected that the present procedures are useful for different phenomena governed by statistical differential equations. In the next section, the first steps for an application in velocity-velocity interactions are presented.

Fig. 7a. Predictions of *n* for *n*"(0) = 3.056, and �~0.40. Fourth order Runge-Kutta.

Fig. 7b. Predictions of *n* for �=0.003 and 2.99812 ≤ n''(0) ≤ 3.2111. Third order Runge-Kutta
