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

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* depend on *z*.

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 oscillations justifies these corrective parcels.

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

$$n = \frac{t \text{ at } (F\_p - P)}{\Delta t \text{ of the observation}} \tag{5}$$

This definition also implies that

$$1 - n = \frac{t \text{ at } (F\_n + N)}{\Delta t \text{ of the observation}} \tag{6}$$

*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):

One Dimensional Turbulent Transfer

Isolating *n,* equation (8) leads to

using equations (5) through (7). It follows that

is calculated similarly to equation (8), furnishing

Thus, a reduction coefficient ߙ is defined here as

profiles, all profiles are interrelated.

*f*

minimum value in this region).

**for scalars -** 

Using Random Square Waves – Scalar/Velocity and Velocity/Velocity Interactions 9

 1 *P n <sup>N</sup>*

> *n p n F F*

*F F*

Thus, the partition function *n* previously defined by equation (5) coincides with the normalized form of *F* given by equation (9). Note that *n* is used as weighting factor for any statistical parameter that depends on *F*. For example, the mean value *Q* of a function *Q*(*F*)

As a consequence, equations (9) and (9a) show that any new mean function *Q* is related to the mean function *F* . Or, in other words: because *n* is used to calculate the different mean

From the above discussion it may be inferred that any new variable added to the problem will have its own partition function. In the present section of scalar-velocity interactions,

**2.4 Bimodal square wave: Adjusting amplitudes using a reduction coefficient function** 

The sketch of figure 2c shows that the parcel *P* is always smaller or equal to *F F <sup>p</sup>* . As already mentioned, this parcel shows that the amplitude of the fluctuations is reduced.

0 1 *P FF*

where ߙ is a function of *z* and quantifies the reduction of the amplitude due to interactions between parcels of liquid with different *F* values (described here as a measure of diffusion effects, but which can be a measure of any cause that inhibits fluctuations). Using the effect of diffusion to interpret the new function, values of ߙ close to 1 or 0 indicate strong or weak influence of diffusion, respectively. Considering this interpretation, Schulz & Janzen (2009) reported experimental profiles for ߙ in the mass concentration boundary layer during airwater interfacial mass-transfer, which showed values close to 1 in both the vicinity of the surface and in the bulk liquid, and closer to 0 in an intermediate region (giving therefore a

 *f p <sup>f</sup>* (10)

two partition functions are described: *n* for *F* (scalar) and *m* for *V* (velocity).

*n*

The mean value of *F* is obtained from a weighted average operation between *Fp*-*P* and *Fn*+*N*,

*<sup>n</sup>* (7)

(1 ) *F nF n F <sup>p</sup> <sup>n</sup>* (8)

(1 ) *Q nQ F P n Q F N p n* (9a)

(9)

Fig. 2. *a*) Sketch of the *F* record of the gray plane of figure 1, at *z*, *b*) Simplified record alternating *F* between *Fp* and *Fn*, *c*) Simplified record with amplitude damping. Upper and lower points do not superpose at the discontinuities (the *F* segments are open at the left and closed at the right, as shown in the detail).

$$N = \frac{P n}{\left(1 - n\right)}\tag{7}$$

The mean value of *F* is obtained from a weighted average operation between *Fp*-*P* and *Fn*+*N*, using equations (5) through (7). It follows that

$$
\overline{F} = nF\_p + (1 - n)F\_n \tag{8}
$$

Isolating *n,* equation (8) leads to

8 Hydrodynamics – Advanced Topics

Fig. 2. *a*) Sketch of the *F* record of the gray plane of figure 1, at *z*, *b*) Simplified record alternating *F* between *Fp* and *Fn*, *c*) Simplified record with amplitude damping. Upper and lower points do not superpose at the discontinuities (the *F* segments are open at the left and

closed at the right, as shown in the detail).

$$m = \frac{\overline{F} - F\_n}{F\_p - F\_n} \tag{9}$$

Thus, the partition function *n* previously defined by equation (5) coincides with the normalized form of *F* given by equation (9). Note that *n* is used as weighting factor for any statistical parameter that depends on *F*. For example, the mean value *Q* of a function *Q*(*F*) is calculated similarly to equation (8), furnishing

$$
\overline{Q} = n \left. Q \left( F\_p - P \right) + (1 - n) \left. Q \left( F\_n + N \right) \right. \tag{9a}
$$

As a consequence, equations (9) and (9a) show that any new mean function *Q* is related to the mean function *F* . Or, in other words: because *n* is used to calculate the different mean profiles, all profiles are interrelated.

From the above discussion it may be inferred that any new variable added to the problem will have its own partition function. In the present section of scalar-velocity interactions, two partition functions are described: *n* for *F* (scalar) and *m* for *V* (velocity).

#### **2.4 Bimodal square wave: Adjusting amplitudes using a reduction coefficient function for scalars -** *f*

The sketch of figure 2c shows that the parcel *P* is always smaller or equal to *F F <sup>p</sup>* . As already mentioned, this parcel shows that the amplitude of the fluctuations is reduced. Thus, a reduction coefficient ߙ is defined here as

$$P = \alpha\_f \left[ F\_p - \overline{F} \right] \tag{10} \\ \qquad \qquad 0 \le \alpha\_f \le 1 \tag{10}$$

where ߙ is a function of *z* and quantifies the reduction of the amplitude due to interactions between parcels of liquid with different *F* values (described here as a measure of diffusion effects, but which can be a measure of any cause that inhibits fluctuations). Using the effect of diffusion to interpret the new function, values of ߙ close to 1 or 0 indicate strong or weak influence of diffusion, respectively. Considering this interpretation, Schulz & Janzen (2009) reported experimental profiles for ߙ in the mass concentration boundary layer during airwater interfacial mass-transfer, which showed values close to 1 in both the vicinity of the surface and in the bulk liquid, and closer to 0 in an intermediate region (giving therefore a minimum value in this region).

One Dimensional Turbulent Transfer

time as shown in equations (5) and (6).

Using Random Square Waves – Scalar/Velocity and Velocity/Velocity Interactions 11

Fig. 3. Juxtaposed fluctuations of *f* and ߱, showing a compact form of the time fractions *n* and (1-*n*), and the use of the superposition function ߚ. The horizontal axis represents the

An advantage of using random square waves as shown in Figure 2 is that they generate only two fluctuation amplitudes for each variable, which are then used to calculate the wished statistical quantities. Of course, the functions defined in sections 2.3 through 2.5 (partition, reduction and superposition functions) are also used, and they must "adjust" the statistical quantities to adequate values. From equations (8), (10), and (11), the two instantaneous

In figure 1 the scalar variable is represented oscillating between two homogeneous values. But nothing was said about the velocity field that interacts with the scalar field. It may also be bounded by homogeneous velocity values, but may as well have zero mean velocities in the entire physical domain, without any evident reference velocity. This is the case, for example, of the problem of interfacial mass transfer across gas-liquid interfaces, the application shown by Schulz et al. (2011a). In such situations, it is more useful to use the rms

case, with null mean motion, all equations must be derived using only the vertical velocity

fluctuations (like equations 14 and 15 for f) considering the random square waves approximation. An auxiliary velocity scale *U* is firstly defined, shown in figure 4, considering "downwards" (߱ௗ) and "upwards" (߱௨) fluctuations, which amplitudes are

as reference, as commonly adopted in turbulence. For the one-dimensional

*<sup>f</sup>* (positive) (14)

*<sup>f</sup>* (negative) (15)

and for the velocity

**2.6 The fluctuations around the mean for bimodal square waves** 

*f F PF nF F* <sup>1</sup> *p p* (1 ) 1 *<sup>n</sup>*

*f F N F nF F* <sup>2</sup> *<sup>n</sup> <sup>p</sup> <sup>n</sup>* 1

fluctuations ߱. It is necessary, thus, to obtain equations for <sup>2</sup>

scalar fluctuations are then given by equations (14) and (15)

**2.7 Velocity fluctuations and the RMS velocity** 

velocity <sup>2</sup>

functions of *z*.

From equations (7), (8) and (10), *N* and *P* are now expressed as

$$\begin{aligned} N &= \alpha\_f \, n \left( F\_p - F\_n \right) \\ P &= \alpha\_f \left( 1 - n \right) \left( F\_p - F\_n \right) \end{aligned} \qquad \begin{aligned} 0 &\le \alpha\_f \le 1 \end{aligned} \tag{11}$$

As for the partition functions, any new variable implies in a new reduction coefficient. In the present section of scalar-velocity interactions, only the reduction coefficient for *F* is used (that is, ߙ( . In the section for velocity-velocity interactions, a reduction coefficient ߙ௩ for *V* (velocity) is used.

#### **2.5 Bimodal square wave: Quantifying superposition using the superposition coefficient function -**

Let us now consider the two main variables of turbulent scalar transport, the scalar *F* and the velocity *V*, oscillating simultaneously in the interval *z*1<*z*<*z*2 of Figure 1. As usual, they are represented as *FFf* and *V V* , where *F* and *V* are the mean values, and *f* and ߱ are the fluctuations. The correlation coefficient function ߩ)*z*) for the fluctuations *f* and ߱ is given by

$$\rho\left(z\right) = \frac{1}{\Delta t} \int\_{t\_1}^{t\_2} \rho\left(z, t\right) dt = \frac{1}{\Delta t} \int\_{t\_1}^{t\_2} \frac{\alpha \, f}{\sqrt{\alpha^2} \sqrt{f^2}} \, dt = \frac{\overline{\alpha \, f}}{\sqrt{\alpha^2} \sqrt{f^2}} \tag{12}$$

If the fluctuations are generated by the same cause, it is expected that the records of ߱ and *f* are at least partially superposed. As done for *F*, it is assumed that the oscillations ߱ can be positive or negative and so a partition function *m* (a function of *z*) may be defined. If we consider a perfect superposition between *f* and ߱, it would imply in *n*=*m*, though this is not usually the case. Aiming to consider all the cases, a superposition coefficient ߚ is defined so that ߚ=1.0 reflects the direct superposition (*m*=*n*), and ߚ=0.0 implies the inverse superposition of the positive and the negative fluctuations (*m*=1-*n*) of both fields.

The definition of ߚ is better understood considering the scheme presented in figure 3. In this figure all positive fluctuations of the scalar variable were put together, so that the nondimensional time intervals were added, furnishing the value *n*. As a consequence, the nondimensional fraction of time of the juxtaposed negative fluctuations appears as 1-*n*. The velocity fluctuations also appear juxtaposed, showing that ߚ=1 superposes *f* and *n* with the same sign (++ and --), while ߚ=0 superposes *f* and *n* with opposite signs (+- and - +). The positive and negative scalar fluctuations are represented by *f*1 and *f*2, respectively. The downwards and upwards velocity fluctuations are represented by ߱ௗ and ߱௨, respectively.

Thus, *m*, which defines the fraction of the time for which the system is at ߱ௗ, is expressed as

$$m = 1 - \left(\beta + n - 2\,\beta \, n\right) \tag{13}$$

ߚ is a function of *z*. Also here any new variable implies in new superposition functions. In the present section of scalar-velocity interactions only one superposition coefficient function is used (linking scalar and velocity fluctuations).

0 1 

, where *F* and *V* are the mean values, and *f* and

22 22

(12)

 

 

*<sup>f</sup>* (11)

As for the partition functions, any new variable implies in a new reduction coefficient. In the present section of scalar-velocity interactions, only the reduction coefficient for *F* is used (that is, ߙ( . In the section for velocity-velocity interactions, a reduction coefficient ߙ௩ for *V*

Let us now consider the two main variables of turbulent scalar transport, the scalar *F* and the velocity *V*, oscillating simultaneously in the interval *z*1<*z*<*z*2 of Figure 1. As usual, they

߱ are the fluctuations. The correlation coefficient function ߩ)*z*) for the fluctuations *f* and ߱ is

*f f z z t dt dt*

If the fluctuations are generated by the same cause, it is expected that the records of ߱ and *f* are at least partially superposed. As done for *F*, it is assumed that the oscillations ߱ can be positive or negative and so a partition function *m* (a function of *z*) may be defined. If we consider a perfect superposition between *f* and ߱, it would imply in *n*=*m*, though this is not usually the case. Aiming to consider all the cases, a superposition coefficient ߚ is defined so that ߚ=1.0 reflects the direct superposition (*m*=*n*), and ߚ=0.0 implies the inverse

The definition of ߚ is better understood considering the scheme presented in figure 3. In this figure all positive fluctuations of the scalar variable were put together, so that the nondimensional time intervals were added, furnishing the value *n*. As a consequence, the nondimensional fraction of time of the juxtaposed negative fluctuations appears as 1-*n*. The velocity fluctuations also appear juxtaposed, showing that ߚ=1 superposes *f* and *n* with the same sign (++ and --), while ߚ=0 superposes *f* and *n* with opposite signs (+- and - +). The positive and negative scalar fluctuations are represented by *f*1 and *f*2, respectively. The downwards and upwards velocity fluctuations are represented by ߱ௗ and ߱௨,

Thus, *m*, which defines the fraction of the time for which the system is at ߱ௗ, is expressed

*m nn* 1 2 

ߚ is a function of *z*. Also here any new variable implies in new superposition functions. In the present section of scalar-velocity interactions only one superposition coefficient function

(13)

*t t f f* 

From equations (7), (8) and (10), *N* and *P* are now expressed as

(velocity) is used.

given by

respectively.

as

**coefficient function -** 

are represented as *FFf* and *V V*

is used (linking scalar and velocity fluctuations).

 1

*f pn*

**2.5 Bimodal square wave: Quantifying superposition using the superposition** 

2 2

 

1 1 , *t t*

1 1

superposition of the positive and the negative fluctuations (*m*=1-*n*) of both fields.

*t t*

*f pn*

*P nF F*

*N nF F*

Fig. 3. Juxtaposed fluctuations of *f* and ߱, showing a compact form of the time fractions *n* and (1-*n*), and the use of the superposition function ߚ. The horizontal axis represents the time as shown in equations (5) and (6).
