**4. Challenges**

26 Hydrodynamics – Advanced Topics

Figure 9 shows the measured *u'2* values together with the curve given by 27.6 *n n v v* 1 . As can be seen, the curve 27.6 *n n v v* 1 leads to a peak close to the wall. In this case, the function is normalized using the friction velocity, so that the peak is not limited by the value of 0.5 (which is the case if the function is normalized using *Up*-*Un*). It is interesting that the forms of 2 *u u\** / and 27.6 *n n v v* 1 are similar, which coincides with the conclusions of Janzen (2006) for mass

Figure 10 shows the cloud of points for 1*-*ߙ௨ obtained from the data of Wei & Willmarth (1989), following the procedures of Janzen (2006) and Schulz & Janzen (2009) for mass transfer. As for the case of mass transfer, ߙ௨ presents a minimum peak in the region of the

Fig. 9. Comparison between measured values of *u*'/*u*\* and /\* 1 *Uu n n <sup>p</sup> v v* . The gray

Fig. 10. 1*-*ߙ௨ plotted against *n*, following the procedures of Schulz & Janzen (2009). The gray

cloud envelopes the points calculated using the data of Wei & Willmarth (1989).

transfer, using *ad hoc* profiles for the mean mass concentration close to interfaces.

boundary layer (maximum peak for 1-ߙ௨).

cloud envelopes the data from Wei & Willmarth (1989).

After having presented the one-dimensional results for turbulent scalar transfer using the approximation of random square waves, some brief comments are made here, about some characteristics of this approximation, and about open questions, which may be considered in future studies.

As a general comment, it may be interesting to remember that the mean functions of the statistical variables are continuous, and that, in the present approximation they are defined using discrete values of the relevant variables. As described along the paper, the defined functions (*n*, , , RMS) "adjust" these two points of view (this is perhaps more clearly explained when defining the function ). This concomitant dual form of treating the random transport did not lead to major problems in the present application. Eventual applications in 2-D, 3-D problems or in phenomena that deal with discrete variables may need more refined definitions.

In the present study, the example of mass transfer was calculated by using constant reduction coefficients (), presenting a more detailed and improved version of the study of Schulz et al. (2011a). However, it is known that this coefficient varies along *z*, which may introduce difficulties to obtain a solution for *n*. This more complete result is still not available.

It was assumed, as usual in turbulence problems, that the lower statistical parameters (e.g. moments) are appropriate (sufficient) to describe the transport phenomena. So, the finite set of equations presented here was built using the lower order statistical parameters. However, although only a finite set of equations is needed, this set may also use higher order statistics. In fact, the number of possible sets is still "infinite", because the unlimited number of statistical parameters and related equations still exists. A challenge for future studies may be to verify if the lower order terms are really sufficient to obtain the expected predictions, and if the influence of the higher order terms alter the obtained predictions. It is still not possible to infer any behavior (for example, similar results or anomalous behavior) for solutions obtained using higher order terms, because no studies were directed to answer such questions.

In the present example, only the records of the scalar variable *F* and the velocity *V* were "modeled" through square waves. It may eventually be useful for some problems also to "model" the derivatives of the records (in time or space). The use of such "secondary records", obtained from the original signal, was still not considered in this methodology.

The problem considered in this chapter was one-dimensional. The number of basic functions for two and three dimensional problems grows substantially. How to generate and solve the best set of equations for the 2-D and 3-D situations is still unknown.

Considering the above comments, it is clear that more studies are welcomed, intending to verify the potentialities of this methodology.
