**3. Velocity-velocity interactions**

22 Hydrodynamics – Advanced Topics

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

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*\*)

*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.

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

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

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

, a range based on the � experimental values of

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

*<sup>f</sup>* is a function of

closely follows the form of the measured curve. Because it is known that

figure 6a were obtained for 0.001 0.005

Fig. 6a. Predictions of *n* for *n*"(0) = 3.056.

simpler schemes can be used to obtain adequate results.

application in velocity-velocity interactions are presented.

Fourth order Runge-Kutta.

*A*=0.90.

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

$$\frac{\partial \overline{V}\_{j}}{\partial t} + \overline{V}\_{i} \frac{\partial \overline{V}\_{j}}{\partial \mathbf{x}\_{i}} = \frac{\partial}{\partial \mathbf{x}\_{i}} \left( \nu \frac{\partial \overline{V}\_{j}}{\partial \mathbf{x}\_{i}} - \overline{v\_{i} \mathbf{w}\_{j}} \right) - \frac{1}{\rho} \frac{\partial \overline{p}}{\partial \mathbf{x}\_{j}} + B\_{i} \,\prime \qquad \qquad \text{i, j} = 1, 2, 3. \tag{54}$$

*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:

$$\frac{1}{\rho} \frac{\partial \overline{p}}{\partial x} = \frac{\partial}{\partial z} \left( \nu \frac{\partial \overline{U}}{\partial z} - \overline{\rho u} \right) \tag{55}$$

This equation is similar to equation (2) for one dimensional scalar fields. As for the scalar case, the mean product *u* appears as a new variable, in addition to the mean velocity *U* . 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 the partition, reduction and superposition functions for this velocity field.

One Dimensional Turbulent Transfer

Or, normalizing the RMS value (*u*'2):

Using Random Square Waves – Scalar/Velocity and Velocity/Velocity Interactions 25

The second order central moment for the *x* component of the velocity fluctuations is given by:

<sup>2</sup> 22 2 <sup>2</sup> 1 2 1 11 *u un u n n n U U v v v v u pn*

*p nv v u*

<sup>1</sup> ln 5.2

*vv u vv u*

(65)

2

' 1 1 *vv u*

Equation 63 shows that the relative turbulence intensity profile is obtained from the mean velocity profile *nv* and the reduction coefficient profile ߙ௨. As done by Schulz & Janzen

<sup>1</sup> *<sup>u</sup>*

As can be seen, the functional form of ߙ௨ is obtainable from usual measured data, with exception of the proportionality constant given by 1/*Up*, which must be adjusted or conveniently evaluated. Figure 9 shows data adapted from Wei & Willmarth (1989), cited by Pope (2000), and the function *n n v v* 1 is calculated from the linear and log-law profiles

To obtain a first evaluation of the virtual constant velocity *Up*, the following procedure was adopted. The value of the maximum normalized mean velocity is *U*/*u\**~24.2 (measured), where *U* is the mean velocity and *u\** is the shear velocity. The value of the normalized RMS *u* velocity, close to the peak of *U,* is *u'*/*u\*~*1.14. Considering a Gaussian distribution, 99.7% of the measured values are within the range fom *U*/*u\**-3 *u'*/*u\**. to *U*/*u\**+3 *u'*/*u\**. A first value of *Up* is then given by *U*+3*u',* furnishing *Up*/*u\*~*24.2+3\*1.14~27.6. Physically it implies that patches of fluid with *Up* are "transported" and reduce their velocity while approaching

> 0.41 27.6 27.6 *<sup>v</sup> <sup>y</sup> <sup>u</sup>*

The value 0.41 is the von Karman constant and the value 5.2 is adjusted from the experimental data. The notation *u*+ and *y*+ corresponds to the nondimensional velocity and

the kinematic viscosity of the fluid. Equation (65) is the well-known logarithmic law for the velocity close to surfaces. It is generally applied for *y+*>~11. For 0<*y+*<~11, the linear form *u+*=*y+* is valid so that equation (65) is then replaced by a linear equation between *nv* and *y+*.

1 1 27.6 1 1 \* \*

*n n n n u u* 

*UU n n*

(2009), the profile of ߙ௨ can be obtained from experimental data, using equation (63).

(negative) (61)

(62)

(63)

(64)

, where

(66)

is

<sup>2</sup> 1 *u nU U vp n u*

2

*p n u u n n U U*

2

1

close to the wall, also measured by Wei & Wilmarth (1989).

the wall. With this approximation, the partition function is given by:

*n*

From equation (63) it follows that:

2

*u U*

*p*

distance, respectively, used for wall flows. In this case, *u+*=*U*/*u\** and *y+*=*zu\**/

Both the upper and the lower parts of the flow sketched in figure 8 may be considered. We consider here the lower part, so that it is possible to define a zero velocity (*Un*) at the lower surface of the flow, and a "virtual" maximum velocity (*Up*) in the center of the flow. This virtual value is constant and is at least higher or equal to the largest fluctuations (see figure 8), allowing to follow the analogy with the previous scalar case.

Fig. 8. The flow between two parallel planes, showing the reference velocities *Un* and *Up*. The partition function *nv*, for the longitudinal component of the velocity, is defined as:

$$m\_v = \frac{\text{t at (}\text{U}\_p - P\text{)}}{\text{At of the observation}}\tag{56}$$

It follows that:

$$1 - n\_v = \frac{t \text{ at } (\mathcal{U}\_n + N)}{\Delta t \text{ of the observation}} \tag{57}$$

Equation (7) must be used to reduce the velocity amplitudes around the same mean velocity. It implies that the same mass is subjected to the velocity corrections *P* and *N*. As for the scalar functions, the partition function *nv* is then also represented by the normalized mean velocity profile:

$$m\_v = \frac{\overline{\mathcal{U}} - \mathcal{U}\_n}{\mathcal{U}\_p - \mathcal{U}\_n} \tag{58}$$

To quantify the reduction of the amplitudes of the longitudinal velocity fluctuations, a reduction coefficient function ߙ௨ is now defined, leading, similarly to the scalar fluctuations, to:

$$\begin{aligned} N &= \alpha\_u \, n\_v \left( \mathsf{U}\_p - \mathsf{U}\_n \right) \\ P &= \alpha\_u \left( \mathsf{1} - n\_v \right) \left( \mathsf{U}\_p - \mathsf{U}\_n \right) \end{aligned} \qquad \begin{aligned} 0 &\le \alpha\_u \le 1 \end{aligned} \tag{59}$$

It follows, for the *x* components, that:

$$
\mu\_1 = (1 - n\_v) \left( \mathcal{U}\_p - \mathcal{U}\_n \right) \left( 1 - \alpha\_u \right) \tag{60}
$$

$$
\mu\_2 = -n\_v \left( \mathcal{U}\_p - \mathcal{U}\_n \right) (1 - \alpha\_u) \tag{61}
$$

The second order central moment for the *x* component of the velocity fluctuations is given by:

$$
\overline{\mu^2} = \mu\_1^2 n\_v + \mu\_2^2 \left(1 - n\_v\right) = n\_v \left(1 - n\_v\right) \left(1 - \alpha\_u\right)^2 \left(\mathcal{U}\_p - \mathcal{U}\_n\right)^2 \tag{62}
$$

Or, normalizing the RMS value (*u*'2):

24 Hydrodynamics – Advanced Topics

Both the upper and the lower parts of the flow sketched in figure 8 may be considered. We consider here the lower part, so that it is possible to define a zero velocity (*Un*) at the lower surface of the flow, and a "virtual" maximum velocity (*Up*) in the center of the flow. This virtual value is constant and is at least higher or equal to the largest fluctuations (see figure

Fig. 8. The flow between two parallel planes, showing the reference velocities *Un* and *Up*. The partition function *nv*, for the longitudinal component of the velocity, is defined as:

( ) <sup>1</sup>

Equation (7) must be used to reduce the velocity amplitudes around the same mean velocity. It implies that the same mass is subjected to the velocity corrections *P* and *N*. As for the scalar functions, the partition function *nv* is then also represented by the normalized mean

> *<sup>n</sup> <sup>v</sup> p n U U*

To quantify the reduction of the amplitudes of the longitudinal velocity fluctuations, a reduction

*U U*

*<sup>n</sup> <sup>v</sup>*

*t*

*n*

coefficient function ߙ௨ is now defined, leading, similarly to the scalar fluctuations, to:

 1

*u vpn*

*uv p n*

*P nUU*

*u nU U* <sup>1</sup> (1 ) 1 *<sup>v</sup> <sup>p</sup> n u*

*N nU U*

It follows, for the *x* components, that:

*v*

*n*

*t*

*n*

It follows that:

velocity profile:

( ) of the observation *p*

of the observation

*t at U N*

(56)

(57)

0 1 

(58)

*<sup>u</sup>* (59)

(positive) (60)

*t at U P*

8), allowing to follow the analogy with the previous scalar case.

$$
\mu'\_2 = \frac{\sqrt{\mu^2}}{\left(\mathcal{U}\_p - \mathcal{U}\_n\right)} = \sqrt{n\_v \left(1 - n\_v\right)} \left(1 - \alpha\_u\right) \tag{63}
$$

Equation 63 shows that the relative turbulence intensity profile is obtained from the mean velocity profile *nv* and the reduction coefficient profile ߙ௨. As done by Schulz & Janzen (2009), the profile of ߙ௨ can be obtained from experimental data, using equation (63).

$$1 - \alpha\_u = \frac{\sqrt{u^2}}{\left(\mathcal{U}\_p - \mathcal{U}\_n\right)\sqrt{n\_v\left(1 - n\_v\right)}}\tag{64}$$

As can be seen, the functional form of ߙ௨ is obtainable from usual measured data, with exception of the proportionality constant given by 1/*Up*, which must be adjusted or conveniently evaluated. Figure 9 shows data adapted from Wei & Willmarth (1989), cited by Pope (2000), and the function *n n v v* 1 is calculated from the linear and log-law profiles close to the wall, also measured by Wei & Wilmarth (1989).

To obtain a first evaluation of the virtual constant velocity *Up*, the following procedure was adopted. The value of the maximum normalized mean velocity is *U*/*u\**~24.2 (measured), where *U* is the mean velocity and *u\** is the shear velocity. The value of the normalized RMS *u* velocity, close to the peak of *U,* is *u'*/*u\*~*1.14. Considering a Gaussian distribution, 99.7% of the measured values are within the range fom *U*/*u\**-3 *u'*/*u\**. to *U*/*u\**+3 *u'*/*u\**. A first value of *Up* is then given by *U*+3*u',* furnishing *Up*/*u\*~*24.2+3\*1.14~27.6. Physically it implies that patches of fluid with *Up* are "transported" and reduce their velocity while approaching the wall. With this approximation, the partition function is given by:

$$m\_v = \frac{\mu^+}{27.6} = \frac{\frac{1}{0.41} \ln y^+ + 5.2}{27.6} \tag{65}$$

The value 0.41 is the von Karman constant and the value 5.2 is adjusted from the experimental data. The notation *u*+ and *y*+ corresponds to the nondimensional velocity and distance, respectively, used for wall flows. In this case, *u+*=*U*/*u\** and *y+*=*zu\**/, where is the kinematic viscosity of the fluid. Equation (65) is the well-known logarithmic law for the velocity close to surfaces. It is generally applied for *y+*>~11. For 0<*y+*<~11, the linear form *u+*=*y+* is valid so that equation (65) is then replaced by a linear equation between *nv* and *y+*. From equation (63) it follows that:

$$\frac{\sqrt{\mu^2}}{\mu^\*} = \frac{\mathcal{U}\_p}{\mu^\*} \sqrt{n\_v \left(1 - n\_v\right)} \left(1 - \alpha\_u\right) = 27.6 \sqrt{n\_v \left(1 - n\_v\right)} \left(1 - \alpha\_u\right) \tag{66}$$

One Dimensional Turbulent Transfer

**4. Challenges** 

future studies.

coefficients (

defining the function

, 

Using Random Square Waves – Scalar/Velocity and Velocity/Velocity Interactions 27

As a last observation, the conclusion of section 2.7, valid for the scalar-velocity interactions, are now also valid for the transversal component of the velocity. The mean transversal velocity is null along all the flow, leading to the use of the RMS velocity for this component.

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

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*,

lead to major problems in the present application. Eventual applications in 2-D, 3-D problems or

In the present study, the example of mass transfer was calculated by using constant reduction

(2011a). However, it is known that this coefficient varies along *z*, which may introduce

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

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

Considering the above comments, it is clear that more studies are welcomed, intending to

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

in phenomena that deal with discrete variables may need more refined definitions.

difficulties to obtain a solution for *n*. This more complete result is still not available.

higher order terms, because no studies were directed to answer such questions.

best set of equations for the 2-D and 3-D situations is still unknown.

verify the potentialities of this methodology.

**5. Conclusions** 

, RMS) "adjust" these two points of view (this is perhaps more clearly explained when

). This concomitant dual form of treating the random transport did not

), presenting a more detailed and improved version of the study of Schulz et al.

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

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 boundary layer (maximum peak for 1-ߙ௨).

Fig. 9. Comparison between measured values of *u*'/*u*\* and /\* 1 *Uu n n <sup>p</sup> v v* . The gray cloud envelopes the data from Wei & Willmarth (1989).

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).

As a last observation, the conclusion of section 2.7, valid for the scalar-velocity interactions, are now also valid for the transversal component of the velocity. The mean transversal velocity is null along all the flow, leading to the use of the RMS velocity for this component.
