**3.4 Turbulence phenomena analysis**

The turbulence phenomena can be studied from different parameters. In the present study, we propose the description of the Reynolds number, the isotropic turbulence, the Q field, the Lambda2 fields and the vorticity.

Considering the impeller diameter for the characteristic length, the Reynolds number is 1326 at 20 rpm, 2651 at 40 rpm, 3977 at 60 rpm and 6629 at 100 rpm.

**Figure 7.** *Kinematic pressure (m2 .s<sup>2</sup> ) at 40 rpm (z = 0.06 m, y = 0).*

**Figure 8.** *Kinematic pressure (m2 .s<sup>2</sup> ) at 100 rpm (z = 0.06 m, y = 0).* *CFD Simulations in Mechanically Stirred Tank and Flow Field Analysis: Application… DOI: http://dx.doi.org/10.5772/intechopen.93926*

Therefore, local turbulence phenomena can occur around the impeller, however, the turbulence phenomena dissipate rapidly away from the impeller. If the isotropic turbulence k is very low, we consider that we are in a laminar regime. The maximum k value is 2.2 � <sup>10</sup>�<sup>16</sup> <sup>m</sup><sup>2</sup> .s�<sup>2</sup> at 20 and 40 rpm, 3.0 � <sup>10</sup>�<sup>11</sup> <sup>m</sup><sup>2</sup> .s�<sup>2</sup> at 60 rpm and 3.9 � <sup>10</sup>�<sup>8</sup> m2 .s�<sup>2</sup> at 100 rpm. Therefore, we observe these local important values at the impeller level.

Q and Lambda2 fields provide a precise description of the turbulence and local rotation. The Lambda2 (s�<sup>2</sup> ) function object computes the second largest eigenvalue of the sum of the square of the symmetrical and anti-symmetrical parts of the velocity gradient tensor. Q iso-surfaces are good indicators of turbulent flow structures. The Q function object computes the second invariant of the velocity gradient tensor (s�<sup>2</sup> ):

$$Q = \frac{1}{2} \left[ \left( \text{tr}(\nabla u) \right)^2 - \text{tr}(\nabla u \cdot \nabla u) \right] \tag{29}$$

**Figure 9** presents the velocity flood with the Lambda2 contours. The Lambda2 and velocity profiles are similar in zones between the blades but different at the impeller extremities.

**Figure 10** shows the Q field at the impeller level at 40 and 100 rpm. The extremum values of the Q field are obtained at the impeller extremities. The Q field varies from – 5171 to 39,405 s�<sup>2</sup> at 40 rpm and from �27,000 to 289,000 s�<sup>2</sup> at 100 rpm. The value is close to zero in areas away from the impeller. The Q field varies from �1383 to 8924 s�<sup>2</sup> at 20 rpm and from �9314 to 94,137 s�<sup>2</sup> at 60 rpm.

The Lambda2 field at the impeller level at 40 rpm is shown in **Figure 11**. It varies from – 3677 to 31,177 s�<sup>2</sup> at 40 rpm and from – 15,561 to 226,008 s� <sup>2</sup> at 100 rpm. Both the Q and Lambda2 field maximum values are about seven times higher at 100 rpm than at 40 rpm. The Lambda2 field varies from �1125 to 8449 s�<sup>2</sup> at 20 rpm and from �5234 to 73,482 s�<sup>2</sup> at 60 rpm.

**Figure 12** presents the vorticity module (s�<sup>1</sup> ) at different digester heights in continuous flow regime at 40 rpm (z = 0.06 m, z = 0.08 m, z = 0.259 m and y = 0). The vorticity is a pseudo vector field that describes the local spinning motion (the curl of the velocity). The maximum vorticity module is 314.8 s�<sup>1</sup> at 20 rpm, 621.7 s�<sup>1</sup> at 40 rpm, 959.4 s�<sup>1</sup> at 60 rpm and 1680.7 s�<sup>1</sup> at 100 rpm.

The extremum values are observed at the impeller level. We notice that the maximum value of vorticity is multiplied by 2.7 from 40 to 100 rpm. We can

**Figure 9.** *Velocity (m.s*�*<sup>1</sup> ) flood and Lambda2 (s*�*<sup>2</sup> ) contour at 40 rpm (z = 0.06 m).*

**Figure 10.**

*Q field (s<sup>2</sup> ) at the impeller level at 40 and 100 rpm.*

**Figure 11.** *Lambda2 field (s<sup>2</sup> ) at the impeller level at 40 rpm.*

*CFD Simulations in Mechanically Stirred Tank and Flow Field Analysis: Application… DOI: http://dx.doi.org/10.5772/intechopen.93926*

#### **Figure 12.**

*Vorticity module (s<sup>1</sup> ) in continuous flow regime at 40 rpm (z = 0.06 m, z = 0.08 m, z = 0.259 m and y = 0).*

consequently note that the variations of the Q and Lambda2 fields are more significant than the variation of the vorticity with the increase of the impeller mixing speed.

**Figure 13** summarizes the variation of maximum vorticity, pressure, Q and Lambda2 values in function of mixing speed. The Q maximum value variation is the most pronounced among the parameters observed.
