The flow-field in the vessel is highly dynamical and turbulent, which is why the flow is unsteady. To describe the flow properly, the statistical methods are to be applied first. Then, the special methods such as OPD to analyze spatio-temporal data are to be used.
3.1. Statistical Analysis
The measuring planes defined in
Figure 3 were explored. The acquired time-series of velocity fields were subjected to statistical analysis.
First, the time mean velocity fields are to be presented. The mean flow topologies in the planes of measurement A-A and B-B are shown in
Figure 4. The vector fields are shown together with arbitrarily placed vector-lines.
In
Figure 4, the positions of Primary Circulation Loop (PCL), Macro-Vortex (MV), and Macro-Vortex-Upper (MVU) are shown.
The flow patterns in the planes A-A and B-B are rather different. The common characteristic is the downwards pointing jet above the impeller towards the impeller center, with an angle of about 30° from the vessel axis, and rising flow close to the vessel casing PCL (i.e., major vertically oriented vortex with chaotic core area) in the channel beside the impeller, with its center in position (1.5,1.5). In addition, the B-B plane shows smaller vortex MVU in the upper part in the region of the jet formation. In the plane B-B, the above-mentioned strong vortex MV with opposite orientation to the PCL is located at the position (0.3,0.7) below the impeller central part.
In
Figure 5 and
Figure 6, the horizontal and vertical mean velocity components U’ and V’ distributions are shown, respectively. The black broken line denotes vanishing quantity, at zero value.
The upper part of the vessel flow patterns in the planes A-A and B-B are rather different, which will be described in detail later. The PCL is not symmetrical, its axial part is significantly more intensive that its radial ones, which is due to the influence of the ascent wall current and sucking impeller current. A strong ascent wall current with an intensity that significantly exceeds the adjacent PCL is significant for the A-A plane flow pattern.
The mean flow pattern is represented in
Figure 4,
Figure 5 and
Figure 6, where the mean height of the PCL reached up to z′ = 3.76, i.e., close to 2/3 of the liquid surface height z′ = 5, and a similar mean height of the PCL was reported in previous work [
14].
Strong wall ascent flow is detected in the plane A-A, see
Figure 6. It reaches the liquid surface vicinity, where it bends inwards. Concerning its radial position here, the mean radial coordinate is almost identical as that in the lower part (around r′ = 2.19), while close to the wall, it fades away. This finding is in good agreement with the results of a previous paper [
12], where analysis of liquid surface swelling was carried out. The authors claimed that the mean maximal liquid swelling is present within the interval of dimensionless radial coordinate r′ = 2.5 − 0.375 = 2.125 to 2.5 − 0.75 = 1.75, while close to the wall, it is significantly smaller.
In the plane of measurement B-B, the ascent wall flow is present only in the lower part, while in the upper part, it vanishes, even descent flow is detected here.
Further, it is possible to detect lower velocity magnitude around the vertical axe of PCL (bending from upwards to downwards), as well as the expected higher velocity magnitude at the right upper part of PCL influenced by impeller sucking.
The radial velocity component distribution, U′, in
Figure 5 shows that intense outward flow appears predominantly within the impeller discharging area. On the other hand, significant inward flow is present mainly within the wall ascent current (gradually from bottom to top, bending inwards) and as expected, at the upper margin of primary circulation.
Figure 7 shows the detail of the flow pattern close to the baffle, for the plane of measurement C-C.
The flow close to the baffle is strongly ascending on both sides, and behind the baffle (the left-hand side), the back-flow wake is distinct.
The dynamical activity is indicated by the local value of the TKE, and its distributions in planes A-A, B-B, and C-C are shown in
Figure 8.
The dynamic activity is dominant bellow the impeller, and the main source of turbulence could be located to the spot below the impeller tip (red spot in
Figure 8). In the upper vessel part above the impeller, the secondary dynamics activity is located within the suction jet-flow.
In
Figure 8, A-A shows moderate dynamics activity within the area of strong ascent wall current, and this fact suggests predominantly uniform axial flow in this region. Dynamics activity at the wall region within the plane B-B is even smaller as the descent wall current dominates here. Concerning the area above the impeller, there is essentially analogical planar distribution of variations as within the A-A plane, only the descent wall current expresses slightly lower values of variations compared to the ascent wall current in plane A-A.
The next areas with very low dynamics activity are under the impeller hub, near the bottom close to the middle of the vessel, at the wall bottom corner, just under the liquid level close to the middle of the vessel, and just above the impeller hub.
The turbulence production could be assessed using the correlation coefficient value, which estimates the cross component of the Reynolds stress. In
Figure 9, distributions of the correlation coefficient (CC) are shown.
The correlation coefficient distributions indicate more intensive turbulence production in the vessel’s upper part within the jet than below the impeller.
3.2. Frequency Analysis at Selected Points
The next part of the investigation deals with frequency analysis of velocity components at selected points, which are shown in
Figure 10. The points were selected within the areas occupied by the dominant low-frequency flow macro-structures (described already in previous works [
3,
9,
10]), such as the PCL, the impeller discharge jet, the MV near the tank bottom center, and axial wall ascent (descent) currents.
Resulting values of dominant frequencies, F′
1, are presented in
Table 1 along with the corresponding points coordinates, evaluated velocity components, and mean velocity magnitudes. The velocity component used for frequency analysis is indicated in the fourth column. All quantities are shown in dimensionless form. If a second, quite different significant frequency was found in the spectra, its value F′
2 is presented as well. The spectral analysis was performed for the two velocity components U and V, as indicated in
Table 1,
Table 2 and
Table 3, the component L represents the resultant of the U and V components. The velocity component U is radial and V is axial component, respectively. In
Table 3, the symbol T denotes tangential component. Then, the mean values of the velocity components are to be shown. The F′
1 stands for the first dominant frequency, the F′
2 stands for the second dominant frequency, and A
1/A
2 is the ratio of the corresponding amplitudes. Similar analyses were carried out in the measuring planes B-B, see
Table 2, and C-C, see
Table 3, where frequencies near the baffle are given. In
Table 2 and
Table 3, only the first dominant frequency F′
1 is shown.
The data in
Table 1,
Table 2 and
Table 3 allow us to draw the following implications. As for the dynamics of PCL behavior, the value of F′
1 (L) 0.06 at the point PCD1 within the lower part of the PCL, close to the core, can be attributed to the frequency of PCL generation and therefore to the average frequency of the PCL cycle itself, very probably. The second represents the frequency with approximately half value, F′
2 (L) = 0.033, suggesting that there is an average of the two different alternating cycles, resulting in smaller or bigger developed PCL. It is worthy to note that the value of F′
1 (L) is reasonably close to the value presented for PCL oscillation frequency in previous work [
2]: 0.0945.
The F′1 (L) for points PCD2 and PCD3 at the outer lower part of the PCL is very low, about 0.0045 and 0.0054. This implies that the PCL reaches this area in just less than 10% of its cycle. Values of F′2 (L) for these points are 0.1065 and 0.114. These quite higher values can be attributed to PCL lower border oscillation within this area.
The frequencies F′1 (L) in the points UL and UR within the upper outer part are 0.105 and 0.126 and can be attributed to the frequencies of upper border PCL oscillation around given positions. The value of F′2 (L) for point UL 0.045 can be similarly assigned to the PCL frequency reaching this position.
The frequency F′1 (L) value 0.0525 in the point on the very upper margin location PCU3 can be similarly attributed to frequency of PCL reaching to this position with oscillation frequency around this height, F′2 (L) = 0.102.
The frequency F′1 (V) in the points PCD4 and PCD 5 with values 0.1185 and 0.1215 can be then analogically attributed to the lower PCL margin oscillation frequency.
Next, 4 points within the impeller discharge jet, IDJ1, IDJ2, IDJ3, IDJ4, are to be presented. The value of F′1 (V) in the point IDJ1 0.1005 is possible to attribute to the frequency of impeller discharge jet pulsation. Regarding the same F′1 (V) for lower point IDJ2 and just a little higher F′1 (V) = 0.11 for the even lower IDJ3 point, it can be stated that this jet reaches lower axial position z′ = 0.63 in all pulsation cases.
Frequency of the strong impinging jet to vessel bottom can be attributed to F′1 (V) in the lowest point IDJ4 0.012, which is significantly lower than discharge jet pulsation frequency (within IDJ1), as expected.
Next, investigated points are related to the above-mentioned macro-vortex at the bottom middle vessel under the impeller. The F′1 (L) detected in the points MVD1 and MVD2 at the upper vortex margin has values of 0.1425 and 0.126.
Finally, the points within the ascent wall current with radial position r′ = 2.09 and 2.2 were analyzed. They respond approximately to the inner margin of the baffle within the interval of the axial positions above the bottom, just under the mean height of PCL z′ = 3.18 to close to liquid level z′ = 5.04, with these results:
Point AC1 at z′ = 3.19, which can be still considered within the PCL area, expresses an expectable close value of F′1 (V) 0.051 to the above-noted F′ for PCL cycle, 0.06.
For the point AC2 at z′ = 3.64, under the average maximal height of PCL, its value is F′1 (V) 0.165.
Regarding its axial position, this value can be attributed to the frequency of the ascent current separating from PCL, identified in the introduction as FMF. An identical value of significant F′2 (V) for AC1 suggests oscillation of primary circulation within the point connected with this ascent current separating.
As point AC2 also manifested a significant F′2 (V) with a value analogically identical to F′1 (V) 0.051, it is possible to suppose that macro-dynamics within these points are connected with the generation of the PCL and affect near the wall ascent current.
In previous work [
15], the average value of all types of FMF frequency generation was 0.174, and therefore close to our findings. Similarly, in Reference [
6], the authors detected a close value of dominant frequency of 0.186 within the area upstream of baffle. The frequency F′
1 (V) of the third point AC3 at z′ = 4.48 has the value 0.0525. As its axial position according to the mentioned work [
15] corresponds to already separated FMF, it is possible to state that approximately 32% of the generated ascent current creates it.
The identical value of F′
1 (V) in the next point AC4 at z′ = 4.71 implies that all separated FMF reach this axial position. The F′
1 (V) value in the point AC5 at z′ = 4.88 (therefore close to liquid level) can be attributed to the frequency of FMF crashing the surface and the subsequent liquid level MS (manifesting as the uplift of liquid surface by approximately Δh = 0.01 m for the given conditions). A previous contribution [
1,
17] dealing with MS detection with the help of the liquid level horizontal visualization and conducting probe presenting for the same conditions (rpm = 400, PBT 4, C/H = 0.4), indicated a very close value of MS frequency 10 min
−1, i.e., dimensionless value 0.0255. Also, the contribution [
12] dealing with liquid level dynamic mapping within the net of selected points, presenting for dimensionless MS frequency localized tangentially in front of the inner baffle margin, corresponds to interval of values 0.0225–0.03.
Overall, the lower values of F′1 (V) for the upper points AC3–AC5 can be attributed to the transition regime (not fully turbulent) within this area, as mentioned above in the Section “Statistical Analysis”.
We also made a comparison of U′ and V′ velocity components for the selected points presented in
Table 1 with those detected in Reference [
22] for similar relative positions, i.e., rc′ = 2r/D, zc′ = z/C. Authors in Reference [
22] used the same PBT 6 impeller and the setup D = T/2, C = T/4, and N
imp = 345 rpm. As the authors positioned the vertical measured plane mid-way between the baffles (at maximal distance from them), this comparison can reveal the influence of the baffle distance for the vertical flow-field as well. The results of this comparison are presented in
Table 4, where rc1′, zc1′, U1′, and V1′ are our coordinates and velocity magnitudes for selected points, and rc2′, zc2′, U2′, and V2′ are the coordinates of the nearest positions and corresponding velocity magnitudes from Reference [
22] (Figure 3 and Figure 11).
Next follows a description of the flow pattern within the plane B-B intersecting the center of the vessel with angle 22.5° (see
Figure 3). In the plane B-B, it can be supposed that there is minimal baffle influence on the flow pattern.
The time-averaged flow vector field with contour of dimensionless U′ and V′ velocity magnitudes are presented in
Figure 4,
Figure 5 and
Figure 6.
In
Figure 4 (and
Figure 6 as well), within the lower part of the vessel, there is predominantly a detectable significantly smaller wall ascent current area compared to the plane A-A. This finding is expected because of the baffle absence, as well as the significantly smaller average maximal height of the PCL, reaching within this plane approximately just z′ = 1.88. Concerning PCL center position, there is no significant difference in the plane A-A (approximately (1.5;1.5)). There is also no detectable difference of the average radial localization of the most intensive wall ascent current area.
However, there is a significant difference of the flow pattern within the area above the impeller (see
Figure 4) due to the absence of the wall ascent current, implying that this one reaches average maximal height at just z′ = 2.08−2.7. This finding agrees with the previous results [
1,
12], which report that the MS of liquid level is present exclusively in the baffle vicinity. Within this plane, the weaker wall descent current is differently detectable. This descent current between baffles was already reported in Reference [
11], where the authors further observed collision of that with the wall upward current within the lower part of the vessel, and subsequent origin of the above-mentioned smaller vortex MVU with counterclockwise orientation, due to the PCL. The area of this vortex appeared quite weak within the main flow field (marked in
Figure 4, B-B), with the center at approximately (1.7;4). At axial position, approximately z′ = 2.08, this descent current becomes adjacent to the PCL upper part and subsequently, close to the middle of the sucking impeller jet. The counterclockwise MV near the bottom middle vessel is again marked in
Figure 4, B-B.
The similar frequency analysis of selected points within the plane B-B follows. Corresponding values of dominant frequencies F′, along with corresponding points coordinates, evaluated velocity components, and mean velocity magnitudes are shown in
Table 2.
Almost identical F′
1 (L) values of point PCD1 (within the middle of the PCL close to the core) and PCU1 (diagonally opposite point), 0.021 and 0.023 respectively (see
Figure 10), can be again attributed to the average frequency of generation (oscillation) of the PCL, and this significantly lower frequency in the plane A-A implies longer phases of developed PCL sustaining and shorter chaotic phases, or vice-versa.
Here, it is worthy to note that a previous contribution [
2] analyzing the PCL oscillations using flow visualization within the plane A-A presented a 400 rpm asymptotic value of the PCL relative incidence for identical impeller revolution (i.e., phase of growing and sustaining PCL), for approximately 80% of the whole cycle period on average.
The value F′1(L) of point PCU2 within the upper part of PCL 0.042 is again possible to attribute to the PCL upper margin oscillation frequency, and the value F′1(L) of the diagonally opposite point PCD2 0.029 analogically to the PCL lower margin oscillation frequency. Based on these results, it is possible to state that the planar development of the PCL within this plane is almost symmetrical. Further, the upper margin exhibits two significant oscillations in one PCL cycle, while the lower margin shows two significant oscillations during 70% of the PCL cycles.
Subsequently, the evaluation of the Dominant Frequency (hereinafter DF) axial component velocity course within the impeller discharge jet is to be discussed. The first selected point IDJ1 just below the impeller (the same coordinates as the point IDJ1 in the plane A-A) expresses F′1(V) 0.021, which is the same as the PCL oscillation frequency. The second selected point IDJ2 close to the vessel bottom (r′ = 0.94, z′ = 0.73) has a lower F′1 (V) value of 0.006, as expected.
Concerning the counterclockwise vortex near the bottom middle vessel, the two selected points, MVD1 at vortex upper margin and MVD2 at vortex lower margin, indicate F′1 (L) values of 0.04 and 0.1, respectively.
A further two points near the wall within the lower segment were selected: AC1 (r′ = 2.29, z′ = 0.94) and AC2 (r′ = 2.29, z′ = 1.58). The AC1 value of F′
1 (U) 0.027 (i.e., at the position where the dominant influence of primary circulation or impeller discharge jet is located) is close to PCL oscillation, as expected, while the significantly lower value F′
1 (V) of the higher point AC2, 0.012 (i.e., at the position where the wall ascent current separates from the PCL), implies significantly lower relative incidence of the wall ascent current at this location compared to the plane A-A. Subsequently, two points were selected at similar radial positions, near the wall within the upper area: points DC1 (r′ = 1.98, z′ = 3.29) and DC2 (r′ = 2.29, z′ = 4.17), see
Figure 10. Quite low DF values of both points, F′
1 (V) = 0.03 for DC1 and F′
1 (V) = 0.03 for DC2, confirm that the upper flow segment is, in terms of macro-dynamics, significantly more stationary than the lower one. Moreover, these low DF values along with the low DF values within the wall ascent current area compared to plane A-A imply an interaction of the ascent and descent currents within this plane.
Finally, the plane C-C was analyzed, and the flow is perpendicular to the baffle through its radial center (i.e., in tangential direction, see
Figure 3). The main reason for this choice was to compare the flow fields in the area just in front of the baffle (i.e., dominance of the wall ascent current) and the area just behind the baffle. However, as shown in
Figure 3, within this plane, it is possible to visualize the tangential profile of the flow field within the interval from a large distance down to close to the baffle. Within the plane A-A, the radial profile of the flow field is visualized tangentially in front of the baffle.
In other words, comparing the results within the wall ascent current area for the planes A-A and C-C, we could assess horizontal homogeneity of the geometry, and macro-dynamics of the wall ascent current. Corresponding time-averaged flow vector fields with contours of dimensionless U´ and V´ velocity components are presented in
Figure 5. It is apparent that the flow field is totally different in front of and behind the baffle.
The dominant wall ascent current in front of the baffle is replaced by a significantly weaker decent current behind the baffle, pointing from the baffle diagonally downwards and near the baffle turned upwards. It almost fades close the liquid level. The geometry of the ascent wall current within the plane C-C shows that it is adjacent to the baffle (tangential dimensionless coordinate t′ = 0.075), while mean tangential position of the most intense wall ascent current area is at tangential coordinate of approximately t′ = 0.208, in front of the baffle.
The similar frequency analysis of instantaneous velocity components at selected points for the plane C-C was presented with corresponding values of dominant frequencies, F′, along with corresponding points coordinates, see
Table 3.
The comparison of F′ values at selected points within this plane and corresponding points within the plane A-A had the following outcomes.
The lowest point below the level of mean height of PCL, AC1 (t′ = 0.208, z′ = 3.29), expressed an F′1 (V) value of 0.036, thus lower but with close relation to the F′1 (V) value of the corresponding point AC1 within the plane A-A.
The F′1 (V) value of the next point AC2 (t′ = 0.26, z′ = 3.67), 0.114, which is attributed to the ascent current origin, is also lower than the F´1 (V) of corresponding point AC2 within the plane A-A. However, it is possible to state the proportionality of the points AC1 and AC2 taken for both planes A-A and C-C.
Points AC3–AC6 (t′ = 0.26, z′ = 4–4.58) show the identical F′1 (V) value of 0.0345. This fact implies that the macro-behavior of the wall ascent current in axial position z′ = 4.58 is essentially uniform with the given average frequency of oscillations.
With respect to the significantly lower F′1 (V) value of 0.003 for the highest point AC7 (t′ = 0.26, z′ = 4.88), it is possible to state that the liquid level MS appeared at this position with significantly lower frequency than in the plane A-A (effect of the baffle on the MS average frequency surely exists).
Concerning the dynamical activity within the plane C-C (see the TKE distribution in
Figure 8), there is no detectable difference of planar distribution of this quantity within the ascent wall current area compared to the plane A-A.
For the area of “wake” behind the baffle, two points were selected close to the vessel wall (
Table 3,
Figure 5), UDC (t′ = −0.875, z′ = 4.02) and LDC (t′ = −0.875, z′ = 3.5). The value of F′
1 (U) for point UDC, 0.065, is closely related (in the middle of the interval) to the values of F′
1 (V) for points within the wall ascent current for both planes A-A and C-C. The F′
1 (U) frequency value in the point LDC is 0.012, and it is significantly lower, as is the mean axial velocity within this point.