4.2.1. Total Enstrophy Analysis
Enstrophy is a new quantity derived from energy. The higher the value of enstrophy, the faster the energy is dissipated. In turbulence, there are various scales and shapes of vortices, which move, stretch, and deform continuously. Stretching of the vortex tube length results in smaller−scale motions. The process of energy transfer from large scales of motion to small scales is known as the “energy cascade.” Large, low−frequency eddies obtain their energy from the mean flow and each other. Small, high−frequency eddies, on the other hand, lose energy through viscous dissipation. Conversely, enstrophy can be used to characterize the average elongation of the vortex tube, and it is proportional to the turbulent energy dissipation rate
. Therefore, as the vortex tube stretches, the vortex strength increases. However, the increased vortex strength, in turn, leads to enhanced energy dissipation. The enstrophy was defined as the volume integration of the scalar
[
20].
where
is the fluid density and
is the flow vorticity.
is a necessary and sufficient condition for the flow to be irrotational. By integrating the dissipation function, the enstrophy is associated with energy dissipation. We can further analyze the distribution of total enstrophy in different pumps, reveal the trend of mechanical energy dissipation, and finally construct a quantitative correlation between the internal vortex and external characteristics, such as the impeller head.
Figure 12 shows the values of total enstrophy in the volute domain in one impeller cycle. One data file was extracted every 12°, for a total of 30. It can be found that the total enstrophy curves of the four model pumps have seven peaks and troughs. This was because the model pump had seven blades. When the blade sweeps across the tongue area, the energy dissipation of the flow increases and reaches a peak, which corresponds to the crest position in the figure. Comparing the total enstrophy of the flow in each model pump, it was found that the curves of the bionic pumps were lower than those of the prototype pump, and the STVT−3 profile had the lowest total enstrophy value.
Table 5 shows the time−averaged total enstrophy of each model pump in the
impeller cycle and the reduction compared with the prototype pump. It can be observed that STVT−3 exhibited the largest reduction, reaching approximately 10%. The reductions in STVT−1 and STVT−3 were slightly lower than those in STVT−3, both of which were approximately 9%. Compared to the original tongue, the sinusoidal structure can reduce the energy dissipation of the centrifugal pump, and STVT−3 demonstrates the most significant effect.
To visually display the enstrophy of the flow in each model pump, the distribution of enstrophy per unit volume (i.e.,
, simply called enstrophy density) in the flow is analyzed
.
Figure 13 shows the isosurface distribution of the enstrophy density in the four model pumps, and the value of the isosurface was 65,000. It can be observed that in the impeller domain, the enstrophy of the flow is primarily distributed on the blade pressure side and near the leading edge of the blade. In the volute domain, the enstrophy of the flow was mainly distributed near the tongue and impeller outlet. Higher enstrophy was observed near the tongue. Because the clearance between the volute wall and blade was the smallest, the energy dissipation in this area was the largest. There is little difference in the enstrophy distribution of the four model pumps in the impeller, which shows that the bionic tongue cannot significantly reduce energy dissipation in the impeller. Near the tongue, the enstrophy of bionic pumps was much lower than that of the OVT profile. The OVT profile has extremely high−energy dissipation at the downstream tongue. The energy loss of the STVT profiles was concentrated only on the tongue. The enstrophy profiles of STVT−1 and STVT−2 involve several small areas in this region, whereas the STVT−3 profile has almost no enstrophy. This is similar to the conclusions drawn from
Figure 12.
The above results show that the bionic tongue has a significant impact on the flow in the tongue area, particularly downstream of the tongue, and has little impact on the impeller domain. As shown in
Figure 14, four sections with
y+ of 15, 20, 25, and 30 in the downstream region of the tongue were selected to analyze the enstrophy distribution.
As shown in
Figure 12, the enstrophy curve of the flow shows seven blade−passing periods when each impeller rotates by one revolution. Among them, the flow enstrophy of the profile of STVT−3 has the maximum decrease. Therefore, the distribution of the STVT−3 and the OVT profiles enstrophy at six evenly distributed times (T1–T6) in approximately one blade passing period was analyzed, and the rotation angle from Tn−1 to Tn is 9°, as shown in
Figure 15.
The enstrophy in the flow of the OVT and STVT−3 profiles is concentrated around the tongue, where large shedding vortices exist and a large energy loss is experienced. In
Section 1 and
Section 2, the high−energy loss area of the OVT profile first shows an increasing trend and then decreases with time, as shown in
Figure 15a,b. At time T1, the energy loss was the smallest. After that, the high−energy loss area extended in the downstream direction of the tongue, reached a maximum at time T4, and then decreased in the reverse direction. This is because in the entire period (T1–T6), the blade sweeps through the tongue, and the distance between the blade and tongue is the shortest at time T3. The clearance between the blade and tongue was the smallest, and the interaction between the rotor and the stator has the greatest influence. This leads to the maximum energy loss at this moment. In
Section 3 and
Section 4, the energy dissipation of the OVT is significantly weakened in the tongue region. At time T1, only a large area of concentrated energy loss was observed. As time passes, the high−energy loss area moves backward as a whole and slowly splits into several small pieces until it disappears. This is because vortices flow in the downstream direction of the tongue near the wall (
y+ < 10). Fluctuations occurred at
y+ = 15−20. From time T1 to T4, the vortices moved in the vertical direction to the high
y+ section. As the distance from the wall becomes larger, the turbulent fluctuation decreases and the energy loss is further reduced, which results in a much smaller energy loss area in
Section 3 and
Section 4 than in
Section 1 and
Section 2. In the horizontal direction, as the blade sweeps across the tongue area, the streamwise vortices move downstream to the tongue and then disappear slowly. This results in a downstream movement of the high−energy dissipation zone of
Section 3 and
Section 4.
Compared to the enstrophy distribution of the OVT profile, the high−energy dissipation region of the STVT−3 profile is reduced to varying degrees in each cross−section. In
Section 1, the high−energy dissipation region of the STVT−3 profile extends along the groove of the sinusoidal tongue, occupying a smaller area than the prototype pump. In
Section 2, this phenomenon is even more obvious, with the energy loss area of the STVT−3 profile splitting from the entirety of the OVT profile into a slender strip of area. As described above, enstrophy is directly related to the kinetic energy that corresponds to the dissipation effects in the fluid. The sinusoidal structure of the bionic tongue makes the high dissipation area near the downstream and of the tongue much smaller, resulting in lower energy loss. In
Section 3, the high−energy loss area of the STVT−3 profile disintegrates into several small pieces. Compared to the OVT profile, the total area occupied by this region was significantly reduced. In particular, at times T3, T4, and T5, there are almost no high−energy loss areas. However, in
Section 4, the high energy loss area of the STVT−3 profile is much weaker than that of the OVT profile in all instances, and large high−energy loss areas exist. This indicates that the bionic tongue has a significant impact on reducing pressure fluctuations in the tongue region, which will also reduce energy loss. Furthermore, this improvement has a relatively small effect in
Section 1 and
Section 2 near the wall and a significant effect in
Section 3 and
Section 4.
4.2.2. Enstrophy Transport Analysis
The enstrophy transport equation can be derived according to the vortex dynamics equation. Ignoring the expansion and viscosity fluctuations, it takes the following form [
20]:
The left side of the above formula represents the growth rate of enstrophy along the particle track. The right−hand side represents the production of enstrophy, where is the tensor of the strain rate. If the deformation of the flow stretches the vortex tube, the entropy increases, which accelerates the energy dissipation and loss of mechanical energy. is an enstrophy and molecular diffusion mechanism. is the molecular viscosity dissipation of enstrophy, which always increase dissipation. All three terms are then given as time−averaged variables from the transient simulation results. As indicated above, the STVT−3 profile had the best improvement effect. Therefore, we studied the energy−loss characteristics of the OVT and STVT−3 profiles according to the enstrophy transport equation.
Figure 16 shows the three terms on the right−hand side of the enstrophy transport equation at the intersection between the midplane and outlet of the impeller (illustrated as a red circle in the pump sketch). It can be observed that all the curves have seven peaks, corresponding to the positions of the seven blades of the model pumps. The first peak was the largest, and then it decreased sequentially. In addition, the peak of the production
is significantly higher than that of
and
, which shows that the net enstrophy change is larger than 0, which means there is a mechanical energy loss. Throughout the three figures, the significant reduction in amplitude of all three terms shows that the structure of the STVT−3 bionic tongue can stabilize the flow and decrease the mechanical energy loss near the tongue.
Figure 17 shows the three terms at the intersection between the midplane and volute wall (illustrated as a red curve in the pump sketch). In contrast to the seven peaks of each curve in
Figure 16, there was only one peak near the tongue, and the fluctuation near the tongue was much greater. Compared with the impeller outlet area, the amplitude of each peak increased significantly. This can be explained by the vortex tube stretching and the vorticity of the wake vortex shedding from the trailing edge of the blade increasing, which increases mechanical energy dissipation. Compared with OVT, STVT−3 had the largest reduction in the peak of the production term near the tongue. The results indicate that the STVT−3 bionic tongue structure can most significantly reduce the energy loss near the tongue.
Figure 18 shows the isosurface distribution of the enstrophy production in the volute domain of the OVT pump (colored by the velocity magnitude), and the value of the isosurface is 65000. It can be observed that the vortex tube stretching is primarily distributed in the impeller outlet and the area near the partition tongue. Among them, the vortex tube stretching in the tongue area was the most obvious. There is a much higher energy loss in this region than in other areas. Therefore,
Figure 19 shows the iso−surface distribution of the product terms for both STVT−3 and OVT at different times (T1−T6) near the tongue to study the time−evolution characteristics of the enstrophy production in the two pumps.
From the figure, the stretching of the vortex tube primarily occurs at the impeller outlet and tongue region, which implies a large amount of energy loss. Observing the distribution of the enstrophy production of the OVT at T1–T6 times, it can be noticed that the stretching of the vortex tube primarily starts at the trailing edge of the blade. The vortex is thrown out of the impeller owing to the inertia of the blade rotation and then hits the tongue region, and this cycle occurs repeatedly. This leads to a general elongation of the average length of the vortex tube near the tongue, which in turn generates a large amount of energy loss.
Comparing the production of the two model pumps at various times, it was found that the production term of the profile of the STVT−3 is much shorter than that of the OVT. Additionally, the distribution of production at the impeller outlet is reduced, which leads to a reduction in the total vortex tube stretch in the tongue region. This shows that the bionic tongue can decrease the stretching of the vortex tube, lessen enstrophy, and finally reduce mechanical energy loss.