2.2. Boundary Conditions and Operating Points
The rated output was set to 100%. Three operating conditions (i.e., 20%, 100%, and 120%) of the output at the rated and maximum heads were simulated. These operating conditions covered the common operating conditions of a turbine, i.e., an extremely small-load operating point, a high-efficiency operating point, and a large-load operating point.
CFX software was used to numerically calculate the entire flow field of the Francis turbine. The calculation result of the flow field under pure water conditions was used as the initial calculation condition. Moreover, the SST
k-ω [
28,
29,
30,
31] turbulence model was used to calculate the internal flow conditions at different GVOs. Pure water and steam (25 °C) were set as the gas-liquid two-phase medium [
32,
33], respectively, and the continuous phase was set as the phase state. The total pressure inlet was used as the inlet condition, and the inlet pressure was converted to the head. For the inlet condition, the volume fraction of pure water and steam were set to 1 and 0, respectively. In addition, the outlet was set as the static pressure outlet. The SIMPLE algorithm with strong convergence and fast convergence speed was adopted for the coupling between velocity and pressure in the whole calculation process. The non-slip wall was adopted for the solid wall surface, the wall function was considered as “automatic”, and the residual standard was set at 10
−5. The operating point parameters are shown in
Table 4. The unsteady calculation time step was set to the time it takes for the runner to rotate by 2° (0.0006667 s). The verification of the time step independence is shown in
Figure 2.
The dimensionless pressure coefficient,
Cp, are reference values for quantifying the intensity of pressure pulsations in turbines, as shown in Equation (1).
where
Cp is the dimensionless pressure pulsation coefficient (%),
Pi is the corresponding pressure at point
i (Pa),
is the time-averaged pressure,
H is the head (m),
ρ is the density of water, and g is the gravitational acceleration.
Figure 2 shows the time domain diagram of the pressure pulsation at different time steps for the monitoring points in the vaneless space. As shown, the pressure pulsation waveforms obtained at time steps corresponding to 1° and 2° are basically the same, while the waveforms obtained at time steps corresponding to 5° have more missing shapes. Considering the calculation accuracy and calculation time, 2° was chosen as the calculation time step, that is, the number of calculation steps in one rotation cycle is 180.
2.3. Numerical Simulation Results
Both the flow rate and flow circulation of the fluid were adjusted to varying degrees after flowing through the stay and guide vanes. After the energy conversion occurred in the runner, it flowed into the draft tube. The flow characteristics in the diversion parts, stay vanes, guide vanes, and runners affect the draft tube. The flow in the draft tube is relatively complicated compared to flows in other components of the turbine, which is affected by the flow characteristics at the runner outlet and the shape of the draft tube. Abraham et al. [
34] verified that a slight adjustment of the inlet angle facilitated the formation of the optimally sized vortex rope in the center of the draft tube using CFD analysis.
The flow velocities and directions on the cross-sections at different blade heights were observed so as to facilitate an analysis of the flow characteristics when the fluid flows into the draft tube. SPAN = (0–1) represents the distance from the runner band to the runner crown. The velocity vector distribution on the internal axial plane of the turbine was observed at SPAN = 0.9.
Figure 3 presents the velocity vectors of the flow components at the rated head. Here, the flow characteristics are considered good when fluid flows through the stay vanes under different working conditions, and the velocity is uniform without a sudden increase. At a small load operating point, the variation in the velocity vector is uneven, the kinetic energy that is gained is small, and the direction of the relative velocity does not match the flow direction in the runner, leading to serious impact loss at the runner inlet. Meanwhile, the relative velocity at the runner outlet is small, and the cross-flow phenomenon occurs in the flow passage. Therefore, the fluid cannot flow out of the runner smoothly. At the high-efficiency operating point, the flow velocity at the guide vanes increased uniformly along the flow passage, the streamline was smooth without impact, and there was no evident flow separation or vortex in the runner passage. On the other hand, the overall flow situation at the large-load operating point is similar to that at the high-efficiency operating point, and the runner outflow velocity is the highest.
The velocity vectors in the flow components at the maximum head are shown in
Figure 4. The head increased and the velocity around the airfoil increased in the runner region due to the change in inflow conditions. The turbulence characteristics of the three GVOs were similar to those of the rated head. Based on the velocity triangle at the runner inlet, the circumferential velocity is a fixed value, the absolute velocity increases at the inlet, the absolute flow angle is related to the GVO, and the relative velocity angle increases, reducing the impact loss at the blade inlet.
The cavitation volume fractions on the flow section at the blade heights of SPAN = 0.1, SPAN = 0.5, and SPAN = 0.9 at the rated head and maximum head, respectively, are provided, as shown in
Figure 5, to demonstrate the cavitation situation in the runner intuitively and clearly. At the small load operating point, serious cavitation occurs in the blade outlet near the crown (SPAN = 0.1). In addition, cavitation bubbles almost occupy the entire flow passage in this area, whereas there is almost no cavitation bubble in the runner passage in the middle area and near the band (SPAN = 0.5−0.9). On the other hand, there were a few cavitation bubbles at the high-efficiency operating point, and cavitation only occurred near the band (SPAN = 0.9) at the blade outlet edge. Meanwhile, the cavitation area at the blade outlet at the large-load operating point is similar to that under a 100% output, but the cavitation degree is relatively high. In general, the effect of cavitation generated at a small-load operating point is completely different from that at the high-efficiency and the large-load operating points, which has a significant influence on the state of the fluid flowing into the draft tube. The effect of cavitation at the maximum head is slightly more significant than that at the rated head.
In summary, the GVO affected the velocity at the runner outlet, resulting in a change in the direction and magnitude of the velocity at the inlet of the draft tube. At the small-load operating point, the flow state in the passage was found to be mainly affected by flow impact loss, exit cavitation conglomeration, and other factors. Our analysis of the velocity circulation at the runner outlet shows that the entrainment velocity of the inflow is large, and the relative velocity is small, resulting in a large angle of inclination between the direction of the outflow velocity and the inlet. In the high-efficiency zone and under the high GVO conditions, a large outflow results in a large relative velocity. The interference between the vortex and cavitation bubbles was not apparent, the flow in the runner was smooth, and the streamline at the inlet of the draft tube was evenly distributed.
The streamline in the draft tube was random due to the influence of various factors. Therefore, it was not ideal to observe the streamlines of the entire draft tube. A total of 12 cross-sections were selected on the conical section, elbow section, and diverging section as monitoring sections to observe the flow characteristics, velocity distribution, and other features of the fluid at different sections and to obtain the influence pattern of the GVOs.
The velocity vectors in the monitoring sections at various GVOs at the rated and maximum heads are shown in
Figure 6. Based on
Figure 6a,b, at the small-load operating point, the outflow velocity on the blade near the crown was found to be small due to severe conglomeration by cavitation bubbles. A large area of dead fluid was formed in the central area of the draft tube, as shown in the enlarged conical section, and the backflow phenomenon appeared in this section. After leaving the runner, the fluid near the band still exhibited a circumferential velocity similar to the direction of the velocity at the inlet of the draft tube. Under the action of inertia and gravity, the fluid flows diagonally downward along the wall, and the inclination angle gradually decreases with the downward movement of the fluid. By comparing the two conditions at small GVOs, we found that the inflow under the maximum head increases compared to that under the rated head, the area of dead fluid contracts slightly, and the inclination angle at the inlet improved. However, the overall flow characteristics are still chaotic and disordered.
Based on
Figure 6c,d, at the high-efficiency operating point, the low-energy fluid flowing out of the runner is concentrated at the center of the conical section, and the central low-velocity region in the center expands gradually with the conical section. The velocity distribution was no longer axially symmetric when flowing through the diverging section, and the flow characteristics became complex. The distribution of the velocity vectors on various sections of the diverging section was uneven, and a small area of backflow occurred in the outlet section.
In
Figure 6e,f, it can be seen that the outflow velocity is dominated by the relative velocity due to the increase in the flow rate. The inlet velocity in the draft tube was uniform, and its direction was perpendicular to the inlet surface. The low-velocity region is concentrated along the axis near the sidewall. There are 13 regular velocity changing cycles “increasing first and decreasing later” on the circumference, which is consistent with the number of runner blades. This indicates that the circumference is affected by the rotor-stator interaction, leading to an uneven velocity circulation at the inlet of the draft tube. This phenomenon disappears when the fluid flows downward, and the velocity on the sidewall is relatively low at the location where the curvature of the elbow section is large. There is an apparent low-velocity region in the diverging section, which may be resultant of the action of the vortex rope in the draft tube.
The flow characteristics at the center of the section of
y = 0 of the draft tube were observed (see
Figure 7) to further analyze the flow inside the draft tube. It was found that the flow characteristics at the inlet of the conical section have the characteristics of low central velocity, high velocity near the wall, and vortices between the two flow characteristics, which are marked in the boxes in
Figure 7a,b, based on the observations of the small-load operating point condition. This shows that flow separation occurred in the conical section under this working condition, and the hydraulic loss was significant. As can be seen when observing the high-efficiency operating point condition, there is no apparent flow separation in the draft tube. However, there are vortices at the transition from the conical section to the elbow section, and the size is asymmetrical, as shown by the boxes in
Figure 7c. Under this working condition, vortices with a certain eccentricity were generated. Vortices still exist under the maximum head condition, are concentrated on both sides of the axial line of the draft tube, and their distribution form is similar to Karman vortices, as indicated by the boxes in
Figure 7d. A spiral vortex rope can exist under these conditions. Based on the observations of the large-load operating point condition, the absolute velocity of the fluid was dominant, and the effect of the rotating motion was small. The vortices mainly appeared in the elbow section, which may have been caused by the sudden expansion of the flow passage.
In general, the overall flow condition at the maximum head was better than that at the rated head. As the head increased, the inlet flow increased, the influence of peripheral velocity decreased, and the flow became more stable. Among them, the best performance condition is the condition in the high-load area at the maximum head, and there is almost no vortex.
The physical quantity vorticity, which describes the rotation of the velocity vector, was used to express the position and shape of the vorticity, as shown in
Figure 8. Based on the figure, at the small-load operating point, vortices fill the entire draft tube. Moreover, at the high-efficiency operating point, there are eccentric rotating vortex ropes. In addition, at the large load operating point, a columnar vortex rope appears, and the influence of the vortex rope decreases as the head increases. The vorticity in the conical section of the vortex rope was high based on the observations of the shape of the vortex rope. This type of high-vorticity area (i.e., the bright yellow area in the figure) can be approximated as the vortex trajectory. The vortex ropes almost remained in the central area in the conical section, shifted to the wall with a small curvature near the elbow section, and became irregular towards the diverging section, causing interference to the outflow of the draft tube.
The vorticity distribution in the section of
y = 0 in the draft tube was analyzed, as shown in
Figure 9. The characteristics and development direction of the eddy current can be clearly observed during the flow process. At the small-load operating point, the vorticity distributions in the dead fluid area under the rated and maximum heads are symmetrical. This symmetry is damaged by diffusion and an uneven velocity distribution of the flow field at the elbow section, and the impact of the fragmentation vorticity on the flow instability is large. At the high-efficiency operating point and the large-load operating point, the vorticity has two forms (i.e., jet shape and alternating distribution around the axis), corresponding to the non-eccentric columnar vortex and spiral vortex zones, respectively. Meanwhile, the vortex rope in the conical section rotates in the opposite direction to the rotating direction of the runner under large-load conditions based on the positive and negative values of the high-vorticity area.
The vortex rope structures in the draft tube are connected to the hub in the upper part and have a certain eccentricity at the elbow section [
35], which is consistent with the calculation results in this study. The area with regular variation patterns is mainly concentrated in the conical section, as shown in
Figure 8 and
Figure 10. After passing through the elbow section, the flow became complex and irregular, and the vortex rope in the draft tube and its influence only appeared before the elbow section.
The vortex rope rotates in cycles. The vortex rope rotation for each operating point at a 91 m head is shown in
Figure 11.
Under OP1, the runner outlet backflow is significant, and a large area of stagnant water forms in the draft tube of the central area. Furthermore, due to the rotating effect of the runner, together with the stagnant area, the peripheral fluid has a circular velocity similar to the speed direction at the exit of the runner. Under the action of inertia and gravity, the fluid flows downward along the wall and the flow interferes with the stagnant zone in the center of the straight cone section of the draft tube, making it appear to rotate. Here, the flow state is turbulent, and the interaction with the draft tube wall mainly produces random pressure pulsations.
Under OP2, the draft tube vortex rope is also a short, columnar vortex rope, rotating in the same direction as that of the runner, with the radius of the vortex rope contracting and becoming thinner as the draft tube runner moves downwards. Over time, the overall shape of the vortex rope remains essentially unchanged, with rotation causing the draft tube’s elongated vortex rope to contract and become slightly eccentric. The center of rotation of the vortex rope is on the same axis as the straight tapered section of the draft tube. The eccentricity of the vortex rope at the draft tube does not cause any significant disturbance in the flow field in the draft tube section, and the magnitude of the pressure pulsation in the draft tube is small for this opening condition.
Under OP3, the draft tube vortex rope is a columnar vortex rope; its direction of rotation is opposite to that of the runner. The shape of the vortex rope is slightly distorted, but there is no eccentric deformation of the vortex rope with time or other phenomena. The disturbance of the straight cone section of the draft tube area is limited; furthermore, it is observed that there is an inertial potential energy of the fluid near the lower ring outlet of the runner, which interacts with the wall of the draft tube to produce a broken vortex rope on the wall. The vortex propagates downwards and disappears after interacting with the wall of the curved elbow section of the draft tube.