Study of the effects of modified draft tube with inclined conical diffuser on draft tube and upstream region

Various countermeasures including geometry optimization have been proposed and proved to have a huge impact on flow filed within the draft tube in order to mitigate the vortex rope and the induced pressure fluctuations for Francis turbine under part load operation. However, the effect of these approaches on the hydro-dynamics in upstream region still remains unclear, which is of great significance for the overall performance of the unit. This study aims to explore the influences of modified draft tube with inclined conical diffuser on the pressure fluctuations in whole flow passage. The results reveal that Generation-1 and Generation-2 draft tubes are effective in alleviating pressure fluctuations resided in the draft tube, but the former one would trigger a low-frequency pressure fluctuation with higher amplitude in the runner zone. In addition, the modified draft tube has a very limited effect on the high-frequency pressure fluctuations in the guide vane and vaneless areas. To eliminate the adverse effect of inclined conical diffuser, a design with a transitional section is put forward to smoothly connect the runner zone and inclined conical diffuser. This further developed Generation-2 draft tube presents a reasonably good performance in terms of improving flow stabilities, i.e. alleviating the pressure fluctuations not only in draft tube but also in upstream region such as runner zone. This study provides reference for obtaining a better mitigating effect on pressure fluctuations in the whole turbine flow passage.


Introduction
It is well known that the vigorous development of renewable energy across the world has played a positive role in climate mitigation and carbon emission reduction [1].Over the past few decades, hydropower has shown dominance in the total renewable energy production, since hydropower owns the advantages of mature technology, relatively low cost, as well as the good quality of generated power [2].However, the proportion of hydropower has been decreased in recent years; on the one hand, the economically exploitable capacity is limited for a certain locality, on the other hand, the worldwide electricity market has witnessed an increasing penetration of new energy sources.Due to the inherent intermittency and volatility presented by wind, solar, geothermal, among others, the quality of produced power is poor, even worse, their large-scale integration into the power grid will impact the stability of power grid, resulting in the oscillations of power system and splitting of the power generator.Confronted with this challenge, the fast peak and frequency regulation provided by hydropower has been given full play, that is, main tasks that hydropower performs have been substituted by being a regulatory energy source, including pumped storage and cascade hydropower stations in various basins [3,4].
During the normal operation of hydro unit, the turbine is usually operated with the highest efficiency under rated conditions, thus the stability of hydropower station can be guaranteed.Nevertheless, when hydropower is utilized as a regulatory role, hydro turbine is required to switch back and forth and operates over a wide operating range, which gives rise to two main issues: transient instability during the change of guide vane opening and flow instability under off-design operating conditions.With regard to the former, many researchers have conducted extensive investigations on hydrodynamic behaviours such as pressure fluctuation under transient processes, and put forward the optimization of closure law of guide vanes to improve transient characteristics [5,6].For the latter one, many countermeasures have been proposed and studied, such as air admission or water jet, installation of fins, baffles or J-groove on the conical diffuser of draft tube, modified runner cone and so on, which can be further divided into active, passive control methods and draft tube shape optimization [7][8][9][10][11][12][13][14].
As hydraulic turbines are forced to operate under off-design conditions more often than before, it is pressing to change the traditional design concept of draft tube which only aims to improve the peak efficiency [15].In other words, in this context it is indispensable for the designer to comprehensively increase the efficiency of turbine in wide range of operating regimes and to extend the range of stable operation to part load and full load conditions [16].In an earlier study, the authors have proposed a novel modified draft tube with inclined conical diffuser, their results indicated that this kind of modified draft tube was effective in mitigating vortex rope and consequent pressure fluctuations [17,18].However, the optimum configuration of draft tube has not been determined yet, in addition, the influence of modified draft tube on the flow instabilities upstream has not been explored either.
In this paper, the goal is to alleviate the flow instability within the draft tube under partial load conditions by virtue of optimizing the geometry of draft tube.The traditional elbow-type draft tube was redesigned and was labelled as Generation-1 and Generation-2, the difference between the two modified draft tubes lies in the addition of a transitional section for the latter design to smoothly connect the runner zone and inclined conical diffuser, so as to ease or eliminate the adverse effect of inclined conical diffuser.To study this type of modified draft tube in depth, a series of pressure monitoring points were installed in the modified draft tube wall to verify their effect on the mitigation of pressure fluctuations.Other recording points were set up in guide vane, vaneless and runner areas to investigate the influence of modified draft tube on flow instabilities in upstream region.

Geometry of turbine and modified draft tube
A Francis turbine with medium specific speed is investigated as the research object, which consists of four sub-domains including spiral case, tandem cascade, runner, and draft tube.The whole threedimensional geometry of turbine is displayed in Figure 1 and its main geometrical specifications are listed in Table 1.The traditional elbow draft tube was redesigned and two modified draft tubes are labelled as Generation-1 and Generation-2 after conducting geometric optimization.With regard to Generation-1, the concave and convex sides of the elbow wall could be described by a uniform sigmoidal curve, as given by Eq. (1) and Eq. ( 2), respectively, which were obtained in previous numerical optimization [18].The latter configuration of draft tube is proposed for the first time, which focuses on the smooth connection between runner outlet and draft tube inlet with a small circular arc on the upstream side, in order to guide the water flow through a transitional symmetrical section before entering the inclined conical diffuser.Hybrid meshes consisting of 4.12×10 6 elements were generated for different sub-domains by means of commercial code ANSYS ICEM, the minimum quality of the elements adopted in the following numerical simulation was kept above 0.25 to provide relatively reliable results.
Where the origin of the local coordinate system was placed at the starting point of each curve.

Simulation set-up
The widely used computational fluid dynamics (CFD) software, Fluent, was adopted in this study to resolve three-dimensional single-phase incompressible flow in the turbine passage, numerical simulations were performed using Reynolds-averaged Navier-Stokes equations supplemented with RNG k-H turbulence model.The reason of adopting RNG k-H turbulence model consists in a relatively small error in the calculation results while demanding less computational resources for large-scale numerical simulations.It can be observed from grid sensitivity analysis that the difference between the results from RNG model and SST k-ω is negligible and acceptable.The partial-load operating point under rated head was selected for simulation and subsequent investigation of flow instability within flow passage, the boundary conditions at spiral case inlet and outlet of the draft tube were set as total pressure (1.18 MPa) and static pressure (0.04 MPa) respectively to obtain a constant rated head of 116 m.The partial-load conditions could be achieved by varying guide vane opening from the rated flow condition (Qbep), the studied volume flow rates was 56.38 m 3 /s, i.e. 64%Qbep.A non-slip boundary condition was applied to the walls [19,20].The SIMPLEC algorithm was chosen to achieve the coupling of velocity and pressure equations.A second-order upwind scheme was used for convection discretization as well as a central difference scheme for the diffusion terms in the momentum equations.In the course of numerical simulations, steady and unsteady calculations were performed successively, the results of the steady simulations were adopted as the initial conditions for the subsequent unsteady simulations.The multiple-reference-frame model and sliding mesh approach were employed to simulate the rotor-stator interaction between runner and guide vanes (and draft tube) in the steady and unsteady simulations, respectively.In the unsteady simulations, a time step of 0.0005 s equal to 0.75 of runner rotation was chosen to capture a wide frequency band of pressure fluctuations.The total calculation time of unsteady simulations was set to 10 s, equivalent to about 41 revolutions of runner.

Validation of numerical results
As displayed in table 2, the accuracy and reliability of numerical simulation carried out in this study is verified by comparing the efficiency calculated from simulation with that of the model test, which is obtained from the hill chart of the model turbine.It can be discovered that the relative error γ is 1.69% for the traditional draft tube, which is acceptable in the numerical simulation of large-scale hydraulic machinery.For the two different modified draft tubes, the relative deviation of efficiency against the simulation results is less than 2%, note that Generation-2 of modified draft tube proposed in this study exhibits better performance in terms of turbine efficiency.

Vortex rope and internal flow field
The iso-surfaces of Q-criterion were utilized to illustrate the evolution of vortex rope in this paper, The definition of Q in Cartesian coordinate is defined as follows: Where u, v and w refer to velocity components in x, y and z directions.Typical vortex ropes with periodic precession based on the iso-surface of Q=120 2 s are presented in Figure 2 for the three different draft tubes.As the turbine is operated under part-load condition, the rotation direction of the vortex rope is clockwise, being consistent with runner rotation.Based on the pattern of vortex rope presented in the same time interval, its period of precession equals to 0.75s, corresponding to a frequency of 1.3Hz for the traditional type.Among the three types of draft tube, vortex rope demonstrates significant difference in terms of shape and volume.With regard to the traditional draft tube, the thick vortex rope dominates a large portion of draft tube around the central axis and there exists vortex breakdown in one cycle.In contrast, although the change of length of vortex rope for Generation-1 draft tube is negligible, the radius of precessing vortex core is smaller, resulting in a reduction in the volume of vortex rope.Regarding the Generation-2 draft tube, the length of vortex rope is significantly decreased, leading to a smaller vortex volume, while with more induced vortex shedding in the elbow region.To further demonstrate the impact of the modified draft tube on internal flow, the turbulent kinetic energy on the meridian plane of draft tube is shown in Figure 3.It can be found out that the distribution of turbulent kinetic energy in the conical diffuser of traditional draft tube is not as uniform as that in the modified ones, accompanied by chaotic streamlines.In the elbow section, the turbulent kinetic energy is evenly distributed in Generation-1 draft tube, while the values for Generation-2 and traditional draft tubes are large and uneven.Acquired frequency and amplitude of pressure fluctuations were normalised using the following two Equations: Where E p denotes the pressure fluctuation factor, ( ) p t ( ) p( is the unsteady pressure in Pa, ( ) p t is the timeaverage pressure in Pa, U is the water density in  5.It is evident that the pressure fluctuations within the two modified draft tubes are prominently less intense than those within the traditional type.For the traditional type, the dominant frequency factor of pressure fluctuations is 0.32, which agrees with the value of 0.2-0.4(Rheingans frequency) in previous studies.While with regard to the Generation-1 and Generation-2 draft tubes, the frequency factor appears to be 0.16 or 0.08 for certain recording points, being 1/2 or 1/4 of 0.32, which could be explained by a weaker effect of vortex precession on the downstream sides and resultant lower frequency of pressure fluctuation.One interesting phenomenon is that the frequency factor of pressure fluctuations on upstream draft tube wall remains 0.32 for Generation-1 and Generation-2, the difference between the pressure fluctuation on the upstream and downstream walls is probably resulted from the disturbance of asymmetric wall boundary conditions.In general, In terms of pressure pulsation amplitude, the value for modified draft tube is significantly reduced by comparison with traditional one, although the weakening degree is different for different monitoring points.In order to have a better understanding of the comprehensive influence on pressure fluctuation for the modified draft tube, the maximum amplitudes of pressure fluctuations for all monitoring points are displayed in Figure 6.Among the three types of draft tube, maxima value of pressure fluctuations belong to the traditional draft tube, followed by the Generation-2 and Generation-1 draft tubes.It should be noted that for T12, the amplitudes of pressure fluctuations within the Generation 1 and Generation-2 draft tubes are larger than that within the traditional one.In terms of maximum amplitudes, the reduction is 64.73% and 42.26% for Generation-1 and Generation-2 draft tubes, respectively.Considering all the monitoring points, although the mitigating effect on pressure fluctuation of Generation-2 is not as good as Generation-1 on downstream side, a better alleviation of pressure fluctuations can be observed on the upstream wall.It can be concluded that the Generation-1 and Generation-2 draft tubes share a similar mitigating effect on mitigating pressure fluctuations within the draft tube.

On the upstream region
As shown in Figure 7, a series of monitoring points were set up in the upstream area of draft tube and they can be classified as follows: (1) Three monitoring points were created in the volute, which were uniformly distributed along the flow direction.
(2) A total of 8 monitoring points were installed in the tandem cascade, among which two were located at the top and bottom of the stay vane (FV1, FV2), other two were arranged on the two sides of the bottom part of stay vane (FV3, FV4).For guide vane, the principle of setting monitoring points is similar to that for stay vane, including 4 points (GV1, GV2, GV3, GV4).
(3) Two monitoring points were created on the vaneless zone, which were located at the upper and lower parts of vaneless zone (VL1, VL2).
(4) A total of 10 monitoring points were set up on the runner: 6 points on the pressure side (R1p, R2p, and R3p) and suction side (R1n, R2n, and R3n) located from the leading edge to the trailing edge; 4 points on the hub (H1 and H2) and shroud (S1 and S2)).The maximum pressure fluctuation amplitudes of all monitoring points on upstream area of draft tube are displayed in Figure 8 to gain more insight into the overall impact of modified draft tube on the upstream flow instabilities.For different types of draft tube, the difference of pressure fluctuation amplitudes in spiral case and tandem cascade is relatively small, indicating that the influence of draft tube modification on the pressure fluctuations within these two regions which are far away from draft tube is negligible.By contrast, the modified draft tube does have a strong influence on pressure fluctuations resided in the vaneless area and runner regions, whose dominant frequency corresponds to high frequency ( ) respectively.It can be discovered that the modified draft tube has a stronger influence on the low-frequency pressure fluctuations compared to the high frequency one, as the influence of draft tube modification on the upstream flow instabilities is achieved by weakening the harmonics of pressure fluctuations generated by the precession of vortex rope, furthermore, the decrease in high-frequency fluctuation amplitude is resulted from the modulation with the already weakened low-frequency pressure fluctuations.Generation-2 performs better than Generation-1 in the respect of alleviating pressure fluctuations in the runner zone, since Generation-1 triggers low-frequency pressure fluctuations with higher amplitude than traditional draft tube; while the further developed Generation-2 draft tube presents lower pressure fluctuation amplitude than the other two types of draft tube.In conclusion, Generation-2 draft tube with a transitional section to smoothly connect the runner zone and inclined conical diffuser exhibits a reasonably good performance in terms of improving flow stabilities within the overall flow passage, more accurately, is able to mitigate the pressure fluctuations not only in draft tube but also in upstream regions such as runner zone.

Conclusions
The mitigating effect of modified draft tube with inclined conical diffuser on vortex rope and consequent pressure fluctuations in comparison with traditional elbow-type draft tube is investigated in this paper.The two modified draft tubes are labelled as Generation-1 and Generation-2, the latter is highlighted with a transitional section to smoothly connect the runner zone and inclined conical diffuser.The relative error γ in turbine efficiency is 1.69% for the traditional draft tube, which is acceptable in the numerical simulation of large-scale hydraulic machinery and served as a validation of presented simulation results.The periodic precession of vortex ropes based on iso-surface of Q indicates that the precession period of vortex rope is 0.75s, corresponding to the low frequency of pressure fluctuations within draft tube.Regarding the volume and length of vortex rope, the vortex rope in Generation-2 is significantly decreased, accompanied by evenly distributed turbulent kinetic energy and uniform streamlines.The unsteady pressure measurements were performed on both draft tube and upstream region, and the results reveal that Generation-1 and Generation-2 are effective in alleviating pressure fluctuations resided in the draft tube.For Generation-1, better alleviation of pressure fluctuations can be observed on the downstream side; while a larger reduction in upstream region accompanied with a fairly large increase in pressure fluctuation amplitude on certain spot is discovered in Generation-2.Taken pressure fluctuations in upstream regions into account, low-frequency pressure fluctuations with higher amplitude than traditional draft tube are triggered for Generation-1; whereas the further developed Generation-2 draft tube presents lower pressure fluctuation amplitude than the other two types of draft tube.It can be concluded that Generation-2 draft tube is superior to the Generation-1 in mitigating pressure fluctuation not only in draft tube but also in upstream regions such as runner zone.

Figure 1 .
Figure 1.3D geometry of turbine and configuration of modified draft tubes.

Figure 2 .
Figure 2. Periodic behaviour of vortex rope in one period (T=0.75 s) for three different draft tubes.

Figure 3 .
Figure 3. Turbulent kinetic energy on the meridian plane of draft tube.

4 .
Pressure fluctuations within flow passage4.1In the draft tubeA total of 14 recording points were installed on the draft tube wall, in which seven points were located on the upstream and downstream sides, respectively.The locations of monitoring points for different types of draft tube are shown in Figure4.The principle of setting monitoring points is to keep the elevation and spacing of monitoring points the same, for a better comparison between different configuration of draft tube.

2 Figure 4 .
Figure 4. Recording points in traditional and modified draft tubes.

3 /
kg m , E f refers to the frequency factor, f is the frequency in Hz, runner f is the runner rotational frequency and equals to 4.167 for the studied turbine.The obtained time domain pressure fluctuations and corresponding frequency domain diagram with Fast Fourier Transform for the four monitoring points T1 ~ T5 on the downstream side of three types of draft tube are presented in Figure

Figure 5 .
Figure 5. Pressure fluctuations at different recording points for traditional and modified draft tubes.

Figure 6 .
Figure 6.Comparison of maximum pressure fluctuation amplitudes at different points along flow direction.

Figure 7 .
Figure 7. Layout of pressure recording points on upstream area.

Figure 8 .
Figure 8.Comparison of pressure fluctuation amplitudes at different points on upstream regions.

Table 1 .
Geometrical specifications of the studied turbine.

Table 2 .
Hydraulic efficiency of turbine with traditional and modified draft tubes under rated water head.