Prediction of the precessing vortex core in the Francis-99 draft tube under off-design conditions by using Liutex/Rortex method ＊

Cong Trieu Tran , Xin-ping Long Bin Ji Chaoqun Liu

1.State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan 430072, China

2. School of Power and Mechanical Engineering, Wuhan University, Wuhan 430072, China

3.Facullty of Hydraulic Engineering National University of Civil Engineering, 55 GiaiPhong, Hanoi, Vietnam

4. Department of Mathematics, University of Texas at Arlington, Arlington 76019, USA

Abstract: The turbulent flow in the draft tube of a Francis turbine is very complicated while working under off-design conditions.Although the off-design conditions were widely studied, the vortex core line in the draft tube of a Francis turbine with splitter blades is not well understood, especially the vortex rope property.This letter presents a prediction of the behavior of the vortex rope in the draft tube of the Francis-99 turbine obtained by the computational fluid dynamics (CFD), where the Liutex/Rortex method, as the most recent vortex definition, is applied to analyze the periodical precession of the vortex rope in the draft tube cone.The advantage of this Liutex/Rortex method is shown by its enhanced ability to represent the vortex rope structurewith the vortex-core lines.Furthermore,since it seems to be very hard to define a sharp boundary surface for the whole vortex structure, it is advantageousfocusing only on the vortex core line,by which different vortex structures can be clearly differentiated.The evolution of the vortex core and the process of the vortex breakdown in the draft tube are revealed,which might help to comprehend the development of the turbulent flow in the draft tube.

Key words: Cavitating vortex rope, Francis turbine, Liutex/Rortex method, off-design condition, vortex core line

The Francis turbine operating far from the regime of the best efficiency is characterized by the abnormal flow in the draft tube and the appearance of a spiral vortex or columnar vortex, called the vortex rope.Arpe et al.[1]found that the dominant frequency of a vortex rope lies between 0.2 and 0.4 times of the runner frequency.A good understanding of the periodical precession of this vortex, as well as the vortex rope structure in the draft tube, is essential for preventing structural vibrations and increasing the number of operation hours under off-design conditions.Nevertheless, detailed characteristics of the vortex structures with accurate visualizations are a challenging task.

Several attempts were made to capture the vortex structures in the draft tube of the Francis turbine.Gavrilov et al.[2]focused on detecting and analyzing the vertical structures and the evolution of the vortex core at deep partial-load points (with a flow rate of only 35%), using two URANS models and a hybrid LES/RANS method, where the vortex structures are visualized by theλ2[3]andQ-criterions[4].However,their results are strongly influenced by the choice of the threshold values, which will result in different vortex structures for different threshold selections.In particular, using theλ2andQ-criterions, one may obtain non-physical vortex structures, and the “vortex breakdown” for some large thresholds while obtaining no“vortex breakdown” for some smaller thresholds[5].These will make the physical explanation of the turbulent flow futile.Liu et al.[6]proposed the Omega method (Ωmethod), which is not sensitive to the threshold selection and can successfully capture both strong and weak vortices at the same time.However, theΩmethod has also some limitations like the introduction of an uncertain parameter of epsilon （ε）[6-7].

In this letter, the vortex structures are identified by different methods, with several methods for extracting a line feature called the vortex core line.For instance, the vorticity is a traditional and common indicator for the presence of vortices.However, this technique has some limitations including: (1) sensitive to other non-local vector features, (2) producing no contiguous lines[8].Recently, a new vector called“Liutex/Rortex” is introduced by Liu et al.[9-10]to describe the local rigid rotation of fluids.This method,which provides not only Liutex/Rortexiso-surfaces but also the different strength along with the Liutex cores,can be used to analyze the process of the vortex generation and development[11-12].The Liutex/Rortex method has not yet been applied to investigate the complex fluid flows such as the cavitation flow in the Francis turbine in literature.Hence, to overcome the difficulty of a more accurate visualization of the vortex structures, a well-defined method such as the Liutex/Rortex method helps the vortex identification in the Francis turbine.

Inspired by the above idea, this letter focuses on identifying and clarifying the precessing vortex rope in the draft tube of the Francis-99 turbine[13]under the off-design conditions.By using the shear-stress transport turbulence model (SST)[14]and the Zwart-Gerber-Belamri (ZGB) cavitation model[15], the cavitating flow in the draft tube is simulated, and the vortex structures in the draft tube cone are visualized by the Liutex/Rortex method.The periodical evolution of the vortex rope in the draft tube is further revealed with the Liutex core line.

The numerical configurations are set according to the Francis-99 model turbine, consisting of a runner with 15 long blades and 15 splitter blades, a spiral casing, 28 guide vanes, and a draft tube (as shown in Fig.1).The three operating points: the high load (HL),the best efficiency point (BEP), and the partial-load(PL) are created with ANSYS ICEM CFD using the ICEM files provided by the second workshop[13].The mesh size for the complete model at the BEP is about 20×106elements.The quality of the mesh satisfies the common industrial standard, as reported by Trivedi et al.[16], Goyal et al.[17].In this study, the complete turbine is simulated in two steps, including the steady and unsteady simulations.In the unsteady simulations,the initial field is obtained from the steady simulations.

The mass flow inlet boundary is set at the casing inlet, and the static pressure is set at the draft tube outlet.The runner is assumed as a rotating part while the casing, the stay vanes, the guide vanes, and the draft tubes are assumed as stationary parts.The rotational speed of the runner is set as 332.59 rpm and the components are connected with others by the domain interface.The general grid interfaces (GGI)connect the stationary domains to the rotating domain.

Fig.1 Simulation domain of Francis-99 turbine

The SST turbulent model is a widely used turbulence model for the turbo machinery[18-19].In this study, the SST turbulence model and the ZGB cavitation model are adopted for the simulation of the unsteady cavitating flow ina Francis-99 turbine.To investigate the cavitating vortex rope, the unsteady simulation is performed for ten complete rotations of the runner, which takes a total computation time of 1.8 s.The time step is set ast=5× 10-4s (10th of the runner rotation per time step).The convergence criterion is set to a root-mean-square (rms) value maximum 10-5.

To validate the simulation method used in this study, Table 1 shows a comparison of the hydraulic efficiency and the torque obtained by the simulation with the SST model and those obtained by experiments under three operating conditions.The maximum discrepancies between the simulated and experimental efficiencies are 4.24% under the PL operating condition, 3.05% at the BEP and 2.65% at the HL.The numerical torque is set as 465 Nm (PL),706 Nm (BEP), and 820 Nm (HL), higher than the experimental torque and the ratio of the experimental torque to the numerical torque is 11.7%, 14.6%, and 10.7%, respectively.During the simulation, the numerical efficiency is higher than the experimental efficiency at all operating points because the flow leakage losses and other losses during the measurements are not considered in the numerical simulation.In the numerical simulation, the mesh quality, the vortex, and the flow separation may be the cause of inaccuracy in the torque calculation.Taking into account the above comparison, the overall accuracy of the simulation is acceptable.

For further clarification of the reliability of the Liutex/Rortex method for the vortex definition in the turbulent flow, in the present study, the vortex rope frequency is investigated.The unsteady pressure at two levels DT5 and DT7 (see Fig.2) are plotted in Fig.3.The analysis of the vortex rope morphology is made during a low-frequency period in order to examine itsdynamics.As a result, a low-frequency cycle of 0.6 s(1.66 Hz) is observed.And to reveal the time evolution of the vortex rope, six snapshots with a time step of 0.1s are plotted in Fig.4 by using the Liutex/Rortex iso-surface.The pressure amplitudes are consistent to a vortex rope frequency of 1.66 Hz(about 0.3 times of the runner frequency).

Table 1 Comparison between numerical and experimental values of turbine energy characteristics

Fig.2 (Color online) Side view of the Francis-99 draft tube cone

Fig.3 (Color online) Unsteady pressure at two levels DT5 and DT7 on the cone, based on 3-D numerical simulation with SST model

Here, the vortex rope is compressed during the first phase of the low-frequency cycle, after that, one sees its stretching, breakdown, shedding, and moving downstream.The period in the low-frequency case is related with the pressure fluctuations associated with the precession of the vortex rope.

On the other way, by makinga fast Fourier transform (FFT) of the results, the dominant frequency of the pressure fluctuations can be obtained.The frequency spectrum obtained from the present simulations under the PL condition at the DT5 pressure monitoring point is shown in Fig.5.The vortex rope frequency is found to be about 0.3 times of the runner frequency.The result is consistent with the value numerically obtained by Arpe et al.[1]and is in very good agreement with the value of 0.294 as observed in the experimental studies[20].The frequency of the vortex rope is obtained by the pressure fluctuations and the Liutex method, see Table 2, and compares well with the experimental result.

Fig.4 (Color online) The distribution of uy of solid wall vary with time changing

Fig.5 (Color online) Pressure fluctuation frequency under the PL condition from cavitating flow analysis at DT5 monitor pressure point

Table 2 Comparisons of the frequency of the vortex rope obtained by the pressure fluctuations and Liutex method

The vortex rope structure is composed of two different parts: the vortex core centre line and itssurrounding regime.The dynamics of the vortex core line will further illuminate the mechanisms behind these observations.According to Liu et al.[9-10],the vortex core line is defined as a Liutex line which passes the points satisfying the condition of?R×r=0,R＞0 whererrepresents the direction of the Liutex vector.This definition is used to find the Liutex (vortex) core lines in the flow field,which is uniquely defined without any threshold requirement[11-12].

Under the PL condition, Fig.6 visualizes a series of snapshots of the vortex core lines obtained by temporal evolutions in one cycle.The vortex core lines are colored by the Liutex magnitude, chosen as an indicator of the vortex strength.The picture shows significant motions of the vortex core as it rotates with the precession frequency and it is described by a stream line.The vortex core line is shown as a conical spring with a variable helix angle Therefore, the vortex structure and the precessing vortex core are represented as a unique morphology by the Liutex core lines.

Under the HL condition, Fig.7 visualizes a series of snapshots of the vortex core lines obtained by temporal evolutions in one cycle.When the vortex rope occurs under the HL condition, the core is centered in the draft tube cone.Figure 7 shows that the Liutex vortex core line move sover time from the runner outlet centre to the downstream with different vortex strengths (red/blue color), described by a stream line.The movement of the vortex core segments with different strengths reflects the change of the pressure distribution in the draft tube cone,which causes the pressure fluctuation with a smaller frequency.These pictures are clearly shown in the FFT analysis of the unsteady wall pressure signals measured at the DT5, as shown in Fig.8, which would not be observed if the traditional vortex definition methods are used.

In addition, since it seems very hard to define a sharp boundary surface for the whole vortex structure,we focus only on the vortex core line with the advantage that different vortex structures can be clearly distinguished.With the Liutex methods, the vortex core lines are more stable, as they are at the center of the vortex where it could be very clearly identified.

Based on the above visualization illustrated by the Liutex core line for the turbulent flow in the draft tube of the Francis-99 turbine and the following conclusions can be made:

First, the Liutex/Rortex method is verified to be able to successfully represent the structure and the process of the vortex rope in the turbulent flow in the Francis turbine.The process of the vortex breakdown under the off design conditions operation is shown with the use of the Liutex/Rortex method

Fig.6 (Color online) Vortex core structure evolution in one cycle under the PL condition

Fig.7 (Color online) Vortex core structures evolution in one cycle under the HL condition

Fig.8 (Color online) Pressure fluctuation frequency underthe HL condition from cavitating flow analysis at DT5 monitor pressure point

Second, from the variation in time of the precessing Liutex core, the vortex core line in a draft tube under the off-design conditions can be described by stream lines properly.

Finally, by properly extracting the Liutex core line from the unsteady 3-D velocity field, it is revealed that a periodic sequence of the vortex precessing movement is observed at the draft tube cone under the off design conditions.

Acknowledgments

The numerical calculations in this study were done on the supercomputing system in the Supercomputing Center of Wuhan University, Wuhan,China.The authors would like to thank the organizers of the Francis-99 second Workshop for the use of their CAD, meshing, and experimental data.