Spatial distribution of rigid vorticity in pump turbine under turbine mode with different heads

Pump turbines under off-design operating conditions can generate significant vortex structures that cause hydraulic instability in the unit. The aim of this study is to investigate the spatial distribution of large-scale rigid vortex structures in the runner blade channels of a model pump turbine operated under different head conditions with fixed guide vane opening in turbine mode. A new method is proposed that utilizes a relative streamline coordinate system based on mean camber line and provides a unique perspective to quantify the spatial distribution and intensity of various physical variables within the runner blade channels from a novel perspective along the streamwise, circumferential, and spanwise directions. Ultimately, an analysis is conducted on the causes of rigid vortex structures. This paper provides a novel and advanced research technique to analyse unstable flow structures in pump turbines.


Introduction
As a central component of pumped storage power stations [1], the hydraulic performance of fluid machinery such as pump turbines has been extensively investigated with model tests and numerical simulation methods [2][3][4][5].Under off-design conditions in turbine mode, pump turbines are prone to unstable flow structures in the runner region, leading to the hydraulic instability characteristics of the unit.Therefore, it is necessary to investigate the vortex structures and their spatial distribution in the runner blade channels under different operating conditions.
Vortices are one of the key aspects in the study of turbulence, and accurately definition and identification of vortices remains an important research topic.In the classical theory of vortex dynamics, the rotational motion of a fluid is represented by vorticity [6].However, research has shown that there is no necessary relationship between the magnitude of vorticity and the strength of fluid rotation [7].Therefore, based on vortex decomposition theory [8], Liu et al. [9,10] extracted the part that characterizes the rigid rotational motion of the fluid from the vorticity and named it rigid vorticity, while the remaining part that characterizes the anti-shear motion of the fluid called shear vorticity.Furthermore, Wang et al. [11] have provided an explicit formula for rigid vorticity, and several scientists have verified the reliability of the vortex identification method based on rigid vorticity [12,13].
In addition, under the ideal assumption of infinitely thin and infinitely numerous blades in runner, the streamlines in the runner blade channels can be approximated as blade mean camber lines of the blade that are uniformly and axially symmetrically distributed along both spanwise and circumferential directions within the blade channels.Therefore, this paper proposes a relative streamline coordinate system based on the mean camber line for analysis of unstable flow structures, which can quantitatively describe the spatial distribution and intensity of rigid vortices and related physical variables in streamwise, circumferential and spanwise directions of the blade channels from a novel perspective.Thus, rigid vorticity and the relative streamline coordinate system can provide a new and advanced theoretical tool and research technique for the characterization and analysis of unstable flow structures in pump turbines.

Governing equations
The continuity equation and the momentum equation for steady, incompressible fluid flow are as follows: ( ) Where  �  、  �  are the components of the time-averaged velocity, m/s;   、  are the Cartesian coordinate components, m; ρ is the density, kg/m 3 ; t is the physical time, s;  � is the time-averaged pressure, Pa; μ represents the dynamic viscosity, Pa•s;   represents the subgrid-scale stress, Pa.

Vorticity decomposition theory
According to the vorticity decomposition theory, the vorticityωcan be decomposed into the rigid vorticityωR, which characterizes the rigid rotational motion of the fluid, and the shear vorticityωS, which characterizes the fluid's anti-shear motion: (3) The direction ofωR is the real eigenvector r of the velocity gradient tensor ▽V, which represents the local rotation axis of the fluid parcel.Its magnitude is twice the angular velocity of the fluid packet's rigid rotation, which is the absolute rotational strength.Wang et al. [10] provided an explicit formula for theωR: Whereλci is the imaginary part of the complex conjugate eigenvalues of ▽V.

Geometric Model
The computational domain of the model pump turbine is shown in Figure 1, and its main parameters in turbine mode are presented in Table 1.

Grid Generation
The runner and guide vane regions were meshed using ANSYS-TurboGrid commercial software, while the rest of the computational domain was meshed using ANSYS-ICEM commercial software.Structured hexahedral meshes were used in all domains, and the boundary layer regions were refined for accurate simulation.The y+ values of the runner blades and guide vanes are maintained around 30 for the design condition.Rated discharge Qr(m3 /s) 0.32

Experimental Validation
The guide vane opening was kept at 24mm, which is the optimum opening for the pump turbine in turbine mode.By varying the inlet flow rate, simulations were performed to compute the hydraulic performance at different heads and they were validated against the experimental results of the full properties of the model.The comparison is shown in Figure 2. Q represents the mass flow rate, H is the head, and η donates the efficiency.The subscript BEP represents the best efficiency operating condition.
The relative errors in the simulated head and efficiency at each flow rate were all less than 5%, indicating that the numerical results are reliable.

The relative streamline coordinate system
As shown in Figure 3, local coordinate systems were established along the streamwise, circumferential, and spanwise directions.For each blade channel, the streamwise coordinate (s) was set from 0.0 to 1.0 along a streamline from runner inlet to outlet; the spanwise coordinate (span) was set from 0.0 to 1.0 from hub to shroud; and the circumferential coordinate (c) was set from 0.0 to 1.0 from the suction surface side to the pressure surface side of the blade.This enables for the observation of changes and spatial distributions of physical quantities along the streamline.

Spatial distribution of rigid vortex structures
The analysis was conducted for the low head condition represented by 0.6QBEP, the optimum condition represented by 1.0QBEP, and the high head condition represented by 1.4QBEP.The distribution of vorticity and rigid vorticity along the streamline for each condition is shown in Figures 4 to 6  (d) Vorticity in span0.2;(e) Vorticity in span0.5;(f) Vorticity in span0.8.In the optimal condition, the amplitude of the rigid vorticity is significantly lower.When the head deviates from the optimal condition and decreases, a large fluctuation in vorticity strength and rigid vorticity strength appears along the streamline, which indicates the existence of numerous local vortex structures between the runner blade channels.When the head deviates from the optimal condition and increases, the flow influence on the blade pressure surface intensifies, and strong vortex structures are generated near the blade suction surface.The vorticity distribution and the contour of the rigid vorticity distribution of the selected turbine blade channel are shown in Figures 7 to 9. When the head deviates from the optimal operating condition and decreases under the mode of the turbine, the smoothness of the spatial distribution of vorticity decreases due to increased number of low-pressure regions inside the blade channel, which leads to the occurrence of strong flow separations and small-scale vortex structures.
Figure7.Vorticity distribution within the runner blade channel at low head operating condition.In the optimal operating condition, the angle between the direction of flow and the blade angle is small, in which leads to a weaker flow separation.As a result, the total strength of the vorticity and the rigid vorticity is less, and the distribution of the vorticity is smoother and more even.There is a distinct high-amplitude vorticity and low-amplitude rigid vorticity on the blade wall region, reflecting the wall shear effect.On the side of the blade pressure surface e near the inlet edge, the vorticity is in the mediumamplitude range, while the rigid vorticity is in the high-amplitude range, reflecting the rigid rotation of the fluid.When the head increases, according to the velocity triangle of hydraulic machinery, the water flow meets the blade pressure surface near the inlet edge, resulting in a high-speed flow region, while the suction surface side creates low-pressure regions and flow separations.High vorticity regions appear on both the pressure and suction surface of the blade near the inlet edge, while high-amplitude rigid vorticity only appears only on the suction surface side, indicating that rigid vorticity is those formed by flow separation near the suction surface vertebrae can be precisely identified.The high-speed flow near the pressure surface will form a certain shear velocity gradient, but will not form vortices.Vortices accompany shear flow to develop and disappear within a certain range, while non-uniform shear flow can propagate a long distance in space.

Conclusions
Based on the theory of rigid vorticity, this article investigates the spatial distribution of large-scale rigid vortex structures in blade channels of a model pump turbine under different head conditions in turbine mode.The following conclusions are drawn: • The use of a relative streamline coordinate system based on the blade mean line of curvature allows for the observation of changes in the physical variables of the flow field from the perspective of the streamline, providing a novel and advanced research technique for the analysis of unstable flow structures in pump turbines.• Vorticity tends to have areas of high amplitude regions near the wall, and their overall distribution within the channels is characterized by large and uniform amplitudes.This is attributed to the shear flow caused by the boundary layer or convective flow.Conversely, the rigid vorticity approaches zero in the vast majority of the near-wall regions and exhibits a clearer and more distinct distribution within the channels, significantly avoiding the influence of shear flow.This allows for a clear visualization of the vortex structure and its rigid rotational strength.• Under optimal operating conditions, both the vorticity and the rigid vorticity have small low amplitudes and their distribution is smoother and more uniform.However, when the head deviates from the optimum condition and decreases, a large amount of fluctuation in the vorticity and the rigid vorticity occurs along the streamline, and the smoothness of the spatial distribution decreases.This leads to strong flow separation and local vortex structures within the runner blade channels.Moreover, when the head deviates from the optimal condition and increases, the impact of the flow on the blade pressure surface is increased, and the strong vortex structures are generated on the blade suction surface near the inlet.Regardless of the operating conditions, the vortex structures within the runner blade channels generally appear near the hub side of the runner and the pressure surface side near the inlet, which can provide a reference for optimizing of the hydraulic performance of the runner.

Figure 1 .
Figure 1.Entire computational domain of the model pump-turbine operated in turbine mode.Table 1.Main parameters of the model pump-turbine operated in turbine mode.Parameter Value Runner inlet diameter D1(m) 0.443 Runner outlet diameter D2(m) 0.250 Number of runner blades 9 Number of guide-vanes 20 Specific speed nq 41 Rated speed nr(rpm) 1462 Rated working head Hr(m) 55

Figure 2 .
Figure 2. Validation of the external characteristics.

Figure 3 .
Figure 3. Schematic diagram of the relative streamline coordinate system. .

Figure 4 .
Figure 4.The distribution of vorticity in the relative streamline coordinate system at low head condition (a).Rigid vorticity in span0.2;(b).Rigid vorticity in span0.5;(c).Rigid vorticity in span0.8;(d)Vorticity in span0.2;(e) Vorticity in span0.5;(f) Vorticity in span0.8.In the optimal condition, the amplitude of the rigid vorticity is significantly lower.When the head deviates from the optimal condition and decreases, a large fluctuation in vorticity strength and rigid vorticity strength appears along the streamline, which indicates the existence of numerous local vortex structures between the runner blade channels.When the head deviates from the optimal condition and increases, the flow influence on the blade pressure surface intensifies, and strong vortex structures are generated near the blade suction surface.

Figure 8 .
Figure 8. Vorticity distribution within the runner blade channel at optimal operating condition.When the head increases, according to the velocity triangle of hydraulic machinery, the water flow meets the blade pressure surface near the inlet edge, resulting in a high-speed flow region, while the suction surface side creates low-pressure regions and flow separations.High vorticity regions appear on both the pressure and suction surface of the blade near the inlet edge, while high-amplitude rigid vorticity only appears only on the suction surface side, indicating that rigid vorticity is those formed by flow separation near the suction surface vertebrae can be precisely identified.The high-speed flow near the pressure surface will form a certain shear velocity gradient, but will not form vortices.Vortices accompany shear flow to develop and disappear within a certain range, while non-uniform shear flow can propagate a long distance in space.

Figure 9 .
Figure 9. Vorticity distribution within the runner blade channel at high head operating condition.