Tip clearance on pressure fluctuation intensity and vortex characteristic of a mixed flow pump as turbine at pump mode

Accepted Manuscript Tip clearance on pressure fluctuation intensity and vortex characteristic of a mixed flow pump as turbine at pump mode

Yabin Liu, Lei Tan PII:

S0960-1481(18)30675-X

DOI:

10.1016/j.renene.2018.06.032

Reference:

RENE 10188

To appear in:

Renewable Energy

Received Date:

08 March 2018

Accepted Date:

11 June 2018

Please cite this article as: Yabin Liu, Lei Tan, Tip clearance on pressure fluctuation intensity and vortex characteristic of a mixed flow pump as turbine at pump mode, Renewable Energy (2018), doi: 10.1016/j.renene.2018.06.032

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ACCEPTED MANUSCRIPT

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Tip clearance on pressure fluctuation intensity and vortex characteristic of a mixed flow pump as turbine at pump mode

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Yabin LIU, Lei TAN*

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State Key Laboratory of Hydroscience and Engineering, Department of Energy and Power Engineering, Tsinghua University, Beijing 100084, China.

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* Author to whom correspondence should be addressed; E-Mail: [email protected]; Tel.: +86-10-6278-0605.

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Received: / Accepted: / Published:

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Highlights:

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Pressure fluctuation intensity presents to be a triangular shape.

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Tip leakage vortex (TLV) can be divided into four categories.

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Primary TLV, secondary TLV, entangled TLV and dispersed TLV.

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Relative vorticity transport equation is used for the vortex derivation.

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Relative vortex stretching item, Coriolis force item and viscous diffusion item.

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Abstract: The present work investigates the pressure fluctuation intensity and vortex characteristic of a mixed flow pump as turbine at pump mode with a tip clearance. The tip clearance between the blade tip and shroud can induce tip leakage flow and interact with main flow, which causes the unstable flow structure and complex vortex in the passage. The external characteristics of experimental results and numerical simulation are in agreement. With tip clearance increasing, the head and efficiency of pump decrease by 10.8% and 6.26%, respectively. The distribution of pressure fluctuation intensity is presented as a triangular shape under design flow rate. Results show that the tip leakage vortex (TLV) can be divided into four categories, namely, primary TLV, secondary TLV, entangled TLV, and dispersed TLV. The flow rate has a significant influence on the TLV structure and trajectory, and the starting point of the primary TLV shifts to approximately 20% of the blade chord at large flow rate. The relative vorticity transport equation is introduced to analyze the vortex derivation by using the relative vortex stretching, Coriolis force and viscous diffusion.

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Keywords: mixed flow pump; pump as turbine; tip clearance; pressure fluctuation intensity; vortex characteristic

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1. Introduction

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Hydropower is one of the reliable and renewable sources of energy worldwide. In the past decades, pumps as turbines (PATs) are widely used in miniature hydropower [1] and pump storage plants [2] owing to their effective utilization sufficiency and high flexibility. Therefore, the energy performance and operation stability of a PAT should be guaranteed for energy conservation and environment-friendly engineering [3, 4]. Many studies investigated the energy performance and flow patterns of PATs in pump and turbine modes by considering guide vane angle and impeller diameter [5–7]. However, tip clearance in PAT has been rarely reported and its effect and mechanism has remained unclear. The tip clearance between the blade tip and shroud is inevitable in the rotating component of the PATs, as shown in Figure 1. The tip clearance can induce leakage flow and interact with the mainstream in the impeller despite its small size, which causes problems, such as performance drop and cavitation that are relatively common in hydraulic machineries [8–10]. Shroud

Shroud

Blade tip Blade tip

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Figure 1. Tip clearance between blade tip and shroud

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A considerable number of studies investigated the influence of tip clearance on the energy performance of pumps and turbines. An experimental investigation on a mixed flow pump [11, 12] indicated that pump efficiency and head are significantly affected by tip clearance size and the relationship between them is linear near the design condition, and these findings coincide with the numerical results of Liu et al. [13]. A similar study on an axial flow waterjet pump [14] with different tip clearances demonstrated that the system efficiency decreases by 25% when tip clearance of the impeller diameter increases from 0.7% to 1.5%. Meanwhile, a study on impulse turbine [15] illustrated that increase in tip clearance affects turbine performance in which the tip clearance of 1% axial chord led to 4% efficiency drop. Tip clearance in Wells turbines [16] causes a negative effect on the turbine performance and increase in tip clearance results in a wide stable operation range. The above results indicate that tip clearance affects the energy performance of pumps and turbines, and the principle underlying its influence is complicated and associated with machinery type and operating conditions. Operation stability is a key indicator used for evaluating the performance of pumps and turbines. According to previous studies [17–19], pressure fluctuation and fluid force are the main factors associated with operation stability in pumps and turbines, and the main origination is the rotor–stator interaction. However, the flow pattern becomes unstable, and the coherent mechanism becomes complex when the influence of tip clearance is considered. A numerical research revealed that the influence of tip clearance on pressure fluctuation in impellers is more serious than that in diffusers for axial pumps [20]. Another study on a low specific speed mixed flow pump indicated that leakage vortex slightly affects pressure fluctuation in impellers when tip clearance is below a certain value, and increase in tip clearance

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induces significant pressure fluctuation in impellers [21]. Generally, the strongest pressure fluctuation occurs at the blade tip near the leading edge, and the domain frequency of pressure fluctuation is no longer the blade passing frequency under the influence of large tip clearance [13]. Cavitation occurs in the same pump at increased tip clearance, and the amplitudes of pressure fluctuations are considerably enhanced under cavitation condition [22]. Regarding radial force, unsymmetrical tip clearances cause the trajectory center shift of radial force, and average radial force sharply increases. Moreover, the fluctuations of radial force are immensely intensified [23]. Therefore, tip clearance significantly affects the operation stability of pumps and turbines in terms of pressure fluctuation, radial force, and cavitation. To determine the mechanism underlying the influence of tip clearance on energy performance and operation stability, intensive studies on the internal flow characteristic near a tip clearance region were conducted. The tip leakage vortex started at approximately 30% blade chord, gradually shifted from the blade tip to the flow passage, expanded in size, and broke when approaching the neighboring blade based on particle image velocimetry methods and high-speed video images in an axial water-jet pump. The swirling direction of peripheral filaments was contrary to that of tip leakage vortex, and several vortices shed from the blade tip in the middle blade chord sucked in the main leakage vortex [24–27]. The pressure difference between the blade pressure side (PS) and suction side (SS) was the main cause for the leakage jet flow in a turbomachinery by using large eddy simulation (LES). Violent turbulence intensity was induced by the tip leakage vortex and leakage jet flow, and then significantly intensified the viscous loss near the tip clearance was significantly intensified. The distributions of mean pressure and pressure fluctuation demonstrated that the minimum mean pressure and maximum pressure fluctuation occurred at the center of tip leakage vortex on 30% chord section. Thus, so it was crucial to optimize the flow pattern at approximately 30% blade chord for the prevention of cavitation and vibration [28–30]. Recent numerical studies also showed that the tip leakage trajectory is closely related to flow rate, and the impact of tip clearance on flow pattern in the impeller weakens at a high flow rate. When approaching the neighboring blade, the tip leakage broke because of the tip clearance jet flow and interaction with the main flow. And an interaction occurred between the tip leakage vortex cavitation and sheet cavitation in the flow passage under cavitation condition [31–33]. These studies have revealed a deep understanding on the flow characteristic of tip leakage flow and proposed several optimization schemes, including tip rounding [34–35] and tip thickening [36], to improve the performance of pumps and turbines with tip clearance. Systematic investigation on the energy performance, operation stability, and flow characteristic of a mixed flow PAT is rarely performed. In the present work, the influence of tip clearance on the pressure fluctuation intensity of the mixed flow PAT is investigated, and the vortex characteristic is revealed. The relative vorticity transport equation is introduced to analyze the vortex derivation by using the relative vortex stretching, Coriolis force, and viscous diffusion.

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2.1. Physical model of pump

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The pump in the present work is designed on the basis of a direct and inverse iterative design method [37]. Figure 2 shows the five blades of the impeller and three interfaces of the computation domain. The three interfaces are the interface between suction pipe and impeller, the interface between impeller and guide vane, and the interface between guide vane and outlet pipe, respectively.

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Table 1 lists the main design parameters of the pump. Blade tip clearance TC is 1 mm, which is approximately 1% of the mean blade span. Specific speed ns is defined as the following equation:

ns 

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4

3.65  n  Q

(1)

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H where the units of all variables are consistent with those in Table 1.

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(a) Pump

(b) Impeller

Flow direction

Interface 3 Interface 2 Interface 1 115 116

(c) Computation domain

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Figure 2. Tested pump: (a) pump; (b) impeller; (c) computation domain.

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Table 1. Parameters of pump Parameter

Value

ACCEPTED MANUSCRIPT Design flow rate Q (m3/s)

5 0.54

Design head H (m)

17

Rotational speed n (r/min)

1450

Specific speed ns

464

Number of Impeller Blade Zi

5

Number of Guide Vane Blade Zg

6

Mean blade span (mm)

102

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2.2. Mesh arrangement

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In the present work, hexahedral structured meshes are adopted for the entire computation domain. ICEM-CFD is used for the generation of a hexahedral structured mesh for each computation domain. For the accurate simulation of the flow structure near the blade, the O topology method and local mesh refinement are applied around the blade [38], as shown in Figure 3(a). A total of 20 nodes are arranged from the PS to the SS and 15 nodes from the blade tip to the impeller shroud in the tip clearance region, as shown in Figure 3(b), for the capturing of flow details in this region.

Blade

PS

Blade tip

SS

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(a) Near blade

(b) Tip clearance region

Figure 3. Local mesh refinement: (a) near blade; (b) tip clearance region.

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3. Numerical method and setting

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3.1. Numerical method

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The computational fluid dynamics code of CFX 14.5 is employed in the present work. The SST k-ω

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turbulence model is used in steady simulation. The boundary conditions are set as the total pressure at

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the inlet, the mass flow at the outlet, and no slip wall at the walls. The convergence criterion is defined

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as that when the root mean square (RMS) residual is below 1×10−5, the calculation is considered to be

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convergent. The advection scheme and turbulence numeric are both set as the high resolution. For the

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coupling of the rotational and stationary domains, frozen rotor and transient rotor stator methods are

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applied for steady and unsteady simulations, respectively [39]. For the transient simulation, the time step

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is set to 2.5862×10−4 s, which corresponds to 1/160 T (T is the revolution period of the impeller). The

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independence of time step is already validated in previous research (see [13] for details).

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In the initialization of the steady calculation result, an unsteady simulation with LES is conducted for the acquisition of flow structure and associated turbulence characteristic in the pump. The governing equations for LES are given by filtering the time-dependent Navier–Stokes equations in the physical space. The filtered incompressible momentum equation can be expressed as follows:

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U i  1  p    U i U j    ij  U iU j       (2)   t t  xi x j   x j xi   x j where U i denotes the filtered velocity component, p denotes the filtered pressure, ν denotes the

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kinematic viscosity. τij denotes the subgrid-scale stress, with considering the influence of the small scales. The large scale turbulent flow is solved directly and the influence of the small scales is taken into account by Smagorinsky model.

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3.2. Independence test of mesh density

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In the numerical simulation, the mesh density is an important factor that may influence the accuracy of simulation results. To evaluate the influence of mesh density on simulation results, four sets of meshes with grid numbers from 4,715,842 to 9,230,887 (Meshes 1 to 4) are selected to conduct the simulation. Figure 4 shows that the variations of head and efficiency are significantly reduced with the increase of mesh number, and the difference between Mesh 3 and Mesh 4 is less than 0.02%, which can be ignored. Therefore, Mesh 3 with 7,827,832 elements is used in the present research due to the calculation cost and accuracy.

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



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Figure 4. Head and efficiency versus grid number

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3.3. Simulation accuracy validation

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The experiment of the pump is conducted on the test apparatus in Beifang Investigation, Design &Research Co., Ltd, and the comprehensive measurement error is evaluated to be in the range of ±0.28%

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based on uncertainty analysis of pump system. 17 flow rates from 400 kg/s to 720kg/s are selected to calculate the corresponding head and efficiency. Figure 5 shows that good agreement between the numerical results and the experiment results is obtained, especially near the design flow rate. In the range of 420 kg/s to 660 kg/s, the maximum relative errors of pump head and efficiency between experiment and numerical results are 1.64% and 2.88% respectively. Under the design flow rate of 540 kg/s, the relative errors of pump head and efficiency are 1.59% and 2.41%, respectively. Therefore, the selected mesh arrangement and numerical methods are qualified to guarantee the accuracy and reliability of the present work.

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Fig. 5 Comparison of energy performance between experimental and numerical results.

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4. Result and Discussion

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4.1 Parameter definition

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Taking the steady calculation result as the initial field, a transient simulation is conducted for 16 revolutions of the impeller. A total of 1280 sets of transient data are obtained based on the results of last 8 revolutions. To evaluate the pressure fluctuation intensity in the pump, pressure fluctuation intensity p ' is defined by the RMS method as follows:

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p p' 

1 N  pi N i 1

1 N

N

 ( p  p) i 1

i

(3) (4)

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where N is the sample number, pi is the pressure at each time step, and p is the arithmetic average of

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pressure. The dimensionless intensity of pressure fluctuation IPF is defined as:

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I PF 

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p'

(5)

1 U Tip 2 2

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where U Tip is the tip velocity at blade leading edge with a value of 22.78 m/s, and ρ is the fluid density.

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The flow parameters are analyzed in several perspectives to obtain the flow instability characteristics in the impeller with tip clearance. Figure 6 shows the sketch of monitoring points and sections, where Hb denotes the blade height (from hub to shroud) and λ denotes the ratio of blade chord (from leading edge to trailing edge). A total of 6 axial sections along the blade chord are set and named as leading edge section, 20%, 40%, 60%, 80% blade chord sections, and trailing edge section, respectively, as shown in Figure 6(a). A total of 11 points (PS0–PS11) along the blade chord on the blade PS at 98% of blade height are set, as shown in Figure 6(a), and another 11 corresponding points are set on the blade SS. The same points are also set at 75% and 50% of blade height, respectively. Figure 6(b) shows the location of 5 points on one axial section in the tip clearance, including TPS (at PS of blade tip), TSS (at SS of blade tip), TM (blade tip middle), SM (on shroud), and MM (midpoint between TM and SM).

SM

PS

MM TPS

TM

TSS

Hb SS

λ

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PS0

(a) On blade

(b) In tip clearance

Figure 6. Monitoring points and sections: (a) on blade; (b) in tip clearance.

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4.2 Energy performance

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Figure 7 shows the comparison between the energy performance levels of PATs with 1 or 0 mm tip clearances. The variation trends of performance curves with 1 or 0 mm tip clearances are nearly the same. The head and efficiency of the PATs at the pump and turbine modes remarkably decrease when the tip clearance increases from 0 mm to 1 mm. For example, the efficiency decreases by 6.26%, and the head reduces by 10.8% at pump mode under the design flow rate. The results show that tip clearance has a significant effect on the energy performance of PATs despite its relatively small size. In addition, the efficiency drop under high flow rates is larger than that under low flow rates. The highest efficiency of PAT at pump mode with 0 mm tip clearance is 87.84% under the flow rate of 520 kg/s. For the turbine mode, the efficiency drop under low flow rate is serious and the head drop constantly increases with the increase of flow rate. The highest efficiency of PAT at turbine mode with 0 mm tip clearance is 89.40% under the flow rate of 620 kg/s. The influence of tip clearance on the energy performance of PAT is

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closely related to the flow rate, and the present study concentrates on the flow characteristic at pump mode.

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(a) Pump mode

(b) Turbine mode

Figure 7. Performance curve of head H and efficiency η versus flow rate

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4.3 Flow pattern of TLV

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Figure 8 shows the three dimensional vortex structure induced by tip clearance for different flow rates, and the vortex is represented by the Q criterion, which is the second invariant of the velocity gradient

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tensor defined as Q  1 ((  W ) 2  tr (W 2 )) . For the design flow rate shown in Figure 8(b), the vortex

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can be divided into four categories, namely primary TLV, secondary TLV, entangled TLV, and dispersed TLV. The primary TLV starts at the tip corner of the blade leading edge, gradually develops toward the middle flow channel, and entrains the surrounding fluid. The secondary TLV is induced by the interaction of tip clearance jet flow and mainstream in the blade-to-blade channel, and its size and strength increase along the blade chord. In the early stage, the secondary TLV is relatively weak and imbibed in the primary TLV, which enhances the primary TLV. The secondary TLV interacts with the primary TLV and generates a helical entangle effect with its continuous development. This helical entangle effect terminates the primary TLV and leads to entangled TLV, which crashes as the dispersed TLV. Subsequently, the dispersed TLV diffuses in the blade-to-blade channel and dissipates. Therefore, the spatio-temporal evolution of TLV can be divided into four stages, namely, stage A: inception and development of primary TLV; stage B: inception and development of secondary TLV; stage C: entangled TLV induced by helical entangle interaction of primary TLV and secondary TLV; and stage D: diffusing and dissipation of dispersed TLV. At low flow rate, the primary TLV is disordered and develops in the direction perpendicular to the mainstream. The starting point of TLV shifts toward the hub close to the blade leading edge. The secondary TLV becomes short near the blade trailing edge and becomes thick near the leading edge. Generally, the secondary TLV shrinks in the circumferential direction. The entire TLV occupies a large volume of the blade-to-blade channel that blocks the normal flow and reduces the work capability and efficiency of the pump. At large flow rate, the start point of primary TLV moves downstream along the blade chord. The

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primary TLV keeps coherent up to the next blade tip. The secondary TLV mainly recedes to the blade trailing edge. The helical entangle interaction weakens and delays, and few dispersed TLV can be observed. At large flow rate, the head will not sharply drop because the block effect of TLV on normal flow becomes weak. As shown in Figure 8, the intensity of the primary TLV becomes strong and affects a large region in the impeller although the vortex interaction under large flow rate is weak. Thus, many streamlines in the flow passage are entrained in the primary TLV and deteriorate the flow pattern. This phenomenon also indicates that suppressing the primary TLV is crucial to improve the energy performance of the pump with tip clearance. Secondary TLV

Secondary TLV

Secondary TLV

Entangled TLV Primary TLV Dispersed TLV Primary TLV

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Dispersed TLV

Start point

Primary TLV

Start point

Start point

(a) 0.8Q (b) 1.0Q (c) 1.2Q 6 -2 Figure 8. Vortex structure defined by Q criterion (Q=1.5×10 s ) for three flow rates.

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4.4 Pressure fluctuation intensity

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Figure 9 shows the variation of pressure fluctuation intensity from the leading edge to the trailing edge with different blade heights and in the tip clearance region at design flow rate. The results show that the maximum pressure fluctuation appears at the blade SS near leading edge of 98% Hb, and the pressure fluctuation on 50% Hb is relatively weak in the PS and SS. For PS, the position of maximum pressure fluctuation for different blade heights shifts from middle λ to the trailing edge, as shown by the dashed line arrow. This condition can be attributed to the development of TLV, especially when the entangle interaction develops toward the middle mainstream channel along the blade chord, which makes the flow pattern turbulent with the increase of λ. For SS, the values of IPF at different blade heights are close to zero, except for the points at the leading edge of 98% Hb, especially for the first point IPF that is greater than that of other points. This condition is attributed to that the primary TLV incepts at this position and induces violent turbulent flow at the tip of the blade SS. The TLV enters the middle channel and approaches the blade PS when it develops along the blade chord. Thus, the pressure fluctuation intensity at the PS drastically varies. As shown in Figure 9(b), the curves of TM, MM, and SM have a similar trend, which illustrates that the tip clearance height has a slight influence on the distribution of pressure fluctuation. Their maximum value appears at 20%–30% blade chord because this position is the start point of secondary TLV, as shown in Figure 8(b). However, a remarkable difference of pressure fluctuation is observed between the PS and SS, as shown in the lines of TPS and TSS. The maximum value on the TPS line appears at 50% of the blade chord because the flow pattern on TPS is mainly affected by the dispersed TLV from the former blade

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tip clearance. The above analyses indicate that the most severe pressure fluctuations appear on the SS at the leading edge (98% Hb) and on the PS at the middle chord (98% Hb). Thus, the flow pattern around the blade tip has a significant influence on the operating stability of the pump.

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(a) On blade (b) In tip clearance Figure 9. Pressure fluctuation intensity on blade from leading edge to trailing edge. Figure 10 shows the velocity vector through the tip clearance from the blade PS to SS and the distribution of IPF at the six axial sections (Figure 6) under design flow rate. For all sections, the fluid on the PS is sucked in the tip clearance, and this fluid (label B) straightly jets out of tip clearance at SS. For the section near the blade leading edge, the primary TLV (label A) arises near the blade SS and moves away from the SS along the blade chord. Meanwhile, the secondary TLV (label C) forms in the tip clearance with strong pressure fluctuation presented by the red contour. As shown in the section close to the blade trailing edge (Figure 10(f)), the reverse flow appears near at the blade tip surface moving from the SS to PS and reverses again at the corner of the blade PS. The distributions of pressure fluctuation at the sections from the leading edge to the trailing edge gradually become even, as shown by the contour, and the maximum pressure fluctuation at the section near the leading edge is nearly 15 times that at the section near the trailing edge. From the leading edge to the trailing edge, the secondary TLV is squeezed and expands to the mainstream channel from the PS to SS, which corresponds to Figure 8(b).

B

Tip

C

PS

SS

Tip

A 296 297

SS

PS

(a) Leading edge

(b) 20%λ

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Tip

SS

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Tip

PS

SS

PS

(c) 40%λ

(d) 60%λ

Tip

PS

SS

D Tip

SS

300 301

(e) 80%λ

PS

(f) Trailing edge

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Figure 10. Velocity vector at axial sections, contoured by pressure fluctuation intensity.

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Figure 11 shows the distribution of IPF on the blade pressure surface for three flow rates. Under the design flow rate, the pressure fluctuation on the pressure surface shows a triangle distribution, and influence angle αp (related to IPF) is defined. The value of influence angle αp is measured to be 15°. In this triangle region, the strongest pressure fluctuation appears at 50% chord at 98%Hb, and 60%–70% chord at 75%Hb, which corresponds to the trend in Figure 9(a). Under low flow rate, the distribution of strong pressure fluctuation is no longer a triangle and it mainly appears in the region near the blade leading edge. This phenomenon is related to the flow impact at the leading edge and the blocking effect of secondary flow under low flow rate. Under large flow rate, the pressure fluctuations are significantly weakened, which illustrates that the increase of flow rate restrains the flow instability induced by the tip clearance for the pump. As shown in Figures 8 and 11, strong pressure fluctuations are observed at the regions on the pressure surface where the dispersed TLV appears. For large flow rate, few vortex interactions and dispersed TLV occur. Therefore, the pressure fluctuation on the blade pressure surface declines with the increase of flow rate.

αp

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(a) 0.8Q

(c) 1.2Q

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Figure 11. Distribution of pressure fluctuation intensity on blade pressure surface.

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4.5 Vortex analysis of TLV

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In the present work, the relative vorticity transport equation [40–42] in the relative coordinate system is given in Equation (6). This equation is applied to analyze the inception, evolution, and dissipation of tip leakage vortex near the tip clearance region to obtain a profound understanding of the flow mechanism.         p  D r m (6)  ( r )W   r (  W )  2  (  W )    2  r 2 Dt m

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



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where  r denotes the relative vorticity, W denotes the relative velocity, and ν denotes the kinematic

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viscosity. The left   item in Eq. (6) represents the variation rate of vorticity. On the right hand, the first item ( r )W is the relative vortex stretching item (RVS), which is related to the relative velocity gradient.

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The second item is the relative  vortex dilation item, which is related to the relative velocity divergences. The third item 2  (  W ) is the effect of Coriolis force (CORF), which is related to the rotational

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motion. The fourth item

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viscos diffusion item (VISD) due to fluid viscosity [39]. The second and fourth items can be ignored due to the incompressibility in the present work. To comprehensively obtain the flow pattern near the tip clearance region, five circumferential sections that correspond to 92% Hb, 94% Hb, 96% Hb, 98% Hb, and 99% Hb (on the blade tip) are set to present the above items under design flow rate. Figure 12 shows the distribution of dimensionless RVS. Compared with the TLV structure shown in Figure 8, the RVS item is related to all the four structures of TLV, namely, primary TLV, secondary TLV, entangled TLV, and dispersed TLV, especially at the sections of 99% Hb and 98% Hb. The function of RVS on four TLV is still working with the decrease of blade height. In addition, the RVS related to the dispersed TLV maintains a relatively high value for different blade heights. This phenomenon shows that the RVS is the predominant driving power on the inception and development of TLV.

  m p 2    is attributed to the baroclinic torque. The last item r is the m 2

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(a) 92%Hb

(b) 94%Hb

(c) 96%Hb

(d) 98%Hb

(e) 99%Hb

Fig. 12 Distribution of relative vortex stretching item on different circumferential sections. Figure 13 shows the distribution of dimensionless CORF. Compared with the TLV structure shown in Figure 8, the CORF item is mainly related to the secondary TLV. The Coriolis force is common in rotational machineries and is constantly the main reason for pressure difference between the PS and SS under the same radius. Thus, the Coriolis force can drive the development of the secondary TLV toward the blade-to-blade channel. Therefore, the CORF item plays an important role on the inception and development of secondary TLV.

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(a) 92%Hb

(b) 94%Hb

(c) 96%Hb

(d) 98%Hb

(e) 99%Hb

Fig. 13 Distribution of effect of the Coriolis force on different circumferential sections.

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Figure 14 shows the distribution of dimensionless VISD. On the basis of the TLV structure shown in Figure 8, the high value of RVS item appears in the region where the helical entangle interaction occurs. The helical entangle interaction induces significant shearing effect and makes the primary TLV and secondary TLV tangle into the entangled TLV. Therefore, the VISD item is related to the inception and development of the entangled TLV. The VISD item nearly disappears with the decrease of blade height, which demonstrates that this item mainly affects the TLV near the blade tip.

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(a) 92%Hb

(b) 94%Hb

(c) 96%Hb

(d) 98%Hb

(e) 99%Hb

Fig. 14 Distribution of viscos diffusion item on different circumferential sections.

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5. Conclusions

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A systematic investigation on the energy performance and flow characteristic of a pump with tip clearance is conducted. The conclusions based on detailed analyses can be drawn as follows: (1) Compared with no tip clearance, the 1 mm tip clearance remarkably decreases the energy performance in which the efficiency is reduced by 6.26% and the head is reduced by 10.8% at design flow rate. (2) In terms of the TLV evolution, the vortex around the tip clearance can be divided into four categories, namely, primary TLV, secondary TLV, entangled TLV, and dispersed TLV. The helical entangle interaction of primary TLV and secondary TLV is observed, which forms the entangled TLV. The starting point of primary TLV is near the leading edge, and the inception position of secondary TLV is approximately 20%–30% blade chord, which results in the maximum pressure fluctuations in the corresponding tip clearance regions. (3) The results also show that the flow rate has a significant effect on flow instability and TLV trajectory. Flow instability is restrained when the flow rate increases from 0.8 Qd to 1.2 Qd. Under high flow rate, the starting point of primary TLV shifts from the leading edge to approximately 20% blade chord, and the helical entangle interaction diminishes and delays. In addition, the turbulent flow characteristic is remarkably influenced by the tip clearance, which makes the distribution of pressure

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fluctuation intensity on blade pressure side presents a triangle pattern with an influence angle of 15°. This interesting finding indicates that the influence scope of TLV has a potential correspondence with the TLV trajectory (4) The TLV derivation is analyzed by using the relative vorticity transport equation. The results show that the relative vortex stretching item is related to four TLV structures, the Coriolis force item is mainly related to the secondary TLV, and the viscous diffusion item is mainly related to the entangled TLV. These findings reveal the dominant factors for vorticity transportation and provide beneficial foundation for the suppression of TLV and improvement of pump performance.

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Acknowledgments

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This work has been supported by the National Natural Science Foundation of China [Grant number 51741906], the State Key Laboratory of Hydroscience and Engineering[Grant number 2018-KY-02], the Open Research Fund Program of State Key Laboratory of Hydroscience and Engineering[Grant number sklhse-2018-E-01] , the Key Laboratory of Fluid and Power Machinery (Xihua University), Ministry of Education [Grant number szjj-2017-100-1-004].

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