Numerical Study on the Internal Flow Characteristics of an Unshrouded Centrifugal Impeller with Splitter Blades at Off-design Conditions*

Yong-shun Zeng Lian-qiang Liu Xiao-lin Hao Zhi-feng Yao,3 Fu-jun Wang,3 Min Yang

（1.College of Water Resources&Civil Engineering,China Agricultural University,2.Beijing Aerospace Petrochemical Technology and Equipment Engineering Corporation;3.Beijing Engineering Research Center of Safety and Energy Saving Technology for Water Supply Network System)

Abstract：In order to investigate the complex flow characteristics inside an unshrouded centrifugal impeller with splitter blades at off-design conditions,and analyze its influence on pump operation stability,a numerical simulation study was carried on using the curvature-corrected SST-CC turbulence model;the head and efficiency accorded with experimental results.The pressure fluctuation,unsteady radial force and velocity were analyzed quantitatively and the numerical results indicate this:the peak to peak value of pressure fluctuation in the impeller channel gradually increases in the flow direction and at 0.49Qn,the slope of peak to peak value to normalized impeller channel behind the splitter blade is 8.57 times greater than that before the splitter blade.The greater the flow rate deviates from the design condition,the larger the peak to peak value of the pressure fluctuation and radial force;in particular at 0.27Qn,the maximum radial force is 194.29% greater than that of the design condition.When the operating flow rate is smaller than 0.83Qn,the stall occurs and the stall vortex could block the impeller discharge;as the flow rate decreases further,the pressure amplitude at rotational frequency gradually increases in the impeller channel and the prevailing frequency changes from the blade passage frequency(BPF)to the rotating stall frequency in the diffuser.The tip leakage vortex(TLV) is generated in the tip region and rotated move downstream in the impeller flow channel,and the backflows appear on the blade suction side and in the tip and the tongue regions;the smaller the flow rate,the more serious the TLV and backflow phenomenon.The rotating stall causes uneven flow in the impeller channel,increasing the pressure fluctuationandtheradialforce,and resulting in an imbalance of the impeller rotation.

Keywords：Off-design Condition；Splitter Blade；Unshrouded Centrifugal Impeller；Pressure Fluctuation；Radial Force

0 Introduction

The unshrouded centrifugal impeller is not easy to block,easy to assemble and dismantle.It is widely used in aerospace,chemical and urban drainage fields [1].For the purposes of increasing the head and avoiding an excessive jam in the impeller inlet,widespread interest has been shown in using the unshrouded impeller with the splitter blades [2].Due to the effect of tip leakage flow over both the long and splitter blades,the unsteady and spatial asymmetry flow characteristics in the impeller channel and vane diffuser would be more complicated at the off-design condition.More attention should be paid to the excessive pressure fluctuation and radial force,especially at small flow rates,with a rotating stall and backflow phenomena.

Pressure fluctuation characteristics are closely related to spatial position.In the case of an impeller without splitter blades,the pressure fluctuation in the impeller channel increased along the flow direction,and the pressure fluctuation on the pressure side was higher than that on the suction side[3].Due to the rotor and stator interference(RSI),the prevailing frequency in the impeller channel was dependent on the number of stator blades [4],and the prevailing frequency in the volute chamber was the BPF [5].The pressure fluctuation characteristics of the unshrouded centrifugal impeller with splitter blades are more complicated.In the impeller,the tip leakage flow interacted with the non-rotational free stream before the splitter blade,while it interacted with the rotating channel flow behind the splitter blade.[6].The interaction of the tip leakage flow and rotating channel flow was more intense.Jaatinen-Värri et al.[7]suggested that the flow on the splitter blade suction side was less uniform than that on the long blade suction side.This would result in significantly different pressure fluctuation characteristics in the long blade channel and the splitter blade channel.For the centrifugal pump with a vane diffuser,the pressure fluctuation is highest in the vaneless region due to the RSI between the impeller blades and the vane blades [8].Generally,the closer to the vane blade,the larger the peak value of the pressure fluctuation [9].Due to the rotation of the impeller,the distance of the impeller blade relative to the vane blade was not frozen in time,and this results in the spatial-varied characteristic of pressure fluctuation in the vaneless region [10].The simulation conducted by Kergourlay et al.[11]showed that the radial thrust of the impeller with splitter blades was greater than that without splitter blades.This also suggested that more focus should be given to the pressure fluctuation characteristic in the region affected by the splitter blade.

Internal flow characteristics at off-design flow rates are more complicated than that at design flow rate.Both the experiment[12]and the numerical simulation[9]indicated that the pressure fluctuation was the lowest at the design flow rate,while it increased gradually when the operating conditions deviated from the design flow rate.At small flow conditions,in particular,an additional low frequency component of pressure fluctuations may be generated as a result of the rotating stall in the impeller flow channel [13].The vortex separation on the blade suction side due to the effect of the rotating stall,and the pressure fluctuation amplitude could reach two to three times of that in the design condition [14].Based on the experimental and numerical simulation methods,Zhou et al.[15]investigated the periodic process of stall vortex generation,development and shedding,indicating that the stall vortex shedding frequency was around 0.24 times of the rotational frequency.The rotating stall may result in the imbalance of the impeller,and simulation results[16]indicated that the maximum radial force at 0.31 times design flow rate increasing to four times that in the design condition.Tip leakage flow was inevitable due to the pressure difference over the blade tip of the unshrouded impeller and was strongly affected by flow rate;the trajectory of the leakage flow interacts with the adjacent blade and causes recirculation at the inlet tip,when the flow rate was sufficiently low[17].As the tip leakage flow and back flow mix into the wake,the jetwake characteristics at impeller discharge will be more complicated[18].

Due to the effect of Coriolis force and centrifugal force,the traditional eddy viscosity model in the rotating coordinate system may not be accurate enough to predict the internal flow of the pump.By modifying the kinetic energy generation term in the governing equation,Spalart and Shur [19]improved the simulation accuracy of the rotation and curvature properties.Smirnov and Menter [20]coupled the Spalart-Shur curvature correction method with the SSTk-ωmodel,and then formed the SST-CC model.Tao et al.[21]applied the SSTk-ωmodel and SST-CC model to the numerical simulation of a centrifugal pump.The calculated results showed almost no variation at the design conditions,while the relative velocities predicted by the SST-CC model were more precise at small flow rates.In fact,the curvature correction coefficient had a significant influence on the calculation accuracy.If the coefficient was too small or too large,this will lead to an excessive and lower prediction of turbulent energy,respectively[22].In the case of a centrifugal impeller,the investigation of Tao et al.[22]indicated that when the coefficient was 0.399,the simulation results accord with the experimental results.

In order to analyze the complex internal flow characteristics of an unshrouded centrifugal impeller with splitter blades at off-design conditions,unsteady simulations were carried out,based on the SST-CC model at a flow rate range of between 0.27 and 1.12Qn.The relationship between pressure fluctuations and different spatial positions were quantitatively reported;the influence of the rotating stall on pressure fluctuation and radial force characteristics were analyzed;and the influences of tip leakage flow and backflow on the flow characteristics were described.

1 Theory background

1.1 Turbulence model

The SSTk-ωmodel realized the switching of the Wilcoxk-ωmodel in the near wall region and standardk-εmodel in the free shear flow region[23].Moreover,this model considers the influence of turbulent shear stress transmission and improves the prediction accuracy of the boundary layer flow separation under the adverse pressure gradient,compared to the traditionalk-εmodel [24].The SSTk-ωmodel was used to close the Reynolds averaged Navier-Stokes(RANS)equations,as follows[24]:

wherekandωare the turbulence kinetic energy equation and specific dissipation rate equation,respectively.PkandDkare the turbulence kinetic energy production term and destruction term,respectively.PωandDωare the specific dissipation rate production term and destruction term,respectively.uiis the flow velocity,μis the dynamic viscosity coefficient,μtis the eddy viscosity,F1is the model switching mixed equation,ρis the density of fluid,σk,σωandσω2are the constant coefficients.

1.2 Curvature correction equation

The turbulence kinetic energy production termPkof the SSTk-ωmodel was corrected,as follows:

whereCsis the empirical correction coefficient;Cr1=1,Cr2=2 andCr3=1 are the constant coefficients.Sis the strain rate,Ωis the rotation rate,Ωrotis the rotation rate of the reference frame.

1.3 Definition of other parameters

The specific speed was defined as:

wherenis the rotational speed of the impeller [r/min];Qnis the design flow rate[m3/s];Hnis the design head[m].

Pressure fluctuation was defined as being dimensionless,as follows[25]:

whereCpis the pressure coefficient,[-];piis the transient pressure,[Pa];pˉis the time-averaged pressure,[Pa];uis the circumferential speed in the impeller discharge,[m/s];is the peak to peak pressure,[-];Cpmaxis the maximum pressure coefficient,[-]andCpminis the minimum pressure coefficient,[-].

Flow rate and frequency were defined as being dimensionless,as follows:

whereQcis the relative flow rate,[m3/s];Qis the inlet flow rate,[m3/s];fcis the relative frequency,[-];fis the frequency of pressure fluctuation,[Hz]andfnis the rotational frequency,[Hz].

The hydraulic efficiency is defined as follows:

wheregis the gravity acceleration,[m/s2];His the head,[m]andMis the torque of the impeller,[N∙m].

The total efficiency can be calculated as follows[26]:

whereηt,ηvandηmare the pump total efficiency[%],volumetric efficiency [%]and mechanical efficiency [%],respectively.ηvandηmare related to pump volume loss and mechanical loss and these can be estimated empirically based on specific speed and pump type,according to the reference[26].

The vortex recognition method was based on the Q-Criterion,as follows[27]:

whereΩijandSijare the strain rate tensor [s-1]and rotation rate tensor[s-1],respectively.

The impeller tip Reynolds number is defined as follows:

whereρis the fluid density,[kg/m3];vis the tangential velocity of the impeller,[m/s];D1is the impeller outlet diameter，[m]andμis the dynamic viscosity，[kg·s/m].

2 Calculation settings

2.1 Physical model

The three-dimensional calculation domain as showed in Fig.1,includes the inlet extension domain,impeller,diffuser,volute chamber and outlet extension domain.As for the unshrouded impeller,the tip between the upper wall and the blade is 1mm.The long blades and the split blades are alternately distributed,and the specific speed is 69.The other key geometrical and hydraulic parameters are shown in Tab.1.

Fig.1 Physical model

Tab.1 Key geometrical and hydraulic parameters

2.2 Mesh independence analysis

As showed in Fig.2,the structured mesh of the impeller is generated using the commercial software Turbo-grid.In order to improve the orthogonality,expansion and aspect ratio of the mesh elements,the mesh scales in the location of the leading edge,trailing edge and near wall region are locally refined.The first layer thickness in the near wall region of the blade is 0.2mm,and there are 10 mesh nodes in the tip region.

Fig.2 Mesh of impeller

Element number independence analysis of the impeller is performed at design condition,as showed in Fig.3.When the element number increases from 3.87×106to 4.81×106,the relative variations of the head and hydraulic efficient are 0.14% and 0.18%,respectively,and the impeller elements of 3.87×106are used for the subsequent simulation.At the design flow rate,the averagedy+is 60.93,and the auto wall function is used for the near wall solution.

Fig.3 Grid independence analysis

2.3 Time step independence analysis

Time step analysis is carried out at design condition,as showed in Fig.4.When time steps are less than 5.59×10-5s and 1.12×10-4s,they have almost no effect on head and hydraulic efficiency,respectively.In order to balance the calculation time and calculation accuracy,the time step of 5.59×10-5s is used for the subsequent unsteady simulation and the ratio of this time step to the rotational period of the impeller is 1/360.

Fig.4 Time step independence analysis

2.4 Monitoring location

A large number of pressure fluctuation monitoring points are placed in the impeller flow channel and diffuser,as showed in Fig.5.On the suction side of the long blade,the monitoring points A1-A11 are arranged in the flow direction.On the pressure side of the long blade,the monitoring points B1-B11 are arranged in the flow direction.In the diffuser,from one tongue to the next,the monitoring points C1-C15 are arranged in the flow direction;C1-C11 are defined as the vaneless region.The axial projection of the cylindrical surface is showed in Fig.5,which is an imaginary surface used to detect the radial velocity distribution.

Fig.5 Monitoring location

2.5 Solve control

Numerical simulation is based on ANSYS CFX,the advection scheme option is high resolution,the transient scheme option is the second order backward Euler method,the residual type is root mean square (RMS) and the residual target is 1×10-4with a maximum iteration number of five.The frame change options in the rotor-stator interface for steady and unsteady simulation are frozen rotor and transient rotor stator,respectively.The steady simulation results are set as the initial file for the unsteady simulation.The inlet condition is normal speed and the outlet condition is opening with 1.5MPa pressure.Since the impeller has no shroud,the upper wall of the tip region is set as the counter rotating wall,and the other boundary conditions are no slip wall.The curvature correction coefficient is set as 0.399,which is consistent with the simulation setting of Tao et al.[22].

3 Results and discussions

3.1 Calculation accuracy verification

As shown in Fig.6,after 2000-time steps of steady simulation,the unsteady simulation was carried out.The residual curves history showed that all the RMSs of continuity equation and momentum equation in three directions converged to 1×10-4.

Fig.6 Residual curves history

The hydraulic performance test of the investigated pump was conducted in water;the experiment and simulation results are showed in Fig.7.As for the simulation method,the head is calculated based on the total pressure in the inlet and outlet of the pump,while the volume loss,mechanical loss and flow leakage,etc.are not considered by comparison with real operating conditions.Therefore,the head and hydraulic efficiency obtained by the simulation are higher than those obtained by the experiment method.The maximum relative deviations for head and efficiency are 15.84% and 10.54%,respectively.Fortunately,the volume loss and mechanical loss can be estimated empirically according to the specific speed [26],i.e.,ηv=0.96,ηm=0.89,then the total efficiency can be calculated,based on equation (17).The total efficiency accorded with the experimental results and the relative deviations are below 4%.

Fig.7 Performances of experiment and simulation

3.2 Pressure fluctuation characteristics

3.2.1 Pressure fluctuation in the impeller channel

At the flow rates of 0.49Qn,0.83QnandQn,the peak to peak value of pressure fluctuation in the impeller channel is showed in Fig.8.In the flow direction,the peak to peak value of pressure fluctuation gradually increased,and the increase rate behind the splitter blade is significantly greater than that before the splitter blade.At the flow rate of 0.49Qn,on the pressure side,the peak to peak value of pressure fluctuations before and behind the splitter blade are linear fitted,using the least squares method,and the curve fitting function can be written asrespectively.The slope behind the splitter blade is 8.57 times larger than that before the splitter blade.At design flow rate,the peak to peak value of pressure fluctuations on the pressure and suction sides indicated no significant difference.However,at the flow rate of 0.49Qn,the peak to peak value of pressure fluctuations on the pressure side are greater than those on the suction side before the splitter blade,yet this phenomenon is reversed behind the splitter blade.The pressure fluctuation is lowest at design flow rate,while it gradually increased when the operating condition deviated from the design condition.This conclusion is consistent with previous experimental results[25].

In the impeller channel,there is maximum pressure fluctuation in the location ofA11.As showed in Fig.9,the frequency components in this location include rotational frequency,five times of rotational frequency (prevailing frequency) and their harmonic frequencies.The rotational frequency is induced by the imbalance of impeller,and the five times of frequency is induced by the RSI between the impeller and the five tongue blades.The pressure amplitude at rotational frequency under the design flow rate is the minimum,and is increased gradually when the operating flow rates deviate from the design conditions,especially at small flow rates.At the flow rate of 0.27Qn,in particular,the amplitude of rotational frequency can reach 24.94% of the prevailing frequency.This indicates that the imbalance of the impeller at the small flow rates may be result in high-level radial force.

Fig.8 Peak to peak value of pressure fluctuations in the impeller channel

Fig.9 Pressure fluctuation spectrum characteristics in A11at different flow rates

3.2.2 Pressure fluctuation in the diffuser

Fig.10 Peak to peak value of pressure fluctuation in the diffuser

At the flow rates of 0.49Qn,0.83Qnand 1Qn,the peak to peak value of pressure fluctuations in the diffuser are showed in Fig.10.In the location of the tongue region and in the outlet of the diffuser,there are maximum and minimum peak to peak values of pressure fluctuations,respectively.In the vaneless region,high-level pressure fluctuation always exist,which may enhance the instability of the pump system.The peak to peak value of pressure fluctuation gradually increases when the operating flow rate deviate from the design condition,and this result is consistent with the result in the impeller channel.It is worth noting that the peak to peak value of pressure fluctuation in the diffuser outlet at different flow rates,demonstrate little change.

In the diffuser,there is maximum pressure fluctuation in the location ofC11,and the pressure fluctuation spectrum characteristics in this location at different flow rates are showed in Fig.11.At the flow rate of 1.0 and 1.12Qn,due to the effect of RSI,the prevailing frequency and the secondary frequency are six times and twelve times that of the rotational frequency,and they have long blades passage frequency and total blades passage frequency,respectively.When the flow rate is less than 0.83Qn,a frequency component less than the rotational frequency occurs.Especially,at the flow rate of 0.49Qn,the prevailing frequency is the low frequency and its amplitude is 11.91 times greater than that of long blades passage frequency.The spectrum characteristics indicated that there may be a severe rotating stall phenomenon at small flow rates.

Fig.11 Pressure fluctuation spectrum characteristics in the location of C1at different flow rates

3.3 Radial forces characteristics

Fig.12 shows the components of the transient radial force in thexandydirections at different flow rates.Due to the work of six long blades,the radial forces distribution is a regular hexagon and is uniformly distributed in four quadrants.When the flow rate is smaller than 0.83Qn,the radial forces significantly increase.Especially,at the flow rate of 0.27Qn,the maximum radial force is 194.29% times greater than the design flow rate.

3.4 Internal flow characteristics in the impeller

When the long blade passes the tongue and the next tongue,the times are defined ast=0Tandt=1/5T,respectively,whereTis the rotational period.In the single impeller channel at different flow rates,the velocity and total pressure distributions att=1/10Tare showed in Fig.13.Rotatingblades work on fluids under the action of centrifugal force,and the total pressure in the flow channel gradually increased in the flow direction.Since the pump did not cavitate during the test,the cavitation model was not opened during the simulation.Due to the influence of the outlet boundary condition with 1.5MPa pressure,negative pressure occurs in the impeller inlet,however,this will not affect the internal flow characteristics.As showed in Fig.13 (a)～(c),there are large scale vortices in the flow channel at small flow rates.In particular,whenQ=0.27Qnand 0.49Qn,there may be a severe flow separation phenomenon.The vortices first appear at the entrance of the long blade,and the flow channel between the long blade and the splitter blade is blocked by the vortex after it moves downstream,which may even result in backflow in the impeller discharge.As showed in Fig.11,there is a high-level pressure fluctuation with low frequency,and this may be induced by the rotating stall.

Fig.12 Radial force characteristics at different flow rates

Fig.13 Streamline in the impeller channel at different flow rates

At the flow rate of 1.0Qn,the pressure and velocity vector distributions in the tip region are shown in Fig.14.From the leading edge to the trailing edge,the pressure gradually increased,and this result is consistent with that in the impeller channel.In the tip region close to the blade,there is a flow separation point,which indicated that there may be vortices generated in the tip region.

Fig.14 Flow separation point in the tip region

The internal flow characteristics in the tip region,at different flow rates,are shown in Fig.15.The vortices are recognized by the Q-Criterion,according to equation (20).The black and white streamlines star from the tip region of the long and the splitter blades,respectively.Even under the design condition,as showed in Fig.15(a),the streamline from the tip region is rotated in the flow direction,and the existence of the TLV can be identified by the vorticity distribution.At the flow rate of 0.49Qn,there are TLVs on the suction sides of both the long and the splitter blades,as showed in Fig.15(b).As showed in Fig.15(c),at the flow rate of 0.27Qn,the TLVs generated by the long and short blades are connected,and the volume of backflow vortices in the impeller discharge is significantly greater than that of 0.49Qn.

At design and small flow rates,whent=1/30T,the radial velocity distributions in the cylindrical surface are showed in Fig.16.The radial velocity distribution is closely related to the spatial position.Simulation results indicated that the radial velocities on the blade pressure side are greater than that on the suction side.There are negative radial velocities in the tongue and tip region,which proved that there are backflow phenomena.At design flow rate,the radial velocity is uniformly distributed between the long and splitter blades in the flow direction,as showed in Fig.16(b).In the case of the small flow rate conditions,positive and negative radial velocities occur simultaneously in the flow channel between the long and the splitter blades,which also indicates that there are backflow vortexes in the impeller discharge.As showed in Fig.16(b) and (c),the areas where backflows occur at small flow conditions are greater than the design condition.Moreover,at the same location,the backflow velocities at small flow conditions are greater than those at the design condition.It is noteworthy that there are high radial velocities distributed on the blade pressure side and close to the hub in all flow rates,which may result in a jet-wake pattern.

Fig.15 TLV and backflow vortices in the impeller channel at different flow rates

Fig.16 Radial velocity distribution in the impeller discharge at different flow rates

4 Conclusion

Based on the SST-CC turbulence model,the unsteady characteristics of the flow field are investigated at design and off-design conditions for the three-dimensional unshrouded centrifugal impeller with splitter blades.The total efficiencies obtained by simulation accord with the experimental results,and the deviations are within 4%.The main conclusions are as follows:

1) Spatial positions in the impeller flow channel and vane diffuser have significant influence on pressure fluctuations.In the impeller channel,the peak to peak value of pressure fluctuations increase in the flow direction,while the slope of peak to peak value to normalized impeller channel behind the splitter blade is 8.57 times greater than that before the splitter blade,at the flow rate of 0.49Qn.In the vane diffuser,the maximum pressure fluctuation appears at the tongue region.

2) The pressure fluctuation and radial force are lowest at design condition,while they increase as the operating flow rate deviated from the design condition.When the flow rate is smaller than 0.83Qn,the rotating stall occurs;the pressure amplitude at rotational frequency increased significantly,and the prevailing frequency in the diffuser change from BPF to the rotating stall frequency.At 0.49Qn,the amplitude of the rotating stall frequency is 11.94 times greater than that of the BPF;at 0.27Qn,the maximum radial force is 194.29% times greater than that at design flow rate.

3) The mix of tip leakage flow and backflow complicates the internal flow characteristics of the investigated pump at small flow rates.The TLV is generated in the tip region and rotated move downstream in the impeller flow channel,and the backflow phenomenon may occur on the blade suction side and in the tip and the tongue regions.At small flow rates,the TLV and backflow are strengthened,and this is more likely to cause uneven distribution of flow velocity in the impeller flow channel.