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Research ArticleA Comparative Assessment of Spalart-Shur Rotation/CurvatureCorrection in RANS Simulations in a Centrifugal Pump Impeller
Ran Tao, Ruofu Xiao, Wei Yang, and Fujun Wang
College of Water Resources and Civil Engineering, China Agricultural University, Beijing 100083, China
Correspondence should be addressed to Ruofu Xiao; xrf@cau.edu.cn
Received 10 June 2014; Accepted 29 August 2014; Published 30 September 2014
Academic Editor: Shaofan Li
Copyright ยฉ 2014 Ran Tao et al.This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
RANS simulation is widely used in the flow prediction of centrifugal pumps. Influenced by impeller rotation and streamlinecurvature, the eddy viscosity models with turbulence isotropy assumption are not accurate enough. In this study, Spalart-Shurrotation/curvature correction was applied on the SST ๐-๐ turbulence model. The comparative assessment of the correction wasproceeded in the simulations of a centrifugal pump impeller. CFD results were compared with existing PIV and LDV data underthe design and low flow rate off-design conditions. Results show the improvements of the simulation especially in the situationthat turbulence strongly produced due to undesirable flow structures. Under the design condition, more reasonable turbulencekinetic energy contour was captured after correction. Under the low flow rate off-design condition, the prediction of turbulencekinetic energy and velocity distributions became much more accurate when using the corrected model. So, the rotation/curvaturecorrection was proved effective in this study. And, it is also proved acceptable and recommended to use in the engineeringsimulations of centrifugal pump impellers.
1. Introduction
Reynolds-averaged Navier-Stokes (RANS) simulation pro-vides effective solutions for numerical simulations in engi-neering. By solving the time-averaged N-S equations, RANSsimulation reduces the consumption of computing resourceson the premise of keeping sufficient accuracy. Based on theBoussinesq assumption [1, 2], eddy viscosity models (EVM)directly established the relationship between Reynolds stresstensor and the traceless mean strain rate tensor. Becauseof the simplicity and stability, EVMs are widely used inengineering simulations. However, because of the coordinateinvariance, EVMs are insensitive to streamline curvature andsystem rotation [3]. Moreover, with the turbulence isotropyassumption, the rotation effect on turbulent flow is difficultto describe. However, under the influences of the nonlinearwave induced by the Coriolis force, the flow structure willchange [4]. So, a proper correction for EVM is necessary inthe RANS simulation of rotating and swirling flow.
To describe the effects of rotation or streamline curvatureon a turbulent flow, Bradshaw [5] first proposed the gradient
Richardson number ๐ ๐. Combined with the low-Reynolds-number ๐-๐model, Khodak and Hirsch [6] introduced a newform of ๐ ๐ number which allows including the influence ofcurvature and rotation on the three-dimensional turbulentflow. It improved the prediction accuracy of mean velocityand Reynolds stresses in verification cases. On the basis ofGalilean invariance, Spalart and Shur [3] differentiated theproperty between streamline curvature and system rotationand also introduced a new ๐ ๐ number. By establishing inter-mediate variables, this new๐ ๐ numberwas used to correct theturbulence production term of EVM.This correctionmethodwas applied to Spalart-Allmaras model by Shur et al. [7]and proved to be much more accurate than the original S-Amodel.Moreover, Dufour et al. [8, 9] corrected the ๐-๐modeland compared it with the Spalart-Shur corrected S-A modelthrough a compressor case. Improvements were obtainedafter the rotation and curvature correction for the corrected๐-๐model. Smirnov andMenter [10] also applied the Spalart-Shur correction to SST ๐-๐ model that is known as theSST-RC model. The computational accuracy was proved tobe significantly improved with just a little increase of the
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014, Article ID 342905, 9 pageshttp://dx.doi.org/10.1155/2014/342905
2 Mathematical Problems in Engineering
time costs. Then, Dhakal and Walters [11] corrected the SST๐-๐ model based on the correction method by York et al.[12]. The new model was compared with the SST-RC modeland was proved accurate without compromising stabilityand efficiency. With the modified mean-flow time scale,Hellsten [13] also introduced a new ๐ ๐ number to correctthe ๐ equation of SST ๐-๐model. This new corrected model,which is called SST-RC-Hellstenmodel, was sensitized on theeffects of system rotation and streamline curvature with slightimprovement on the numerical behavior. Based on Spalart-Shur correction method and Hellstenโs time scale, Zhang andYang [14] corrected the S-A model and got reasonable resultsin a U-turn duct case.
In general, all the correction methods mentioned abovewere theoretically feasible for swirling turbulent flow. Forcentrifugal pumps, the internal flow regime varies with theoperating condition [15]. Under off-design conditions, theturbulent flow in the impeller is much more complicated dueto the vortex, backflow, and other secondary flow structures.Hence, it is necessary to ensure the simulation accuracy ofcentrifugal pumps under the influence of system rotation andstreamline curvature. However, in the vast majority of RANSsimulations of centrifugal pumps, the rotation and curvatureeffects are not considered when using EVM. In order toevaluate the Spalart-Shur rotation/curvature correction in thecentrifugal pump cases, RANS simulations were conductedwith both the original and the corrected SST ๐-๐models.Thesimulation results were compared with the experimental databy particle image velocimetry and laser Doppler velocimetry[16, 17].
2. Turbulence Modeling
As mentioned above, the SST-RC turbulence model [10] wasused in the turbulent flow simulation. Compared with theoriginal SST ๐-๐ model [18], in consideration of the tur-bulence anisotropy, the rotation/curvature correction coeffi-cient๐
๐1was introduced as amultiplier of the turbulence pro-
duction term ๐ in the turbulence kinetic energy ๐ equationand specific dissipation rate ๐ equation. So, the productionterm ๐ is defined as follows:
๐ = ๐๐1๐๐๐
๐๐ข๐
๐๐ฅ๐
, (1)
where๐๐1is given empirically with specific limiters as follows:
๐๐1= max [min (๐rot, 1.25) , 0] ,
๐rot = (1 + ๐ถ๐1)
2๐โ
1 + ๐โ[1 โ ๐ถ
๐3tanโ1 (๐ถ
๐2๐)] โ ๐ถ
๐1,
(2)
where๐ถ๐1,๐ถ๐2, and๐ถ
๐3are constants valued as 1.0, 2.0, and 1.0
respectively. The remaining functions are defined as follows:
๐โ
=๐
๐,
๐ =2๐๐๐๐๐๐
๐๐ท3[D๐๐๐
D๐ก
+ (๐๐๐๐
๐๐๐+ ๐๐๐๐
๐๐๐)ฮฉ
rot๐] ,
๐2
= 2๐๐๐๐๐๐,
๐2
= 2๐๐๐๐๐๐,
๐ท2
= max (๐2, 0.09๐2) ,(3)
where ฮฉrot is the rotation rate of the reference frame andthe term D๐
๐๐/D๐กrepresents the Lagrangian derivative of the
strain rate tensor. ๐๐๐is the strain rate tensor and ๐
๐๐is the
rotation rate tensor by
๐๐๐=1
2(๐๐ข๐
๐๐ฅ๐
+๐๐ข๐
๐๐ฅ๐
)
๐๐๐=1
2(๐๐ข๐
๐๐ฅ๐
+๐๐ข๐
๐๐ฅ๐
) + ๐๐๐๐ฮฉ
rot๐.
(4)
The ๐๐๐in (1) is the Reynolds stress tensor. In the Boussi-
nesq assumption [2], ๐๐๐is proportional to the strain rate
tensor ๐๐๐by
๐๐๐= ๐๐ก(2๐๐๐โ2
3
๐๐ข๐
๐๐ฅ๐
๐ฟ๐๐) , (5)
where ๐๐กis the turbulent eddy viscosity. By applying the rota-
tion/curvature correction, anisotropic effects were consid-ered in the simulationwhen solving the turbulence equations.
3. Numerical Simulation
To assess the Spalart-Shur rotation/curvature correction inRANS simulation of centrifugal pump, the internal flow in theimpeller was numerically simulated. Based on the numericalresults and available experimental data [16], the flow detailswere comparatively investigated under both the design andlow flow rate off-design conditions.
3.1. Pump Impeller Model. The scheme of the investigatedcentrifugal pump impeller is shown in Figure 1. The specific-speed๐
๐ of this impeller is about 26.3 calculated by
๐๐ = 3.65
๐โ๐
๐ป3/4, (6)
where ๐ is the rotational speed of 725 r/min, ๐๐is the design
flow rate of 3.06 ร 10โ3m3/s, and ๐ป
๐is the design head of
1.75m. The geometrical parameters of impeller are shown inTable 1 and illustrated in Figure 1. Two operating conditionsincluding ๐ = 1.0๐
๐and ๐ = 0.25๐
๐were simulated and
performed in this study.
3.2. Flow Domain Discretization. The flow domain consistedof the impeller only. For a better geometric adaptability, tetra-hedral mesh elements were used to discretize the impellerdomain. For the usage of wall functions, ๐ฆ+ of the firstelement outside the walls should be set in the log-layer. So,prism boundary layers were used in the near wall region.
Mathematical Problems in Engineering 3
b1
b2
tb
Rb
D1
D2
Figure 1: Scheme of the investigated centrifugal pump impeller.
Table 1: Geometrical parameters of impeller.
Parameter ValueInlet diameter๐ท
171.0mm
Outlet diameter๐ท2
190.0mmInlet height ๐
113.8mm
Outlet height ๐2
5.8mmNumber of blades ๐ 6Blade thickness ๐ก
๐3.0mm
Inlet blade angle ๐ฝ1
19.7 degreesOutlet blade angle ๐ฝ
218.4 degrees
Blade curvature radius ๐ ๐
70.0mm
Table 2: Detailed parameters of mesh scheme.
Parameter ValueNodes 406012Elements 2046670Prism boundary layers 5The first layer height 0.1mmBoundary layer growth rate 1.2
Then, a mesh-size independence check was conducted tocompromise the accuracy and costs in the simulation. Bymodifying the mesh size, the residuals of head and hydraulicefficiency was ensured to be less than 1 ร 10โ3. By modifyingthe boundary layer height, the ๐ฆ
+ values were controlledwithin the range from 1.53 to 26.97 so that the near-wallregion could be solved with wall functions. The final meshscheme is shown in Table 2 and Figure 2.
3.3. Simulation Settings. In this study, transient numericalsimulations were conducted.Three dimensional incompress-ible N-S equations were solved in the simulation process.The fluid medium was set as water at 25 degree centigrade(density ๐ = 997 kg/m3 and dynamic viscosity ๐ = 8.899 ร
10โ4 kg/mโ s). The reference frame was set as rotational with
the speed of 725 r/min. The reference pressure was 1 Atm.Mass flow inlet was set at the impeller inflow with the
Figure 2: Schematic map of the mesh.
velocity normal to the boundary. Pressure outlet was set at theimpeller outflow with a value of 0 Pa. No-slip wall conditionwas given on the solid wall boundaries including hub, shroudand blades.
4. Results and Discussion
In order to fully assess the impacts of Spalart-Shur correction,as mentioned above, the simulations were proceeded underthe design (1.0๐
๐) and off-design (0.25๐
๐) conditions. The
design condition is the most important operating conditionof pump. A correct simulation of the flow details is obviouslysignificant. The off-design condition is also crucial. Unde-sirable flow structures make the flow hard to predict. Forthis reason, the improvements of simulation accuracy arenecessary. Hence, based on the CFD results, the comparativeassessment of the rotation/curvature correction and discus-sions are given as follows.
4.1. Flow under the Design Condition. Under the designcondition (1.0๐
๐), the velocity field on the spanwise 50%
surface was simulated and compared with the LDV data [16]as shown in Figure 3. It can be seen that the flow regime issmooth, stable, and uniform among all the impeller passages.
The correction coefficient๐๐contour on the spanwise 50%
surface is shown in Figure 4. It indicates the enhancement orreduction of local turbulence production under the influenceof rotation and curvature. As the multiplier of turbulenceproduction term, the value of ๐
๐was almost 1.0 in the
vast majority of impeller domain. However, small scale ofreduction was detected at the blade leading edge (LE) and inthe near suction surface (SS) region.
Figure 5 shows the comparisonmap of turbulence kineticenergy ๐
2๐ท. In the impeller, fluid separated from the blade
surface while flowing around the LE. High ๐2๐ท
regionoccurred due to the small scale local separation. Also, in thenear SS region, high turbulence occurred due to the localunattached flow. After the rotation/curvature correction, therange of high ๐
2๐ทregion became smaller than before because
4 Mathematical Problems in Engineering
(a) SST ๐-๐ (b) SST-RC (c) LDV
Figure 3: Velocity field on spanwise 50% surface under 1.0๐๐.
1.25
0.00
Figure 4: Correction coefficient ๐๐on spanwise 50% surface at
1.0๐๐.
of the turbulence reduction. As shown in Figure 5, by com-paring the ๐
2๐ทvalue and pattern, the result by SST-RCmodel
was more consistent with the PIV data [16].Additionally, the relative velocity distributions on the
spanwise 50% surface are analyzed and shown as follows.The radial component (๐
๐) and tangential component (๐
๐ก)
in three different radiusโ positions were plotted, respectively,in Figure 6 (๐
2= 0.5๐ท
2). Considering the flow uniformity
among all the passages, the velocity distributions were com-pared based on just a single passage. Seen from the velocitycurves, the velocities magnitude by SST-RC model was justslightly bigger than that by SST ๐-๐ model. In considerationof the deviations between the PIV andLDVdata [16], both theoriginal and the corrected turbulence models get the similarvelocity distribution regularities.
In general, the impact of rotation/curvature correctionon the simulation accuracy was not obvious under 1.0๐
๐.
Nevertheless, there was no obvious extra time cost whenusing SST-RC model instead of SST ๐-๐ model. So, it would
be reasonable to use the SST-RC model in the simulation ofcentrifugal pump under the design condition.
4.2. Flow under the Low Flow Rate Off-Design Condition.Under the low flow rate off-design condition (0.25๐
๐), the
velocity field on the spanwise 50% surface was also simu-lated and compared with the LDV data [16] as shown inFigure 7. With the flow rate decreasing, flow regime in theimpeller became undesirable with secondary flow structures.As shown in the velocity field map, fluid did not flow alongthe direction of blade geometry. Back flow fromoutlet to inletoccurred in the passage. Lateral secondary flow from bladepressure surface (PS) to suction surface (SS) also occurred.Under the influence of all the disordered flow structures,the flow uniformity among all the passages disappeared.Some passages were blocked by secondary flow, but someother passages were smoother. As shown in the LDV exper-iment, the blocked passage and unblocked passage occurredalternately in the impeller [16]. This phenomenon was alsocaptured by numerical simulations. In this situation, thestreamline curvature under 0.25๐
๐became more obvious
than that under 1.0๐๐.
The correction details under 0.25๐๐
are shown inFigure 8. As plotted in the ๐
๐contour, reductions of tur-
bulence production were detected in the vast majority ofimpeller domain. But influenced by the flow regime, therotation/curvature correction was different in each passage.The passages marked โAโ and โBโ in Figure 8 represented theunblocked and blocked passages, respectively. In passage-A,the low ๐
๐region occurred at the whole blade SS.The high ๐
๐
region occurred at the PS near LE and the midpassage neartrailing edge (TE). In passage-B, the low ๐
๐region occurred
at SS near LE and midpassage near TE. The high ๐๐region
occurred at SS near TE.Influenced by the differences of coefficient ๐
๐, the tur-
bulence kinetic energy ๐2๐ท
had also changed after thecorrection. Figure 9 shows the ๐
2๐ทcontour on the spanwise
50% surface under 0.25๐๐. As shown in the contours, the
turbulence kinetic energy was low in the unblocked โAโpassages andwas high in the blocked โBโ passages. Due to theflow separation at LE and backflow at TE, the high ๐
2๐ทregion
Mathematical Problems in Engineering 5
0.15
0.00
(m2/s
2)
(a) SST ๐-๐
0.15
0.00
(m2/s
2)
(b) SST-RC
0.15
0.00
(m2/s
2)P1
P2
(c) PIV
Figure 5: Contour of turbulence kinetic energy ๐2๐ท
on the spanwise 50% surface under 1.0๐๐.
occurred at LE and TE in โBโ passages. On the contrary,because of the well-behaved flow regime, the ๐
2๐ทvalue was
lower in โAโ passages. Compared with the PIV data [16], thelocation and intensity of ๐
2๐ทwere more reasonable after cor-
rection. Moreover, the alternately blocked phenomena weremore obvious by using the SST-RC model.
To assess in details, the relative velocity distributions onthe spanwise 50% surface are also plotted in Figure 10. Con-sidering the differences of flow regime among all the passages,two adjacent passages (โAโ and โBโ) were analyzed. The dis-tributions of velocity components changed after correction.Compared with the experimental data [16], it can be seenthat the results became more accurate. In particular, flowseverely separated at blade LE with a stronger streamlinecurvature and higher turbulence. So, corrected by reducingthe production of turbulence, radial and tangential velocitiesbecamemore consistent with experiments especially at 0.5๐
2.
All in all, the correction impact was obvious under0.25๐
๐. So, it is strongly recommended to use the corrected
model under the low flow rate off-design condition.
5. Conclusions
By comparatively assessing the Spalart-Shur correction inthe RANS simulations in centrifugal pump impeller under
different operating conditions, conclusions can be drawn asfollows.
(1) In the RANS simulations, the isotropous descriptionof turbulence model is not perfect enough. The flowin a centrifugal pump impeller is strongly affected bythe system rotation and streamline curvature. Withthe pump rotation, separation flow occurred at bladeleading edge. Under different operating conditions,the scale of separation is also different. Particu-larly under low flow rate off-design conditions, fluiddoes not flow along the blade geometry; secondaryflow structures become more and more obvious andoccurred everywhere in the pump impeller passages.
(2) By the supplements of descriptions of turbulenceanisotropy, turbulence production term is corrected.Verified by comparing the CFD results with exper-imental data, improvements are found after correc-tion. Under the design condition, the impact ofcorrection is not obvious but theoretically reasonable.Under the low flow rate off-design condition, sim-ulation accuracy is significantly improved especiallyin the strong separation region. Moreover, there isno obvious extra time cost when using the correctedmodel. Hence, in the RANS simulations of centrifugal
6 Mathematical Problems in Engineering
0
0.1
0.2
0.3
0 20 40 60
Wr/U
2
0
0.1
0.2
0.3
0.4
0.5
0 20 40 60
Wt/U2
Theta angle (โ) Theta angle (โ)
(a) Radius ๐ = 0.50๐ 2
0
0.05
0.1
0.15
0.2
0 20 40 600
0.1
0.2
0.3
0.4
0 20 40 60
Wr/U
2
Wt/U2
Theta angle (โ) Theta angle (โ)
(b) Radius ๐ = 0.75๐ 2
0
0.05
0.1
0.15
0.2
0 20 40 600
0.1
0.2
0.3
0.4
0.5
0 20 40 60
LDVSST k-๐SST-RC
PIVLDV
SST k-๐SST-RC
PIV
Theta angle (โ)Theta angle (โ)
Wr/U
2
Wt/U2
(c) Radius ๐ = 0.98๐ 2
Figure 6: Relative velocity distributions on the spanwise 50% surface at radius of 0.50, 0.75, and 0.98๐ 2under 1.0๐
๐.
Mathematical Problems in Engineering 7
(a) SST ๐-๐ (b) SST-RC (c) LDV
Figure 7: Velocity field on spanwise 50% surface under 0.25๐๐.
1.25
A
B
0.00
Figure 8: Correction coefficient ๐๐on spanwise 50% surface at 0.25๐
๐.
0.00
0.25
(m2/s
2)
A
B
(a) SST ๐-๐
0.00
0.25
(m2/s
2)
A
B
(b) SST-RC
0.00
0.25
(m2/s
2)
A
B
(c) PIV
Figure 9: The contour of turbulence kinetic energy ๐2๐ท
on the spanwise 50% surface under 0.25๐๐.
8 Mathematical Problems in Engineering
0
0.1
0.2
0 20 40 60 80 100 120
0
0.1
0.2
0.3
0 20 40 60 80 100 120
Theta angle (โ) Theta angle (โ)
Wr/U
2
Wt/U2
โ0.1
โ0.2
โ0.1
โ0.2 โ0.3
(a) Radius ๐ = 0.50๐ 2
0
0.05
0.1
0 20 40 60 80 100 120
0
0.05
0.1
0.15
0 20 40 60 80 100 120
Theta angle (โ) Theta angle (โ)
Wr/U
2
Wt/U2
โ0.05
โ0.05
โ0.1
โ0.1
โ0.15
(b) Radius ๐ = 0.75๐ 2
0
0.1
0 20 40 60 80 100 120
0
0.1
0.2
0.3
0.4
0.5
0 20 40 60 80 100 120
Theta angle (โ) Theta angle (โ)
Wr/U
2
Wt/U2
โ0.1
โ0.2
โ0.1
โ0.2
SST k-๐SST-RC
SST k-๐SST-RCLDV
PIVLDVPIV
(c) Radius ๐ = 0.98๐ 2
Figure 10: Relative velocity distributions on the spanwise 50% surface at radius of 0.50, 0.75, and 0.98๐ 2under 0.25๐
๐.
Mathematical Problems in Engineering 9
pumps, if the flow regime is undesirable with strongsecondary flow structures, it will be very appropriateto apply the Spalart-Shur rotation/curvature correc-tion to the SST ๐-๐ model and other eddy viscosityturbulence models.
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper.
Acknowledgments
The authors would like to acknowledge the financial supportgiven by the National Natural Science Foundation of China(no. 51139007) and National โTwelfth Five-Yearโ Plan forScience & Technology Support (no. 2012BAD08B03).
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