Investigation on the Flow in a Rotor-Stator Cavity with Centripetal Through-Flow †

International Journal of Turbomachinery Propulsion and Power

Article Investigation on the Flow in a Rotor-Stator Cavity with Centripetal Through-Flow †

Bo Hu *, Dieter Brillert, Hans Josef Dohmen and Friedrich-Karl Benra ID

Department of Mechanical Engineering, University of Duisburg-Essen, 47057 Duisburg, Germany; [email protected] (D.B.); [email protected] (H.J.D.); [email protected] (F.-K.B.) * Correspondence: [email protected]; Tel.: +49-203-379-1703 † This paper is an extended version of our paper in Proceedings of the European Turbomachinery Conference ETC12, 2017, Paper No. 97.

Academic Editor: Claus Sieverding Received: 31 July 2017; Accepted: 6 October 2017; Published: 19 October 2017

Abstract: Daily and Nece distinguished four flow regimes in an enclosed rotor-stator cavity, which are dependent on the circumferential Reynolds number and dimensionless axial gap width. A diagram of the different flow regimes including the respective mean profiles for both tangential and radial velocity was developed. The coefficients for the different flow regimes have also been correlated. In centrifugal pumps and turbines, the centripetal through-flow is quite common from the outer radius of the impeller to the impeller eye, which has a strong influence on the radial pressure distribution, axial thrust and frictional torque. The influence of the centripetal through-flow on the cavity flow with different circumferential Reynolds numbers and dimensionless axial gap width is not sufficiently investigated. It is also quite important to convert the 2D Daily and Nece diagram into 3D by introducing the through-flow coefficient. In order to investigate the impact of the centripetal through-flow, a test rig is designed and built up at the University of Duisburg-Essen. The design of the test rig is described. The impact of the above mentioned parameters on the velocity profile, pressure distribution, axial thrust and frictional torque are presented and analyzed in this paper. The 3D Daily and Nece diagram introducing the through-flow coefficient is also organized in this paper.

Keywords: rotor-stator cavity; centripetal through-ﬂow; axial thrust; frictional torque

1. Introduction In radial pumps and turbines, the leakage flow (centripetal through-flow) is quite common from the outer radius of the impeller to the impeller eye, which has a major impact on the pressure distribution, axial thrust (Fa) and frictional torque. Von Kármán [1] and Cochran [2] gave a solution of the ordinary differential equation for the steady, axisymmetric, incompressible flow. Daily and Nece [3] examined the flow of an enclosed rotating disk both analytically and experimentally. They distinguished the four flow regimes, shown in Figure1, by correlating different empirical equations of the moment coefficients. Kurokawa et al. [4–6] studied the cavity flow with both centrifugal and centripetal through-flow. Schlichting and Gersten [7] organized an implicit relation based on the results of Goldstein [8] for the moment coefficient under turbulent flow conditions. Poncet et al. [9] studied the centripetal through-flow in a rotor-stator cavity and obtained an equation of the core swirl ratio K based on the local flow rate coefficient (Cqr) for Batchelor type flow [10]. For Batchelor type flow, the centrifugal disk boundary layer and the centripetal wall boundary layer are separated by a central core. Debuchy et al. [11] derived an explicit equation for K which is valid over a wide range of Cqr. Launder et al. [12] provided a review of the current understanding of instability pattern that are created in rotor-stator cavities leading to transition and eventually turbulence. Recent experimental

Int. J. Turbomach. Propuls. Power 2017, 2, 18; doi:10.3390/ijtpp2040018 www.mdpi.com/journal/ijtpp Int. J. Turbomach. Propuls. Power 2017, 2, 18 2 of 17 thatInt. are J. Turbomach.created Propuls.in rotor-stator Power 2017 , cavities2, 18 leading to transition and eventually turbulence. Recent2 of 17 experimental investigations up to circumferential Reynolds numbers Re = 5 × 108 with and without through-flow have been conducted by Coren et al. [13], Long et al. [14] and Barabas et al. [15]. The 8 scopeinvestigations of the present up study to circumferential is shown in Figure Reynolds 1. numbers Re = 5 × 10 with and without through-ﬂow have been conducted by Coren et al. [13], Long et al. [14] and Barabas et al. [15]. The scope of the present study is shown in Figure1.

Figure 1. Flow regimes according to Daily and Nece [3]. Figure 1. Flow regimes according to Daily and Nece [3].

TheThe main main dimensions dimensions of the of therotor-stator rotor-stator cavity cavity are areillustrated illustrated in Figure in Figure 2. This2. This study study focuses focuses on on the theinfluence influence of non-pre-swirl of non-pre-swirl centripetal centripetal through-flow through-flow on the on thecavity cavity flow. flow. Based Based on the on theresults results from from 0 previousprevious studies, studies, CD (through-flow(through-flow coefficient)coefficient) maymay have have a largea large influence influence on the on moment the moment coefficient, 0 coefficient,noted as notedCM. Uncertaintiesas . Uncertainties still exist still in exist the effectin the ofeffectCD ofon CM onwith different with different values values of Re and of G Re and(dimensionless G (dimensionless axial axial gap). gap). This This study study is aimed is aimed to provideto provide more more results results to to increase increase the the dataset dataset and andto to better better understand understand the the influence influence of of above above parameters parameters on onC p(pressure (pressure coefficient), coefficient),CF (axial (axial thrust thrustcoefficient) coefficient) and andCM . The. The definitions definitions of the of significantthe significant dimensionless dimensionless parameters parameters in this in study this study are given are ingiven Equations in Equations (1a)–(1k). (1a–k).

Figure 2. Main dimensions of the test rig. a: Hub radius; b: outer radius of the disk; r: radial Figure 2. Main dimensions of the test rig. a: Hub radius; b: outer radius of the disk; r: radial coordinate; coordinate; s: axial gap of the front chamber; sb: axial gap of the back chamber; t: thickness of the disk; s: axial gap of the front chamber; : axial gap of the back chamber; t: thickness of the disk; z: axial z: axial coordinate. coordinate.

∙Ω · b2 Re = = (1a) (1a) v 2 ∙Ω · r Re=ϕ = (1b) (1b) v s G = (1c) b Int. J. Turbomach. Propuls. Power 2017, 2, 18 3 of 17

= (1c)

Int. J. Turbomach. Propuls. Power 2017, 2, 18 3 of 17 = (1d) ∙ . 0 m 2∙∙(CD = −)∙d (1d) = µ· b (1e) ∙ ∙ b Z 2 · π · (pb.− p) · rdr CF = ∙ (1e) = ρ · ω2 · b 4 (1f) a 2∙∙∙ 0.2 Q· Reϕ Cqr == (1g)(1f) 2 ·π · Ω · r3 z ζ = (1g) = s (1h) r x = (1h) 2∙|b | = (1i) ∙2·∙|M | C = (1i) M ρ · Ω2 · b5 ∗ = p p∗ =∙ ∙ (1j)(1j) ρ · Ω2 · b2 ∗ ∗ ∗ ∗ C=p = p(=1(x =)1−) − (p)( x) (1k)(1k)

2. Theoretical Analysis 2. Theoretical Analysis 0 In this study, the values of Q (volumetric through-flow rate), CD and Cqr are negative for In this study, the values of Q (volumetric through-flow rate), and are negative for centripetal through-flow. Using a two-component Laser Doppler Anemometer (LDA) system, centripetal through-flow. Using a two-component Laser Doppler Anemometer (LDA) system, Poncet Poncet et al. [9] correlated Equation (2a) to evaluate the core swirl ratio K (the ratio of the angular et al. [9] correlated Equation (2a) to evaluate the core swirl ratio K (the ratio of the angular velocity of velocity of the fluid to that of the disk at ζ = 0.5) with centripetal through-flow when Cqr ≥ −0.2. the fluid to that of the disk at =0.5) with centripetal through-flow when −0.2. Debuchy et Debuchy et al. [11] determined Equation (2b) to calculate the values of K for a wider range Cqr ≥ −0.5 al. [11] determined Equation (2b) to calculate the values of K for a wider range −0.5 with a with a two-component LDA system. The results from Equation (2b) are smaller than those from two-component LDA system. The results from Equation (2b) are smaller than those from Equation Equation (2a) at large values of Cqr , compared in Figure3. (2a) at large values of , compared in Figure 3. 5 7 K = 2 · −5.9 · Cqr + 0.63 − 1(2a) (2a) =2∙−5.9∙ + 0.63 −1

5 −8.85 · C + 0.5 4 −8.85 ∙ qr+0.5 K = (2b) = (−1.45Cqr) (2b) e.

Figure 3. Comparison of results from Equation (2a,b). K: core swirl ratio at =0.5; : local flow Figure 3. Comparison of results from Equations (2a) and (2b). K: core swirl ratio at ζ = 0.5; Cqr: local rateﬂow coefficient; rate coefﬁcient; Eq: equation. Eq: equation.

A number of studies, such as those by Kurokawa et al. [6], Poncet et al. [9], Coren et al. [13], and Barabas et al. [15], show that the pressure distribution along the radius of the disk can be Int. J. Turbomach. Propuls. Power 2017, 2, 18 4 of 17 estimated with the core swirl ratio K both with and without through-ﬂow. Will et al. [16–18] determined Equation (3) to evaluate the pressure distribution along the radius of the disk for the incompressible, steady ﬂow. It is obtained directly from the radial momentum equation when the turbulent shear stress is neglected. In a rotor-stator cavity, the cross sectional area changes in radial direction. Consequently, the pressure must also change since the mean velocity changes in radial direction according to the continuity equation.

2 ! 2 ∂p vϕ ∂vr 2 2 ρ · Q = ρ − vr = ρ · K · Ω · r + (3) ∂r r ∂r 4 · π2 · s2 · r3

The difference of the force on both sides of the disk is the main source for axial thrust, noted as Fa, calculated with Equation (4a). Fa f (force on the front surface of the disk; calculated with Equation (4b)) and CF f (CF on the front surface) respectively represent the force and the thrust coefﬁcient on the front surface of the disk (in the front chamber, shown in Figure2), while Fab (force on the back surface of the disk; calculated with Equation (4c)) and CFb are those on the back surface of the disk (in the back chamber). a and pb represent the radius of the hub (see Figure2) and the pressure at x = 1, respectively. The back chamber (G = 0.072), shown in Figure2, is supposed to be an enclosed cavity. The values of 0 CFb (CF on the back surface) are obtained when CD = 0 and the axial gaps of both cavities have the same size for different Re (under that condition CF f = CFb). After obtaining those values, the values of 0 CF f with different values of CD can be calculated with Equation (5).

Fa = Fab − Fa f (4a)

2 2 4 Fa f = π · pb · b − CF f · ρ · Ω · b (4b)

2 2 2 4 4 Fab = π · pb · b − a − CFb · ρ · Ω · b − a (4c)

2 4 4 2 Fa + C · ρ · Ω b − a + πp a C = Fb b (5) F f ρ · Ω2 · b4

3. Test Rig Design and Experimental Set-Up The design of the test rig is shown in Figure4. The cross section of the test rig is depicted in Figure4a. The centripetal through-flow (volumetric through-flow rate Q), shown by the black arrows in Figure4a, is supplied with water by a pump system. The view along the “ A” direction is sketched in Figure4b. The shaft sealing at the back cavity is depicted in Figure4c ( rseal = 10 mm). A picture of the test rig is shown in Figure4d. The shaft is driven by an electric motor. A frequency converter is used to adjust the speed of rotation (0~2500/min) with the absolute uncertainty of 7.5/min. In this study, only the axial gap of the front chamber is changed by installing six sleeves with different length. There are 24 channels in the guide vane (instead of entirely open at the periphery) to get more uniform centripetal through-flow, shown in Figure4b. The area of each channel is 4 × 10−6 m2. In this study, the channels in the guide vane are radial directed (see Figure4b). Other parameters of the experiments in this study are given in Table1. The transducers in the test rig include two pressure transducers (36 pressure tubes), a torque transducer and three tension compression transducers. A thrust plate is fixed by a ball bearing and a nut from both sides to convey the axial thrust to the tension compression transducers. A linear bearing is used to minimize the frictional resistance during the axial thrust measurements. During the measurements of axial thrust, the calibrations of the axial thrust transducers are performed when changing the axial gap width of the front chamber. When measuring the torque, the values of the shaft without the disk rotating at different speeds of rotation are subtracted. The measured Rz of the disk is 1 µm. The values of Rz on all the other surfaces of the test rig are below 1.6 µm. Int. J. Turbomach. Propuls. Power 2017, 2, 18 5 of 17 Int. J. Turbomach. Propuls. Power 2017, 2, 18 5 of 17

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Figure 4. Test rig design. : axial thrust; : radius of shaft seal. Figure 4. Test rig design. Fa: axial thrust; rseal: radius of shaft seal.

Table 1. Parameters of the experiments. Table 1. Parameters of the experiments. 3 b (mm) n (/min) Q (m /s) s (mm) (mm) a (mm) t (mm) Figure 4. Test rig design. : axial thrust; : radius of shaft seal. 110 0~2500 −5.563 × 10−4~0 2~8 8 23 10 b (mm) n (/min) Q (m /s) s (mm) sb (mm) a (mm) t (mm) n: speedTable of rotation;1. Parameters Q: volumetric of the experiments. through-flow rate. 110 0~2500 −5.56 × 10−4~0 2~8 8 23 10 The measurements of the axial thrust3 include two steps. The cavities for each step are depicted b (mm) n n(/:min) speed of rotation;Q (m /s)Q: volumetrics (mm) through-flow (mm) a rate.(mm) t (mm) in Figure 5. The first110 step0~2500 is to measure−5.56 × the10−4 ~0axia l force2~8 imposed 8 by the 23 drive end 10 of the motor when the shaft without the diskn: isspeed rotating of rotation; at different Q: volumetric speed ofthrough-flow rotation in rate. the air. For the second step, all Thethe measurementsresults are modified of the by axialsubtracting thrust above include values two obtained steps. The at the cavities first step for according each step to are the depicted speed in The measurements of the axial thrust include two steps. The cavities for each step are depicted Figureof5. rotation. The first The step shaft is tohead measure in Figure the 5b axialis consid forceered imposed as a part byof front the drivesurface end (with of the area motor of ∙ when). the in Figure 5. The first step is to measure the axial force imposed by the drive end of the motor when shaft withoutThe geometry the disk of the is nut rotating is ignored. at different Then, the speed values of of rotation thrust coefficient in the air. on For a single the second surface step, can be all the the shaft without the disk is rotating at different speed of rotation in the air. For the second step, all resultscalculated are modified with Equation by subtracting (5). above values obtained at the first step according to the speed of the results are modified by subtracting above values obtained at the first step according to the speed · 2 rotation.of rotation. The shaft The shaft head head in Figure in Figure5b 5b is consideredis considered as as aa part of of front front surface surface (with (with the area the areaof ∙ of).Ω b ). The geometryThe geometry of theof the nut nut is is ignored. ignored. Then, Then, the values of of thrust thrust coefficient coefficient on ona single a single surface surface can be can be calculatedcalculated with with Equation Equation (5). (5).

Figure 5. Cavities to measure the axial thrust.

FigureFigure 5. 5.Cavities Cavities to measure measure the the axial axial thrust. thrust.

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The relative error, noted as eT, of the pressure transducers is 1% (full scale (FS)). The value of eT for the torque transducer is 0.1% (FS). The value of eT for the axial thrust transducers is 0.5% (FS). All the experimental results are the ensemble average of 1000 samples. The uncertainties of the measured results, noted as ∆N, are the differences between the real values and the measured values. They are estimated with the root sum squared method. The uncertainties are calculated with Equation (6). NT is the uncertainty due to the transducers. ND is the uncertainty due to the data acquisition system. The measuring range (Mr) of the torque meter is 0~10 Nm. The measured range of the pressure transducer is 0~2.5 bar (absolute pressure). The measured range of the thrust transducers is −100~100 N. The input voltage signals are the following ranges: 0~10 V for the pressure transducers and the torque transducer, −10 V~10 V for the axial thrust transducers. The absolute accuracy of the data acquisition system (with NI USB-6008) is 4.28 mV in this study. The random noise and zero order uncertainty are neglected because they are very small. The distributions of the results are considered as normal distributions and the normal distribution coefﬁcient is selected as 1.96 (95% conﬁdence level). nT represents the number of the transducers used to obtain one result together. To evaluate Fa 0 and M on the front surface, the total results, the results on the back surface (obtained when CD = 0 and s = sb) and the results when the shaft is rotating without disk are measured with the transducers. Hence, the measuring times to obtain one result, noted as nM, is 3 for the axial thrust, the frictional torque (M) and Re (by measuring n). The uncertainties of the measurements are summarized in Table2. q q q n · n · (e · M )2 n · n · (e · M )2 2 2 T M T r T M D r ∆N = NT + ND ; NT = √ ; ND = √ (6) 1.96 · 1000 1.96 · 1000

Table 2. Uncertainty analysis for the measurements.

0 p (Bar) Fa (N) M (Nm) Re CD ∆N 4.04 × 10−4 2.43 × 10−2 3.00 × 10−4 9.01 × 104 4.1 nT 1 3 1 1 1 nM 1 3 3 3 1

∆N: uncertainty of the measured results; nT : number of transducers; nM: measuring times to obtain one result; 0 p: pressure; M: frictional torque; Re: global circumferential Reynolds number; CD : through-ﬂow coefﬁcient.

4. Numerical Simulation To predict the cavity ﬂow, numerical simulations are carried out using the ANSYS CFX 14.0 code [15]. There are 24 channels in the guide vane. Considering the axial symmetry of the problem, a segment (15 degree) of the whole domain is modeled and a rotational periodic boundary condition is applied. Structured meshes are generated with ICEM 14.0. The domain for the numerical simulation when G = 0.072 is depicted with yellow color in Figure6. The simulation type is set as steady state. Barabas et al. [15] found that the simulation results from the shear stress transport (SST) k − ω turbulence model in combination with the scalable wall functions are in good agreement with the measured pressure in a rotor-stator cavity with air. The deviations of the pressure measurement are less than 1%. Hence, in this study, the same turbulence model and wall functions are used. The boundary conditions at the inlet and at the outlet are pressure inlet and mass ﬂow outlet, respectively. The values of the pressure at the inlet are set according to the pressure sensor at the pump outlet. The convergence criteria for all the numerical simulations are set as 10−5 in maximum type. The turbulent numeric is set as second order upwind. The maximum value of y+ in all the simulation models is 13.4. Int. J. Turbomach. Propuls. Power 2017, 2, 18 7 of 17 Int. J. Turbomach. Propuls. Power 2017, 2, 18 7 of 17

FigureFigure 6. 6.Domain Domain forfor numericalnumerical simulation simulation when whenG G= = 0.072.0.072.

5.5. ResultsResults andand DiscussionDiscussion

5.1.5.1. SimulationSimulation ResultsResults ofof VelocityVelocityDistributions Distributions TheThe velocityvelocity profilesprofiles are are sensitive sensitive to to the the boundary boundary condition condition at at the the inlet. inlet. TheThe profilesprofiles inin thethe frontfront chamberchamber are are discussed discussed in in this this part part becausebecause therethere isis aa smallsmall jetjet flowflow throughthrough each eachchannel channelin inthe the guideguide vane. All the velocities are made dimensionless by dividing them by Ω · b. The values of Vz are positive vane. All the velocities are made dimensionless by dividing them by ∙. The values of are whenpositive they when have they a direction have a from direction the disk from to the wall.disk to The the velocity wall. The profiles velocity at three profiles radial at positions three radial for × 6 Repositions= 1.9 for10 Reand = 1.9G =× 0.072106 and (wide G = 0.072 gap) are(wide shown gap) inare Figure shown7. in The Figure dimensionless 7. The dimensionless radial velocities radial arevelocities not exactly are not zero exactly in the zero central in the cores, central shown cores, in Figure shown7a–c. in Figure From the7a–c. distribution From the distribution of tangential of velocity,tangential there velocity, are central there cores are central at all the cores investigated at all the radialinvestigated positions radial where positions the values where of V theϕ are values almost of constant, are almost shown constant, in Figure shown7d–f. in At Figurex = 0.955 7d–f. and At x = 0.955 0.79, theand values x = 0.79, of the values tangential of the velocity tangential are 0 smaller at ζ = 0.5 when CD increase from 1262 to 3787 and 5050, depicted in Figure7d,e. The velocity are smaller at =0.5 when | | increase from 1262 to 3787 and 5050, depicted in Figure profiles are special, because according to the former results (such as from Poncet et al. [9] and Debuchy 7d,e. The profiles are special, because0 according to the former results (such as from Poncet et al. [9] et al. [11]), the increase of CD will result in an increase of the core swirl ratio K for centripetal and Debuchy et al. [11]), the increase of | | will result in an increase of the core swirl ratio K for through-flow. Then, the tangential velocity should increase instead of decrease. Probably this can be centripetal through-flow. Then, the tangential velocity should increase instead of decrease.0 Probably attributed to the jet flow through the channel at the inlet, which is stronger at large CD . At x = 0.955,| | this can be attributed to the jet flow through the0 channel at the inlet, which is stronger at large . the values of |Vz| become smaller| | for larger CD in general, shown| | in Figure, 7g. The direction of Vz is At x = 0.955, the values of become 0 smaller for larger0 in general shown in Figure 7g. The from the disk towards the wall for CD = 0 and CD| =|1262=0 at x =| 0.955,| = while it is from the wall direction of is from 0 the disk towards 0the wall for and 1262 at x = 0.955, while it towards the disk at CD = 3787 and CD = 5050. There are axial circulations of the fluid in the front is from the wall towards the disk at | | = 3787 and | | = 5050. There are axial circulations of chamber. The directions of the axial circulations, however, are strongly influenced by the values of the 0 fluid in the front chamber. The directions of the axial circulations, however, are strongly CD , depicted in Figure7g–i. influenced by the values of | |, depicted in Figure 7g–i. × 6 TheThe velocityvelocity profilesprofiles at at the the three three radial radial coordinates coordinates for forRe Re= = 1.9 1.9 × 10106 andand GG == 0.0180.018 (small(small gap)gap) are shown in Figure8. The dimensionless radial velocities Vr vary along ζ, shown in Figure8a–c. are shown in Figure 8. The dimensionless radial 0velocities vary along , shown in Figure 8a–c. The values of Vr decrease with the increase of CD in general. The tangential velocity Vϕ decreases The values of decrease with the increase of | | in general. The tangential velocity decreases constantly with the increase of ζ, which is the characteristic of the regime III, shown in Figure8d–f. constantly with the increase of , which is the characteristic of the regime III, shown in 0Figure 8d–f. At x = 0.955 and x = 0.79, the values of tangential velocity are much smaller at large CD , shown in At x = 0.955 and x = 0.79, the values of tangential velocity are much smaller at large | |, shown in FigureFigure8 d.8d. The reason is is that that the the impact impact of ofthe the jet jetflow flow at the at theinlet inlet becomes becomes greater greater for smaller for smaller G. The G. The profiles of Vz are quite different at x = 0.955 in Figure8g, compared with those in Figure7g. profiles of are quite different at x = 0.955 in Figure 8g, compared with those in Figure 7g. The The values of Vz are less than those from Figure7h,i. The results of Vz indicate that the axial circulations values of are less than those from Figure 7h,i. The results of indicate that the axial circulations ofof fluidfluid areare sensitive sensitive to to the the values values of ofG G. .

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6 6 FigureFigure 7. 7. VelocityVelocity profiles proﬁles for for ReRe == 1.9 1.9 × ×1010 andand G G= 0.072= 0.072 (wide (wide gap). gap). G: G:dimensionlessdimensionless axial axial gap. gap.

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Figure 8. Velocity profiles for Re = 1.9 × 106 and G = 0.018 (small gap). Figure 8. Velocity proﬁles for Re = 1.9 × 106 and G = 0.018 (small gap). 5.2. Pressure Distributions 5.2. Pressure Distributions Due to the construction of the geometry, there is no pressure tube at x = 1. The closest tube is at Due to the construction of the geometry, there is no pressure tube at x = 1. The closest tube is x = 0.955. The pressure values at x = 1 (reference pressure) are taken from numerical simulation. Since at x = 0.955. The pressure values at x = 1 (reference pressure) are taken from numerical simulation. the values of p from numerical simulation at x = 0.955 are close to those from experiments, the small Since the values of p from numerical simulation at x = 0.955 are close to those from experiments, errors are neglected. the small errors are neglected. The pressure coefficient are positive values because the pressure drops towards the shaft. In The pressure coefﬁcient Cp are positive values because the pressure drops towards the shaft. Figure 9, the values of are plotted versus the dimensionless radial coordinates for three values of In Figure9, the values of Cp are plotted versus the dimensionless radial coordinates for three values of Re and G. The experimental results show that the values of increase with increasing 0 | |. The Re and G. The experimental results show that the values of Cp increase with increasing C . The values values of decrease with the increase of Re and G in general. When Re = 1.9 ×10 Dand Re = 2.79 × of C decrease with the increase of Re and G in general. When Re = 1.9 ×106 and Re = 2.79 × 106, 6 p −4 −4 10 , the uncertainties of the are respectively− 2.74 × 10 and− 1.34 × 10 , which are very small the uncertainties of the Cp are respectively 2.7 × 10 and 1.3 × 10 , which are very small compared compared with the measured results. Hence, they are neglected in Figure 9d–i. with the measured results. Hence, they are neglected in Figure9d–i.

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0 FigureFigure 9. 9. InfluenceInﬂuence of of CD on Cp (pressure(pressure coefﬁcient)coefficient) inin dependencedependence ofof ReRe andand GG..

5.3. Axial Thrust Coefficient Coefﬁcient Based on the axial thrust measurements (direction of the axial thrust see Figure4 4a),a), aa correlationcorrelation for is determined, given in Equation (7). The results from Equation (7) are consistent with the for CF f is determined, given in Equation (7). The results from Equation (7) are consistent with the | 0| experimental results, plotted in Figure 1010.. With With the the increase increase of of CD , ,the the values values of of CF f increase, which can be attributed to the drop of p. The The values values of of CF f are are smallersmaller forfor higherhigher valuesvalues ofof GG andand ReRe.. .∙ ∙ =6.6∙10h ∙ln() − 0.113i ∙ −4 0 ∙ 0.122 ∙ ln() −0.67 (7) −3 (1.2·10 ·|CD |) CF f = 6.6 · 10 · ln(Re) − 0.113 · e · [0.122 · ln(G) − 0.67] (7)

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Figure 10. Mean -0 curves in dependence of Re and G. : axial thrust coefficient; : on Figure 10. Mean C -CD curves in dependence of Re and G. CF: axial thrust coefﬁcient; C : CF on theFigure front 10. surface. Mean F f - curves in dependence of Re and G. : axial thrust coefficient;F f: on thethe front front surface. surface. 5.4. The 3D Daily and Nece Diagram 5.4.5.4. The The 3D 3D Daily Daily and and Nece Nece Diagram Diagram The typical tangential velocity profiles for regime III (merged disk boundary layer and wall The typical tangential velocity proﬁles for regime III (merged disk boundary layer and wall boundaryThe typical layer) andtangential regime velocity IV (separated profiles disk for bo reundarygime III layer (merged and walldisk boundary boundary layer) layer areand given wall boundary layer) and regime IV (separated disk boundary layer and wall boundary layer) are given in inboundary Figure 11. layer) and regime IV (separated disk boundary layer and wall boundary layer) are given Figurein Figure 11. 11.

Figure 11. Typical tangential velocity profiles for: (a) regime III and (b) regime IV. Figure 11. Typical tangential velocity profiles for: (a) regime III and (b) regime IV. Figure 11. Typical tangential velocity profiles for: (a) regime III and (b) regime IV. BasedBased on on the the simulation simulation results results of of the the tangential tangential velocity, velocity, part part of of the the Daily Daily and and Nece Nece diagram diagram (see(see Figure FigureBased1) 1)on is is extendedthe extended simulation into into 3D results3D by by distinguishing distinguishingof the tangential the the tangentialvelocity, tangential part velocity velocity of the profiles profilesDaily (seeand (see FigureNece Figure diagram 11 11)) at at (see Figure 1) is extended into 3D by distinguishing the tangential velocity profiles (see Figure0 ≤ 11) at xx= = 0.955, 0.955,x x= = 0.790.79 andandx x= = 0.57.0.57. The scope ofof thisthis studystudy isis thethe followingfollowing parameterparameter ranges:ranges: C|D | 5050 ,, 6 7 0.380.38x = ×0.955, × 106 ≤x≤ Re=Re 0.79 ≤ ≤3.17 and3.17 × x10× =7 100.57.andand 0.018The 0.018 scope ≤ G ≤ of0.072.G this≤ 0.072. Theystudy are Theyis the categorized arefollowing categorized into parameter two into regimes, two ranges: regimes, namely | | namely regime 5050, regimeIII0.38 (below × 10 III6 (below≤ the Re ≤distinguishing 3.17 the × distinguishing 107 and 0.018lines) ≤ lines)and G ≤ 0.072.regime and regime They IV (aboveare IV categorized (above the distin the distinguishingintoguishing two regimes, lines). lines). Currently,namely Currently, regime five 0 fivedistinguishingIII distinguishing(below the distinguishinglines lines are are found found lines) for for different and different regime | C DIV|, shown (aboveshown in inthe Figure Figure distin 1212b.guishingb. TheThe distinguishingdistinguishinglines). Currently, lineline five at at distinguishing0= lines are found for different | |, shown in Figure 12b. The distinguishing line at CD =00 is is almost almost equal equal to to that that from from Daily Daily and and Nece Nece [ 3[3].]. The The distinguishing distinguishing lines lines become become steeper steeper for for =0 is almost equal0 to that from Daily and Nece [3]. The distinguishing lines become steeper for higherhigher values values of of C|D |.. TheThe approximate distinguishing distinguishing surface surface is is drawn drawn through through the the lines, lines, shown shown in inFigurehigher Figure values12a. 12a. Below of Below | and| and. The above above approximate the the surface surface distinguishing are are re regimegime surfaceIII and is regimeregime drawn IV,throughIV, respectively.respectively. the lines, Near Nearshown thethe in Figure 12a. Below and above the surface are regime III and regime IV, respectively. Near the

Int. J. Turbomach. Propuls. Power 2017, 2, 18 12 of 17 Int.Int. J.J. Turbomach.Turbomach. Propuls.Propuls. PowerPower 20172017,, 22,, 1818 1212 ofof 1717 distinguishingdistinguishingdistinguishing surface,surface,surface, theretherethere isisis aaa mixingmixingmixing zonezonezone wherewhwhereere regimeregimeregime III IIIIII and andand regime regimeregime IV IVIV coexist coexistcoexist in inin the thethe front frontfront chamber.chamber.chamber. In InIn this thisthis study, study,study, it itit is isis not notnot plotted plottedplotted in inin Figure FigureFigure 12 12.12..

FigureFigureFigure 12. 12.Part12. PartPart of the ofof thethe 3D 3D Daily3D DailyDaily and andand Nece NeceNece diagram: diagram:diagram: (a) Distinguishing((aa)) DistinguishingDistinguishing surface surfacesurface and andand (b) Distinguishing ((bb)) DistinguishingDistinguishing lines. lines.lines. 5.5. Moment Coefﬁcient 5.5.5.5. MomentMoment CoefficientCoefficient According to the experimental results from Han et al. [19], the moment coefﬁcient on the cylinder surfaceAccordingAccording of the disk toto ( CthetheMcyl experimentalexperimental) can be estimated resultsresults with fromfrom Equation HaHann etet (8) al.al. for [19],[19], smooth thethe disks.momentmoment coefficientcoefficient onon thethe cylindercylinder surfacesurface ofof thethe diskdisk (()) cancan bebe estimatedestimated withwith EquationEquation (8)(8) forfor smoothsmooth disks.disks.

2 · M 2∙2∙ cyl 0.0840.084· ∙ ∙π · ∙ ∙t CMcyl=== == (8) ρ∙∙· Ω2·∙∙b5 ∙∙2 1.5152 (8)(8) ∙(b∙(· lg Ω·b )).. υ WhenWhen 0 =0=0,, thethe valuesvalues ofof forfor GG == 0.0180.018 (regime(regime III)III) andand GG == 0.0720.072 (regime(regime IV)IV) areare When CD = 0, the values of CM for G = 0.018 (regime III) and G = 0.072 (regime IV) are compared incomparedcompared Figure 13 inina,b, FigureFigure respectively. 13a13a andand The FigureFigure differences 13b,13b, respectirespecti betweenvely.vely. the TheThe experimental differencesdifferences resultsbetweenbetween and thethe those experimentalexperimental from the resultsresults andand thosethose fromfrom thethe correlationscorrelations byby DailyDaily andand NeceNece [3][3] forfor bothboth regimeregime IIIIII andand regimeregime IV,IV, areare correlations by Daily and Nece [3] for both regime III and regime IV, are colossal. ks is the equivalent colossal.colossal. is is thethe equivalentequivalent surfacesurface roughness,roughness, defineddefined inin EquationEquation (11).(11). AccordingAccording toto SchlichtingSchlichting surface roughness, deﬁned in Equation (11). According to Schlichting et al. [7], ksl (the limitation for hydraulicetet al.al. [7],[7], smooth) (the (the limitationlimitation wall can for befor estimatedhydraulichydraulic smooth) withsmooth) Equation wallwall cancan (12). bebe estimated Theestimated disk therefore withwith EquationEquation can be (12).(12). considered TheThe diskdisk thereforetherefore cancan bebe consideredconsidered hydraulichydraulic smoothsmooth whenwhen 38.2 μm38.2 μm.. ToTo explainexplain thethe widewide gap,gap, thethe hydraulic smooth when Rz ≤ 38.2 µm. To explain the wide gap, the results for a rougher disk resultsresults forfor aa rougherrougher diskdisk (( == 22 μm 22 μm)) areare alsoalso plottedplotted forfor comparison.comparison. TheThe resultsresults areare muchmuch closercloser (Rz = 22 µm) are also plotted for comparison. The results are much closer to those from the equation to those from the equation by Daily and Nece [3]. Hence, the differences can be attributed to the byto Dailythose andfrom Nece the [equation3]. Hence, by the Daily differences and Nece can [3]. be attributed Hence, the to differences the difference can of be surface attributed roughness. to the differencedifference ofof surfacesurface roughness.roughness. EquationsEquations (9)(9) anandd (10)(10) areare thereforetherefore determineddetermined toto satisfysatisfy thethe Equations (9) and (10) are therefore determined to satisfy the experimental results (Rz = 1 µm for experimental results ( = 1 μm for the disk). theexperimental disk). results ( = 1 μm for the disk).

0 FigureFigureFigure 13. 13.13. Comparison ComparisonComparison of ofof the thethe results resultsresults of ofof C M (moment (moment (moment coefﬁcient) coefficient)coefficient) for forforG GG= == 0.018 0.0180.018 and andandG GG= == 0.072 0.0720.072 at atat C D ==0=00..

Int. J. Turbomach. Propuls. Power 2017, 2, 18 13 of 17 Int. J. Turbomach. Propuls. Power 2017, 2, 18 13 of 17

(.∙ ∙ ) = 0.011 ∙ ∙ ∙ (9) 1 1 −4 0 − − (0.8·10 ·|CD |) CM3 = 0.011 · G 6 · Re 4 · [e ] (9) = 0.014 ∙ G ∙Re ∙(.∙ ∙ ) (10) 1 1 −4 0 − (0.46·10 ·|CD |) CM4 = 0.014 · G 10 · Re 5 · [e ] (10) ∙ =π · ε, = 0.978 ∙ (11) k = ,·ε = 0.978 · R (11) s 8 z 100100· ν ∙ k== (12)(12) sl (1(1−− K))· ∙∙r · Ω To introduce the influence of 0 on the moment coefficient, the results of from both the To introduce the influence of CD on the moment coefficient, the results of CM from both the experimentsexperiments andand thethe equationsequations areare plottedplotted versusversus ReRe inin FigureFigure 1414.. WithWith thethe increaseincrease ofof ReRe,, thethe flowflow regimeregime may may change change from from regime regime III III to regimeto regime IV (seeIV (s theee distinguishingthe distinguishing lines lines in Figure in Figure 12). For 12).G For= 0.018 G = and0.018G =and 0.036, G = most 0.036, of most the flow of the regimes flow areregimes regime are III regime and the III results and arethe closeresults to thoseare close from to Equation those from (9) inEquation general, (9) shown in general, in Figure shown 14a,b. in Figure The flow 14a,b. regimes The flow change regimes from change regime from III to regime regime III IV to with regime the IV with the increase 0of Re for | |0 =0 and | | = 1262 at G = 0.036 in Figure 14b. For G = 0.054 increase of Re for CD = 0 and CD = 1262 at G = 0.036 in Figure 14b. For G = 0.054 and G = 0.072, mostand G of = the 0.072, flow most regimes of the are flow regime regimes IV and are theregime results IV areand close the results to those are from close Equation to those (10)from in Equation general, depicted(10) in general, in Figure depicted 14c,d. Thein Figure results 14c,d. of C Thefrom results the equationsof from are the in goodequations agreement are in withgood thoseagreement from M 0 | | with those from experiments.C The values of increase withC the increase of , while decrease experiments. The values of M increase with the increase of D , while decrease with the increase with the increase of Re. At large values of Re,0 the impact of on becomes lesser. At the same0 of Re. At large values of Re, the impact of CD on CM becomes lesser. At the same values of CD , | | thevalues intersection of , pointsthe intersection of the curves points from of Equation the curves (9) from and thoseEquation from (9) Equation and thos (10)e from are closeEquation to those (10) inare Figure close 12to b.those The in difference Figure 12b. can The be attributed difference to can the be existence attributed of to the the mixing existence zone. of the mixing zone.

0 Figure 14. Curves for CM in dependence of CD for different values of Re and G. Figure 14. Curves for in dependence of for different values of Re and G.

TheThe resultsresults fromfrom EquationsEquations (9)(9) andand (10)(10) atat thethe distinguishingdistinguishing lineslines shouldshould bebe equal.equal. TheThe resultsresults ofof CM3//CM4 (CM for regime III/III/ CM forfor regime regime IV) IV) for for a non-dimensionednon-dimensioned gap widthwidth GG atat thethe distinguishingdistinguishing lineslines (see(see FigureFigure 1212b)b) areare presentedpresented inin FigureFigure 15 15.. TheThe differences,differences, attributedattributed toto thethe

Int.Int. J. J. Turbomach. Turbomach. Propuls. Propuls. Power Power 20172017,, 22,, 18 18 1414 of of 17 17 Int. J. Turbomach. Propuls. Power 2017, 2, 18 14 of 17 existence of the mixing zone, are less than 4%. The results indicate that the distinguishing lines (in existence of the mixing zone, are less than 4%. The results indicate that the distinguishing lines (in Figureexistence 12), ofEquations the mixing (9) and zone, (10) are are less reasonable. than 4%. The results indicate that the distinguishing lines Figure 12), Equations (9) and (10) are reasonable. (in Figure 12), Equations (9) and (10) are reasonable.

Figure 15. Results of / ( for regime III/ for regime IV) at the distinguishing lines. / FigureFigure 15. 15.Results Results of ofC M3/CM4 ((CM for for regimeregime III/III/CM forfor regime regime IV) at the distinguishing lines. An example is provided on the applications of the results in this paper. Shi et al. [20] studied the An example is provided on the applications of the results in this paper. Shi et al. [20] studied the axial Anthrust example of a single is provided stage well on thepump applications based on both of the numerical results in simulati this paper.on and Shi experiments. et al. [20] studied The axial thrust of a single stage well pump based on both numerical simulation and experiments. The pressurethe axial acting thrust at of the a single impeller stage is shown well pump in Figure based 16a. on Based both numericalon Equation simulation (7), the thrust and coefficient experiments. at pressure acting at the impeller is shown in Figure 16a. Based on Equation (7), the thrust coefficient at surfaceThe pressure 1 and actingsurface at 2the can impeller be calculated is shown when in Figurethe leakage 16a.Based flow is on estimated. Equation The (7), thevolumetric thrust coefficient leakage surface 1 and surface 2 can be calculated when the leakage flow is estimated. The volumetric leakage through-flowat surface 1 and rate surface is considered 2 can be calculatedas 5% of the when flow the ra leakagete of the flow pump. is estimated. The force Theat the volumetric impeller leakageeye (at through-flow rate is considered as 5% of the flow rate of the pump. The force at the impeller eye (at surfacethrough-flow 3) with rate numerical is considered simulation. as 5% Then, of the the flow axial rate thrust of the of pump.the impeller The forcecan be at calculated the impeller when eye surface 3) with numerical simulation. Then, the axial thrust of the impeller can be calculated when the(at surfaceaxial force 3) with of the numerical shaft is simulation.estimated. They Then, pred the axialicted thrustthe force of the on impellerall the surfaces can be calculatedof the impeller when the axial force of the shaft is estimated. They predicted the force on all the surfaces of the impeller andthe axialthe shaft force to ofcalculate the shaft the is axial estimated. thrust. TheyThe axial predicted force of the the force shaft on is allobtained the surfaces at different of the flow impeller rate and the shaft to calculate the axial thrust. The axial force of the shaft is obtained at different flow rate fromand thethe shaftsimulation to calculate results the by axial Shi thrust.et al. [2 The0]. The axial maximum force of thedifference shaft is obtainedbetween their at different simulation flow from the simulation results by Shi et al. [20]. The maximum difference between their simulation resultsrate from and the measurements simulation results of the byaxia Shil thrust et al. [is20 5.9%.]. The The maximum values of difference − between are plotted their versus simulation results and measurements of the axial thrust is 5.9%. The values of − are plotted versus 0 inresults Figure and 16b. measurements The experimental of the axialresults thrust of is 5.9%.−The are valuesobtained of F byab − subtractingFa f are plotted the forces versus onCD thein in Figure 16b. The experimental results of − are obtained by subtracting the forces on the restFigure of the 16b. surfaces The experimental (from numerica resultsl simulation). of Fab − Fa The f are results obtained from by Equation subtracting (7) are the in forces better on agreement the rest of rest of the surfaces (from numerical simulation). The results from Equation (7) are in better agreement withthe surfaces the experimental (from numerical results simulation).than those from The the results equation from Equationby Kurokawa (7) are et inal. better[4] when agreement ranges with with the experimental results than those from the equation by Kurokawa et al. [4] when0 ranges fromthe experimental 1150 to 4630. results than those from the equation by Kurokawa et al. [4] when CD ranges from from 1150 to 4630. 1150 to 4630.

FigureFigure 16. 16. AxialAxial thrust thrust in in a acentrifugal centrifugal single single stage stage well well pump pump [20]: [20 (a]:) Pressure (a) Pressure distribution distribution and and(b) Figure 16. Axial thrust in a centrifugal single stage well pump [20]: (a) Pressure distribution and (b) Comparison(b) Comparison of of F−ab − F a[20]. f [20 ]. Fab: force: force on on the the back back surface surface of of the the disk; disk; Fa f: :forceforce on on the the front front Comparison of − [20]. : force on the back surface of the disk; : force on the front surfacesurface of of the the disk. disk. surface of the disk. There are still some limitations of this work. The results and correlations are limited to non-pre- ThereThere areare stillstill somesome limitationslimitations ofof thisthis work.work. TheThe results and correlations are limited to non-pre- swirl centripetal through-flow. The centripetal leakage flow in the radial pumps and turbines, swirlswirl centripetalcentripetal through-flow. through-flow. The The centripetal centripetal leakage leakage flow inflow the radialin the pumpsradial andpumps turbines, and however,turbines, however, contains a certain amount of angular momentum, which deserves further investigation. All containshowever, a contains certain a amount certain ofamount angular of angular momentum, momentum, which which deserves deserves further further investigation. investigation. All theAll the experimental results are obtained with the smooth disk ( =1 μm). The applications of the experimentalthe experimental results results are obtained are obtained with the with smooth the smooth disk (Rz disk= 1 µ(m ).=1 μm The applications). The applications of the equations of the equations are still limited because all of the results are influenced by the surface roughness of the areequations still limited are still because limited all of because the results all areof the influenced results byare the influenced surface roughness by the surface of the disks.roughness Some of more the disks. Some more results will be presented with rough disks and the equations will be modified by disks. Some more results will be presented with rough disks and the equations will be modified by

Int. J. Turbomach. Propuls. Power 2017, 2, 18 15 of 17 results will be presented with rough disks and the equations will be modiﬁed by introducing the impact of the surface roughness of the disks in the next step. Currently, the distinguishing lines for regime III and regime IV are obtained by evaluating the tangential ﬂow component based on numerical simulation. This will be put on an experimental level by measuring the velocity components in both tangential and radial direction with a Laser Doppler Velocimetry (LDV) system in the future.

6. Conclusions The influence of centripetal through-flow on the velocity, radial pressure distribution, axial thrust and frictional torque in a rotor-stator cavity with different axial gaps is illustrated to be strong. 0 A correlation is determined, which enables to predict the influence of G, Re and CD on the thrust coefficient CF for a smooth disk (Rz = 1 µm). For the first time, part of the 3D Daily and Nece diagram is obtained by distinguishing the tangential velocity profiles. Currently, the flow regimes are catagorized into two regimes, namely regime III and regime IV. Five distinguishing lines and the approximate distinguishing surface are 0 presented. Two correlations are determined to predict the influence of CD on CM for the two regimes with good accuracy for the smooth disk (Rz = 1 µm). At the distinguishing lines, the results from the two equations are very close. Using the equations for the axial thrust coefficient and the moment coefficient, the influence of the centripetal through-flow can be better considered when designing radial pumps and turbines with smooth impellers. Some more attention will be drawn in the future to the impact of the disk roughness. The 3D Daily and Nece diagram will be modified based on the velocity measurements with a LDV system.

Acknowledgments: This study is funded by China Scholarship Council and the chair of turbomachinery at University of Duisburg-Essen. Author Contributions: Bo Hu, Dieter Brillert, Hans Josef Dohmen and Friedrich-Karl Benra devised and designed the experiments; Hans Josef Dohmen guided the design and the construction of the test rig; Dieter Brillert was responsible for the lab safety and the risk assessment; Friedrich-Karl Benra contributed the funds for the experiments and was supervisor during the research; Bo Hu did all the measurements and wrote the paper while Friedrich-Karl Benra reviewed the paper. Conﬂicts of Interest: The authors declare no conﬂict of interest.

Nomenclature

Latin Symbols a Hub radius b Outer radius of the disk 0 CD Through-flow coefficient CF Axial thrust coefficient CF f CF on the front surface CFb CF on the back surface CM Moment coefficient CMcyl Moment coefficient on the cylinder surface of the disk CM3 CM for regime III CM4 CM for regime IV Cp Pressure coefficient Cqr Local flow rate coefficient eT Relative error of the transducer eD Relative error due to the data acquisition device Fa Axial thrust Fa f Force on the front surface of the disk Fab Force on the back surface of the disk Int. J. Turbomach. Propuls. Power 2017, 2, 18 16 of 17

G Dimensionless axial gap K Core swirl ratio at ζ = 0.5 ks Equivalent surface roughness ksl Limitation of ks for hydraulic smooth wall M Frictional torque Mcyl Frictional resistance on the cylinder surface of the disk Mr Measured range . m Mass ﬂow rate ND Uncertainty of the data acquisition system NT Uncertainty of the transducer ∆N Uncertainty of the measured results n Speed of rotation nT Number of transducers nM Measuring times to obtain one result p Pressure pb Pressure at r = b p∗ Dimensionless pressure Q Volumetric through-ﬂow rate Re Global circumferential Reynolds number Reϕ Local circumferential Reynolds number r Radial coordinate rseal Radius of shaft seal s Axial gap of the front chamber sb Axial gap of the back chamber t Thickness of the disk Vr Dimensionless radial velocity Vz Dimensionless axial velocity Vϕ Dimensionless tangential velocity x Dimensionless radial coordinate z Axial coordinate Greek Symbols ε Diameter of spheres ζ Dimensionless axial coordinate µ Dynamic viscosity of water v Kinematic viscosity of water ρ Density of water Ω Angular velocity of the disk Abbreviations FS Full scale LDA Laser Doppler Anemometer LDV Laser Doppler Velocimetry SST Shear Stress Transport

References

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