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International Journal of Turbomachinery, Propulsion and Power
, Volume 5 (2) – Apr 17, 2020

/lp/multidisciplinary-digital-publishing-institute/impact-of-leakage-inlet-swirl-angle-in-a-rotor-stator-cavity-on-flow-yuQOztbAjR

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International Journal of Turbomachinery Propulsion and Power Article Impact of Leakage Inlet Swirl Angle in a Rotor–Stator Cavity on Flow Pattern, Radial Pressure Distribution and Frictional Torque in a Wide Circumferential Reynolds Number Range Tilman Raphael Schröder * , Hans-Josef Dohmen, Dieter Brillert and Friedrich-Karl Benra Chair of Turbomachinery, University of Duisburg-Essen, 47057 Duisburg, Germany; hans-josef.dohmen@uni-due.de (H.-J.D.); dieter.brillert@uni-due.de (D.B.); friedrich.benra@uni-due.de (F.-K.B.) * Correspondence: tilman.schroeder@uni-due.de † This paper is an extended version of that published in the Proceedings of 13th European Conference on Turbomachinery Fluid Dynamics & Thermodynamics, ETC13, Lausanne, Switzerland, 8–12 April 2019; Paper No. 171. Received: 30 January 2020; Accepted: 1 April 2020; Published: 17 April 2020 Abstract: In the side-chambers of radial turbomachinery, which are rotor–stator cavities, complex ﬂow patterns develop that contribute substantially to axial thrust on the shaft and frictional torque on the rotor. Moreover, leakage ﬂow through the side-chambers may occur in both centripetal and centrifugal directions which signiﬁcantly inﬂuences rotor–stator cavity ﬂow and has to be carefully taken into account in the design process: precise correlations quantifying the effects of rotor–stator cavity ﬂow are needed to design reliable, highly efﬁcient turbomachines. This paper presents an experimental investigation of centripetal leakage ﬂow with and without pre-swirl in rotor–stator cavities through combining the experimental results of two test rigs: a hydraulic test 5 6 rig covering the Reynolds number range of 4 10 Re 3 10 and a test rig for gaseous 7 8 rotor–stator cavity ﬂow operating at 2 10 Re 2 10 . This covers the operating ranges of hydraulic and thermal turbomachinery. In rotor–stator cavities, the Reynolds number Re is deﬁned as Re = W /n with angular rotor velocity W, rotor outer radius b and kinematic viscosity n. The inﬂuence of circumferential Reynolds number, axial gap width and centripetal through-ﬂow on the radial pressure distribution, axial thrust and frictional torque is presented, with the through-ﬂow being characterised by its mass ﬂow rate and swirl angle at the inlet. The results present a comprehensive insight into the ﬂow in rotor–stator cavities with superposed centripetal through-ﬂow and provide an extended database to aid the turbomachinery design process. Keywords: rotor–stator cavity; leakage ﬂow; centripetal through-ﬂow; disc torque; axial thrust; radial pressure distribution 1. Introduction In rotor–stator cavities, which are found in all radial turbomachines, complex ﬂow patterns occur and inﬂuence axial thrust on the shaft as well as disc-friction torque. During the design phase of radial turbomachinery, to design axial bearings reliably we have to know the maximum axial thrust on the shaft with sufﬁcient precision. Moreover, if the geometry of the cavities is designed carefully, losses due to friction can be minimised and efﬁciency maximised. The ﬁrst thorough experimental study of turbulent rotor–stator cavity ﬂow was conducted by Daily and Nece [1], who designed a test rig able to reach Reynolds numbers up to Re = 10 . Based on their torque and velocity distribution measurements, they identify two turbulent ﬂow regimes in which Int. J. Turbomach. Propuls. Power 2020, 5, 7; doi:10.3390/ijtpp5020007 www.mdpi.com/journal/ijtpp Int. J. Turbomach. Propuls. Power 2020, 5, 7 2 of 17 rotor and stator boundary layers are either merged or separated. When these layers are separated, velocity measurements indicate the existence of a ﬂuid core in the cavity middle, where radial velocity is negligible and circumferential velocity varies only with radius, not with axial coordinate. The existence of these ﬂow regimes depends on circumferential Reynolds number Re and relative axial gap width, deﬁned as the ratio of axial gap width to disc outer radius. They also provide empirical correlations for the torque coefﬁcient c for each ﬂow regime, allowing turbomachinery designers to estimate disc torque and machine efﬁciency during the design phase. In [2], models for rotor–stator cavity ﬂow with through-ﬂow are introduced which are based on assumed velocity proﬁles and wall shear stress measurements. The models allow for the calculation of radial pressure distribution and axial thrust for the cases of closed cavity and centripetal through-ﬂow. Additionally, in the through-ﬂow case, they take into account the angular momentum ﬂux into the cavity. This paper tests the validity of these models at a large Reynolds number range. Radtke and Ziemann [3] experimentally investigated not only closed rotor–stator cavities, but also centripetal and centrifugal through-ﬂow with and without preswirl alongside variations of the cavity geometry. We compare our measurement results to theirs. Poncet et al. [4] investigated turbulent ﬂow with separated boundary layers in a rotor–stator cavity with and without centripetal through-ﬂow in the Reynolds number range of 1.04 10 Re 4.2 10 at relative cavity widths of 0.024 G 0.048. In their paper, Laser Doppler Anemometry measurements of core swirl ratio are compared to predictions based on measurements of radial pressure distribution. Using the /7 power law for the boundary layer velocity proﬁles, they derived a theoretical model for the core swirl ratio K (r) which is calibrated using the measurement results. The validity of their model is reported for a wide range of Reynolds numbers and through-ﬂow mass ﬂuxes. Poncet et al. [5] proposed a new Reynolds stress turbulence model suitable for numerical investigations of rotor–stator cavity ﬂow that is reported to be superior to k-# turbulence models for this application. They investigated the mean ﬂow structure, the core swirl ratio, the radial pressure distribution and turbulence statistics for closed cavities, and centripetal as well as centrifugal through-ﬂow. Their paper gives a comprehensive summary of ﬂow structures with separated boundary layers inside rotor–stator cavities. Will [6] investigated centripetal and centrifugal ﬂow in a rotor–stator cavity, using analytical, experimental and numerical methods. He developed a new one-dimensional ﬂow model of core swirl ratio, assuming separated boundary layers and using the logarithmic law of the wall. Wang et al. [7] performed particle image velocimetry measurements in a water-ﬁlled microscale rotor–stator cavity made of glass with smooth compared to hydrophobic discs at low Reynolds numbers. They found a reduction of torque of more than 50 % when using hydrophobic rotors, which was attributed to a thin air layer between disc and liquid in the case of hydrophobic rotor surfaces and to a reduction of turbulence intensity. In the present study, experimental results of two rotor–stator test rigs are combined to study how disc torque, radial pressure distribution and axial thrust are inﬂuenced by Reynolds number, cavity width, centripetal through-ﬂow and angular momentum ﬂux into the cavity. The small test rig is operated with water, can be equipped with different discs with an outer radius of b = 110 mm each 5 6 and is made of PMMA (Plexiglass). It covers a Reynolds number range of 3.8 10 Re 3.2 10 and an axial gap width range of 0.0182 G 0.0728. A more detailed description can be found in [8–10]. The large test rig, ﬁrst introduced by Barabas et al. [11], can reach Reynolds numbers up to Re = 2.5 10 since it has a large disc radius and is operated with carbon dioxide. Relative gap widths of 0.0125 G 0.0375 are investigated in this test rig. Both rigs allow the superimposition of centripetal through-ﬂow with different inlet swirl angles. Only corotating inlet swirl is investigated. 2. Rotor–Stator Cavity Test Rigs, Measurement Procedures and Uncertainty Calculation A schematic of the large rotor–stator test rig used in this study is given in Figure 1. The rotor has angular velocity W, outer radius b and hub radius a. Cavity width is given by distance s. Through-ﬂow Int. J. Turbomach. Propuls. Power 2020, 5, 7 3 of 17 with mass ﬂow rate m ˙ enters the cavity through a centripetal swirl generator, which has inlet width d, and exits axially at the hub. Swirl generators are changed to create through-ﬂow with different inlet swirl angles a. In the case of a closed cavity, the swirl generator is replaced by a ring structure, resulting in a cylindrical shroud without any opening. The rig is cooled by water to remove the heat Version March 27, 2020 submitted to Int. J. Turbomach. Propuls. Power 3 of 17 generated by ﬂuid friction. swirl generator p (b), T (b) Stator back Figure 1. A schematic of the p r ( ) large rotor–stator cavity test Back rig. The dashed lines indicate cavity the pressure measurements; measured torque coefﬁcients c p r p, min correspond to the torque acting m ˙ p, min on the surface indicated by the Rotor red line. Figure 1. A schematic of the large rotor–stator cavity test rig. The dashed lines indicate the pressure 77 2. Rotor–Stator Cavity Test Rigs, Measurement Procedures and Uncertainty Calculation measurements; measured torque coefﬁcients c correspond to the torque acting on the surface indicated by the red line. 78 A schematic of the large rotor–stator test rig used in this study is given in Figure 1. The rotor has 79 angular velocity Ω, outer radius b, and hub radius a. Cavity width is given by distance s. Through-ﬂow A detailed drawing of the small test rig can be seen in ([10], Figure 33), including the sensors used. 80 with mass ﬂow rate m ˙enters the cavity through a centripetal swirl generator, which has inlet width It is constructed similarly to the large test rig but does not have a hub in its front cavity. Additionally, 81 d, and exits axially at the hub. Swirl generators are changed to create through-ﬂow with different it does not need to be cooled. 82 inlet swirl angles α. In the case of a closed cavity, the swirl generator is replaced by a ring structure, To determine the axial force resulting from the radial pressure distribution in the front cavity 83 resulting in a cylindrical shroud without any opening. The rig is cooled by water to remove the heat of the small test rig, the following measurement technique was used; greater detail is given in [10]. 84 generated by ﬂuid friction. 85 A detailed drawing of the small test rig can be seen in Hu [10, Fig. 33], including the sensors used. Tension–compression force transducers and a linear ball bearing are used to measure the axial thrust 86 It is constructed similarly to the large test rig but does not have a hub in its front cavity. Additionally, on the rotor and the shaft. The sum of the axial forces on both parts is given by 87 it does not need to be cooled. Z Z h h i i 88 To determine the axial force resulting 2 from the radial 2 pressur 4 e distribution 4 4 in the front cavity F = F F = p (r) dA p (r) d A = p a p (b) pW r c b a c b (1) ab af b f Fb Ff 89 of the small test rig, the following measurement technique was used; greater detail is given in [10]. A A b f 90 Tension-compression force transducers and a linear ball bearing are used to measure the axial thrust 91 on the rotor and the shaft. The sum of the axial forces on both parts is given by where F is the axial force measured by tension–compression force transducers, and F and F are the ab af axial forces due to the pressure p in the back and the pressure p in the front cavity, respectively. The b f Z Z h h i i radial pressure distribution in the back and front cavity is characterised by the axial thrust coefﬁcients 2 2 4 4 4 F = F − F = p (r) dA− p (r) d A = −π a p (b)− πΩ ρ c b − a − c b (1) ab af b f Fb Ff c and c , respectively, which are deﬁned as Fb Ff A A b f 2 2 p b a p (r = b) F ab 92 where F is the axial force measured by tension-compression force transducers, and F and F are the a ab af c = (2) Fb 2 4 4 prW [b a ] 93 axial forces due to the pressure p in the back and the pressure p in the front cavity, respectively. The b f 94 radial pressure distribution in the back and front cavity is characterised by the axial thrust coefﬁcients pb p (r = b) F af c = . (3) Ff 95 c and c , respectively, which are2deﬁned 4 as Fb Ff prW b In the case of a closed cavity without through-ﬂow and equal widths of the front and back cavity 2 2 π b − a p (r = b)− F ab (s = s ), identical ﬂow structures in the front and back cavity are assumed, which leads to the c = (2) back Fb 2 4 4 πρΩ [b − a ] assumption c = c . In this test rig conﬁguration, the axial thrust coefﬁcient Fb Ff πb p r = b − F ( ) af c = . (3) Ff 2 4 πρΩ b F + p a p (r = b) c = (4) Fb 2 4 prW a 96 In the case of a closed cavity without through-ﬂow and equal widths of the front and back cavity 97 (s = s ), identical ﬂow structures in the front and back cavity are assumed, which leads to the back 98 assumption c = c . In this test rig conﬁguration, the axial thrust coefﬁcient Fb Ff Int. J. Turbomach. Propuls. Power 2020, 5, 7 4 of 17 of the back cavity is calculated from measurements of the net axial thrust F . The measured coefﬁcient c is transformed into a correlation, which is then used to calculate the axial thrust coefﬁcients c in the Fb Ff front cavity from net axial force F measurements using Equation (1) in different test rig conﬁgurations. The large test rig has nine taps to measure radial pressure distribution, to which differential pressure transducers are connected to measure the pressure differences p r = b p r directly ( ) ( ) (see Figure 1). In addition, temperature and absolute pressure are measured at the outer radius b. From these measurements, density r and kinematic viscosity n are calculated. The radial pressure distribution in the small test rig is measured at twelve distinct radial positions; density and kinematic viscosity are assumed to be constant [10]. The pressure coefﬁcient p (r = b) p (r) c (r) = (5) 2 2 rW b is a dimensionless measure of static pressure p (r) relative to that at disc outer radius b. The axial thrust coefﬁcient c can be calculated using the pressure coefﬁcient c by F p c = c (r) r dr (6) b r p, min p, min and this procedure is employed for the large test rig. The minimal pressure measurement radius is denoted by r (see Figure 1). The integral is approximated using the trapezoidal rule, i.e., p, min pressure coefﬁcient c is assumed to vary linearly with the radius r between two adjacent pressure measurement taps. The torque coefﬁcient front c = 2 (7) 2 5 rW b measures the ﬂuid friction torque M on the front-facing side of the disc, indicated by a red line front in Figure 1. It is calculated as follows for both test rigs (see [10]): the overall torque M is measured all between the motor and the coupling, and includes the friction torque M generated by the bearings friction and the ﬂuid friction on the shaft. The ﬂuid friction in the back cavity is modelled by a correlation, c , and the friction on the cylindrical outer rotor surface is given by the correlation c . M, back M, cylindrical This leads to M M (r, W) all friction c = 2 c c (8) M M, back M, cylindrical 2 5 rW b where M (r, W) is a correlation derived from measurements with no disc installed. In the large friction test rig, this is conducted at different CO pressure levels p and the shaft rotating at varying angular velocities W. In the small test rig, only the angular velocity W is varied [10]. The correlation c is found by setting the front and back cavity widths equal, s = s , with M, back back the test rigs in the closed cavity conﬁguration and by assuming c = c . The overall torque M, back M, front M at varying angular velocities W and pressure levels p in the large test rig is measured, then c all M, back is found from M M (r, W) all friction 2c = 2 c . (9) M, back M, cylindrical 2 5 rW b In [3], a correlation c is given which predicts the friction on the cylindrical outer rotor M, cylindrical surface in rotor–stator cavities. It is assumed that this correlation holds for the friction on this surface in the large test rig, although extrapolation to Reynolds numbers higher than those investigated by Radtke and Ziemann [3] is required. This correlation is used in the torque coefﬁcient calculation in the large test rig case. The friction on the cylindrical outer rotor surface is neglected in the small test rig [10]. Int. J. Turbomach. Propuls. Power 2020, 5, 7 5 of 17 For the large test rig, the calculation of torque coefﬁcient uncertainties includes measurement uncertainties for the following quantities: overall torque M , breakaway torque, temperature T, all absolute static pressure p, angular rotor velocity W and manufacturing tolerances. The uncertainty of the correlation c is not known and not included in the calculation, meaning that the M, cylindrical torque coefﬁcient uncertainties calculated here provide a lower bound; in reality they may be higher. To calculate these uncertainties, the ﬁrst-order error propagation method is used as presented in [12]. Uncertainties of measurements in the small test rig are taken from [10]. The curve ﬁt and model parameter uncertainty calculations are carried out as presented in [13]; the method is brieﬂy summarised here. Let x = (x , , x ) be a column vector of measured values 1 n with associated covariances u x , x and y = (y , , y ) be a column vector of unknown model i j 1 m parameters. The column vector of model equations is given by M (x, y) = 0. The measured values x will most likely not fulﬁl the model M (x, y) = 0 for any vector of model parameters y; therefore, the vector x is replaced by a column vector z = (z , . . . , z ) of free parameters that should fulﬁl the 1 n model equations M (z, y) = 0 while minimising the squared error norm c . This error norm assigns a high weight to measurement results with low uncertainties and vice versa. The problem is solved, T T T using a column vector b of Lagrange multipliers, for the unknown column vector w = b , y , z by ﬁnding a Karush–Kuhn–Tucker point w of the Lagrange function 1 1 2 T 1 T L (x, w) = c + M (z, y) b = [z x] U [z x] + M (z, y) b. (10) xx 2 2 Here, U = u x , x are components of the matrix U of measurement covariances. The ( ) xx i j xx i j Karush–Kuhn–Tucker conditions for this problem, which are solved for w, read 0 1 0 1 ¶L M z, y ( ) (x, w) ¶b B C B C ¶L ¶M (x, w) (z, y) b M (x, w) = = = 0. (11) @ A @ A ¶y ¶y ¶L ¶M x, w ( ) z x + U (z, y) b xx ¶z ¶z A solution w to these conditions includes the vector y of optimal model parameters. The covariances u w , w , including the covariances u y , y of optimal model parameters i j i j y , are derived by ﬁrst order error propagation and are given by U = Q (w = w ) U Q (w = w ) (12) w w xx ¶M ¶M Q (w) = (x, w) (x, w) . (13) ¶w ¶x Here, (U ) = u w , w are the components of the matrix U of covariances of the vector w . w w w w i j i j The standard uncertainties u y are then given by u y = u y , y . i i i i If the uncertainties of the measured values are unknown, the curve ﬁt method reduces to calculating the Karush–Kuhn–Tucker point y that minimises the squared error norm /2 M (x, y ) M (x, y ). In this case, all measured values x are equally weighted. 3. Closed Cavity The ﬂow in a closed cavity with a cylindrical shroud but without preswirl guide vanes is taken as the reference conﬁguration, the conﬁguration originally studied by Daily and Nece [1]. In this section, the dependency of the torque coefﬁcient, the pressure coefﬁcient and the axial thrust coefﬁcient on the circumferential Reynolds number and the relative axial gap width are investigated. In later sections, the inﬂuence of centripetal through-ﬂow is compared to this reference conﬁguration. Int. J. Turbomach. Propuls. Power 2020, 5, 7 6 of 17 3.1. Torque Coefﬁcient In the radial turbomachinery design process, frictional losses in side chambers should be minimised. The gap width has a signiﬁcant inﬂuence on frictional losses; the correlations by Daily and Nece [1] showed that disc torque is minimal in the transition zone between the two ﬂow regimes. Moreover, they found that for merged and separated boundary layers, the torque coefﬁcient c is 0.25 0.20 proportional to Re and Re , respectively. Figure 2 compares torque measurements of closed cavities with hydraulically smooth discs from the small, water-operated test rig from [10], the large carbon dioxide test rig, and the test bench used by Radtke and Ziemann [3], and gives the correlations provided by Daily and Nece [1]. All uncertainties shown as error bars in ﬁgures are 99 % (2.576 standard deviations) conﬁdence intervals; all those quoted in text are 68 % (1 standard deviation) intervals. Figure 2. Measured torque coefﬁcients (see Equation (7)) of closed rotor–stator cavities with hydraulically smooth disc. The lines belonging to measurement points are best curve ﬁts. “Hu (2018)” refers to [10], “R., Z. (1982)” to [3], and “D., N. (1960)” to [1]. The low torque coefﬁcient values of the small test rig are a result of the hydrophobic properties of the material used and its low surface roughness: PMMA is used for both the disc and the casing; its transparency implies that its surface is very smooth. In the large, carbon dioxide operated test rig, the installed disc is polished and hydraulically smooth at all Reynolds numbers investigated. Optical measurements of the disc surface do not show any signiﬁcant roughness. In contrast, the shroud and the casing are rough, the roughness of the latter being approximately 10 μm tip-to-tip. Moreover, holes are drilled in the cylindrical outer disc surface for balancing purposes, which increases the torque. All these aspects inﬂuence torque magnitude and make a direct comparison of the different test rig results difﬁcult. Therefore, only the behaviour of the torque coefﬁcients with varying Reynolds number and different axial gap widths is investigated. For the small test rig, the change in torque coefﬁcient with increasing Reynolds number shows a good correlation with the results of Daily and Nece [1], with torque coefﬁcient being proportional 0.253 0.010 0.223 0.009 to Re and Re for the small cavity width G = 0.0182 and the large cavity width G = 0.0728, respectively. According to Daily and Nece [1], measurements at the small cavity width Int. J. Turbomach. Propuls. Power 2020, 5, 7 7 of 17 G = 0.0182 are in the merged boundary layer regime, while those at the large cavity width G = 0.0728 feature separated boundary layers. In this range of Reynolds number up to Re = 4 10 and with incompressible ﬂuids, no deviation from the predictions made by Daily and Nece [1] can be seen. The large test rig and that of Radtke and Ziemann [3] are operated with compressible gases, carbon dioxide and air, respectively. Furthermore, they operate in Reynolds number regions of Re 3 10 . Torque data from these test rigs show that by increasing the Reynolds number Re, the torque coefﬁcient c decreases signiﬁcantly more slowly than in the Reynolds number range of Re 4 10 : the 0.134 0.003 largest torque coefﬁcient decrease is proportional to Re (large test rig, G = 0.0125), while 0.07 the slowest is proportional to Re ([3], G = 0.0125), as shown by the curve ﬁts in Figure 2. This suggests that the regimes introduced by Daily and Nece [1] cannot reliably be applied to rotor–stator cavities at Reynolds numbers Re 10 . It is important to note that the behaviour of the torque coefﬁcient differs for each test rig with respect to the relative cavity width: the data by Radtke and Ziemann [3] show that with increasing relative cavity width, the exponent c in c µ Re decreases, while the data from the large test rig show the opposite: with increasing relative cavity width, the exponent c increases. 3.2. Radial Pressure Distribution A radial pressure gradient develops in rotor–stator cavities, which exerts axial thrust on the rotor disc or on the shaft in a radial turbomachine. To predict this precisely, the radial pressure distribution must be known for the operating point of the machine. Kurokawa and Sakuma [2] proposed two models to calculate the core swirl ratio K = /Wr (with circumferential velocity u at the cavity middle z = /2) and the pressure coefﬁcient c (r). Both j p models are applicable to rotor–stator cavity ﬂow without through-ﬂow as well as with centripetal through-ﬂow. Each model is based on the characteristics of one of the two ﬂow regimes identiﬁed by Daily and Nece [1]. The interference gap model assumes the absence of a rotating core, the presence of boundary layers on the rotor and the stator with a thickness of half the cavity width each, and velocity distributions according to the /7 power law. In the non-interference gap model, a rotating core with axially constant circumferential and vanishing radial velocity is assumed to exist between the boundary layers. The models take into account the circumferential Reynolds number Re; additionally, the interference gap model includes the relative cavity width G, but the non-interference gap model does not. In the case of closed cavity ﬂow, no other variables are taken into account. Figure 3 shows the variation of the radial pressure distribution for a range of circumferential Reynolds numbers at a small cavity width. Measurement data for the small test rig are taken from [10]. According to Kurokawa and Sakuma [2], only the small test rig operates in the non-interference gap region and all other measurements are at operating conditions with interfering boundary layers. 6 7 Measurements in the Reynolds number range of 1.5 10 Re 2.6 10 show that with increasing Reynolds number, the radial pressure gradient increases, and at Re 2.6 10 the radial pressure gradient is greatest. However, although the Kurokawa and Sakuma models predict similar behaviour with increasing Reynolds numbers in this range, their predictions deviate signiﬁcantly from the observed data, especially at low relative radii /b. In addition, a further increase of the Reynolds number Re to the range of Re 3 10 leads to a decrease of the radial pressure gradient, while the models by Kurokawa and Sakuma [2] predict a continued increase. Note that in the small test rig, the changes in radial pressure gradient with Reynolds number are small for the relative cavity width G = 0.0182 [10], thus only one representative set of measurements is shown. Int. J. Turbomach. Propuls. Power 2020, 5, 7 8 of 17 Figure 3. Radial pressure distribution c (r) (see Equation (5)) at small cavity widths with closed cavities. The markers denote measurements; lines are predictions using the models proposed by Kurokawa and Sakuma [2]. /b is the radial position relative to the discs’ outer radius. “Hu (2018)” refers to [10], and “R., Z. (1982)” to [3]. 3.3. Axial Thrust Coefﬁcient In Figure 4, axial thrust coefﬁcients c are plotted against Reynolds numbers Re for different relative cavity widths G alongside predictions by the models given in Kurokawa and Sakuma [2]. The axial thrust coefﬁcient decreases as cavity width increases and as Reynolds number increases. For the small test rig, c decreases signiﬁcantly faster with increasing Reynolds number than for the large test rig. Moreover, with small relative cavity width G = 0.0182, the decrease in axial thrust coefﬁcient slows down for Reynolds numbers Re 2 10 , and thus its slope in this range is much closer to the slope of the axial thrust coefﬁcients obtained from the large test rig. This change in slope indicates a change in ﬂow characteristics, for example, an increasing inﬂuence of turbulence. Further research is needed to investigate this phenomenon. 5 8 Considering the whole Reynolds number range of 3.8 10 Re 2.5 10 and all axial gap widths investigated, a continuous decrease of axial thrust coefﬁcients with increasing Reynolds numbers is apparent. There is no hint of discontinuities or other phenomena in the Reynolds number range where no measurement data are available. Fitting the axial thrust coefﬁcients c measured in the small and the large test rig to a curve results in the empirical approximation h i 0.505 2 2 7 c = 3.76 10 + 2.28 10 10 Re . (14) This curve is plotted in Figure 4 as a brown dotted line. It is again apparent that the ﬂow models by Kurokawa and Sakuma [2] fail to capture the variation of axial thrust with Reynolds number, but the inﬂuence of different cavity widths is reproduced qualitatively. Measurements published by Radtke and Ziemann [3] are much closer to predictions by Kurokawa and Sakuma [2] than to the results of the small and the large test rig, and do not show any signiﬁcant inﬂuence of the cavity width. Int. J. Turbomach. Propuls. Power 2020, 5, 7 9 of 17 Figure 4. Change of axial thrust coefﬁcient c with Reynolds number Re. Markers are measurements of the large test rig, the small test rig by Hu [10], and the test rig by Radtke and Ziemann [3]. Lines of the same colour as a set of markers are predictions by the models given in [2]. Lines and markers of the same colour share operating conditions, i.e. their axial cavity widths G are the same for both the large and small test rig. The ﬁnal curve is a best ﬁt to the measurement results of the large test rig and the data of Hu [10]. Uncertainties are neglected in the ﬁt. 4. Centripetal Through-Flow In a radial turbomachine, superposed centripetal through-ﬂow can occur in a side chamber, signiﬁcantly inﬂuencing the coefﬁcients investigated earlier. The mass ﬂow through the cavity is given by the through-ﬂow coefﬁcient m ˙ c = (15) rWb with c < 0 for centripetal through-ﬂow. The small and large test rigs can operate at through-ﬂow 3 3 coefﬁcient ranges of 6.64 10 c 0 and 1.45 10 c 0, respectively. D D A signiﬁcant amount of angular momentum can be carried into the rotor–stator cavity if the inﬂow has a circumferential velocity component; this angular momentum ﬂow is given by the coefﬁcient m ˙ tan a c = (16) 2 2 5 2 4p r b W d where a is the preswirl guide vane angle relative to the radial direction. With a preswirl angle of a = 0°, radial inﬂow occurs and no angular momentum ﬂows into the cavity, resulting in an angular momentum coefﬁcient of c = 0. Corotational swirl corresponds to preswirl angles of a > 0°, with the angular momentum coefﬁcient c being positive. The small test rig covers the range of 5 5 0 c 9.6 10 and the large one 0 c 1.6 10 . L L It is important to note that computational ﬂuid dynamics simulations along with measurements in the small test rig not presented here show a signiﬁcant inﬂuence of centripetal inlet area on the results: a jet develops at the inlet, with smaller inlet areas generating higher velocity jets for the same 3 2 mass ﬂow rate. The inlet gap widths d are 3.75 10 and 1.82 10 times the disc radius b for the large and small test rig, respectively. Results are presented separately. 4.1. Large Test Rig The large test rig has four different conﬁgurations: in addition to the closed cavity, one of three different preswirl guide vanes for centripetal through-ﬂow may be installed. The preswirl angles investigated are 0°, 26° and 52°. Int. J. Turbomach. Propuls. Power 2020, 5, 7 10 of 17 Figure 5 illustrates the torque coefﬁcients c measured with and without centripetal through-ﬂow 7 7 for a relative gap width of G = 0.0125 and in the Reynolds number range of 7.5 10 Re 9.6 10 . It is apparent that with increasing centripetal mass ﬂow (c decreasing below zero), the torque coefﬁcient c ﬁrst decreases, reaches a minimum and then increases to higher values than that of the closed cavity conﬁguration. The torque coefﬁcient minima are all found at through-ﬂow mass ﬂux coefﬁcients of c 6.1 10 , but at different angular momentum ﬂow coefﬁcients c . This D L indicates that the point where the minima occur depends only on mass ﬂow, and not on inlet swirl angle. However, the torque coefﬁcient values strongly depend on the angular momentum ﬂow into the cavity, with high preswirl angles leading to lower torque coefﬁcients at the same mass ﬂux coefﬁcients. Figure 5. Change of measured torque coefﬁcients c with through-ﬂow coefﬁcient c and angular M D momentum coefﬁcient c at the relative gap width of G = 0.0125 and in the Reynolds number range of 7 7 7.5 10 Re 9.6 10 : the closed cavity conﬁguration is compared to centripetal through-ﬂow at preswirl angles of 0°, 26° and 52°. For large Reynolds numbers of Re 1.8 10 shown in Figure 6, the decline of torque coefﬁcients from radial inﬂow to a preswirl angle of 26° is much larger than for the moderate Reynolds numbers in Figure 5, but the torque coefﬁcients at preswirl angles 26° and 52° differ signiﬁcantly only at high through-ﬂow mass ﬂux (small through-ﬂow coefﬁcient values c < 0). Minimum torque coefﬁcients 4 4 are found at c 3.7 10 for radial inﬂow and the 26° preswirl angle, but at c 5 10 D D for the 52° preswirl angle. For this gap width and Reynolds number range, the angular momentum ﬂux entering the cavity plays an important role: for radial inﬂow, where by deﬁnition c = 0, the minimum torque coefﬁcient is c = 8 10 , but with increasing angular momentum inﬂow in the case of corotating preswirl, the torque coefﬁcient quickly drops to values below 6.5 10 . Int. J. Turbomach. Propuls. Power 2020, 5, 7 11 of 17 Figure 6. Torque coefﬁcients c measured at the relative gap width of G = 0.0125 and in the Reynolds 8 8 number range of 1.8 10 Re 2.9 10 : The closed cavity conﬁguration is compared to centripetal through-ﬂow at three different preswirl angles. 7 7 With a relative cavity width of G = 0.0375 and a Reynolds number of 7.5 10 Re 9.6 10 , the torque coefﬁcient behaves as shown in Figure 7. The only difference in experimental set-up from that of Figure 5 is that the cavity width is three times larger. Again, torque coefﬁcients are signiﬁcantly higher with radial inﬂow (a = 0°, no preswirl) than with preswirl, but while a difference between preswirl angles of 26° and 52° is observed at the smaller cavity width, no signiﬁcant difference is observed here. For this larger cavity width, the minima of the torque coefﬁcient seem to be shifted to smaller through-ﬂow mass ﬂux coefﬁcients compared to the small cavity width, but this conclusion is not certain since the uncertainties of the measured torque coefﬁcient are about as large as the change in the torque coefﬁcients itself. The explanation for greater torque coefﬁcients in the case of radial inﬂow, which has been observed, is as follows. If through-ﬂow enters the cavity radially, i.e., with a preswirl angle of a = 0°, it creates a large gradient in the axial direction between the circumferential velocities of the ﬂuid and disc close to its outer radius. This gives rise to a large drag, resulting in torque coefﬁcients signiﬁcantly higher than in the closed cavity or the centripetal through-ﬂow case with corotating preswirl. The difference in circumferential velocity between the inﬂow and the disc is smaller with corotating preswirl, therefore the velocity gradient, drag and torque coefﬁcient decrease. The development of axial thrust coefﬁcients with centripetal through-ﬂow is shown in Figure 8 7 7 for the same measurements that are shown in Figure 5 (G = 0.0125, 7.5 10 Re 9.6 10 ): centripetal through-ﬂow signiﬁcantly increases the thrust coefﬁcient as essentially it is an accelerated ﬂow which results in a larger radial pressure gradient. With increasing centripetal mass ﬂow (decreasing c ), the axial thrust coefﬁcient reaches a maximum for preswirl angles of 26° and 52° at a through-ﬂow coefﬁcient of c 1.1 10 . For radial inﬂow, it seems that the thrust coefﬁcient converges towards a maximum, too, which is outside of the investigated through-ﬂow mass ﬂux range. For this gap width and Reynolds number, centripetal through-ﬂow with preswirl angles of 26° and 52° generates higher axial thrust coefﬁcients compared to radial inﬂow with a preswirl angle of 0°. The small differences in the axial thrust coefﬁcients between measurements with preswirl angles of 26° and 52° are not considered signiﬁcant, because the experiment at the preswirl angle of 52° was Int. J. Turbomach. Propuls. Power 2020, 5, 7 12 of 17 conducted at a slightly higher Reynolds number than those for 26°. The angular momentum coefﬁcient c alone is again insufﬁcient to describe the axial thrust behaviour because multiple different axial thrust coefﬁcient values are observed for one angular momentum coefﬁcient value. Figure 7. Torque coefﬁcients c measured at the relative gap width of G = 0.0375 and in the Reynolds 7 7 number range of 7.5 10 Re 9.6 10 : The closed cavity conﬁguration is compared to centripetal through-ﬂow at three different preswirl angles. Figure 8. Change of measured axial thrust coefﬁcients c with through-ﬂow coefﬁcient c and angular F D momentum coefﬁcient c at the relative gap width of G = 0.0125 and in the Reynolds number range of 7 7 7.5 10 Re 9.6 10 : the closed cavity conﬁguration is compared to centripetal through-ﬂow at preswirl angles of 0°, 26° and 52°. Int. J. Turbomach. Propuls. Power 2020, 5, 7 13 of 17 To investigate the inﬂuence on axial thrust of mass and angular momentum ﬂow into the cavity, the radial pressure distribution of the closed cavity conﬁguration and the centripetal through-ﬂow case with 52° preswirl angle is plotted in Figure 9. To provide a better view of the pressure gradients, markers and error bars are omitted. With increasing centripetal through-ﬂow, the pressure at relative radii of /b < 0.7 decreases as indicated by larger pressure coefﬁcient values c . Simultaneously, the pressure at relative radii of 0.75 < /b < 1 increases, leading to lower pressure coefﬁcients compared to the closed cavity conﬁguration. Since in the computation of axial thrust coefﬁcients the pressure coefﬁcients are weighted with the radial position r, a small pressure increase at large radii can balance larger pressure drops at small radii. This happens at the maxima of axial thrust coefﬁcients as at this point a change in the radial pressure distribution does not cause a change in c . Figure 9. Measured radial pressure distribution at the relative cavity width of G = 0.0125 and in the 7 7 Reynolds number range of 7.5 10 Re 9.6 10 : the closed cavity conﬁguration is compared to centripetal through-ﬂow at the preswirl angle of 52°. 4.2. Small Test Rig In Figure 10, axial thrust coefﬁcients c and torque coefﬁcients c measured in the small test rig F M with the cavity closed are compared to those obtained with radial inﬂow for the small relative cavity width of G = 0.0182. The result shows an almost linear increase of thrust and torque coefﬁcient with decreasing through-ﬂow coefﬁcient c , with the slope depending on the circumferential Reynolds ¶ c ¶ c number: as Reynolds number increases, the slopes M/¶ c and F/¶ c decrease. D D In comparison to the results of the large test rig, it appears that the torque coefﬁcient minima, observed in the large test rig for moderate through-ﬂow mass ﬂow rates and radial inﬂow, do not appear. The axial thrust coefﬁcients of the large test rig with radial inﬂow, plotted in Figure 8, increase with decreasing through-ﬂow coefﬁcients, just as in the small test rig. However, in the large test rig, this increase decelerates with decreasing through-ﬂow coefﬁcients, while, in the small test rig, no deceleration is observable. In Figure 11, the inﬂuence of preswirl angle on torque and axial thrust coefﬁcient for the small test rig is presented. It is important to note that, in this ﬁgure, the Reynolds number of the data points is found on the lower horizontal axes while the through-ﬂow coefﬁcient is found on the upper ones. The torque coefﬁcient behaviour with varying preswirl angles is similar to that observed in the large Int. J. Turbomach. Propuls. Power 2020, 5, 7 14 of 17 test rig: when the preswirl angle increases from 0° (radial inﬂow) to 26°, the torque coefﬁcient drops sharply, while a further increase of the preswirl angle to 52° only leads to a marginal decrease of the torque coefﬁcient compared to the results with a preswirl angle of 26°. Regarding the thrust coefﬁcient, it is found that an increase of the preswirl angle increases the thrust coefﬁcient, as in the large test 6 3 rig. Interestingly, this increase is very small at the operating point Re 2 10 , c 3 10 , but larger at all other points. Figure 10. Axial thrust coefﬁcients c and torque coefﬁcients c measured in the small test rig F M (compare [10]) at the small relative cavity width of G = 0.0182: shown are results of a closed cavity (c = 0) and of radial inﬂow (a = 0°, c = 0, c < 0). D L D Figure 11. The inﬂuence of the preswirl angle on torque (c ) and axial thrust coefﬁcients (c ) in the M F small test rig (compare [10]) with the relative cavity width set to G = 0.0182. For the results shown here, the product of through-ﬂow coefﬁcient c and Reynolds number Re is constant. The horizontal error bars refer to the Reynolds number uncertainties. Int. J. Turbomach. Propuls. Power 2020, 5, 7 15 of 17 5. Conclusions In the closed cavity conﬁguration and in the Reynolds number range of Re 3.2 10 , the decrease of torque coefﬁcient with increasing Reynolds number behaves in the same way as found experimentally by Daily and Nece [1] and as predicted by Kurokawa and Toyokura [14]. For Reynolds numbers Re 3 10 , torque coefﬁcient decreases signiﬁcantly more slowly with increasing Reynolds number. This behaviour is observed in two different test rigs with two different compressible ﬂuids (the large carbon-dioxide-operated test rig used here and the air-operated test rig built by Radtke and Ziemann [3]). In both the small and large test rigs and in the closed cavity case, axial thrust coefﬁcient is found to decrease with increasing Reynolds number. Neither of the models proposed by Kurokawa and Sakuma [2] predicts this behaviour. In the large test rig at the small cavity width, the radial pressure gradient decreases with increasing Reynolds number. However, the model by Kurokawa and Sakuma [2] applicable to this case predicts the opposite behaviour. However, these models capture the inﬂuence of the relative axial gap width qualitatively: with increasing Reynolds number, axial thrust coefﬁcient decreases; with increasing cavity width, axial thrust coefﬁcient increases. In the large test rig with centripetal through-ﬂow, as mass ﬂow coefﬁcient decreases, torque coefﬁcient ﬁrst decreases to a minimum and then increases. The torque coefﬁcient minima are found at different mass ﬂow coefﬁcients, depending on the speciﬁc Reynolds number, relative axial gap width, and preswirl angle. Such minima are not found in the small, water-operated test rig, where in contrast centripetal through-ﬂow with radial inﬂow increases torque coefﬁcients. In both the large and small test rigs, increasing the preswirl angle from 0° to 26° leads to a reduction of torque coefﬁcient. No signiﬁcant differences between preswirl angles of 26° and 52° were observed, 7 7 with one exception: in the large test rig, at Reynolds numbers in the range of 7.5 10 Re 9.6 10 , at the small cavity width of G = 0.0125, and when the preswirl angle was increased to 52°, a further reduction of torque coefﬁcient was observed. 7 7 For the small gap width G = 0.0125 and a Reynolds number range of 7.5 10 Re 9.6 10 , a mass ﬂow coefﬁcient of c < 0 with any preswirl angle investigated lead to a larger axial thrust coefﬁcient than that of the closed cavity conﬁguration without through-ﬂow. Characteristic maxima of axial thrust coefﬁcients are observed for preswirl angles of 26° and 52°. For radial inﬂow, the coefﬁcients seem to converge towards a maximum, which lies outside the test rig’s operating range. Almost the same behaviour of axial thrust coefﬁcients is observed in the small test rig with radial inﬂow, where centripetal through-ﬂow increases the axial thrust coefﬁcients, but no signs of maxima are found. Preswirl angle was also found to inﬂuence axial thrust coefﬁcient, with an increase in preswirl angle resulting in an increase in the coefﬁcient in both the large and small test rigs. Finally, the increase of axial thrust coefﬁcient with centripetal through-ﬂow and a preswirl angle of 52° observed in the large test rig is a result of the pressure increasing at relative radii of /b > 0.75 and decreasing at relative radii of /b < 0.7 compared to the radial pressure distribution of the closed cavity case. Author Contributions: Conceptualization, methodology, software, validation, formal analysis, data curation, investigation, visualization and writing—original draft preparation, T.R.S.; resources, H.-J.D.; supervision and writing—review and editing, H.-J.D., D.B. and F.-K.B.; project administration, F.-K.B.; and funding acquisition, D.B. and F.-K.B. All authors have read and agreed to the published version of the manuscript. Funding: The research conducted with the large test rig was funded jointly by BMWi (Bundesministerium für Wirtschaft und Technologie of the German government) and Siemens Power and Gas, Compressors. The authors are grateful for the ﬁnancial support under contract number 03ET7071C. Conﬂicts of Interest: The authors declare no conﬂict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results. Int. J. Turbomach. Propuls. Power 2020, 5, 7 16 of 17 Abbreviations The following abbreviations and mathematical symbols are used in this manuscript: PMMA Polymethyl Methacrylate a Hub radius b Disc radius b, b Column vector of Lagrange multipliers, used in curve ﬁts c Mass ﬂow coefﬁcient c Axial thrust coefﬁcient as a function of c (large test rig) F p c , c Axial thrust coefﬁcient in back and front cavity, respectively (small test rig) Fb Ff c Through-ﬂow angular momentum coefﬁcient c Torque coefﬁcient for one disc side c Pressure coefﬁcient d Width of centripetal through-ﬂow inlet gap F , F , F Axial thrust, axial thrust in back and front cavity ab af G Relative axial gap width i, j Indices of matrix components k Turbulence kinetic energy K Core swirl ratio m ˙ Mass ﬂow rate of through-ﬂow M x, y , M x, w Column vector of model functions/extended model functions ( ) ( ) p (r) Pressure at radial position r p (r), p (r) Pressure at radial position r in back and front cavity, respectively b f Q (w) Sensitivity matrix, used in curve ﬁt error propagation r Radial position 0 r b Re, Re Circumferential Reynolds number, local Reynolds number s, s Axial gap width, axial gap width of back cavity back M , M , M Overall torque, friction torque and torque on front disc face all friction front U , U , u Matrices of covariances, covariance components xx w w i j u Circumferential ﬂuid velocity w, w Column vector of all unknown values, used in curve ﬁts r Minimal radial pressure measurement position p, min x = (x , , x ) Column vector of measured values, used in curve ﬁts 1 n y = (y , , y ) , y Column vector of model parameters, used in curve ﬁts z Axial coordinate 0 z s z = (z , . . . , z ) , z Column vector of ﬁtted values, replaces x in model function a Swirl angle of the preswirl generator # Dissipation rate of the turbulent kinetic energy k n Kinematic viscosity r Density c Squared error norm, used in curve ﬁts W Angular velocity of the rotor L Lagrange function References 1. Daily, J.W.; Nece, R.E. Chamber Dimension Effects on Induced Flow and Frictional Resistance of Enclosed Rotating Disks. J. Basic Eng. 1960, 82, 217–230. [CrossRef] 2. Kurokawa, J.; Sakuma, M. Flow in a Narrow Gap Along an Enclosed Rotating disk with Through-Flow. JSME Int. J. Ser. 2 Fluids Eng. Heat Transf. Power Combust. Thermophys. Prop. 1988, 31, 243–251. [CrossRef] 3. Radtke, F.; Ziemann, M. Scheibenreibung - Vorhaben Nr. 213 - Experimentelle und Theoretische Untersuchungen des Reibungseinﬂusses an Rotierenden Scheiben - Abschlussbericht; Technical Report; Institut für Dampf- und Gasturbinen, Rheinisch-Westfälische Technische Hochschule Aachen: Aachen, Germany, 1982. 4. Poncet, S.; Chauve, M.P.; Le Gal, P. Turbulent rotating disk ﬂow with inward throughﬂow. J. Fluid Mech. 2005, 522, 253–262. [CrossRef] 5. Poncet, S.; Chauve, M.P.; Schiestel, R. Batchelor versus Stewartson ﬂow structures in a rotor-stator cavity with throughﬂow. Phys. Fluids 2005, 17, 075110. [CrossRef] 6. Will, B.C. Theoretical, Numerical and Experimental investigation of the Flow in Rotor Stator Cavities with Application to a Centrifugal Pump. Ph.D. Thesis, University of Duisburg-Essen, Duisburg, Germany, 2011. Int. J. Turbomach. Propuls. Power 2020, 5, 7 17 of 17 7. Wang, C.; Tang, F.; Li, Q.; Wang, X. Experimental investigation of the microscale rotor–stator cavity ﬂow with rotating superhydrophobic surface. Exp. Fluids 2018, 59, 47. [CrossRef] 8. Hu, B.; Brillert, D.; Dohmen, H.J.; Benra, F.-K. Investigation on the Inﬂuence of Surface Roughness on the Moment Coefﬁcient in a Rotor-Stator Cavity With Centripetal Through-Flow; Volume 1A, Symposia: Keynotes; Advances in Numerical Modeling for Turbomachinery Flow Optimization; Fluid Machinery; Industrial and Environmental Applications of Fluid Mechanics; Pumping Machinery. In Proceedings of the Fluids Engineering Division Summer Meeting, Waikoloa Beach, HI, USA, 30 July–3 August 2017. [CrossRef] 9. Hu, B.; Brillert, D.; Dohmen, H.J.; Benra, F.K. Investigation on the ﬂow in a rotor-stator cavity with centripetal through-ﬂow. In Proceedings of the 12th European Conference on Turbomachinery Fluid Dynamics and Thermodynamics, Stockholm, Sweden, 3–7 April 2017. 10. Hu, B. Numerical and Experimental Investigation on the Flow in Rotor-Stator Cavities. Ph.D. Thesis, University of Duisburg-Essen, Duisburg, Germany, 2018. 11. Barabas, B.; Clauss, S.; Schuster, S.; Benra, F.K.; Dohmen, H.J.; Brillert, D. Experimental and Numerical Determination of Pressure and Velocity Distribution Inside a Rotor-Stator Cavity at Very High Circumferential Reynolds Numbers. In Proceedings of the 11th European Conference on Turbomachinery Fluid Dynamics and Thermodynamics, Madrid, Spain, 23–25 March 2015. 12. Bevington, P.; Robinson, D.K. Data Reduction and Error Analysis for the Physical Sciences, 3rd ed.; McGraw-Hill Higher Education; McGraw-Hill Education: New York, NY, USA, 2003. 13. Weise, K.; Wöger, W. Meßunsicherheit und Meßdatenauswertung; Forschen - Messen - Prüfen; Wiley-VCH: Weinheim, Germany, 1999. 14. Kurokawa, J.I.; Toyokura, T. Study on Axial Thrust of Radial Flow Turbomachinery. In Proceedings of the The Second International JSME Symposium Fluid Machinery and Fluidics, Keidanren Kaikan, Tokyo, 4–9 September 1972. © 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution NonCommercial NoDerivatives (CC BY-NC-ND) license (https://creativecommons.org/licenses/by-nc-nd/4.0/).

International Journal of Turbomachinery, Propulsion and Power – Multidisciplinary Digital Publishing Institute

**Published: ** Apr 17, 2020

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