## Abstract

We discuss the static characteristics, i.e., the induced force on the rotor, the attitude angle and the pressure difference, of off-centered annular gaps with an axial flow component without cavitation. The paper focuses on turbulent flow inside the annulus. Instead of modeling the inertia effects using the bulk-flow approach, the presented model uses an integro-differential approach in combination with power-law ansatz functions for the velocity profiles and a Hirs’ model to calculate the resulting pressure field. For an experimental validation of the model and comparison to the bulk-flow approach, an annular gap test rig is presented using magnetic bearings to inherently measure the position as well as the force on the rotor induced by the flow field inside the gap. The experimental investigations are performed using three different annuli with three different lengths. The impact of a variation of a modified Reynolds number, the flow number and the preswirl ratio on the resulting force on the rotor, the attitude angle and the pressure difference across the annulus is investigated and compared to the presented model and the bulk-flow approach.

## 1 Introduction

In modern turbomachinery, such as centrifugal pumps, multiple narrow annular gaps like media-lubricated journal bearing or annular seals exert hydrodynamic forces on the rotor. In general, the fluid flow within an annulus is three-dimensional and either laminar, transitional, or turbulent. The circumferential flow component driven by viscous forces is often superimposed by an axial flow component caused by an axial pressure difference. Due to design and operation parameters, the flow at the gap entrance is also superimposed by a preswirl, which is convected into the gap by the axial flow component.

*θ*. Here, the tilde $\u25fb~$ declares variables with dimensions, whereas dimensionless variables are written without it. On dimensional ground the induced dimensionless pressure field $p:=2p~/(\u03f1~\Omega ~2R~2)$ is only a function of six dimensionless measures: (i) the relative gap clearance $\psi :=h\xaf~/R~$, (ii) the dimensionless annulus length $L:=L~/R~$, (iii) the relative eccentricity $\epsilon :=e~/h\xaf~$, (iv) the Reynolds number in circumferential direction $Re\phi :=\Omega ~R~h\xaf~/\nu ~$, (v) the flow number $\varphi :=C\xaf~z/(\Omega ~R~)$, and (vi) the preswirl $C\phi |z=0:=C~\phi |z=0/(\Omega ~R~)$

_{φ}> 1000 [6,7]. Neither are the inertia effects always negligible [5,8–11], i.e., the characteristic parameter [12–14] is in the order of magnitude

*ψ*Re

_{φ}∼ 1. For example, for typical media-lubricated journal bearings and annular seals, the Reynolds number is in the order of magnitude Re

_{φ}∼ 10

^{3}· · · 10

^{4}. Paired with relative clearance in the order of magnitude

*ψ*∼ 10

^{−3}and significant axial pressure gradients, the characteristic parameter is in the order of magnitude

*ψ*Re

_{φ}∼ 1 · · · 10. Therefore, the convective terms in the momentum equations have to be taken into account. As stated above, this can be done either by elaborately solving the Navier–Stokes equations or by using simplified calculation methods like a modified Reynolds’ equation [11,15] or the bulk-flow approach [6,16–18]. Within those methods, the governing equations are integrated across the film thickness $h~$ leading to an integro-differential approach, cf. [19,20]

However, what distinguishes the methods from each other is, apart from the treatment of the wall shear stresses $\tau ~yi|0h~$ [21] and their application on transitional flow [7,22], the treatment of the integrals in the equations. By averaging the velocity profiles over the gap height $C\xaf~i(\phi ,z~):=1/h~\u222b0h~c~i(\phi ,z~,y~)dy~$ the previously mentioned bulk-flow model is derived. Although the bulk-flow model is one of the most widely used models for calculating the characteristics of annular seals, the assumption of averaged velocity profiles is by no means uncritical [23,24]. Strictly speaking, the assumption is only valid when the real velocity profiles are close to block-shaped ones, i.e., Re_{φ} → ∞.

To overcome this drawback, Launder and Leschziner [19,20] as well as Constantinescu and Galetuse [9,25] and Simon and Frêne [26] used parabolic ansatz functions $\u222b0h~c~i2dy:=a~C\xaf~i2+b~C\xaf~i+d~$ to describe the integrals of the velocity profiles within the momentum equations. This type of modeling assumes that the velocity profiles are unaffected by fluid inertia. Here, the disadvantage lies in the complex calibration of the describing coefficients $a~$, $b~$, and $d~$. The integrals in the continuity equation are treated by means of gap height averaged velocities.

To address the use of bulk-velocities in general, Brunetière and Tournerie [11,22] recently presented a model suitable for laminar, transitional, and turbulent film flows by using flowrates instead of bulk, i.e., averaged, velocity profiles to describe the integrals. Following the work of Constantinescu and Galetuse, however, the shape of the velocity profile is considered to be unaffected by inertia effects.

As presented before, available models in the literature are based on one specific assumption about the shape of the velocity profiles. However, dependent on the operating conditions, the specific assumption of each available model might be more or less suitable. To overcome this drawback, a general calculation framework, the clearance-averaged pressure model (CAPM) based on the work of Lang [14,27] and Robrecht et al. [28], was developed. In this calculation framework, the shape of the velocity profiles is incorporated into the solution procedure in a very modular way. Specifically, the considered velocity profiles can be changed, e.g., between laminar, turbulent, and bulk-velocity profiles as appropriate for the flow regimes to be calculated.

Here, the paper focuses mainly on turbulent flows with moderate Reynolds numbers as one important field of application for the calculation framework. For this, power-law ansatz functions are chosen as velocity profiles due to the fact that they are well established and widely used to model turbulent velocity profiles throughout the literature [29,30]. This is specifically advantageous as many authors reported numerical values for the empirical exponents of the power-law functions [30–32].

The clearance-averaged pressure model is based on the integro-differential approach Eq. (2), i.e., the system of nonlinear partial differential equations, is solved by using a SIMPLE-C (semi-implicit method for pressure linked equations–consistent) algorithm. The model is then prevalidated by comparison with numerical data based on the bulk-flow approach by Nelson and Nguyen [33] and San Andrés [34]. For further validation, an annular gap test rig is presented using magnetic bearings to inherently measure the position as well as the force on the rotor induced by the flow field inside the gap. The experimental investigations are performed using three different annuli with three different lengths. The impact of a variation of a modified Reynolds number, the flow number and the preswirl ration on the resulting force on the rotor, the attitude angle and the pressure difference across the annulus is investigated and compared to the presented model and the bulk-flow approach.

## 2 The Clearance-Averaged Pressure Model

In the following, the basic equations and the boundary conditions for calculating the pressure field inside the annulus are derived. Furthermore, an analysis of the describing equations leads to a further reduction of dimensionless variables.

In addition to the dimensionless variables introduced in Sec. 1, the dimensionless wall shear stresses are given by $\tau y\phi |01:=2\tau ~y\phi |0h~/(\u03f1~\Omega ~2R~2)$ and $\tau yz|01:=2\tau ~yz|0h~/(\u03f1~\Omega ~2R~2)$. The integrals in the equations are solved analytically by using power-law ansatz functions to describe the velocity profiles in circumferential and axial direction, cf. [14,27]. Due to the modular integration of the velocity profiles, the ansatz functions can be adapted to gap flows at arbitrary Reynolds numbers. The requirement Re_{φ} → ∞ for block-shaped velocity profiles in the bulk-flow approach is thus eliminated. In particular, the power-law ansatz functions can be used to treat laminar, transitional, and turbulent gap flows in a unified model framework. Throughout the literature, the modeling of the velocity profiles either by the bulk-flow approach or the assumptions of unaffected profiles has so far been assumed to be fully developed, cf. [10,19,20,26].

*L*is much larger than the hydrodynamic entrance length $Lhyd:=L~hyd/R~$, i.e.,

*L*≫

*L*

_{hyd}. Stampa [35] and Herwig [36] show that the hydraulic entrance length is a power-law function of the Reynolds number Re

_{φ}, the flow number

*ϕ*, the relative gap clearance

*ψ*, and an empirical constant

*k*

_{hyd}

*k*

_{hyd}< 8.8 and power-law exponents 1/6 <

*n*

_{hyd}< 1/4. Considering typical Reynolds numbers being in the order of magnitude of Re

_{φ}∼ 10

^{3}· · · 10

^{4}as well as flow numbers being in the order of magnitude

*ϕ*∼ 1 and typical relative clearances being in the order of magnitude

*ψ*∼ 10

^{−3}, the hydrodynamic entrance length is in the order of magnitude

*L*

_{hyd}∼ 10

^{−3}· · · 10

^{−1}. This is good agreement with the numerical results obtained by Lang [14]. Within this paper, we focus on generic annular gaps with lengths being in the order of magnitude

*L*∼ 1, i.e., it is reasonable to assume fully developed velocity profiles. Therefore, the power-law ansatz functions reads

*y*

_{rot}: = 1 −

*y*is introduced starting at the rotor surface.

*C*

_{φ}and

*C*

_{z}are the centerline velocities at half gap height and

*n*

_{φ}= 5,

*n*

_{z}= 6.5 the exponents of the power-law ansatz functions. The exponents were obtained by an extensive numerical study using a three-dimensional CFD model with approximately 10.9 million cells and an RSM turbulence model. The model was solved using the commercial software ansys fluent, resulting in a good comparison to the exponents reported by Sigloch [30] and Reichardt [31]. Therefore, the integrals for the continuity and the momentum equations yield

*τ*

_{yi,stat}and

*τ*

_{yi,rot}read

*f*

_{rot}and

*f*

_{stat}, the dimensionless effective relative velocity between the wall (rotor rot, stator stat) and the fluid $Ci:=C\phi ,i2+\varphi 2Cz,i2,i=rot,stat$. Here, the components

*C*

_{φ,i}and

*C*

_{z,i}are boundary layer averaged velocities between the wall and the corresponding boundary layer thickness

*δ*assuming fully developed boundary layers throughout the annulus. The boundary layer averaged velocities yield

*m*

_{f}and

*n*

_{f}as well as the Reynolds number Re

_{φ}. The empirical constants describe arbitrary lines within the double logarithmic Moody diagram. By careful selection of the coefficients, it is possible to model both laminar and turbulent frictions. In addition, it is possible to model hydraulically smooth and hydraulically rough friction behavior. Furthermore, a representation of the transition area between laminar and turbulent flows is supported. Modeling the wall shear stresses by using the fanning friction factor in combination with the Hirs’ model, the wall shear stresses are of the order

*ψ*Re

_{φ}. For laminar flow, i.e.,

*m*

_{f}= −1, the modified Reynolds number reduces to the characteristic parameter. Therefore, the induced pressure is a function of five dimensionless measures

*p*is the pressure difference across the annulus and

*ζ*describes the inlet pressure loss coefficient. In general,

*ζ*is a function of the inlet geometry as well as the relative gap clearance and operating parameters including the Reynolds number Re

_{φ}, the flow number

*ϕ*and the relative shaft displacement, i.e.,

*ɛ*

Strictly speaking, this constant swirl boundary condition is only valid at a concentric rotor position. However, due to the complex flow pattern at an eccentric rotor position a constant swirl boundary condition is an acceptable approximation.

The nonlinear partial differential equation system cannot be solved analytically. The solution of the partial differential equation system, i.e., the induced pressure field inside the annulus *p*(*x*, *z*) and the centerline velocities *C*_{φ}, *C*_{z}, are obtained by using a SIMPLE-C algorithm based on the work of van Doormaal and Raithby [39]. Here, a finite difference scheme at a two-dimensional uniform structured grid with a resolution of 51 × 51 cells is used. The velocities are determined at the grid nodes and the pressure is determined at the cell center. In order to determine the velocity corrections for the nodes of the grid which lie directly at the gap outlet, an extrapolation of the velocity field from the interior to the gap outlet and a subsequent mass flow correction to maintain global mass conservation is applied, cf. Versteeg [40]. The nonlinearity in the integrals is handled by applying the Picard-iteration and the spatial derivatives are approximated by hybrid schemes.

Here, *F*_{X} and *F*_{Y} are the induced forces in *X* and *Y* direction, whereas *F*_{res} is the resulting force and *θ* the attitude angle. Typically, the wall shear stresses do not significantly contribute to the induced forces, cf. Scharrer [41]. It can be shown that the forces from the wall shear are in the order of magnitude $O(\psi )$. Therefore, they are typically neglected in most of the literature and the induced forces are dominated by the pressure field inside the annuls.

## 3 Prevalidation

Prior to validating the clearance-averaged pressure model using the experimental setup presented later on, the static characteristic, i.e., the resulting force *F*_{res} on the rotor and the attitude angle *θ* are compared to the numerical results by Nelson and Nguyen [33] and San Andrés [34]. Here, the resulting force on the rotor, as well as the attitude angle, is investigated. Nelson and Nguyen use a fast Fourier transform to solve the bulk-flow equations and calculate the static characteristics of an annular seal with length *L* = 2.05, a preswirl *C*_{φ}|_{z=0} = 0.3, and three dimensionless pressure differences $\Delta p:=2\Delta p~/(\u03f1~\Omega ~2R~2)=57;21;10$ corresponding to the three modified Reynolds numbers Re_{φ}* = 0.065; 0.073; 0.080. In contrast, San Andrés uses a form of the formerly presented bulk-flow model to calculate the resulting force on the rotor of an annulus with length *L* = 2, a modified Reynolds number Re_{φ}* = 0.047, a pressure difference Δ*p* = 4.5, and a preswirl *C*_{φ}|_{z=0} = 0.2.

Figure 3 shows the comparison of the resulting force acting on the rotor and the attitude angle. The lines are the results by Nelson and Nguyen [33] and the markers represent the calculation results by the CAPM. Furthermore, the thickness of the lines as well as the size of the Marker correlates to the modified Reynolds number. The thicker the line thickness and the larger the marker, the higher the modified Reynolds number. The calculations of the resulting force acting on the rotor *F*_{res} as well as the attitude angle *θ* show a good agreement with the results obtained by Nelson and Nguyen. The predicted forces acting on the rotor by the clearance-averaged pressure model are slightly higher than the resulting forces by Nelson and Nguyen. Regarding the attitude angle, the trend is reserved. Here, the CAPM results are slightly below the calculations by Nelson and Nguyen.

Figure 4 shows the comparison of the resulting force acting on the rotor determined by the CAPM and the data published by San Andrés [34]. Due to the lack of data, a comparison of the attitude angles similar to the results by Nelson and Nguyen is not possible. Again, the forces acting on the rotor predicted by the clearance-averaged pressure model are in good comparison to the results obtained by the bulk-flow model by San Andrés. Similar to the comparison with the predictions by Nelson and Nguyen, the CAPM results are slightly above the ones of the bulk-flow model.

## 4 The Experimental Setup

For further validation purpose and experimental investigation of the induced forces on the rotor, a worldwide unique test rig is designed. Additional information on the test rig can be found in the work of Kuhr et al. [42,43]. Figure 5 shows the annular gap flow test bench at the laboratory of the Chair of Fluid Systems at the Technische Universität Darmstadt.

The test bench essentially consists of two magnetic bearings supporting the rotor. They also serve as an inherent displacement and force measurement system. Compared to existing test rigs where the shaft is supported by ball or journal bearings, cf. [44–46], magnetic bearings have the advantage of being completely contactless and thus frictionless. In addition, the ability to displace and excite the shaft at user-defined frequencies makes them ideal for determining the static and dynamic characteristics of annular gap flows.

Figure 6 shows a technical drawing of the main part of the test rig. The fluid path is indicated by the black arrows. The test rig consists of five components: (i) two active magnetic bearings to bear the rotor and measure the induced hydrodynamic forces on the shaft; (ii) the inlet to guide the flow and to measure the preswirl as well as the static pressure at the entrance of the annulus; (iii) the gap module; (iv) the outlet; and (v) the mechanical seals.

Due to the dependence of the magnetic flux density on the position of the rotor inside the magnetic bearing, the hall sensors have to be calibrated. This is done by using an iterative procedure initially developed by Krüger [47]. The calibration is performed using the known rotor mass and its center of gravity as a reference force $F~ref$. For an unloaded shaft, the force measurement of the magnetic bearing with a rotor positioned eccentrically in the bearing must output both the mass as well as the center of gravity of the rotor. The measurement uncertainty after calibration reduces to $\delta ~F~,hall<\xb10.035F~$. The displacement of the rotor within the magnetic bearing is measured using four circumferentially distributed eddy current sensors per bearing, with an uncertainty of $\delta ~x,AMB<\xb10.75\mu m$. To monitor the temperature of each bearing, each is equipped with two PT100 temperature probes.

The inlet is specifically designed to generate preswirled flows in front of the annulus. It is well known that the pressure field inside the annulus is influenced by the preswirl $C\phi |z=0$. To investigate and quantify the influence on the static characteristics, 12 circumferentially distributed tubes introduce water tangentially into the inlet. By dividing the flow into two parts, the gap and the bypass volume flow, it is possible to continuously vary the circumferential velocity component upstream of the annulus. A pitot tube connected to a pressure transmitter ($\delta ~\Delta p<0.01bar$) is used to measure the circumferential velocity component. The test rig is capable of generating a preswirl in the range of $C\phi |z=0=0\u20261.7$.

In order to determine the position of the rotor within the gap module, the position of the shaft is measured at two planes: (i) at the entrance and (ii) exit of the lubrication gap. For this purpose, two eddy current sensors with a user-defined measuring range of 1 mm and an absolute uncertainty of $\delta ~x,GAP<\xb12.4\mu m$ are used in an 90° arrangement.

Due to the modular design of the test rig, it is possible to vary the length of the annulus as well as the relative clearance by changing the stator inlay and the rotor diameter within the gap module. The relative length can be varied in a range of 0.2 ≤ *L* ≤ 1.8 and the relative clearance can be modified in a range of 10^{−3} ≤ *ψ* ≤ 10^{−2}.

The supply pressure is measured at the inlet of the gap module by using a pressure transmitter connected to four wall pressure taps equally space around the annulus with an uncertainty of $\delta ~p,z=0<\xb10.033bar$. The pressure difference across the annulus is measured by a differential pressure sensor with an absolute uncertainty of $\delta ~\Delta p<\xb10.04bar$.

The test rig is designed to investigate pressure differences of up to 13 bar. The volume flow through the gap is measured using an electromagnetic flowmeter with an absolute uncertainty of $\delta ~Q\u02d9,GAP<\xb10.04Q~GAP$. To avoid cavitation within the annulus, the test rig can be pressurized up to 15 bar. The test rig is operated using water at a constant temperature of 35 °C and is fed by a 10 staged 55 kW centrifugal pump resulting in flow numbers up to *ϕ* ≤ 5. Although the paper focuses only on the static characteristics of annular gaps, the test rig is also capable of identifying the dynamic characteristics of the annulus, i.e., the stiffness, damping, and inertia coefficients.

## 5 Experimental and Simulation Results

The experimental investigations are performed using three different annuli with lengths $L~=71.5mm;92.95mm;114.4mm$. The three lengths correspond to the dimensionless lengths *L* = 1.0; 1.3; 1.6. Besides the influence of the annulus length, the influence of the modified Reynolds number Re_{φ}*, the influence of the flow number *ϕ*, and the influence of the preswirl *C*_{φ}|_{z=0} are investigated. Here, the variation of the modified Reynolds number is achieved by keeping the relative gap clearance constant at *ψ* = 4.2 ‰, i.e., $h\xaf~=0.3mm$, and modifying the Reynolds number in a range of Re_{φ} = 3000; 4000; 5000, i.e., $\Omega ~=103.671/s;138.231/s;$$172.791/s$. This is reasonable because the Reynolds number and the relative gap clearance only occur as a product in the system of nonlinear partial differential equations, cf. Sec. 2. All experiments are conducted using water at a temperature of 35 °C over an eccentricity range 0.1 ≤ *ɛ* ≤ 0.8. The experimental results are compared to the ones obtained by the clearance-averaged pressure model as well as the bulk-flow model, i.e., averaged velocity profiles, with regard to the resulting force on the rotor $Fres:=2F~res/(\u03f1~\Omega ~2R~3L~)$, the attitude angle *θ*, and the pressure difference across the annulus $\Delta p:=2\Delta p~/(\u03f1~\Omega ~2R~2)$.

For the calculations, a fully turbulent flow within the annulus is assumed. The empirical constants of the Hirs’ wall friction model are *n*_{f} = 0.0645 and *m*_{f} = −0.24, whereas the inlet pressure loss coefficient is *ζ* = 0.25.

### 5.1 Annulus Length.

Figure 7(a) shows the influence of the annulus length on the resulting force versus eccentricity. In contrast to the prevalidation results, in the following, the solid lines represent the calculation results obtained by the clearance-averaged pressure model, whereas the dashed lines represent the results by the bulk-flow model. The markers represent the experimental data. Again, the line thickness as well as the marker size correlates with the varied parameter. The thicker the line thickness and the larger the marker, the larger the varied parameter. The modified Reynolds number as well as the flow number and the preswirl were kept constant over each measurement within a range of $\xb11%$. The influence of the annulus length is investigated, choosing a modified Reynolds number Re_{φ}* = 0.031, a flow number *ϕ* = 0.7, and a preswirl *C*_{φ}|_{z=0}. It is found that the resulting forces on the rotor are in very good agreement with the predicted ones obtained by the CAPM and the bulk-flow model. Although the differences between the two model approaches are almost negligible, the results of the CAPM are somewhat closer to the results of the test rig. The force displacement curves are linear at small eccentricities *ɛ* < 0.5 and become increasingly nonlinear with increasing eccentricity. This is due to the fact that the induced forces at small eccentricities are mainly caused by the Lomakin effect, while the forces induced by the hydrodynamic effect become more dominant as the eccentricity increases.

Figure 7(b) shows the corresponding attitude angle versus eccentricity. It shows good agreement between the measured and predicted attitude angle for both, the bulk-flow model and the CAPM. Again, the differences between the modeling approaches are almost negligible, although in terms of the attitude angle, the results by the bulk-flow model are somewhat closer to the experimental data. The differences between measured and calculated results decrease with increasing length. Here, the attitude angle increases with increasing annulus length. In contrast to a classical journal bearing, the attitude angle does not follow the Gümbel curve, i.e., a semicircular displacement of the rotor loci with increasing eccentricity. This is reasonable because cavitation is not considered in the CAPM and it is prevented during the tests by increasing the pressure level of the entire test rig. The prevention of cavitation leads to a resulting force dominated by the *Y* force component, resulting in a mainly horizontally moved rotor.

Figure 8 shows the pressure difference across the annulus versus eccentricity. Here, the most significant differences between the two modeling approaches as well as between the models and the experimental data can be seen. Compared to the resulting force acting on the rotor and the attitude angle, the difference between the CAPM and the bulk-flow model is not negligible. Here, the predictions of the CAPM are more consistent with the experimental data. Regarding the pressure difference versus eccentricity, it is shown that the pressure difference decreases with increasing eccentricity and increases with an increasing annulus length. The decreasing pressure difference with increasing eccentricity is due to the fact that the flow resistance opposed by the annulus decreases with increasing eccentricity.

### 5.2 Modified Reynolds Number.

In the following, the influence of the modified Reynolds number on the force, attitude angle, and pressure difference is investigated. Figure 9(a) shows the influence of the modified Reynolds number on the resulting force on the rotor versus eccentricity for an annulus length *L* = 1.3, a flow number *ϕ* = 0.7, and a preswirl *C*_{φ}|_{z=0} = 0.5. Due to the constant clearance *ψ* = 4.2 ‰ the modified Reynolds number is controlled by controlling the angular frequency of the rotor. Here, the three modified Reynolds numbers Re_{φ}* = 0.029, 0.031, and 0.033 correspond to the three Reynolds numbers Re_{φ} = 3000, 4000 and 5000. Figure 9(a) shows a good agreement between the experimental results and the predictions of both, the clearance-averaged pressure model and the bulk-flow model. By increasing the modified Reynolds number, the force on the rotor slightly decreases.

This seems contradictory at first, as one would expect an increase in the force on the rotor with an increased angular frequency. This is indeed the case when considering the dimensional force $F~res$. Therefore, one could assume that the decrease in the resulting force therefore results solely from the definition of the dimensionless force being inversely proportional to $F\u221d1/\Omega ~2$. If the definition of the modified Reynolds number $Re\phi *:=\psi Re\phi \u2212mf=(h\xaf~/R~)(\Omega ~R~h\xaf~/\nu ~)\u2212mf$, it becomes clear that the variation of the same can be achieved either by a variation of the angular frequency $\Omega ~$, a variation of the mean gap height $h\xaf~$, a variation of the rotor radius $R~$, or a variation of the kinematic viscosity $\nu ~$. However, the result of a decreased dimensionless resulting force on the rotor would be the same. On dimensional grounds, the modified Reynolds number is a measure for the relevance of inertia effects within the annuls. Therefore, an increased modified Reynolds number is equivalent to increased inertia forces or decreased viscous. However, this also means that by increasing the modified Reynolds number, the pressure buildup within the annular gap is reduced. This is reflected in a reduction of the resulting force on the rotor.

Figure 9(b) shows the influence of the modified Reynolds number on the attitude angle versus eccentricity. Here, the attitude angle predicted by the CAPM and the bulk-flow model slightly decreases by increasing the modified Reynolds number. However, due to the small angle variations, the experimental data do not show a clear trend toward a decreasing or increasing attitude angle. Nevertheless, the experimental data and the simulations show a good agreement.

Figure 10 shows the influence of the modified Reynolds number on the pressure difference across the annulus versus eccentricity. Similar to the variation of the annulus length, the predictions of the clearance-averaged pressure model are somewhat closer to the experimental results. It shows a decreasing curve with increasing eccentricity. In addition, by increasing the modified Reynolds number, the pressure difference decreases.

The explanation is equivalent to the one for the reduction of the resulting force on the rotor. As the modified Reynolds number increases, the inertia forces become stronger respectively the viscous forces become smaller. Similar to the pressure buildup inside the annulus, the dissipation across the annular gap, i.e., the pressure difference, decreases accordingly.

### 5.3 Flow Number.

In the following, the influence of the flow number on the force, the attitude, and the pressure difference across the annulus is investigated. Figure 11(a) shows the force versus eccentricity for an annulus length *L* = 1.3, a modified Reynolds number Re_{φ}* = 0.031, and a preswirl of *C*_{φ}|_{z=0} = 0.5. It shows a good agreement between the simulation results by the CAPM and the bulk-flow model and the experimental data. The induced force on the rotor is increasing by increasing the flow number. This is reasonable because an increase in flow number results in an increased pressure difference across the annulus, cf. Fig. 12. Therefore, the Lomakin effect is increased, resulting in an increased force on the rotor.

Figure 11(b) shows the corresponding attitude angle versus eccentricity. The predictions of the CAPM and the bulk-flow model are again in good agreement with the experimental data. The difference between the two modeling approaches becomes visible only at high relative eccentricities, i.e., *ɛ* > 0.8. Here, the attitude angle decrease when the flow number is increased. This is reasonable, as mentioned before, the prevention of cavitation inside the annulus leads to a dominating *Y* force component acting on the rotor. Therefore, the rotor is mainly moved horizontally. The increased Lomakin effect due to the increasing flow number results in an increasing force on the rotor direction pointing to the annulus center, i.e., a force in negative *X* direction. Therefore, the attitude angel decreases with an increasing flow number.

Figure 12 shows the influence of the flow number on the pressure difference across the annulus. As mentioned above, by increasing the flow number the pressure difference increases, resulting in an increased Lomakin effect. The experimental data are in good agreement with the results calculated by the clearance-averaged pressure model and the bulk-flow model. Again, the predictions of the CAPM are closer to the results obtained by the annular gap test rig.

### 5.4 Preswirl.

Finally, the influence of the preswirl on the force on the rotor, the attitude angle, and the pressure difference across the annulus is investigated. Figure 13(a) shows the influence of the preswirl on the resulting force for an annulus length *L* = 1.3, a modified Reynolds number Re_{φ}* = 0.031, and a flow number *ϕ* = 0.7. Both, the experimental data and the simulations by the CAPM and the bulk-flow model show an increasing force due to the increased preswirl in front of the annulus. This is reasonable because an increase in preswirl mainly increases the force in circumferential direction, i.e. *F*_{Y}, resulting in an increased resulting force and an increasing attitude angle. Figure 13(b) shows the attitude angle versus eccentricity. As mentioned, the angle is increased by increasing the preswirl in front of the annulus. The predicted attitude angles are in good agreement with the experimental results. However, the results by the bulk-flow model are somewhat closer to the experimental data. Comparing the two modeling approaches, it is evident that the difference between the two is small in terms of the resulting force and the attitude angle.

Figure 14 shows the influence of preswirl on the pressure difference across the annulus. Here, the CAPM and the bulk-flow model predicts an increase in the pressure difference with an increasing preswirl. The increase in pressure difference obtained by the bulk-flow model is more pronounced than the increases in pressure difference by the CAPM. A distinct tendency in the measurement data is not apparent. Therefore, the predictions by clearance-averaged pressure model are closer to the experimental data than the results by the bulk-flow model, especially for high preswirls. For the results obtained by the CAPM, the deviations between the measurements and the model are within the measurement uncertainty. Regarding the increasing pressure difference with an increasing preswirl predicted by the model, this is due to the increase in circumferential velocity which leads to an increased friction in the annulus, which in return results in an increased pressure difference across the annulus.

In summary, the following facts can be derived for the comparison of the presented model, i.e., the CAPM, and the bulk-flow model:

The experimental data are in good agreement with the simulations performed by the clearance-averaged pressure model and the bulk-flow model.

The differences between the CAPM and the bulk-flow model regarding the resulting force and the attitude angle are small. Regarding the resulting force acting on the rotor, the results obtained by the CAPM are somewhat closer to the experimental data. However, regarding the attitude angle, the bulk-flow predictions are slightly better.

The differences between the CAPM and the bulk-flow model regarding the pressure difference across the annulus are more significant than the ones regarding the force and attitude angle. Here, the clearance-averaged pressure model is clearly superior to bulk-flow model.

For the experimental and simulation results, i.e., the variation of the annulus length, the modified Reynolds number, the flow number, and the preswirl, the following facts can be stated:

By increasing the annuls length the force on the rotor, the attitude angle, and the pressure difference across the annulus is increased.

By increasing the modified Reynolds number the force on the rotor, the attitude angle, and the pressure difference is decreased.

By increasing the flow number, the Lomakin effect increases, resulting in an increased force on the rotor and an increased pressure difference across the annulus. In contrast, the attitude angle decreases when the flow number is increased.

By increasing the preswirl the force on the rotor, the attitude angle and the pressure difference are increased.

## 6 Conclusions

In the presented paper, the static force characteristics of annular gaps with an axial flow component are discussed. First, a new calculation framework, the clearance-averaged pressure model (CAPM) is presented. Here, the CAPM used an integro-differential approach in combination with power-law ansatz functions for the velocity profiles and a Hirs’ model to calculate the resulting pressure field. Due to the modular integration of the velocity profiles, the ansatz functions can be adapted to gap flows at arbitrary Reynolds numbers. The requirement Re_{φ} → ∞ for block-shaped velocity profiles in the bulk-flow approach is thus eliminated. In particular, the power-law ansatz functions can be used to treat both laminar and turbulent gap flows in a unified model framework. Second, an experimental setup is presented using magnetic bearings to inherently measure the position as well as the force induced by the flow field inside the gap. Compared to existing test rigs where the shaft is supported by ball or journal bearings, magnetic bearings have the advantage of being completely contactless and thus frictionless. In addition, the ability to displace and excite the shaft at user-defined frequencies makes them ideal for determining the static and dynamic characteristics of annular gap flows. Third, an extensive parameter study is carried out, focusing on the characteristic load behavior, attitude angle, and pressure difference across the annulus. The experimental results are compared to the results of the clearance-averaged pressure model and the bulk-flow approach. It is shown that the results of the CAPM and the bulk-flow model are in good agreement with the experimental data. Here, the difference between the modeling approaches is small regarding the force and the attitude angle, whereas the CAPM is superior to the bulk-flow approach regarding the pressure difference across the annulus. It is shown that by increasing the annuls length the force on the rotor, the attitude angle, and the pressure difference across the annulus increase. In addition, by increasing the modified Reynolds number the force on the rotor, the attitude angle, and the pressure difference decrease. Due to the Lomakin effect, increasing the flow number will result in an increased force on the rotor as well as an increased pressure difference. In contrast, the attitude angle decreases. By increasing the preswirl upstream of the annulus, the force on the rotor, the attitude angle, and the pressure difference increase.

## Acknowledgment

We gratefully acknowledge the financial support of the industrial collective research program (IGF no. 19225/BG 2), supported by the Federal Ministry for Economic Affairs and Energy (BMWi) through the AiF (German Federation of Industrial Research Associations e.V.) based on a decision taken by the German Bundestag. In addition, we kindly acknowledge the financial support of the Federal Ministry for Economic Affairs and Energy (BMWi) due to an enactment of the German Bundestag (Grant No. 03ET7052B) and KSB SE & Co. KGaA. Special gratitude is expressed to the participating companies and their representatives in the accompanying industrial committee for their advisory and technical support.

## Conflict of Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

## Nomenclature

### Dimensional variables

- $e~$ =
eccentricity (m)

- $h~$ =
gap function (m)

- $h\xaf~$ =
mean gap height (m)

- $p~$ =
pressure (Pa)

- $A~$ =
pole surface (m

^{2}) - $B~$ =
flux density (T)

- $F~$ =
force on the rotor (N)

- $L~$ =
annulus length (m)

- $R~$ =
shaft radius (m)

- $c~\phi $ =
velocity in

*φ*direction (m/s) - $c~z$ =
velocity in

*z*direction (m/s) - $C~\phi $ =
centerline velocity in

*φ*direction (m/s) - $C\xaf~\phi |z=0$ =
swirl velocity at the annulus entrance (m/s)

- $C~z$ =
centerline velocity in

*z*direction (m/s) - $C\xaf~z$ =
mean velocity in

*z*direction (m/s) - $L~hyd$ =
hydrodynamic entrance length (m)

- $\Delta p~$ =
pressure difference (Pa)

- $Q~GAP$ =
volume flow (m

^{3}/s) - $X~,Y~,z~$ =
spatial coordinates (m)

- $\delta ~$ =
boundary layer thickness (m)

- $\eta ~$ =
dynamic viscosity (Pa s)

- $\mu ~$ =
magnetic field constant (N/A

^{2}) - $\u03f1~$ =
density (kg/m

^{3}) - $\tau ~y\phi |0h~$ =
sum of the wall shear stresses on the stator and rotor in

*φ*direction (Pa) - $\tau ~yz|0h~$ =
sum of the wall shear stresses on the stator and rotor in

*z*direction (Pa) - $\nu ~$ =
kinematic viscosity (m

^{2}/s) - $\Omega ~$ =
angular frequency of the rotor (1/s)

### Nondimensional variables

*f*=Fanning friction factor (–)

- $h:=h~/h\xaf~$ =
gap function (–)

- $p:=2p~/(\u03f1~\Omega ~2R~2)$ =
pressure (–)

*y*=radial coordinate for the starting at the stator surface (–)

- $z:=z~/L~$ =
axial coordinate (–)

- $F:=2F~/(\u03f1~\Omega ~2R~3L~)$ =
force on the rotor (–)

- $L:=L~/R~$ =
annulus length (–)

- $c\phi :=c~\phi /(\Omega ~R~)$ =
velocity in

*φ*direction (–) - $cz:=c~z/C\xaf~z$ =
velocity in

*z*direction (–) *k*_{hyd}=empirical constant of the hydrodynamic entrance length (–)

*m*_{f},*n*_{f}=empirical constant in the wall shear stress model (–)

*n*_{hyd}=exponent of the hydrodynamic entrance length (–)

*n*_{φ},*n*_{z}=exponents of the power-law ansatz functions (–)

- $C\phi :=C\xaf~\phi /(\Omega ~R~)$ =
centerline velocity in

*φ*direction (–) - $C\xaf\phi (\phi ,z):=C\xaf~\phi (\phi ,z~)/(\Omega ~R~)$ =
gap height averaged velocity in

*φ*direction (–) - $C\phi |z=0:=C\xaf~\phi |z=0/(\Omega ~R~)$ =
preswirl at the annulus entrance (–)

- $Cz:=C~z/C\xaf~z$ =
centerline velocity in

*z*direction (–) - $C\xafz(\phi ,z):=C\xaf~z(\phi ,z~)/C\xaf~z$ =
gap height averaged velocity in

*z*direction (–) - $Lhyd:=L~hyd/R~$ =
hydrodynamic entrance length (–)

- $\Delta p:=2\Delta p~/(\u03f1~\Omega ~2R~2)$ =
pressure difference (–)

- $Re\phi :=\Omega ~R~h\xaf~/\nu ~$ =
Reynolds number (–)

*y*_{rot}=radial coordinate for the starting at the rotor surface (–)

- $X,Y:=X~/h\xaf~,Y~/h\xaf~$ =
spatial coordinates (–)

- $\delta :=\delta ~/h~$ =
boundary layer thickness (–)

- $\epsilon :=e~/h\xaf~$ =
relative eccentricity (–)

*ζ*=inlet pressure loss coefficient (–)

*θ*=attitude angle (–)

- $\tau y\phi |01:=2\tau ~y\phi |0h~/(\u03f1~\Omega ~2R~2)$ =
sum of the wall shear stresses on the stator and rotor in

*φ*direction (–) - $\tau yz|01:=2\tau ~yz|0h~/(\u03f1~\Omega ~2R~2)$ =
sum of the wall shear stresses on the stator and rotor in

*z*direction (–) *φ*=circumferential coordinate (–)

- $\varphi :=C\xaf~z/(\Omega ~R~)$ =
flow number (–)

- $\psi :=h\xaf~/R~$ =
relative gap clearance (–)

- $Re\phi *:=\psi Re\phi \u2212mf$ =
modified Reynolds number (–)

## References

*Forschungsberichte zur Fluidsystemtechnik*, Shaker Verlag, Aachen.