#### 5.1. Core Swirl Ratio

To evaluate the pressure distribution, the values of

K need to be estimated. Although some correlations are determined to predict the values of

K with centrifugal through-flow, such as Equations (2a)–(2c), it still exists an uncertainty on the impact of

G on

K. In addition, the geometry of the cavity, especially at the outlet, does have large influence on

K. Based on Equation (4), the pressure difference between the two adjacent pressure tubes number

e and number

$e+1$ can be calculated with Equation (8). The parameter

$\overline{K}$ represents the average value of

K between two adjacent pressure tubes. Twelve pressure tubes exist in the front chamber from

$r=0.05$ m (

$x=0.455$) to

$r=0.105\mathrm{m}$ (

$x=0.954$). Since the radial distances between the adjacent pressure tubes are small, the application of the average values of

K results in a small error only. The values of

${C}_{qr}$ for the experimental results are calculated with

$r=\frac{{r}_{e}+{r}_{e+1}}{2}$. Therefore, the values of

K (

$K\approx \overline{K}$) can be verified combining the pressure measurements with Equation (9).

On the basis of the results from both numerical simulations and pressure measurements, Equation (10) is determined to describe the impacts of

G and

${C}_{qr}$ on

K. For the rotation dominant flow, the values of

K follow the analytical law given in Equation (10a). For the through-flow dominant flow, the values of

K decrease exponentially with increasing

${C}_{qr}$ following Equation (10b). In

Figure 8, the experimental results based on pressure measurements by Equation (9) are compared with those from numerical simulations and those from Equation (10). Relatively large errors occur only when

${C}_{qr}>0.01$, which can be attributed to the application of the average values of

K in and around the transition zone between the rotation dominant flow and the through-flow dominant flow. For the future work, more pressure taps are intended to be used at the low radius to reduce the error.

The influence of

G on

K is weak according to the results in

Figure 9. Poncet et al. [

15,

16,

17] and Debuchy et al. [

18,

19] neglected the impact of

G on

K with the results from LDA measurements. In most of the radial pumps and turbines,

G is a variable along the radius. A simplified correlation is required with good accuracy over the parameter range of

G in this study. By pressure measurements, Equation (11) is correlated to predict the values of

K when

G ranges from 0.018 to 0.072. The results from Equation (11) are in good accordance with both the simulation results and the experimental results compared in

Figure 9. In this paper, Equation (11) is applied in the calculation process of pressure instead of Equation (10).

Some values of

K, however, do not fit to the results from Equation (11), especially at

$x=0.955$ for wider gaps. A selection of the results is shown in

Figure 10a. Near the outlet, an area change from the front cavity to the passage in the guide vane for

$G=0.036$, 0.054 and 0.072 occurs. The measured pressure at

$x=0.955$ is strongly influenced by the geometry at the outlet of the test rig (see

Figure 10b). Based on the simulation results, small vortices near the outer radius of the disk exist and therefore the measured values at

$x=0.955$ are used only partially during the calculation of

K.

#### 5.3. Axial Thrust

From the axial thrust measurements, an empirical correlation for the thrust coefficient in a rotor-stator cavity with centripetal through-flow is determined by Hu et al. [

26]. Compared with the experimental results in this paper, the correlation is modified for centrifugal through-flow, written as:

where

The comparison of the results of

${C}_{F}$ for different

G and

${C}_{D}{}^{\prime}$ are shown in

Figure 12. The parameter

Bp, which is the calculated thrust coefficient obtained from the pressure calculation along the radius of the disk, is calculated by combining the measured pressure with Equation (4) according to the values of

K (calculated with

${C}_{qr}$ every 1 mm along the radius for the rotation dominant flow or the through-flow dominant flow) from Equation (11). In the transition zone, the correlation of

K for rotation dominant flow is used. The experimental results of

${C}_{F}$ are in good agreement with those from the pressure calculation and Equation (12). The differences between the results of

${C}_{F}$ from Equation (12) and by axial thrust measurements are less than 5% in most of the measured points. Relatively large errors (smaller than 12.9%) only occur at

Re = 0.38 × 10

^{6}. The values of

${C}_{F}$, which are smaller for wider axial gaps, decrease with increasing

$\left|{C}_{D}{}^{\prime}\right|$. As commonly understood, however, the values of

${C}_{F}$ increase with increasing

$\left|{C}_{D}{}^{\prime}\right|$ in a rotor-stator cavity with centripetal through-flow (

${C}_{D}{}^{\prime}$ is negative) [

26].

#### 5.5. Moment Coefficient

According to the experimental results from Han et al. [

27], the moment coefficient on the cylinder surface of the disk, noted as

${C}_{Mcyl}$, can be estimated with Equation (13) for smooth disks.

By combining the torque measurements with the results from Daily and Nece [

3] and Dorfman [

28], two correlations are determined to predict the moment coefficient for a single surface of the disk, given in Equations (14) and (15). For an enclosed rotor-stator cavity, the results from Equations (14) and (15) are around 0.28 times those from the correlations by Daily and Nece [

3]. Hu et al. [

26] attribute the large gaps to the difference of disk surface roughness.

The experimental results of

${C}_{M}$ are compared with those from Equations (14) and (15), depicted in

Figure 14. In

Figure 13a, most of the flow regimes are regime III at

$G=0.018$. The experimental results of

${C}_{M}$ are in good accordance with those from Equation (14). When

G increases to 0.036, the flow regimes shift from regime III to regime IV with the increase of

Re for

${C}_{D}{}^{\prime}=0$, 1262 and 2525. The regime changes can also be found based on the experimental results of

${C}_{M}$. For example, at

G = 0.036 and

${C}_{D}{}^{\prime}=1262$, the results of

${C}_{M}$ from Equation (15) rather than from Equation (14) are approaching those from torque measurements when

Re ≥ 1 × 10

^{6}. In

Figure 13a, the separating point of regime III and regime IV at the same values of

G and

${C}_{D}{}^{\prime}$ is at

Re ≥ 0.75 × 10

^{6}. The difference can be attributed to the existence of the mixing zone. For

$G=0.054$ and

$G=0.072$, most of the flow regimes are regime IV according to

Figure 13a. The results of

${C}_{M}$ from Equation (15) are more coincident with those from torque measurements. The regime III may occur at small

Re and large

${C}_{D}{}^{\prime}$. The amount of

${C}_{M}$ increases by increasing

${C}_{D}{}^{\prime}$ while it drops faster for higher Reynolds numbers

Re at small

G. Compared with the cases for centripetal through-flow [

26], the centrifugal through-flow will result in larger values of

${C}_{M}$ at the same values of

$\left|{C}_{D}{}^{\prime}\right|$, which is in accordance with the results of Dibelius et al. [

29].

On the distinguishing lines (see

Figure 13a), the results from Equation (14) are supposed to be equal to those from Equation (15). The results of

${C}_{M3}$/

${C}_{M4}$ at the distinguishing lines are presented in

Figure 15. Most of the differences attributed to the existence of the mixing zone are very small and cover an amount lower than 5%. According to

Figure 15, Equations (14) and (15) can be used to predict the transition from regime III to regime IV with good accuracy. All the results show that the moment coefficient can be predicted with Equations (14) and (15) with the 3D diagram.

There are still some limitations of this work. All the experimental results are obtained with the smooth disk $({R}_{z}=1\mathsf{\mu}\mathrm{m})$. The applicability of discussed equations will become broader by introducing the impact of surface roughness of the disks in the next step. The existance of through-flow dominant region is ignored during the calculation of ${C}_{M}$ because it may only occur at small radius for small Reynolds numbers and large through-flow coefficient. Since the frictional torque in that region is proportional to ${r}_{c}{}^{5}$ (the parameter ${r}_{c}$ is the radial coordinate below which the flow is dominant by through-flow), which is of small value compared with ${b}^{5}$, the above simplification results in small errors only. The distinguishing lines will be modified based on the measured velocity components in both tangential and radial directions in the future. The outlet geometry has a relatively large influence on the results of K, which deserves further investigations. The impacts of boundary conditions (at both inlet and outlet) and internal flow structures on ${C}_{F}$ and ${C}_{M}$ need to be investigated further.