Effect of Radial Height of Helical Static Blade on the Cavitation Performance of Inducer

Abstract

Cavitation is a major concern in liquid rocket engine turbopumps, and as an effective measure to improve cavitation quality, an inducer with helical static blades has attracted attention in recent years. In order to study the effect of the radial height of helical static blades on the cavitation performance of the inducer, CFD methods based on the Reynolds-averaged N-S equation, the standard k-ε turbulent model, and the Schnerr and Sauer cavitation model are employed to analyze the cavitation flow characteristics of a certain inducer with different helical static blades. The results show that with the increase in radial height, the backflow in the flow field is enhanced. Affected by this situation, the head is improved, the efficiency is reduced, and the low-pressure zone on the suction surface at entrance is enlarged. The helical static blade can delay the channel blocking of cavitation by providing an extra channel for the extension of bubbles. However, the effectiveness is restricted because the cavitation area enlarges with the radial height of the helical static blade. Although the effect of radial height on the head and the cavitation performance is opposite, there is an optimal radial height from 0.05 to 0.125 that improves both at the same time.

1. Introduction

In liquid rocket engines, an inducer is always installed before the main impeller to increase the inlet pressure of the centrifugal pump sufficiently, so that the influence of cavitation is limited to the inducer passage. Cavitation is a common vaporization process in liquid machinery due to the fact that the partial pressure of fluid is reduced to below the vapor pressure at the local temperature [1]. Cavitation that occurs in an inducer will cause performance deterioration, induce cavitation instabilities, and even lead to launch failure [2,3,4].

Over the years, researchers have been committed to improving the cavitation quality of inducers in two main directions. One is to focus on the performance characteristic of inducer geometry parameters, such as the blade number [5,6], the blade spacing [7], the tip clearance [8,9], and the shape of the blade’s leading edge [10,11,12]. In another direction, casing adjustment schemes attract more and more attention. The findings of Tsujimoto [13,14,15,16] have shown that various types of cavitation instabilities are concerned with stable cavity length, while inlet casing enlargement was confirmed to help to avoid cavitation instability. Kimura [17] studied the effect of inlet casing geometry as well. He found that a slight modulation in inlet casing geometry can distinctly change the inlet vortex structure, which conduces to suppress the cavitation surge effectively. It is precisely because of excellent performance in the cavitation problem that more schemes of casing adjustment have been designed and studied in recent years. Shimiya [18] tested a kind of shallow groove on casing wall, called J-grooves, and the research results indicated that rotating cavitation and asymmetric cavitation are both controlled to some extent. Kang [19] compared the influence of straight casing and circumferential groove casing on the cavitation performance. It was shown that the tip leakage vortex is trapped within the groove to avoid interacting with the next blade.

In addition, Ankudinov [20,21,22] proposed to install helical static blades for the inducer, and he observed that the composite structure, called the axial-vortex stage, has a higher head radio and a better cavitation quality than the separate inducer. The helical static blade is a spiral cascade fixed on casing, with an opposite thread rotation direction to the inducer. More vortex flows appear at the periphery, which increases the complexity of fluid flow. Li [23] compared the cavitation performance of an inducer with and without helical static blades. It was found that the helical static blade channel helps to delay the time in which bubbles block the inducer channel. Timushev [24] analyzed the unsteady flow with the CFD method, and the results showed that when helical static blades are equipped, the pressure pulsation of blade passing frequency in the inducer outlet becomes lower. Ankudinov [25] studied helical static blades with different axial lengths as well, and he pointed out that the pressure coefficient increases almost linearly with the increase in the axial length. He also found low-frequency oscillations provoked by vortex disturbances at inlet, and he proposed to optimize the position and number of vanes of helical static blades.

At present, there is little research on the influence of the radial height of helical static blades on the cavitation performance of inducers. In order to improve the cavitation quality of an inducer, further optimization of helical static blades is necessary. In this study, helical static blades with different radial heights are employed in the cavitation study of inducers, and the analysis of the results provides a theoretical reference for the design of helical static blades.

2. Research Object

The main research object in this study is an equal pitch inducer with helical static blades, as shown in Figure 1. The inducer is designed as follows: the flow rate Q = 5 kg/s, the head H = 7.5 m, and the rotating speed n = 4000 r/min, and the main structure parameters are shown in

Table 1. Main parameters of the inducer.

Parameters	Number of Blades	Inlet Hub Diameter dh1/mm	Outlet Hub Diameter dh2/mm	Tip Diameter Dt/mm
Value	3	24	40	78

. The helical static blade has the same pitch and the opposite turning direction to the inducer. A characteristic parameter k is defined as h_r/D_t, in which h_r is the radial height of helical static blade and D_t is the tip diameter of the inducer, so that h_r can be described more reasonably.

Table 2. Parameters of the radial height of helical static blade in different schemes.

Scheme	k	Radial Height hr/mm
Scheme 1	0.05	4
Scheme 2	0.075	6
Scheme 3	0.1	8
Scheme 4	0.125	10
Scheme 5	0.15	12
Scheme 6	0.175	14
Scheme 7	0.2	16

shows the corresponding relationship between k and h_r in different schemes.

3. Simulation Method

3.1. Geometric Model

The whole calculation fluid domain, as shown in Figure 2, is divided into four parts: inlet section; inducer; helical static blade; outlet section. In order to ensure the flow stability, the inlet and outlet sections are extended to 5 times the pipe diameter.

3.2. Mesh Generation

The polyhedral algorithm with good adaptability and fast solution convergence [26] is used to deal with the complex boundary conditions of the inducer part and the helical static blade part. For the remarkable areas, such as the tip clearance and the inlet edge of blades, the grid is refined locally. In order to eliminate the effect of grid number on the simulation accuracy, a mesh independence study is performed with the separate inducer, and the results are presented in Figure 3. It is shown that the head tends to be stable with the progressive increase in grid number. The last two grid numbers are, respectively, 1.57 million and 2.21 million, and the corresponding heads are 7.536 m and 7.531 m; the change in the head is less than 0.1%. Taking into account accuracy and efficiency, the mesh dimension control scheme of 1.57 million cells is employed for the subsequent calculation, as shown in Figure 2.

3.3. Governing Equations

Cavitation flow is commonly considered to be an incompressible viscous fluid, so in this paper, the Reynolds-averaged Navier–Stokes equations considering the continuity equation are employed as follows:

$$\frac{\partial \rho }{\partial t}+\frac{\partial \left(\rho u_i\right)}{\partial x_i}=0$$

(1)

$$\frac{\partial \left(\rho u_i\right)}{\partial t}+\frac{\partial \left(\rho u_i{u}_j\right)}{\partial x_j}=-\frac{\partial p}{\partial x_i}+\frac{\partial }{\partial x_i}\left[\mu \left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}-\frac{2}{3}{\delta }_{ij}\frac{\partial u_k}{\partial x_k}\right)\right]+\frac{\partial \left(-\rho \overline{u_i^{\prime }{u}_j^{\prime }}\right)}{\partial x_j}$$

(2)

where $\rho$ is the density of the mixed fluid, $u_i$, $u_j$, and $u_k$ are the velocity components, $p$ is the pressure, $\mu$ is the turbulent viscosity, and $-\rho \overline{u_i^{\prime }{u}_j^{\prime }}$ is the Reynolds stress.

3.4. Turbulence Model

As a supplement to the governing equations, the calculation uses the standard k-ε turbulence model to solve fully turbulent flows. As for the near wall area, standard wall function is adopted. The standard k-ε model computes the turbulent viscosity ${\mu }_t$ through the turbulence kinetic energy k and its rate of dissipation ε in the following equations:

$${\mu }_t=\rho C_{\mu }\frac{k^2}{\epsilon }$$

(3)

$$\frac{\partial \left(\rho k\right)}{\partial t}+\frac{\partial \left(\rho k{u}_i\right)}{\partial x_i}=\frac{\partial }{\partial x_j}\left[\left(\mu +\frac{{\mu }_t}{{\sigma }_k}\right)\frac{\partial k}{\partial x_j}\right]+G_k-\rho \epsilon$$

(4)

$$\frac{\partial \left(\rho \epsilon \right)}{\partial t}+\frac{\partial \left(\rho \epsilon u_i\right)}{\partial x_i}=\frac{\partial }{\partial x_j}\left[\left(\mu +\frac{{\mu }_t}{{\sigma }_{\epsilon }}\right)\frac{\partial \epsilon }{\partial x_j}\right]+C_{1\epsilon }\frac{\epsilon }{k}{G}_k-C_{2\epsilon }\rho \frac{{\epsilon }^2}{k}$$

(5)

where $G_k$ is the turbulent kinetic energy generation term, ${\sigma }_k$ and ${\sigma }_{\epsilon }$ are the turbulent Prandtl numbers, and $C_{\mu }$, $C_{1\epsilon }$, and $C_{2\epsilon }$ are constants, the values of which are 0.09, 1.44, and 1.92, respectively.

3.5. Cavitation Model

In the cavitation flow of an inducer, the vapor nuclei form cavities in low-pressure regions, and the cavities collapse in high-pressure regions, with steep density variations in the whole cavitating regions. In this paper, the Schnerr and Sauer cavitation model [27] is used to deal with the evaporation and condensation process:

$$m^{+}=\frac{{\rho }_{\mathrm{v}}{\rho }_{\mathrm{l}}}{\rho }\alpha \left(1-\alpha \right)\frac{3}{r_{\mathrm{b}}}\sqrt{\frac{2}{3}\frac{\left(p_{\mathrm{v}}-p\right)}{{\rho }_{\mathrm{l}}}}\hspace{1em}\hspace{1em}\left(p\le p_{\mathrm{v}}\right)$$

(6)

$$m^{-}=\frac{{\rho }_{\mathrm{v}}{\rho }_{\mathrm{l}}}{\rho }\alpha \left(1-\alpha \right)\frac{3}{r_{\mathrm{b}}}\sqrt{\frac{2}{3}\frac{\left(p-p_{\mathrm{v}}\right)}{{\rho }_{\mathrm{l}}}}\hspace{1em}\hspace{1em}\left(p\ge p_{\mathrm{v}}\right)$$

(7)

where ${\rho }_{\mathrm{v}}$, ${\rho }_{\mathrm{l}}$, and $\rho$ are, respectively, the density of liquid, vapor, and two-phase mixture, $\alpha$ is the volume fraction of vapor, and $r_{\mathrm{b}}$ is the diameter of bubbles.

3.6. Boundary Conditions

In this paper, we use Fluent to calculate the fluid domain, and the Multiple Reference Frame is adopted to deal with different zones, setting the inducer part as a rotating domain, while the others are set as as static domains. The inlet boundary condition is Total Pressure, and the cavitation characteristic curves are obtained by reducing the inlet pressure gradually. The outlet boundary condition is Mass Flow Rate. The solid wall is set as a non-slip wall surface. The fluid medium is 25 °C water, the vapor pressure of which is 3168 Pa, and the reference pressure is set to 0. In the calculation process, the velocity and pressure coupling mode adopts the SIMPLE algorithm, and the convergence criteria are determined by observing that the convergence residual is less than 10⁻⁴, or the outlet pressure tends to be stable.

4. Results Analysis

4.1. Comparison of Simulation and Experiment

In order to verify the reliability of numerical calculation, a water test was conducted for an inducer with and without the helical static blades, whose radial height is k = 0.125. Figure 4 shows the test object and the experimental system [28].

Figure 5 is the comparison of hydraulic performance between calculation results and test results. It can be seen that the variation tendency of the calculated value, whether head or efficiency, is in agreement with the test value. The test value is slightly smaller because the numerical calculation does not take the effects of frictional loss into account. For the separate inducer, the maximum relative errors of head and efficiency between calculated value and test value are 1.7% and 2.8%, while for the inducer with helical static blades, the maximum relative errors are 2.2% and 3.5%, respectively. Therefore, whether helical static blades are equipped or not, the error between the calculated value and the test value is within 5%, which proves the simulated method to be reasonable in the allowable error range.

4.2. Analysis of Simulation Results

Figure 6 plots the cavitation performance curves of both the separate inducer and the inducer with helical static blades, the radial height of which is k = 0.05, 0.1, 0.15, and 0.2, respectively. The curve describes the relationship between the cavitation number and the head, and the cavitation number, a dimensionless number that characterizes inlet pressure, is defined as the following equation:

$$\sigma =\frac{p_{\mathrm{in}}-p_{\mathrm{v}}}{\frac{1}{2}\rho u^2}$$

(8)

where p_in is the inlet pressure of inducer, p_v is the vapor pressure at local temperature, ρ is the fluid density, and u is the characteristic velocity of the flow. With the decrease in inlet pressure, the bubbles generate and gradually increase. However, the performance of inducer is not affected until the cavitation area develops to a certain scale. When the inlet pressure, or cavitation number, decreases to a certain value, the head then drops rapidly, as shown in Figure 6. In general, when the head drops by 10%, the corresponding cavitation number is defined as the critical cavitation number, which is used to evaluate the cavitation performance of inducer.

It can be seen that the similarity among different cases lies in that the head remains basically unchanged under the non-cavitation condition and decreases sharply when the cavitation number is below the critical cavitation number. With the decrease in cavitation number, the scale of bubbles continuously increases and gradually blocks the inducer flow channel, resulting in the sharp drop of the head. When the helical static blade is implemented, a portion of bubbles enters the channel of the helical static blade and squeezes the fluid within, which will reduce the leak of the helical static blade and increase the head as well. Therefore, for the inducer with helical static blades, the head increases slightly before the drop in the early cavitation stage, and the increment gets larger with the increase in radial height in a certain stage.

In order to analyze the effect of radial height of helical static blade, the rest of this section separately discusses the changes in external characteristics, flow field, critical cavitation number, and the cavitation evolution process.

4.2.1. External Characteristic

Figure 7a shows the change of the rated head under different radial heights. With the increase in the radial height, the head increases initially and decreases afterwards. The maximum head is 9.97 m at k = 0.15. When the radial height changes from 0.05 to 0.2, the head increases by 0.9%, 8.4%, 18.8%, 23.3%, 24.3%, 22.4%, and 16.9%, relative to the head of the separate inducer. Figure 7b shows that the rated efficiency gradually decreases with the radial height changing from 0.05 to 0.2; the decreasing amplitude of the efficiency is 6%, 8%, 10%, 13%, 15%, 17%, and 18%, respectively. Therefore, the implement of the helical static blade will improve the head of the inducer but reduce the efficiency at the same time; there is an upper limit for the increase in the head, which is concerned with the radial height of the helical static blade. Because the inducer is not the main pressurizing component in the liquid rocket engine, the loss of efficiency in the inducer will not significantly affect the overall supercharge efficiency of the propulsion system, while the increase in the head will increase the inlet pressure of the centrifugal pump, resulting in the improvement of cavitation performance.

4.2.2. Flow Field

Figure 8 shows the velocity and streamline distribution of axial plane under different radial heights at $\sigma =0.6$. For the separate inducer, a strong backflow area distributing intensively at the leading edge of the inlet can be seen on account of tip leakage and blade rounding. For the inducer with helical static blades, the backflow in the whole flow field is enhanced. Because of the work of the inducer, the liquid passes from inlet to outlet and its pressure increases. The pressure difference between inlet and outlet causes some liquid at the outlet to flow back to the inlet along the channel of the helical static blade, and this liquid then enters into the mainstream. Therefore, backflow is the main flow mode in the channel of helical static blades, accompanied with a large number of vortexes. On the one hand, the backflow causes the additional loss of efficiency; on the other hand, the fluid from the channel of helical static blade has a smaller attack angle because of the opposite turning direction of blades, resulting in the improvement of head to some extent. As the radial height increases, the fluid passing through the channel of helical static blade increases, so that the fluid with a smaller attack angle increases, and the head increases, but the loss of efficiency increases as well. In addition, the installment of a helical static blade will affect the inlet backflow vortex at the same time. When the radial height k = 0.05, the inlet backflow vortex is mainly concentrated around the channel of the helical static blade. With the increase in radial height, the scale of the inlet backflow and the velocity of the tip vortex continues to increase, which aggravates the instability of the flow field at the inlet of the inducer.

Figure 9 is the static pressure distribution of suction surface and pressure surface at $\sigma =0.6$. Owing to the pressurization of inducer, the pressure along the flow direction gradually increases; the low-pressure zone appears in the leading edge of the suction surface; the high-pressure zone appears in the trailing edge of the pressure surface. As the radial height of helical static blade increases, the area of the low-pressure zone at the inlet expands, while the pressure of the high-pressure zone at the outlet rises. The expansion of the low-pressure zone adversely affects the cavitation performance of the inducer, but the rise in the outlet pressure shows that the cavitation is limited to the inlet area. Hence, further research about the effect of the cavitation evolution process on the cavitation performance of inducers is needed, and it is presented in the rest of this section.

4.2.3. Critical Cavitation Number

Figure 10 shows the change in the critical cavitation number under different radial heights. It can be seen that when the radial height k is from 0.05 to 0.125, the corresponding critical cavitation number is below the case of the separate inducer, which means the inducer can operate under the more serious cavitation condition with the help of a helical static blade. As the radial height continues to increase, the critical cavitation number gradually rises, and the cavitation performance continuously declines. Therefore, in a certain range of radial height, the helical static blade helps to improve the cavitation performance of the inducer.

4.2.4. Cavitation Evolution Process

Figure 11 shows the bubble volume distribution in the flow channel of the inducer and helical static blade under different cavitation numbers with different radial heights.

When the cavitation number $\sigma =0.16$, it can be seen in each case that bubbles appear first on the suction surface near the inlet side of the inducer. The analysis of flow field shows that here, the velocity increases dramatically on account of the blade rounding and the backflow vortex, and the local pressure is at its lowest, so that cavitation is most prone to generate in this area. Compared with the separate inducer, a larger scale of cavitation area exists on the suction surface of the inducer with helical static blades under the same cavitation number. As the radial height of the helical static blade increases, the cavitation area gradually enlarges, because the backflow vortex at the inlet enhances with the radial height. Then, the bubbles will collapse rapidly due to the effect of the high-pressure fluid behind, with no significant change in head.

With the decrease in cavitation number, the cavitation becomes more serious, and the cavitation evolution process becomes different for the inducer with and without helical static blades:

For the separate inducer, the cavitation area mainly extends towards the entrance along the axial direction, and at the same time, towards the hub along the radial direction. When the cavitation number decreases to 0.08, the thickness of the bubbles increases to affect the pressure face of the adjacent blade. At this time, the head of the inducer begins to decrease according to Figure 6. When the cavitation number decreases to ${\sigma }_{10\%}$, it can be seen that the bubbles basically cover half of the suction surface, block the channel of the inducer, and cause the head to drop rapidly.

For the inducer with helical static blades, take, for example, the radial height k = 0.1; the channel of the helical static blade is another way in which the cavitation area can extend. At the cavitation number 0.08, although the scale of cavitation area is larger than the same situation in the separate inducer, there are not enough bubbles within the inducer to affect the channel, and the head is manifested as a small increment, as shown in Figure 6. With the increase in the radial height, more space is provided in the channel of the helical static blade for the diffusion of bubbles. Nonetheless, because of the enlarged low-pressure zone, more bubbles generate in the situation of higher radial height under the same cavitation number, which limits the further improvement of cavitation performance.

5. Conclusions

(1)

In a certain range, with the increase in the radial height of helical static blade, the head increases, the efficiency decreases, and the critical cavitation number increases. There is an optimal radial height from 0.05 to 0.125 to improve the head and the cavitation performance of inducer at the same time.

(2)

The backflow and vortex universally exist in the channel of the helical static blade and will gradually strengthen with the radial height. The backflow causes the fluid to have a smaller attack angle because of the opposite turning direction of the blades and helps to improve the head. However, the enhanced flow will increase the low-pressure area at inlet as well.

(3)

The helical static blade can delay the time when bubbles block the channel of inducer, but the effectiveness is limited because the cavitation area prominently enlarges with the radial height of the helical static blade.