## 1 Introduction  
With the continuous development of society, oil field exploitation and transportation technology requirements in industrial applications are continually increasing. The conventional conveying equipment, which includes liquid-phase pumps, compressors and separators, is being replaced by multiphase pumps. Multiphase pumps directly transport the mixture of oil, gas and water from subsea oil wells through a single pipeline instead of two pipelines, which can simplify equipment usage and save capital costs [1]. As the core of oil-gas mixture transportation technology, the design and application of multiphase pumps has become a research hotspot. At present, two main types of multiphase pumps are widely used in the energy industry. The first is the twin-screw multiphase pump; the advantage of this is that the gas-liquid two-phase medium is more stable and the change in gas volume fraction has little effect on the performance of the pump. The disadvantage is that the conveying flow is low, and it is sensitive to particles such as sediment. This type of pump has low reliability, short lifespan and its maintenance is inconvenient [2]. The other pump type is the helical-axial-flow oil-gas multiphase pump; the advantage of this kind of pump is that the structure of the impeller has a long, square channel with a larger flow curvature radius and the ability for large conveying flow, with a higher gas-liquid two-phase transportation capacity than other centrifugal pumps and mixed-flow pumps. When the centrifugal pump conveys the gas-liquid two-phase medium with a certain inlet gas volume fraction, the efficiency will drop sharply and it becomes difficult even to maintain the operation of the pump [3]. The helical-axial multiphase pump can transport a gas-liquid multiphase flow with a gas volume fraction from 0% to 90%; it can continue to work under the condition of high inlet gas volume fraction [4]. The disadvantage is that the performance of the pump will decrease with the increase in the gas volume fraction [5]. Combined with the advantages, the application of multiphase pumps is an ideal choice for gas-liquid two-phase transportation in the petroleum industry. The characteristics of different types of multiphase pumps are listed in Table 1.  
The initial application of the multiphase pump was conducted in the 1990s: the world's first multiphase pump, the Poseidon helical-axial multiphase pump developed by the French Petroleum Institute (IFP), Norwegian National Oil Company (Statoil) and France's Total Oil Company was installed by Shell, which successfully boosted the well effluent from a satellite well to the host platform six miles away [6]. Subsequently, various investigations of multiphase pumps have been carried out. To date, hundreds of multiphase pumps have been used all over the world, including in desert oil fields [7], marginal oil fields [8] and deep-sea oil fields [9; 10]. The helical-axial multiphase pumps are widely used at present, but the gas-liquid two-phase flow is easily affected by many external factors, and relative motions occur between phases and complex interface effects, creating problems for determining the evolution mechanism of two-phase flow [11; 12]. Therefore, the significant issues for improving the performance of multiphase pumps are the design and optimization of multiphase pumps since they are directly related to the pump performance [13]. Many scholars have carried out research into the multiphase pumps since the 1990s, and they have contributed a great deal to the design and internal flow analysis of multiphase pumps. [PERSON] et al. [14] designed an impeller by combining axial pump and axial compressor theory and built a three-dimensional (3D) model. [PERSON] et al. [15] simulated the two-phase 3D turbulence flow characteristics numerically