## 1 Introduction  
Marine pipelines carry fluids for different mechanical equipment. Seawater is an important cooling medium [1] for power systems, and it enters marine pipelines through subsea valves [2]. The flowing seawater erodes the pipeline at different levels, which is one of the main reasons for the failure of the pipe fittings [3]. In serious cases, the main and auxiliary engines in the power system cannot be cooled, threatening the safe navigation of the ship [4]. In addition, unlike low-latitude single-phase seawater, there is a large amount of sea ice floating in polar seas [5]. When a ship breaks through the ice, fine ice slag and ice debris can pass through the filter and enter the seawater pipeline, forming a two-phase flow of seawater-ice in the pipe, and the presence of particles will increase the erosion and wear of the pipeline [6]. At present, pipeline erosion exists in many fields such as petroleum, building materials, energy, shipbuilding, metallurgy, etc., and is a form of local corrosion that can cause great damage [7]. There have been many studies on pipeline erosion, but most of them assume static working conditions. However, for a ship in operation, the impact of ramming during ice breaking, high-load operation of the power system; fluid pulsation in the pipe; and other factors will also cause low-frequency vibration of the pipe [8]. Vibration may change the movement trajectory of particles in the pipe and the impact strength of particles on the pipe wall, thus affecting the degree of abrasion of the pipeline [9]. In addition, due to layout constraints, 90\({}^{\circ}\) elbows are common in ship pipelines [10]. During the transport of fluid media, the fluid experiences sudden changes in flow direction at the corners. The direct erosion of the pipe wall by the particles in the pipe due to inertial force and secondary flow cause erosion at corners to be dozens of times more severe than that of straight pipes [11]. Therefore, it is highly necessary to study the abrasion of elbows under vibration. At present, the erosion of pipelines by multiphase flow is a difficult problem and a hotspot in research [12], especially pipeline wear caused by solid-liquid two-phase flow [13]. The research methods mainly include experimental tests and computational fluid dynamics (CFD) simulation [14]. Through a large number of experiments and repeated observations, early scholars found that when solid particle mixed fluid flowed in the pipe, it had an erosion effect on the wall [15]. Thus, many theoretical foundations were gradually formed to provide reference for future research, such as the micro-cutting theory proposed by [PERSON] [16], the deformation and wear theory used by [PERSON] [17; 18], the secondary erosion theory mentioned by [PERSON] [11], and the forging extrusion theory proposed by [PERSON] [19], which proposed to resolve the deficiencies of the micro-cutting theory. These theoretical studies revealed the basic law of two-phase flow erosion wear. Later, [PERSON] and [PERSON] also found that the degree of wear of the pipeline was related to the impact angle of the solid particles and the flow velocity of the fluid [20]. [PERSON] and [PERSON] [21] studied the erosion effect of two-phase flow on pipes at 90\({}^{\circ}\) elbows and found that the degree of wear was not only related to the flow velocity, but also to the solid particle concentration. In addition, [PERSON] found that the erosion effect of particles on the pipe wall at the elbow was much greater than that of the straight pipe [11]. Later, [PERSON] and others tested the wear process of the 90\({}^{\circ}\) elbow, predicted the wear size, and determined the maximum wear position, but did not consider the effect of secondary flow; thus, the experimental results were limited. With the continuous progress of computer science and the rise of fluid simulation technology, the CFD numerical simulation method [22] is increasingly favored by scholars.