## 1 Introduction  
With the emergence of ultra-high, ultra-deep, ultra-large-scale, and other complex projects, concrete pumping technology has become a key technology in modern building construction [1; 2]. Improving concrete pumping efficiency, reducing wear and tear on pumping pipes, and decreasing energy consumption are essential for enhancing construction efficiency and lowering costs. Generally, excessively high pumping pressure can lead to concrete segregation, increase wear on aggregate particles and the pumping pipeline, and result in higher energy consumption due to greater pumping power requirements. Reasonable planning of the pumping pipeline layout to effectively reduce pressure loss during concrete conveying is crucial for lowering concrete pumping pressure [3]. The pumping pipe of the pump truck comprises various sections, including straight pipes and bends, which are connected by flanges. The straight pipe is widely used because of its low local resistance. In the pumping straight pipe section, the concrete flow exhibits good continuity and a stable flow state. Under the joint action of pump pressure and pipe wall friction in a non-uniform shear field, the aggregates are pushed by larger shear forces towards regions with smaller shear forces, while the mortar is squeezed through the gaps between the aggregates to the pipe wall. As a result, the concrete in the pump pipe is divided into a stable lubrication layer and a plunger layer [4; 5]. [PERSON] et al. used ultrasonic detection to measure the thickness of the lubrication layer at different flow rates. They found that the thickness of the lubrication layer was not significantly affected by the particle size of the coarse aggregate [6; 7]. [PERSON] et al. took the thickness of the lubrication layer into account and derived an equation for calculating both the flow rate and the pressure loss [8]. [PERSON] et al. observed that concrete exhibited shear flow at higher flow rates and established a relationship between the pumping pressure and the flow parameters of the lubrication layer [9]. [PERSON] et al. established the one-dimensional flow model for concrete pumping in straight pipe sections, considering the concrete flow characteristics, and they used Moody's chart to derive the pressure loss value for concrete pumping [10; 11]. The bend pipe can flexibly change the direction of concrete flow to ensure that the concrete is transported to the construction site. Compared to the straight pipe section, the flow of concrete in the bend section is more complex and variable due to interactions such as inertia, wall friction, particle collisions, and particle-avoidance collisions. Numerical simulation techniques, including single-phase flow, discrete element methods, and particle flow simulations, are widely used to model the rheological behavior of concrete. Concrete flow is a very difficult mixture to simulate. In this type of work, there is often a calibration phase that links the simulation with the real behavior of the material [12]. [PERSON] et al. used the CFD method to develop a simulation model to simulate a slider experiment and used single-phase flow to study the general trend of pressure variation with the discharge rate using this model, which yielded satisfactory results [13]. [PERSON] et al. used the \(SST\)\(k\)-\(\omega\) turbulence model to predict the flow behavior and pressure loss of pumped concrete in the pipeline [14]. [PERSON] et al. simulated the flow behavior of pumped concrete in a pipe based on the discrete element method, considering the contact parameters between coarse aggregate and cement paste particles [15]. [PERSON] et al. carried out a study on the effect of coarse aggregate volume fraction on concrete pumping pressure based on the discrete element method [16]. [PERSON] et al. used particle flow simulations to analyze the flow characteristics of concrete during pumping and found that the numerically simulated pressure loss values were close to the experimental results [17].