## 1 Introduction  
When high-speed jets flow within hydraulic valves, the force exerted on the spool due to changes in fluid momentum is referred to as flow force. This force is one of the key factors determining the performance of many hydraulic components, including power output, stability, and dynamic characteristics [1; 2; 3]. For instance, in servo proportional valves used in high-performance applications, the presence of flow force impedes the opening of the valve port, limiting the improvement of both performance and control precision [4]. In the case of widely used single-stage electro-hydraulic valves, flow force significantly reduces spool stability under conditions of high pressure differentials or large flow rates, thus restricting the applicable flow and bandwidth range of such valves [5]. Over the past decades, extensive research has been conducted on the theoretical, numerical, and experimental aspects of valve flow force characteristics, continuously exploring effective ways to compensate or control the flow forces acting on the spool. The flow force acting on the spool can be divided into steady and transient flow forces, with the transient force typically being much smaller than the steady force [6]. There are three main approaches to studying steady flow force: theoretical methods, experimental methods, and numerical simulation methods [7]. Theoretical methods typically separate the calculation of pressure and viscous flow forces, estimating the forces on the spool using the momentum theorem [8]. However, theoretical methods rely on a series of simplifications and assumptions, often leading to discrepancies compared to actual scenarios. The application of simulation environments and optimization procedures (software) gives designers information about the optimal structure. Such approaches allow to one to avoid building many prototypes and save the design costs [9]. In fact, using CFD to optimize the structure of various hydraulic valves before conducting experimental validation has become the mainstream approach for studying spool flow force characteristics and compensation methods. In terms of flow force characteristics, several scholars have investigated the behavior of spool flow forces under different throttling types and operating conditions. [PERSON] et al. [10; 11] compared the peak flow force at the initial opening of spools in open- and closed-center directional control valves. Their study showed that, in open-center valves, the flow force increases with pump flow, with the peak force occurring when recirculation flow disappears, while its position remains constant. In closed-center valves, the position of the flow force peak depends on the pump flow rate and pressure drop. [PERSON] and [PERSON] [5] analyzed the effects of viscosity and non-metering momentum flux on steady flow forces, noting that viscosity increases the steady flow force for positive damping lengths, while it tends to reduce the force for negative damping lengths. [PERSON] et al. [12], considering seven different flow conditions, studied the flow force characteristics of a nozzle-flapper pilot valve. The results showed that during the reattachment process of the radial jet forming a vortex within the pilot valve annulus, undesirable flow forces causing flapper instability arise, in addition to the main flow forces acting on the flapper. [PERSON] et al. [13] examined the influence of groove geometry on the flow characteristics of slide valves, finding that the groove shape significantly affects flow area, jet angle, steady flow forces, and throttling stiffness. In particular, the throttling stiffness of a U-shaped valve port lies between that of spherical and triangular valve ports. [PERSON] et al. [14] conducted numerical studies on the flow forces within three typical conical valve structures used in water hydraulic valves, revealing that cavitation slightly reduces flow forces. However, two-stage throttle valves can effectively suppress cavitation, though their flow forces are much larger than those of other valves.