## 1 Introduction  
When a new gas turbine engine is under development, abundant hardware builds are necessary to measure the performance characteristics, which are much too expensive [1]. Therefore, the simulation of an overall gas turbine engine is extremely needed[PERSON] Thanks to both the advance of the CFD (Computational Fluid Dynamics) and the increasingly developed computer technology, fully three-dimensional (3D) computation is increasingly involved [2, 3]. However, it is very time-demanding. For example, when [PERSON] et al. [4] simulated a 20\({}^{\circ}\) sector gas turbine engine (shown as Figure 1) on the coarse grids containing 14 million cells, a simulation was performed in nearly 2 weeks relying on 700 processors [5, 6, 7, 8]. On the other hand, the zero-dimensional method (0D) commonly used in the performance evaluation cannot present the radial distribution of aerodynamic parameters. Therefore, the throughflow method (2D) is still a significant part of its performance prediction and rapid flow field acquisition [9]. To achieve this ultimate goal of the whole gas turbine engine simulation, the throughflow method applied in compressors is first explored in this paper. The most widely used throughflow method is the streamline curvature method. Comparing with it, the circumferential average method (CAM) first, proposed by [PERSON] [10] is appealing in the following aspects: choke mass flow can be predicted, certain types of shock can be captured, and the same code can be used for subsonic, transonic and supersonic meridional velocities. Much research has been performed based on the CAM in turbomachinery. However, one element indispensably mentioned in every throughout method applied in turbomachinery is that the absence of the description about the circumferential motion leads to the blade force missing (shown as Figure 2), which needs the blade force modeling appended. As for the inviscid blade force model, it was first presented by [PERSON] [11]. In his method, the blade force modeling was proposed based on the fully time-dependent axisymmetric Euler equations. In this theory, the flow conformed to the average stream surface for each blade row, and the blade force must be orthogonal to the average stream surface in order to obtain a loss-free flow turning. Moreover, the blade force, which they thought determined the flow path, was its tangential component. Based on the theory of [PERSON], [PERSON] [12] modified the calculation formula of the inviscid blade force, where the tangential component of the inviscid blade force was substituted by the modulus of the inviscid blade force in the calculation formula. For the design mode of turbomachinery, it was first proposed by [PERSON] and [PERSON] [13], where the target tangential velocity was prescribed through the flow angle at the trailing edge (TE). In their research, how the circumferential velocity evolves inside the blade passage in the streamwise direction is needed to be given. [PERSON] [14] adopted another method in the solution of the inviscid blade force, where the blade force was computed from the circumferential momentum and was not solved anymore in the bladed regions. The aforesaid methods depended on a formulation, which had a constant coefficient determining the convergence effect. Based on these methods above, convergence problems often arose when iteratively solving these inviscid blade force formulations, especially with an inappropriate relaxation factor. Considering this, [PERSON] [15] presented a method similar to the large-particle method to avoid solving the inviscid blade force explicitly. This method consisted of two steps for each time step. First, the effects caused by the inviscid blade force in the momentum equations were neglected. Moreover, then only the effects caused by the inviscid blade force were taken into account in the governing equations. However, all these foregoing methods are dependent on the assumption of the average stream surface.