## 1 Introduction  
Increasing the aerodynamic loading of the compressor is of great significance for shortening the axial length of the aero-engine and reducing the weight of the structure. Therefore, the question, how to improve the compressor loading, has been a hot issue in the field of aerodynamic design [1]. Under such background, the tandem blade technology has become an effective method to break the load limit of conventional aerodynamic configurations, and it is of wide interest by researchers. Earlier studies of a tandem blade showed the potential performance advantages, especially in aerodynamic loaded conditions. In the 1970s, the Pratt and Whitney Group conducted a systematic investigation for highly loaded tandem blades using a single-stage test rig [2, 3]. Comparisons of conventional and tandem blade experiment results indicated that a tandem blade achieved better performance in pressure ratio and efficiency. [PERSON] et al. [4, 5] designed and optimized the tandem blades in a multistage axial compressor, and the flow loss decreased about 18 percent relative to the conventional configuration. By 1990, [PERSON] [6] summarized the different technology roadmaps for highly loaded axial flow compressors, and pointed out the loaded capacity of the tandem blade. The tandem blade can be regarded as a special configuration split out from the conventional blade, and thus more degrees of freedom in blade design and optimization have to be considered. Therefore, a large number of investigations were conducted for exploring the influence of new design variables on tandem blade performance. The key design principles were extracted after many theoretical analyses, numerical simulations, and cascade experiments. [PERSON] et al. [7] established the mathematical model of tandem airfoil loss based on a large number of experimental results, and systematically studied how these key parameters (such as the relative position, the solidity ratio, and the camber ratio of front and rear blade) affect the blade performance. [PERSON] et al. [8] investigated the relative position of the 2D tandem blade (explained in Figure 1) under the subsonic condition, and they pointed out that the tandem blade achieves the best performance when the AO is approximately zero and the PP is close to 0.7-0.8. The study of [PERSON] et al. [9; 10] showed a transonic tandem blade with a large overlap achieved a loss reduction of 30 percent and increase of turning of 10 percent. [PERSON] [11] presented a model to predict the minimum-loss incidence and deviation angles of a splitter blade, and it pointed out a way to guide the splitter blade into conventional design processes. Their efforts provide a reference to develop the empirical model for tandem blade design. Furthermore, the mathematical model [12; 13] used for the analysis of entropy generation and heat transfer [14] also can be referenced to estimate tandem blade performance. The CFD (computational fluid dynamics) combined numerical optimization algorithm rising in recent years has been adapted for tandem blade design. The numerical optimization method can help designers find the optimal solution, and even carry out in-depth analysis to further understand the flow mechanism. [PERSON] et al. [15] optimized the 3D tandem stator relative position with the multiobjective algorithm, and the optimal relative position principle was summarized. [PERSON] et al. [16] optimized a 2D tandem blade with the multiobjective genetic algorithm and artificial neural network method. A massive amount of simulation results [17; 18; 19; 20] indicated that the tandem blade performance should be estimated in design and off-design conditions comprehensively; for instance, a larger pitchwise blade distance will lead to an increase of the loss and the choke margin simultaneously. All the results stated above show the prescribed velocity distribution design concept has become an effective measure in highly loaded axial compressors optimization.