## 1 Introduction  
In recent years, the study of supercritical carbon dioxide (SCO\({}_{2}\)) Brayton cycle and its components has attracted lots of attention. Various heat sources including solar power [1; 2; 3; 4; 5], nuclear power [6; 7] and waste-heat utilization [8] are employed. As the key component in a Brayton cycle, the design and characteristics of turbomachinery deserve to be investigated in depth. SCO\({}_{2}\) has a critical point around room temperature (7.38 MPa, 304.25 K). As a working fluid in a power cycle, SCO\({}_{2}\) has a large number of advantages. First of all, it is environmentally friendly and the critical point is easy to reach. Hence, it is safe and cheap in industrial applications. Additionally, the high density and low viscosity of SCO\({}_{2}\) can result in the high efficiency and compact mechanical structure of turbines and compressors in the power system [9; 10; 11]. The rapid changes of density near the critical point can reduce the compressor work and lead to a higher heat transfer coefficient in heat regenerators and precoolers [12]. Finally, there is little phase change in a SCO\({}_{2}\) power cycle. Thus, it needs fewer valves and no condenser, which results in a concise cycle. To realize this proposed technology, Sandia National Laboratory has developed several SCO\({}_{2}\) power cycles including various power class [13]. The radial turbine and compressor test rigs were established to investigate the key techniques. However, the outcomes of efficiency and rotation speed were far from the expectations. Hence, the design method and flow mechanism need to be further considered before the commercialization of this technique. A traditional design process of the turbomachinery is time-consuming. It needs several engineers to conduct multiple numerical calculation to obtain the case of best performance. Hence, it is vital to establish an automatic process in order to save the manpower and accelerate the design. The flow conditions are rather complex in turbomachinery, which results in the difficulty of turbine design. In the past years, the research of design, influence of key parameters and optimization are widely covered [14; 15; 16; 17]. Solidity, as a design parameter, plays an important role in turbine design. It was proposed by [PERSON] [18] in 1945, in which he estimated the optimum solidity for turbines with large angular deflection. However, [PERSON] [19] pointed out that the prediction of [PERSON]'s optimum solidity correlation was limited to the outlet flow angle of the blade. In the past investigations, the influence of solidity on axial wind turbine performance occupied the vast majority. [PERSON] et al. [20] investigated the effects of flanged diffusers on rotor performance of small wind turbines with different rotor solidities and wind speeds. [PERSON] [21] studied the effect of the turbine solidity and the usage of hybrid system between drag and lift types on small wind turbine performance numerically and experimentally. [PERSON] et al. [22] experimentally investigated the influence of solidity on the performance and flow field aerodynamics of vertical axis wind turbines at low Reynolds numbers. [PERSON] et al. [23] thoroughly analyzed the effects of rotor solidity and leakage flow on the unsteady flow in axial turbine. Some researches focus on the solidity of stator vane in radial-inflow turbines. [PERSON] et al. [24] conducted the numerical and experimental study of the performance effects of varying vaneless space and vane solidity in radial turbine stators. [PERSON] et al. [25] concentrated on the influence of the guide vane solidity on the performance of a radial impulse turbine with pitch-controlled guide vanes. [PERSON] et al. [26] evaluated the effects of outlet blade angle, solidity, blade height, expansion ratio, and surface roughness on the stator velocity coefficient. To sum up, the stator solidity has been considered in radial-inflow turbines while the choice of rotor solidity is rarely investigated, especially its impact in a SCO\({}_{2}\) turbine.