## 1 Introduction
[2] investigated the effects of the distance between the rotors and rotor speed in the CCRR using large eddy simulation (LES). If the distance between the rotors is small, the performance of the FR decreases significantly, and, if the distance is large, the RR generates variable loads due to the wake effects of the FR. Further, if the rotation speed of the RR increases, the performance of the FR decreases significantly. In addition, turbulent regions are also generated by the blade root vortices of the FR. This also showed that a separation of 0.4 times the diameter resulted in a 10% higher efficiency than the single rotor. [PERSON] et al. [3] compared four CFD methods for CCRRs. They reported that the sliding mesh method is the most appropriate for dynamic calculation, but the mixing plane (MP) method can simulate the dynamics of the rotor appropriately. Considering the results, the MP method was used as the reference in the present research. It is explained in a little more detail in (3) below. In addition, studies of the CCRR using full-model CFD have been undertaken by [PERSON] and [PERSON] [4] and [PERSON] [5]. There are no special notes on them as an analysis method. CFD with actuator line model (ALM)  
One of the typical approaches for the simplification of CFD is the ALM, developed by [PERSON] [6]. The ALM calculates the loads on the blades based on the blade element theory. And the flow around the blades is induced by the reaction force. [PERSON] et al. [7] applied the ALM for the calculation of CCRR performance. No significant interference was found between the two rotors if the clearance between the two rotors is 0.5 time of the rotor diameter (\(D\)), which is significantly larger than the CCRR in the present study (0.1 \(D\)). 1. CFD with mixing plane (MP) model  
The MP model transfers the interaction between two numerical regions. The application of the present model is limited for a steady symmetric flow [7], which is assumed in the present study. Each of the FRs and RRs is represented by one blade, periodic boundary conditions are applied in the rotation direction, and the interaction between the two rotors is transmitted at the radial position of each blade [8]. Since the interaction between the rotors is averaged in the azimuth, it is not theoretically possible to analyze the dynamic characteristics as obtained by full-model CFD. However, it has been reported that analyses of the average load can be undertaken with good agreement. Even with this method, three or more calculations are required for each condition to find the point where the torque is balanced, but the analysis time can be significantly reduced in accordance with the number of blades. However, this method is limited when applied to axially symmetrical airflow and for average loads. Details of this method are shown in Section 4. 1. Vortex lattice method  
[PERSON] et al. [9] compared single rotors and the CCRR using the vortex lattice method. They reported that the axial induction coefficient increases as the solidity increases, and the axial induction coefficient of the CCRR is lower than that of the single rotor. The maximum power coefficient of the CCRR is approximately 30% higher than that of the single rotor. A verification of this method was not conducted for the CCRR, but only for a single rotor. In addition, [PERSON] et al. [10] conducted an analysis using the free-wake vortex lattice method, considering the effects of the mutual interference between the FR and RR. The induction coefficient of RR was compared with that of the blade element and momentum (BEM) method, assuming that the RR is completely within the wake of FR; that is, the influence of the RR on FR is not considered. 1. Blade element and momentum (BEM) method  
[PERSON] et al. [11] developed a modified BEM method and investigated the influence of the design parameters on the CCRR. This method considers the interaction of the FR with RR as a flow within a well-developed wake but does not consider the interaction of the RR with FR.