###### Abstract
Basically, those models were based on the quasi-steady assumption that the propeller rotates much faster than the ship's advance speed. Those models only could predict time-averaged propeller flow field. More advanced propeller models have also been proposed to distribute body force on 3D propeller geometry to consider the blade-to-blade effect or blade rotation. [PERSON] et al. [34] combined a RANS solver and a vortex lattice method. The blade geometry was described by the B-spline surface for easy manipulation in design purpose. The RANS cells inside the 3D blade shape were first identified. The force per volume solved by the vortex lattice method was imposed at the cell center. The propeller panel code panMARE was utilized by [PERSON] et al. [35] in RANS code FreSco+. A plane was located at an upstream propeller to transfer the flow velocities along the propeller's angular and radial directions. A cell search algorithm was proposed to find the RANS cell center closest to the panel for panMARE. Later, the free surface effect was considered [36]. The singularities were distributed on the free surface geometry extracted by Fourier transform from RANS simulations with the propeller model. In the presented work, the cell was only searched in the 2D propeller plane. The side/lateral geometry of the propeller blade was neglected. In the above-mentioned propeller models, generally, the strength of the free and bound vortex was solved inside the propeller program. The bound vortex was distributed on the propeller surface or along the foil sections. The free vortex was shedding from the trailing edge of the propeller or foil to the far downstream. Therefore, not only the propeller-induced velocity subtraction was required, but the direction of the free vortex was also assumed. However, in CFD simulations and real situations, especially when the propeller operates behind the ship hull, the free vortex is not necessarily guaranteed to be the same as assumed in the propeller program. It is much more complicated and involves the propeller tip vortex and the influence from hull-propeller-rudder interaction. To resolve this issue, a propeller model based on a quasi-steady assumption and blade element theory (BET) had been proposed [37] and has had some applications [38; 39; 40; 41; 42]. In the following discussion, it is called the \"average model\", i.e., the body forces are distributed in average on the circular plane or actuator disk. The total velocity from the CFD flow solution is directly used as propeller inflow, so that the propeller-induced velocity subtraction is no longer needed. As a result, only the free vortex is solved in the propeller model. The bound vortex imbedded inside the CFD flow solution is used in the propeller computation, i.e., the propeller model and the CFD share the same bound vortex. The purpose of the presented work is to develop the average model [37] further to consider blade rotation and more realistic geometry. It is still based on BET using total velocity and coupled with the RANS solver CFDSHIP-IOWA V4.5 implicitly and interactively. The innovation of the proposed propeller model is summarized as follows. The blade geometry in the 2D propeller plane is constructed by chord length, pitch, and skew distribution. The grid points inside the blade's leading and trailing edges are identified by a numerical search algorithm. The propeller's local thrust and torque are calculated using the lift and drag coefficients. The total velocities at 25% of chord length along foil section are provided via the inverse distance weighting interpolation from a RANS solution. The body force is distributed linearly along the chord length with a peak value at the leading edge and a value of zero at the trailing edge. Several actual applications of the proposed model are conducted successfully. A five-bladed KP505 propeller is simulated in open water for advance coefficient \(J=0.3\) to \(1.0\). A grid independence test is conducted for \(J=0.5\), \(0.7\), and \(0.8\).