###### Abstract  
The purpose of this study is to consider propeller geometry and blade rotation in the propeller model in a CFD code. To predict propeller performance, a body force propeller model was developed based on blade element theory and coupled with URANS (unsteady Reynolds-averaged Navier-Stokes) solver CFDSHIP-IOWA V4.5 both implicitly and interactively. The model was executed inside the flow solver every inner iteration. The grid points inside each 2D blade geometry were identified by a numerical search algorithm. To calculate the lift coefficient, the total flow velocities at 25% foil chord length were obtained using the inverse distance weighting interpolation from the RANS solution. The body forces were distributed linearly along the chord length with the maximal value located at the leading edge and zero at the trailing edge. The main achievements are: (1) for a KP505 propeller in an open water condition, the error of the thrust coefficient generally is around or less than 3%, which is a better prediction than the previous model. (2) For a behind-hull condition, the error is about 1%. (3) For an E1619 propeller in an open water condition, the error is around 6%. (4) The blade-to-blade effect and unsteady flow field between blades are sufficiently resolved by the model. CFD; URANS; body force propeller model; open water propeller; behind-hull condition; propeller rotation +
Footnote †: journal: Applied Sciences  
Meanwhile, several body force propeller models have been developed to save computational time and grid generation effort. Generally, propeller programs are based on the small disturbance method and potential flow theory. Thus, a numerical procedure or coupling method is required to subtract the propeller-induced velocity or obtain the effective wake from a viscous flow solution. Force distribution is predicted by propeller program and then transferred to RANS as body force, which is the source term of momentum equations. Normally, a propeller/actuator disk or a specific propeller plane is utilized to exchange those velocities and body forces between RANS and propeller code. A body force propeller model can be a prescribed model as well, e.g., a Hough and Ordway circulation distribution [21]. Based on known propeller open water data, a given thrust and torque can be transformed to a body force distribution in CFD grids. In T2015, the model was implemented by [PERSON] et al. [22] in CNR-INSEAN code, [PERSON] et al. [23] for CFDSHIP-IOWA, and [PERSON] et al. [24] in OpenFOAM and Star-CCM+. In fact, body force propeller models coupled with RANS code have been under development, have a long history of use, and have been described in many publications. Until now, it has been very useful, effective, and efficient for ship applications. [PERSON] et al. [25] coupled PUF-2 propeller code with a RANS code based on the finite analytical method. [PERSON] and [PERSON] [26] introduced a linear-induced velocity subtraction between the Yamazaki propeller model [27] and CFDSHIIIP-IOWA. The propeller model was based on the lifting line method. PUF-14 code was coupled with CFDSHIP-IOWA later, as well in [PERSON] et al. [28]. [PERSON] and [PERSON] [29] implemented MPUF3A propeller code inside a Chimera RANS solver to study propeller cavitation in propeller-hull interaction. Viscous nominal wake was obtained first, and then, the wake was updated by solving body forces in RANS. In the end, an effective wake for MPUF3A would be obtained. In a similar manner, [PERSON] et al. [30] used MIT-PSF-2 propeller code in a RANS code UVW. The propeller inflow for MIT-PSF-2 is circumferentially averaged from the RANS solution. Using the circumferentially averaged inflow and updating the nominal wake to an effective one, [PERSON] et al. [31] coupled an unsteady propeller panel code developed by [PERSON] [32] with Star-CCM+. [PERSON] et al. [33] applied the blade element momentum theory in OpenFOAM to model a propeller behind a ship.