## 3 Propeller-Induced Jet Model  
The propeller-induced jet represents a complex flow condition that is difficult to simulate using analytical or numerical models. In working circumstances, the blades of the propellers will generate tremendous backward pushing force to drive the momentum of the ship forward. To represent this complicated flow phenomenon, [PERSON] et al. [24] and [PERSON] and [PERSON] [9] have proposed the use of their respective analytical approaches. Their theoretical models have been proven to represent the propeller-induced flow well while allowing fast computation (i.e., as compared to a numerical approach). The efflux velocity, \(\mathrm{V_{ef}}\), is one of the key products of the jet flow caused by the propeller's rotation. To model it, [PERSON] et al. [11] proposed  
\[\mathrm{V_{ef}}=1.22\mathrm{n^{1.01}D_{P}^{0.84}C_{t}^{0.62}} \tag{12}\]  
where \(\mathrm{D_{P}}\) is the propeller diameter, \(\mathrm{n}\) is the rotational velocity by the propeller, and \(\mathrm{C_{t}}\) is the thrust coefficient. However, the maximum axial velocities while the propeller is operational, \(\mathrm{V_{max}}\), should be reduced at a location further behind the propeller. [PERSON] [8] suggested that it should be determined via the ratio of blade area to the total circular area covered by propeller rotation, \(\mathrm{\beta}\); the distance from the propeller, \(\mathrm{X}\); and \(\mathrm{D_{P}}\) as shown below. \[\frac{\mathrm{V_{max}}}{\mathrm{V_{ef}}}=0.87\bigg{(}\frac{\mathrm{X}}{ \mathrm{D_{P}}}\bigg{)}^{-\frac{\mathrm{\beta}}{4}} \tag{13}\]  
According to a theoretical suggestion by [PERSON] et al. [12], and later by the practical model of [PERSON] [25], the axial velocity distribution, \(\mathrm{V_{r}}\), measured from the centre of the propeller, can be modelled using the Gaussian probability relationship. Judging the complex flow movement, overlapping probability functions can also occur, which leads to the following suggested model [9]:  
\[\frac{\mathrm{V_{r}}}{\mathrm{V_{max}}}=\exp\left[-\frac{1}{2}\frac{(\mathrm{ r}-\mathrm{R})^{2}}{\sigma^{2}}\right] \tag{14}\]  
where \(\mathrm{r}\) is the radial distance from the centre of propeller to the end of the blade (or the blade length), and \(\mathrm{R}\) is the radial distance from the centre of the propeller to the location of \(\mathrm{V_{max}}\). The standard deviation within the probability distribution model, \(\sigma\), can be represented according to the following relationship (provided that \(\mathrm{X}/\mathrm{D_{P}}>0.5\)):  
\[\sigma=\frac{1}{2}\mathrm{R}+0.075\bigg{(}\mathrm{X}-\frac{\mathrm{D_{P}}}{2} \bigg{)} \tag{15}\]  
In Figure 5a, the axial velocity model in Equations (14) and (15) is used to calculate and compare with the laboratory measurements by [PERSON] [8]. The test conditions for that experiment were as follows: \(\mathrm{D_{P}}=7.5\) cm, \(\mathrm{n}=1000\) RPM (revolutions per minute), and \(\mathrm{X}/\mathrm{D_{P}}=2.67\). From Figure 5a, there is a clear discrepancy between the measured data and the model. The in-balance bell shape presented by the measured data can be observed on two sides separated by the propeller axis of \(\mathrm{r}=0\); however, the theoretical model presents well-balanced calculated points on the positive and negative sides from \(\mathrm{r}=0\). In the experiment, the thrust spinning momentum was clearly magnified sideward (i.e., due to the installed angle of the propeller), where this effect has not been reproduced in the modelled result. However, in comparison, both experimental and calculated data exhibit a similar curve shape and characteristic, which verifies that the model can represent propeller-induced jets reasonably. Further, the laboratory test conducted by [PERSON] and [PERSON] [9] was also used to compare with the model. The physical test conditions were as follows: \(\mathrm{D_{P}}=7.6\) cm and \(\mathrm{X}/\mathrm{D_{P}}=1.00\), where \(\mathrm{n}\) was tested on multiple RPMs.