**4. Results**

The directions of the drag force, trim angle, and sinkage are defined as shown in Figure 9.

**Figure 9.** The directions of the drag force, trim angle and sinkage.

#### *4.1. Verification and Validation*

To ensure the reliability of numerical simulation, verification and validation study is conducted for the model-scale ship and the full-scale ship, in deep-water condition.

#### 4.1.1. Model Scale (Fr = 0.26, Re = 1.477e+7)

In the previous work [32], the sensitivity of grid spacing and time step was studied for a KCS scaled model. The velocity of the scaled model is 2.197 m/s and the length between perpendiculars is 7.2786. Three grid cases and three time-step schemes were conducted for the KCS scaled model, as a preparation step; it is a key step to ensure the accuracy of the results. Table 5 shows the results obtained by the different grid cases and Table 6 shows the results obtained for different time step cases. Table 7 shows the results of numerical uncertainty of the KCS scaled model. Three different grids were generated with a grid refinement ratio <sup>√</sup> 2 and three time-step schemes were generated with a refinement ratio of 2.


**Table 5.** Total resistance coefficients of different grid cases of the 1:31.6 scaled model.

**Table 6.** The total resistance coefficients of different time steps of the 1:31.6 scaled model.




For both grid-spacing and time-step, the convergence factor R was between 0 and 1, which meant that monotonic convergence was achieved and the generalized Richardson extrapolation method could be used. From Table 7, it can be observed that the uncertainties of the grid and time-step for the total resistance coefficient of the 1:31.6 scaled model were 1.984% *SC* and 0.025% *SC*, respectively, where *SC* is the bench mark experimental data obtained from Equation (25) and the total numerical uncertainty is 1.984% *SC*.

#### 4.1.2. Full Scale (Fr = 0.26, Re = 2.84e+9)

Three grid cases and three time-step schemes were conducted for the full-scale KCS ship. Table 8 shows the results obtained for different grid cases and Table 9 shows the results obtained for different time step cases. Table 10 shows the results of numerical uncertainty. Three different grids were generated with a constant refinement ratio <sup>√</sup><sup>3</sup> 2 and three time-step schemes were generated with a constant refinement ratio <sup>√</sup> 2. The reference EFD value comes from the literature [29].

**Table 8.** The total resistance coefficients of different grids of the full-scale KCS.


**Table 9.** The total resistance coefficients of the different time steps of the full-scale KCS.



**Table 10.** The numerical uncertainties of the total resistance coefficients of the full-scale KCS.

For both grid-spacing and time-step, the convergence factor R was between 0 and 1, which meant that monotonic convergence was achieved and the generalized Richardson extrapolation method could be used. From Table 10, it can be observed that the uncertainties of grid and time-step for the total resistance coefficient of the 1:31.6 scaled model were 4.926%*SC* and 0.924%*SC*, respectively, where *SC* is the bench mark experimental data obtained from Equation (25) and the total numerical uncertainty is 5.012%*SC*.

#### 4.1.3. Grid Spacing Sensitivity

For ships passing through the shallow-water channels, the flow parameters under the keel can change significantly, so the grid spacing between the bottom of the tank and the keel may significantly affect the calculation results.

As a reference for subsequent calculations, the grid spacing of the model scaled at H/T = 4 is chosen to study. Tables 11 and 12 show the results of the convergence study.


**Table 11.** The grid spacing convergence analysis.

**Table 12.** The numerical uncertainties of the grid spacing under the keel.


Since the calculated resistance coefficients of the middle and fine grid spacing are very close, the subsequent calculations are carried out using the middle spacing to ensure accuracy and to reduce the number of grid cells for computation processing.

#### 4.1.4. Validation Based on EFD data

Assuming that the benchmark experimental value is D, the comparison error E can be defined as:

$$\mathbf{E} = \mathbf{D} - \mathbf{S}\_{\mathbb{C}} \tag{28}$$

in which, S*<sup>C</sup>* is the benchmark numerical result from Equation (25).

The validation uncertainty *UV* is given by:

$$\mathcal{U}\_V^2 = \mathcal{U}\_D^2 + \mathcal{U}\_{\mathcal{SN}}^2 \tag{29}$$

Where, *UD* = 1% is the uncertainty of the experimental data provided for the KCS towed resistance. The results of the validation study are given in Table 13.


**Table 13.** Validation study.

#### *4.2. Force and Attitude*

#### 4.2.1. Model Scale (Fr = 0.26, Re = 1.26e + 7)

The velocity of the scaled model is 2.197 m/s and the length between perpendiculars is 7.2786 m, which are same as presented with the previous cases of uncertainty analysis. The only difference is that the experimental kinematic viscosity leads to a slightly different Reynolds number from previous studies, however, previous studies can still be used to prove that the selected grid and time step are appropriate. As shown in Figure 10 and Table 14, the total resistance coefficient increases with a decrease of water depth; the increase of ship resistance in shallow water is shown intuitively. With the decrease of water depth, the total resistance coefficient increases gradually, but the total resistance coefficient of the three calculation cases with draft ratios H/T = 6, H/T = 8, and H/T = 10 does not increase significantly compared with the deep-water resistance measured in the model test. According to the International Towing Tank Conference (ITTC), the water depth H which can ignore the shallow water effect shall meet the following requirements:

$$H > 3\sqrt{BT} \tag{30}$$

$$H > 2.75 \frac{V^2}{\mathcal{S}} \tag{31}$$

where B is the maximum width of the ship; T is the draft; V is the ship speed; and g is the acceleration of gravity.

**Figure 10.** The total resistance coefficients of the KCS scaled model with different depth/draft ratios (H/T).



A further calculation shows that the calculation cases of H/T = 6, H/T = 8, and H/T = 10 are in this range, which indicates that the shallow-water effect is not so obvious.

Table 14 shows the detailed data of the resistance coefficients, sinkage, and trim in different calculation cases. The experimental fluid dynamics (EFD) value is obtained from the Tokyo 2015 workshop website.

As shown in Figure 11, the sinkage of the KCS increases monotonically with the decrease of H/T. There are many empirical formulas for predicting sinkage and the calculation results were compared with Raven's method in the following section. Figure 12 shows the changing trend of the trim angle in different water depths; the negative value means trim by the bow. As the H/T decreases, the trim angle gradually becomes larger.

**Figure 11.** The dimensionless sinkage of the KCS scaled model with different depth/draft ratios (H/T).

**Figure 12.** The trim angle with different depth/draft ratios of KCS scaled model (H/T).

4.2.2. Full Scale (Fr = 0.26, Re = 2.84e + 9)

As shown in Figure 13, the total resistance coefficient increases with the decrease of water depth; the increase of ship resistance in shallow water is shown intuitively. For a KCS, with the decrease of water depth, the total resistance coefficient increases gradually, but the total resistance coefficient of the two working conditions with draft ratio H/T = 6.7 and H/T = 10 does not increase significantly compared with the deep-water working condition (H/T = 21.3), for which the water depth is equal to the length between perpendiculars of the KCS.

**Figure 13.** Total resistance coefficients of the full-scale KCS with different depth/draft ratios (H/T).

4.2.3. Comparison with the Existing Experimental Results

Some calculations were conducted for a 1:75 KCS scaled model to compare with the experimental study of Khaled Elsherbiny [36]; a low-speed point (FnH = 0.32) and a high-speed point (FnH = 0.8) are chosen for comparison. The comparison results of the total resistance coefficient and sinkage are given in Table 15. The EFD value is obtained from the results in the literature by interpolation.

**Table 15.** The comparison with the experimental value.


#### *4.3. Wave Properties*

As we know, the dispersion relation for limited water depth is:

$$c = \sqrt{\frac{\xi^{\lambda}}{2\pi} \tanh\left(\frac{2\pi h}{\lambda}\right)}\tag{32}$$

or

$$\frac{c}{\sqrt{\wp^h}} = \sqrt{\frac{\lambda}{2\pi h} \tanh \frac{2\pi h}{\lambda}}\tag{33}$$

Evidently, for a larger λ/h, the factor introduces a dependence on the ratio of wavelength to water depth. When the water depth h further decreases and the ratio λ/h becomes large, as Equation (27) shows, the propagation speed of waves will reach a limiting value of *c* = - *gh*. This indicates that there is an upper limit to the wave propagation speed in shallow water.

The different propagation speeds of waves in different water depths led to the differences in wave properties among the calculation cases. To show the wave properties, the nondimensionalized wave height on the hull surface and the Y = 0.1509 Lpp section are extracted.

#### 4.3.1. Model Scale

As shown in Figure 14, the wave height curves for the cases of H/T = 6, H/T = 8, and H/T = 10 almost coincide, while for the H/T = 4 case, there is a clear departure from the other curves. It can also be seen that the wave height distribution characteristics on the Y = 0.1509 Lpp section are very similar to those on the hull surface, but tend to be flat on the whole. The further away from the ship, the more horizontal the free surface becomes, because the kinetic energy of the wave gradually changes into potential energy in the process of propagation.

**Figure 14.** The model-scale wave profiles at different water depths: (**a**) on the hull surface and (**b**) on the Y = 0.1509 Lpp section.

#### 4.3.2. Full Scale

It can be seen from Figure 15 that the wave height distribution characteristics of the full-scale ship are very similar to the wave properties of the scaled model.

**Figure 15.** The full-scale wave profiles at different water depths: (**a**) on the hull surface and (**b**) on the Y = 0.1509 Lpp section.

#### *4.4. Comparison with the Method of Raven*

To get a proportion of viscous resistance, the estimation method of the form factor is introduced in this study.

According to the literature, "the most popular empirical formula for determining the form factor is attributed to Watanabe" [37].

$$k = -0.095 + 25.6 \cdot \frac{\mathcal{C}\_{\rm B}}{\left(\sharp\right)^{2} \sqrt{\mathfrak{F}}} \tag{34}$$

The calculated value of KCS is close to 0.1.

#### 4.4.1. Correction Process

After studying the effect of shallow water on viscous resistance, Raven began to do further research on the shallow water effect. A complete set of shallow water resistance correction procedures was proposed in his paper [38]. The correction steps are following as:

1. The correction of the viscous resistance coefficient is

$$r\_{vfc} = 1 + 0.57 \left(\frac{r}{7}\right)^{1.79} \tag{35}$$

in which T represents the draft; H represents the water depth; and *rv f ac* represents the ratio of shallow-water viscous resistance coefficient to deep-water viscous resistance coefficient.


$$\frac{d\_{\text{single}}}{L} = \max\left\{ 1.46 \frac{\text{V}}{L^3} \left[ \frac{F\_{\text{nH}} \text{\textdegree\space }}{\sqrt{1 - F\_{\text{nH}} \text{\textdegree }}} - \frac{F\_{\text{nHD}} \text{\textdegree\space }}{\sqrt{1 - F\_{\text{nHD}} \text{\textdegree }}} \right] 0 \right\} \tag{36}$$

in which L is the ship length, <sup>∇</sup> = *LBTCB* is the displacement volume; and *FnHD* = <sup>√</sup> *<sup>V</sup>* 0.3*gL* is the Froude depth number.

4. Estimate the resistance increase due to additional sinkage

$$
\sigma\_{\rm sink} = (1 + \delta \mathcal{V})^{\frac{2}{3}} \tag{37}
$$

in which *rsink* is the factor that represents the effect of the sinkage increase caused by shallow water and <sup>δ</sup><sup>∇</sup> <sup>=</sup> *dsinkage*·*Awp* <sup>∇</sup> is the additional displacement volume due to additional sinkage.

5. The total resistance coefficient increase factor caused by shallow water is

$$r\_{tfac} = \left[r\_{visc} \cdot r\_{vfac} + (1 - r\_{visc}) \cdot r\_{wfac}\right] \cdot r\_{sink} \tag{38}$$

in which *rt f ac* is the correction factor of total resistance; *rvisc* is the relative contribution of viscous resistance in deep water at the same ship speed with a computational case; and *rw f ac* = 1, which restricts the use of this method to the case where *FnH* < 0.65.

	- *FnH* < 0.65, the Froude depth number should be less than 0.65. Otherwise, a significant increase of wave-making resistance will occur.
	- *<sup>T</sup> <sup>H</sup>* < 0.5, the ratio of the draft to the water depth T/H should be less than 0.5. For a higher value, the viscous resistance could not be accurately estimated using a simple formula.
	- δ∇ < 5%, the increase of displacement volume δ∇ should be no more than 5%, otherwise, a better method may be needed to estimate the effect of the draft difference.

#### 4.4.2. Model Scale

Table 16 gives the brief process of resistance correction using Raven's method and compares the results with CFD results; a maximum difference of 5.8771% appears when H/T = 4.


**Table 16.** The difference of total resistance coefficients between Raven's method and CFD results.

As shown in Figure 16, the calculation results show quite good agreement with Raven's method when the water depths are deeper, but as the Froude depth number becomes larger with the decreasing of water depth h, a larger difference between the estimation and calculation occurs. In the case of the shallowest water depth, there is a maximum difference, and we find the Froude number, in this case, is close to the critical value 0.65. In Raven's correction method, the increase of wave resistance was ignored when FnH < 0.65, however, a significant increase in wave resistance may occur when FnH > 0.65, which could be the reason why a small difference is observed.

**Figure 16.** The comparison of the model-scale total resistance coefficients, estimated from Raven's equations (**Black**) and computed by CFD code (**Red**).

As can be seen in Figure 17, a clear difference appears in the case of H/T = 4: we cannot make sure here whether it is caused by a deficiency of the CFD simulation or a deficiency of the empirical formula. However, when considering the resistance increase caused by an additional sinkage, Raven adopted the method of constant admiralty coefficient, which can be defined as:

$$\frac{\Delta\_1^{\frac{2}{3}\cdot u\_0^{-3}}}{P\_1} = \frac{\Delta\_2^{\frac{2}{3}\cdot u\_0^{-3}}}{P\_2} \tag{39}$$

in which, Δ<sup>1</sup> and Δ2, *P*<sup>1</sup> and *P*<sup>2</sup> are the displacement and effective power at different water depths. As we know, the effective power can be expressed as a product of resistance and ship speed, i.e., *P*1,2 = *R*1,2*u*0. As a result, the ratio of ship resistance at the same speed in different water depths can be derived as:

$$\frac{R\_1}{R\_2} = \left(\frac{\Delta\_1}{\Delta\_1}\right)^{\frac{2}{3}}\tag{40}$$

**Figure 17.** The comparison of model-scale dimensionless sinkage between the predicted value obtained by Raven's formula (**Black**) and the calculated value obtained by CFD simulations (**Red**).

Assumed *R*<sup>1</sup> is the resistance at a finite water depth and *R*<sup>2</sup> is the resistance in unrestricted water, Equation (31) can be derived.

Although the difference between the predicted sinkage and the value calculated by the CFD code is up to 26%, the value can as low as 0.5% when comparing the *rsink*, which means it has little influence on estimating the increase of total resistance.

#### 4.4.3. Full Scale

Table 17 gives the brief process of resistance correction using the Raven method and compares the results with the CFD results; a maximum difference of 20.942% appears when H/T = 3.3.


**Table 17.** The difference of total resistance coefficients between Raven's method and CFD results.

As shown in Table 11, the difference between the CFD results and the prediction results are within the acceptable range at the water depth to draft ratios H/T = 6.7 and H/T = 10, but an unacceptable difference occurs when H/T = 3.3.

Even though the comparison differences between the CFD results and the prediction results are larger than the model-scale comparison differences, a similar trend is observed: the largest difference occurs in the case of the minimum water depth to draft ratio. Actually, with the increase of the Froude depth number, the increase of wave resistance in shallow water may be significant, however, it was ignored in the process of resistance correction, which may cause an underestimate of the total resistance coefficient. In addition, there is no experimental data of full-scale trials, so the reference value is extrapolated using the value measured in model test, which may further increase the differences between the CFD and prediction results.

### **5. Discussion**

#### *5.1. Shallow-Water E*ff*ect on Ship Resistance*

#### 5.1.1. Viscous Resistance

According to fluid dynamics, when encountering a channel of limited water depth, the flow velocity between the tank bottom and the ship keel will increase significantly. As a result, the static pressure of particles will decrease, which results in the dynamic sinkage of ships, as well as the increase of frictional resistance. In addition, the increase of the relative velocity between the ship and the water results in an increase of the pressure gradient, which increases the viscous pressure resistance.

Figure 18 shows the model-scale dimensionless flow velocity in the x-axis direction between the ship keel and the tank bottom. The number marked on the figure represents the overspeed ratio γ, which is defined as:

$$
\gamma = \frac{U}{u\_0} \tag{41}
$$

where U represents the dimensionless flow velocity in the x-axis, and *u*ˆ0 is the nondimensionalized ship speed, i.e., *u*ˆ0 = 1. As a result, γ = U can be derived.

**Figure 18.** The overspeed ratios at different water depths for the 1:31.6 scaled model of the KCS (water depth decreases from top to bottom, corresponding to H = 4T, H = 6T, H = 8T, H = 10T respectively).

It is evident from the figure that both the peak value of the overspeed ratio and the gradient of flow velocity increases with a decrease of water depth, which is in line with the above analysis.

#### 5.1.2. Wave Resistance

The influence factor of wave resistance is the Froude depth number FnH. The critical value of the Froude depth number is FnH = 1, where FnH < 1 represents the interval of subcritical speed, and FnH > 1 corresponds to the interval of supercritical speed.

According to the wave theory, the wave resistance increases significantly with the increase of the Froude depth number in the subcritical interval. However, the wave resistance decreases abnormally with the increase of FnH after the ship speed exceeds the critical value.

Figure 19 shows the wave patterns of the full-scale simulations. It can be seen from the figure that a higher Froude depth number indicates a higher wave height contour density. It also means that a higher Froude depth number indicates a larger wave resistance.

**Figure 19.** The wave patterns of full-scale simulations for different Froude depth numbers: (**a**) FnH = 0.66, (**b**) FnH = 0.47 and (**c**) FnH = 0.38.

#### **6. Conclusions**

Taking the full-scale and model-scale KCS as study objects, numerical simulations were conducted to calculate the ship resistance at different water depth/draft ratios. The hydrodynamic force, sinkage, trim angle, and wave properties at different water depths are presented and discussed. The in-house URANS CFD solver, based on the finite difference method (FDM), is used for this study. Two right-handed Cartesian coordinate systems are established to predict the 2-DOF motion of the forward ship and the single-phase level-set method is used to capture the change of the free surface. Lots of previous applications of HUST-Ship show quite a good accuracy. All results of resistance, trim angle, sinkage, and wave patterns show differences among different water depths, which indicates that the HUST-Ship solver can well express the effect of shallow water.

Verification study in terms of grid and time step sensitivity was performed to make sure that the numerical method was reliable; the Richardson extrapolation method was used in the process. A validation study was then conducted to judge the availability of the numerical results. Furthermore, the sensitivity of the grid spacing between the keel and the bottom of the towing tank was studied to obtain a proper grid spacing of the computational domain.

The results of total resistance coefficients and dynamic sinkage obtained by the CFD simulations were compared with the predicted value obtained by Raven's method. The comparison indicates that the differences between the CFD results and Raven's estimation results are extremely small in larger water depths. When the water depth becomes shallower, the differences between CFD and Raven's estimation increased rapidly. As Raven claimed in his paper, the increase of wave resistance can be ignored when F*nH* < 0.65, however, some classical literatures support different critical values. As the results of this paper show, a lower critical value of FnH may be more appropriate, since the difference between the estimation and the CFD goes up to 20% when FnH is over 0.65 (0.66) in the full-scale simulation, and a maximum difference of 5.8771% occurs in the model-scale simulation when the FnH is close to 0.65 (0.61). Comparing the model-scale simulation results with the full-scale simulation results, a similar trend for the change of resistance and altitude with a decrease of water depths is evident.

Even though the conclusion can be roughly obtained, the drawback of this paper is that only the designed speed of the KCS is considered, and the important impact index FnH can only be controlled by changing the water depth. Future works will contain the simulation of different Froude numbers to get a more persuasive conclusion.

**Author Contributions:** D.F.: methodology, investigation, resources, writing–original draft, funding acquisition. B.Y.\*: software, data curation, visualization, validation, formal analysis. Z.Z.: writing–review and editing, Project administration, funding acquisition. X.W.: supervision, funding acquisition. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Acknowledgments:** The research work comes from the ITTC workshop in 2019. This research was sponsored by the Advanced Research Common Technology Project of CHINA CMC (41407010401, 41407020502). The essential support is greatly acknowledged.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **List of Symbols**

