**Water depth is a crucial factor that affects a ship’s resistance, and thus influences fuel consumption and even determines the maximum navigating speed. Therefore, for ships designed to operate in shallow water, it is essential to consider its effects on ship resistance. A new method now makes this a lot easier.**

*For his PhD research, Qingsong Zeng made a case study for an inland barge to better predict ship resistance in shallow water. Below is a summary of his work.*

Shallow water effects on ship resistance were recorded about 200 years ago. In the summer of 1834, John Scott Russell observed that the resistance of a vessel in shallow water attained a local maximum at a certain velocity, as shown in figure 1, and immediately after this point, it dropped to a local minimum after which the increment of the resistance was recovered. He published this observation on the transactions of the Royal Society of Edinburgh (Russell, 1837). This finding is now wellknown, and researchers have continued the study of ship resistance in shallow water to this day.

*Figure 1. Instead of being a parabola, the curve of resistance against ship velocity in shallow water shows a different trend (Russell, 1837). *

A systematic and well-known study was proposed by O. Schlichting (1934). Based on a large number of model tests, several graphs were provided to show how a ship’s resistance in shallow water deviates from that in deep water. Lackenby (1963) improved Schlichting’s method by modifying the method of speed correction and enlarging the range of application. However, Schlichting admitted that his method is lacking physical bases and not valid in extremely shallow water.

Jiang (2001) proposed a mean effective speed based on the effective hydraulic blockage including the concept of the mean sinkage. In his case studies, this method made the total resistance to be a unit function of the effective speed and independent of the water depth. These methods are based on a correction of the resistance in deep water and reliant on the accuracy of the deep water prediction. However, deep water should be seen as an exception from shallow water and not vice versa.

Better resistance prediction can benefit all hydrodynamics-related research

A correct understanding of the resistance of ships in shallow water from the very basis is necessary to build a more robust approach to improve resistance prediction. It can benefit all further hydrodynamics-related ship researches, for example, a reliable performance prediction, truly valid rules for ship design, and even future work on understanding ship propulsion in (extremely) shallow water. This approach also allows the further application of the well-accepted extrapolation method while taking into account the inherent deviations caused by shallow water.

*Figure 2. A 3D geometry of the inland barge.*

## Ship information

An inland barge with a transom is applied in this study, as shown in figure 2. The sketch of the under-water sections of the ship is shown in figure 3. Since the length of the ship is 86 metres and it sails in the Rhine, it is named as “Rhine Ship 86” in this study. The stern shape is simplified into a transom for research purposes. The main dimensions of the ships are listed in table 1.

*Figure 3. The sketch of the under-water sections of the inland barge.*

*Table 1. The main dimensions of the Rhine Ship 86 (Lpp: length between the perpendiculars; B: beam; CB: block coefficient; S: area of the wetted surface).*

## Method

In this study, all Computational Fluid Dynamics (CFD) calculations were performed in an FVM (Finite Volume Method) code ANSYS Fluent (version 18.1). The SST k-ω model is chosen as the turbulence model. The scheme of the pressure-velocity coupling is “coupled” and the discretisation of the gradient is “least squares cell-based”. For the discretisation of pressure, PREssure STaggering Option (Presto!) is used, and “second order upwind” is applied for the discretisation of momentum, turbulent kinetic energy, and specific dissipation rate.

Ship resistance can be decomposed into different components based on the corresponding physical meanings. A popular decomposition according to (ITTC, 2017) is where three components are addressed: frictional resistance, viscous pressure resistance, and wave-making resistance.

The viscous pressure resistance is considered by a so-called form factor (Hughes, 1954). This factor was initially considered as constant and can be determined by the method of Prohaska (1966). Based on this decomposition, the frictional resistance can be calculated by a ship-model correlation line; the viscous pressure resistance can be achieved through low-speed model tests; the wave-making resistance can be obtained by ship model tests. By this means, ship resistance at full scale can be predicted.

However, it is now well-known that the form factor varies with the ship scale, even in deep water (García-Gómez, 2000). In shallow water, as will be shown in this research, the shear stress, pressure distribution along the ship hull, as well as the characteristics of ship-generated waves are influenced by the limited water depth. No strong relationship can be found among shallow water effects on these physical phenomena, from which the three resistance components are generated. Therefore, corrections on each component for shallow water should be made individually. In the following three sections, shallow water effects on each resistance component will be discussed in sequence.

## Frictional resistance

As a relatively low speed is applied for most ships sailing in shallow water, the frictional resistance takes the majority of total resistance. Therefore, accurately predicting the frictional resistance is of the highest importance for those ships. After CFD simulations, the results of Cf in various water depths for the Rhine Ship 86 illustrated in figure 4.

It can be derived from figure 4 that for h/T >= 1.2, the inland barge has a higher value of Cf when the water is shallower. Simultaneously, Cf is inversely proportional to both h/T and lg(Re). Based on this, a formula can be fitted through Matlab, which is shown as follows:

Physically, frictional resistance is an integral of all the local shear stresses on the hull surface projected into the longitudinal direction. In shallow water, the presence of water bottom forces a thinner ship boundary layer resulting in higher frictional resistance.

*Figure 4. The frictional resistance coefficient (Cf) against lg(Re) in different water depth for the Rhine Ship 86.*

Furthermore, the water around the ship changes its local orientation and, thus, affects the projected shear stress.

## Viscous pressure resistance

Through double-body computations, the viscous pressure resistance can be obtained approximately, but separately by integrating the pressure on the ship surface in the longitudinal direction. Due to the influence of viscosity, the water loses kinetic energy when it passes from the bow to the stern and a pressure difference is formed. In this section, a new defi nition of the form factor is proposed searching for a stronger physical basis.

If one keeps using a flat plate friction line in the definition of form factor, shallow water effects on both the friction and the viscous pressure resistance are included simultaneously in the form factor, which weakens the physical basis of this factor, that is, a part of shear force is transferred into the form factor. To remedy this, the computed friction (Cf_c) of the ship is recommended to define the form factor in shallow water, by which all shear forces can be kept in the friction. To distinguish it from the conventional way, an asterisk is used:

By this definition, the factor k* clearly represents the viscous pressure resistance. Meanwhile, the 1+k* is not expected to be constant with ship scales, since its Reynolds number dependency will be observed to be even more pronounced in shallow water.

In principle, it is required to keep the influence of transom outside the form factor (ITTC, 2017), but it is hard to put into practice. For a large number of inland ships, the immersed transom is commonly found, as well as the backward-facing tunnel endings. Their effects are already included in the form factor derived from, for example double-body computations. There are, at least to the author’s knowledge, no reliable methods to separate it precisely from form factor.

The results of 1+k* against the lg(Re) for the Rhine Ship 86 in different water depths are displayed in figure 5.

*Figure 5. The form factor against lg(Re) for the Rhine Ship 86 with different water depths.*

From figure 5, it can be seen that:

- When the value of lg(Re) increases, 1+k* first decreases, but encounters an increase starting at lg(Re) 6.6, which might be caused by the vortices behind the stern, as depicted in figure 6.
- Shallower water will lead to a higher 1+k*. For h/T = 1.20, the 1+k* increases by 19.7 per cent at lg(Re) = 5.8 and 8.2 per cent at lg(Re) = 9.2 compared to the corresponding deep water cases.

A vertical vortex and a horizontal vortex, as shown in figure 6, are generated and their cores are interconnected, which provides a low-pressure region behind the stern. For a relatively high Reynolds number, the pressure in the vortex core is even lower, both for deep water and shallow water. This can lead to a larger pressure difference between the bow and the stern, which determines a larger 1+k* at a certain higher Reynolds number.

*Figure 6. The vortices generated after the stern for lgRe = 6.4 and lgRe = 9.2 in deep water (top) and shallow water (bottom) for the Rhine Ship 86. *

The ship’s form, the water depth, and the transom can all be counted as form effects, therefore, the “form factor” is still nominally appropriate to describe the viscous pressure resistance for a ship with a transom. An adjusted formula considering shallow water effects and the transom is suggested for Rhine Ship 86 for h/T 1.2:

The first part of is the form factor of a slender ship (the Wigley hull, see the full PhD thesis for more details), which shows the basic effects of the boundary layer caused by a mildly curved ship. The remaining part of represents the effects of ship form, such as an altered boundary layer, ship-generated vortices, and so on. It should be noticed that the remaining part of can be improved by including more parameters of the ship, such as the geometric parameters of the transom, but a study of various hull forms are subject to further investigations.

Physically in shallow water, the hull together with the fairway bottom form a restricted space for water to pass through. The presence of water bottom provides an additional boundary layer and will interact with the ship’s boundary layer when the water is shallow enough. According to the Bernoulli equation and the continuity equation, a decreased cross-section around a ship will cause an increase of fl ow speed and a decrease of pressure. The pressure distribution is therefore influenced, resulting in a different viscous pressure resistance.

## Wave-making resistance

In this section, the scale effects, expressed by the extended range of the lg(Re) (= 6.0 ~ 9.2), are studied for the Rhine Ship 86, and the Reynolds dependency of wave-making resistance is also discussed. The wave-making resistance coefficient (Cw) is obtained by comparing the results of two types of computations: with free surface and without free surface (double-body computation). It provides insight into the scale effects on wave-making resistance qualitatively. The difference of the frictional resistance coefficients (^Cf) between these two types of computation was found to be one order less than Cw. In the simulations, the same value of y+ is guaranteed for the same speed in the calculations with and without free surface. Results of Cw of all cases are shown in figure 7.

Based on figure 7:

- When Frh <= 0.6261, the wave-making resistance coefficient (Cw) can still be seen as independent of Reynolds number for the applied ship;
- When Frh >= 0.6725, ship form plays a role in the scale dependency of Cw: the values of Cw at model scale show larger values than full scale;
- Scale effects on the wave-making resistance coefficient are decreasing with an increasing Reynolds number. This is in line with common sense that the effect of viscosity is smaller when the Reynolds number is larger.

*Figure 7. The wave-making resistance (Cw) against base10 logarithm of Reynolds number (lg(Re)) with different depth Froud number (Frh) the Rhine Ship 86. *

Therefore, the assumption that the wave-making resistance coefficient is independent of ship scales is challenged and needs to be revised. Based on the results shown in fi gure 7, a modification for the Cw can be given as follows. A factor β is defined to represent the difference of Cw between full and model scale.

where the subscripts s and m denote full scale and model scale, respectively. For the Rhine Ship 86 at Frh > 0.6261, and 6.0 lg(Re) 7.4:

For other cases: β = 1.0

Physically, when the water is sufficiently shallow, water particles can no longer move freely, and the oscillating movements adjacent to the bottom will be affected by the bottom friction. An oscillatory boundary layer is formed above the bottom, and consequently, a part of wave energy is dissipated in the boundary layer (H. Schlichting, 1979) and wave characteristics, such as wave height and wave length, will be influenced accordingly (Putnam & Johson, 1949).

## Conclusions

This study describes an approach to improve the extrapolation of ship resistance that is to be applied to model scale resistance results carried out in shallow water. The effects of limited water depths on the three components of ship resistance (frictional resistance, viscous pressure resistance, and wave-making resistance) have been studied individually. Empirical formulas have been developed for an inland barge in various water depths.

For all three components, both the water depth and the Reynolds number are important factors that determine a ship’s resistance in shallow water. The conventional extrapolation of ship resistance from model to full scale, where shallow water effects on ship’s friction and scale effects on wave-making resistance are not discussed, needs to be reconsidered.

A method to correct the conventional way is suggested for the specific inland barge. This can hopefully be of use to a researcher working on ship hydrodynamics with limited water depth for a ship type different from the Rhine Ship 86.

*This article appeared as part of SWZ|Maritime’s April 2020 inland navigation special. It was written by Dr Qingsong Zeng. Zeng obtained his doctor’s degree at Delft University of Technology in 2019 and is specialised in CFD and ship hydrodynamics in shallow water.*

## References

This article is a simplified version of the author’s PhD thesis: “A method to improve the prediction of ship resistance in shallow water” (DOI).

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