**1. Introduction**

Computational fluid dynamics (CFD) has become widely accepted as a useful tool to predict the flow around a ship. This is facilitated by the increase in available computational power, which has allowed practitioners to re-create the flow around a vessel even on a standard computer. Thus, the number of cells or, more generally, the computational effort required to perform a numerical simulation in model-scale is not thought to be prohibitive for practical applications.

Regardless of the advances in every field of numerical modelling, CFD is not yet considered a replacement of model-scale experimentation. This is because it is not possible to guarantee that a particular numerical model will perform with the same level of accuracy across all possible case studies. For example, new energy-saving devices, or novel underwater shapes may require research into the best applicable modelling approaches. Additionally, the consequences of implementing modelling assumptions may not be fully understood. Specifically, although in model-scale computations, a significant portion of the ship hull is covered by a laminar layer, most turbulence models assume the flow is fully turbulent. Yet, results with accuracy of a few percentage error can be found in the open literature [1–7].

There are also different aspects of the problem of modelling ship flows that can be validated with different levels of confidence. For instance, the resistance of a ship can be measured accurately. However, velocities in the wake of the ship or free surface elevations require complex and expensive equipment. Thus, in the course of validating a numerical result, researchers typically analyse the error in observed integral quantities (resistance, motions, etc.) but tend to assume that other flow features are also accurately modelled as a consequence. Although this may be the true in many cases, an approach to validate aspects of the flow around a ship, such as the generated wave field, is necessary. Ideally, such a method would not rely on expensive equipment nor complex mathematics; in other words, it should be accessible. It is important to mention that some experimental campaigns report on a wide range of features of the flow around the ship, for instance, the flow properties in the wake [8,9].

The research presented herein is motivated primarily by the manner in which the problem of ship resistance is typically solved. That is, the principle of Galilean relativity is invoked (also called frame invariance; further information can be found in [10]). Namely, the water is flowing over a stationary ship (in the direction of the incoming flow). This assumption has several consequences. Those particularly important to the naval architect are:

#### (1) Levels of inlet turbulence.

This can have an impact on the overall properties of the flow [11] and may require calibration in some cases. For example, according to Lopes et al. [12], the onset of transition from laminar to turbulent boundary layer is strongly dependent on the level of free-stream turbulence. Some two-equation models, such as the SST *k–*ω model (which is widely used in marine hydrodynamics), are known to predict excessive decay of free-stream turbulence, which may affect the results. More recently, Lopes et al. [13] examined the same topic. According to them, even if one were to employ a more advanced eddy-viscosity model, capable of accounting for transition, the location of where laminar–turbulent transition occurs is highly dependent on the level of inlet turbulence.

(2) Wave reflections and their damping.

In cases where the volume of fluid (VOF) method is used [14], a damping length is often prescribed. That is, a length over which all waves are damped, extending from the boundary it is applied to in the normal direction. Setting an inappropriate damping length can have severe consequences to the predicted parameters [15].

#### (3) The temporal dependency of free surface flows.

The simulation of free surface flows via CFD cannot be solved using steady-state solvers (except in rare academic cases), because they require that properties are convected through the domain [14]. Theoretically, in the frame of reference of the ship, the flow—once converged—is steady. Therefore, ship resistance is frequently classed as a pseudo-steady problem. In reality, towing takes place over time and is a fundamentally unsteady process. Here, the presence of turbulence, which is by definition time-dependent [16], should also be kept in mind.

A second aspect inspiring this work partly stems from point (2) above. Although these may be of less interest to the naval architect, they carry their own importance nonetheless. Specifically, this concerns the destruction of ship waves, regardless of whether or not damping is prescribed. Once a ship-generated wave reaches the outlet, it is irreversibly destroyed, and the information it carries is lost. In shallow and restricted waters, ship waves are of great importance because they cause bank erosion and may even lead to destruction of coastal features/infrastructure [17,18]. In extreme cases, they may even be the cause of loss of life, as stated by Soomere [19]. Therefore, the accurate modelling of ship waves and their interactions with riverbeds or canal sides is important. Soomere [19] also advances a criticism of ship-induced flow predictions, pointing out that the flow is only described at a distance of few ship lengths.

Clearly, ship waves are both of practical and research interest. Therefore, the validation of numerical ship-generated waves is of high importance. In this respect, the work Caplier et al. [20], Fourdrinoy et al. [21], and Gomit et al. [22] is important to mention. The authors of the

aforementioned references systematically developed and implemented a technique to capture and analyse ship-generated waves from a model experiment. Of interest to the present researchh is the fact that in their studies, the authors proved the dispersion relation in deep and shallow water and demonstrated its validity for ships experimentally. Since the developed technique relies primarily on spectral representation of the wave field, it is thought prudent to attempt its application to a numerically generated free surface disturbance caused by a ship. It is expected that, if applied correctly, it is possible to validate a numerical wave field simply by means of processing a virtual free surface, which would be undoubtedly of practical use. Such a method has the potential to change how numerical solutions of surface piercing bodies are treated.

The present paper will attempt to apply the aforementioned spectral technique on a different type of numerical towing tank. Instead of relying on Galilean relativity, the present paper will present a numerical replica of a towing tank, where the ship advances over a stationary fluid. This is achieved via the overset domain method, where the ship is encased in what is essentially a moving box. To perform the numerical simulations, the commercial Reynolds-averaged Navier–Stokes (RANS) solver, Star-CCM+ version 13.06, is used. The specific case studies adopted in this paper are selected to maximise the practical relevance of the study. Specifically, the New Suez Canal is replicated, alongside a standard rectangular canal, which were investigated experimentally by Elsherbiny et al. [23]. The KRISO container ship (KCS) with a scale factor of 1:75, following the available experimental data, is used for all simulations.

The aim of this paper is primarily to demonstrate that it is possible to create a virtual towing tank where the ship is towed using the overset method, i.e., a virtual towing tank that does not rely on Galilean relativity. The generated wave field will then be used to estimate the Kelvin half-angle for an example case. The adopted approach also allows one to split the near- and far-field wave systems, which is used on the fully nonlinear disturbance, generated by the KCS at a variety of speeds in two different canals.

This work is organised as follows. Section 2 contains a description of the adopted case studies, while Section 3 explains the adopted methodology, which is split into the two techniques used in this paper, namely the computational set-up and the spectral representation techniques. Section 4 is dedicated to results and their discussion, whereas Section 5 contains conclusions and summary.

#### **2. Case Studies**

As mentioned in the previous section, the case studies adopted for this work are taken from the experimental work of by Elsherbiny et al. [23]. The rationale behind this choice relates to the particular objective of this study. To elaborate, shallow-water studies are a natural choice for the examination of ship-generated waves. This is because they present several features that are absent in deep-water ship-generated waves. Shallow-water waves are nonlinear, and their Kelvin half-angle is speed-dependent [24,25]. This is illustrated in Figure 1, which is constructed via Havelock's [17] linear method. Here, the Kelvin wake angle increases from its deep-water value of ≈19.47◦ to 90◦. The theory predicts that at a depth Froude number (*Fh* = *U*/ - *gh*, where *U* is the ship speed in m/s, *g* is the gravitational acceleration and *h* is the water depth) of one, *Fh* = 1, the ship-generated waves will travel at the same speed as the disturbance, indicating the Kelvin wedge fills the entire half-plane aft of the disturbance, i.e., at a half-angle of θ = 90◦. The relationships derived by Havelock [17] are omitted in the present work, as they are available in the open literature.

**Figure 1.** Kelvin half-angle of ship-generated waves in shallow waters as a function of the depth Froude number.

Ship-generated waves are also of greater concern in restricted areas than in deep waters, because they may affect the surrounding environment detrimentally. In navigational fairways, bank erosion is of particular concern, which has led to authorities restricting the speed with which vessels are legally allowed to operate [26]. Such a restriction simultaneously guards against groundings.

The case studies adopted herein are chosen to reflect the aforementioned points. In this respect, the recent work of Elsherbiny et al. [24] is used as a benchmark. From their experimentally investigated cases, two different canal cross-sections are selected: the New Suez Canal and a standard rectangular canal. These are graphically depicted in Figure 2.

**Figure 2.** Graphical depiction of the cross-section of the selected case studies. Top: New Suez Canal; bottom: rectangular canal.

The ship used in this study also follows from the experimental campaign of Elsherbiny et al. [23]. Namely, the KCS hull form is used, scaled by a factor of 1:75. This translated into a depth-to-draught ratio of 2.2, based on the ship's design draught. In order to ensure that a well-defined Kelvin wake is simulated, the chosen depth Froude numbers are towards the high end of the experimentally available

conditions. The ship's particulars are given in Table 1, whereas test matrix alongside the predicted Kelvin half-angles (via Havelock's [17] method) are described in Table 2. It should be noted that Havelock's [17] method was originally devised for point sources and is therefore not expected to be perfectly accurate for nonlinear three-dimensional surface piercing bodies. Nonetheless, it is a useful starting point. Additionally, turbulence, viscosity and vorticity may influence ship-generated waves, particularly in the near-field [27].


**Table 1.** Ship characteristics.



The relatively high depth Froude numbers ensure the numerically generated wave field will be discernible. The spectral method used also performs best at high speeds, where the near- and far-field disturbances generated by the ship are well visible. This can be seen by consulting the results of Caplier et al. [20].

#### **3. Methodology**

This section is presented in two major parts. These reflect the methodologies used in this work. The first section presents the numerical set-up, which is followed by an explanation of the spectral method in the second subsection.
