**2. Methods**

The original axial momentum theory of Rankine is based on the axial motion of the water flowing through the propulsor (propeller) actuator disc. The geometry of the propulsor is irrelevant, and its rotational effects on the water flow are neglected. Therefore, this theory is suitable to describe and study the characteristics of water flow through an ideal propulsor (Figure 1) [19].

This theory is based on the following assumptions:


**Figure 1.** Stream tube of ideal propulsor based on the momentum theory. Source: Authors based on [19].

Three typical cross sections are introduced to study the acceleration of the flow between the upstream and the downstream caused by the pressure jump of the actuator disc:


The power absorbed by the propulsor is given by:

$$P\_{\rm D} = \frac{\dot{m}\_{\rm B}}{2} \cdot \left(v\_{\rm C}^{-2} - v\_{\rm A}^{-2}\right) \tag{1}$$

It is also equal to:

$$P\_{\rm D} = T\_{\rm D} \cdot \upsilon\_{\rm B} = T\_{\rm D} \cdot \frac{\dot{m}\_{\rm B}}{\rho \cdot A\_{\rm B}}.\tag{2}$$

The thrust generated by the propulsor is given by:

$$T\_{\rm D} = \dot{m}\_{\rm B} \cdot (v\_{\rm C} - v\_{\rm A})\_{\prime} \tag{3}$$

$$T\_{\rm D} = \frac{\dot{m}\_{\rm B}}{\rho} \cdot \left(\frac{\dot{m}\_{\rm C}}{A\_{\rm C}} - \frac{\dot{m}\_{\rm A}}{A\_{\rm A}}\right). \tag{4}$$

When examining fluid flow, the basic laws of physics can be applied, i.e., conservation law of mass, conservation law of momentum, and conservation law of energy. All these laws, as well as viscous phenomena in real (Newtonian) fluid, are reflected in the Navier–Stokes equations that describe both laminar and turbulent flow.

The Navier–Stokes partial differential equations are applicable for all unit actions on the fluid particle in three basic directions of space, i.e., weight, pressure, and friction and inertia. They can be simplified to the form:

$$X - \frac{1}{\rho} \cdot \frac{\delta p}{\delta x} + \upsilon \cdot \left(\frac{\delta^2 v\_x}{\delta x^2} + \frac{\delta^2 v\_x}{\delta y^2} + \frac{\delta^2 v\_x}{\delta z^2}\right) = a\_{x\prime}$$

$$Y - \frac{1}{\rho} \cdot \frac{\delta p}{\delta y} + \upsilon \cdot \left(\frac{\delta^2 v\_y}{\delta x^2} + \frac{\delta^2 v\_y}{\delta y^2} + \frac{\delta^2 v\_y}{\delta z^2}\right) = a\_{y\prime} \,\,\,\,\tag{5}$$

$$Z - \frac{1}{\rho} \cdot \frac{\delta p}{\delta z} + \upsilon \cdot \left(\frac{\delta^2 v\_z}{\delta x^2} + \frac{\delta^2 v\_z}{\delta y^2} + \frac{\delta^2 v\_z}{\delta z^2}\right) = a\_z \,\,\,\,\,\,$$

These equations can be interpreted as the specific form of Newton's second law for the flow of viscous incompressible fluid per unit mass, on the right with the product of acceleration and weight and on the left with the sum of mass and surface (pressure and viscous) forces [20,21].

The CFD method is the most commonly used in computer modeling of fluid flow. Several mesh-based methods have been developed in this area, where the geometry under investigation is replaced by a 2D or 3D mesh and the flow problem is solved using the Navier–Stokes equations (Figure 2). The basic principle of CFD is to create a computational domain that consists of a geometric model of the actual and discretized form (mesh), a definition of boundary conditions, a set of physical properties and calculation methods, and possibly external geometry boundaries of the flow area (external flow).

In the CFD simulation of sailing, the geometry under investigation consists of the outer surface of the hull, surrounded by a flow area, mostly of hexahedral shape. This is a typical case of external flow where the flow takes place in the surrounding environment and not within the computational geometry. The investigated physical phenomena take place in a multiphase environment, at the boundary of two phases (water–air), which considerably increases the computational complexity of these tasks.

The most serious limitation in CFD analysis is the number of mesh elements (and nodes). In each iteration of the calculation, the hydrodynamic state of the elements is evaluated individually, and their excessive number can enormously increase computational complexity and machine time. For this reason, it is necessary to keep the number of elements as low as possible, but not to the detriment of the accuracy of the calculation.

We call it a quality mesh when the elements have the same size, are geometrically regular, and their distribution is also regular in the discretized area. A suitable choice of element size ensures that the hydrodynamic properties of the flow are captured, but velocities are decisive for dimensions [20,21].

Many numerical methods have been developed to address particular physical problems. Their application depends both on the suitability of the method for solving the issue and on the history of development.

**Figure 2.** 3D geometry of a typical CFD domain (Layout E, deep water). Source: Authors.

By replacing the geometry of the examined area with a mesh of generated nodal points, the flow calculation domain is discretized, thus, allowing the flow equations to be converted into algebraic equations.

The finite volume method (FVM) is a method that, in a discretized form, retains very reliably the principles of conservation laws of balanced physical quantities in the control volume and is, therefore, the most widely used CFD simulation apparatus for solving the Navier–Stokes equations [22].

Numerical modeling of turbulent flows is still in the process of research and development, supported by the latest knowledge of mathematics, physics, and technical computational methods. However, there is no universal model of turbulence that is generally and effectively applicable in all cases. In order to choose the most suitable model for a particular calculation case, it is necessary to consider the possibilities and limitations of individual numerical models.

The time-averaging method (RANS—Reynolds-averaged Navier–Stokes equations), which has relatively low computational capacity requirements and provides acceptable accuracy, is becoming increasingly widespread in engineering simulations. It consists of parametric modeling of turbulent flow by time-averaged values of physical quantities using the Reynolds method. Several different RANS methods have been developed for various specific task types, which simplify the modeling of swirls using added transport equations [23].

For simulation purposes, a typical Danube cargo ship hull has been chosen with a pushed barge type bow (to be able to push barges) and with a simplified aftship suitable for all the examined layouts (Figure 3). The main particulars of the ship hull are shown in Table 1.



The dimensions of the hexahedral calculation domain are shown in Table 2.


**Table 2.** Dimensions of the hexahedral calculation domain.

The layout cases of the CFD domains were designed as follows:


The CFD analyses have been performed with identical set-up parameters (Table 3). Only the boundary conditions and the initializing values vary from case to case.


**Table 3.** CFD set-up parameters.

The following hull/propulsion layout cases have been analyzed for discrete flow speed values taken from the vessels working in the range 0–6 m/s:


The actuator disc was simulated using a fan boundary condition in the ANSYS Fluent CFD package. A boundary condition was applied on a circular surface whose diameter was set so that the total area of the propellers was constantly 2.4 m2. Diameters and locations of actuator disk centers in individual layouts are shown in Table 4.



Special attention has been paid to cases C, D, and E because they can be considered as distributed ship propulsion systems, which were the main focus of interest.

For comparison reasons, all the studied cases have been examined under restricted draft conditions in shallow water, keeping a minimum of 0.3 m UKC, and also, under unrestricted draft conditions in deep water.

**Figure 3.** Different hull/propulsion layouts examined by CFD analyses. Source: Authors.

A total of 50 CFD flow analyses have been performed for unrestricted and restricted draft conditions. The mass flow rate values recorded for the actuator disc and the upstream/downstream monitors at every propulsion position have been used in Equation (4) of the Rankine momentum theory to obtain the increase in thrust generated by the propulsor. The power absorbed by the propulsors has been calculated by means of Equation (2) from the already known thrust value.

A visual check of the water flow field around the ship hull and the propulsors, the free surface of the water, and the wave pattern is possible in the graphical output generated by the CFD system (Figure 4). Some special variations in the values of the reported hydrodynamic quantities can be easier understood in this simple way (Figure 5).

**Figure 4.** Velocity magnitude on the free water surface (Layout E, deep water). Source: Authors.

**Figure 5.** Dynamic pressure at the propulsors' center plane (Layout E, deep water). Source: Authors.
