Coordinate systems and motion models

The description of a mathematical problem can often be simplified by choosing a suitable coordinate system. In this work two global Cartesian and two local curvilinear coordi-nate systems are used for the modelling purposes. They are presented in the first part of this section, Chapter 2.1.1. The second part, Chapter 2.1.2, deals with different types of propeller motion and their mathematical description.

2.1.1 Reference coordinate systems

In the present work, the flow past a single body or multiple bodies with an individual rotational and/or translational speed is considered. For that purpose two different refer-ence frames are defined: a Cartesian referrefer-ence frame that is fixed in space and a Cartesian reference frame moving in unison with the considered body. Both Cartesian systems are defined by the base unit vectors e_i, i = 1,2,3. The reference frame fixed in space is described by the coordinatesX= (X, Y, Z), whereas the body-fixed Cartesian reference frame is identified by the coordinatesx = (x, y, z)and thez-axis is positive upwards (s.

Figure 2.1).

For different applications, for example for the treatment of the sheet cavitation model, to be introduced later, it is more convenient to use a local surface-fitted curvilinear coordi-nate system. Thus, two additional local body-fixed coordicoordi-nate systems are introduced.

The first one consists of orthogonal base unit vectors τ_i, i = 1,2,3 and local coordi-natesξ = (ξ, η, ζ). The second one is composed of non-orthogonal base unit vectorst_i, i = 1,2,3 and the local coordinates are described by the vector s = (s₁, s₂, s₃). In a

Chapter 2. Governing equations

e₃ Z Y e₂ X

e₁

z e₃

y e₂ x e₁

Figure 2.1: Space and body xed coordinate systems

local orthogonal coordinate system, all unit vectors are aligned perpendicular as shown in Figure 2.2a on an example grid. In a local non-orthogonal coordinate system, the tan-gential vectors are perpendicular to the normal vector but the tantan-gential vectors are not orthogonal amongst each other, as shown in Figure 2.2b.

τ₁ ξ ητ₂

ζ τ₃

(a) Local orthogonal coordinate system

s₁ t₁ s₂ t₂

s₃ t₃

(b) Local non-orthogonal coordinate sys-tem

Figure 2.2: Local surface-tted coordinate systems

For the construction of a local coordinate system, a quadrilateral with the edge points c_i, i = 1, . . . ,4and the centroidc₀ is considered (s. Figure 2.3). For both systems it is as-sumed that the origin of the system lies at the centroid of the quadrilateral. The orthogonal system is then defined by the following unit vectors:

τ₁ = (c₁−c₄) + (c₂−c₃)

|(c₁ −c₄ +c₂−c₃)|, τ₃ = (c₄−c₂)×(c₁−c₃)

|(c₄−c₂)×(c₁−c₃)|, τ₂ = τ₃×τ₁

|τ₃×τ₁|. (2.1) The local non-orthogonal coordinate system is defined in the way that the tangential vectorst₁andt₂pass through the midpoints of the quad edgesc₁¯c₂andc₄¯c₁, respectively.

The normal vector is the vector that is perpendicular to both tangential vectors:

t₁ = (c₁−c₄) + (c₂−c₃)

|(c₁−c₄+c₂−c₃)|, t₂ = (c₄−c₃) + (c₁−c₂)

|(c₄−c₃+c₁−c₂)|, t₃ = t₁×t₂

|t₁×t₂|. (2.2)

Chapter 2. Governing equations

c₀ c₁

c₂ c₃

c₄

Figure 2.3: Edge points and the midpoint of a quad

The transformations from a global to a local system and vice versa are described in detail in Appendix B.

2.1.2 Motion models of a propeller

There are three different types of motion of a fluid element: translation, rotation and deformation. In this work, no deformation of the fluid particles is considered, the body is assumed to be rigid. Thus, for each considered body its translational velocity and/or its angular velocity and the axis about which it rotates must be defined. The translation of fluid particles is described by the vector:

V_{inf low} = (U, V, W), (2.3)

whereU, V andW are constant velocity components in thex, y andz direction, respec-tively. For a propeller, the body-fixed reference system translates with an inflow velocity V_{inf low} =V_ship -V_wake, whereV_ship = (V_ship,0,0)is the ship speed and V_wake is the ship wake field. The wake describes the velocity field at the aft-ship and is the result of the interaction between several effects. There is a potential induced part, a viscous part and a part which is induced through the waves. Since the ship propeller operates in the wake of a ship, the actual inflow velocity seen by the propeller is lower than the ship speed. The percentage of the ship speed seen by the propeller in the axial direction is described by the wake fraction: w= 1−^VA,x/Vship, where V_A is the advance velocity. In ship theory there are two different definitions of the wake: the effective wake and the nominal wake.

The nominal wake is the wake field that can be observed without the existence of a pro-peller. In contrast, the effective wake is the wake field that develops due to the interaction between the propeller inflow velocity field and the propeller presence (s. Carlton, 2007, p.

68ff.).

For a non-uniform inflow, i.e.V_wake6≡0, the velocity vectorV_wakeis defined as follows:

V_wake =V_ship−V_A. (2.4)

In case of a uniform inflow to the body, the fluid particle’s motion does not depend on time in the body-fixed Cartesian reference frame, i.e. the flow is time-independent. Since the

Chapter 2. Governing equations

inflow velocity of the propeller is not uniform in general, an unsteady flow is considered in this work.

In addition to the translational velocity, the rotation of a propeller plays an important role.

The rotation velocity of a propeller is described by the vector:

V_rot(x) = ω×x, (2.5)

where x is the position vector and ω is the angular velocity. The overall undisturbed velocity relative to the propeller is then given by (s. Figure 2.4):

V_∞(x) =−V_{inf low}(x)−V_rot(x). (2.6) In addition, the flow of a propeller can be subject to a pitch, heave, surge, yaw, roll or

V^{inf low}

axis of rotation V_rot

Figure 2.4: Motion model of a marine propeller

sway motion. Heave, surge and sway motions are three aspects of translation, whereas the components of the rotational motion are pitch, yaw and roll. In order to model this motion, the propeller is allowed to move according to a prescribed periodical linear or rotational ship motion along the x, y andz-axis, respectively. The resulting translational and rotational velocities are then defined as functions of time:

V_{inf low}(t) = (U(t), V(t), W(t)), V_rot(x, t) =ω(t)×x. (2.7) For example, in heaving, a ship is accelerated upwards and downwards along the vertical axis, such that for the parallel velocity of the propeller it holds:

V_{inf low}(t) = (U,0, W(t)), (2.8)

where U is constant andW(t)describes the time-dependant periodical heave motion of the ship in thezdirection.

Chapter 2. Governing equations