Free surface modelling and wave generation

In most of the simulations presented later on, the ship or the propeller operates in the vicinity of the free surface. Hence, the location of the free surface has to be determined during the simulation. Therefore, a volume-of-fluid (VoF) method is applied. In this group of methods, the computational domain contains multiple phases, i.e. the water phase as well as the air phase. The distribution of water (c = 0) and air (c = 1) in the computational domain is then determined by an additional transport equation for the mixture fractionc∈[0,1]:

Computational models for ship propulsion in waves

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Afterwards, the fluid properties are calculated according to the mixture fraction:

ρ=ρw·c+ρa·(1−c),

µ=µw ·c+µa·(1−c). (3.12)

Within this, the indexa denotes the air phase and the index wrefers to the water phase.

The interpolation of the face values from the cell centres is more complex than for other properties and requires a special interpolation scheme. The method used in FreSCo⁺, is the downwind biased interpolation scheme HRIC (High Resolution Interface Capturing), proposed by Muzaferija and Peric [64], which predicts the face values of the mixture fraction without significant amounts of numerical diffusion and in a bounded range. The interpolation is sensitive to the dimensionless time step, specified in eq. 3.5 . In the HRIC scheme the respective maximum values are limited toCo≤0.7, otherwise large amounts of numerical diffusion are introduced. Moreover, the HRIC scheme is supplemented by an additional interface-sharpening approach, which prohibits an unphysical mixture of water and air in a conservative way. The limitations on the time step are necessary for the VoF equation, but not for the other equations. To accelerate the solution process, a subcycling technique is applied, wherein smaller time steps are used for the VoF equation than for the others. Thus, within one time step of the momentum equation, the distribution ofcis updated several times. In combination with a variable time-step size, a remarkable speed up of the solution procedure can be achieved as illustrated by Manzke et.al.[57]. The major benefit of VoF-type interface capturing schemes is the larger flexibility compared to interface tracking systems in terms of large deformations of the free surface. As the interface between water and air is not a domain boundary, also ruptured deformations of the interface including breaking waves or splashes, can be simulated.

To account for the seaway, waves have to be generated at the inflow boundaries. Conse-quently, different wave theories (i.e. Airy, Stokes), external programs or a moving body simulating a wave maker, can be used. For the calculations in this thesis the Airy theory [27] is applied. The Airy theory is based on potential flows, thus no viscous effects are considered. Furthermore, the theory is linear and only valid for waves with a small wave height / wave length ratio. Linearity in this context means that the combined boundary condition (combination of kinematic and dynamic BC) at the free surface is linearised and fulfilled for the still water level. Thus, strong limitations on the wave steepness have to be accepted. On the other hand, a superposition of single wave components is possible, which makes the theory highly flexible. Accordingly, the Airy theory is also often used in potential flow methods to model a natural seaway by a superposition of single Airy waves.

Figure 3.3 illustrates the coordinate system that is used for the definition of the waves. In this example, the waves are travelling in the positive x-direction, while the z-coordinate points upwards. Furthermore, the definition of the wave length (λ) and the wave amplitude

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Computational models for ship propulsion in waves

Figure 3.3: Coordinate system used for the description of waves.

(ζ) are given in this figure. The resulting wave elevation is a harmonic function andˆ depends on the actual location (x= (x, y, z)^T) and the time:

z(t) = zSW + ˆζcos(ωt−kxcosµ−kysinµ). (3.13) Here,z_SW is the still-water height of the free surface andωdenotes to the wave frequency.

The angle µ is the encounter angle between the direction of the ship speed and the direction of the travelling waves, which is defined according to figure 3.4.

The Cartesian coordinates of the velocity vector follow from the equations for the orbital velocities:

vx =vsx +ωζeˆ ^{−k(|z−zSW|)} cos(ωt−kxcosµ−ky sinµ) cosµ , v_y =v_sy +ωζeˆ ^{−k(|z−zSW|)} cos(ωt−kxcosµ−ky sinµ) sinµ , v_z =−ωζeˆ ^{−k(|z−zSW|)} sin(ωt−kxcosµ−ky sinµ).

(3.14)

For the velocities in air the following relations can be used:

v_x =v_sx −ωζeˆ ^{−k(|z−zSW|)} cos(ωt−kx cosµ−ky sinµ) cosµ , v_y =v_sy −ωζeˆ ^{−k(|z−zSW|)} cos(ωt−kx cosµ−ky sinµ) sinµ , v_z =−ωζeˆ ^{−k(|z−zSW|)} sin(ωt−kx cosµ−ky sinµ).

(3.15)

In these equationsk= ^2π_λ denotes to the wave number. The quantitiesv_sx andv_sy repre-sent the Cartesian coordinates of an optional domain or ship velocity. Equations (3.13)-(3.15) are valid for deep water waves with ^λ₂ ≤water depth.

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Figure 3.4: Definition of the encounter angle between the wave direction and the ship’s course.

The equations are then used to determine the boundary variables for the transport variables in eq. 3.8 and 3.11. Alternatively to the Airy theory, also the Stokes theory can be used to compute the boundary variables. In the Stokes theory, the boundary condition at the free surface is fulfilled at the actual free surface position. This makes the theory non-linear, but valid for a larger wave steepness compared to the Airy theory. The available implementation inFreSCo⁺ refers to third order stokes waves. The required equations for the velocity components and the height of the free surface can be found in Cieslawski [20].

For offshore applications the Stokes theory of 5th order is used sometimes to compute impact loads of extreme waves (see Kleefsman [42] for details). Another possibility for generating the waves is to use an external program. Then, waves of arbitrary shape or a superposition of wave components can be generated at the inflow of the computational domain as demonstrated in Clauss [21]. Here, the author couples a potential-flow based finite-element method with a VoF-based RANS solver to compute wave loads. This coupling allows the use of very compact domains in the RANS solver and therefore, reduces the computational times.

Im Dokument Evaluation of the Unsteady Propeller Performance behind Ships in Waves (Seite 54-57)