Hydrodynamic problem decomposition

Under the hypotheses of LPT, the wave field surrounding a floating body can be described as a superposition of an incident, a diffracted and a radiated wave field [26] [74]. The incident wave field,φi, is defined as the wave propagating in the absence of the floating body. The diffracted wave field, φd, is a consequence of the interaction between the incident wave and the floating body, when the body is kept fixed at its equilibrium position. For the generation of the radiation wave field,φra, the floating body is forced to oscillate due to an external force, in the absence of an incident wave field. Therefore, the total velocity potential,φ, can be obtained as:

φ(x,y,z,t) = φi(x,y,z,t) +φd(x,y,z,t) +φra(x,y,z,t) (2.54) As explained in Section 2.2.1, in absence of viscous stress components, the force and moment applied from the fluid to the body are given by the integration of the total pressure over the wetted body surface, as described in (2.11) and (2.12). In LPT, the surface S corresponds to the motionless wetted body surface in calm water. Introducing equations (2.29) and (2.54) into (2.11), it follows [26] that: ff l(t) =ρ Z S ∂φi(x,y,z,t) ∂t + ∂φd(x,y,z,t) ∂t + ∂φra(x,y,z,t) ∂t +gz  n · dS (2.55) =ff k(t) +fd(t) +fra(t) +fb(t) (2.56) where ff k(t) =ρ Z S ∂φi(x,y,z,t) ∂t n dS (2.57)

is the Froude-Krylov force,

fd(t) =ρ Z

S

∂φd(x,y,z,t)

∂t n dS (2.58)

is the diffraction force,

fra(t) =ρ Z

S

∂φra(x,y,z,t)

∂t n dS (2.59)

is the radiation force andfb is the linearised buoyancy force given by (2.34). It is important to underline that, in (2.57), (2.58) and (2.59), S is the wetted surface of the fixed body, considered below the undisturbed mean FSE, as shown in Fig. 2.4 [9]. Similarly, introducing equations (2.29) and (2.54) into (2.12), it follows [26] that:

τf l(t) =ρ Z S ∂φi(x,y,z,t) ∂t + ∂φd(x,y,z,t) ∂t + ∂φra(x,y,z,t) ∂t +gz  (r × n) dS (2.60) = τf k(t) + τd(t) +τra(t) +τb(t) (2.61) where τf k(t) =ρ Z S ∂φi(x,y,z,t) ∂t (r × n) dS (2.62)

is the Froude-Krylov moment,

τd(t) =ρ Z

S

∂φd(x,y,z,t)

∂t (r × n) dS (2.63)

is the diffraction moment,

τra(t) =ρ Z

S

∂φra(x,y,z,t)

∂t (r × n) dS (2.64)

is the radiation moment.

The summation of Froude-Krylov and diffraction forces is called the excitation force:

therefore, the total force applied from the fluid to the body is given by:

ff l(t) =fe(t) +fra(t) +fb(t). (2.66) Similarly, the summation of Froude-Krylov and diffraction moments is called the excitation mo- ment:

τe(t) = τf k(t) + τd(t) (2.67)

and the total moment applied from the fluid to the body is given by:

τf l(t) = τe(t) + τra(t) + τb(t) (2.68) In the following Sections 2.4.1.1 and 2.4.1.2, more details regarding the excitation, radiation and buoyancy forces are provided where, for simplicity, the general 6 DoF problem has been reduced to a heave single DoF.

2.4.1.1 Excitation force

The excitation force is the force acting on the body, when it is held fixed in the presence of waves [75] [69]. In the context of LPT, the excitation force is the superposition of the Froude-Krylov force (2.57) and of the diffraction force (2.58) [9], where the Froude-Krylov force is obtained by integrating the pressure, due to the undisturbed incident wave field, over the mean wetted surface of the fixed body, as explained in Section 2.4.1. Therefore, the Froude-Krylov force can be con- sidered as the force interaction between the waves and a ‘ghost’ body, which feels and reacts to the incident wave field, but does not alter it [9]. In the case of a body, having dimensions significantly smaller than the wavelength, the diffraction force is not significant compared to the Froude-Krylov force [9], indicating that the Froude-Krylov force is a reasonable approximation of the excitation force, with the computational advantage of avoiding the resolution of the diffraction problem. In the literature, the definition of the Froude-Krylov force is not unique, it can be calculated consid- ering only the dynamic pressure, due to the incident field [69] [76] or it can be calculated utilising the total pressure (static plus dynamic pressure) [77] [78] [79]. In the present work, the excitation force counts only the dynamic part of the pressure; in this way, the excitation force is zero if there are no incident waves (otherwise, the excitation force would be equal to the buoyancy force, in the absence of incident waves). This definition will be useful in Section 6.2, where the excitation force represents the output of the system under investigation. In the case of a single DoF body, moving in heave, the excitation force is given by [74]:

fe(t) =

Z _∞

−∞he(t − ζ)η(ζ)dζ (2.69)

where he(t) is the excitation impulse response function. It is important to note that, in (2.69), the upper limit of the convolution integral is +∞, indicating that it is necessary to know the future values ofη, in order to calculate fe at the present instant t. The fact that the relationship between η and fe is noncausal can be intuitively understood in the case whereη is defined with respect to a point placed in the centre of the body; indeed, the body will experience a force before the wave crest has arrived to the body centre [80] [81]. Moreover, the causality is not guaranteed even ifη is considered in a location on the upstream side and outside the body [82]. The noncausal relationship betweenη and fe becomes particularly important for real-time WEC control strategies, where wave forecasting is required [83] [84]. Taking the Fourier transform of (2.69), it follows that:

Fe(ω) = He(ω)η(ω) (2.70)

2.4.1.2 Radiation force

In the case of a body oscillating in the absence of incident waves, the hydrodynamic force, applied from the fluid to the body, is called the radiation force. It is important to underline that, in LTP, the calculation of the radiation force has an inconsistency, since the body is supposed to be moving to generate radiated waves but, at the same time, the wetted body surface is supposed to be unaltered in (2.59). In LPT, the body motion generates a time-changing fluid pressure, which is integrated on a constant surface, S, creating a time-changing radiation force. In the case of a single DoF body moving in heave, the radiation force is given by [85]:

fra(t) = −m∞¨z(t) −

Z t

−∞hra(t − ζ)˙z(ζ)dζ (2.71)

where z(t) is the position of the body, m∞ the high-frequency asymptote of the added-mass and hra(t) the reduced radiation impulse response function. The shape of the wetted body surface determines the hydrodynamic radiation force felt by the body, when it moves in the fluid. The convolution integral, in (2.71), describes the water memory effect and it is evaluated from minus infinity to the present time instant t, indicating that the relationship between the body velocity, ˙z, and the radiation force, fra, is causal. In the frequency domain, the relationship between the body’s velocity and the radiation force is given by the radiation impedance, Zra(ω) [69]:

Fra(ω) = −Zra(ω)iωZ(ω) (2.72)

where Fra(ω) and iωZ(ω) are the Fourier transform of fra(t) and ˙z(t) respectively and

Zra(ω) = N(ω) + iωma(ω) (2.73)

The real part of the radiation impedance, N(ω), is the radiation resistance, also called the hydrody- namic damping coefficient, which describes the dissipative effect of the energy, transmitted from the oscillating body to the waves (the waves propagate away from the body). The imaginary part of the radiation impedance is the radiation reactance, given by the product of the frequency,ω, and the added mass, ma(ω). The radiation reactance refers to the alternate exchange of kinetic energy, associated with the velocity of the water, and the gravitational potential energy related to the lifted water [9]. The added mass represents the additional inertial effect due to the acceleration of the water, which moves together with the body. At infinite frequency, the added mass tends to the finite constant, m∞, which is utilised to form the reduced radiation impedance, Hra:

Hra(ω) = Zra(ω) −iωm∞=N(ω) + iωma(ω) −m∞. (2.74) The inverse Fourier transform of Hra(ω) is the reduced radiation impulse response function, hra(t). Hra(ω) and hra(t) satisfy the following properties [86]:

lim ω→0Hra(ω) = 0 (2.75) lim ω→∞Hra(ω) = 0 (2.76) hra(t) = _π2 Z _∞ 0 N(ω)cos(ωt)dω (2.77) N(ω) ≥ 0 ∀ ω (2.78) lim t→0+hra(t) 6= 0 (2.79) lim t→∞hra(t) = 0 (2.80)

Equation (2.77) derives from the causality of hra(t) [69] [74]. Equation (2.78) is a consequence of the system passivity (passivity describes an intrinsic characteristic of systems that can store and

dissipate energy, but not create it). For linear time-invariant systems, a necessary and sufficient condition for passivity is that the real part of the system transfer function is positive for all fre- quencies [86] [87]. Equation (2.79) is a consequence of equations (2.77) and (2.78) [88]. In the frequency domain, equation (2.71) becomes:

Fra(ω) = m∞ω2Z(ω) −Hra(ω)iωZ(ω). (2.81)