# A Lagally formulation of the wave drift force

Alain Ledoux, Bernard Molin, Gérard Delhommeau, Fabien Remy

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Alain Ledoux, Bernard Molin, Gérard Delhommeau, Fabien Remy. A Lagally formulation of the wave drift force. 21st International Workshop Water Waves and Floating Bodies, Apr 2006, Loughborough, United Kingdom. pp.1-4. [hal-00456766](https://hal.science/hal-00456766v1)

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# A Lagally formulation of the wave drift force

A. Ledoux*, B. Molin**, G. Delhommeau***, F. Remy**

* Principia RD, 215 Voie Ariane, Zac Athelia I, 13705 La Ciotat

** École Généraliste d'Ingénieurs de Marseille, 13451 Marseille cedex 20

*** École Centrale de Nantes, 1 rue de la Noé, 44300 Nantes

(corresponding author: bernard.molin@egim-mrs.fr)

*There is no theorem like the Lagally theorem (Georg Weinblum<fnref n="1" />)*

## 1 Introduction

There are two well-known formulations of the wave drift force: the "far-field method", based on momentum considerations, introduced by Maruo (1960) and later extended by Newman (1967), and the "near-field method", based on direct pressure integration and first proposed by Pinkster & Van Oortmerssen (1977). The far-field method is considered as more accurate but it delivers the horizontal components only and it cannot provide individual drift forces in a multiple body configuration. The near-field method does not suffer from these restrictions but its numerical accuracy is poor in some cases.

In the software Diodore the drift force is computed following another way, based on the Lagally theorem. This method was first proposed by Guével & Grekas (1981). In their paper the (lengthy) derivations deal with an unbounded fluid domain and the drift force expressions with a free surface are given *ex abrupto*, without justifications. Moreover they are ambiguous. Subsequently, in his 1986 monograph on wave energy recovery, Guével gives the same expressions (10) (11) as we obtain here (and as have been coded in Diodore for many years), but again without proof.

In this paper we re-establish rigorously the Lagally formulation of the drift force. As compared with the classical formulations it appears to offer several advantages.

## 2 Theoretical developments

We consider a floating body submitted to incoming regular waves with frequency $\omega$. The linearized velocity potential is $\Phi(x, y, z, t) = \Phi_I + \Phi_P$ where $\Phi_P$ combines diffracted and radiated components.

Be $S$ the wetted surface of the body, $\Sigma$ a surrounding fictitious control surface and $F$ the free surface in-between. We denote by $\Gamma_S$ and $\Gamma_\Sigma$ the respective intersections of $S$ and $\Sigma$ with the free surface. The normal vector $\vec{n}$ is taken in the outward direction from the enclosed fluid domain.

The following identity holds:

$$ \iint_{S \cup \Sigma \cup F} \left( \frac{1}{2} \nabla \Phi \cdot \nabla \Phi \vec{n} - \nabla \Phi \nabla \Phi \cdot \vec{n} \right) dS \equiv \vec{0}. \quad (1) $$

Restricting ourselves to the horizontal components, the integral over the free surface $F$ can be transformed as follows:

$$ \iint_F \left( \begin{pmatrix} \Phi_x \\ \Phi_y \end{pmatrix} \right) \Phi_z dS = - \frac{\omega^2}{g} \iint_F \left( \begin{pmatrix} \Phi_x \\ \Phi_y \end{pmatrix} \right) \Phi dS = - \frac{\omega^2}{2g} \int_{\Gamma_S \cup \Gamma_\Sigma} \Phi^2 \vec{n}_0 d\Gamma \quad (2) $$

with $\vec{n}_0$ the unit normal vector to $\Gamma_S$ and $\Gamma_\Sigma$ within the free surface plane (thus different from $\vec{n}$ if $S$ and/or $\Sigma$ are not vertical at the free surface).

So we can write

$$ \begin{aligned} & -\frac{\omega^2}{2g} \int_{\Gamma_S} \Phi^2 \vec{n}_0 d\Gamma + \iint_{\Sigma} \left( \frac{1}{2} \nabla\Phi \cdot \nabla\Phi \vec{n} - \nabla\Phi \nabla\Phi \cdot \vec{n} \right) dS = \\ & -\frac{\omega^2}{2g} \int_{\Gamma_S} \Phi^2 \vec{n}_0 d\Gamma + \iint_S \left( \frac{1}{2} \nabla\Phi \cdot \nabla\Phi \vec{n} - \nabla\Phi \nabla\Phi \cdot \vec{n} \right) dS \end{aligned} \quad (3) $$

<footnote n="1">¹quoted from Landweber (1967)</footnote>
where the normal vector $\vec{n}$ over S is now into the fluid.

Taken as a time-averaged value, either side of this equation is nothing but the horizontal drift force as obtained from momentum consideration. Taking $\Sigma$ to infinity yields the Maruo expressions of the drift force. Equation (3) means that the control surface can also be taken coinciding with the body itself (Molin, 1979).

We now assume that the velocity potential $\Phi_P$ is generated by a source density $\sigma$ over the wetted surface S. This induces a flow within the body as well, with velocity potential $\Phi_1 = \Phi_I + \Phi_{P1}$. For clarity we index with the subscript 2 the velocity potential outside the body: $\Phi_2 = \Phi$.

Identity (1) also holds inside the body:

$$ \iint_{S \cup F_1} \left( \frac{1}{2} \nabla \Phi_1 \cdot \nabla \Phi_1 \vec{n} - \nabla \Phi_1 \nabla \Phi_1 \cdot \vec{n} \right) dS \equiv \vec{0} \quad (4) $$

with $F_1$ the internal free surface. Restricting ourselves again to the horizontal components, we deduce

$$ \iint_S \left( \frac{1}{2} \nabla \Phi_1 \cdot \nabla \Phi_1 \vec{n} - \nabla \Phi_1 \nabla \Phi_1 \cdot \vec{n} \right) dS - \frac{\omega^2}{2g} \int_{\Gamma} \Phi_1^2 \vec{n}_0 d\Gamma \equiv \vec{0}. \quad (5) $$

Subtracting (5) from the right-hand side of equation (3), the drift force is obtained as

$$ \begin{aligned} \vec{F}_d = & -\rho \frac{\omega^2}{2g} \int_{\Gamma_S} (\Phi_2^2 - \Phi_1^2) \vec{n}_0 d\Gamma \\ & +\rho \iint_S \left[ \frac{1}{2} (\nabla\Phi_2 \cdot \nabla\Phi_2 - \nabla\Phi_1 \cdot \nabla\Phi_1) \vec{n} - \nabla\Phi_2 \nabla\Phi_2 \cdot \vec{n} + \nabla\Phi_1 \nabla\Phi_1 \cdot \vec{n} \right] dS \end{aligned} \quad (6) $$

where time-averaged value is implied.

With a source singularity distribution, we have $\Phi_1 = \Phi_2$ over S and $\nabla\Phi_2 - \nabla\Phi_1 = \sigma \vec{n}$.

As a result the integral over the floatation contour $\Gamma_S$ in (6) is zero identically. Replacing $\nabla\Phi_1$ with $\nabla\Phi_2 - \sigma \vec{n}$ in the integral over S, we obtain the simple expression:

$$ \vec{F}_d = -\rho \iint_S \sigma \left( \nabla \Phi_2 - \frac{1}{2} \sigma \vec{n} \right) dS. \quad (7) $$

The velocity potential $\Phi_2$ being expressed as

$$ \Phi_2 = \Phi_I + \frac{1}{4\pi} \iint_S \sigma(Q,t) G(P,Q) dS_Q, \quad (8) $$

one has

$$ \nabla\Phi_2 - \frac{1}{2}\sigma \vec{n} = \nabla\Phi_I + \frac{1}{4\pi} \iint_S \sigma(Q,t) \nabla_P G(P,Q) dS_Q. \quad (9) $$

Further it appears that the Rankine term $1/PQ$ in the Green function $G(P,Q)$ does not contribute. This would take the form

$$ \begin{aligned} \vec{F}_{dR} &= -\frac{\rho}{4\pi} \iint_S \sigma(P) dS_P \iint_S \sigma(Q) \nabla_P \left( \frac{1}{PQ} \right) dS_Q \\ &= -\frac{\rho}{4\pi} \iint_S dS_P \iint_S dS_Q \sigma(P) \sigma(Q) \frac{\overline{PQ}}{PQ^3} \\ &= \frac{\rho}{4\pi} \iint_S dS_P \iint_S dS_Q \sigma(P) \sigma(Q) \frac{\overline{QP}}{PQ^3} \\ &= -\vec{F}_{dR} \end{aligned} $$

So $\vec{F}_{dR} \equiv \vec{0}$ and, finally:

$$ \vec{F}_d = -\rho \left\langle \iint_S \sigma \vec{V} dS \right\rangle \quad (10) $$

with

$$ \vec{V} = \nabla\Phi_I + \frac{1}{4\pi} \iint_S \sigma(Q,t) \nabla_P H(P,Q) dS_Q \quad (11) $$
and $G = 1/PQ + H(P, Q)$. In an unbounded fluid domain with imposed incident kinematics, $H \equiv 0$ and the usual Lagally expression is recovered.

It is straight-forward to establish that the associated expression for the drift moment in yaw is

$$M_{dz} = -\rho < \iint_S \sigma \overrightarrow{OP} \wedge \overrightarrow{V} dS > \cdot \overrightarrow{k} \quad (12)$$

## 3 Comments

Alike Maruo's and Newman's formulations, in the case of a surface piercing body, these expressions are restricted to the horizontal components. Extensions to the vertical components would involve a line integral over $\Gamma_S$ (see Molin & Hairault, 1983) and an integral over the interior free surface. In the case of a fully submerged body there are no corrections and the vertical components are delivered as well (which is not true with the far-field method).

Another advantage over the classical far-field method is that individual drift forces are obtained in multiple body situations. Equation (7) directly applies to this case. The Rankine part can also be removed, but only the Rankine contribution of the considered body. All velocity components due to the presences of the other bodies must be added to the incoming wave velocity when computing (10). Numerical calculations have shown that there is no decisive accuracy gain in using equations (10) (11) in place of equation (7), which is much simpler to implement.

Advantages over the pressure integration method is that the obtained expressions are less prone to numerical inaccuracies, especially in the case of a body with sharp corners.

## 4 Results

First we consider two spheres half-immersed with a separation distance, from center to center, equal to 3 times their radius $a$. The waterdepth is infinite. The spheres are free to respond to the waves, which propagate in a direction perpendicular to the arrangement. Figure 1 shows the obtained drift force in $x$ (wave direction) and $y$, for the sphere located in $x = 0, y = 1.5a$. The pressure integration method and the Lagally formulation are used, with the Lagally results being obtained from equation (7) (without removing the Rankine part), implemented in the ECN code Aquadyn. For the $x$-component, the far-field result is also given (by symmetry it is half the force on the total arrangement). It can be seen that, except in the vicinity of the first irregular frequency, the agreement between the different formulations is excellent. The transverse force $F_{dy}$ is mostly positive, meaning a repulsive force from the other sphere. It can also be observed that the peak value of the in-line drift force is much larger than for an isolated sphere (Kudou, 1977).

<figure><img src="image_2.png" /><figcaption>Figure 1: Normalized drift force acting on one sphere in a twin arrangement. $x$- (left) and $y$- (right) components.</figcaption></figure>
Next we consider the two ships case studied by Kashiwagi (2004) — see also Kashiwagi et al., 2005. One ship is a modified Wigley hull model (ship A) while the other is a rectangular barge model (ship B). They are both 2 m in length with a separation distance *S* equal to 1.797 m. The waves are coming from abeam with ship A on the weather side of the arrangement, held motionless.

In figure 2 we present the normalized drift forces in sway, as obtained with the Lagally formulation, and as calculated by Kashiwagi (2004) with the pressure integration method (and a higher-order boundary element method — HOBEM). The results are plotted vs. the wavelength λ divided by the length L of the two ships. It can be seen that the agreement is excellent, except at small λ/L values where our meshing becomes too coarse.

<figure><img src="image_3.png" /><figcaption>Figure 2: Normalized sway drift forces on ships A (left) and B (right). Comparison between the Lagally formulation and the near-field results of Kashiwagi.</figcaption></figure>

## 5 References

GUÉVEL P. 1986 La récupération de l'énergie des vagues, in *Sciences et techniques de l'armement*, tome 60, 17-207 (in French).

GUÉVEL P. & GREKAS A. 1981 Le théorème de Lagally généralisé et ses applications en hydrodynamique navale, *Proc. ATMA Conf.*, Paris, 523-550 (in French).

KASHIWAGI M. 2004 Wave drift forces and moments on two ships with side-by-side arrangement, *Proc. 19th Int. Worksop Water Waves & Floating Bodies*, Cortona.

KASHIWAGI M., ENDO K. & YAMAGUCHI H. 2005 Wave drift forces and moments on two ships arranged side by side in waves, *Ocean Engineering*, **32**, 529-555.

KUDOU K. 1977 The drifting force acting on a three dimensional body in waves, *J. Soc. Nav. Arch. Japan*, **141**.

LANDWEBER L. 1967 Lagally's theorem for multipoles, *Schiffstechnik*, **14**, 19-21.

MARUO H. 1960 The drift force on a body floating in waves, *J. Ship Res.*, **4-3**, 1-10.

MOLIN B. 1979 Computations of wave drift forces, *Proc. OTC Conf.*, Houston, paper 3627 (alternatively Second-order drift forces upon large bodies in regular waves, *BOSS'79*, London, 363-370).

MOLIN B. & HAIRAULT J.P. 1983 On second-order motions and vertical drift forces for three-dimensional bodies in regular waves, *Proc. Int. Workshop on Ship and Platform Motion*, Berkeley, 344-357.

NEWMAN J.N. 1967 The drift force and moment on ships in waves, *J. Ship Res.*, **11-1**, 51-60.

PINKSTER J.A. & VAN OORTMERSSEN G. 1977 Computation of the first- and second-order wave forces on oscillating bodies in regular waves, *Proc. 2nd Int. Conf. Num. Ship Hydrodynamics*, Berkeley, 136-156.