## Abstract

Almost 4 years after the appearance of Salvesen–Tuck–Faltinsen (*STF*) strip theory (Salvesen et al., 1970, “Ship Motions and Sea Loads,” Annual Meeting of the Society of Naval Architecture and Marine Engineers (SNAME), New York, Nov. 12–13), Salvesen in 1974 published his popular method for calculation of added resistance (Salvesen, 1974, “Second-Order Steady State Forces and Moments on Surface Ships in Oblique Regular Waves,” Vol. 22; Salvesen, 1978, “Added Resistance of Ships in Waves,” J. Hydronautics, 12(1), pp. 24–34). His method is based on an exact near-field formulation; however, he applied the long-wave and the weak-scatterer assumptions to present his approximate method using the *integrated quantities* (hydrodynamic and geometrical coefficients). Considering the available computational powers in the 1970s, both of these assumptions were absolutely justifiable. The intention of this paper is to disseminate the results of a recent study at the Technical University of Denmark, whereby the Salvesen’s formulation has been revisited and the added resistance is computed from the original exact equation without invoking the weak-scatterer or the long-wave assumptions. This is performed using the solutions of the radiation and the scattering problems, obtained by a low-order boundary element method and the two-dimensional free-surface Green function inside our in-house STF theory implementation (Bingham and Amini-Afshar, 2020, DTU_Strip Theory Solver). The weak-scatterer assumption is then removed through a direct calculation of the x-derivatives of the velocity potentials and the normal vectors along the body. Knowing the velocity potentials over each panel, the long-wave assumption is also avoided by a piece-wise analytical integration of sectional Kochin Function (Kochin, 1936, “On the Wave Resistance and Lift of Bodies Submerged in Fluid,” Transactions of the Conference on the Theory of Wave Resistance, Moscow.). The presented results for five ship geometries testify that the correct treatment of the original equation is achieved only after both of the above-mentioned assumptions are removed. Implemented in this manner, Salvesen’s method proves to be relatively more accurate and robust than has been generally perceived during all these years.

## 1 Introduction

Alongside computational fluid dynamics (CFD) [1,2] and three-dimensional (3D) weakly or fully nonlinear potential flow models [3–6], state-of-the-art methods for calculation of second-order wave drift force and added resistance also include *Enhanced Unified Theory* [7–10]. Considering the computational effort and accuracy of the results, this method that is based on *Unified Slender-Body Theory* [11–18], the Kochin Function and Maruo’s method [19,20], is extremely efficient and mature. The unified slender-body theory considers a pure two-dimensional (2D) strip-theory solution, together with a solution which takes into account the interaction of the adjacent two-dimensional sections in the vicinity of the body (so-called the *inner field*). This interaction solution is calculated by a *matching* process between the solution in the inner-field and a three-dimensional solution in the *outer-field*, where the Laplace equation is solved subject to the free-surface and the radiation boundary conditions all in their 3D form. In fact, this theory unifies “ordinary” slender body theory at low limit of the encounter frequency where the *Froude–Krylov* exciting forces are dominant, with strip theory that is applicable at high values of the encounter frequency.

Two popular and classic examples of the methods for calculation of wave drift force and added resistance using strip theory are the method of Gerritsma–Beukelman [21], and the method of Salvesen [22,23]. The method by Gerritsma–Beukelman [21] is based on the *far-field* approach where the drift force is made proportional to the amplitude of the *radiated* waves (due to both body motion and scattering) in the far field. According to this method and based on their own strip theory presented in Ref. [24], the far-field radiated energy can be calculated by a line integral over the length of the vessel. As mentioned in the abstract, the Salvesen’s method is formulated using the *STF* strip theory. He derives the original 3D formulation for the hydrodynamic force starting from the *near-field* pressure integration over the body surface. After some manipulation, the equation for the mean second-order force is presented based on two terms. One is the Kochin Function and incorporates the interaction of the incident wave with the disturbance waves, and the other is due to the interaction of the disturbance waves with each other. He adopts the weak-scatterer assumption and neglects the second part, but mentions that this assumption might lead to inaccurate results in zero-speed or beam- and near-beam cases. The Kochin Function part of the equation is computed by the long-wave approximations, and the formulation is derived in terms of the integrated quantities like added mass, damping, and some geometrical coefficients. In the late 1970s, there was also some research conducted to improve the methodologies for force calculation by strip theories. For example, the work presented in Ref. [25] is an extension of the Gerritsma–Beukelman method to oblique seas. In other research in 1976, Lin and Read [26] aimed to complement Salvesen’s method by incorporating the body-generated waves (disturbance waves). They derived another set of exact equation for calculation of mean second-order forces and presented the final formulations based on the Kochin Function inside of the strip theory. Unfortunately, no calculations were presented in that paper. Lin and Read in Ref. [26] mentioned that the results would be published in a follow-up paper, but to the knowledge of the author this has not taken place. This fact has been noted also by Salvesen [23] and Loukakis and Sclavounos [25].

Similar to the enhanced unified theory, the Kochin Function and Maruo’s methodology can be applied inside the *STF* strip theory to obtain the added resistance without the above-mentioned assumptions. Clearly, the results will not be as accurate as those from the enhanced unified theory. However, it can be implemented with relative ease since for the strip theory no *matching* is required between the *inner solutions* and the *outer solution*, and the added resistance can be expressed with respect to the 2D sectional Kochin Function. The details of this implementation are presented in a recent paper [27], where it is also shown how this methodology predicts the added resistance by *STF* strip theory more accurately than the *approximate* Salvesen’s method in Refs. [22,23].

In the present paper and as an another step toward utilizing all capacities of strip theory for calculation of wave drift force and added resistance, it is demonstrated that in fact Salvesen’s method performs significantly better when evaluated based on its original exact formulation compared to the widely used approximate form. An implementation of the *STF* theory is used, where both the radiation and the scattering problems are solved based on a low-order boundary element method and the two-dimensional free-surface Green function [28]. The knowledge of the velocity potentials over the sections allows us to avoid the weak-scatterer and the long-wave assumptions. The former by a direct calculation of *x*-derivatives of the potentials and the normal vectors along the ship length, and the latter through a piece-wise exact integration over the two-dimensional sections. Calculations are then conducted for five ship geometries, and the results are compared with experiments and other reference solutions. The results confirm that the weak-scatterer part of the equation should not be neglected, even for a slender geometry like the classic Wigley hull. Moreover, it is demonstrated that the Kochin Function, which has been computed by Salvesen using the long-wave approximations, should be calculated instead based on the direct integration of the velocity potentials over the 2D sections. This is necessary as the long-wave approximations are not applicable in a majority of cases. By adopting both of these strategies, it is shown that the original equation in its exact form is relatively powerful in predicting the added resistance and wave drift forces accurately. It is important here to mention two major contributions where Salvesen’s method has been considered without the long-wave assumptions. Both of these works are due to Ming-Chung Fang in Refs. [29,30]. The first work is based on strip theory, and the second is based on the 3D boundary element method. In both cases, the disturbance velocity potentials are ignored, which means that he has also invoked the weak-scatterer assumption. No attempt has been made in these studies to calculate the body potential terms, and it is argued that they have no significant contribution to the final results. So, the main novelty of the present paper is the clear demonstration of the influence of the disturbance potential terms on the wave drift force and the added resistance.

In the coming sections, first the theory and the relevant formulations are introduced, then the numerical methods are described in some detail, and finally the results, discussions, and conclusions are presented.

## 2 Theory

*U*is the forward speed,

*ω*is the encounter frequency,

*ξ*

_{k}is the complex amplitude of the body motion in the

*k*th mode, and

*ϕ*

_{k}is the corresponding radiation velocity potential due to unit-amplitude motion. Note that in

*STF*theory, the surge mode is neglected. The scattering potential is denoted by

*ϕ*

_{s}, and the velocity potential of the incident wave

*ϕ*

_{0}with amplitude

*A*, heading angle

*β*, wave number in deep water $K=\omega 02/g=2\pi /\lambda $, wave frequency

*ω*

_{0}and wavelength

*λ*is defined by

*g*. The relation between the wave frequency and the encounter frequency

*ω*is defined by

*ω*

_{0}=

*ω*+

*K U*cos

*β*. See also Fig. 1 for further notations. Inside the

*STF*theory, the 3D velocity potentials for the radiation modes are decomposed into

*speed-dependent*and

*speed-independent*parts as $\varphi k=\varphi k0+U/(i\omega )\varphi kU$, with $\varphi kU=0$ for

*k*= 2, 3, 4, $\varphi 5U=\varphi 30$ and $\varphi 6U=\u2212\varphi 20$. The corresponding 2D sectional potentials are defined by

*ψ*

_{2},

*ψ*

_{3},

*ψ*

_{4},

*ψ*

_{5}=

*ψ*

_{3}(−

*x*+

*U*/(i

*ω*)), and

*ψ*

_{6}=

*ψ*

_{2}(

*x*−

*U*/(i

*ω*)). Therefore, only three velocity potentials are unknown. All these velocity potentials satisfy the Laplace equation $\u22072\psi k=0$ and the zero-speed free-surface condition

*y*−

*z*plane. In addition to the free-surface and the radiation conditions, the scattering velocity potential satisfies the body boundary condition

*STF*theory, refer to Ref. [31].

*n*denotes the 3D normal vector given by (

*n*

_{1},

*n*

_{2},

*n*

_{3}). Here, $\Phi B=\u211c{\varphi Bei\omega t}$ is the sum of the unsteady potentials due to all disturbance waves (radiation and scattering), and $\Phi 0=\u211c{\varphi 0ei\omega t}$ is the unsteady potential due to the incident wave.

*ρ*is the density of the fluid. This equation is composed of the second-order pressure terms due to the product of the first-order velocity potentials. Typically, each first-order quantity with harmonic time variation can be represented as

*A*

_{j}cos(

*ωt*+

*θ*

_{j}), where the amplitude and the phase are denoted by

*A*

_{j}and

*θ*

_{j}, respectively. In the frequency domain, this quantity can be expressed by $\u211c{Xjei\omega t}$ in which $Xj=Ajei\theta j$ is the

*complex phasor*. Therefore, the second-order term

*p*

^{(2)}arising from the product of two first-order quantities can be written as

*ϕ*

_{B}and

*ϕ*

_{0}), to calculate the added resistance as follows. First consider the interaction between the incident and the disturbance waves, i.e., only the part containing $\u2207\Phi 0$ on the right-hand side of Eq. (6). Inserting the complex phasors of the velocity potentials leads to the mean second-order force as

*harmonic*potentials

*ϕ*

_{i},

*ϕ*

_{j}(meaning $\u22072\varphi i,j=0$), which satisfy the free-surface condition, applying Green’s theorem leads to

_{B}satisfy all of the above-mentioned conditions. Except for the radiation condition, the same statement is also true in the case of $\u2207\Phi 0$ and $\u2207\varphi B*$. Therefore, the mean force can be expressed instead as an integral over the body surface

_{B}≪ Φ

_{0}, and accordingly neglects the second contribution $F\xaf\varphi B$, which is a weak-scatterer assumption. Considering only Eq. (11) and inserting the incident wave potential, he obtains

*STF*strip theory. As mentioned before, these equations are in fact the Kochin Function [33]. Here,

*ψ*

_{B}represents the sectional velocity potentials of the disturbance waves, and the integration over a 2D section is denoted by

*C*

_{x}. The body length is shown by

*L*. In addition Salvesen adopts the following long-wave assumptions for computation of the sectional integrals of the Kochin Functions in Eqs. (13) and (14)

*b*is the sectional beam,

*d*is the sectional draft, and

*s*is the sectional area coefficient calculated by normalizing the sectional area with

*bd*. Adopting the Haskind relation, the added resistance and the mean drift force then is expressed with respect to line integrals along the body comprising only of the

*integrated*quantities such as sectional added mass, damping, and the above-mentioned geometrical coefficients. See the details of this derivation and the final

*approximate*formulations in Sec. 3.1 of his paper [22]. In a quite recent paper by the author [27], using the long-wave assumptions (15) and (16), the approximate relation for the Kochin Function is presented for calculation of the added resistance and drift force inside

*STF*theory using Maruo’s methodology [19,20]. In that paper, it is shown that these long-wave assumptions are only valid for zero- and very low speeds, and therefore, for accurate results, the Kochin Function should be computed

*directly*based on the knowledge of the sectional velocity potentials

*ψ*

_{B}. Therefore, it sounds logical to investigate the impact of these long-wave assumptions also on the approximate equations presented by Salvesen for calculation of Eq. (13) or (14). As mentioned in the Introduction section, an implementation of the

*STF*strip theory is performed where both the radiation and the scattering potentials are computed based on the boundary element method and the 2D free-surface Green Function. See Refs. [27,28] for more details. This allows the Kochin Function in Eq. (13) or (14) to be computed directly, using a piece-wise analytical integration over the two-dimensional section. As is shown in Sec. 4, these long-wave assumptions break down in forward-speed cases also here for the calculation of the added resistance based on Salvesen’s approximate formulations for Eq. (13) or (14).

*x*component of the force is

*x*derivatives of the velocity potentials and the normal vectors along the length of the body. Inserting the potentials and the body boundary conditions from their 2D definitions in

*STF*theory, Eq. (17) in its expanded form is

^{x}denotes the

*x*derivative. The results shown in Sec. 4 confirm that in fact the body disturbance term is an important part of Eq. (6) and has a profound effect on the added resistance. In Sec. 4, it is demonstrated that a direct computation of the Kochin Function in Eq. (13) is not enough to get accurate solutions. It is only after incorporating Eq. (18) and treating Eq. (6) as a whole that accurate results are achieved. The reason that Salvesen’s approximate method provides reasonable results is that the errors introduced by the long-wave assumption largely cancel the neglect of the disturbance potential terms. This behavior is also shown in Sec. 4.

## 3 Numerical Methods

In this section, some details of the numerical methods are described. An in-house implementation of the *STF* theory is used [28], whereby the solutions for the radiation and the scattering potentials are provided for each two-dimensional section along the ship length (see Ref. [27] for further details).

### 3.1 Evaluation of the Kochin Function Term.

*STF*strip theory in Ref. [28],

*ψ*

_{B}is calculated and known. Having the geometry of each panel including

*N*

_{2},

*N*

_{3}, and replacing the normal derivative of

*ψ*

_{B}with the body boundary condition from Eq. (4) or (5), the line integrals over

*C*

_{x}in Eq. (13) or (14) can be computed. As the velocity potential and the normal vector are assumed to be constant over each panel, these integrals can be evaluated easily by a series of piece-wise exact integrations along all panels. So, the part of the sectional integral related to the gradient of the potentials can be calculated as

*δ*is the angle of the normal vector to each panel (

*N*

_{2}= sin

*δ*,

*N*

_{3}= −cos

*δ*). Here the integrals are evaluated for each panel extending between the two vertices at (

*y*

_{m},

*z*

_{m}) and (

*y*

_{m+1},

*z*

_{m+1}). The number of panels on each 2D section is denoted by

*N*

_{p}. Similarly, the line integral over the vessel’s length

*L*is also computed using a piece-wise analytical integration procedure. In the present framework, this integration is carried out simply by assuming a linear relation between the values of two consecutive sectional integrals evaluated at

*x*

_{j}and

*x*

_{j+1}as

*N*

_{x}is number of 2D sections along the vessel length.

### 3.2 The Disturbance Potential Terms.

*x*-derivatives of the disturbance velocity potentials

*ψ*

_{B}and the normal vectors, which appear in the terms

*Q*and

*M*in Eq. (18) are computed by the method proposed by [34]. This scheme applies a relation from differential geometry (see for example Ref. [35]) to express the Cartesian gradient of the potential on the body surface as a combination of the surface gradient and the derivative in the normal direction. Taking

*u*and

*v*as parametric coordinates in the longitudinal and vertical directions along the hull surface, we write

*E*,

*F*,

*G*are the coefficients of the first fundamental form of the surface given by

The sectional integrals over *C*_{x} in Eq. (18) are calculated using simple summations along the two-dimensional sections. The convergence of this integration is also quite fast and is achieved at 20–30 panels. In addition the line integral along the body is performed using the trapezoidal rule.

## 4 Results

According to the formulations introduced in Sec. 2, and for the purpose of comparison, results for added resistance are presented for the following three cases:

The results based on only Eq. (13), together with the long-wave assumptions (15) and (16). Note this method is the one which has been introduced by Salvesen and is being used currently by the community. In this paper, these results are denoted by

**Salvesen (App.)**. Here “App.” indicates the fact that the Kochin Function in Eq. (13) is calculated using the long-wave approximations.The results based on only Eq. (13), when the related Kochin Function is calculated instead by

*direct*piece-wise analytical integration of the disturbance potentials over the sections. These results are denoted by**Salvesen (Dir.)**.The results based on both Eqs. (13) and (18), where the Kochin Function is computed by

*direct*piece-wise integration (as in case 2), and in addition, the body disturbance term is taken in to account by the direct calculation of the*x*derivative of the potentials and the normal vectors. The results using this procedure are denoted simply as**Salvesen (New)**.

The results according to above-mentioned three methods are presented for five geometries, all shown in Fig. 2. Wherever possible, they are compared with experimental measurements or with solutions based on other methods. The results using Maruo’s formulation inside the *STF* strip theory [27] are denoted as **STF-Maruo**. For the wave drift force *Fr* = 0, the solution by wamit [36] is also shown. Note also for the results here, only the heave and pitch modes are included.

### 4.1 Geometry 1—Wigley Hull Type I.

Figure 3 presents the results for the wave drift force (*Fr* = 0) and the added resistance (*Fr* = 0.2, 0.3), for the Wigley hull Type I. The results based on the 3D boundary element method by Shao and Faltinsen in Ref. [37] is shown for *Fr* = 0.2. For *Fr* = 0.3, the added resistance calculated using the 3D finite-difference model of Amini-Afshar and Bingham (OWD3D-Seakeeping) [38–40] is also presented for the comparison.

### 4.2 Geometry 2—Modified Wigley.

### 4.3 Geometry 3—KVLCC2.

### 4.4 Geometry 4—RIOS Bulk Carrier.

The results for the **RIOS** bulk carrier [43] are presented in Fig. 6, for the wave drift force *Fr* = 0 and the added resistance at *Fr* = 0.18. The experimental measurements in the plot are denoted as Kashiwagi (2012), Kashiwagi [43] and Kashiwagi (2017), see Ref. [44]. The most recent experimental data which is based on a novel approach and uses the *unsteady pressure measurement* technique are due to Kashiwagi in Ref. [45].

### 4.5 Geometry 5—DTC Container.

Next the Duisburg Test Case (DTC) container ship is considered [46]. The dimensions of the container ship in this case are *L* = 355 m, *B* = 51.0 m and the design draft *d* = 14.5 m. The results are for *Fr* = 0.05, *Fr* = 0.11 and *Fr* = 0.20, and are presented in Fig. 7 (see also Fig. 2(e) for the shape and the strip discretization of this vessel). The measurements and the CFD calculations are by Sigmund and el Moctar from Ref. [47].

### 4.6 Discussion.

First note that in order to find the contribution of the disturbance potential term, the results denoted by **Salvesen (Dir.)** and **Salvesen (New)** should be compared. In addition, to find the validity of the long-wave assumption, the results named **Salvesen (App.)** should be compared with the results denoted as **Salvesen (Dir.)**.

The results presented for these five geometries all depict the same picture about the essence of the terms in Eq. (6). As mentioned earlier and can be observed from the plots, the long-wave assumptions are generally valid only for the zero-speed case in the head-seas condition. This is why the head-seas drift forces calculated based on **Salvesen (Dir.)** and **Salvesen (App.)** are more or less the same, though both tend to dramatically over-predict the force compared with the reference solution from wamit. In addition, as shown in the case of the Wigley Hull for *Fr* = 0, *β* = 140 deg, the long-wave assumption also breaks down for zero-speed problems if the heading angle is other than 180 deg. For all of these zero-speed cases, the computation based on the exact implementation, i.e., **Salvesen (New)** are remarkably closer to the wamit or other reference solutions. Moreover, a comparison between the results from **Salvesen (Dir.)** and **Salvesen (New)** reveals that the body disturbance term (17) has a significant contribution to the final zero-speed results. For the forward-speed cases (added resistance), as also recently demonstrated by Ref. [27], the long-wave assumption is no longer valid. This again can be seen by comparing the original results based on Salvesen’s method (denoted as **Salvesen (App.)**) with the results using the direct computation of the Kochin Function in Eq. (13), (i.e., those denoted by **Salvesen (Dir.)**). The role of the body disturbance term is also important here in order to be able to predict the added resistance accurately. Ignoring these body potential terms leads to over-prediction of the results especially for the high-frequency ranges. The results based on the complete form of Eq. (6) with no long-wave or weak-scatterer assumptions (denoted as **Salvesen (New)** in the plots) agree considerably better with the experimental measurements. As is well known, the approximate Salvesen’s method over-predicts the zero-speed drift forces, while it under-predicts the added resistance. It is shown in this paper that both of these shortcomings are remedied by using the complete implementation of Salvesen’s original formulation. Moreover, as can be seen from the figures, the **Salvesen (New)** results compare with the experiments even better than those based on **STF-Maruo** from Ref. [27]. No experimental measurements were available to present here for the added resistance of these geometries in beam- or near-beam seas conditions. But from the figures, it can be seen that neither the long-wave nor the weak-scatterer assumption are acceptable also in these cases.

## 5 Conclusions

Salvesen in Refs. [22,23] mentioned that the neglect of the body disturbance term related to $\u2207\Phi B$ would lead to less accurate results in zero-speed and beam- or near-beam seas conditions; however, no serious question was raised about the validity of the long-wave assumptions (15) and (16) in computing the wave drift forces and added resistance. In addition, to the knowledge of the author, no attempt has been made to investigate the influence of the body potentials on the wave drift force and added resistance. Therefore, in this study, the original equation derived by Salvesen in 1974 has been revisited. The objective was to investigate the validity of both of these assumptions, in terms of the *approximate* expressions for computing the second-order quantities. An implementation of *STF* strip theory using a low-order boundary element method and the free-surface Green function has been used. Using the resultant solutions for the radiation and the scattering velocity potentials over the two-dimensional sections along the body length, Eqs. (13) and (17) have been calculated in their original form. The results confirm that neither the long-wave nor the weak-scatterer assumptions are generally acceptable, and good agreement with measurements is achieved when the body disturbance part is included, and at the same time, the Kochin Function is computed without a long-wave approximation. In fact it is proved in this paper that the role of the body disturbance term is not a matter of relative importance for some wave conditions or geometries. The body disturbance term related to $\u2207\Phi B$ is in reality a fundamental part of the equation for the added resistance (6), and regardless of the geometry or the wave condition, the correct results are attained only after incorporating this part into the equation. More importantly Salvesen’s approximate relations provide reasonable results, only because the long-wave assumption tends to partially correct for the neglect of the body disturbance terms. If Salvesen had computed the Kochin Function in Eq. (13) without the long-wave assumptions, he would have achieved the results denoted here by **Salvesen (Dir.)**, which considerably over-predict the forces, especially in the short-wave range.

The Kochin Function related to the horizontal drift force in Eq. (14) can also be calculated directly and without the long-wave assumption. In addition, the related body disturbance term $\u2207\Phi B$ can be included by the direct computation of the *y* derivatives of the velocity potentials and the normal vectors. A similar picture about the accuracy of the results for the horizontal drift force can be expected. The exact equation for the mean second-order yaw moment has also been derived by Salvesen in Ref. [22]. Based on the long-wave and weak-scatterer assumptions, he presented *approximate* relations for computing also these second-order quantities. Therefore, the same methodology shown in this paper for the wave drift force and added resistance can be adopted for accurate computation of the mean second-order yaw moments. Maruo’s formulation, as applied to *STF* theory in Ref. [27], requires multiple direct calculations of the Kochin Function integral at each frequency, while the presented methodology in this paper needs only one calculation. However, it requires that the body disturbance terms are taken in to account via a direct computation of the derivatives of the potentials and the normal vectors. This extra computational burden is worth considering the influence of the disturbance velocity potentials on the accuracy of the results denoted by **Salvesen (New)** in this paper.

## Footnotes

## Acknowledgment

I am deeply indebted to Professor Harry B. Bingham (my Ph.D. and post-doc advisor at the Technical University of Denmark) from whom and in whose school of thought, I have learned the fascinating subject of marine hydrodynamics. In addition, I am especially grateful to him for his valuable support, comments, and suggestions regarding this paper. This work has been funded by the ShippingLab project,^{1} and its financial support is highly appreciated.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The data and information that support the findings of this article are freely available online.^{2}