On the numerical accuracy of the wave load distribution on a ship advancing in short and steep waves
Authors
 First online:
 Received:
 Accepted:
DOI: 10.1007/s0077301101568
Abstract
Defining numerical uncertainty is an important part of the practical application of a numerical method. In the case of a ship advancing in short and steep waves, little knowledge exists on the solution behaviour as a function of discretisation resolution. This paper studies an interfacecapturing (VOF) solution for a passenger ship advancing in steep (kA = 0.24) and short waves (L _{w} /L _{pp} = 0.16). The focus is to estimate quantitative uncertainties for the longitudinal distributions of the first–third harmonic wave loads in the ship bow area. These estimates are derived from the results of three systematically refined discretisation resolutions. The obtained uncertainty distributions reveal that even the uncertainty of the first harmonic wave load varies significantly along the ship bow area. It is shown that the largest local uncertainties of the first harmonic wave load relate to the differences in the local details of the propagating and deforming encountered waves along the hull. This paper also discusses the challenges that were encountered in the quantification of the uncertainties for this complex flow case.
Keywords
Numerical uncertainty Volumeoffluid method First–third harmonic ship wave loads Short and steep waves1 Introduction
In recent years, a ship advancing in waves has become a popular flow case for the users and the developers of interfacecapturing methods, e.g. [1–14]. In such a flow case, the interfacecapturing methods are advantageous, because they enable modelling the effect of arbitrary freesurface behaviour. In practice, the development of both the interfacecapturing methods and the computational resources has been required to run computations on a ship advancing in waves.
The previous publications cover several examples on the computational modelling of a ship advancing in waves. These studies have considered both global [1–9] and more local wave loads [2, 10–13]. As for the harmonic content of these wave loads, the focus has mainly been on the zeroth [3–6, 9] and on the first [2–6] harmonic wave loads, but some examples also on higher harmonic results exist: second and third harmonic wave loads in [11] and spectral analysis of wave loads in [5, 8].
In order to have confidence in computational predictions, they are compared with measured results. However, a computational solution depends on the selected discretisation resolution. Therefore, the behaviour of the numerical solution in this respect should be studied before validating the computations against measured results.
When analysing the solution behaviour as a function of the discretisation, the computations need to be repeated with several discretisation resolutions (usually three in minimum) in order to find out the dependence of solution on the selected resolutions. Previous numerical studies on a ship advancing in waves have been based on the results of three discretisation resolutions [2, 5, 6, 9–11]. Most of these studies have considered the solution behaviour by simply giving or comparing the three results [2, 9–11]. Such comparisons give an idea on the variation of the results within the selected discretisation range and may show sufficiently similar solution behaviour on the selected discretisations.
However, these kinds of qualitative comparisons do not give a quantitative estimate for the difference between the obtained solution and the exact solution of the continuous mathematical model. The difference between the obtained solution and the exact solution of the numerical method is called numerical error. In practice, the numerical error is taken into account as an uncertainty. Estimating this error is important when judging the capability of the method to predict flow behaviour. This is especially important when considering, if the selected models within a method are adequate for predicting certain flow behaviour.
In the case of a ship advancing in waves, little results have been published on the numerical uncertainty. In the studies [5, 6], quantitative uncertainty has been analysed for one very moderate wave condition L _{w} /L _{pp} = 1.5 and kA = 0.025 in the case of a surface combatant. This uncertainty estimation has been restricted to the zeroth harmonic global wave loads and that of the first harmonic results has been omitted due to oscillating results. The choice of the authors of [5, 6] not to give uncertainty estimates for oscillating results relates to the difficulty of treating nonmonotonically converging results. The different uncertainty estimation approaches have different attitudes towards such results.
The present study considers the solution behaviour of first–third harmonic wave load distributions on a ship bow area as a function of the discretisation using three discretisation resolutions. The results are studied both by simple comparisons and by estimating the numerical uncertainties of the solution of the fine discretisation resolution. In this case, the encountered waves are steep kA = 0.24 and short in comparison to the ship length L _{w} /L _{pp} = 0.15. Almost similar flow conditions (kA = 0.24, L _{w} /L _{pp} = 0.16) have been applied for a less full bow form in [11] to estimate total forces on the bow area. The focus of this study is in the estimation of the quantitative uncertainty for first–third harmonic wave load distributions and in the encountered challenges of quantifying the uncertainty.
Little knowledge exists on similar computational cases. Both the flow conditions and the load parameters of interest are different from most of the previously published simulations on a ship advancing in waves. It is assumed that the different flow conditions and the interest in higher harmonic components require higher time resolution than in the case of the zeroth and first harmonic wave loads studied previously. Furthermore, the numerical behaviour of the distributions of the predicted wave loads as a function of discretisation resolution has been seldom presented. As regards different harmonic components, there are some indications that having converging results for even the first harmonic global wave loads in moderate conditions can be challenging, see [5, 6]. In the case of the second and third harmonic components, there is even less previous knowledge.
The analysis of the present results differs from the previous similar studies, because of the motivation on the selected wave conditions. This flow case is interesting because it can cause springing vibration. This fact affects the analysis of the numerical uncertainty of the wave loads. Firstly, the wave load is analysed as a longitudinal distribution in the ship bow area. Analysis of the distributions is important when studying springing, because the actual vibratory excitation results from both the longitudinal wave load distribution and the longitudinal distributions of the hull eigenmodes. Secondly, the present analysis of the wave load distribution covers the first–third harmonic components. The second harmonic wave load is the actual load that causes springing in the selected flow case, but the first and the third harmonic components are included to get a more general idea of the solution behaviour. Thirdly, we compare the behaviour of the first–third harmonic single frequency components with the behaviour of the respective components that include the effect of their surrounding frequency components. This is reasonable from the point of view of springing, because several frequencies around the critical frequency contribute to the vibratory excitation. This is reasonable also from the point of view of studying numerical uncertainty, because the wave energy may spread differently in the frequency domain with different discretisation resolutions. Fourthly, the local uncertainties of the load distributions are compared with the uncertainty of the respective global load. This is done to study whether the uncertainty level of a global quantity can represent the uncertainty level of a local quantity.
In this paper, the interfacecapturing solution method applied is presented in Sect. 2. The computational case of the study is described in detail in Sect. 3. The approaches used in the analysis of the results are presented in Sect. 4. The results are presented in Sect. 5 and discussed in Sect. 6. Finally in Sect. 7 the conclusions are given.
2 Numerical method
The computations were performed with the commercial flow solver ISISCFD. The solver is an unstructured finite volume solver. It includes a volumeoffluidtype interfacecapturing method to simulate freesurface flows. The flow is treated as incompressible and without surface tension. The flow solution (velocity \( \vec{U} \), pressure p and volume fraction c distributions) is obtained for each time step by iterating the solution of the momentum equations, the pressure equation and the volume fraction concentration equation. The numerical method is published in [15] and some further and updated details on it are given in [16]. In the present study, the solver is used as an Eulersolver: in other words the viscosity of the fluid is ignored.
In the present study, a second order backward discretisation was chosen for the time derivatives. For the velocity and the volume fraction discretisations the second order GDSscheme and the BICSscheme [16] were respectively selected. In the numerical method, a special discretisation is used for the pressure to take into account the discontinuity of the density on the interface of the air and water [15].
3 Simulation case
The selected simulation case was chosen because of an interest in the secondorder springing excitation. In practice this means that the encounter frequency of the ship and the waves was chosen such that the second harmonic wave load could excite the vertical twonode mode of the fullscale hull. The wave was chosen to be very steep, which should ensure significant higher harmonic excitation. As the main purpose of this paper is to study the behaviour of the numerical solution as a function of the discretisation resolution, the simulation was repeated three times with systematically refined discretisations to enable the uncertainty estimation.
3.1 Case conditions
Ship and wave particulars in the model scale
Length L _{pp} 
6.69 m 
Wave length L _{w} 
1.05 m 
Breadth 
1.10 m 
Wave height H _{w} = 2A 
0.08 m 
Draught 
0.18 m 
Wave steepness kA 
0.24 
Block coefficient 
0.72 
Encounter period T _{e} 
0.38 s 
3.2 Spatial domain
Locations of the grid boundaries
Coarse 
Medium 
Fine  

x _{FPP} − x _{min}/L _{w} 
14.70 
14.70 
14.70 
x _{FPP} − x _{max}/L _{w} 
2.86 
2.86 
2.86 
y _{FPP} − y _{min}/L _{w} 
0.00 
0.00 
0.00 
y _{FPP} − y _{max}/L _{w} 
6.63 
6.63 
6.63 
z _{FPP} − z _{min}/L _{w} 
7.14 
8.57 
7.95 
z _{FPP} − z _{max}/L _{w} 
2.61 
1.18 
1.80 
Boundary conditions
x _{min} 
Imposed velocity 
x _{max} 
Numerical wave boundary condition 
y _{min} 
Symmetry condition 
y _{max} 
Imposed velocity 
z _{min} 
Wall with slip condition 
z _{max} 
Imposed pressure 
Hull 
Wall with slip condition 
Locations of the boundaries of the refinement boxes
b1 
b2 
b1 
b2  

x _{FPP} − x _{bi,1}/L _{w} 
8.12 
1.93 
x _{FPP} − x _{bi,2}/L _{w} 
2.86 
0.13 
y _{FPP} − y _{bi,1}/L _{w} 
0.00 
0.00 
y _{FPP} − y _{bi,2}/L _{w} 
6.63 
0.95 
z _{FPP} − z _{bi,1}/L _{w} 
0.12 
0.06 
z _{FPP} − z _{bi,2}/L _{w} 
0.23 
0.38 
Cell sizes in the refinement boxes, time steps and total number of cells
L _{w} /L _{c} ^{a} 
H _{w} /H _{c} ^{b} 
T _{e}/Δt ^{c} 
Number of cells (M)  

Coarse 
58.32 
8.00 
245.16 
2.06 
Medium 
72.90 
10.00 
306.45 
3.41 
Fine 
87.49 
11.94 
367.86 
6.09 
The discretised domain moved during the simulation with the ship velocity along the positive xaxis. The ship hull and the domain were kept fixed in the other degrees of freedom, because the waveinduced ship motions are insignificant in the waves that are very short in comparison to the ship length (about L _{w} /L _{pp} = 0.16).
3.3 Time domain
The length of the simulation in the time domain was chosen such that it ensures a sufficiently long analysis period in the regular waves with the selected ship speed.
The duration of the simulation (0.0–10.8 s) includes three parts: the acceleration ramp for the ship speed with a one half sinusoidal ramp profile (0.0–3.0 s), the time required for the propagation of the waves to the ship bow area (0.0–7.0 s) and the analysed time interval (7.0–10.8 s = 10*T _{e}).
The time steps were chosen such that there are at least 80 time steps per third harmonic period, Table 5. They were refined systematically with the same ratios as the computational volumes.
Within each time step, the convergence of the results is controlled by two inputparameters: maximum number of iterations (10) and orders of magnitude (2) by which the residual is reduced.
3.4 Computational resources
Average CPU time/one time step and number of time steps
CPU (s) 
Time steps  

Coarse 
406.8 
6967 
Medium 
2509.4 
8700 
Fine 
6886.9 
10500 
4 Analysing the computational results
The results presented in this paper are mainly derived from the pressures p and volume fractions c on the hull. The wall values of the pressure are the same as the ones in the closest computational volume, in other words the pressure gradient over a wall is set to zero, (Queutey P, personal communication, September 2008). In addition to the pressure and volume fraction values on the faces, the information on the locations of the face central points (x, y, z) is used. The data to be analysed consists of the values at an unstructured set of points without information on the locations of the cells with respect to each other and without information on the location of the corners of faces. For the calculation of the total pressure, the information on the surface area of each face on the wall is also utilised.
4.1 Wave excitation
4.1.1 Frame force
Vertical force is analysed on a set of ship frames. As unstructured grids are used, the grid points are not in practice located on vertical intersections that present frames. Instead, points N _{p,f} within thin vertical sections are selected and used to calculate the instantaneous force on a frame.
4.1.2 Force on the total observation area
N _{p,a} indicates the number of cell centres that are situated within the observation area.
4.2 Harmonic components
The force histories are subjected to discrete Fourier transformation DFT (e.g. [18]) to obtain the first–third harmonic amplitudes. The denotation F is used here for a force history from which its mean value F _{mean} has been subtracted.
From the point of view of the signal analysis, the time histories given by the computations are data sequences F = F(n) of discrete times n = 1,2,…,N _{ t }. The length of the time history L _{ t } = N _{ t }Δt defines the spacing Δω ([Δω] = rad/s) of the frequency domain by Δω = 2π/L _{ t }. The total number of points N _{ ω } in the frequency domain is limited by the Nyquist frequency π/Δt.
4.3 Uncertainty estimation
Generally speaking, the numerical uncertainty consists of contributions from the iteration number, the grid resolution, the time step, the roundoff and the other parameters, e.g. [19]. In the present study, the presented uncertainties include the effect of the grid resolution and of the time step. These two uncertainty sources are studied simultaneously as the Courant number is fixed in the computations. The effect of roundoff is assumed to be negligible in comparison to the other sources of uncertainty.
The convergence conditions are:

Monotonic convergence: 0 < R < 1

Oscillatory convergence: −1 < R < 0

Divergence: R > 1 or R < −1.
The applied uncertainty estimation approach is presented in [20]. Its application in the present study differs from that in the study [20] by using only three discretisation resolutions. The approach utilises the order of accuracy q, the difference δ_{RE,1} between the fine grid solution ϕ _{1} and the estimated exact solution ϕ _{0} and the data range Δ_{M} to estimate the uncertainty.

$$ {\text{For }}0. 9 5\le q \le 2.0 5,U_{\phi } = \, 1.25\delta_{{{\text{RE}},1}} . $$(15)

$$ {\text{For }}0 \le q \le 0. 9 5,U_{\phi } = { \min }(1.25\delta_{{{\text{RE}},1}} ,1.25\Updelta_{\text{M}} ). $$(16)

$$ {\text{For}}q \ge 2.0 5,U_{\phi } = { \max }(1.25\delta_{{{\text{RE}},1}}^{*} ,1.25\Updelta_{\text{M}} ). $$(17)
\( \delta_{{{\text{RE}},1}}^{*} \) is obtained with Richardson extrapolation using q equal to the theoretical value.
4.4 Freesurface levels on the hull
For the interpolation of the freesurface levels, the hull surface is constructed from the unstructured set of points using Triangle, [21, 22]. (The ycoordinates were ignored during the construction and their effect was added afterwards.) As a result, there is a surface that consists of Delaunay triangulations with corner points that are the central points of the original surface grid. The interpolation of the freesurface levels on the hull is done with the postprocessor tool Ensight.
5 Results
Section 5.1 presents the vertical force distributions given by the three discretisation resolutions, while their uncertainties are given in Sect. 5.2. In addition, Sect. 5.3 studies the source of the largest first harmonic uncertainties.
The observation area is limited between x _{1} = 5.20 m = 0.78L _{pp} (close to the ship fore shoulder) and x _{2} = 6.63 m = 0.99 L _{pp} (close to the ship fore perpendicular). The length of this observation area is 1.4 times the length of the encountered waves. To have the force distribution as a function of x, 36 equally spaced frames are selected within the observation area. One frame consists of the points, which are within ±0.67 times the cell length on the coarse grid from the specified xcoordinate. This ensures that there are enough points on each frame with each discretisation resolution.
5.1 Force amplitude distributions with the three discretisation resolutions
The results in Fig. 4 include both the first–third harmonic amplitudes ξ_{span,i } describing the energy within the frequency spans and the amplitudes ξ_{single,i } describing the energy of the single frequency components. The differences between the discretisation resolutions given by these two widths of frequency span do not deviate significantly, even if the effect of the frequency span becomes more distinct in some locations the higher the harmonic component is. The amplitude distributions show that the use of the wider frequency span slightly increases the amplitudes in a rather systematic manner for all the three discretisation resolutions.
5.2 Uncertainty of force
The monotonic or nonmonotonic convergence of each local result is denoted in Fig. 5. Most (ξ_{span,i } 56% and ξ_{single,i } 53%) of the local first harmonic amplitudes converge monotonically. The nonconverging results are mainly located near the ship fore perpendicular (x _{FPP} = 6.69 m). Some (ξ_{span,i }: 33% and ξ_{single,i }: 31%) of the second harmonic amplitudes converge monotonically. The converging results are located around x ≈ 5.7 m and in the vicinity of x ≈ 6.3 m. As for the third harmonic results, the results with the monotonic (ξ_{span,i } 50%, ξ_{single,i } 47%) and the nonmonotonic convergence are spread over the observation area.
The local values of the uncertainty distributions of the harmonic amplitudes vary significantly around their average values, Fig. 5. In the case of all three harmonics, the smallest values are located in the vicinity of x = 6.1–6.2 m. The largest values are located near the ship fore perpendicular and towards the rear end of the observation area.
The uncertainty distributions are compared with the respective uncertainties of the harmonic amplitudes of the vertical force integrated over the total observation area in Fig. 5. The first harmonic amplitude of the integrated quantity does not converge monotonically, whereas the second and third harmonic amplitudes do, except ξ_{single,2 }. The uncertainty of the first harmonic component of the integrated quantity (nonconverging) is closest to the average of the uncertainty distribution, but the difference (calculated: (integrated quantity minus average of the distribution) divided by average of the distribution) is still significant (ξ_{span,i } 36% and ξ_{single,i } 41%). The uncertainties of the second and the third harmonics of the integrated quantity (converging except ξ_{single,2}) are smaller than almost all the local uncertainties of the respective uncertainty distribution.
5.3 Impact of local flow detail on local uncertainties
The results in Sect. 5.2 show that the uncertainties of the first harmonic force components are especially large around x = 5.7 m. The results in Sect. 5.1 show that this is a consequence of the diminution of the first harmonic force amplitudes with discretisation refinements in this area. This section studies why the first harmonic uncertainties are especially large there.
6 Discussion
The aim of this study was to estimate the quantitative numerical uncertainty for the first–third harmonic wave load distributions. This section discusses the challenges that were encountered when estimating the uncertainties.
It was decided that the iterative error is ignored, because its definition would have required such significant computational resources. In a time accurate case, estimating iterative error is challenging, because, on one hand, it accumulates from the previous time steps during some time spans of the solution. On the other hand, the oscillatory behaviour of the solution can also diminish it during some other time spans of the solution. In practice, its estimation would have required repeating the computations with several iterative parameters. The comparison of the obtained results would have revealed the iterative error. The omission of the iterative error can make the uncertainty estimates too small.
The reliability of an uncertainty estimate is affected by the convergence behaviour of the solution. The current uncertainty estimation approaches focus mainly on the converging solutions while the estimates for nonconverging results are less well justified. The number of nonconverging results in this study and the previous observation in e.g. [5, 6] indicate that it is very difficult to avoid nonconverging results in a case of a ship advancing in waves. This underlines the need for having uncertainty estimation approaches that would pay more attention to nonmonotonically converging results.
It was also demonstrated that having the results of only three discretisation resolutions leaves some uncertainty on the obtained convergence conditions. The present monotonically converging results relates to the largest data ranges while some nonmonotonically converging results are even very close to each other. Both in theory and in practice, it is possible that three results that are close to each other oscillate or even diverge, but a larger set of results could show that the general trend is converging. Similarly, three converging results can be a small part of a larger oscillating or diverging set of results. It was also observed that the present monotonically converging results vary significantly from the theoretical order of accuracy. Furthermore, when considering small changes between the results, it should be noted that the smaller the differences between the results of the discretisation resolutions, the larger the effect of small disturbance on the solution behaviour. In the present case, one disturbing factor is the simplistic implementation of the wave boundary condition. Another is that the hexahedral grids may not be fully topologically identical despite the systematic grid refinement.
The abovementioned issues indicate that the present results may not fully disclose the solution behaviour of this flow case. The only way of confirming the findings would be to repeat the computation with more discretisation resolutions. Then, the solution behaviour could be studied for a larger scale of resolutions and the convergence behaviour could be confirmed. The problem with this kind of option is that a lot of computational resources would be required in order to reveal for instant an oscillatory behaviour of the solution.
A further purpose for increasing the number of the computations would be to confirm the obtained uncertainty estimates. For this purpose, the computations that are used for the uncertainty estimation should be in or close to the asymptotic range due to the uncertainty estimation approach. As already mentioned, reaching the asymptotic range can be very challenging in the case of shipwave interaction.
In this study, one practical challenge is the requirements for the computational resources, which limited the number of runs to three. In this case, the computational requirements are affected by both the spatial and the time discretisation resolutions. Both of these resolutions were systematically refined between the selected discretisations to keep the Courant number fixed. Often, the spatial discretisation—the number of grid points—is used as a measure for the accuracy level of the computation and for the demand of the computational resources. In the present case, the time discretisation is equally an important measure, because a very fine discretisation (245–368 time steps/encounter period) is used to ensure a meaningful analysis of the second and third harmonic wave loads. From the point of view of running computations, the high time resolution may be a more demanding requirement than the high spatial resolution. With a high spatial resolution, adding number of processors can make the computations faster, even if the increasing interprocessor communications with increasing number of processors limits this benefit. In the case of a high time resolution, it is not possible to accelerate the time stepping by an approach like using multiprocessors.
The complexity of the studied flow case, on its part, makes the understanding of the variation of the obtained uncertainty levels challenging. The present results demonstrate that the resolution dependency of an apparently small flow detail can affect significantly local uncertainty levels. In this respect, the amount of the falling mixture of air and water from a water splash was a critical factor that made the first harmonic uncertainties locally significantly larger than elsewhere.
7 Conclusions
The quantitative uncertainty was estimated for ship forward speed diffraction problem in short and steep waves. The present flow case is characterised by a strong deformation of the encountered waves on the hull and by rapidly varying longitudinal excitation distribution on the ship bow area.
The presented results show that the uncertainty levels of the force amplitudes vary significantly along the hull. It was also noticed that the energy around the first and the second harmonic components was quite strictly focused on the main components with all the three discretisations. Thus, the use of a larger frequency span did not have a significant effect on the estimated uncertainties. The local uncertainties are poorly presented by the uncertainties of the global quantities. Firstly, in the case of all the observed harmonic components, any constant would represent the local uncertainties poorly because of the large variation. Secondly, in the case of the second and the third harmonic components, the uncertainties of the global quantities are much smaller than most of the local uncertainties.
It was noticed that estimating quantitative uncertainty is challenging in the present case. From the point of view of the current uncertainty estimation approaches, having several nonmonotonically converging results left some uncertainty on the obtained uncertainties. The straightforward solution to this would be to repeat the computations with more discretisation resolutions.
From the point of view of practical application of the present results, their usability can be judged on the basis of the obtained uncertainty levels. In this respect, the conclusions are different for the foremost half and the rearmost half of the observation area. In the foremost half, the uncertainty levels of the first and the second harmonic results are low enough to assess the magnitudes and the ratios of the first and the second force amplitudes. As a practical example, the uncertainties in this area are low enough in order to validate the results against measurements and to take the conclusion whether the selected modelling approach is reasonable for this flow case. In the second half of the observation area, the uncertainties are larger. Even if they are low enough to determine the order of magnitude of the load, which may be sufficient in some cases, decreasing the uncertainty in this area is relevant. Then, the most straightforward task would be to continue refining systematically the discretisation resolutions. However, this could lead to unreasonable computational efforts. Furthermore, it is reasonable to keep in mind that the present results show that the splash behaviour has an important effect on the first harmonic uncertainties. Therefore, we think that one reasonable option is to further study the effect of the spatial discretisation on the splash behaviour. In this connection, the present modelling assumptions, e.g. omitting fluid viscosity and surface tension, should be considered, too.
Acknowledgments
This study was carried out mainly within a research project funded by Tekes, the Finnish Funding Agency for Technology and Innovation, and Aker Yards (now STX Europe). Part of it was done within a research project funded by the Academy of Finland. Some of the work done by the first author was also funded by The Finnish Graduate School in Computational Fluid Dynamics. The financial support is gratefully acknowledged. The computational resources provided by CSC—the Finnish IT Center for Science is also gratefully acknowledged. The authors are thankful to Prof. Michel Visonneau and the CFDteam of ECNCNRS for the discussions and the development of ISISCFD.
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.