fully nonlinear numerical wave tank computations of wave

Fully nonlinear numerical wave tank computations of wave loads on an array of bottom mounted circular cylinders

F. D'Este, R. Codiglia and G. Contento DINMA - University of Trieste, 10 - 34127 Trieste, Italy

Abstract

The aim of this study is to analyse the nonlinear effects in the interaction between regular waves and a 4 column structure by means of the fully nonlinear 3D Numerical Wave Tank method. In particular the attention is focused on the frequency analysis of the results in the ka range where Malenica et al. [ I ] have shown the possibility of occurrence of the second order near trapping phenomenon. The specific case analysed corresponds to a radius to wavelength ratio or ka parameter 4 times smaller than that causing first order near trapping. It will be shown that even in small steepness waves, the Fourier analysis applied to the wave elevation in the zone between the cylinders reveals the presence of large magnification factors of the second harmonic component.

Introduction

The problem of the interaction between waves and an array of circular cylinders has been studied by many authors in the last ten years. The design of the deck height on the calm sea level and the estimate of high frequency loads from relatively short waves on multicolumn structures are the basic motivations of these works. McCamy and Fuchs [Z] derived their analytical solution for a single cylinder in linear diffraction regime. In 1990 Linton and Evans [3] have shown that large magnification factors of the global loads compared to the isolated cylinder case can be achieved for specific ratios of the wave length h to

Transactions on the Built Environment vol 56, © 2001 WIT Press, www.witpress.com, ISSN 1743-3509

292 Fluid Structure Interaction

the cylinder radius a and of the wave length h to the distance between cylinder axis d. Their study is to be applied for cylinders laying on the corner of a regular polygon. In that work the magnification of forces is explained with the occurrence of the near trapping phenomenon. Recently Contento et al. [4] have shown some experimental results on first order near trapping occurring on a 4 column structure in regular waves. The data presented regard the wave elevation and wall pressure at the cylinder surface for a ka parameter approximately equal to 2.0, where k is the incident wave number. It has been shown that in that condition the pressure and the wave amplitude can be magnified by a factor 4 if compared to the isolated cylinder case. Most available results in the literature refer in any case to theoretical studies in the frame of inviscid flow and linearised free surface boundary conditions. Higher order solutions by Scolan & Malenica [5] are said to be still applicable in the frame of small amplitude waves. Malenica et al. [ l ] have shown analytically the possibility of second-order near trapping when the incident wavelength is four times the wavelength causing first order near-trapping. In this paper, we analyse the fully nonlinear behaviour of the interaction between regular waves and a 4 column structure in the situation studied by Malenica et al. [ l ] . It is expected that ultra-harmonic oscillations could take place in the free surface zone between the cylinders, leading to an evident magnification of the double-frequency loads and run-up. The fully nonlinear wavy flow is obtained by the 3D Numerical Wave Tank method. The rest of the paper is organised as follows: the geometric and hydrodynamic properties of the physical problem studied are described first. Afterwards the fundamentals of the mathematical model and the numerical scheme adopted for the simulation are shown. Finally the results obtained are described and discussed thoroughly.

Geometric configuration and hydrodynamic regime

In the following, reference is made to Figure (1). Four bottom mounted circular cylinders with diameter D=2a are placed with their axis at the corner of a square. The side of the square is d. The depth of the sea bed is h. The incoming waves direction corresponds to one of the diagonals of the square. In this specific study the ratio d d is set to 0.275.


Fluid Structure Interaction 293

Incornmg waves direction

Figure 1 Geometric configuration of the structure and reference frames

The physical problem we are facing with is characterised by diameter to wave length ratios D/A ranging between 1 and 6 approximately. Actually, for the specific case analysed here the ka parameter is around 0.5, where k is the incident wave number and a is the cylinder radius. Moreover, since the magnification factor of the higher order terms of the free surface elevation are expected to be large in the zone between the cylinders (from now on called "internal zone"), the incident wave steepness is kept to relatively low levels, i.e. WA = 1/50. According to ChaIcrabarti [6], these hypotheses allow us to study the phenomenon in the frame of incompressible/inviscid fluid with irrotational flow.

Mathematical model and numerical scheme

The fluid flow is described in an inertial frame of reference 0-xyz, the x/y-axis laying on the undisturbed free surface and the z-axis being upwardly oriented. The velocity potential @(X, y,z,t) yields in the fluid domain D and the fluid

velocity is written as F = -V@. The ftee surface profile is assumed to be single valued so that it can be written as z = ~ ( x , y,t) , where V is taken positive upwardly from the still water level. Laplace equation in the unknown velocity potential 4(x, y,z,t) is solved by a simple layer potential approach (Jaswon & Symm, [7]). Fully nonlinear kinematic and dynamic boundary conditions are applied on the free surface. A semi-lagrangian approach is chosen, the markers on the free surface being allowed to move over vertical trajectories only. The following two equations are thus derived (Contento et al, [g]):



v(x, y ) is a suitable artificial damping factor used in the wave absorbing zone

on the free surface to prevent substantial reflection from the vertical walls of the tank where the no cross-flow condition is applied It has been shown in the literature that for regular 2D waves v(x, y = const.) depends on the space variable, on the wave length A and on the frequency m . Usually a quadratic function of the generalised space variable s is used as follows

SO is the starting point of the damping layer, whose length must be at least one

wave-length. Finally p is a fine tuning factor. Here p is assumed equal to 1 . This value has been adopted in a previous application [8] leading to a reflection coefficient CR I 3% in a wide non-dimensional frequency range. Waves are generated at one end of the boundary of the domain D by a piston type wavemaker W extending from above the calm water level down to the bottom of the tank. A Neumann boundary condition is applied on W as follows

where pp is the local velocity of the paddle.

As far as the numerical scheme is concerned, the simple layer potential approach has been appropriately modified to include the symmetry conditions given by the flat bottom of the tank and lateral walls. The solution of the final integral equation has been achieved by a constant quadrilateral panel method (Hem & Smith, [g]). Eqs. ( 1 ) at each node of the free surface are time stepped by a 4th order Runge-Kutta scheme. The boundary of the domain D is subdivided into structured patches with consistent subdivisions at their neighbouring sides. Interpolation, when needed, is performed by a surface spline in tension (Nielsen, [lOl). The free surface is regridded at each sub-step. In principle, the regridding scheme can be chosen arbitrarily and in this specific application we used a constant projection of the mesh in the X-y plane. This is extremely advantageous



when the mixed eulerian-lagrangian approach (vertical trajectories) is used even though accuracy could become poor in zones with a large gradient. The mesh on the cylinders, on the wavemeker and on the vertical wall is derived according to the wetted portion of their surface. No smoothing or filtering is applied. Further details on the numerical scheme, 3D regridding apart, can be found in Contento et al. [8].

Results and discussion

As mentioned before, the results shown here refer to the following hydrodynamic and geometric parameters k ~ 0 . 5 , a e 0 . 2 7 5 and H/il= 1/50. Figure (2a-b) shows the computational domain and the mesh adopted for the simulation (half physical domain only). The wavetank is 6.61 wavelengths long, 1.5 wavelengths half-wide and 0.33 wavelengths deep. The number of panels on the free surface is 3425; the cylinders 1 and 3 are subdivided in 360 surface elements, cylinder 2 in 720 elements and both the wavemaker and the end-wall

are represented with 150 panels. The time step has been set to %O , T being the

incident wave period. Even though it is not strictly consistent with the nonlinear features of the simulation, the motion of the piston wavemaker has been derived from the linear theory (Dean and Dalrymple, [ll]). This assumption seems a reasonable compromise since the incident wave steepness is rather small H/A =

1/50. Figure (3a-d) show a sequence of snapshots taken from the simulation over a whole wave period at the steady state, i.e. &er the wave front generated by the wavemaker has reached the structure and the wave-structure interaction is fully

developed and steady. The time elapsed between two snapshots is %. The

contour plot represents both the free surface elevation and the dynamic pressure on the cylinders' wetted surface. Quantitative results can be obtained from the Fourier analysis of the wave elevation at the grid points. Figure 4a-b show the contour plots of the first and second order components of the free surface elevation in a zone around the cylinders. Specifically the contour variable of the plots is the ratio (or magnification factor) between the Fourier component of the wave elevation at the position examined and the corresponding Fourier component of the undisturbed incident wave. It can be observed that both harmonic components show a large magnification factor in the internal zone. The first Fourier component is magnified up to 1.8 times and obviously its radial and angular distribution is completely different from that of the isolated cylinder (not shown here). In some sense this is just an expected linear result. On the other hand, a much larger magnification factor is found in the second harmonic component. Here we find large peak values in the internal zone mostly at the cylinder 3 weather-side. A large peak is found also at the surface of cylinder 1 and 2 facing cylinder 3. The space distribution of these peaks in the internal zone shows a close correlation with that of first order near trapping for an incident wave four



time shorter than this, i.e. ka=2.0 shown in Figure 5. From this comparison it can be argued that the nonlinearities of the free surface have induced a large ultra-harmonic oscillation in the internal zone and this could be an index of occurrence of a second order near-trapping. The nonlinear behaviour of the free surface has obvious consequences on the pressures and forces acting on the cylinders. Figure 6a-b shows the time series of the X-and y-components of the forces on the cylinders, normalised by the linear in-line force given by MacCamy and Fuchs [2] for the isolated cylinder. As a general comment, the X-component on both cylinders 1 and 3 is approx. 15% higher than in the h e a r isolated case. Moreover the y-component on cylinder 2 is almost 30% of the X-component in the linear isolated case. This lateral y-component exhibits a strongly nonlinear behaviour (Figure 6b). Less evident is the harmonic composition of the xcomponent on the cylinders 1 and 3. In Figure 7 the first and second Fowier components of the forces (X- and y- components) are shown. Even in small steepness waves like this ( W A = 1/50), the free surface induced nonlinearities make the in-line term of the second order force on cylinder 3 approximately 10% of the first order term and mostly the second order y-component on cylinder 2 is more than 30% of the first order y- component. The presence of near trapping is witnessed by the fact that the second order X-component on cylinder 3 and y-component on cylinder 2 have similar values. Indeed this phenomenon has been already observed at first order for ka=2.0 (Contento et al. [4]).

Conclusions

In this paper the ultra-harmonic behaviour of the free surface elevation in the interaction of a train of regular waves with a four column - bottom mounted structure has been studied. The analysis has been conducted by means of a numerical simulation based on the inviscid fluidhrotational flow assumptions. The fully nonlinear free surface conditions have been applied. The results obtained follow the indications given by other authors (Malenica et al. [l]) in the sense that they give evidence of large ultra-harmonic oscillations induced by long waves. Specifically the condition here analysed corresponds to wavelengths four times longer than those exciting the first order near trapping (Linton and Evans [3]; Contento et al. [4]). Further investigations are obviously in progress to clearly capture the nonlinear near-trapping showing the feature of the nonlinear wave-structure interaction as a function of the wave steepness and ka parameter.



Acknowledgements

This research has been supported by MURST 60%, 2000, Project: "Studio nel dominio del tempo dell'interazione onde-strutture marine in presenza di fiontiera libera non lineare" and by CETENA SpA - Research plan 2000-2002.

References

Malenica, S., Eatock Taylor, R., Huang, J.B., 1999, 'Second order water wave diffrizction by an array of vertical cylinders', Journal of Fluid Mechanics, Vol. 390. Mac Camy, R.C., Fuchs, R.A., 1954, 'Wave Forces on Piles: A diffraction Theory', U.S. Army Corps of Engineering, Beach Erosion Board, Tech. Memo, No. 69, Washington D.C. Linton, C.M., Evans, D.V., 1990, 'The interaction of waves with arrays of vertical circular cylinders', Journal of Fluid Mechanics, Vol. 215, pp. 549- 569. Contento, G., D'Este, F., Calcagno, G., Penna, R., 2000, Experimental Evidence of Near Trapping in an Array of Bottom Mounted Circular Cylinder in Regular Waves, Int. Conference of Offshore and Polar Engineering - ISOPE, The Int. Society of Offshore and Polar Engineers, Seattle, Maggio-Giugno 2000, Vol. 3, pp. 281-287. Scolan, Y.M., Malenica, S., 1998, 'Experimental and numerical second order difhcted waves around an array of 4 cylinders', Proceeding, Int. Workshop on Water Waves and Floating BodiesJ98. Chakrabarti, S.K., 1990, Nonlinear Methods in Offshore Engineering, Developments in Marine Technology, Elsevier Sc. Jaswon, M.A., Symm, G.T., 1977, Integral equation methods in potential theory and elastostatics, Academic Press Inc, London. Contento, G., Codiglia, R., D'Este, F., 2001, Nonlinear effects in transient non-breaking waves in a closed basin, accepted for publication on Int. Jou. Applied Ocean Research. Hess, J.L., Smith, A.M.O., 1962, Calculation of non-lifting potential flow about arbitrary three-dimensional bodies, Report No. E.S. 40622, Douglas Aircraft Co., Long, Beach.

[l01 Nielsen, G.M., Franke, R., 1984, A method for construction of surfaces under tension, Rocky Mountain Journal of mathematics, Vol. 14, No. 1, pp. 203-221.

[l l ] Dean, R.G., Dalrymple, R.A., 1984, Water waves mechanics for scientists and engineers, Prentice Hall, Englewood Cliffs, New Jersey.



Figure 2a Physical domain and mesh used.

Figure 2b Details of the grid at the cylinder surface.



Figure 3 Snapshots of the free surface during the simulation. The time lag between them is T/4.


Fluid Structure Interaction 30 1

Figure 4a-b Contour plot of the first (a) and second (b) Fourier term of the free surface elevation normalised by the corresponding Stokes' component of the incident wave (ka=0.5).



Figure 5 Contour plot of the amplitude of the free surface elevation from a linear simulation with ka=2.0 (amplitude not normalised).

Figure 6a In-line force on the cylinders (1: thick solid line; 2: thin solid line; 3: dash-dotted line) vs time and normalised by the linear in-line force given by MacCarny and Fuchs for the isolated cylinder.



I l l l

-1.w l 1 1 I I I I I I I I 1 ... .

15 16 17 18 19 20

tlT Figure 6b y-component of the force acting on the cylinder 2 vs time and

normalised by the linear in-line force given by MacCamy and Fuchs for the isolated cylinder.

1.60 Q)

2 2 econd Fourier component Q)

c, k 0 1.20 m u 8 C 0 a E 00 0.80

z *g g

0.40

3 € ::

0.00 Fx - cylinder 1 Fx - cylinder 2 Fx - cylinder 3 Fy - cylinder 2

Figure 7 First and second Fourier components of the force acting on the cylinders. Normalisation is based on the MacCamy and Fuchs solution for the isolated cylinder.


fully nonlinear numerical wave tank computations of wave

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