accuracy was verified through convergence test, mass and energy conservations .... The integration domain includes the free surface, the bot- tom surface, the ...
APOR 354
Applied Ocean Research 20 (1998) 309–321
Fully nonlinear interactions of waves with a three-dimensional body in uniform currents M.H. Kim, M.S. Celebi, D.J. Kim Department of Civil Engineering, Offshore Technology Research Center, Texas A&M University, College Station, TX 77843-3136, USA Received for publication 18 May 1998
Abstract Fully nonlinear wave interactions with a three-dimensional body in the presence of steady uniform currents are studied using a Numerical Wave Tank (NWT). The fully nonlinear NWT simulations are compared with perturbation-based time-domain solutions. The threedimensional NWT uses an indirect Desingularized Boundary Integral Equation Method (DBIEM) and a Mixed Eulerian–Lagrangian (MEL) time marching scheme. The Laplace equation is solved at each time step and the fully nonlinear free surface boundary conditions are integrated with time to update its position and boundary values. A regridding algorithm is devised to eliminate the possible saw-tooth instabilities. The incident waves are generated by a piston-type wavemaker and the current is introduced in the whole fluid domain at the start of simulations. The outgoing waves are dissipated inside a damping zone by using spatially varying artificial damping on the free surface. Computations are performed for the nonlinear diffractions of steep monochromatic waves by a truncated vertical cylinder in the presence of uniform coplanar or adverse currents. The NWT simulations are also compared favorably with the experimental results of Mercier and Niedzwecki [1] and Moe [2]. q 1998 Elsevier Science Ltd. All rights reserved. Keywords: Numerical Wave Tank; Three dimensional; Wave-current-body interactions; Non linear diffractions
1. Introduction Accurate predictions of wave loads and run-up on large offshore or coastal structures in combined wave and currents are of practical importance in the design and operation of these structures. When waves and currents coexist, the overall diffraction pattern and the resultant loading can be significantly different from the wave-only case. Most recent publications on this topic have been based on the perturbation-based potential theory in the frequency domain (e.g. Nossen et al. [3], Emmerhoff and Sclavounos [4], Teng and Eatock Taylor [5], Zhao et al. [6]). Alternative timedomain solutions have also been obtained in the context of linearized theory by Isaacson and Cheung [7], Prins [8], and Kim and Kim [9]. In particular, Kim and Kim [9] used a THOBEM (Time-domain Higher-Order Boundary Element Method) to compute the spatial derivatives of velocity potentials more accurately. The time-domain analysis is in general mathematically much simpler and can be more easily extended to fully nonlinear problems compared to the frequency-domain analysis. The perturbation approach in general neglects the short ship–wave systems generated by currents, and thus may not accurately predict the run-up around a structure as current speed increases. Until now, the
study of fully nonlinear wave–current–body interactions in the time domain is very limited (e.g. Ferrant [10]). The major difficulties associated with the nonlinear freesurface simulations are (i) the complicated nonlinear free surface boundary conditions that have to be satisfied on the instantaneous free surface not known a priori, (ii) various types of numerical instabilities, (iii) appropriate numerical open-boundary conditions which can simulate open-sea conditions, and (iv) substantial CPU time unless some approximations are used. The use of fully nonlinear timedependent free surface boundary conditions for twodimensional surface waves was first introduced by Longuet-Higgins and Cokelet [11], who adopted a Mixed Eulerian–Lagrangian (MEL) time stepping technique. The two-dimensional MEL method was subsequently extended by Dommermuth and Yue [12] to study the motion-induced nonlinear waves by axisymmetric bodies. Later, Cao, Schultz and Beck [13][14] and Cao, Lee and Beck [15] used similar time-stepping techniques and the so-called Desingularized Boundary Integral Equation Method (DBIEM) to study various nonlinear free-surface problems including the nonlinear waves generated by free-surface pressure disturbances. Alternatively, Yang and Ertekin [16] and Boo et al. [17] used a Eulerian approach for nonlinear
0141-1187/98/$ - see front matter q 1998 Elsevier Science Ltd. All rights reserved PII: S0 14 1 -1 1 87 ( 98 ) 00 0 25 - X
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NWT simulations. Recently, Celebi et al. [18][19] investigated fully nonlinear wave interactions with a stationary vertical cylinder in a three-dimensional NWT with side and end beaches and their numerical results for a series of higher harmonics compared reasonably with Mercier and Niedzwecki’s [1] experiments. In this paper, the MEL time-stepping technique and the DBIEM are used to investigate the fully nonlinear diffraction of stationary three-dimensional bodies by steep regular waves and steady uniform currents inside a numerical wave tank equipped with a wavemaker and end beach. The numerical wavemaker and end beach are designed to allow the passage of currents and maintain the volume of water constant inside the NWT. The fully nonlinear simulations with currents are validated through comparison with the perturbation approach of Kim and Kim [9]. It is seen that the fully nonlinear simulations become close to the linear results when wave steepness decreases. For robust simulations of fully nonlinear interactions between waves, currents and three-dimensional bodies, a number of important problems need to be resolved including (i) robust numerical implementation of a wavemaker and a far-field closure, (ii) robust treatment of the body and free surface intersection line, (iii) development of a stable numerical algorithm for the integration of the free surface, and (iv) development of an efficient matrix solver for large, dense and unsymmetrical matrices. The robust numerical implementation of the upstream and downstream boundary conditions is of great importance for the overall success of fully nonlinear numerical wave tank simulations. In many cases, the limitation in memory space and CPU time forces us to use a relatively small computational domain with a transparent downstream boundary. A number of researchers have suggested different ideas and there exist pros and cons for each method compared to the others. The frequently used methods are the Orlanski condition (e.g. Orlanski [20], Yeung et al. [21], Boo et al. [17], Contento et al. [22]), matching to simplified outer solutions (e.g. Dommermuth and Yue [12]), periodic boundary conditions (e.g. Longuet-Higgins and Cokelet [11]), absorbing beach (or artificial damping, sponge layer) (e.g. Israeli and Orzsag [23], Baker et al. [24]), active absorber by a piston (e.g. Clement [25]), and piston–beach hybrid absorber (e.g. Clement [25]). In this paper, an optimized, adaptive f n-type numerical beach (artificial damping) is used. The time marching of the nonlinear free surface conditions was carried out by using a robust fourth/fifth-order Runge–Kutta–Fehlberg scheme. The possible saw-tooth instability on the free surface or similar numerical instabilities near the free surface/body intersection line were controlled by a specially devised regridding scheme. When this technique is used, it was found that no additional smoothing techniques were necessary. In particular, the desingularization technique is known to be robust in dealing with the region of the confluence of boundary conditions (Beck
[26]), such as free-surface/body intersection lines. To solve the matrix equation at each time step, a specially devised domain-decomposition iterative technique is used, which performs well, especially for the diagonally dominant large matrix. The matrix is partitioned into many submatrices and the unknowns on each row are updated simultaneously using the line Jacobi method. The details are given in Celebi et al. [19] and Celebi [27]. When proper discretizations are used, the developed numerical wave tank computer program is robust, accurate and free of numerical instabilities, except when incident waves are too steep or current velocities are too large. Its accuracy was verified through convergence test, mass and energy conservations, and comparisons with the experiments of Mercier and Niedzwecki [1] and Moe [2] as well as perturbation-based computations (Kim and Yue [28], Kim and Kim [9]). It is seen that the present method can reliably produce nonlinear free surface elevation, run-up, pressure and forces including a spectrum of higher harmonics. As current speed increases, the flow separation will be developed around a body and the potential theory becomes less meaningful. As an indication of the applicability of the potential theory, flow separation effects may be neglected when the relevant Keulegan–Carpenter (KC) number is small (say less than 2), which is also partly verified by comparing the potential NWT with an independently developed viscous-flow-based NWT (Kim et al. [29]).
2. Fully nonlinear simulations 2.1. Governing equation and boundary conditions An ideal, irrotational fluid is assumed so that a velocity potential F(x,y,z,t) exists and the fluid velocity is given by its gradient. It is also assumed that the surface tension on the free surface can be neglected. A Cartesian coordinate system (Fig. 1) is chosen such that the z ¼ 0 plane corresponds to the calm water level and z is positive upwards. The total velocity potential F including steady uniform current U can then be expressed as F ¼ Ux þ f(x, y, z, t)
(1)
The velocity potentials F and f describe fluid motions and they satisfy the Laplace equation in the computational domain: =2 F ¼ =2 f ¼ 0
(2)
On the instantaneous free surface h(x,y,t), both the kinematic and dynamic boundary conditions must be satisfied. The kinematic free surface boundary condition requires that the velocity of the fluid and the free surface should be equal: ]h ]F ¼ ¹ =F·=h þ on SF ]t ]z
(3)
The dynamic free surface condition requires that the
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Fig. 1. A schematic representation of a numerical wave tank.
pressure on the free surface should be the same as the ambient pressure P a: 2 ]F 1 P ¼ ¹ gh ¹ =F ¹ a on SF (4) ]t 2 r where r is the fluid density and g the gravitational acceleration. In addition, the zero-normal-flux condition at the bottom, wall and body boundaries as well as the proper radiation condition at far field has to be satisfied. In the MEL approach, a time stepping procedure is used in which a mixed boundary value problem is solved at each time step. At each step, the value of the potential is given on the free surface (Dirichlet boundary condition) and the value of the normal derivative of the velocity potential (Neumann boundary condition) is known on the body, wall, wavemaker and bottom surfaces. After the mixed boundary value problem is solved, the free surface elevation, potential and its normal derivative are updated. For the time integration, a Runge–Kutta–Fehlberg (fourth/fifth-order) method was used and proved to be stable and accurate. The most common approach to time march the free surface boundary conditions in fully nonlinear simulations is the material node approach in which the nodes or collocation points follow the individual fluid particles, i.e. DXF (t) ¼ Ui þ =f Dt
(5)
Assuming that P a ¼ 0, one obtains Df 1 ¼ ¹ gh þ =f·=f Dt 2
(6)
where XF ðtÞ ¼ ðxF ðtÞ; yF (t),z F(t)) is the position vector of a fluid particle on the free surface and D ] ¼ þ =F·= (7) Dt ]t is the usual material derivative. The use of the above kinematic and dynamic free surface conditions allows the value of the elevation and potential to be stepped forward in time. Using the desingularized method, the spatial derivative of the velocity potential =f can be determined analytically after solving the boundary value problem for f, which is further explained in the next section. 2.2. Indirect desingularized boundary integral equation method The mixed boundary value problem must be solved at each time step for the unknown velocity potential f. With the given potential on the free surface and the known normal velocity on the body, wall, wavemaker and bottom surfaces, the potential at any point in the fluid domain is given by the distribution of Rankine sources: ZZ 1 )dQ (8) j(xs )( f(x) ¼ Q x ¹ xs where Q is the integration surface outside the fluid domain and j is the source strength to be determined. Applying the relevant boundary conditions, the desingularized indirect boundary integral equations that must be solved to
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determine the unknown source strengths are ZZ 1 )dQ ¼ fo (xc ) (xc [ Gd ) j(xs )( Q xc ¹ xs and ZZ
j(xs )
Q
] 1 )dQ ¼ x(xc ) (xc [ Gn ) ( ]n xc ¹ xs
(9)
(10)
where x s ¼ (x9,y9,z9): a source point on the integration surface, x c ¼ (x,y,z): a field point on the real boundary, f 0 ¼ the given potential value at x c, G d ¼ surface on which f 0 is given, x ¼ the given normal velocity at x c, G n ¼ surface on which x is given. The integration domain includes the free surface, the bottom surface, the body surface, the wavemaker, the side wall and the truncation surfaces. The discretized version of the above integral equation is solved for j by an iterative solver. Subsequently, f can be evaluated by Eq. (8). Its spatial derivative =f can be obtained from the derivative of Eq. (8), where the differentiation of the Rankine source can be done analytically. In the case of the flat bottom, image sources can be used to eliminate the integration over the bottom surface. In the case of the numerical wave tank simulation with two vertical side walls, an infinite array of image sources can alternatively be used to simulate the side-wall effects. For convenience, it is assumed that the water depth is constant and the body is symmetric with respect to the xz plane. Then, we can use image sources with respect to the bottom (z ¼ ¹ h) and the symmetry plane (y ¼ 0), and the integration surfaces can be reduced to half fluid domain. In this case, the Green function is given by 1 1 1 1 þ þ þ (11) G¼ R1 R2 R3 R4 where q R1 ¼ (x ¹ x9)2 þ (y ¹ y9)2 þ (z ¹ z9)2 R2 ¼
q (x ¹ x9)2 þ (y þ y9)2 þ (z ¹ z9)2
R3 ¼
q (x ¹ x9)2 þ (y ¹ y9)2 þ (z þ z9 þ 2h)2
R4 ¼
q (x ¹ x9)2 þ (y þ y9)2 þ (z þ z9 þ 2h)2
(12)
In the desingularized method, the source distribution is outside the fluid domain so that the source points never correspond to the field (or collocation) points, and the resulting integrals are non-singular and can be straightforwardly integrated by numerical quadratures. The isolated sources are distributed a small distance above each of the nodes. The desingularized distance is in general given by L d ¼ l d(D m) b9, where D m represents the local mesh size and l d and b9 are the constant parameters to be selected. If the singularity is located too far from the boundary, the resulting influence coefficient matrix is likely to be poorly conditioned, which results in slow convergence in solving the matrix iteratively. If the singularity is too close to the boundary, the numerical integration of the kernel may not be robust. Therefore, appropriate l d and b9 values need to be determined. In this paper, b9 ¼ 0.5 and l d ¼ 0.5,1 (depending on wavelength) are chosen after numerical testing. A detailed study with regard to the performance of DBIEM with the desingularization parameters is reported in Cao et al. [14]. The hydrodynamic forces and moments on the body can be calculated by integrating the pressure over the instantaneous wetted body surface ZZ PnB dS (13) F¼ SB
and ZZ M¼
P(r 3 nB ) dS
(14)
SB
where nB ¼ (n1 , n2 , n3 ) r 3 nB ¼ (n4 , n5 , n6 ) r and n B are the position vector and unit normal vector (out of fluid) of the body surface, respectively. The pressure on the body surface is given by the following Bernoulli’s equation P 1 df ]f ¼ ¹ gz ¹ =f·=f ¹ þ v:=f ¹ U r 2 dt ]x
(15)
where df dt is the time derivative of the velocity potential following a moving node on the body and v is the velocity of the node relative to the Oxyz system. A numerical differentiation (e.g. backward differencing) for df dt can lead to poor estimates of the derivative and possible instabilities. Therefore, in this paper, a new boundary value problem df (BVP) was set up for df dt using the given dt on the free ] df surface and ]n( dt ) ¼ 0 on the body surface. The df dt on the free surface was calculated from the dynamic free-surface condition. The time derivative of the velocity potential was then obtained by solving the second matrix equation at each time step with the same influence coefficient matrix. This scheme can also be directly applied to the floating-body simulation.
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2.3. Saw-tooth instability and regridding Longuet-Higgins and Cokelet [11] first found the socalled saw-tooth instability during their two-dimensional breaking wave simulations. They employed a smoothing technique to suppress the development of the saw-tooth instability. Later researchers pointed out that the possible cause of the instability was the concentration of Lagrangian points in the region of higher gradients and the minimum grid size cannot be effectively controlled for given time step. The time evolving deformation of the Lagrangian grid system in general results in the numerical degradation and eventual breakdown. In this regard, a regridding algorithm similar to that of Dommermuth and Yue [12] is used to eliminate such instabilities. At each time step, the distorted material nodes tracing water particles are curve-fitted by applying B-spline curves in both the x and y directions. After determining the coefficients of the B-spline, a new set of uniformly spaced Lagrangian points are created on the free surface and body surface. Subsequently, a linear interpolation technique is employed to redistribute the boundary values (velocity potentials and their derivatives) on the new set of nodal points. Then, the time stepping is continued with the fourth/fifth-order Runge–Kutta– Fehlberg time-integration scheme. The same regridding scheme was also applied at the body-free surface intersection line. Using this algorithm, the instabilities are effectively removed and no artificial smoothing is required. The present material node approach and regridding scheme can be successfully applied as long as the current speed is not large. If current speed is large, the horizontalmovement-of-nodes technique is more appropriate to avoid crossing of Lagrangian points. The details are given in Celebi [27]. The main disadvantage of regridding is the potential loss of resolution which is usually provided by more closely spaced Lagrangian points in the areas of large gradients. The advantages of regridding over artificial smoothing particularly in the present context are, however, substantial in that the smoothing cannot be straightforwardly applied at the body–free surface intersection line and the crossing of Lagrangian points can be more easily controlled. 2.4. Wave generation and absorption The incoming waves are generated by the prescribed motion of a piston-type wavemaker. In the beginning, the wavemaker velocity is gradually increased by using a linear ramp function. The wavemaker condition can easily produce large-amplitude nonlinear incident waves. It can also generate large asymmetric or breaking waves by controlling the wavemaker motion. The uniform current is then introduced in the entire fluid field to generate the wave-current environment. The end beach is designed to allow the passage of currents. The fluid particles (the nodes) on the wavemaker are
313
forced to move with the wavemaker in the normal direction, while the particles are allowed to move freely in the other directions. Fully nonlinear free surface boundary conditions are applied along the free surface and wavemaker intersection line. During the simulation, the intersection line is updated and regrided like the body waterline, which is shown to be effective in preventing possible numerical instability there. Inside the downstream damping zone, the artificial damping is applied only to the dynamic free-surface condition so that waves are damped out as much as possible before they reach the truncation boundary. On the truncation boundary, only uniform currents are passed through. Israeli and Orzsag [23] showed that a broader spectrum of waves can be effectively damped by this kind of beach and the absorbing efficiency can be improved by allowing the damping coefficient to be gradually increased in the direction of wave propagation. Both f- and f n-type beaches have been tested, and the following f n-type beach was shown to be the most efficacious and subsequently used: Df 1 ]f ¼ ¹ gh þ =f·=f ¹ n (16) Dt 2 ]n where n is a damping coefficient. The use of the f n-type beach ensures the energy absorption rate is always positive. Our numerical experiments show that the choice of proper damping coefficient n(x,y) is important to achieve maximum wave dissipation. This coefficient is set to be zero outside the damping zone. Inside the damping zone, it is chosen to be smooth and gradually increasing. On the transition boundary between the damping zone and the inner solution domain, C 2 or higher continuity have to be satisfied to guarantee high performance. The damping coefficient used in the present NWT calculation is defined as ( ) 0 ¹ L # x # x0 n(x) ¼ n0 [1 ¹ cos p((x ¹ x0 )=(2L ¹ x0 ))] x0 , x # L (17) where 2L is the length of the tank and x 0 is the starting point of the damping zone. In addition, the performance of the artificial beach depends on the ratio of the beach length to wave length. After a series of tests, it is found that the beach length has to be at least one wave length to guarantee minimal reflection. In view of this, the length of the damping zone is adjusted depending on the pertinent wave length. 3. Perturbation method The problem formulated in the preceding section is fully nonlinear. Solving this fully nonlinear problem directly in the time domain requires tremendous computing time because the influence matrix should be set up and inverted at each time step as the free surface moves to a new position. Assuming small wave amplitudes, a perturbation procedure may be introduced so as to make this problem more
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manageable. In other words, all the physical variables are perturbed with respect to the mean free surface so that the computational domain can be fixed. Then, the calculation of influence matrix coefficients is required only once and it can be used repeatedly for all time steps and different wave conditions, which makes the current approach efficient and more computationally manageable. By introducing two independent small parameters, i.e. the wave steepness parameter e ¼ H/l (wave height/wave length) andpthe current speed parameter (or Froude number F n) d ¼ U= gl, where l ¼ characteristic body length, the velocity potential and wave elevation can be expressed by the following perturbation series:
and
where f¯ is the zeroth-order steady disturbance potential and f (1) and f (2) the time-dependent first-order and second-order potentials. Using Taylor expansion about the mean free surface, a series of new boundary value problems at each order can be obtained and the terms higher than O(e,ed) and O(d) are neglected under the assumption that both parameters are small (Nossen et al. [3], Emmerhoff and Sclavounos [4]). The boundary value problem for the steady disturbance potential f¯ can be obtained by collecting terms of order O(d). Subsequently, the boundary value problem for the first-order unsteady wave potential f (1) can be obtained by collecting terms of order O(e) and O(ed), i.e. up to the first order in wave steepness. The numerical solutions can then be obtained in the time domain by the direct application of the Green theorem with Rankine sources. In this paper, the higher-order boundary element method (HOBEM) is used in preference to the constant panel method to compute the second-order spatial derivatives of f¯ more accurately. The procedure for the numerical implementation of the above boundary value problems in the time domain using HOBEM is detailed in Kim and Kim [9].
computations are compared with the perturbation results (or experimental results, if available) for various wave steepness and current speeds. For the perturbation method, 288 nine-node quadratic boundary elements are used in the half computational domain (Fig. 2). More specifically, 144 elements are used on the free surface, 48 elements on the body surface, and 96 elements on the open boundary. The total number of nodes used is 1318. On each element, quadratic interpolation functions are used to describe the variation of velocity potentials. After convergence testing and comparison with analytical solutions, it is shown that the force results can be obtained with less than 1% error for given discretizations and time step Dt ¼ T/100. For the present fully nonlinear free surface calculations, the numerical error can be controlled by the number of node points on the solution domain, the desingularization distance, the error tolerances for the Runge–Kutta–Fehlberg integration scheme and the iterative matrix solver, and beach performance parameters, such as beach length and damping coefficient. The desingularization distance was selected as explained in the preceding section. The time step Dt ¼ 0.1 s was selected based on the test for the shortest wave and used for other frequencies. To check the convergence with the number of nodes, three different free-surface discretizations, N F ¼ 1246, 2420 and 3840, were used, while the nodes on the other boundaries remain fixed. qThe convergence rate was measured by P Erms ¼ N1T Ni ¼T 1 (jiM ¹ jiM ¹ 1 )2 , where N T is the total number of nodes and M denotes the number of iterations. For M ¼ 12; E rms ¼ 10 ¹3, 10 ¹6 and 10 ¹8, respectively for NF ¼ 1246, 2420 and 3840, which shows that more accurate results can be obtained with finer discretization. For the ensuing examples, 2420 nodes were used to discretize the free surface, 186 nodes on the body, 175 nodes on the wavemaker, 175 nodes on the end wall and 330 nodes on the side wall in a half domain. For the selected spatial and temporal discretizations and the open-boundary scheme used, it is shown that the mass and energy are
4. Numerical results and discussions To demonstrate the usefulness and accuracy of the numerical wave tanks for wave–current–body interactions, we present results here for nonlinear diffractions by a truncated vertical cylinder inside a rectangular tank with side walls (Fig. 1) in the presence of monochromatic waves and uniform coplanar or adverse currents. The fully nonlinear
Fig. 2. Schematic view of truncated cylinder.
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Fig. 3. (a) Wave probe (x, y) ¼ ( L4, 0), w ¼ 2:0 rad/sec., l ¼ 15:4 m. (b) Wave probe (x; yÞ ¼ ð0:95L;0), w ¼ 2.0 rad/sec., l ¼ 15.4 m.
conserved within 2% error (Celebi et al. [19]) for typical wave periods and wave slopes used here. Fig. 3 shows the free-surface profiles generated by a harmonic wavemaker motion at two different locations (total tank length 2L). The waves shown in Fig. 3(a) exhibit typically nonlinear features, such as higher crests and smaller troughs. The wave profile is only slightly contaminated by the small reflection of transient long waves. To completely
suppress those waves, a longer beach is needed. Near the end of the damping zone, we can see in Fig. 3(b) that the wave height is sufficiently reduced by viscous dissipation. In the following, the nonlinear NWT simulations are carried out for a stationary vertical truncated cylinder (radius a ¼ 0:23 m, draft p ¼ 1.33 m in model scale), and the results for Fn ¼ U= ga ¼ 0 are compared with Mercier and Niedzwecki’s [1] experiments conducted in the OTRC
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Fig. 4. First-order weather (v ¼ p) side run-up on a uniform vertical truncated cylinder for F n ¼ 0.
Fig. 6. Third-order weather (v ¼ p) side run-up on a uniform vertical truncated cylinder for F n ¼ 0.
Fig. 5. Second-order weather (v ¼ p) side run-up on a uniform vertical truncated cylinder for F n ¼ 0.
Fig. 7. Third-order horizontal force for a vertical truncated cylinder.
wave tank at Texas A&M University. For F n Þ 0, the NWT simulations are compared with the perturbation results. The principal dimensions of the NWT used for this simulation are depth/draft ¼ 4.2, tank length (without damping zone)/
depth ¼ 1.2, and tank width/depth ¼ 1.5. The cylinder was located at the center of the NWT. For the NWT simulations, the incident waves were generated by a harmonic motion of the piston-type wavemaker. The wave condition used in Mercier and Niedzwecki’s experiments was e9 ¼ H/gT 2 ¼
Table 1 First and second harmonic surge forces and pitch moments Surge force (N)
Second-order diffraction DBIEM–NWT
Pitch moment (N m)
First harmonic
Second harmonic
First harmonic
Second harmonic
112.2 113.2
1.5 3.7
44.8 46.1
1.5 2.1
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317
Fig. 8. Weather-side run-up for a truncated vertical cylinder by perturbation theory.
0.0025, 0.005 and 0.0075, which corresponds to wave slopes H/l ¼ 0.016, 0.031 and 0.047. In both NWT simulations and the experiments, the wave steepness was maintained constant for all wave frequencies. To quantitatively observe the nonlinear contributions, a series of higher harmonics were generated by an FFT algorithm. The length of time series used for FFT was 18T. In Figs 4–6, the first three harmonics of non-dimensional wave run-up (where A ¼ H/2) at the weather side (v ¼ 1808) are presented for two different wave steepness, e9 ¼ 0.005 and 0.0075. The experimental results were read from Mercier and Niedzwecki [1]. The NWT computations generally agree well with experiments for the first harmonic (wave frequency) component. As for the second and third harmonic components, which are very small compared to the first harmonic component, the trend looks similar but there is some discrepancy in magnitude between the predicted and
measured values. It is pointed out in Mercier and Niedzwecki [1] that the actual wave heights used for the high-frequency cases are very small, and thus the corresponding measurements of higher harmonics are expected to be less accurate. The discrepancy between the NWT simulations and experiments can also be attributed to the difference in incident waves and viscous effects. It is interesting to see in both experiments and NWT simulations that the dimensionless second- and third-order run-ups are smaller in steeper waves, which cannot be predicted by the perturbation theory. The first and second harmonic surge forces and pitch moments (with respect to the center of waterplane) on the same vertical cylinder obtained from the NWT simulations are also compared in Table 1 with those calculated from the second-order diffraction theory of Kim and Yue [28]. There is a discrepancy in second harmonic quantities but the
Fig. 9. Weather-side run-up for a truncated vertical cylinder by NWT.
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Table 2 First, second and third harmonic run-up
(1)
v¼0
v ¼ p/2
v¼p
h /(H/2)
F n ¼ 0.00 F n ¼ 0.02 F n ¼ 0.05
0.87 0.89 0.81
1.43 1.40 1.39
2.06 2.26 2.46
h (2)/(H 2/4a)
F n ¼ 0.00 F n ¼ 0.02 F n ¼ 0.05
0.53 0.37 0.33
0.67 0.60 0.63
1.14 0.76 0.92
h (3)/(H 3/8a 2)
F n ¼ 0.00 F n ¼ 0.02 F n ¼ 0.05
0.36 0.33 0.33
0.65 0.53 0.57
0.91 0.77 1.13
overall agreement between the two looks reasonable. To further confirm the accuracy of higher harmonics, the simulated third harmonic horizontal forces are compared in Fig. 7 with the theoretical and experimental results presented in Malenica and Molin [30], where the theoretical results (third-order semi-analytical solutions) are for draft/radius ¼ 10 and the experimental results of Moe [2] are for draft/radius ¼ 3. The trend and magnitude of the present NWT simulations correlate well with the other results. The reasonable prediction of third and higher harmonic forces and moments are important for some applications, such as the ringing of a tension leg platform (TLP). Next, the effects of uniform currents on the weather-side run-up are investigated for e9 ¼ 0.0075 and q ¼ 5.1 rad/s (in model scale or q 2a/g ¼ 0.61). In the following, positive Froude number corresponds to coplanar currents. Fig. 8 shows perturbation results and Fig. 9 fully nonlinear simulations. From these long time histories, we can see that the open boundary condition works well and wave reflection is very small. The shift of wave periods (or wave frequencies) in currents is due to the Doppler effect. It is seen in both figures that the run-up amplitude is increased in coplanar currents and decreased in adverse currents, which is also shown by Ferrant [10]. The rates of increase and decrease are greater in fully nonlinear simulations. The change of the first three harmonics of run-up at three different locations due to uniform coplanar currents is summarized in Table 2. For all harmonics, the magnitudes at the weather side are much greater than those at the lee side. The variation of runup as a function of azimuthal angle (q ¼ 5.1 rad/s) can be observed more clearly in Fig. 10(a) and (b). Fig. 10(a) is for F n ¼ 0 and e9 ¼ 0.0075 and Fig. 10(b) for F n ¼ 0.02 and e9 ¼ 0:0025 and 0.0075. In both figures, the NWT results are compared with perturbation results. The NWT results generally give larger values except near the lee side and migrate toward perturbation results as the wave steepness decreases. In Figs 11 and 12, the horizontal forces on the vertical cylinder for e9 ¼ 0.0075 and q ¼ 5.1 rad/s are plotted for three different current speeds, F n ¼ ¹ 0.02, 0 and 0.02. The trend of NWT and perturbation results is in general similar and the NWT results show greater influence by currents.
Fig. 10. (a) Maximum wave run-ups on a truncated vertical cylinder. (b) Maximum wave run-ups on a truncated vertical cylinder.
Fig. 13 shows the change of the first three harmonics of horizontal forces with various current velocities. The first harmonic components monotonically increase with Froude number and the rate of increase/decrease is larger than that of perturbation theory. The difference between NWT and perturbation results is reduced as F n decreases. It can also be seen that higher harmonic forces are more sensitive to the change in F n. Interestingly, the third harmonic components increase in both coplanar and adverse currents. Table 3 shows the change in the first harmonic forces, moments and run-up with wave steepness when F n ¼ 0.02. As expected, the NWT results become close to the perturbation results as wave slope decreases. Table 4 also shows the
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Fig. 11. Horizontal force in adverse and coplanar currents by perturbation theory.
Fig. 12. Horizontal force in adverse and coplanar currents by NWT.
second and third harmonic forces and moments normalized by wave amplitude squared and cubed for two wave slopes. We can see that the present nonlinear simulations produce different values for different wave slopes, which cannot be predicted by the perturbation theory. Finally, Fig. 14 presents the wave run-up around a bottom-mounted vertical cylinder in uniform coplanar currents and a monochromatic incident wave of ka ¼ 0.374, kH ¼ 0:402 and F n ¼ 1.036. The current speed used in this example is F n ¼ 0.083. The first-order and second-order numerical results are taken from Buchmann et al. [31]. It can be observed that the second-order results generally correlate better with the NWT simulations than the first-order results. The second- and higher-order corrections on maximum wave run-up in this case are shown to be appreciable (up to 50%).
5. Concluding remarks The three-dimensional fully nonlinear wave–body interactions in steady uniform currents are studied in a numerical wave tank using a desingularized boundary integral equation method (DBIEM) and mixed Eulerian and Lagrangian (MEL) time marching scheme. The fully nonlinear kinematic and dynamic free-surface conditions were integrated with time by the fourth/fifth-order Runge–Kutta–Fehlberg method. The incident waves were generated by a piston-type wavemaker and the waves downstream were dissipated by the adaptive f n-type damping layer on the free surface inside the damping zone. The uniform current is introduced in the entire fluid and the boundary conditions at the wavemaker, truncation boundary downstream and the body are modified accordingly. The
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Fig. 13. Horizontal force components for vertical truncated cylinder.
Table 3 First harmonic wave forces and run-up for F n ¼ 0.02 and q ¼ 5.1 (rad/s)
F x (1)/rga 2A F z (1)/rga 2A M y (1)/rga 3A h (1)/A(v ¼ 0) h (1)/A(v ¼ p/2) h (1)/A(v ¼ p)
THOBEM
NWT («9 ¼ 0.0025) NWT («9 ¼ 0.0075)
5.87 0.06 8.29 1.01 0.94 1.68
5.91 0.06 8.53 0.95 1.32 1.98
6.17 0.07 9.87 0.89 1.40 2.26
artificial end beach was shown to be effective in minimizing wave reflections and passing currents to maintain the water volume constant. The possible saw-tooth instability on the free surface was prevented by a robust regridding scheme without using artificial smoothing. The resulting matrix equation was solved by an efficient block-iterative method. The typical CPU time for a 10-period simulation at a particular wave frequency is in the order of 10 h on a singleprocessor CRAY J90. The developed NWT was used to study fully nonlinear interactions of monochromatic waves with a truncated vertical cylinder in uniform coplanar or adverse currents. The nonlinear NWT simulations were compared with the perturbation-based numerical results calculated by a Timedomain High-Order Boundary Element Method (THOBEM).
Fig. 14. Run-up on a vertical cylinder for ka ¼ 0.374, kh ¼ 1.036 and F n ¼ 0.083.
The NWT simulations were conducted for several Froude numbers and wave slopes and the simulation results became close to the perturbation results as both wave slopes and currents were decreased. The NWT results were also compared favorably with the experimental results of Mercier and Niedzwecki [1] and Moe [2] as well as the secondorder theory (with currents) of Buchmann et al. [31] and the third-order diffraction theory (without currents) of Malenica and Molin [30]. The effects of currents are shown to be important to the prediction of maximum runup around an ocean structure. It is also shown that higher harmonic forces are more sensitive to the change in current speed than the wave-frequency forces.
Acknowledgements This research was supported by the Offshore Technology Research Center through the National Science Foundation Engineering Research Centers Program, Grant Number CDR 8721512, and NSF ERC Connectivity Award #9720427, and Texas Advanced Research Program. Computations were made in part using a CRAY Grant from UC San Diego Supercomputer Center.
Table 4 Second and third harmonic wave forces and moments for F n ¼ 0.02 and q ¼ 5.1 (rad/s) Second-order (F x (2)/rgaA 2)
Fx Fz Mx
Third-order (F x (3)/rgaA 3)
«9 ¼ 0.0025
«9 ¼ 0.0075
«9 ¼ 0.0025
«9 ¼ 0.0075
2.68 0.05 6.77
2.71 0.05 7.66
3.12 0.09 7.62
3.77 0.10 8.61
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