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inside porous seabed by a two-domain boundary element method
Applied Ocean Research 22 (2000) 255–266 www.elsevier.com/locate/apor
Wave interactions with 2D structures on/inside porous seabed by a two-domain boundary element method M.H. Kim*, W.C. Koo, S.Y. Hong 1 Department of Civil Engineering (Ocean Engineering Program), Texas A&M University, College Station, TX 77843, USA Received 8 November 1999; revised 14 July 2000
Abstract The interaction of water waves with porous seabed or partially/fully buried porous/rigid structures is studied based on the potential theory and Darcy’s law. A two-domain boundary element method (BEM) based on a desingularized integral equation is developed to fully account for the interactions at the soil–water interface and other structural boundaries. The numerical solutions are obtained by using two different methods, iteration method and global-matrix method. Both are shown to be accurate compared to the analytic solution in the case of flat porous seabed of finite thickness. The developed computer program was utilized to study various ocean-engineering applications, such as wave propagation over flat porous seabed, wave scattering by a series of rigid/porous submerged mounds, and wave induced seepage force on a fully/partially buried pipeline. In the case of fully buried pipeline, the exact numerical solution was compared with the so-called no interaction (NI) approximation, which was previously used by many authors. The limitation of the NI approximation is discussed. A longwave (LW) approximation analogous to the inertia–force expression in Morison’s formula is newly developed and proven to be practically very useful in predicting wave induced seepage forces on arbitrarily shaped, fully submerged slender structures. Based on the NI–LW approximation, analytic expressions for the second-order difference-frequency seepage force by bichromatic incident waves are developed. 䉷 2000 Elsevier Science Ltd. All rights reserved. Keywords: Wave interactions; Porous seabed; Boundary element method
1. Introduction It is of great practical interest to coastal/ocean engineers to investigate the interactions of water waves with porous boundaries or porous structures. One example is how the wave-induced pressure affects the seepage in the seabed and the resulting forces on partially/fully buried pipelines or other coastal structures. The porous seabed or submerged mound can also be effective in reducing wave transmission. Most water-wave theories for use in coastal/offshore design assume rigid, impervious sea bottom. In reality, however, the seabeds are sandy and porous, thus the modification of surface waves by such boundaries has to be taken into consideration in some ocean engineering applications. Reid and Kajiura [1] and Dean and Dalrymple [2] developed analytic solutions for the propagation/attenuation of linear monochromatic waves over infinitely thick porous seabed. They used Darcy’s law under the assumption of incompressible fluid and flat, non-deformable seabed. Liu [3] included * Corresponding author. 1 Currently at Korea Research Institute of Ships and Ocean Engineering (KRISO).
the effect of laminar boundary layers at the fluid–soil interface so as to eliminate the discontinuity in the horizontal velocity. He then developed an approximate expression for the combined damping due to porous media and boundary layer. The result shows that the damping due to bottom boundary layer is smaller than that due to porosity in many practical applications. In the cases of granular seabed with the assumption of non-deformable bed and incompressible fluid, the interaction of waves with porous boundaries and the resulting seepage flow in the seabed can be adequately described by Darcy’s law, which was also experimentally verified by Sleath [4] and Monkmeyer et al. [5]. Sleath [4] directly measured the wave induced seepage pressure in coarseand fine-sand bed, the permeability K (m 2) of which ranges from 10 ⫺9 to 10 ⫺10. He found good agreement between the measurement and his prediction based on potential theory with Darcy’s law. Monkmeyer et al. [5] also showed that the predicted wave induced seepage forces on a fully buried pipeline by using the potential theory with Darcy’s law reasonably well correlated against his measurement. The Darcy’s law is known to be most useful for incompressible fluid, non-deformable porous field, and small Reynolds’
0141-1187/00/$ - see front matter 䉷 2000 Elsevier Science Ltd. All rights reserved. PII: S0141-118 7(00)00019-5
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number. In particular, as pointed out by Sleath [4], the Reynolds numbers based on grain size and superficial bed-flow velocity are typically small compared to the critical Reynolds number [6] and consequently Darcy’s law will apply in most practical situations. For much larger Reynolds number, the viscous effects inside porous domain can be more accurately accounted for through Brinkman equation or Forchheimer equation. For more details of these equations, see Chwang and Chan [6]. On the other hand, when both water and soil are considered linearly elastic, Biot’s theory may be employed (e.g. Refs. [7–10]). Water waves passing over a permeable seabed can cause substantial fluctuations in pore-water pressures and the resulting seepage forces on partially or fully buried pipelines are of great engineering concern. MacPherson [11] presented an analytic solution for wave induced seepage forces on completely buried pipelines in an infinitely thick permeable layer based on potential theory and Darcy’s law. Monkmeyer et al. [5] later extended the theory of MacPherson [11] to finite soil thickness. Liu and O’Donnell [12] and Lennon [13] also obtained numerical solutions for the same problem by using boundary integral equation methods. The papers by MacPherson [11], Monkmeyer et al. [5], Liu and O’Donnell [12], and Lennon [13] all assumed that the wave induced sea-bottom pressure is not influenced by seabed permeability to be able to use the ordinary linear wave theory of impermeable bottom in calculating the waveinduced dynamic pressure along the seabed. The pre-calculated pressure was then used as a boundary condition for the boundary-value problem of porous domain. Although this kind of approach seems reasonable for ordinary sands [4], the true interactions between waves and porous media need to be accurately accounted for when porosity is large, and the error is expected to increase as porosity increases. In this paper, the actual wave interactions with porous soil and structures inside fluid/soil layers are investigated by solving the water- and soil-domain boundary-value problems simultaneously by a two-domain desingularized integral equation method. Based on this approach, the validity of the no interaction (NI) approximation used by MacPherson [11] and many other researchers is tested. It is shown that no-interaction theory can be used when porosity parameter K is small. Another useful long-wave (LW) approximation is also devised for simple engineering calculations. Using the LW approximation, the effects of second-order difference-frequency seepage forces on buried pipelines by bichromatic incident waves are also discussed. In addition, the analytic solution of Reid and Kajiura [1] is extended to finite-thickness porous bed (see Appendix A) and it is used to check the two-domain BEM program. Using the developed BEM program, other practically interesting problems, such as seepage forces on partially buried pipelines or wave transmission over multiple porous mounds, are also studied.
2. Boundary-value problem In this paper, we solve the wave interactions with arbitrary porous and rigid boundaries based on potential theory and Darcy’s law. For this, it is assumed that the porous seabed is non-deformable, the fluid is incompressible, and the permeability is not too high (say less than K 10 ⫺7 m 2). In order to solve the boundary value problems in fluid and soil regions simultaneously, the computational domain is separated into two domains: water domain and porous media. The governing equation in each region is given by Laplace equation. First, assuming ideal fluid and harmonic motion of angular frequency v , we can write F ~x; t Re {f ~x e ⫺ivt }: The governing equation for the complex velocity potential f in water is given by 7 2f 0
1
For a fully saturated incompressible soil, the conservation of mass leads to 7·~u 0
2
where u~ is the discharge velocity or the average velocity across a given area of soil. In the porous medium, Darcy’s law relates the velocity to the pressure gradients P S ~x; t Re {pS ~x e⫺ivt } : u~ ⫺
K 7P S m
3
where K is the constant called permeability (typically 10⫺8 m2 for clean small gravel and 10⫺10 m2 for clean fine sand) and m is the dynamic viscosity of fluid. The subscript s denotes the soil region. Actually Eq. (3), which neglects the acceleration (or inertia) terms, assumes that the flow can be treated quasi-statically. The governing equation for the fluid in the soil is obtained by substituting u~ of Eq. (3) into the conservation of mass, Eq. (2): K 4 7· ⫺ 7p S 0 m When m and K are homogeneous in space, 7 2 pS 0:
5
In order to match the two solutions, f and pS ; along the soil–water interface, the required boundary condition is that the pressures and normal velocities should be continuous i.e. irvf pS and
2f K 2pS ⫺ at z ⫺h 2z m 2z
6
where h is the water depth. In the fluid domain, the linearized free-surface condition is 2f v2 ⫺ f0 2z g
on z 0
7
When an obstruction exists either in fluid or soil, we decompose the velocity potential (or pore pressure) into two parts:
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257
Fig. 1. (a) Numerical modeling for wave propagation over porous flat seabed. (b) Numerical modeling for the diffraction by a fully buried pipeline.
incident and diffraction components. The incident part is for the flat interface and porous bottom (without obstruction) and can be obtained analytically in a priori, as described in Appendix A. On the other hand, the effect of scattering of incident velocity potential and pore pressure field by a stationary object has to be calculated to obtain a complete solution. The component representing scattering effects is called diffraction potential (or pressure). The radiation boundary condition for the diffraction potential at the truncated vertical boundaries is 2f ikf 2n
8
where n is the direction of outward unit normal vector. Similarly, the radiation condition for the diffraction part of pS can be written as 2pS ikpS : 2n
9
When the truncation boundaries are located far from the body, the seepage pressure will sufficiently attenuate to be close to zero. In such a case, pS ! 0 may be used as a
boundary condition. Otherwise, Eq. (9) has to be used to ensure that the scattered components are outgoing with nonzero amplitude. For efficient computation, the truncation boundary needs to be located not far from the body but typically 4h–5h away from the obstruction to minimize the effect of local standing waves (or evanescent waves). The bottom of the porous media is assumed to be rigid, and the corresponding no-flux condition is given by 2pS 0 2n
at z ⫺ h ⫹ d
10a
where d is the thickness of soil layer. If there are other rigid boundaries, the same no-flux condition as condition (10a) has to be imposed there. The most general boundary condition along the arbitrarily shaped porous boundary is 2f D 2f K 2pSD 2p ⫺ I ⫺ ⫹ I 10b 2n 2n m 2n 2n pSD ⫹ pSI ivr fD ⫹ fI
10c
where fD and pSD are diffraction potential and pore pressure, respectively. When porosity is zero in condition (10b)
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(or rigid boundary, K 0), the given condition is reduced to that of the diffraction problem with rigid boundary. For the given boundary conditions, the diffraction part can be solved by a specially devised two-domain BEM, fully accounting for the interactions of water and soil, which will be described in the following section. 3. Two-domain boundary element method In this section, a two-domain BEM (desingularized integral equation method) is employed to solve the mixed boundary value problems introduced in the preceding section. In the desingularized method, the source is distributed outside the fluid domain so that the source points never correspond to the field (or collocation) points. The resulting integrals are nonsingular and can be straightforwardly integrated by numerical quadratures. The details of the desingularized method are given, for example, in Koo [14], Celebi [15], and Cao et al. [16]. In particular, Koo [14] showed that the desingularized method can produce equally accurate diffraction/radiation solutions compared to the conventional BEM, in which source and field points are on the same boundary. In contrast to Celebi et al.’s [17] source-based formulation, the present desingularized method is based on a direct potential-based formulation, which means that there is no additional computational step for the source strength. The integral equation to be solved can be written as Z Z 2G 2f ⫺G f ds 0 11 2n 2n V where G is a Green function. For two-dimensional problems, the simple source G is given by G x; z; x i ; zi ln R1
12
where R1 is the distance between the source and field points [18]. To solve the integral equation, isolated sources are distributed to a small distance above each node. The desingularization distance is in general given by Ld ld Dm v
13
where Dm represents the local mesh size and ld and v are the constant parameters to be optimized. With this scheme, the source point becomes closer to the boundary as mesh size decreases. In this paper, v 0:5 and ld 0:5 are chosen after comprehensive numerical testing [14]. First, we explain how the water–soil interactions can be fully accounted for by considering the simplest case i.e. wave propagation over flat porous seabed (see Fig. 1a). Imposing all the boundary conditions, the integral equations in each domain can be written as follows:
• Fluid domain
! 2G v2 G ·ff dG 1 ⫺ g 2n G1 f ZZ 2G 2f ⫹ fr1 ⫺ G r1 dG 1 2n 2n G1 r1 ZZ 2G ⫺ ikG ·fr2 dG 1 ⫹ 2n G1 r2 Z Z 2G K 2psb ⫹ fb ⫹ G dG 1 2n m 2n G1 b
ZZ
0
14
where subscripts f, r1, r2 and b represent free surface, left-end side (incident), right-end side (radiation), and soil–water interface (water bottom), respectively. • Porous domain Z Z 2G 2pb irvfb ⫺ G dG 2 2n 2n G2 b ZZ 2G 2pr1 p ⫺G ⫹ dG 2 2n 2n r1 G2 r1 ZZ 2G ⫺ ikG ·pr2 dG 2 ⫹ 2n G2 r2 Z Z 2G 2p pB ⫺ G B dG 2 ⫹ 2n 2n G2 B 0
15
where subscript B is the rigid bottom of the porous media, thus 2pB =2n 0 there. In the case of no obstruction, diffraction problems need not be solved and the flux at the input boundary (left-side vertical boundary) 2fr1 =2n 2pr1 =2n needs to be prescribed. In the case of diffraction components, the outgoing wave conditions (8) and (9) have to be imposed at both vertical truncation boundaries. The incident wave potential at r1 is given by
fr1 ⫺
igA cosh k z ⫹ h ikx e v cosh kh
16
where A is the initial incident wave amplitude at r1, g the gravitational acceleration, and k the complex wavenumber to be calculated from the analytic dispersion relation (see Appendix A). The corresponding incident pressure on the left side of the porous domain can be written as follows (see Appendix A): pS C·cosh k z ⫹ h ⫹ d eikx
17
where C
⫺1 A rg irvK 1⫺ tanh kh·tanh kd m cosh kh·cosh kd 18
M.H. Kim et al. / Applied Ocean Research 22 (2000) 255–266 Table 1 Amplitude attenuation of a propagating regular wave over 150-m flat porous seabed of thickness 20 m (water depth 2 m, wave frequency 1 rad/s, porosity K 10 ⫺8 m 2)
BEM (Iteration, N 992) BEM (global dom. N 992) BEM (global dom. N 464) Analytic (Newton) Analytic (Approx.)
0m
50 m
100 m
150 m
1.000 1.000 1.000 1.000 1.000
0.890 0.890 0.893 0.890 0.890
0.800 0.800 0.814 0.793 0.793
0.709 0.709 0.719 0.707 0.707
and r fluid density. Applying the conditions (6) at the water–soil interface, the two integral Eqs. (14) and (15) can be inter-connected. To solve the above integral equations, we discretized the entire boundary into many constant elements on which the potential and pressure are assumed to be constant. The integral equation is then converted to an algebraic matrix equation. There are two ways to solve the coupled two-domain integral equations. One is iteration method and the other is global-matrix method. If the iteration method is used, we first assume the initial value of 2pb =2n (usually 0 or no interaction solution to reach convergence faster) in the last integral of Eq. (14), and solve the velocity potential in the water domain using the second condition of (6). Then, the calculated velocity potential fb at the interface boundary is used in the first integral of Eq. (15) via the first condition of (6). Subsequently, the integral Eq. (15) is solved in the soil domain, and we obtain new 2pb =2n at the interface. This procedure is repeated until satisfactory convergence is achieved. If global-matrix method is used, the two integral equations are combined using the given interface conditions (6), and then the resulting global matrix is solved once without iteration. We confirmed that the two different methods produce equally accurate results.
4. Numerical results and discussions First, the amplitude attenuation of a propagating regular wave over 150-m flat porous seabed of finite thickness is calculated by the two-domain BEM, and the results are Table 2 Amplitude attenuation of a propagating regular wave over 150-m flat porous seabed of thickness 20 m for various wave frequencies (water depth 2 m, porosity K 10 ⫺8 m 2) W (rad/s)
0m
50 m
100 m
150 m
0.3 0.7 1.0 1.5
1.000 1.000 1.000 1.000
0.969 0.915 0.890 0.858
0.940 0.844 0.800 0.737
0.912 0.775 0.709 0.628
259
compared in Table 1 (where N total number of boundary elements) with the analytic solutions given in Appendix A. To show the convergence of the BEM, two different discretizations were used. In the analytic solution, the real and imaginary parts of the pertinent wavenumbers were calculated from Eq. (A8) using Newton–Raphson method. The wavenumbers were also obtained from the approximations (A11) and (A12), and the corresponding analytic results are also shown for comparison. In the above, it is seen that the two different numerical methods, iteration and global-matrix methods, generate equally reliable results. It can also be observed that the BEM solutions converge as the number of element increases. The approximate analytic solution based on approximations (A11) and (A12) seems reliable as long as K remains small (say K ⬍ 10⫺8 m2 : In the next tables, the variation of the amplitude-attenuation rate with different wave frequencies and soil-porosity parameters is summarized. In Table 2, it can be seen that the shorter the wavelength, the larger the wave attenuation for the given distance, which can be seen from approximation (A12). In the case of v 1.5 (rad/s), 37% of wave attenuation is possible over 150 m distance. In Table 3 the above, the wave attenuation increases as porosity increases. If the porosity is smaller than K 10⫺10 ; there is almost no decay. From now on, the wave interaction with various porous/rigid boundaries will be studied using the iteration-BEM program. 4.1. Diffraction by a rigid half cylinder mounted on flat porous bed As the second example, let us consider a rigid half cylinder mounted on flat porous seabed under the action of monochromatic waves. The water will penetrate into the soil and cause harmonic seepage force on the bottom of the structure. On the other hand, the top portion is directly affected by wave motions. Along the rigid boundary, two boundary conditions are to be specified i.e. 2fD =2n ⫺2fI =2n in the fluid domain and 2pSD =2n ⫺ 2pSI =2n in the porous domain. The other flat porous regions satisfy the interface boundary conditions (10b) and (10c). After obtaining the diffraction potential and diffraction pressure, we can calculate the wave force and seepage force on the half cylinder. The wave reflection and transmission coefficients by the submerged mound are also of practical interest and shown in Figs. 2 and 3 for various wave conditions and porosity parameters. When the seabed is porous, the reflection and transmission coefficients depend on the distance from the body. In the present case, the reflection and transmission coefficients are evaluated at the truncation boundaries. The truncation boundaries used for this example are located 60r from the cylinder. It is seen that the curves approach the result of rigid seabed as porosity decreases. The figures typically show that the porous seabed can provide
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Table 3 Amplitude attenuation of a propagating regular wave over 150-m flat porous seabed of thickness 20 m for various K values (water depth 2 m, wave frequency 1 rad/s) K (m 2) ⫺10
10 10 ⫺9 10 ⫺8
0m
50 m
100 m
150 m
1.000 1.000 1.000
0.999 0.988 0.890
0.999 0.981 0.800
0.999 0.968 0.709
significant damping if porosity is not small. If there is no porous media and no viscous dissipation, the original energy of the incident wave will be conserved i.e. R2 ⫹ T 2 1: Otherwise, there exists some energy loss and their sum should be less than 1. The transmission coefficient becomes minimal near kh 0.9 ⫺ 1 as a combined effect of diffraction and damping by porosity. In Fig. 4, the pore pressure force on the bottom of the half cylinder is plotted against kh for various K values. The contribution of diffraction component is also separately plotted for comparison. For this example, the incident component is more important than the diffraction component unless kh ⬎ 3.5. 4.2. Wave transmission over multiple porous half-circular mounds Next, the interaction of monochromatic waves with multiple porous half-circular mounds mounted on a flat porous seabed is investigated. As an example, four mounds of radius r with the center-to-center spacing 5r in the water depth of 2r are considered. The porosity of the mound is assumed to be the same as that of seabed. The truncation boundaries are located 10r from the edge of the first and last mounds. Fig. 5 shows the reflection and transmission coefficients for various kr and K values. It is interesting to see that minimum transmission occurs near kr ⬇ 0:6 (or wavelength ⬇ 10r) i.e. when the wavelength is about twice the effective wavelength/spacing of the mounds. This is a well-known phenomenon called Bragg resonance scattering [19] in the study of wave reflection by sand bars. In this case, the additional wave attenuation is caused by interactions with porous boundaries.
Fig. 2. Reflection coefficients of a bottom-mounted half cylinder at 60 m ahead of the structure (domain length 122r, h 2r, d 20r).
Fig. 3. Transmission coefficients of a bottom-mounted half cylinder at 60 m behind the structure (domain length 122r, h 2r, d 20r).
4.3. Wave induced seepage force on fully or partially buried pipelines Wave-induced forces and pressures on partially or fully buried pipelines can be obtained in a similar manner by using the boundary conditions as described in Fig. 1b. In the case of partially buried pipeline, the boundary condition for the top portion in the fluid domain is 2fD =2n ⫺ 2fI =2n while that for the buried part is 2pD =2n ⫺ 2pI =2n: Figs. 6 and 7 show vertical and horizontal forces on a fully buried circular pipeline for various K values as function of h/L (L wavelength). Also shown is the MacPherson’s analytic solution, which was obtained by the diffraction theory assuming NI at the interface. As pointed out earlier, the NI approximation was also used in many other papers, such as Monkmeyer et al. [5], Liu and O’Donnell [12], and Lennon [13]. It is seen from this figure that the NI approximation is very good if K ⬍ 10⫺9 and practically useful up to K 10⫺8 : For larger porosity than K 10⫺8 ; the interactions at the soil–water interface need to be taken into consideration. The seepage force generally increases as waterdepth decreases. However, when K is large, the force may decrease for small values of h/L. In Figs. 6 and 7, the thickness of soil layer is large, and the vertical and horizontal seepage forces are about the same. In the next figure (Figs. 8 and 9), the seepage force magnitudes are plotted against various depths of soil layer. Interestingly, the horizontal force remains almost the same in the range d ⬎ 0.2L, and then rapidly increases as the soil thickness decreases below 0.2L. The sudden increase is mainly due to the increase of diffraction component. On the contrary, the vertical force starts to decrease in the range d ⬍ 0.4L. This kind of effect of soil-layer thickness was also pointed out in Liu and O’Donnell [12]. In Fig. 10, the seepage pressure variation along the circumference (starting from the top) of the pipe is plotted. The pressure distribution is almost symmetric with respect to the vertical line passing through the center. It is noteworthy that the diffraction pressure is greater than the incident pressure around the lower part of the circular pipe. Figs. 11 and 12 show the change of seepage force
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261
Fig. 4. Total (thick lines) and diffraction (thin lines) pore-pressure forces on the bottom of a rigid half-circular mound (K 1 × 10 ⫺10 (solid line), K 1 × 10 ⫺9 (bar line) and K 1 × 10 ⫺8 (dashed line); domain length 122r, h 2r, d 20r).
magnitudes on a partially buried pipeline for various burial depths (or locations of the center point). Fig. 11 shows the force component on the water portion and Fig. 12 is that on the soil portion. For the horizontal force on the water part (Fig. 11), the diffraction component is about the same order of magnitude as incident component. In the case of vertical forces, however, the incident component is much greater than the diffraction component, and thus the so-called Froude–Krilov approximation (estimation by incident component only) may be applied. In Fig. 12, we see similar trend for the force on the soil portion. Once the pipe is fully buried, the difference between incident and diffraction contributions becomes smaller as burial depth increases. This implies that a reasonable approximation can be developed for a fully buried pipeline, as explained in the next section. In Fig.13, the total horizontal and vertical forces on the same pipe normalized by buoyancy force rgpr2 are plotted against various burial depths. For partially buried pipe, the horizontal force is larger than the vertical force. The difference between the two forces continues to decrease as burial depth increases. The two forces become almost the same once it is fully buried. Figs.14 and 15 show the wave induced seepage forces on a pipe as function of diameter to wavelength ratio D/L for
Fig. 5. Reflection and transmission by four porous half-circular mounds mounted on flat porous seabed (K zero (triangle), K 1 × 10 ⫺9 (rectangle), and K 1 × 10 ⫺8 (circle); h 2r domain length 37r, center-tocenter distance of each mound 5r, d 20r).
various K values. The force tends to decrease as porosity increases. Also shown is the diffraction component multiplied by 2 for K 10⫺8 : We can see that the diffraction component is about half of the total force, which leads to useful approximation as shown in the next section.
4.4. Approximation for long waves When the relevant wavelength is large compared to the size of the pipeline, the spatial variation of the seepage pressure field is expected to be very small near the pipeline. In this case, we can develop a LW approximation as follows. First, the wave induced seepage force on the pipe can be ~ obtained from the surface integral RR R R of the pressure RRR F p~n dS: Using Gauss theorem p~n dS 7p dV and assuming that pressure variation around the cylinder is ~ small, the volume integral can be approximated by F ᭙7p; where ᭙ is the displaced volume of a body. Then, the diffraction effect can approximately be added to the ~ ᭙ ⫹ m a =r7p; Froude–Krilov-like integral to yield F where m a is the added mass of the body. The derived formula is similar to the inertia-force expression in Morison formula, which is used for wave-force calculation for slender bodies by long waves. Using the LW approximation, the case K 10⫺8 of Fig. 14 is recalculated and compared with the exact solution in Fig. 16. Also presented is the NI plus LW approximation. It has already been seen in Figs. 6 and 7, that the NI approximation is reasonable if K ⬍ 10⫺8 : The relative errors of the two approximations are plotted in Fig. 17. For the given K value, both LW and NI–LW approximations are practically useful for a wide range of D/L. The relative error of LW approximation increases monotonically, as expected, as the size of diameter to wavelength (D/L) becomes larger. We have already seen in Fig. 8 that the diffraction component of seepage force can be much greater than the incident component when the soil layer thickness is small. In such a case, the validity of the LW approximation has to be carefully checked.
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Fig. 6. Vertical seepage forces on a fully buried pipeline for various permeabilities (ds/D 0.5, D/L 0.1, d/L 1, and H/L 0.07).
Fig. 8. Horizontal wave induced seepage forces (normalized by buoyancy force rgpD2 =4 on a fully buried pipe for K 1 × 10 ⫺8. (total force diamond, incident component rectangle, and diffraction component circle; dp r, h 4r, L 20r, H/L 0.07, and domain length 20r).
4.5. Effects of second-order difference-frequency pressure In Kim and Yue [20], it was pointed out that the secondorder difference-frequency pressure field can attenuate slowly with depth so that, under certain wave conditions, its effect on seabed pressure and seepage force can be more important than that of exponentially decaying linear components. In this section, we investigate the possibility of the above premise by comparing first-order and second-order seepage forces in various bichromatic wave conditions. The depth attenuation of the second-order sum-frequency pressure field is faster than that of the first-order components, and thus its effects on seepage forces are expected to be negligible. The amplitude, frequency, and wavenumber associated with the second-order difference-frequency incident wave potential are Aj ; Al ; vj ⫺ vl ; and kj ⫺ kl ; respectively. The pertinent wavelength is typically much greater than those of constituent first-order waves. Therefore, the NI–LW approximation developed in the preceding section can be used assuming K ⬍ 10⫺8 (m 2). The expression of the second-order difference-frequency incident wave potential is given by gAj Al f x; z; t 2 ⫺
"
C1 C ⫺ 2 vj vl
#
cosh k⫺ z ⫹ h cosh k⫺ h
sin k⫺ x ⫺ v⫺ t
Fig. 7. Horizontal seepage forces (legend same as Fig. 6).
19
where C1
kj2 1 ⫺ tanh2 kj h ⫺ 2kj kl 1 ⫹ tanh kj h·tanh kl h n⫺ ⫺ k⫺ tanh k⫺ h
C2
kl2 1 ⫺ tanh2 kl h ⫺ 2kl kj 1 ⫹ tanh kl h·tanh kj h n⫺ ⫺ k⫺ tanh k⫺ h
n⫺
v⫺2 ; g
k ⫺ kj ⫺ kl ;
and
v⫺ vj ⫺ vl
When k1 h Ⰷ 1 and k2 h Ⰷ 1 and k1 ⫺ k2 h O 1; Ogilvie [21] derived a simpler expression as follows:
f⫺ x; z; t
2Aj Al vj vl cosh k⫺ z ⫹ h v⫺ ⫺ vj ⫹ vl tanh k⫺ h cosh k⫺ h sin k⫺ x ⫺ v⫺ t
20
The corresponding wave induced pressure at the interface by expressions (19) and (20) can be obtained from p⫺ S riv⫺ f⫺ at z -h. Then, the application of the NI–LW ~ ⫺ ᭙ ⫹ ma =r7p S⫺ ; leads to the analytic approximation, F expression for the second-order difference-frequency seepage force. In Fig. 18, the horizontal seepage forces on a pipeline due to the second-order difference-frequency potential are presented as function of mean dimensionless wavenumber. It can be seen that the numerical results obtained from the
Fig. 9. Vertical wave induced seepage forces (legend same as Fig. 8).
M.H. Kim et al. / Applied Ocean Research 22 (2000) 255–266
Fig. 10. Pressure variation along the circumference of a buried pipe (K 1 × 10 ⫺9, D 8 m, dp 1 m, h 4 m, d 20 m, domain length 20 m, L 20 m, frequency 1.62 rad/s, P_o rgA=cosh kh:
exact second-order potential (19) tend to agree with those obtained from Ogilvie’s approximation (20) as individual wavelengths become shorter. It is also seen that the effects of the second-order difference-frequency pressure gradually increase as constituent wavelengths become longer. In Tables 4 and 5, the first-order and second-order seepage forces are compared for various combinations of bichromatic incident waves of unit amplitudes. The second-order contribution in the case of Dv 0:5 is more significant than that of Dv 0:2: For several low-frequency pairs, the magnitudes of the second-order difference-frequency seepage forces can be comparable or even larger than those of first-order forces, as shown in the tables. Therefore, for certain pipeline applications, the second-order effects need to be taken into consideration.
263
Fig. 11. Wave induced forces on the water part of a partially buried pipe: incident horizontal force (diamond), diffraction horizontal force (rectangle), incident vertical force (x) and diffraction vertical force (*). (D 2 m, h 4 m, d 20 m, L 20 m, frequency 1.62 (rad/s), domain length 20 m, H/L 0.07. K 1 × 10 ⫺9). Center of pipe 0 means half of the pipe is buried).
5. Concluding remarks The interaction of water waves with porous seabed and partially/fully buried porous/rigid structures is studied based on the potential theory and Darcy’s law. A two-domain BEM, which fully accounts for the interactions at the soil–water interface and other structural boundaries, is developed by using a desingularized integral equation with simple sources. Two different BEMs, iteration method and global-matrix method, are independently developed for crosschecking. The numerical solutions compared satisfactorily with the analytic solutions of flat porous seabed. The developed computer program was utilized to study
Table 4 Comparison of the first and second-order seepage force on the same pipeline as Fig. 18 (first- and second-order forces by unit-amplitude bichromatic waves Dv 0:2 are normalized by buoyancy force) Frequency pair v1 ; v2
First-order horizontal force
Second-order horizontal force
First-order vertical force
Second-order vertical force
0.7 0.5 0.9 0.7 1.1 0.9 1.3 1.1
0.15 0.12 0.18 0.15 0.19 0.18 0.18 0.19
0.16
0.17 0.13 0.19 0.17 0.2 0.19 0.2 0.2
0.14
0.09 0.06 0.04
0.08 0.05 0.04
Table 5 Comparison of the first- and second-order seepage force on the same pipeline as Fig. 18 (first- and second-order forces by unit-amplitude bichromatic waves Dv 0:5 are normalized by buoyancy force) Frequency pair v1 ; v2
First-order horizontal force
Second-order horizontal force
First-order vertical force
Second-order vertical force
0.9 0.4 1.1 0.6 1.3 0.8 1.5 1.0
0.18 0.10 0.19 0.14 0.18 0.17 0.17 0.18
0.34
0.19 0.11 0.20 0.15 0.20 0.18 0.19 0.20
0.34
0.19 0.13 0.09
0.19 0.13 0.09
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Fig. 12. Wave induced forces on the soil part of a partially buried pipe (legend same as Fig. 11).
Fig. 13. Horizontal (solid line) and vertical (diamond) total wave induced forces (computational condition same as Fig. 11).
various ocean-engineering applications, such as wave propagation over a flat porous seabed, Bragg scattering by a series of rigid/porous submerged mounds, and wave induced seepage force on a fully/partially buried pipeline. It is shown that the wave induced seepage force can be appreciable (e.g. about 20% of the buoyancy force) when waterdepth and soil thickness are small. The wave-induced force can be even greater when the pipeline is partially buried. In the case of fully buried pipeline, the exact numerical solution was compared with the so-called NI approximation, which was used previously by many authors. It is
Fig. 14. Horizontal wave induced seepage forces on a fully buried pipeline for various permeability parameters K: Circle (K 0.5 × 10 ⫺8,) Diamond (K 1 × 10 ⫺8), Rectangle (K 1 × 10 ⫺9), x (K 1 × 10 ⫺10), and Triangle (diffraction component of K 1 × 10 ⫺8, multiplied by 2); h 4r, d 20r, dp r, and domain length 20r.
Fig. 15. Vertical wave induced seepage forces (legend same as Fig. 14).
Fig. 16. Horizontal wave induced seepage forces on a fully buried pipeline. (dp r, h 4r, d 20r, domain length 20r, and K 1 × 10 ⫺8) Numerical result (diamond), LW approximation (rectangle), and NI–LW approximation (triangle).
shown that the error is small if K ⬍ 10⫺8 (m 2) but it can be large for larger K values. A LW approximation analogous to the inertia force in Morison’s formula is newly developed after comparing the role of incident and diffraction components. The LW approximation is proved to be practically very useful to predict, in a simple manner, the wave induced seepage force on an arbitrarily shaped fully submerged slender structure. Based on the NI–LW approximation, the analytic expressions for the second-order differencefrequency seepage force in bichromatic waves (or possibly random waves) are derived. The second-order seepage force
Fig. 17. Relative error for the case of Fig.16: LW approximation (diamond), and NI–LW approximation (circle).
M.H. Kim et al. / Applied Ocean Research 22 (2000) 255–266
265
Finally, the application of the linearized kinematic freesurface condition yields the following dispersion relationship, K rv 2 v ⫺ gk tanh kh ⫺i · gk ⫺ v2 tanh kh·tanh kd m A8
Fig. 18. Second-order difference-frequency horizontal seepage forces on a fully buried pipe calculated by complete equation (diamond) and Ogilvie’s approximate formula (rectangle). The frequency difference between two waves is 0.5 rad/s. (D 2 m, dp 1 m, h 4 m, d 40 m, A(constituent wave amplitude) 1 m, K 1 × 10 ⫺9, and domain length 20 m).
is shown to be more important for lower-frequency wave pairs. In such a case, its magnitude can be even greater (e.g. 30% of the buoyancy force for mean wave slope 0.03) than the constituent first-order forces. Appendix A. Analytic solution for a wave propagating over flat porous seabed of finite thickness The complex wave potential and pore pressure satisfying the Laplace equation can be written as
f x; z B1 cosh k z ⫹ h ⫹ B2 sinh k z ⫹ h eikx
A1
and pS x; z C cosh k z ⫹ h ⫹ d ⫹ D sinh k z ⫹ h ⫹ d eikx A2 where h is the water depth, d the porous-domain depth, k the wave number and v the wave frequency. From the no-flux condition at the bottom of the soil, we have D0
This dispersion relationship yields complex k, which may be written as k kr ⫹ iki : The real wave number is related to the wavelength, while the imaginary component determines the spatial damping rate. It can be seen more clearly from the h x; t Ae⫺ki x Re{ei kr x⫺vt }; free-surface expression where A is the original amplitude of incident wave at the entrance of computational domain. The quantity K rv=m in Eq. (A8) is generally small. For instance, the K value of sand ranges from about 10 ⫺9 to 10 ⫺12 (m 2), while the kinematic viscosity m=r of water is O(10 ⫺6); Therefore, K rv=m ranges from 10 ⫺2 to 10 ⫺5. Considering this, an approximation of (A8) can be developed. In the case of intermediate water depth we can write cosh kh cosh kr h ⫹ iki h sinh kr h and tanh kh tanh kr h ⫹ iki h=cosh2 kr h assuming ki h p 1: Similar approximations can also be used for sinh kh and tanh kd: Substituting the above expressions into Eq. (A8), we can separate the real and imaginary parts as follows: Real : v2 ⫺ gkr tanh kr h ⫹
!" K rv gkr ki d v2 ki d tanh kr h ⫺ ⫹ gki tanh kr d · m cosh2 kr d cosh2 kr d # v2 ki h tanh kr d ⫺ cosh2 kr h A9
irvB1 C cosh kd
A4
v2 ki2 hd ⫹ v tanh kr h tanh kr d ⫺ cosh2 kr h cosh2 kr d
K C sinh kd m
A5
We now apply linearized dynamic free-surface condition h ⫺ 1=g 2F=2t at z 0; and using conditions (A3), (A4), and (A5), we obtain ⫺1 A rg irvK 1⫺ tanh kh·tanh kd A6 C m cosh kh·cosh kd and pS C·cosh k z ⫹ h ⫹ d eikx
A7
A10 #
2
From the second interface condition of (6) (Darcy’s law), B2 ⫺
gkr ki h ⫺ gki tanh kr h cosh2 kr h !" K rv gki2 d · ⫺ gkr tanh kr d ⫹ m cosh2 kr d
Imaginary : ⫺
A3
From the first interface condition of (6) i.e. pS x; ⫺h p x; ⫺h; we obtain
gki2 h cosh2 kr h
Neglecting the small terms containing K rv=mki and ki2 in conditions (A9) and (A10), we obtain
v2 ⬇ gkr tanh kr h and
2
ki ⬇
K rv kr tanh kr d m 2kr h ⫹ sinh 2kr h
A11
A12
When the thickness of soil layer becomes infinity, the above expressions are reduced to those of Reid and Kajiura [1].
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References [1] Reid RO, Kajiura K. On the damping of gravity waves over a permeable sea bed. Trans Am Geophys Union 1957:38. [2] Dean RG, Dalrymple RA. Water wave mechanics for engineers and scientists, Adv Series Ocean Engng. Singapore: World Scientific, 1991. [3] Liu PLF. Damping of water waves over porous bed. J Hydraul Div, ASCE 1973;99:2263–71. [4] Sleath JFA. Wave induced pressure in beds of sands. J Hydraul Div, ASCE 1970;96(HY2):367–78. [5] Monkmeyer PL, Mantovani P, Vincent H. Wave induced seepage effects on a buried pipeline. Proc Coastal Struct 1983;83:519–31. [6] Chwang AT, Chan AT. Interaction between porous media and wave motion. Annu Rev Fluid Mech 1998;3:53–84. [7] Yamamoto T, Koning HL, Sellmeiher H, Van Hijum EB. On the response of a poro-elastic bed to water waves. J Fluid Mech 1978;87:193–206. [8] Madsen OS. Wave-induced pore pressures and effective stresses in porous bed. Geotechnique 1978;28:4. [9] Garrison CJ. Pore pressure loads on gravity structures. Appl Ocean Res 1989;11:4. [10] Silvestri V, Soulie M, Marche C, Louche D. Some considerations on the wave-induced instability of the seafloor. Fourth Int OMAE 1985:1. [11] MacPherson H. Wave forces on pipelines buried in permeable seabed. J Waterways, Port, Coastal Ocean Engng Div, ASCE 1978;104(WW4):407–20.
[12] Liu PLF, O’Donnell TP. Wave-induced forces on buried pipeline in permeable seabeds. Proc Civil Engng Oceans 1979;IV(1):111–21. [13] Lennon GP. Wave-induced forces on buried pipelines. J Waterways, Port, Coastal Ocean Engng 1985;111:3. [14] Koo WC. Simulation of surface waves with porous boundaries in a 2D numerical wave tank. MS dissertation, Department of Ocean Engineering, Texas A&M University, College Station, Texas, 1991. [15] Celebi MS. Computation of transient fully nonlinear free surface waves and wave-body interactions. PhD dissertation, Department of Ocean Engineering, Texas A&M University, College Station, Texas, 1997. [16] Cao Yusong, Schultz W, Beck RF. Three-dimensional desingularized boundary integral methods for potential problems. Int J Numer Meth Fluids 1991;12:785–803. [17] Celebi MS, Kim MH, Beck RF. Fully nonlinear 3D numerical wave tank simulations. J Ship Res 1998;42(1):33–45. [18] Brebbia CA, Dominguez J. Boundary elements: an introductory course, Computational mechanics publications. Southampton, UK: McGraw-Hill, 1992. [19] Mei CC. Resonant reflection of surface water waves by periodic sand bars. J Fluid Mech 1985;152:315–35. [20] Kim MH, Yue DKP. The complete second-order diffraction solution for an axisymmetric body Part.2 Bichromatic incident waves and body motions. J Fluid Mech 1990;211:557–93. [21] Ogilvie TF. Second-order hydro-dynamic effects on ocean platforms. Proc Int Workshop Ship & Ocean Platforms, Berkeley 1983:205–65.
Wave interactions with 2D structures on/inside porous seabed by a two-domain boundary element method M.H. Kim*, W.C. Koo, S.Y. Hong 1 Department of Civil Engineering (Ocean Engineering Program), Texas A&M University, College Station, TX 77843, USA Received 8 November 1999; revised 14 July 2000
Abstract The interaction of water waves with porous seabed or partially/fully buried porous/rigid structures is studied based on the potential theory and Darcy’s law. A two-domain boundary element method (BEM) based on a desingularized integral equation is developed to fully account for the interactions at the soil–water interface and other structural boundaries. The numerical solutions are obtained by using two different methods, iteration method and global-matrix method. Both are shown to be accurate compared to the analytic solution in the case of flat porous seabed of finite thickness. The developed computer program was utilized to study various ocean-engineering applications, such as wave propagation over flat porous seabed, wave scattering by a series of rigid/porous submerged mounds, and wave induced seepage force on a fully/partially buried pipeline. In the case of fully buried pipeline, the exact numerical solution was compared with the so-called no interaction (NI) approximation, which was previously used by many authors. The limitation of the NI approximation is discussed. A longwave (LW) approximation analogous to the inertia–force expression in Morison’s formula is newly developed and proven to be practically very useful in predicting wave induced seepage forces on arbitrarily shaped, fully submerged slender structures. Based on the NI–LW approximation, analytic expressions for the second-order difference-frequency seepage force by bichromatic incident waves are developed. 䉷 2000 Elsevier Science Ltd. All rights reserved. Keywords: Wave interactions; Porous seabed; Boundary element method
1. Introduction It is of great practical interest to coastal/ocean engineers to investigate the interactions of water waves with porous boundaries or porous structures. One example is how the wave-induced pressure affects the seepage in the seabed and the resulting forces on partially/fully buried pipelines or other coastal structures. The porous seabed or submerged mound can also be effective in reducing wave transmission. Most water-wave theories for use in coastal/offshore design assume rigid, impervious sea bottom. In reality, however, the seabeds are sandy and porous, thus the modification of surface waves by such boundaries has to be taken into consideration in some ocean engineering applications. Reid and Kajiura [1] and Dean and Dalrymple [2] developed analytic solutions for the propagation/attenuation of linear monochromatic waves over infinitely thick porous seabed. They used Darcy’s law under the assumption of incompressible fluid and flat, non-deformable seabed. Liu [3] included * Corresponding author. 1 Currently at Korea Research Institute of Ships and Ocean Engineering (KRISO).
the effect of laminar boundary layers at the fluid–soil interface so as to eliminate the discontinuity in the horizontal velocity. He then developed an approximate expression for the combined damping due to porous media and boundary layer. The result shows that the damping due to bottom boundary layer is smaller than that due to porosity in many practical applications. In the cases of granular seabed with the assumption of non-deformable bed and incompressible fluid, the interaction of waves with porous boundaries and the resulting seepage flow in the seabed can be adequately described by Darcy’s law, which was also experimentally verified by Sleath [4] and Monkmeyer et al. [5]. Sleath [4] directly measured the wave induced seepage pressure in coarseand fine-sand bed, the permeability K (m 2) of which ranges from 10 ⫺9 to 10 ⫺10. He found good agreement between the measurement and his prediction based on potential theory with Darcy’s law. Monkmeyer et al. [5] also showed that the predicted wave induced seepage forces on a fully buried pipeline by using the potential theory with Darcy’s law reasonably well correlated against his measurement. The Darcy’s law is known to be most useful for incompressible fluid, non-deformable porous field, and small Reynolds’
0141-1187/00/$ - see front matter 䉷 2000 Elsevier Science Ltd. All rights reserved. PII: S0141-118 7(00)00019-5
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number. In particular, as pointed out by Sleath [4], the Reynolds numbers based on grain size and superficial bed-flow velocity are typically small compared to the critical Reynolds number [6] and consequently Darcy’s law will apply in most practical situations. For much larger Reynolds number, the viscous effects inside porous domain can be more accurately accounted for through Brinkman equation or Forchheimer equation. For more details of these equations, see Chwang and Chan [6]. On the other hand, when both water and soil are considered linearly elastic, Biot’s theory may be employed (e.g. Refs. [7–10]). Water waves passing over a permeable seabed can cause substantial fluctuations in pore-water pressures and the resulting seepage forces on partially or fully buried pipelines are of great engineering concern. MacPherson [11] presented an analytic solution for wave induced seepage forces on completely buried pipelines in an infinitely thick permeable layer based on potential theory and Darcy’s law. Monkmeyer et al. [5] later extended the theory of MacPherson [11] to finite soil thickness. Liu and O’Donnell [12] and Lennon [13] also obtained numerical solutions for the same problem by using boundary integral equation methods. The papers by MacPherson [11], Monkmeyer et al. [5], Liu and O’Donnell [12], and Lennon [13] all assumed that the wave induced sea-bottom pressure is not influenced by seabed permeability to be able to use the ordinary linear wave theory of impermeable bottom in calculating the waveinduced dynamic pressure along the seabed. The pre-calculated pressure was then used as a boundary condition for the boundary-value problem of porous domain. Although this kind of approach seems reasonable for ordinary sands [4], the true interactions between waves and porous media need to be accurately accounted for when porosity is large, and the error is expected to increase as porosity increases. In this paper, the actual wave interactions with porous soil and structures inside fluid/soil layers are investigated by solving the water- and soil-domain boundary-value problems simultaneously by a two-domain desingularized integral equation method. Based on this approach, the validity of the no interaction (NI) approximation used by MacPherson [11] and many other researchers is tested. It is shown that no-interaction theory can be used when porosity parameter K is small. Another useful long-wave (LW) approximation is also devised for simple engineering calculations. Using the LW approximation, the effects of second-order difference-frequency seepage forces on buried pipelines by bichromatic incident waves are also discussed. In addition, the analytic solution of Reid and Kajiura [1] is extended to finite-thickness porous bed (see Appendix A) and it is used to check the two-domain BEM program. Using the developed BEM program, other practically interesting problems, such as seepage forces on partially buried pipelines or wave transmission over multiple porous mounds, are also studied.
2. Boundary-value problem In this paper, we solve the wave interactions with arbitrary porous and rigid boundaries based on potential theory and Darcy’s law. For this, it is assumed that the porous seabed is non-deformable, the fluid is incompressible, and the permeability is not too high (say less than K 10 ⫺7 m 2). In order to solve the boundary value problems in fluid and soil regions simultaneously, the computational domain is separated into two domains: water domain and porous media. The governing equation in each region is given by Laplace equation. First, assuming ideal fluid and harmonic motion of angular frequency v , we can write F ~x; t Re {f ~x e ⫺ivt }: The governing equation for the complex velocity potential f in water is given by 7 2f 0
1
For a fully saturated incompressible soil, the conservation of mass leads to 7·~u 0
2
where u~ is the discharge velocity or the average velocity across a given area of soil. In the porous medium, Darcy’s law relates the velocity to the pressure gradients P S ~x; t Re {pS ~x e⫺ivt } : u~ ⫺
K 7P S m
3
where K is the constant called permeability (typically 10⫺8 m2 for clean small gravel and 10⫺10 m2 for clean fine sand) and m is the dynamic viscosity of fluid. The subscript s denotes the soil region. Actually Eq. (3), which neglects the acceleration (or inertia) terms, assumes that the flow can be treated quasi-statically. The governing equation for the fluid in the soil is obtained by substituting u~ of Eq. (3) into the conservation of mass, Eq. (2): K 4 7· ⫺ 7p S 0 m When m and K are homogeneous in space, 7 2 pS 0:
5
In order to match the two solutions, f and pS ; along the soil–water interface, the required boundary condition is that the pressures and normal velocities should be continuous i.e. irvf pS and
2f K 2pS ⫺ at z ⫺h 2z m 2z
6
where h is the water depth. In the fluid domain, the linearized free-surface condition is 2f v2 ⫺ f0 2z g
on z 0
7
When an obstruction exists either in fluid or soil, we decompose the velocity potential (or pore pressure) into two parts:
M.H. Kim et al. / Applied Ocean Research 22 (2000) 255–266
257
Fig. 1. (a) Numerical modeling for wave propagation over porous flat seabed. (b) Numerical modeling for the diffraction by a fully buried pipeline.
incident and diffraction components. The incident part is for the flat interface and porous bottom (without obstruction) and can be obtained analytically in a priori, as described in Appendix A. On the other hand, the effect of scattering of incident velocity potential and pore pressure field by a stationary object has to be calculated to obtain a complete solution. The component representing scattering effects is called diffraction potential (or pressure). The radiation boundary condition for the diffraction potential at the truncated vertical boundaries is 2f ikf 2n
8
where n is the direction of outward unit normal vector. Similarly, the radiation condition for the diffraction part of pS can be written as 2pS ikpS : 2n
9
When the truncation boundaries are located far from the body, the seepage pressure will sufficiently attenuate to be close to zero. In such a case, pS ! 0 may be used as a
boundary condition. Otherwise, Eq. (9) has to be used to ensure that the scattered components are outgoing with nonzero amplitude. For efficient computation, the truncation boundary needs to be located not far from the body but typically 4h–5h away from the obstruction to minimize the effect of local standing waves (or evanescent waves). The bottom of the porous media is assumed to be rigid, and the corresponding no-flux condition is given by 2pS 0 2n
at z ⫺ h ⫹ d
10a
where d is the thickness of soil layer. If there are other rigid boundaries, the same no-flux condition as condition (10a) has to be imposed there. The most general boundary condition along the arbitrarily shaped porous boundary is 2f D 2f K 2pSD 2p ⫺ I ⫺ ⫹ I 10b 2n 2n m 2n 2n pSD ⫹ pSI ivr fD ⫹ fI
10c
where fD and pSD are diffraction potential and pore pressure, respectively. When porosity is zero in condition (10b)
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M.H. Kim et al. / Applied Ocean Research 22 (2000) 255–266
(or rigid boundary, K 0), the given condition is reduced to that of the diffraction problem with rigid boundary. For the given boundary conditions, the diffraction part can be solved by a specially devised two-domain BEM, fully accounting for the interactions of water and soil, which will be described in the following section. 3. Two-domain boundary element method In this section, a two-domain BEM (desingularized integral equation method) is employed to solve the mixed boundary value problems introduced in the preceding section. In the desingularized method, the source is distributed outside the fluid domain so that the source points never correspond to the field (or collocation) points. The resulting integrals are nonsingular and can be straightforwardly integrated by numerical quadratures. The details of the desingularized method are given, for example, in Koo [14], Celebi [15], and Cao et al. [16]. In particular, Koo [14] showed that the desingularized method can produce equally accurate diffraction/radiation solutions compared to the conventional BEM, in which source and field points are on the same boundary. In contrast to Celebi et al.’s [17] source-based formulation, the present desingularized method is based on a direct potential-based formulation, which means that there is no additional computational step for the source strength. The integral equation to be solved can be written as Z Z 2G 2f ⫺G f ds 0 11 2n 2n V where G is a Green function. For two-dimensional problems, the simple source G is given by G x; z; x i ; zi ln R1
12
where R1 is the distance between the source and field points [18]. To solve the integral equation, isolated sources are distributed to a small distance above each node. The desingularization distance is in general given by Ld ld Dm v
13
where Dm represents the local mesh size and ld and v are the constant parameters to be optimized. With this scheme, the source point becomes closer to the boundary as mesh size decreases. In this paper, v 0:5 and ld 0:5 are chosen after comprehensive numerical testing [14]. First, we explain how the water–soil interactions can be fully accounted for by considering the simplest case i.e. wave propagation over flat porous seabed (see Fig. 1a). Imposing all the boundary conditions, the integral equations in each domain can be written as follows:
• Fluid domain
! 2G v2 G ·ff dG 1 ⫺ g 2n G1 f ZZ 2G 2f ⫹ fr1 ⫺ G r1 dG 1 2n 2n G1 r1 ZZ 2G ⫺ ikG ·fr2 dG 1 ⫹ 2n G1 r2 Z Z 2G K 2psb ⫹ fb ⫹ G dG 1 2n m 2n G1 b
ZZ
0
14
where subscripts f, r1, r2 and b represent free surface, left-end side (incident), right-end side (radiation), and soil–water interface (water bottom), respectively. • Porous domain Z Z 2G 2pb irvfb ⫺ G dG 2 2n 2n G2 b ZZ 2G 2pr1 p ⫺G ⫹ dG 2 2n 2n r1 G2 r1 ZZ 2G ⫺ ikG ·pr2 dG 2 ⫹ 2n G2 r2 Z Z 2G 2p pB ⫺ G B dG 2 ⫹ 2n 2n G2 B 0
15
where subscript B is the rigid bottom of the porous media, thus 2pB =2n 0 there. In the case of no obstruction, diffraction problems need not be solved and the flux at the input boundary (left-side vertical boundary) 2fr1 =2n 2pr1 =2n needs to be prescribed. In the case of diffraction components, the outgoing wave conditions (8) and (9) have to be imposed at both vertical truncation boundaries. The incident wave potential at r1 is given by
fr1 ⫺
igA cosh k z ⫹ h ikx e v cosh kh
16
where A is the initial incident wave amplitude at r1, g the gravitational acceleration, and k the complex wavenumber to be calculated from the analytic dispersion relation (see Appendix A). The corresponding incident pressure on the left side of the porous domain can be written as follows (see Appendix A): pS C·cosh k z ⫹ h ⫹ d eikx
17
where C
⫺1 A rg irvK 1⫺ tanh kh·tanh kd m cosh kh·cosh kd 18
M.H. Kim et al. / Applied Ocean Research 22 (2000) 255–266 Table 1 Amplitude attenuation of a propagating regular wave over 150-m flat porous seabed of thickness 20 m (water depth 2 m, wave frequency 1 rad/s, porosity K 10 ⫺8 m 2)
BEM (Iteration, N 992) BEM (global dom. N 992) BEM (global dom. N 464) Analytic (Newton) Analytic (Approx.)
0m
50 m
100 m
150 m
1.000 1.000 1.000 1.000 1.000
0.890 0.890 0.893 0.890 0.890
0.800 0.800 0.814 0.793 0.793
0.709 0.709 0.719 0.707 0.707
and r fluid density. Applying the conditions (6) at the water–soil interface, the two integral Eqs. (14) and (15) can be inter-connected. To solve the above integral equations, we discretized the entire boundary into many constant elements on which the potential and pressure are assumed to be constant. The integral equation is then converted to an algebraic matrix equation. There are two ways to solve the coupled two-domain integral equations. One is iteration method and the other is global-matrix method. If the iteration method is used, we first assume the initial value of 2pb =2n (usually 0 or no interaction solution to reach convergence faster) in the last integral of Eq. (14), and solve the velocity potential in the water domain using the second condition of (6). Then, the calculated velocity potential fb at the interface boundary is used in the first integral of Eq. (15) via the first condition of (6). Subsequently, the integral Eq. (15) is solved in the soil domain, and we obtain new 2pb =2n at the interface. This procedure is repeated until satisfactory convergence is achieved. If global-matrix method is used, the two integral equations are combined using the given interface conditions (6), and then the resulting global matrix is solved once without iteration. We confirmed that the two different methods produce equally accurate results.
4. Numerical results and discussions First, the amplitude attenuation of a propagating regular wave over 150-m flat porous seabed of finite thickness is calculated by the two-domain BEM, and the results are Table 2 Amplitude attenuation of a propagating regular wave over 150-m flat porous seabed of thickness 20 m for various wave frequencies (water depth 2 m, porosity K 10 ⫺8 m 2) W (rad/s)
0m
50 m
100 m
150 m
0.3 0.7 1.0 1.5
1.000 1.000 1.000 1.000
0.969 0.915 0.890 0.858
0.940 0.844 0.800 0.737
0.912 0.775 0.709 0.628
259
compared in Table 1 (where N total number of boundary elements) with the analytic solutions given in Appendix A. To show the convergence of the BEM, two different discretizations were used. In the analytic solution, the real and imaginary parts of the pertinent wavenumbers were calculated from Eq. (A8) using Newton–Raphson method. The wavenumbers were also obtained from the approximations (A11) and (A12), and the corresponding analytic results are also shown for comparison. In the above, it is seen that the two different numerical methods, iteration and global-matrix methods, generate equally reliable results. It can also be observed that the BEM solutions converge as the number of element increases. The approximate analytic solution based on approximations (A11) and (A12) seems reliable as long as K remains small (say K ⬍ 10⫺8 m2 : In the next tables, the variation of the amplitude-attenuation rate with different wave frequencies and soil-porosity parameters is summarized. In Table 2, it can be seen that the shorter the wavelength, the larger the wave attenuation for the given distance, which can be seen from approximation (A12). In the case of v 1.5 (rad/s), 37% of wave attenuation is possible over 150 m distance. In Table 3 the above, the wave attenuation increases as porosity increases. If the porosity is smaller than K 10⫺10 ; there is almost no decay. From now on, the wave interaction with various porous/rigid boundaries will be studied using the iteration-BEM program. 4.1. Diffraction by a rigid half cylinder mounted on flat porous bed As the second example, let us consider a rigid half cylinder mounted on flat porous seabed under the action of monochromatic waves. The water will penetrate into the soil and cause harmonic seepage force on the bottom of the structure. On the other hand, the top portion is directly affected by wave motions. Along the rigid boundary, two boundary conditions are to be specified i.e. 2fD =2n ⫺2fI =2n in the fluid domain and 2pSD =2n ⫺ 2pSI =2n in the porous domain. The other flat porous regions satisfy the interface boundary conditions (10b) and (10c). After obtaining the diffraction potential and diffraction pressure, we can calculate the wave force and seepage force on the half cylinder. The wave reflection and transmission coefficients by the submerged mound are also of practical interest and shown in Figs. 2 and 3 for various wave conditions and porosity parameters. When the seabed is porous, the reflection and transmission coefficients depend on the distance from the body. In the present case, the reflection and transmission coefficients are evaluated at the truncation boundaries. The truncation boundaries used for this example are located 60r from the cylinder. It is seen that the curves approach the result of rigid seabed as porosity decreases. The figures typically show that the porous seabed can provide
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Table 3 Amplitude attenuation of a propagating regular wave over 150-m flat porous seabed of thickness 20 m for various K values (water depth 2 m, wave frequency 1 rad/s) K (m 2) ⫺10
10 10 ⫺9 10 ⫺8
0m
50 m
100 m
150 m
1.000 1.000 1.000
0.999 0.988 0.890
0.999 0.981 0.800
0.999 0.968 0.709
significant damping if porosity is not small. If there is no porous media and no viscous dissipation, the original energy of the incident wave will be conserved i.e. R2 ⫹ T 2 1: Otherwise, there exists some energy loss and their sum should be less than 1. The transmission coefficient becomes minimal near kh 0.9 ⫺ 1 as a combined effect of diffraction and damping by porosity. In Fig. 4, the pore pressure force on the bottom of the half cylinder is plotted against kh for various K values. The contribution of diffraction component is also separately plotted for comparison. For this example, the incident component is more important than the diffraction component unless kh ⬎ 3.5. 4.2. Wave transmission over multiple porous half-circular mounds Next, the interaction of monochromatic waves with multiple porous half-circular mounds mounted on a flat porous seabed is investigated. As an example, four mounds of radius r with the center-to-center spacing 5r in the water depth of 2r are considered. The porosity of the mound is assumed to be the same as that of seabed. The truncation boundaries are located 10r from the edge of the first and last mounds. Fig. 5 shows the reflection and transmission coefficients for various kr and K values. It is interesting to see that minimum transmission occurs near kr ⬇ 0:6 (or wavelength ⬇ 10r) i.e. when the wavelength is about twice the effective wavelength/spacing of the mounds. This is a well-known phenomenon called Bragg resonance scattering [19] in the study of wave reflection by sand bars. In this case, the additional wave attenuation is caused by interactions with porous boundaries.
Fig. 2. Reflection coefficients of a bottom-mounted half cylinder at 60 m ahead of the structure (domain length 122r, h 2r, d 20r).
Fig. 3. Transmission coefficients of a bottom-mounted half cylinder at 60 m behind the structure (domain length 122r, h 2r, d 20r).
4.3. Wave induced seepage force on fully or partially buried pipelines Wave-induced forces and pressures on partially or fully buried pipelines can be obtained in a similar manner by using the boundary conditions as described in Fig. 1b. In the case of partially buried pipeline, the boundary condition for the top portion in the fluid domain is 2fD =2n ⫺ 2fI =2n while that for the buried part is 2pD =2n ⫺ 2pI =2n: Figs. 6 and 7 show vertical and horizontal forces on a fully buried circular pipeline for various K values as function of h/L (L wavelength). Also shown is the MacPherson’s analytic solution, which was obtained by the diffraction theory assuming NI at the interface. As pointed out earlier, the NI approximation was also used in many other papers, such as Monkmeyer et al. [5], Liu and O’Donnell [12], and Lennon [13]. It is seen from this figure that the NI approximation is very good if K ⬍ 10⫺9 and practically useful up to K 10⫺8 : For larger porosity than K 10⫺8 ; the interactions at the soil–water interface need to be taken into consideration. The seepage force generally increases as waterdepth decreases. However, when K is large, the force may decrease for small values of h/L. In Figs. 6 and 7, the thickness of soil layer is large, and the vertical and horizontal seepage forces are about the same. In the next figure (Figs. 8 and 9), the seepage force magnitudes are plotted against various depths of soil layer. Interestingly, the horizontal force remains almost the same in the range d ⬎ 0.2L, and then rapidly increases as the soil thickness decreases below 0.2L. The sudden increase is mainly due to the increase of diffraction component. On the contrary, the vertical force starts to decrease in the range d ⬍ 0.4L. This kind of effect of soil-layer thickness was also pointed out in Liu and O’Donnell [12]. In Fig. 10, the seepage pressure variation along the circumference (starting from the top) of the pipe is plotted. The pressure distribution is almost symmetric with respect to the vertical line passing through the center. It is noteworthy that the diffraction pressure is greater than the incident pressure around the lower part of the circular pipe. Figs. 11 and 12 show the change of seepage force
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261
Fig. 4. Total (thick lines) and diffraction (thin lines) pore-pressure forces on the bottom of a rigid half-circular mound (K 1 × 10 ⫺10 (solid line), K 1 × 10 ⫺9 (bar line) and K 1 × 10 ⫺8 (dashed line); domain length 122r, h 2r, d 20r).
magnitudes on a partially buried pipeline for various burial depths (or locations of the center point). Fig. 11 shows the force component on the water portion and Fig. 12 is that on the soil portion. For the horizontal force on the water part (Fig. 11), the diffraction component is about the same order of magnitude as incident component. In the case of vertical forces, however, the incident component is much greater than the diffraction component, and thus the so-called Froude–Krilov approximation (estimation by incident component only) may be applied. In Fig. 12, we see similar trend for the force on the soil portion. Once the pipe is fully buried, the difference between incident and diffraction contributions becomes smaller as burial depth increases. This implies that a reasonable approximation can be developed for a fully buried pipeline, as explained in the next section. In Fig.13, the total horizontal and vertical forces on the same pipe normalized by buoyancy force rgpr2 are plotted against various burial depths. For partially buried pipe, the horizontal force is larger than the vertical force. The difference between the two forces continues to decrease as burial depth increases. The two forces become almost the same once it is fully buried. Figs.14 and 15 show the wave induced seepage forces on a pipe as function of diameter to wavelength ratio D/L for
Fig. 5. Reflection and transmission by four porous half-circular mounds mounted on flat porous seabed (K zero (triangle), K 1 × 10 ⫺9 (rectangle), and K 1 × 10 ⫺8 (circle); h 2r domain length 37r, center-tocenter distance of each mound 5r, d 20r).
various K values. The force tends to decrease as porosity increases. Also shown is the diffraction component multiplied by 2 for K 10⫺8 : We can see that the diffraction component is about half of the total force, which leads to useful approximation as shown in the next section.
4.4. Approximation for long waves When the relevant wavelength is large compared to the size of the pipeline, the spatial variation of the seepage pressure field is expected to be very small near the pipeline. In this case, we can develop a LW approximation as follows. First, the wave induced seepage force on the pipe can be ~ obtained from the surface integral RR R R of the pressure RRR F p~n dS: Using Gauss theorem p~n dS 7p dV and assuming that pressure variation around the cylinder is ~ small, the volume integral can be approximated by F ᭙7p; where ᭙ is the displaced volume of a body. Then, the diffraction effect can approximately be added to the ~ ᭙ ⫹ m a =r7p; Froude–Krilov-like integral to yield F where m a is the added mass of the body. The derived formula is similar to the inertia-force expression in Morison formula, which is used for wave-force calculation for slender bodies by long waves. Using the LW approximation, the case K 10⫺8 of Fig. 14 is recalculated and compared with the exact solution in Fig. 16. Also presented is the NI plus LW approximation. It has already been seen in Figs. 6 and 7, that the NI approximation is reasonable if K ⬍ 10⫺8 : The relative errors of the two approximations are plotted in Fig. 17. For the given K value, both LW and NI–LW approximations are practically useful for a wide range of D/L. The relative error of LW approximation increases monotonically, as expected, as the size of diameter to wavelength (D/L) becomes larger. We have already seen in Fig. 8 that the diffraction component of seepage force can be much greater than the incident component when the soil layer thickness is small. In such a case, the validity of the LW approximation has to be carefully checked.
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Fig. 6. Vertical seepage forces on a fully buried pipeline for various permeabilities (ds/D 0.5, D/L 0.1, d/L 1, and H/L 0.07).
Fig. 8. Horizontal wave induced seepage forces (normalized by buoyancy force rgpD2 =4 on a fully buried pipe for K 1 × 10 ⫺8. (total force diamond, incident component rectangle, and diffraction component circle; dp r, h 4r, L 20r, H/L 0.07, and domain length 20r).
4.5. Effects of second-order difference-frequency pressure In Kim and Yue [20], it was pointed out that the secondorder difference-frequency pressure field can attenuate slowly with depth so that, under certain wave conditions, its effect on seabed pressure and seepage force can be more important than that of exponentially decaying linear components. In this section, we investigate the possibility of the above premise by comparing first-order and second-order seepage forces in various bichromatic wave conditions. The depth attenuation of the second-order sum-frequency pressure field is faster than that of the first-order components, and thus its effects on seepage forces are expected to be negligible. The amplitude, frequency, and wavenumber associated with the second-order difference-frequency incident wave potential are Aj ; Al ; vj ⫺ vl ; and kj ⫺ kl ; respectively. The pertinent wavelength is typically much greater than those of constituent first-order waves. Therefore, the NI–LW approximation developed in the preceding section can be used assuming K ⬍ 10⫺8 (m 2). The expression of the second-order difference-frequency incident wave potential is given by gAj Al f x; z; t 2 ⫺
"
C1 C ⫺ 2 vj vl
#
cosh k⫺ z ⫹ h cosh k⫺ h
sin k⫺ x ⫺ v⫺ t
Fig. 7. Horizontal seepage forces (legend same as Fig. 6).
19
where C1
kj2 1 ⫺ tanh2 kj h ⫺ 2kj kl 1 ⫹ tanh kj h·tanh kl h n⫺ ⫺ k⫺ tanh k⫺ h
C2
kl2 1 ⫺ tanh2 kl h ⫺ 2kl kj 1 ⫹ tanh kl h·tanh kj h n⫺ ⫺ k⫺ tanh k⫺ h
n⫺
v⫺2 ; g
k ⫺ kj ⫺ kl ;
and
v⫺ vj ⫺ vl
When k1 h Ⰷ 1 and k2 h Ⰷ 1 and k1 ⫺ k2 h O 1; Ogilvie [21] derived a simpler expression as follows:
f⫺ x; z; t
2Aj Al vj vl cosh k⫺ z ⫹ h v⫺ ⫺ vj ⫹ vl tanh k⫺ h cosh k⫺ h sin k⫺ x ⫺ v⫺ t
20
The corresponding wave induced pressure at the interface by expressions (19) and (20) can be obtained from p⫺ S riv⫺ f⫺ at z -h. Then, the application of the NI–LW ~ ⫺ ᭙ ⫹ ma =r7p S⫺ ; leads to the analytic approximation, F expression for the second-order difference-frequency seepage force. In Fig. 18, the horizontal seepage forces on a pipeline due to the second-order difference-frequency potential are presented as function of mean dimensionless wavenumber. It can be seen that the numerical results obtained from the
Fig. 9. Vertical wave induced seepage forces (legend same as Fig. 8).
M.H. Kim et al. / Applied Ocean Research 22 (2000) 255–266
Fig. 10. Pressure variation along the circumference of a buried pipe (K 1 × 10 ⫺9, D 8 m, dp 1 m, h 4 m, d 20 m, domain length 20 m, L 20 m, frequency 1.62 rad/s, P_o rgA=cosh kh:
exact second-order potential (19) tend to agree with those obtained from Ogilvie’s approximation (20) as individual wavelengths become shorter. It is also seen that the effects of the second-order difference-frequency pressure gradually increase as constituent wavelengths become longer. In Tables 4 and 5, the first-order and second-order seepage forces are compared for various combinations of bichromatic incident waves of unit amplitudes. The second-order contribution in the case of Dv 0:5 is more significant than that of Dv 0:2: For several low-frequency pairs, the magnitudes of the second-order difference-frequency seepage forces can be comparable or even larger than those of first-order forces, as shown in the tables. Therefore, for certain pipeline applications, the second-order effects need to be taken into consideration.
263
Fig. 11. Wave induced forces on the water part of a partially buried pipe: incident horizontal force (diamond), diffraction horizontal force (rectangle), incident vertical force (x) and diffraction vertical force (*). (D 2 m, h 4 m, d 20 m, L 20 m, frequency 1.62 (rad/s), domain length 20 m, H/L 0.07. K 1 × 10 ⫺9). Center of pipe 0 means half of the pipe is buried).
5. Concluding remarks The interaction of water waves with porous seabed and partially/fully buried porous/rigid structures is studied based on the potential theory and Darcy’s law. A two-domain BEM, which fully accounts for the interactions at the soil–water interface and other structural boundaries, is developed by using a desingularized integral equation with simple sources. Two different BEMs, iteration method and global-matrix method, are independently developed for crosschecking. The numerical solutions compared satisfactorily with the analytic solutions of flat porous seabed. The developed computer program was utilized to study
Table 4 Comparison of the first and second-order seepage force on the same pipeline as Fig. 18 (first- and second-order forces by unit-amplitude bichromatic waves Dv 0:2 are normalized by buoyancy force) Frequency pair v1 ; v2
First-order horizontal force
Second-order horizontal force
First-order vertical force
Second-order vertical force
0.7 0.5 0.9 0.7 1.1 0.9 1.3 1.1
0.15 0.12 0.18 0.15 0.19 0.18 0.18 0.19
0.16
0.17 0.13 0.19 0.17 0.2 0.19 0.2 0.2
0.14
0.09 0.06 0.04
0.08 0.05 0.04
Table 5 Comparison of the first- and second-order seepage force on the same pipeline as Fig. 18 (first- and second-order forces by unit-amplitude bichromatic waves Dv 0:5 are normalized by buoyancy force) Frequency pair v1 ; v2
First-order horizontal force
Second-order horizontal force
First-order vertical force
Second-order vertical force
0.9 0.4 1.1 0.6 1.3 0.8 1.5 1.0
0.18 0.10 0.19 0.14 0.18 0.17 0.17 0.18
0.34
0.19 0.11 0.20 0.15 0.20 0.18 0.19 0.20
0.34
0.19 0.13 0.09
0.19 0.13 0.09
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Fig. 12. Wave induced forces on the soil part of a partially buried pipe (legend same as Fig. 11).
Fig. 13. Horizontal (solid line) and vertical (diamond) total wave induced forces (computational condition same as Fig. 11).
various ocean-engineering applications, such as wave propagation over a flat porous seabed, Bragg scattering by a series of rigid/porous submerged mounds, and wave induced seepage force on a fully/partially buried pipeline. It is shown that the wave induced seepage force can be appreciable (e.g. about 20% of the buoyancy force) when waterdepth and soil thickness are small. The wave-induced force can be even greater when the pipeline is partially buried. In the case of fully buried pipeline, the exact numerical solution was compared with the so-called NI approximation, which was used previously by many authors. It is
Fig. 14. Horizontal wave induced seepage forces on a fully buried pipeline for various permeability parameters K: Circle (K 0.5 × 10 ⫺8,) Diamond (K 1 × 10 ⫺8), Rectangle (K 1 × 10 ⫺9), x (K 1 × 10 ⫺10), and Triangle (diffraction component of K 1 × 10 ⫺8, multiplied by 2); h 4r, d 20r, dp r, and domain length 20r.
Fig. 15. Vertical wave induced seepage forces (legend same as Fig. 14).
Fig. 16. Horizontal wave induced seepage forces on a fully buried pipeline. (dp r, h 4r, d 20r, domain length 20r, and K 1 × 10 ⫺8) Numerical result (diamond), LW approximation (rectangle), and NI–LW approximation (triangle).
shown that the error is small if K ⬍ 10⫺8 (m 2) but it can be large for larger K values. A LW approximation analogous to the inertia force in Morison’s formula is newly developed after comparing the role of incident and diffraction components. The LW approximation is proved to be practically very useful to predict, in a simple manner, the wave induced seepage force on an arbitrarily shaped fully submerged slender structure. Based on the NI–LW approximation, the analytic expressions for the second-order differencefrequency seepage force in bichromatic waves (or possibly random waves) are derived. The second-order seepage force
Fig. 17. Relative error for the case of Fig.16: LW approximation (diamond), and NI–LW approximation (circle).
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265
Finally, the application of the linearized kinematic freesurface condition yields the following dispersion relationship, K rv 2 v ⫺ gk tanh kh ⫺i · gk ⫺ v2 tanh kh·tanh kd m A8
Fig. 18. Second-order difference-frequency horizontal seepage forces on a fully buried pipe calculated by complete equation (diamond) and Ogilvie’s approximate formula (rectangle). The frequency difference between two waves is 0.5 rad/s. (D 2 m, dp 1 m, h 4 m, d 40 m, A(constituent wave amplitude) 1 m, K 1 × 10 ⫺9, and domain length 20 m).
is shown to be more important for lower-frequency wave pairs. In such a case, its magnitude can be even greater (e.g. 30% of the buoyancy force for mean wave slope 0.03) than the constituent first-order forces. Appendix A. Analytic solution for a wave propagating over flat porous seabed of finite thickness The complex wave potential and pore pressure satisfying the Laplace equation can be written as
f x; z B1 cosh k z ⫹ h ⫹ B2 sinh k z ⫹ h eikx
A1
and pS x; z C cosh k z ⫹ h ⫹ d ⫹ D sinh k z ⫹ h ⫹ d eikx A2 where h is the water depth, d the porous-domain depth, k the wave number and v the wave frequency. From the no-flux condition at the bottom of the soil, we have D0
This dispersion relationship yields complex k, which may be written as k kr ⫹ iki : The real wave number is related to the wavelength, while the imaginary component determines the spatial damping rate. It can be seen more clearly from the h x; t Ae⫺ki x Re{ei kr x⫺vt }; free-surface expression where A is the original amplitude of incident wave at the entrance of computational domain. The quantity K rv=m in Eq. (A8) is generally small. For instance, the K value of sand ranges from about 10 ⫺9 to 10 ⫺12 (m 2), while the kinematic viscosity m=r of water is O(10 ⫺6); Therefore, K rv=m ranges from 10 ⫺2 to 10 ⫺5. Considering this, an approximation of (A8) can be developed. In the case of intermediate water depth we can write cosh kh cosh kr h ⫹ iki h sinh kr h and tanh kh tanh kr h ⫹ iki h=cosh2 kr h assuming ki h p 1: Similar approximations can also be used for sinh kh and tanh kd: Substituting the above expressions into Eq. (A8), we can separate the real and imaginary parts as follows: Real : v2 ⫺ gkr tanh kr h ⫹
!" K rv gkr ki d v2 ki d tanh kr h ⫺ ⫹ gki tanh kr d · m cosh2 kr d cosh2 kr d # v2 ki h tanh kr d ⫺ cosh2 kr h A9
irvB1 C cosh kd
A4
v2 ki2 hd ⫹ v tanh kr h tanh kr d ⫺ cosh2 kr h cosh2 kr d
K C sinh kd m
A5
We now apply linearized dynamic free-surface condition h ⫺ 1=g 2F=2t at z 0; and using conditions (A3), (A4), and (A5), we obtain ⫺1 A rg irvK 1⫺ tanh kh·tanh kd A6 C m cosh kh·cosh kd and pS C·cosh k z ⫹ h ⫹ d eikx
A7
A10 #
2
From the second interface condition of (6) (Darcy’s law), B2 ⫺
gkr ki h ⫺ gki tanh kr h cosh2 kr h !" K rv gki2 d · ⫺ gkr tanh kr d ⫹ m cosh2 kr d
Imaginary : ⫺
A3
From the first interface condition of (6) i.e. pS x; ⫺h p x; ⫺h; we obtain
gki2 h cosh2 kr h
Neglecting the small terms containing K rv=mki and ki2 in conditions (A9) and (A10), we obtain
v2 ⬇ gkr tanh kr h and
2
ki ⬇
K rv kr tanh kr d m 2kr h ⫹ sinh 2kr h
A11
A12
When the thickness of soil layer becomes infinity, the above expressions are reduced to those of Reid and Kajiura [1].
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[12] Liu PLF, O’Donnell TP. Wave-induced forces on buried pipeline in permeable seabeds. Proc Civil Engng Oceans 1979;IV(1):111–21. [13] Lennon GP. Wave-induced forces on buried pipelines. J Waterways, Port, Coastal Ocean Engng 1985;111:3. [14] Koo WC. Simulation of surface waves with porous boundaries in a 2D numerical wave tank. MS dissertation, Department of Ocean Engineering, Texas A&M University, College Station, Texas, 1991. [15] Celebi MS. Computation of transient fully nonlinear free surface waves and wave-body interactions. PhD dissertation, Department of Ocean Engineering, Texas A&M University, College Station, Texas, 1997. [16] Cao Yusong, Schultz W, Beck RF. Three-dimensional desingularized boundary integral methods for potential problems. Int J Numer Meth Fluids 1991;12:785–803. [17] Celebi MS, Kim MH, Beck RF. Fully nonlinear 3D numerical wave tank simulations. J Ship Res 1998;42(1):33–45. [18] Brebbia CA, Dominguez J. Boundary elements: an introductory course, Computational mechanics publications. Southampton, UK: McGraw-Hill, 1992. [19] Mei CC. Resonant reflection of surface water waves by periodic sand bars. J Fluid Mech 1985;152:315–35. [20] Kim MH, Yue DKP. The complete second-order diffraction solution for an axisymmetric body Part.2 Bichromatic incident waves and body motions. J Fluid Mech 1990;211:557–93. [21] Ogilvie TF. Second-order hydro-dynamic effects on ocean platforms. Proc Int Workshop Ship & Ocean Platforms, Berkeley 1983:205–65.