Second-order wave interaction with a vertical plate

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1 Second-order wave interaction with a vertical plate B. Molin, F. Rem, O. Kimmoun École Généraliste d Ingénieurs de Marseille, 3 4 Marseille cedex 2, France Abstract The second-order interaction of monochromatic water waves with a rigid vertical plate is investigated numericall and experimentall. The plate has a finite width and is projected from one of the side-walls of a wave-tank. This geometr reduces the wave cases to normal incidence but permits a semi-analtical resolution based on eigen-function expansions. Obtained numerical results are shown to be insensitive to the width of the tank beond some value, and to converge quickl with the truncation orders of the expansions, in spite of the second-order potential having a logarithmic singularit at the plate edge. Second-order free surface elevations are compared with values derived from experimental measurements at the BGO-First offshore wave-tank. Good agreement is reported. It is with much pleasure and admiration for his achievements that we dedicate this work to Nick Newman. Introduction For the past twent or thirt ears second-order hdrodnamics has received considerable attention. Numerous contributions have been offered to the resolution of the second-order diffraction problem in regular waves. Semi-analtical solutions have been proposed for particular geometries such as vertical circular clinders standing on the sea-bottom (Chau & Eatock Talor 992, Malenica & Molin 99, Newman 996) or arras of vertical clinders (Malenica, Eatock Talor & Huang 999). These are valuable references to be used as benchmarks for numerical models. It is one purpose of this work to offer another set of reference results, for a different tpe of bod, in the sense that it has a sharp corner (an edge), where the velocit potential is singular. This feature induces numerical difficulties that are absent in the case of circular clinders. It also creates theoretical problems since, whatever the wave steepness, the righthand side of the free surface condition verified b the second-order potential becomes infinite at the plate edge. Phsicall the flow separates and it is not clear how much and how far local flow separation undermines the potential flow approach. In this respect comparisons with experimental measurements are valuable. To permit a semi-analtical resolution, we bound the fluid domain b two vertical walls, alike in a wave-tank, with the plate protruding from one of the walls. This geometr limits the applications to normal incidence. When the walls are taken far apart, confinement effects are shown to disappear. The fluid domain is divided into two sub-domains, up and down-wave from the plate. In either sub-domain the velocit potentials at first- and second-orders can be expanded over sets of eigen-functions that satisf the Laplace equation and the no-flow conditions at the horizontal bottom and at the walls. The problem then reduces to ensuring the no-flow condition at the plate and matching the potentials at the common boundar. Classicall, at second-order, the velocit potential is decomposed into a locked component, that satisfies the non-homogeneous free surface condition, and a free component that satisfies the homogeneous free surface condition.

2 The theoretical developments are described in section 2. Illustrative results are shown in section 3, together with convergence considerations. Comparisons are also shown with experimental results obtained at the offshore tank BGO-First. Good agreement is obtained between measured and calculated second-order free surface elevations along the plate. 2 Theor b 2 d x Figure : Geometr. 2. First-order diffraction problem Figure illustrates the geometr. Regular incoming waves, of amplitude A I and frequenc ω, propagate from left to right in a channel of width b and waterdepth h. The bod consists in a rigid plate of zero thickness, protruding from the wall at the origin of the coordinate sstem. The plate is standing on the bottom and its width is d. The diffraction problem is solved in the frequenc domain, on the basis that a stead state has been reached. That is, to second-order, the velocit potential is written { Φ(x,, z, t) = R ε ϕ () (x,, z) e i ω t + ε 2 (x,, z) e 2 i ω t} + ε 2 (x,, z) () with ε k A I the wave steepness. The first-order velocit potential ϕ () obes the Laplace equation in the mean fluid domain b; h z, the linearized free surface equation g ϕ () / z ω 2 ϕ () = at z =, and homogeneous Neumann conditions at the solid boundaries (bottom, walls, plate). On the weather side of the plate it consists in an incoming component, propagating along the channel, and a diffracted component traveling to the left while reflecting on the walls. On the lee side incoming and diffracted components travel to the right. In the left-hand side sub-domain, the first-order velocit potential can be expressed as ϕ () cosh k (z + h) = A cosh kh { e i kx + } B n e i α n x cos λ n n= (2) 2

3 and in the right-hand side sub-domain ϕ () cosh k (z + h) 2 = A cosh kh { e i kx + } C n e i α n x cos λ n n= (3) with A = i A I g/ω, k the wave number linked to the frequenc ω b the dispersion equation ω 2 = g k tanh kh, and α n = k 2 λ 2 n n =, N (4) α n = i λ 2 n k 2 n = N +, () Here λ n = n π/b and N is the largest integer n such that k is larger than λ n. For n N the modes are progressive; for n > N, the are evanescent. Expressions (2) and (3) ensure that the Laplace equation and the no-flow conditions at the bottom and at the walls are fulfilled. The diffracted waves are outgoing from the common boundar x = with reflections on the side-walls. The remaining conditions to be fulfilled, in x =, are the no-flow condition on the plate ( d) and matching conditions from the plate edge to the exterior wall (d b): ϕ () x ϕ () x = ϕ() 2 x = for d (6) ϕ = ϕ 2 for d b (7) = ϕ() 2 x for d b (8) Since ϕ () / x = ϕ() 2 / x all over the width of the tank and the cos λ n functions form an orthogonal set over [ b], it results that C n B n. The remaining conditions reduce to α n B n cos λ n = k d (9) n= B n cos λ n = d b () n= These equations are written under the form B m cos λ m + n m α n α m B n cos λ n = k α m d () B m cos λ m + n m B n cos λ n = d b (2) Both sides of both equations are multiplied b cos λ m, integrated in over their domains of validit, and added up, resulting into the linear sstem: B m ( + δ m ) b 2 + n m ( ) αn α m d B n cos λ m cos λ n d = with δ ij the Kronecker smbol. This sstem is solved b a standard Gauss routine. k α m λ m sin λ m d (3) 3

4 2.2 Second-order diffraction problem The second-order potential at the double frequenc 2 ω obes the free surface condition in z = : [ g z 4 ω 2 = i ω ϕ () 2 x + ϕ () 2 + ϕ () 2 ( ) ] z 2g ϕ() g ϕ () zz ω 2 ϕ () z (4) or g z 4 ω 2 = i ω [ ϕ () x ] 2 + ϕ () k2 T ϕ ()2 with T = 3 tanh 2 kh. We follow the usual technique of decomposing the second-order potential into a particular component that satisfies the non-homogeneous free surface condition and a free component that verifies the no-flow condition on the plate and takes care of the matching conditions at the common boundar Locked potential in sub-domain After calculations the right-hand side of equation () is obtained as g z 4 ω 2 i ω A 2 = 3 k2 2 cosh 2 kh e2 i k x (6) + [k 2 T + 2 kα n ] B n e i (k α n) x cos λ n A particular solution L with I and with n= + { [k 2 ] T 2α m α n 2λ m λ n cos(λm + λ n ) 4 m= n= + [ k 2 ] } T 2α m α n + 2λ m λ n cos(λm λ n ) B m B n e i (α m+α n )x can be looked for as the sum of three terms: the second-order incident potential, given b I = 3 i 8 L2 = n= () L = ϕ(2) I + L2 + ϕ(2) L3 (7) A 2 I ω sinh 4 kh cosh 2k (z + h) e2i k x (8) β n e i (k α n) x cos λ n cosh ν n(z + h) cosh ν n h (9) ν 2 n = 2 k 2 2 k α n (2) and L3 = m= n= + m= n= β n = i ω k A2 [k T + 2 α n ] B n g ν n tanh ν n h 4 ω 2 (2) γ mn e i (α m+α n )x cos(λ m + λ n ) cosh µ mn(z + h) cosh µ mn h γ mn2 e i (α m+α n )x cos(λ m λ n ) cosh µ mn2(z + h) cosh µ mn2 h (22) 4

5 where µ 2 mn = (α m + α n ) 2 + (λ m + λ n ) 2 (23) µ 2 mn2 = (α m + α n ) 2 + (λ m λ n ) 2 (24) γ mn = i ω A2 [k 2 T 2λ m λ n 2α m α n ] B m B n 4 (g µ mn tanh µ mn h 4 ω 2 ) γ mn2 = i ω A2 [k 2 T + 2λ m λ n 2α m α n ] B m B n 4 (g µ mn2 tanh µ mn2 h 4 ω 2 ) (2) (26) It can be noted that the wave numbers ν n and µ mn2 are complex, while µ mn is real Locked potential in sub-domain 2 Similarl a particular solution is obtained as with L22 = n= L2 = ϕ(2) I + L22 + ϕ(2) L23 (27) β n e i (k α n) x cos λ n cosh ν n(z + h) cosh ν n h (28) ν 2 n = (k + α n ) 2 + λ 2 n (29) and L23 = m= n= + m= n= β n = i ω k A2 [2 α n k T ] B n g ν n tanh ν n h 4 ω 2 (3) γ mn e i (α m+α n )x cos(λ m + λ n ) cosh µ mn(z + h) cosh µ mn h γ mn2 e i (α m+α n )x cos(λ m λ n ) cosh µ mn2(z + h) cosh µ mn2 h (3) Free wave components In either sub-domain we must add up free waves in order to fulfill the boundar and matching conditions in x =. In the left-hand side sub-domain the free wave potential is expressed as [ ] F = cos λ m D m e i rm x cosh k 2(z + h) + D mn e rmn x cos k 2n (z + h) cosh k 2 h where and m= n= (32) 4 ω 2 = g k 2 tanh k 2 h = g k 2n tan k 2n h (33) r 2 m = k 2 2 λ 2 m r mn = k 2 2n + λ2 m (34) In the right-hand side sub-domain the free wave potential is expressed as [ ] F 2 = cos λ m E m e i r m x cosh k 2(z + h) + E mn e r mn x cos k 2n (z + h) cosh k 2 h m= n= (3)

6 2.3 Matching To satisf the no-flow condition on the plate and matching conditions in x =, we take advantage that the cosh k 2 (z+h)/ cosh k 2 h, cos k 2n (z+h) functions provide an orthogonal set in z over [ h ]. We project the locked components over this basis. This ields independent two-dimensional problems (in (x, )), similar to the first-order problem. 3 Results 3. Convergence stud We take a geometr inspired from the square clinder studied in Molin et al. (2b). We replace the square clinder b a plate of equal width. We refer to the case kh = 3 in Molin et al. (2b). So we take a plate width d of m, a waterdepth h of m and a wavelength of m b = N = b = N = 2 b = N = 4 b = N = 2 2 RAO elevation Figure 2: First-order free surface elevation. kh = 3; k (2d) = 6. Sensitivit to truncation order and tank width. First we investigate the effects of series truncation and tank width on the first-order elevation along the plate, on its weather side. Figure 2 shows the obtained RAOs for a channel width of m and truncation orders of, 2 and 4, and for a channel width of m and a truncation order of 2. The plots are extended to twice the width of the plate to check that the RAOs are equal to one when d. It can be observed that the curves are hardl distinguishable when the truncation order is larger than 2. We now consider the second-order potential along the same strip (x = ; 2). We take the tank width equal to m, truncation order N = NY in equal to 2 and we var the truncation order in z of the second-order free waves from 2 up to. The same truncation order N Y in is taken for the first-order and second-order potentials. The secondorder potential is made non-dimensional in the same fashion as in Molin et al. (2b), that 6

7 3 NZ = 2 NZ = NZ = Figure 3: Second-order potential. kh = 3; k (2d) = 6. Basin width: m. NY = 2. 3 NY = NY = 2 NY = Figure 4: Second-order potential. kh = 3; k (2d) = 6. Basin width: m. NZ =. is in the form 2 ω (,, ) (2d)/(g A 2 ). Results are shown in figure 3 where it can be observed that and provide quasi-identical curves. In figure 4 we keep the truncation order NZ equal to and we var NY from to 4. It can be observed that NY = 2 and NY = 4 provide quasi identical results, except b the 7

8 plate edge where the second-order potential is singular: as the truncation order is increased, the singularit starts appearing as a sharp peak. It is shown in the Appendix that the singularit is logarithmic. Fortunatel it is quite localized and the NY = 2 and NY = 4 curves coalesce together at short distance from the plate edge. In figures through 8 we show bird s ee views of the (normalized) second-order potential in the vicinit of the plate. First the imaginar parts of the locked (figure ) and free components (figure 6), then their sum (figure 7). Figure 8 shows the real part of the second-order potential, with a change in sign, to make apparent the peak at the plate edge, due to the logarithmic singularit. The truncation orders NY and NZ have been taken equal to 2 and respectivel. It can be checked visuall, in figures 7 and 8, that the matching at x = ; > d is excellent. Finall figures 9 through show similar views as figures, 6 and 7, for a basin width of 2. m. The truncation order NY has been increased from 2 to 2 to keep the same spatial resolution. It must be pointed out here that m and 2. m are the specified values. The are slightl modified in the code so that the be an integer +. times the half wavelength of the second-order free waves, to avoid resonant transverse modes. So m is changed into.793 m, and 2. m is changed into 2.43 m. Even though the first-order elevations along the matching strip (x = ; > d) differ b less than.2 %, the high-frequenc contents of the locked and free components are quite different. This ma be related to the fact that the wave numbers of the participating modes are different. In particular, the number of progressive modes is 9 at the m tank width and at the 2. m one. It must also be borne in mind that the decomposition into locked and free components is not unique. When the locked and free components are added up, the same patterns as in the m width case are obtained, as confirmed b figure 2, where we also show the curve obtained for a width of m (.3 m) and NY = x Figure : Three-dimensional view of the second-order potential at the free surface. Locked component. Imaginar part. Basin width: m. NZ = ; NY = 2. 8

9 x Figure 6: Three-dimensional view of the second-order potential at the free surface. Free component. Imaginar part. Basin width: m. NZ = ; NY = x Figure 7: Three-dimensional view of the second-order potential at the free surface. Locked plus free components. Imaginar part. Basin width: m. NZ = ; NY = 2. 9

10 x Figure 8: Three-dimensional view of the second-order potential at the free surface. Locked + free components. Real part (changed in sign). Basin width: m. NZ = ; NY = Figure 9: Three-dimensional view of the second-order potential at the free surface. Locked component. Imaginar part. Basin width: 2. m. NZ = ; NY = 2.

11 Figure : Three-dimensional view of the second-order potential at the free surface. Free component. Imaginar part. Basin width: 2. m. NZ = ; NY = Figure : Three-dimensional view of the second-order potential at the free surface. Locked + free components. Imaginar part. Basin width: 2. m. NZ = ; NY = 2.

12 3 b = NY = 2 b = 2. NY = 2 b = NY = Figure 2: Second-order potential in x = for different tank widths. kh = 3; k (2d) = Comparisons with experimental measurements We now present some comparisons with second-order elevations derived from tests at BGO- First in la Sene-sur-mer. The plate, attached to one of the side-walls, has a width d of.2 m, a thickness of cm and a draft of. m, in a waterdepth of 3 m. The instrumentation consists in capacitive wave gauges along the weather side of the plate, at, 2, 4, 6, 8 and cm from the wall. The plate is submitted to regular waves of periods.88,.98,.7,.6,.24,.32 and.39 s, meaning wavelengths (for small amplitude waves) of.2,.,.8, 2., 2.4, 2.7 and 3 m. Wave steepnesses H/L (wave-height over wavelength ratio) from 2 up to 6 % are achieved. Further details on the tests can be found in Molin et al. (2a). The double frequenc component was extracted from the time series via Fourier analsis over a time window chosen a short time after arrival of the wave front (to minimize the thirdorder run-up effect). As for the calculations, the theoretical second-order elevation is given b η (2) = g η() Φ () zt 2g Φ() Φ () g Φ(2) t (36) z= The double frequenc component can be expressed as {[ η (2) 2 i ω 2 = R 3 ω4 g 4 g 3 ϕ()2 4 g ( ϕ () x )] } 2 + ϕ () 2 e 2 i ω t z= (37) Figures 3 through 6 present comparisons between the numerical and experimental values at wavelengths of.2 m,.8 m, 2.4 m and 3 m. The double frequenc elevations are normalized 2

13 b k A 2 I, where k is the wave number and A I the fundamental component of the amplitude of the incoming waves. Experimental values are derived from the measurements at the smaller wave steepnesses H/L of 2, 3 and 4 %. At higher steepnesses the run-up effect quickl appears, rendering the analsis impossible. In the numerical computations the basin width was taken equal to 2 m ( times the plate width), the waterdepth equal to half a wavelength, and the truncation orders NY and NZ equal to 2 and, for all wave periods. In the figures calculated results are shown with and without the second-order potential included, to stress out its significance. 4 T =.88 s with second-order potential without second-order potential H / L = 2 % H / L = 3 % H / L = 4 % A(2) / k AI** (m) Figure 3: Second-order elevation along the plate. Comparison between numerical and experimental values. Wave period:.88 s. 4 with second-order potential without second-order potential H / L = 2 % H / L = 3 % H / L = 4 % T =.7 s A(2) / k AI** (m) Figure 4: Same as figure 3. Wave period:.7 s. It can be seen that the agreement between numerical predictions and measurements is rather good. This is true even at the wave gauge closest to the plate edge, showing that flow separation has limited effects on the neighboring free surface. 3

14 4 T =.24 s with second-order potential without second-order potential H / L = 2 % H / L = 3 % H / L = 4 % A(2) / k AI** (m) Figure : Same as figure 3. Wave period:.24 s. 4 T =.39 s with second-order potential without second-order potential H / L = 2 % H / L = 3 % H / L = 4 % A(2) / k AI** (m) Figure 6: Same as figure 3. Wave period:.39 s. Finall figures 7 and 8 show three-dimensional views of the imaginar parts of the secondorder potential, around the plate, at the wave periods of.88 and.7 s. The are normalized in the same fashion as in the previous section. Quite complicated patterns are obtained as such low wavelengths. 4 Concluding remarks It is noteworth that, in spite of the sharp edge that induces flow separation in the phsical tank and a strong singularit in the potential idealization, good agreement has been finall reported between measured and calculated values of the free surface elevations, to second-order, even at rather short distances from the plate edge. This proves that non-potential flow effects 4

15 (m).8.6 x (m) Figure 7: Three-dimensional view of the second-order potential at the free surface. Imaginar part. Wave period:.88 s (m).8.6 x (m) Figure 8: Same as figure 7. Wave period:.7 s.

16 are confined to a rather small neighborhood of the edge. Problems of numerical convergence, with respect to the truncation orders of the series, were expected at the initiation of this stud. The turned out to be less pronounced than feared, when one considers, not the locked or free wave potentials separatel, but their sum. Acknowledgments The model tests at BGO-First were carried out under the GIS-HYDRO organization with financial support from Conseil général du Var. Appendix. Singularit at the plate edge In this appendix we determine the singularit of the second-order potential at the plate edge. The singularit at first-order is well known: the potential ϕ () behaves locall as R with R = x 2 + ( d) 2 the horizontal distance from the plate edge. So the horizontal velocit squared behaves as /R, while the vertical velocit remains finite. Hence, to the leading order, the free surface equation satisfied b in the vicinit of the plate edge can be written g z 4 ω 2 = i ω A 2 α(ω) R = i ω A2 α J (νr) dν (38) with α a complex constant and J the Bessel function of the first kind and order zero. A solution for (assuming infinite waterdepth for the sake of simplicit), is (R, z) = i ω A2 α g J (νr) ν k 2 e νz dν (39) where the integration contour at the pole k 2 must ensure that the radiation condition is satisfied. We get (R, z) = i ω [ A2 α ] J (νr) P.V. e νz dν + i π J (k 2 R) e k 2 z (4) g ν k 2 in agreement with Wehausen & Laitone 96, equation (2.6) (with a correction in sign). This represents a flow which is radial from the plate edge. Hence the plate is transparent to this flow. At the free surface (z = ), the principal valued integral can be transformed into P.V. which is easil shown to behave as ln(k 2 R) for k 2 R. Hence has a logarithmic singularit at the plate edge. 6 References J (u) du (4) u k 2 R Chau F.P. & Eatock Talor R. 992 Second order wave diffraction b a vertical clinder, J. Fluid Mech., 24, Malenica Š. & Molin B. 99 Third harmonic wave diffraction b a vertical clinder, J. Fluid Mech., 32, Malenica Š., Eatock Talor R. & Huang J.B. 999 Second-order wave diffraction b 6

17 an arra of vertical clinders, J. Fluid Mech., 39, Molin B., F. Rem, O. Kimmoun & E. Jamois, 2a The role of tertiar wave interactions in wave-bod problems, J. Fluid Mech., 28, Molin B., Jamois E., Lee C.H. & Newman J.N., 2b Non-linear wave interaction with a square clinder, Proc. 2th Int. Workshop Water Waves & Floating Bodies, Longearben. Newman J.N. 996 The second-order wave force on a vertical clinder, J. Fluid Mech., 32, Wehausen J.V. & Laitone E.V. 96 Surface Waves. Handbuch der Phsik, 9,