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- Permanent Link:
- http://ufdc.ufl.edu/UF00075482/00001
## Material Information- Title:
- A parabolic equation method in polar coordinates for waves in harbors
- Series Title:
- UFLCOEL-TR
- Creator:
- Kaku, Haruhiko (
*Dissertant*) University of Florida -- Coastal and Oceanographic Engineering Dept. Kirby, James Thornton (*Thesis advisor*) - Place of Publication:
- Gainesville, Fla.
- Publisher:
- University of Florida
- Publication Date:
- 1987
- Copyright Date:
- 1987
- Language:
- English
- Physical Description:
- viii, 82 leaves : ill. ; 28 cm.
## Subjects- Subjects / Keywords:
- Amplitude ( jstor )
Approximation ( jstor ) Breakwaters ( jstor ) Coordinate systems ( jstor ) Damping ( jstor ) Data lines ( jstor ) Geometric lines ( jstor ) Spherical coordinates ( jstor ) Wave diffraction ( jstor ) Waves ( jstor ) Coastal and Oceanographic Engineering thesis M.S Dissertations, Academic -- Coastal and Oceanographic Engineering -- UF Water waves -- Mathematical models ( lcsh ) - Genre:
- non-fiction ( marcgt )
## Notes- Abstract:
- This report details the development of a parabolic approximation method for surface water waves in polar coordinates, and its application to the prediction of wave conditions inside a breakwater harbor. The parabolic approximation is developed by employing a coordinate transformation to the linear mild-slope equation, after which the approximation is obtained using standard splitting methods. The linear model is developed in both small-angle and large-angle forms. The basic linear model is modified to include the effect of wave non-linearity (using a Stokes third-order formulation) and the effect of bottom boundary-layer dissipation. The model is employed to model the decay of waves incident on a breakwater harbor formed by two vertical walls lying on constant-angle radii of a circle. model results are compared to laboratory data, and reasonable agreements is found between data and the large-angle nonlinear model.
- Thesis:
- Thesis (M.S.)--University of Florida, 1987.
- Bibliography:
- Includes bibliographical references (pp. 81-82).
- Funding:
- This publication is being made available as part of the report series written by the faculty, staff, and students of the Coastal and Oceanographic Program of the Department of Civil and Coastal Engineering.
- Statement of Responsibility:
- by Haruhiko Kaku.
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- University of Florida
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- University of Florida
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A PARABOLIC EQUATION METHOD IN POLAR COORDINATES FOR WAVES IN HARBORS Haruhiko Kaku and James T. Kirby Technical Report UFL/COEL-TR/075 Coastal and Oceanographic Engineering Department University of Florida Gainesville, FL 32611 May, 1988 Abstract This report details the development of a parabolic approximation method for surface water waves in polar coordinates, and its application to the prediction of wave conditions inside a breakwater harbor. The parabolic approximation is developed by employing a coordinate transformation to the linear mild-slope equation, after which the approximation is obtained using standard splitting methods. The linear model is developed in both smallangle and large-angle forms. The basic linear model is modified to include the effect of wave nonlinearity (using a Stokes third-order formulation) and the effect of bottom boundary-layer dissipation. The model is employed to model the decay of waves incident on a breakwater harbor formed by two vertical walls lying on constant-angle radii of a circle. Model results are compared to laboratory data, and reasonable agreement is found between data and the large-angle nonlinear model. Contents 1 INTRODUCTION 1 1.1 Mild-Slope Equation ........ ............................... 2 1.2 Parabolic Approximation of the Wave Equation .................... 4 2 THE PARABOLIC APPROXIMATION IN ALTERNATE COORDINATE SYSTEMS 9 2.1 General Coordinate Transformation of the Reduced Wave Equations .. 9 2.2 Parabolic Approximation in the Alternate Coordinate System ........... 11 3 APPLICATION OF THE PARABOLIC APPROXIMATION IN THE POLAR COORDINATE SYSTEM 13 3.1 Parabolic Form in Polar Coordinate System ....................... 13 3.2 Mild-Slope Equation in Polar Coordinate System ............... 16 3.3 Additional Physical Effects ....... ........................... 17 3.4 Applicability of the Parabolic Approximations ................. 20 4 NUMERICAL SOLUTION OF THE PARABOLIC WAVE EQUATIONS 22 4.1 Numerical Scheme for the Parabolic Approximations ................ 22 4.2 Examples of Numerical Solution ........................ 27 4.2.1 Effects of Wave Nonlinearity and Laminar Bottom Damping 27 4.2.2 Noise in the Higher-Order Approximation ............... 28 4.2.3 Effects of a Channel between the Breakwaters ................. 28 5 LABORATORY EXPERIMENT 45 5.1 Facility and Apparatus ..................................... 45 5.2 Experimental Procedure ....... ............................. 48 5.3 Wave Record Analysis ....... .............................. 49 6 COMPARISON BETWEEN NUMERICAL SOLUTION AND LABORATORY DATA 53 6.1 Isobe's Experiments .............................. 53 6.2 Comparison with Laboratory Data ...... ....................... 54 7 SUMMARY AND CONCLUSIONS 66 1 List of Figures 1.1 Comparison of apparent wavenumber direction for each approximation with exact value. . . . . 8 3.1 Geometry of physical domain. . . . 14 3.2 Comparisons between the exact forms (solid line) and the asymptotic forms (dot line) for a) the Bessel function of first kind, b) the Bessel function of second kind and c) the absolute value of the Hankel function. . 21 4.1 Geometry of computational domain. . . 23 4.2 Wave amplitude variation in r direction. Solid line, nonlinear with damping effect; fine dot line, nonlinear without damping effect; dot line, linear with damping effect; h = 0.1m, Ao =0.01m . . 29 4.3 Wave amplitude variation in r direction. Solid line, nonlinear with damping effect; fine dot line, nonlinear without damping effect; dot line, linear with damping effect; h = 0.1m, AO = 0.02m . . 30 4.4 Wave amplitude variation in r direction. Solid line, nonlinear with damping effect; fine dot line, nonlinear without damping effect; dot line, linear with damping effect; h = 0.15m, AO = 0.01m. . . 31 4.5 Wave amplitude variation in r direction. Solid line, nonlinear with damping effect; fine dot line, nonlinear without damping effect; dot line, linear with damping effect; h = 0.15m, Ao = 0.02m. . . 32 4.6 Wave amplitude variation in 0 direction. Solid line, nonlinear with damping effect; fine dot line, nonlinear without damping effect; dot line, linear with damping effect; h = 0.1m, AO = 0.01m . . 33 4.7 Wave amplitude variation in 0 direction. Solid line, nonlinear with damping effect; fine dot line, nonlinear without damping effect; dot line, linear with damping effect; h = 0.1m, AO = 0.02m . . 34 4.8 Wave amplitude variation in 0 direction. Solid line, nonlinear with damping effect; fine dot line, nonlinear without damping effect; dot line, linear with damping effect; h = 0.15m, AO = 0.01m. . . 35 4.9 Wave amplitude variation in 0 direction. Solid line, nonlinear with damping effect; fine dot line, nonlinear without damping effect; dot line, linear with damping effect; h = 0.15m, AO = 0.02m. . . 36 4.10 Wave amplitude contour of the nonlinear solution; Ao = 0.01m, Oo = 0*. 37 4.11 Wave amplitude contour of the nonlinear solution; AO = 0.01m, 00 = 45*. 38 4.12 Noise arising in higher-order model with filtering process (Method 1). Solid line, e = 0; fine dot line, E = 0.05; dot line, e = 0.1; h = 0.15m, AO = 0.01m. 39 ii 4.13 Noise arising in higher-order model with filtering process (Method 2). Solid line, e = 0; fine dot line, E = 0.05; dot line, e = 0.1; h = 0.15m, AO = 0.01m. 40 4.14 Depth contour and breakwater configuration. . . 41 4.15 Wave amplitude variations over the channel after a gap of breakwater. Solid line, higher-order nonlinear solution; fine dot line, lowest-order nonlinear solution; dot line, higher-order nonlinear for plane bottom (h = 0.15m); Ao = 0.01m, ro = 0.615m. Arrows indicate the position of the bank of the channel. . . . . 42 4.16 Wave amplitude contour over a channel; lowest-order approximation. ..... 43 4.17 Wave amplitude contour over a channel; higher-order approximation. ..... 44 5.1 Geometry of the wave tank for constant depth case (in metric unit). .... 46 5.2 Geometry of the wave tank with channel (in metric unit). . 47 5.3 Wave gage response and calibration curve fit by L.S.M. . 48 5.4 Flow chart of the experimental procedure. . . 50 5.5 Flow chart of analysis. . . . 51 5.6 Typical wave record and its energy spectrum. . . 52 6.1 Geometry of the wave tank of Isobe's experiments (in metric unit). 55 6.2 Comparison of higher-order nonlinear (solid line), higher-order linear (fine dot line), lowest-order nonlinear (dot line), Isobe's model (dash line), and laboratory data (x). . . . . 56 6.3 Plane bottom: comparison of higher-order nonlinear approximation (solid line), higher-order linear approximation (fine dot line), lowest-order nonlinear approximation (dot line), and laboratory data (x). Test No.1; T = 0.49sec., Ao = 0.0085m . . . . 57 6.4 Plane bottom: comparison of higher-order nonlinear approximation (solid line), higher-order linear approximation (fine dot line), lowest-order nonlinear approximation (dot line), and laboratory data (x). Test No.2; T = 0.49sec., AO = 0.017m . . . . 58 6.5 Plane bottom: comparison of higher-order nonlinear approximation (solid line), higher-order linear approximation (fine dot line), lowest-order nonlinear approximation (dot line), and laboratory data (x). Test No.3; T = 0.74sec., AO = 0.009m . . . . 59 6.6 Plane bottom: comparison of higher-order nonlinear approximation (solid line), higher-order linear approximation (fine dot line), lowest-order nonlinear approximation (dot line), and laboratory data (x). Test No.4; T = 0.74sec., AO = 0.016m . . . . 60 6.7 Plane bottom: comparison of higher-order nonlinear approximation (solid line), higher-order linear approximation (fine dot line), lowest-order nonlinear approximation (dot line), and laboratory data (x). Test No.5; T = 0.74sec., AO = 0.022m . . . . 61 6.8 Channel: comparison of higher-order nonlinear approximation (solid line), higher-order linear approximation (fine dot line), lowest-order nonlinear approximation (dot line), and laboratory data (x). Arrows indicate the positions of the bank of channel. Test No.6; T = 0.49sec., Ao = 0.0085m. 63 iii 6.9 Channel: comparison of higher-order nonlinear approximation (solid line), higher-order linear approximation (fine dot line), lowest-order nonlinear approximation (dot line), and laboratory data (x). Arrows indicate the positions of the bank of channel. Test No.7; T = 0.49sec., AO = 0.0128m. 64 6.10 Channel: comparison of higher-order nonlinear approximation (solid line), higher-order linear approximation (fine dot line), lowest-order nonlinear approximation (dot line), and laboratory data (x). Arrows indicate the positions of the bank of channel. Test No.8; T = 0.74sec., AO = 0.0105m. 65 iv Chapter 1 INTRODUCTION One of the most important tasks for coastal engineers and scientists is to obtain mathematical models which provide accurate predictions of wave conditions in coastal or offshore areas. One such model which has served as the basis for extensive recent investigations is the mild-slope equation for linear, harmonic surface gravity waves, first derived by Berkhoff (1972). The mild-slope equation describes the simultaneous refraction and diffraction of arbitrary surface wave motions in a two-dimensional domain in situations where the local water depth varies slowly with respect to the surface wavelength. In its elliptic form, however, the equation is only suitable for use in regions of small spatial extent due to practical limitations on computing times, and thus further approximate formulations have been developed. Fortunately, waves in open water in the coastal zone may be characterized by a dominant propagation direction, as utilized in ray methods and other refraction schemes. This geometrical property of the waves can be used as an appropriate reference frame for determining a coordinate system in which the wave field is described, and also establishes a framework for developing approximate models. The parabolic approximation method is based on the restriction that forward-scattered waves propagate principally in the pre-specified direction with all transverse phase variation absorbed into variation of the complex wave amplitude. Various types of parabolic approximations for surface wave motion have been developed and verified in comparison with experimental data. Using a WKB-expansion approach, Mei and Tuck (1980) derived a parabolic equation governing the leading-order amplitude for a forward-scattered wave in constant water depth. This equation was extended to include nonlinear effects by Yue and Mei (1980), who studied wave diffraction by a thin wedge and found that nonlinear diffraction of waves at grazing incidence was governed by a cubic Schr6dinger equation. For a nonconstant but slowly varying depth, Radder(1979) and Lozano and Liu (1980) first derived a parabolic equation by using a multiple-scale perturbation method. Kirby and Dalrymple (1983) extended the variable depth approximation to include the cubic nonlinearity for Stokes waves, and showed that the nonlinear model predicted the development of wave-jump conditions and reductions in amplitude in the vicinity of caustics in comparison to linear results. Their model was verified by comparing with the laboratory data of Berkhoff, Booij and Radder (1982) (Kirby and Dalrymple, 1984). A model of similar form to that of Kirby and Dalrymple was obtained by Liu and Tsay (1984), who studied wave focussing over a semicircular shoal. The numerical solutions were compared with the laboratory data of Whalin (1971) and also showed the importance of nonlinear effects in 1 the focusing zone. Alternatively, a higher-order approximation was obtained by splitting method approximation by Booij (1981), which allowed the mathematical model to treat a wider range of wave directions with respect to the principle propagation direction. Further analysis and verification of Booij's method (and extension of his model to include the cubic nonlinearity) can be found in the work of Kirby (1986). The primary purpose of this study is to outline the development of the parabolic approximation in alternate coordinate systems which are either conformal or non-conformal with respect to transformation from the Cartesian coordinates. Advantageous coordinate sytems are those in which long coastal structures lie along a coordinate line. In the case of a conformal transformation, such a boundary then imposes only normal-derivative boundary conditions on the modelled domain, and the domain does not expand or contract with distance along the boundary. Our attention is focused on the mild-slope equation or its reduced form for the constant depth case, and we further consider only applications employing a transformation to polar coordinates. Additional results employing elliptic coordinates are being prepared and will be reported separately (Kirby and Kaku, 1988). 1.1 Mild-Slope Equation In order to lay the foundation for the derivations below, we provide a derivation of the mild-slope equation in this section. For irrotational, harmonic linear waves propagating over the seabed, where the water depth is given by z = -h(x, y), the velocity potential satisfies the Laplace equation, V () z2 = 0 ; -h(x, y) < z < 0, (1.1) where Vh = -+= (x, y) are the horizontal coordinates and z is the vertical coordinate. At the free surface, the combined linearized boundary conditions yield 9 W z = 0, (1.2) where w is the angular frequency of the harmonic wave motion and g is the gravitational acceleration. The bottom boundary condition is given by = .P Vhh ; z = -h(x,y), (1.3) where Vh is the horizontal gradient operator. We assume that the velocity potential (P can be separated at leading order into a vertical part (the cross-space dependence) and a horizontal part with a time dependence (the propagating mode): 4)= F (z, h) t7(x, y, t), where q is the surface displacement which, for a harmonic wave, may be written as = O(X'Y)e-iwt, where 4 is the two-dimensional surface fluctuation. The time dependence may be dropped from the velocity potential, leading to a problem in spatial coordinates alone. We then rewrite the velocity potential as P=P(z, h) O(x, y). (1.4) 2 I The vertical part F is a function related to local depth h which is assumed to vary only slowly in space in accordance with h(x, y). Assuming that the model is being used to represent the propagating components of linear theory, we may take F to be given by F cosh k(h + z) coshkh ' where k(x, y) is wavenumber satisfying the linear dispersion relationship: W2 = gk tanh kh. (1.6) In order to obtain an equation governing the two-dimensional function 4, the Laplace equation (1.1) is integrated over the depth after multiplication with the function F, which gives -(FV2 + F-,,) dz = 0. (1.7) Applying Green's formula to the second component of the integral gives 0 f-he @F) dz = [Ff2 41Fz z-_h. (1.8) After manipulating (1.8) using (1.2) and (1.3) and substituting the result in (1.7), we obtain Vh- F2dZVh +4 FV2Fdz+k20 F2dZ=0. (1.9) -h -h h-fh We now assume that the bottom topography changes gradually so that 2 F 2 VhF = Oh2 (Vhh)2 + V'h << k2F. (1.10) The second term in (1.9) may then be neglected. Using (1.5) together with the dispersion relationship (1.6), the integral of F2 over the depth reduces to 0O F d cosh 2 k(h + z) dz=C2 (1+ 2kh CCg JF2dz=f c 5~zdZ = -1i+ h =C (1.11) fh -h cosh 2 kh 2g sinh 2kh g' where C = w/k and Cg = C(1 + 2kh/ sinh 2kh)/2. C is the wave phase speed and Cg is the group velocity. Substituting these results in (1.9) finally gives V (CCgVO) + k2CC =0, (1.12) where subscript h has been omitted from the gradient operator for further convenience. This derivation is similar to the one given by Smith and Sprinks (1975). Let us consider some limiting cases where the equation is exact. For the constant depth case, C and Cg are constant. Equation (1.12) then becomes the Helmholtz equation, V20+ k20 = 0, (1.13) which governs linear wave diffraction. For the shallow water or long wave case, C = Cg, the mild-slope equation reduces to the long wave equation for a single frequency, gV (WO7) + W20 = 0. (1.14) 3 1.2 Parabolic Approximation of the Wave Equation When we treat the surface wave field in coastal and offshore areas, it is often possible to restrict attention to waves propagating principally in a uniform direction. The parabolic approximation of the reduced wave equation is based on the assumption that the variation of wave amplitude in transverse direction of wave train is much more rapid than in the propagation direction. As a numerical problem, the parabolic equation method has the advantage of being an initial boundary value problem which may be solved by marching over the domain of interest, and relatively rough grid-size may be used. The two-dimensional function O(x, y) in the mild-slope equation (1.12) describes the surface wave motion. 4 can be given by the product of wave amplitude and phase term, 2= Aei- + C.C., f ~,(.5 where A is a wave amplitude, k- is the wavenumber vector and g is the position vector. We assume that the wave propagation direction may be identified as the x direction. Recalling the fact that the wavenumber k = |k| is a function of the slowly varying water depth h(x, y), the phase function may be defined to be only a function of x after defining some modified wave number k, which is allowed to vary only in the x direction. (1.13) is then rewritten as SAe + c.c., dx, (1.16) where A must be a complex wave amplitude to absorb the difference between the modified and actual phase, which gives A = le(-N. (1.17) The parabolic model of the mild-slope equation may be derived by defining a small parameter 6 related to the rate of amplitude modulation and also to the rate of change of depth over the space of a wavelength. We assume that the derivatives in terms of space (x, y) have the following explicit scale relations, ~- +62- a(1.18) ~8 a (1.19) ay OY' where X and Y relate to the slow spacial variations of wave field, which gives X 82X, Y=6y. These scales are based on Yue and Mei's approach. The wave amplitude A thus may be defined as a function of X and Y. Kirby and Dalrymple (1983) chose a scale for Vh as 0(82) in order to restrict a bottom variation to be locally flat up to the second order in 6; consequently, we adopt the following relationship, a ~ 6 (1 .2 0 ) 4 a(CCg) 2a(CCg) (1.21) ax ax Substituting equation (1.16) together with (1.18)-(1.21) into the mild-slope equation gives CCg(k2 E2)A+ 82 [2ikCCgAx + i(kCCg)xA + (CCgAy)y] + 64(CCgAX)X = 0. (1.22) A parabolic approximation may be obtained by neglecting the term of order 0(6"). For the case of constant depth, this equation reduces to 62[2ikAx + Ayy] + 4 Axx = 0. (1.23) The same form as (1.23) may be obtained by substituting equation (1.16) with (1.18) and (1.19) into the Helmholtz equation (1.13). The 0(62) parabolic form of (1.23) is the simplest approximation of the wave equation as given by Mei and Tuck (1980). On the other hand, a number of methods for deriving parabolic models have been proposed to make the models more accurate. Booij (1981) gave a higher-order splitting method to derive a parabolic approximation of the Helmholtz equation and the mild-slope equation with currents. We apply the same procedure here to the mild-slope equation without current. The differential equation, ak (Li) +,v =O, (1.24) Oax -r.x/ can be split exactly into two decoupled equations describing a transmitted wave field and a reflected wave field: -- = ng+,(1.25) ax = -iicr, (1.26) ax where + in which superscript + indicates the transmitted wave and indicates the reflected wave. Considering the special case where the reflected waves may be neglected then gives + (1.27) We rewrite the mild-slope equation as OZZ + (CC) 0#Z + X 20 = 0, (1.28) where the operator r.2 is given by X2 = k2 1 + k2CC a(CC )] (1.29) We define the function as = co, (1.30) 5 where e is initially unknown. Substituting (1.30) into (1.24) gives S+ [2 'x + -. + 2] = 0. (1.31) In the third term of (1.31), the second x derivative of e and the product of ex and r. may be neglected on the assumption that they are second order in the slowly varying quantities. Then, comparing (1.31) and (1.28), the second term of (1.31) gives 2 -X (CC)ICn CC0' or e2 = CC. (1.32) Substituting (1.29) into (1.32) yields a pseudo operator given by = (kCCg)2 1+ k2CC (CC, )] (1.33) Then, equation (1.25) becomes a {(kCC,) [1 + k2CC (CC, ) } iki(CC)2 1+ k2CCga(CC 4) (1.34) The pseudo operators ( and ic are related to the variables k and (CC,) and to the y derivatives. The second term inside bracket for both operators, y(CC4)/(ck2CC), is assumed to be much smaller than unity through the parabolic approximation. Using the power series expansion and retaining the first two terms then gives (kCC)2 [0+ 1A C (CCO ,#y = ik2(CCg)? + (CC2 [0 aX f 4k2CC0g~ vY JkC~ (C9 or (kCC -). + (kCC). (k2CCg)x 3i [ 2kCC, k 8k3(CCg)2 4k'i(CCg)2 4kCCgC +4k2CC (CCkv)y. = 0. (1.35) In this equation, a cross-derivative term (CCck)y. appears which expresses a diffraction effect in the x direction, and thus partially recovers the effect of the O. term. The equation can be approximated for more severe restriction on the size of y-derivative terms. In (1.34), we may consider the approximation for & as = [(kCCg) 21 1 + k2CC (CC )] (1.36) 6 This means the x derivative of [1 + i1-- -(CCg )14 is neglected. Then (1.35) reduces to 01+ 2kCC.=ik 1+ k1CC (cc, a) 2 (1.37) Taking the first two terms of power series expansion for right-hand side of (1.37) then gives 0 + 2kCC," ik] 0 2k (CCgY)y = 0. (1.38) The wave amplitude equation can then be obtained by substituting (1.16) into (1.35) and (1.38). The lower-order approximation (1.38) gives 2ikA. + i(kCC )z + 2k(k 1) A+ CC (CCgAy), = 0. (1.39) This equation has a very similar form to (1.22) after neglecting the term of order O(84). Since the modified phase function V was applied before the process of parabolic approximation in (1.22), the local wavenumber k was replaced by k. For the case of plane bottom, however, if the x coordinate is taken to coincide with the direction of bottom slope, then k = i, and both equations become identical. The higher-order approximation (1.28) yields a higher-order wave amplitude equation: 2ikA + [ikccc. + 2k(k -) A CCg i(kCC)z+ ik, 12k 3k 1)] 4k2(CCg)2 + 2k2CC 2kCCg (3k 1 +2kCC,(CCAy)yX = 0. (1.40) It is noted that, for constant depth, (1.35) reduces to 3i 1 0 iko 3y + 1 = 0, (1.41) and (1.38) reduces to Oz ikO Y, = 0. (1.42) 2k Both equations are the same forms as were derived by Kirby (1986) or Booij (1981). Booij also derived a criterion for the accuracy of each approximation. The wavenumber k is expressed by its x and y components, I and m respectively as k2 12+m2 or = [12- (-)22. (1.43) k mi Assuming that the y component of k is very small; i.e., m << k, (1.43) can be approximated by a lowest-order binomial expansion S ()2](1.44) 7 0 0 1.43 0 1.45 0 L0 1.44 cos~1(k ) CL (n) 0 0 00 30.00 60.00 90.00 sin-1 (7) Figure 1.1: Comparison of apparent wavenumber direction for each approximation with exact value. This is equivalent to (1.42) in the order of approximation. The form equivalent to (1.41) is 1 1 !(-)2 -- ~ (1.45) k 1(71}()2 This is the first Pad6 approximant of the right hand side of (1.43), as shown by Dingemans (1983). The difference between the three equations (1.43), (1.44) and (1.45) can be evaluated for fixed directions of the wavenumber vector k. The larger the direction of k with respect to the x axis, the larger the error in each approximation. Figure 1.1 shows a comparison of each approximation. If we take the maximum error as 5%, 42.6* of wave direction are allowable for (1.44) and i55.9* for (1.45) (Kirby, 1986). 8 Chapter 2 THE PARABOLIC APPROXIMATION IN ALTERNATE COORDINATE SYSTEMS Most existing mathematical models for the spatial evolution of a wave field are carried out in Cartesian coordinates. In practical situations such as the study of wave fields inside a harbor or estuary, some difficulties may arise in applying parabolic models in Cartesian coordinates to the problem because of complicated boundary shapes and also because of a loss of correspondence between the chosen propagation direction and the actual physical propagation direction. There are thus advantages to developing the mathematical models in alternate coordinate systems, which may allow a wide range of problems to be solved easily. Since the coordinate system may be chosen such that boundaries may be set on constant values of the coordinates, the problems can be solved with simple boundary conditions. In addition, the diffraction effects by some obstacles in the wave field cause the diffracted waves to propagate along the boundaries, that is, the propagation direction may also be set on the alternate coordinate. A parabolic approximation thus may be derived reasonably. In this chapter, the mild-slope equation and the Helmholtz equation in alternate coordinate systems will be discussed. 2.1 General Coordinate Transformation of the Reduced Wave Equations We consider the transformation of the mild-slope equation (1.12) into an alternate coordinate system (U, v), related to (x, y) by, (2.1) The transformation performed here is assumed to be one-to-one in the domain of relevant wave motion. The Jacobian matrix of the transformation is defined by dx xU x. du 1 dy Yu yv dvJ (2.2) 9 1 The corresponding Jacobian determinant J is given by J = zuy, yUXt. (2.3) The first derivatives of u and v with respect to x and y are given by z = , xv U, =- .(.4 Y/u =Y x (2.4) =-7. The first and second derivatives in terms of x and y are given by 8 y, av /u8a axJi u J av' a x a x (2.5) 82 1 [ 82 a2 82 = ( Jy2)9 +W (Y2 (2Jyuy,)8o YX_2 3 [( aU8~2 r'IuJ -V auav +(J yu. Juy2 J y..+ JvI/uI/)+a +(JUyU. J./U + JyuYuv J.uI);], (2.6) 82 1 [ 2 2 a2 a2 -= ( Jx2) + ( JX)8v (2Jxuxv)8o +( J x,,X JuXV -J + +(JUXUXV JxVUx + JXUzX JVx2) a]. (2.7) Substituting (2.6) and (2.7) into (1.12) gives the mild-slope equation in the alternate coordinate system (u, v): ce(CCgu)u 1 [(CC0u)v + (CC#v)u] + f(CCg#)v +J2CC, [(V2u)ou + (V2V)St + kq5 = 0, (2.8) where a = y2 y2 + x2, y = yuy + xUXV, V u = u U + u 7,= [x (ayuu 2,yuv + yvv) yv(ax 2Yxuv + Pxvv)], V2v = zz + vy, = [yu(axuu 2yxuv + )xvv) xu(aIyuu 2-yuv + Pyvv)I. 10 For the case of constant depth, (2.8) reduces to ciou 27youv + 60vv + J2 [(V2U)O" + (V2V)5v + k2#1 = 0. (2.9) This is the Helmholtz equation in the alternate coordinate system. 2.2 Parabolic Approximation in the Alternate Coordinate System We now consider a parabolization of (2.8) using the WKB approach of chapter 1. Assuming that waves are propagating in the +u direction so that the phase accumulates along lines of constant v, the phase function 0 can be obtained according to # = k~ ds = k (. 2 + y2) d U = k ,6 d U 2 k. 2 k/ 2u (2.10) in which s is physical distance along a wave ray in the propagation direction. The phase function may alternately be defined to be a function of u alone by adopting an appropriate average over v. Then, the function 4 in new coordinates (u, v) is given by 2Ae'p + c.c., f = du. (2.11) Following the procedure in chapter 1, we set the scales as a ~ +,2 U (2.12) auYu au' ~ a8 (2.13) v 8 v* The wave amplitude A is then given by A = A(UV). (2.14) Substituting (2.11) into (2.8) using (2.12) and (2.13) then gives a wave equation with terms ordered by small parameter 6: aCC,(k )2A+ J2CC (V2U)ikA+ k2A + [ -[ik#3I {2CCgAv + (CCg)vA} + J2CC0(V2v)Av] + 62 lai { (CCgkAjA)u + CCgk,62AU} + 1(CCgAv )v + J2CCg(V2u)Au] + S' [-y {(CCgAv)u + (CCgAu)v}] + S6 [c(CCgAu)u] = 0. (2.15) A parabolic equation may be obtained by neglecting the term of order O(S4) in (2.15). However, the resulting equation has a complicated form because of the existence of (and ambiguous scale position of) cross-derivative terms, and is thus much more difficult to 11 interpret than the usual Cartesian parabolic approximation. On the other hand, for the case of a conformal coordinate transformation, the Cauchy-Riemann conditions are satisfied: MU =yV, x=-y/u. Therefore, a = =J, y = V2U V2V = 0. Then, (2.8) reduces to (CCg4u)u + (CCo.). + k2JCCg4 = 0 (2.16) while (2.9) reduces to Ouu + 4O, + k2Jo = 0. (2.17) This is a variable coefficient Helmholtz equation. Also, (2.15) reduces to CCA(Jk2 (kJ)2)+62 i(CCgkAA) + iCCgkAAu + (CCgAv)v I + 6' {(CCgAu)u} = 0. (2.18) A conformal transformation thus leads to a model equation of simpler form. This result leads to some computational efficiency, and alsoexplicitely maintains the scale relations used to derive the original parabolic approximation in Cartesian coordinates. 12 Chapter 3 APPLICATION OF THE PARABOLIC APPROXIMATION IN THE POLAR COORDINATE SYSTEM 3.1 Parabolic Form in Polar Coordinate System In the remainder of this study, we will consider the physical domain shown in Figure 3.1, where two semi-infinite breakwaters are set on the lines of constant angle 0a and Ob. The waves entering the harbor through the gap are assumed to propagate radially in the +r direction so that the general concept of a parabolic approximation may be applied in a polar coordinate system (r, 0). In a polar coordinate system, r and 0 axe related to Cartesian (x, y) with a coincident origin by r = (a2 +2) 0 = arctan(-), 3; x = rcos0, y= r sin 0. (3.1) The Jacobian determinant and coefficients become J = X'y# yrze = r, o: = r 2 l = 1, -y = 0, V2U= 1 r V2v = 0. From (2.9), the Helmholtz equation written in polar coordinates is r2#rr + ror + ose + (kr)2o = 0. (3.2) 13 Breakwater segment Oa Y. Figure 3.1: Geometry of physical domain. 14 Ob z, r Incident wave crests 0 It is noted that the transformation between Cartesian coordinates and polar coordinates is not conformal. We consider another coordinate system, which represents a conformal transformation, given by 1 X2 + y2 u= In ( 2 x =roe cos 0, 0 = arctan ( y =roeu sin 0, (3.3) where ro is a constant representing a finite minimum value of r. For this choice of coordinates, the Jacobian determinant becomes J =rie2". (3.4) Using (2.17), the Helmholtz equation in (u, 0) plane becomes Ouu + 'ee + (kroeu)20 = 0. (3.5) The coordinate system (3.3) is equivalent to a polar coordinate system which has been stretched in the r direction. Equation (3.5) is transformed into (3.2) by letting r = roeu, (3.6) then, qOu = ru/.r = rr 4'UU = r(rr. + Orb) and (3.2) is recovered by substituting these values into (3.5). We now consider the parabolic approximation of the Helmholtz equation in polar coordinates. For the assumption that the principal wave direction is in the +r or +u direction, the splitting method shown previously can be applied to either (3.2) or (3.5). Here, to show the resulting form briefly, we consider (3.2) rewritten as Orr + -r + ,20 = 0, (3.7) r where ic is a pseudo operator given by [=k 1+ (kr)2 2. (3.8) Following the method of section 1.3, the higher-order approximation is given by O [r+ 1- ik] 0 [ 32+ 3i _ee+ 1 Oroo = 0O (3.9) 12r ik# r_3k2 +4r2k 4 +4(kr)2#,,=0 39 The lowest-order approximation is also obtained by following the same procedure as was done for Cartesian coordinates. The final form is given by Or + I- ik] # r2#60 = 0. (3.10) 15 3.2 Mild-Slope Equation in Polar Coordinate System The transformation of the mild-slope equation from Cartesian to polar coordinates may be carried out in two steps in order to apply the splitting method to a simple equation. First, the coordinates (3.3) are used to develop the transformation in conformal form. Then, after being approximated into a parabolic equation, the final model is reached in polar coordinates. Following the procedure presented in previous chapter for a conformal transformation, the first and second derivatives of 0 and (CCg) are obtained according to = (cos 00. sin 0#e), I =-7(sin 00t + cos 64e), (CC,)z = [cosO(CC)u sinO(CC,)e], (CCg), = [sin O(CCg)u + cos O(CCg e], 1 then, the mild-slope equation in (u, 0) plane becomes (CCgq u)u + (CCgqe)e + k2JCCgO = 0. (3.11) We now utilize the splitting method to derive a parabolic approximation. Equation (3.11) is rewritten as UU + (CC)U u + k2 J 1 + k21 J (CC a )] = 0. (3.12) Under the assumption that the principal wave direction is in +u direction, the lowest-order approximation is given by (kCCgJ 21)u 2 O + 2 ikJd 0 2 (CCe)e = 0. (3.13) 12kCC, J 2 2kCCg J2 The higher-order form is (kCCgJ 2) . 2kCCgJ i + [8(kCCJ)u (k2CCJ)u 3i (CCga)e 8k3(CCg)2j' Ak4(CC)2j2 k + 4k2Cg J(CCe .eu = 0. (3.14) By letting r = roes, the equations are transformed into a polar coordinates space, giving [(kCCg), + k I-ik i0 r 2kCC, 2r 2kCCr2 (CC9~9)9 = 0, (3.15) 16 from (3.13), and [(kCC)7+ 1 .1 #r+ 2k + --sk i [2kCCgq 2r-j + [ (kCC), (k2CCg), 3 3i (CCge)e 8k3(CCg)2r2 4k4(CCg)2r2 8k2CCgr3 4kCCgr2] +4k2CCgr2 (CCgqe)',. = 0, (3.16) from (3.14). We assume that the wave amplitude is proportional to the wave surface motion function 0 with an averaged phase function 4= Aet fdr+ c.c. (3.17) 2 We define the modified wave number k according to S1 r) =k(r, O)dO. (3.18) Substituting (3.17) into (3.15) and (3.16) gives the wave amplitude equations 2ikCCgA, + i(kCCg)r + ikCC + 2kCC,(k -i) A+ I(CCAe)e = 0, (3.19) which is the lowest-order approximation, and 2ikCCgAr + i(kCCg) + ikCCg + 2kCCg(k -T) A 2iC9,.+[~k~X+ r A [ i(kCCg)r ikr 3i 1 4k2(CC)r2 + 2k2r2 + kr3 2kr2 (3k k) (CCgAa)o + y-2 (CCg A$)$, = 0, (3.20) as the higher-order approximation 3.3 Additional Physical Effects The mathematical models discussed previously do not account for various effects which modify a physical wave field, including energy dissipation at the bottom or side-walls, and wave nonlinearity. With regard to the first effect, Booij (1981) has suggested that the term iwwo might be added to the mild-slope equation, in which w is an unspecified damping factor. The equation becomes V (CC9gV) + (k2CC, + iww)o = 0. (3.21) The effect of the extra term can be illustrated by considering a conservation law of wave energy. Letting 0 have the form = ae'l, (3.22) 2 17 where a is the real wave amplitude and 0 is the real phase function. Substituting (3.22) into (3.21) gives CCV2a + 2iCCgV Va + iCCgaV20 CCa(V b)2 + k2CCa + iwwa + V(CCg) Va + iaV(CC) V = 0. (3.23) The real part of (3.23) yields k2 (VO)2 + (CCVa) = 0. (3.24) CCa For the case of small diffraction effects, the third term of (3.24) is small, and the eikonal equation may thus be obtained in the form k2= (vt)2, (3.25) The imaginary part of (3.23) is 2CCgVt Va + CCaV24 + wwa + aV(CC) Vb = 0. (3.26) Using (3.25) together with the vector manipulation: Cgk = kCg = kg, where k = Ii~j, then gives 2d, Va + aV d + wa = 0. (3.27) Multiplying this equation by a yields the energy transport equation: V (a2 -) = -wa (3.28) The term w thus represents the energy dissipation rate. Dalrymple, Kirby and Hwang (1984) have investigated the effect of various types of localized energy dissipation and have provided corresponding expressions for the term w. For the case of energy dissipation resulting from laminar bottom boundary layer damping, the lowest-order parabolic approximation for water of constant depth becomes (following Dalrymple et al.) 2ikA5 + AYY + A= 0, (3.29) sinh 2kh + 2kh 2w( where v is the kinematic viscosity. Equation (3.29) is for constant depth case and is identical to the parabolic form of (1.23) except for the last term, which causes damping of the wave amplitude. However, it is well known that the bottom boundary layer also causes a frequency shift of the waves. Liu (1986) used a perturbation method to derive the viscous damping coefficient, which had both an imaginary part and a real part. The term obtained by Liu's method may be represented in the mild-slope equation by simply letting w be complex. The corrected form of the parabolic approximation becomes 4k3(1+i) 2ikA. + AYY + sinh +A = 0. (3.30) smnh 2kh + 2kh 2w 18 In the third term of (3.30), the real part of the damping coefficient causes a phase shift while the imaginary part causes wave amplitude damping. The magnitude of the viscous term is relatively small compared to the wave motion itself, and thus, for the case of variable depth, the term may be added to the parabolic approximation of the mild-slope equation without producing any extra terms concerned with the depth variation. The amplitude dispersion caused by increasing wave steepness is effective, especially for regions where focussed waves exist, or where the diffraction effects are strong, such as in the vicinity of the gap in a breakwater. Linear wave models tend to overpredict the wave amplitude in those areas. Kirby and Dalrymple (1983) derived a parabolic approximation of wave equation for mildly varying topography and weakly nonlinear waves using a multiplescale perturbation method developed by Yue and Mei (1980). The equation was based on Stokes wave theory and had the form of a cubic Schr5dinger equation given by 2ikCCgAz + 2k(k !)CCgA + i(kCCg).A + (CCgAy)y kCCgK'IA2 A = 0, (3.31) where K, 0 C cosh4kh+ 8 2 tanh2 kh C 8 sinh 4kh (3.31) may be applied within the region where the Stokes theory applies. This region may be delimited in terms of the Ursell number, U,, given by kiAl U (kh)3 U, increases as waves enter shallower water, and, when U,. becomes 0(1), the wave field is described by cnoidal wave theory rather than Stokes theory. In the region U < 0.32, the Stokes theory gives reasonable results (Isobe et al., 1982). The linearized form of (3.31) has the same form as (1.39). In the process of coordinate transformation, since the nonlinear term does not contain any derivative terms, the parabolic models derived previously may be modified to include this term directly. The final parabolic amplitude equations with bottom damping effect and amplitude dispersion terms are 2ikCCg A, + i(kCC), + ikCCg + 2kCC,(k J) A r +-1(CCAe)e kCCgK'|A2A + 4k3CCA 0, (3.32) ; sinh 2kh + 2kh 2w 0 3 as a lowest-order approximation, and 2ikCCgAr + [(kCCg)r + ikCCg + 2kCCg(k A r i(kCCg)r + ik,. N I k J (CCgAo) Ak2CCgr2 +2k2r2 + kr3 2kr2 (k-E Cej + (CC A)e kCCK'A12A + 4k3CCg(1 + i) _vA= 2kr2 sinh 2kh + 2kh 2w 0, (3.33) as a higher-order approximation. These equations will be solved numerically and verified by means of comparisons with laboratory data in following chapters. 19 3.4 Applicability of the Parabolic Approximations The Helmholtz equation in polar coordinates (3.2) reduces to a Bessel's equation when the problem is radially symmetric (1 = 0.). The solution for outgoing waves is given by = cH4)(kr), (3.34) where c is constant and H(1) is the Hankel function of the first kind of order zero, which is given by H((kr) = Jo(kr) + iYo(kr), (3.35) where Jo and Y are the Bessel functions of the first kind and the second kind of order zero respectively. For large kr the Bessel functions may be approximated as Jo cos(kr ), 2. ? Yo sin(kr -). (3.36) 0 r 4 The function 4 is then given by = [ eikr (337) The term inside bracket may be replaced by a wave amplitude, given by A(r) = c e rkr =cr 2. (3.38) This is a solution of the reduced form of (3.19) for constant depth and for radial symmetry, which is ik 2ikAr + -A = 0. (3.39) r It is thus apparent that the parabolic approximation may only be applied at large kr where the asymptotic form of the Hankel function is valid. Figure 3.2 shows the comparisons between the exact forms and the asymptotic forms of the Bessel functions and the Hankel function. In the plots of the Hankel function, for kr less than 2, a considerable amount of error can be recognized in the asymptotic form. This effect appears to interfere with the modelling of some of the longer wavelength cases in Chapter 6, for which kro at the harbor entrance may not satisfy the far field condition for all the angular components involved. 20 CD -IO (a) 01 .6.00 2. 00 '4.00 6.00 8'.00 I'D.00 Ckr fYrO (d(ln)fra h eslfucino is id )teb se ucin O'b 2.00 q'00 6'.o '.0 1 .0 c'J -0 CD 0.0 20.0 600 80100 0k Fiue32 Cmaios ewe heeatfrm sldlie n heaypoi fom dt ie o a h ese ucin ffrt id )th eslfucino seodkidadc)teasouevlu fte aklfucin I I21 Chapter 4 NUMERICAL SOLUTION OF THE PARABOLIC WAVE EQUATIONS 4.1 Numerical Scheme for the Parabolic Approximations The parabolic equations obtained previously may be solved numerically as an initial value problem, with initial values of A specified at r = ro and appropriate boundary conditions applied at 0 = 0a, Ob. Implicit schemes for the diffusion equation such as the Crank-Nicolson scheme are thus applicable. The computation domain, r > ro and 0a > 0 > 0b, is partitioned into a rectangular grid where each section has size Ar in the r direction and AO in the 0 direction. The parabolic equation can then be discretized onto a 6 point segment with 2 points in the r direction and 3 points in the 0 direction (see Figure 4.1). The computation proceeds in the +r direction by solving n 1st order simultaneous equations at each ith row. The Crank-Nicolson scheme is fully space-centered on the middle point marked by x in Figure 4.1. Then the first derivative of A with respect to r and second derivative with respect to 0 are given according to a~j_ (A .+l A ), ar Art ' 2A) 1 (A'+ 2A.+' + A +_t) + (A3+1 2A + A ) a02 2 (AG)2 [(+1' 1 A~ A ~) and A is given by 1 Aj = ( A ++ A3 ). 2 -7 The nonlinear term may be approximated in an iterative manner: |A12 Ay = ( + 12A.+ +|A .2 A). For the first iteration, A,. is used for i'+1. Then, after the first iteration, the result of the former iteration is used as A. 1. The number of iterations may be determined by convergency of A +1. In the higher-order approximation equation, a cross-derivative term, A~er, appears. This may be expected to play a role of diffraction effect in the r direction 22 ++ ro 0- -b ----j-ij+1+1 .... .... -... Ob Figure 4.1: Geometry of computational domain. 23 Ar r r I instead of Ar,, which was neglected in the process of deriving the approximation. The numerical scheme for this term becomes A =AO ) (A + 2A +1 + A'. j) -(A +1 2A + A'-_1)] This term may introduce spurious behavior in the computational results due to the nonphysical behavior of Fourier components with short transverse wave length; see Kirby (1986). As a result, a filtering process may be necessary. We consider two different methods of filtering in the numerical solution. The first one involves including the relationship, A* A =A _1+(1-2c)A +eA,+1, (4.1) in the scheme. Here, the parameter e is selected as the filter weighting. Then, the first derivative with respect to r is written as 8A-1 = 1 [(cA + (1 2e)A1 + CAi)-(EA 1 + (1 2e)A', + eA+) (Method 1). For the second method, we also consider (4.1), but the numerical scheme does not include this relationship. For each grid row, the amplitude variation is obtained using the basic Crank-Nicolson scheme. Then, (4.1) is applied to smooth the result before the next computational step (Method 2). The initial condition at the breakwater entrance, r = ro, is given by a plane incident wave which has uniform amplitude and phase variation corresponding to the physical distance and propagation direction. The phase function is then given by 4 = k(cos Oox + sin 0oy) = kro cos(0 0), where 0o is the propagation direction of the incident wave. For constant depth at the breakwater entrance and under the assumption that phase accumulates along the +r direction after the entrance, the initial amplitude A(ro, 0) may be given according to A(ro, 0)eikro = aeikro cos(O-eo) then, A(ro, 0) = aeikro[cos(e-0o)-1j, where a is the real value of the wave amplitude. For rigid and impermeable breakwaters, the boundary conditions are aA = 0 ;0 = 0., b. In the finite difference scheme, these are approximated by Aj=2'i+ A +l -A~ A=i-'j2l An-1 Grid size may be determined in accordance with the physical wavelength, and the step size of a single grid-space should be significantly smaller than a wavelength. In the region far 24 from origin, because our mathematical models are carried out in a Polar coordinate system, care is necessary with the size of AO, that is, the computation result may not be well enough resolved at large r if AO is taken to be too large at ro. Thus, AO should be chosen carefully in accordance with the overall computational domain. In the numerical computations, the Newton-Raphson technique was used to determine the wave number k from the linear dispersion relationship. The parabolic models obtained previously yield the difference equation, (COE1+1A* +1 COE1 A*) + I (COE2+1A* +1 + COE2'A*') 4(O2 COE3 +1G3'+1Ai+ COE3j+1G2i+1A* +l + COE3 t'G1+1A1 ] + [COE3 +1G3'A +1 COE3'G2A*' + COE3'_1G1iA _j]} + 2(A1)2Ar { COE4 ++G3+1A ++ COE4 +lG2+1A* +l + COE4+i G1+1A+] COE4 +iG3iA +i COE4,IG2A*' + COE4_,G1 A _]} -A* +1j 2 (COE5+lA*+ + COE5 A* ) +1 (COE+1A* + + COE6'A*') =0 where COE1 = 2(kCCg) , i(kCCg)i COE2 = i(kCCg),j + ri ; + 2(kCC,) (kj V), i(kCCg)r1 ik1 3i 3k' 4(k2CCg) (ri)2 2(k1)2(r)2 4k.(ri)3 2k'(ri)2' COE3'. = ; higher-order approximation 1 lowest-order approximation 2 (0)2, higher-order approximation ,C;E4sp 0, lowest-order approximation 25 cosh 4(kh)' + 8 2 tanh2 (kh). (k )4(C))8 sinh4(kh)I COE58 = ; coefficient of non-linear term 0, ;linear solution ( 4(k7)3(1 + i)(CC,)i 2 sinh 2(kh)i + 2(kh)i \ ) COE63 = ; coefficient of bottom damping effect term 0, ; without bottom damping effect G i' = (CCg)- _1 + (CCg)', G2' = (CC9) -1 + 2(CCg) + (CCg) +1, G3' = (CC9); + (CC9)j+1, ( ((kCCg).+1 (kCCg)i (kCCg)rj = Ar k,j = + , EAj-1 + (1 2E)Aj + eAj+1, ; with filtering process of Method 1 A,. ; without filtering process or filtering process of Method 2 For the constant depth case, the corresponding coefficients become, COE1' = 2ik, COE2 = , 4k(ri)3 (ri)2) COE3 = '; higher-order approximation '; lowest-order approximation 26 (2k(ri)2, ; higher-order approximation COE42 = 0, ; lowest-order approximation 4C cosh 4(kh) + 8 2 tanh2(kh) COE5 ~ C2 8 sinh'(kh) COE5i = ; coefficient of non-linear term 0, ;linear solution 4k(1+i) 2 sinh 2kh + 2kh 2w' COE6.= ; coefficient of bottom damping effect term 0, ; without bottom damping effect 4.2 Examples of Numerical Solution In order to illustrate the effects of various terms in the mathematical models, some numerical results are discussed here. 4.2.1 Effects of Wave Nonlinearity and Laminar Bottom Damping Figures 4.2-4.9 show the comparisons of the solutions obtained from the linear model without damping effect, the nonlinear model without damping effect and the nonlinear model with damping effect. The lowest-order of approximation is used and the breakwater configuration follows the experiments for constant depth as explained in chapter 5, with ro = 1.23m and 0a = -Ob = 45*. Water depths h = 0.15m and h = 0.1m are examined. The conditions for the incident waves are T = 0.5sec, and Ao = 0.01m and 0.02m. The kinematic viscosity v = 1.003 x 10-6m2/sec was used for all numerical calculations. Figures 4.2-4.5 show the wave amplitude variations along lines of constant 0, 0 = 00, 0 = 22.5* and 0 = 45*, and Figures 4.6-4.9 show the variations along lines of constant r, r = 1.83m (r/ro = 1.49), r = 2.43m (r/ro = 1.98), r = 3.03m (r/ro = 2.46). The Ursell numbers for each incident wave amplitude and water depth are U,. = 0.0339 in h = 0.1m and U, = 0.0111 in h = 0.15m for AO = 0.01m and U,. = 0.0678 in h = 0.1m and U, = 0.0222 in h = 0.15m for AO = 0.02m. The bottom damping effect is very small in any case. Because the effect is accumulated with the distance of wave propagation, its effect is more apparent in areas far from the breakwater entrance. As is evident in the mathematical model, the damping effect term in the models is a function of depth for constant k and w; then, for the same wave motion, the effect becomes more pronounced in shallower water. This may be explained by the fact that the relative thickness of the bottom boundary layer with respect to water depth increases with decreasing water depth, and thus the damping effect also increases. The effect of wave nonlinearity is effective for both cases of water depth. This effect causes amplitude reductions and phase shifts near the peak and increases amplitude near the breakwaters, indicating an increase in diffraction effects. Especially for higher incident wave 27 amplitude and larger r area, there are considerable amounts of difference in the maximum wave amplitude between linear and nonlinear solutions. The wave amplitude contours of the nonlinear solutions for A0 = 0.01m are also given in Figures 4.10 for the incident wave direction 0 = 0* and Figure 4.11 is for 00 = 45*. 4.2.2 Noise in the Higher-Order Approximation In the higher-order approximation, components with large transverse wave number appear. These components propagate in the 0 direction more rapidly than realistic longer components. Figure 4.12 shows the wave amplitude variations of higher-order approximation along the 0 direction with the solutions including filtering process Method 1. High frequency wave components can be seen as unrealistic noise. These components are shifted in phase by the filtering process, but the magnitudes are not effectively reduced. In the filtering process Method 1, the relationship (4.1) is included in the difference equation. On the other hand, with the filtering Method 2, the effect of relationship (4.1) is accumulated with the number of rows calculated. The high frequency noise is thus filtered (see Figure 4.13). This method, however, also has the effect of reducing the amplitude of low frequency components since the filter is not conservative of wave energy. In the section r/ro = 2.46 in Figure 4.13, it is apparent that in the vicinity of the center line, 0 = 0*, the filtered results fall below the unfiltered ones. The computational results, therefore, tend to underestimate maximum wave amplitude, especially in areas far from the breakwater entrance, where damping due to the filtering process has accumulated. In calculations of our models, the filter weight parameter E is selected between 0 and 0.15 depending on incident wave characteristics. 4.2.3 Effects of a Channel between the Breakwaters In order to examine the effects of varying bottom topography, we introduce a channel along the harbor centerline 0 = 0*. The configuration of the channel and the breakwaters follows the experiments described in chapter 5. Figure 4.14 shows the depth contour and the breakwater configuration (see Figure 5.3 for detail). The wave field after a gap of breakwaters is affected by means of combined effects of refraction and diffraction. Since the mild-slope equation may be applied on slowly varying bottom, a bank of the channel which forms the transition of two different depth regions, needs to be mild. In the parabolic models, where the wave propagation direction is assumed to be principally in the r direction, the above restriction is effectively more severe in the r direction than in the 0 direction. In addition, the waves radially propagating toward the r direction after diffraction by the breakwater edges are refracted toward the shallower region by the channel banks. This causes the wave number component in the 0 direction to be changed somewhat. Figure 4.15 shows the normalized amplitude variations of the lowest-order and the higher-order nonlinear models in the 0 direction. The solution without a channel, h = 0.15m, is also plotted in Figure 4.15. The corresponding amplitude contours are given in Figures 4.16 and 4.17. The plots indicate that the higher-order approximation reduces the wave amplitude near the line of 0 = 0* faster than the lowest-order approximation. This is due to the higher-order model's increased ability to allow refraction of waves towards larger angle 0. 28 C' 0n CD 0 0 = 0* 't '1.00 0.40o 0.60 1.20 2.00 0D U, CD 0o 0t CD &A0 0 0 9 = 22.5* 00 0.40 0.80 1.20 1.60 2.00 0 = 45* Do 0.O 0.80 1.20 1.60 .00 r(m) (after breakwater gap) Figure 4.2: Wave amplitude variation in r direction. Solid line, nonlinear with damping effect; fine dot line, nonlinear without damping effect; dot line, linear with damping effect; h = 0.1m, Ao = 0.01m. 29 '........... ....... . ' ............. 0 A) A0 . An AO 0n ICId 0* 4- 00 0.40 0.80 1.20 1.60 2.00 0 = 22.5* o9 1 'b. 00 A bL U)- 0.40 0.80 1.20 1.60 2.00 0 = 45' 09 1 T.o o 0.40 0.80 1.20 1.60 2.00 r(m) (after breakwater gap) Figure 4.3: Wave amplitude variation in r direction. Solid line, nonlinear with damping effect; fine dot line, nonlinear without damping effect; dot line, linear with damping effect; h = 0.1m, Ao = 0.02m. 30 1 . .-. .... ........................................................................ 0 U, 106 0= 0* .. .. . D. CD If! CD Ao AlI. 0n o9 0.40 0.80 1.20 1.60 2.00 0 = 22.5' ~*1 T. oo 0.40 0.80 1.20 2.00 M U, IAo CD b.00 0 = 45* 0.40 0.80 1.20 1.60 2.00 r(m) (after breakwater gap) Figure 4.4: Wave amplitude variation in r direction. Solid line, nonlinear with damping effect; fine dot line, nonlinear without damping effect; dot line, linear with damping effect; h = 0.15m, Ao = 0.01m. 31 ' ' ............ ... ............ ...................... 0 M 0) 0D A A 0L 0 In 0 =0 b.oo 0o o. 4 0.80 1.20 1.60 2.00o 0 = 22.50 01 0.1 .1~ -b. oo 0 In 0 0 D. tc i0 80 1.20 '.60 2.00 0 = 450 0- I -oo 0. 40 O. 80 1.20 1.60 2.00 r(m) (after breakwater gap) Figure 4.5: Wave amplitude variation in r direction. Solid line, nonlinear with damping effect; fine dot line, nonlinear without damping effect; dot line, linear with damping effect; h = 0.15m, AO = 0.02m. 32 .......... ........... ..................... ................. ............................................................. 00 = 1.49 M ro AO T~o 10.00 20.00 30.00 40.00 50o.00 0 U, 198 0 To A 9.oo ib.oo 20.oo 3b.oo 4'b.oo s'o.oo r 2.46 or (g CO 1000 i.00 20.00 30.00 40b.00 50.00 0 (degree) Figure 4.6: Wave amplitude variation in 0 direction. Solid line, nonlinear with damping effect; fine dot line, nonlinear without damping effect; dot line, linear with damping effect; h = 0.1m, Ao = 0.01m. 33 C' U-, = 1.49 0 r A .oo 10.oo 2b.oo 3b.00 4b.oo sb.oo 0 =1.98 --..... . 93.oo 10.00 20.00 3b.oo 4o.00 so.oo r 2.46 (dr daapn 0 10.0 20.0 30.0 0.1 A =.0.5.00 0 (degree) Figure 4.7: Wave amplitude variation in 0 direction. Solid line, nonlinear with damping effect; fine dot line, nonlinear without damping effect; dot line, linear with damping effect; h = 0.1m, A0 = 0.02m. 34 -I- = 1.49 Aoo 0* 0 9.00 10.00 20.00 3.oo b.oo '0.oo 0 -=1.98 M ro T.oo 10b.0o 20.00 30.oo 40.oo 50.00 0 U, =2.46 or 0 bLn CD ci300 ibo 2bo 3bI bco 'oo 0 (degree) Figure 4.8: Wave amplitude variation in 0 direction. Solid line, nonlinear with damping effect; fine dot line, nonlinear without damping effect; dot line, linear with damping effect; h = 0.15m, Ao = 0.01m. 35 U1.4 CO In ................... 0 U1.9 or A .................. 0 C 0 10.00 210.00 30.00 40.00 50.00 e d gr e Fi u e : a e m li u e a ia i n n B di e ti n S l d i e, n nl n ar w t damping~~~~~~~~~~~~~ efet2iedtlne.olna4ihutdmigefc;do6ie ierwt d a m i n e f e tr o 5 m 0 = O 0 m 36 Figre4.1: avoe ampliue caonourof the anmnonln nanou soluin 0=OOm, 0 =On 7rr 0 Hi ln i 111111111111111ilun i i i s l nnl M M tnin in inni 11 Min l I M I M inIn InnIIll nI H Figure 4.10: Wave amplitude contour of the nonlinear solution; Ao = 0.01m, 0= 0*. 37 d i l ii l il i l il l ll l l il l il i li l l il l ll l l ll l ll l il i l il i ll l i li l li l il l l ll i ll l l ill009 .i l i li i il i l 00. 00. annuniunnn llnltn unin uuu un ulin nu inlinlnn lin lln~tin/inunnl*1l Figure 4.11: Wave amplitude contour of the nonlinear solution; AO = 0.01m, Oo = 450. 38 0 in CD= 1.49 ro Sn TO=1.98 0 r IAO Dh~o ib.oo 2b.0 o_ 3b.o0 Lib.0 oo b.oo Ln A =2.46 in 0 00o 10.00 20.00 30.00 40.00 5'0.00O 0 (degree) Figure 4.12: Noise arising in higher-order model with filtering process (Method 1). Solid line, c = 0; fine dot line, c = 0.05; dot line, E = 0.1; h = 0.15m, AO = 0.O1m. 39 0 1.49 ro T.oo 1'0.00 2'b.0 3'0.00 4b.Oo s'.00 '=1.98 AOO 0 CD 0 '0.oo ib.oo 2'b.o 30.00 4'.00 5.00 0 U) U) 0 (degree) Figure 4.13: Noise arising in higher-order model with filtering process (Method 2). Solid line, E = 0; fine dot line, E = 0.05; dot line, e = 0.1; h = 0.15m, A0 = 0.Olm. 40 h = 0.075m breakwater channel ,- o0 breakwater incident waves Figure 4.14: Depth contour and breakwater configuration. 41 1 h = 0.15m h = 0.075m o ~ 2.22 AOT ... ......... 9.00 10. 00 20.00 30.00o 4b. oo 5n. o 0r 3.44 or 93.00 10 00 20.00 3b~o L0.00 I50.00 bo n 0 TO 4.66 U!) ......... .......................... 93o ib.oo 2b.00 30.00 Lib.oo 'sb.oo) 0 (degree) Figure 4.15: Wave amplitude variations over the channel after a gap of breakwater. Solid line, higher-order nonlinear solution; fine dot line, lowest-order nonlinear solution; dot line, higher-order nonlinear for plane bottom (h = 0.15m); A0 = O.O1m, r= 0.615m. Arrows indicate the position of the bank of the channel. 42 liii II II II I II III III l III I 11111111111111 I IIIIII III] II 111 1111 II III III1 III IHIII I IIH II III 111 I IH M III III III III [II IIIIE il i ll l l il l l il i l ll i li l l ll l i li l i il I ll i l ll l l ll l i il i l il i li l l li l l li l l il l li l l il i l ll l li2l l R lll lili llll lll llll llil ilil ill illl lill llli ill illl llil ilil ill till lill llil lli llil llll llil llIIlH 111111111 1111 111111!1[ Figure 4.16: Wave amplitude contour over a channel; lowest-order approximation. 43 Figure 4.11: Wave amplitude contour over a channel; higher-order approximation. 44 UHnIIuInI II IIIIHIIII111u11111mIHIIIIII IIIH mII II II mIIIIHIII un II H III II mI H M 11] 1111111111111111 HIIIIIIM B i ll ll ll ll li li li li ll il il i ll li ll ll ll li ll il il li li i li li li li il ll li li il il il il l ll li ll il il il il li iliGol i Chapter 5 LABORATORY EXPERIMENT In order to verify the accuracy of the parabolic models described in the previous chapters, a series of experiments were performed at the Coastal and Oceanographic Engineering Laboratory, University of Florida. The experiments fell into two categories depending on the type of topography used. The first category consisted of tests in constant water depth. The wave field in the region between breakwaters is then described by the Helmholtz equation with constant coefficient in Cartesian coordinates. For the second case, a channel was included in the region between the breakwaters, and the mild-slope equation was expected to describe the wave field. 5.1 Facility and Apparatus Figure 5.1 illustrates the wave tank configuration for the tests of the constant depth case. The wave tank was 7.24m x 7.24m (23.8ft x 23.8ft). The tank topography consisted of a horizontal part and a sloping part. The water depth at the wavemaker was 0.45m and decreased with a slope of 1 : 10 up to the horizontal part, which had a water depth of 0.15m. The wooden breakwaters, which were 1in thick, were placed on the horizontal part. The breakwater enclosed a 90* sector with a gap of 1.74m width. The other ends of the breakwaters were connected to the corners of the tank so that wave propagation around the ends of the breakwaters could be prevented. Sand, rocks and glued fiber mats were placed at the end-wall and the corners of side-wall and breakwaters in order to damp reflected waves inside and outside the breakwaters. A flap-type single paddle wavemaker generated the incident waves, whose direction was normal to the gap. A polar coordinate grid was drawn on the bottom. Each breakwater was placed on a line of constant 0, and the origin of the coordinates was on the horizontal part. For a second test of the parabolic model, a submerged channel was built in the wave tank as shown in Figure 5.2. The geometry of the channel was symmetric about the center line 0 = 0*. A 1 : 3 sloping bank was used as the transition between the two constant depth areas h = 0.15m and h = 0.075m. The channel was 0.87m wide at its deepest part, which was also the gap width between the breakwaters. Sample stations were then established at fixed locations on the (r, 0) grid. Wave records were obtained at each predetermined sampling station using capacitance-type wave gages mounted on steel tripods. The analog signal from the wave gage was converted into a digital signal by means of an A/D converter driven by a MICRO PDP-11 digital computer. The computer also controlled the measuring system, including calibration of gages, computa- 45 Breakwater Wavemaker h =0.15 10 3.71 3.0 x 1:10 slope 2.69 0.87 pill // Figure 5.1: Geometry of the wave tank for constant depth case (in metric unit). 46 2.69 4 1.74 2.69 / / / / / *1 Breakwater 7- h = 0.075 1 h = 0.075 Figure 5.2: Geometry of the wave tank with channel (in metric unit). 47 3.185 0.435 0.225 0.225 2.96 0.87 2.96 x 41:10 slope 0 0 0 LL Z (cm) D 0 0 150.00 175.00 200.00 225.00 250.00 A/D OUT x10 Figure 5.3: Wave gage response and calibration curve fit by L.S.M. tion of calibration coefficients by least-square method, timing the sampling frequency and period, and storing the data into a specific file. Since the capacitance-type wave gages used did not give a linear response with variation of water surface elevation, a quadratic approximation was utilized for gage calibration. Figure 5.3 shows plots of values of gage readings versus water surface elevation during static calibration, and the quadratic approximation determined by least-squares method. The calibration curve agreed with gage values with errors less than 0.05mm. 5.2 Experimental Procedure Because of the limited size of the tank, some physical effects caused by the tank boundaries such as cross-wise re-reflected waves and currents on the down wave side of the breakwaters were generated by the incident waves and often grew with time. A modulation of the incident wave amplitude possibly due to the reflected waves or to complex motion of water behind the wave maker was also observed several minutes after turning on the wave maker and, therefore, every sampling was taken within the first minute of each run. Up to 5 minutes with the wavemaker off was then allowed between two consecutive samples in order to let the water calm down. Figure 5.4 shows a flow chart of the experimental procedure. A 20Hz sampling frequency was chosen for each sample and the sampling period was 48 Table 5.1: Summary of experimental conditions Test No. Tank Bottom ro(m) Wave Period(sec) Ao(m) kAo U,. 1 0.49 0.0085 0.144 0.0089 2 0.017 0.288 0.0176 3 Plane 1.23 0.74 0.009 0.077 0.0364 4 0.016 0.137 0.0647 5 0.022 0.188 0.0888 6 0.49 0.0085 0.144 0.0089 7 Channel 0.615 0.0128 0.217 0.0132 8 0.74 0.0105 0.090 0.0425 selected between 15sec and 30sec depending on the period of the incident waves and on the location of the sampling station. The longer the wave period was, the longer the sampling period should be in order to recognize unexpected disturbances in enough number of waves. The stations far from the gap and near the breakwater sometimes required longer sampling period due to the presence of relatively larger amplitude modulations relative to the incident wave amplitude. Table 5.1 shows a summary of conditions for each experiment. 5.3 Wave Record Analysis Because of the existence of long waves at the natural frequency of the tank and higher-order harmonics, the wave record analysis must resolve the extra frequency regions not considered in the theory, although the waves beyond the gap showed quite stable trend in time and space. In analysis of the records, a Fourier transform was applied to isolate the fundamental frequency band, and then wave amplitudes were calculated for each station. Figure 5.5 shows a flow chart of wave record analysis. A typical wave record and its energy spectrum are shown in Figure 5.6. It is apparent that the record includes higher harmonics as well as some broad-banded noise. Since the theory here is for the amplitude of the fundamental frequency component, it is appropriate to filter out the range of frequencies corresponding to bound harmonics of the fundamental component generated by the wavemaker. Selecting the frequency band 1 H. ~ 3 H. in Figure 5.6(b) and transforming into time-domain gives the modified wave record shown in Figure 5.6(c). This procedure is seen to reduce the apparent modulation of the wave amplitude, leading to more stable estimates of uniform wave height. We note that several of the experiments were run with incident wave steepnesses large enough to promote the onset of Benjamin-Feir sideband instabilities. The form of the energy spectrum displayed in Figure 5.6(b) is consistant with the presence of growing side bands; this mechanism could account for most of the remaining amplitude modulation apparent in the filtered wave record shown in Figure 5.6(c). 49 no L start calibration start? Iyes read gage signal surface elevation calculation of calibration coefficients by L.S.M. input sampling freq. 'and period sampling start read gage signal convert digital signal Data File into surface elevation another sampling? yes no stop Wave gage A/D converter Figure 5.4: Flow chart of the experimental procedure. 50 input calibration coefficients start [ read Data File wave record FFT analysis select frequency band inverse FFT calculate average wave height stop Figure 5.5: Flow chart of analysis. 51 A O Co 1 (CM) CID U U U 12.00 1S.00 (b) 12.00 1 .00 1 (c) .00 3. 00 6. 00 9.00 12.00 15.00 T(sec) Figure 5.6: Typical wave record and its energy spectrum. 52 I (a) V V I I I V Y VI V V 3.00 6. 0. CD 0 00 C T(sec) E CD 9 CD I.- *1 b 0o 3.oo 6'.00 -'.00 w (H.) Hill 11 IA I 11 (cm) D 0 I I I Il ' 9 ' 1 ' i 1 , Chapter 6 COMPARISON BETWEEN NUMERICAL SOLUTION AND LABORATORY DATA In this chapter, we discuss the wave phenomena on the down wave side of the breakwaters by means of comparisons of the mathematical models and two sets of laboratory data. The first set of data is taken from the paper of Isobe (1987). The second set of data was obtained in this study, as described in the previous chapter. 6.1 Isobe's Experiments The first set of data considered here was obtained by Isobe (1987), who studied the wave refraction and diffraction problem by employing a ray-front coordinate system which is determined by considering not only refraction effects by a bottom topography but also diffraction effects by structures in the relevant domain. The geometry of the wave tank is shown in Figure 6.1. Isobe derived a parabolic approximation in ray-front coordinates following the method of Tsay and Liu (1982). The resulting equation is equivalent to equation 3.19 in the order of approximation, which is a lowest-order linear approximation. Since we use a polar coordinate system and the boundaries are placed along the lines of constant 0, our computational domain is restricted to have straight boundaries at each side of the breakwater harbor. We thus omit the second parts of the breakwaters, which are perpendicular to the shoreline, from Isobe's experimental configuration. The computational domain has the r distance the same as the first parts of the breakwaters, which are inclined to the shoreline. The domain is shown in Figure 6.1 by dash lines. The conditions of the incident waves are 9.1cm wave height, 0.83sec wave period and 18* wave propagation direction in the deep part of the tank. At the entrance of the breakwaters, the refracted incident wave angle is 16.03* using Snell's law. The incident wave directions in our domain at the first row of the domain are not constant, because the entrance line of the domain is not parallel to the depth contour. The computed results, however, show that assuming a constant value 16.030 as the incident wave direction of the first row makes little difference from the result using values at each point from Snell's law. The results shown in this chapter are obtained by using constant incident wave direction 16.03*. Considering the validity of the non-linear models, at the entrance of the breakwaters where the water depth is approximately 0.2m, the incident wave has the wave steepness and the Ursell number as 53 kAO = 0.3 and U,. = 0.13 respectively using the incident wave amplitude in the deep part of the tank, which is used in the numerical computations. The Ursell number indicates that the non-linear models may be applied to the problem. Figure 6.2 shows the comparison of the parabolic approximations and the laboratory data at the line BB'. The numerical results of our models were obtained by taking corresponding points near the line BB'. The lowest-order nonlinear approximation gives higher wave amplitude than the other approximations and laboratory data for the most part, especially on the left side of the peak, where the assumed wave propagation direction differs from the real one the most causing the worst error in the lowest-order model. The higherorder approximations give reasonable predictions in this area, and the nonlinear solution shows an appropriate amplitude reduction in the vicinity of the peak, where Isobe's model and the present linear higher-order model overpredict the results. These results point out the importance of nonlinear effects in determining the pattern of wave height in a diffracted wave field. 6.2 Comparison with Laboratory Data The second set of experiments was described in chapter 5. The laboratory data and corresponding numerical solutions of the higher-order nonlinear, the higher-order linear and the lowest-order nonlinear models are given in Figures 6.3-6.10. In the higher-order numerical solutions of longer wave period, the large wavenumber components still exist, although the filter weight parameter e = 0.15 for filtering process was used. However, the qualitative features of the solutions are apparent enough to compare to the laboratory data for most cases. Thus, we did not use larger e values in order to avoid losing physical meaning in the solutions. In general, for the constant depth case, results of the higher-order approximation agree well with laboratory data. In the section r/ro = 1.38 (Fig. 6.3-6.7), the laboratory data indicate that the diffraction effects occur at smaller 0 than the values predicted by the lowest-order model and are well predicted by the higher-order approximation except the values at 0 = 0* for T = 0.49sec, where the laboratory data give much smaller amplitude than any numerical solutions. This may be due to the reflected waves from the end wall of the tank and the incident waves forming a modulated wave envelope along the line 0 = 0*. The laboratory data along the line 0 = 0*, thus, may include larger error than that of the other position, even though a long period of time was used for wave record sampling. In the section r/rO = 1.87 for T = 0.49sec (Fig. 6.3 and 6.4), the higher-order approximation predicts the wave amplitude very well within 0* < 0 < 200 and slightly underpredicts the laboratory data at 0 = 30* and 0 = 40*. In the section r/ro = 2.2 for T = 0.49sec, the lower incident wave amplitude case (Fig. 6.3) shows that the laboratory data has milder amplitude variation in the 0 direction in the vicinity of the center than any numerical solutions while the higher incident wave amplitude case (Fig. 6.4) shows reasonable agreement with the higher-order nonlinear or lowest-order nonlinear approximation. In Figures 6.5-6.7 (T = 0.74sec), it is shown that the difference between the lowest-order approximation and the higher-order approximation is more obvious at the section of small r than at the large r. The large modulation of amplitude with 0 at small values of r is thought to be a result of problems with small kro and invalidity of the Hankel function asymptote. For this case (T = 0.74sec), kro = 10.4. The wave propagation direction at the large r is supposed to be closer to the direction of the r coordinate than at the small r so 54 0.60.50.8 :: Bc B$ o-B B Sh.3.6 h=0.4 h=0.24 Figure 6.1: Geometry of the wave tank of Isobe's experiments (in metric unit). 55 2.3 0.9 1.4 1 9.0 0.9 3.5 L n Lfl Lx , o -1.50 -'1.00 -'0.50 0~0 '.s0 1.00 1.s0 B (in) B' Figure 6.2: Comparison of higher-order nonlinear (solid line), higher-order linear (fine dot line), lowest-order nonlinear (dot line), Isobe's model (dash line), and laboratory data (x). that the error involved in the lowest-order approximation becomes small. The laboratory data are well predicted by the higher-order approximation at r/ro = 1.38 and r/ro = 1.87. The effects of wave nonlinearity is not readily apparent for the T = O.74sec cases. When a channel exists behind the breakwaters, the wave propagation direction is affected by refraction. Figures 6.8 and 6.9 show the comparison of the numerical solutions and the laboratory data for T = 0.498ec. In the section r/r0 = 2.22, there are obvious differences between the lowest-order approximation and the higher-order approximation between 0 = 100 and 0 = 250, where the bottom profile changes from the fiat part to the slope part of the channel. The lowest-order approximation shows higher wave amplitude in this area than the laboratory data while the higher-order approximations agree well with the data. The difference of the real wave propagation direction from the assumed one, which is the r direction, is not expressed properly by the lowest-order approximation. In the section 7/to = 3.44, the numerical solutions agree well with the laboratory data for A0 = 0.0085m (Fig. 6.8). Here wave nonlinearity is not effective enough to be examined in comparison with the laboratory data, and neither is the difference between the lowest-order and the higherorder approximation. For Ao = 0.0128m (Fig. 6.9), however, the higher-order nonlinear approximation seems to give the best fit. In the section r/r0 = 4.66 for both incident wave amplitude cases (Fig. 6.8 and 6.9), the laboratory data shows slightly different trend from any- approximations. Data points at 0 = 150,20* and 300 for A0 = 0.0085m, and at 9 = 150 and 200 for Ao = O.0128m have much larger values than those in the section r/r0 = 3.44. This effect is not explained in terms of the forward propagating waves covered by the parabolic approximation. 56 0 U) CX-- ..............38 =1.387 ......... 100 i.00 20.00 30.00 '.00 5'0.00 XOX 0 'b.oo 10.00 2b.0 s.oo 3boo oo s'0.00 C3 0 -1.87 U' AbOO 10.00 2b.o0 3b.= 0.0.o 50.00 0 U) C)=2.2 CD *-*r* o x 0 (dgre Figure 6.3: Plane bottom:. comparison of higher-order nonlinear approximation (solid line), higher-order linear approximation (fine dot line), lowest-order nonlinear approximation (dot line), and laboratory data (X). Test No.1; T = O.49sec., AO = 0.0085m. 57 0 U, = 1.38 Ato X 9j.oo 10.00 20.00 30.00 40.00 50.00 0 * -= 1.87 x xx 0,r 9.oo ib.oo 2b.oo 3'0.00 qb.oo 50.00 0 U, 1- =2.2 0 X. ............................... T.o 10.00 20.0o 310.00 40.oo 50.00 0 (degree) Figure 6.4: Plane bottom: comparison of higher-order nonlinear approximation (solid line), higher-order linear approximation (fine dot line), lowest-order nonlinear approximation (dot line), and laboratory data (x). Test No.2; T = 0.49sec., AO = 0.017m. 58 0 U) 0 =1.38 9oo 1'.00 20.00 3'0.00 q'.oo s'b.oo 1.87 LAL xX In o -- 0X -x ...... .oo ib.oo 2b. oo 3b.oo 'b.oo s'.oo C r-2.2 Ao . 9D3.00 10.00 2b.oo 3b.oo 4b.oo '50.00 0 (degree) Figure 6.5: Plane bottom: comparison of higher-order nonlinear approximation (solid line), higher-order linear approximation (fine dot line), lowest-order nonlinear approximation (dot line), and laboratory data (x). Test No.3; T = O.74sec., A0 = 0.009m. 59 CD U, C X 1.38 to -x 10i'.00 2'0.00 30.00 4'0.00 5'0.00 0 = 1.87 o xt U, x .o !- x 0 .OD 10.00 20.00 30.oo 40.0o 50.oo C U, 0 (degree) Figure 6.6: Plane bottom: comparison of higher-order nonlinear approximation (solid line), higher-order linear approximation (fine dot line), lowest-order nonlinear approximation (dot line), and laboratory data (x). Test No.4; T = O.74sec., A0 = O.016m. 60 -., 1.38 C-3~~~- ......r 0 CiLO .o '.00 20.00 30.oo 40.oo 50.oo -3x 1.87 to Ao =......2.. U) cb.00 10.00 20.00 3b.oo 4b.00 50.00 =2.2 ................... C. C 'bo) 10.00 20.00 30.00 40.00 50.00 0 (degree) Figure 6.7: Plane bottom: comparison of higher-order nonlinear approximation (solid line), higher-order linear approximation (fine dot line), lowest-order nonilinear approximation (dot line), and laboratory data (x). Test No.5; T = O.74sec., AO = 0.022m. 61 Results for the case of wave period T = 0.74sec is given in Figure 6.10. The higher-order approximations give large amplitude modulation at the section r/ro = 2.22 and r/ro = 3.44. For this case kro = 5.27. The parabolic approximation for this condition is thus no longer reliable. In the section r/ro = 4.66, the numerical solution give smooth line, however, the filtering process is accumulated through the computational domain, the result may not be reliable since the earlier results of the domain do not have reasonable predictions. 62 0 In o N. =2.22 o ro 0 x xx 36oo ib.oo 2b.oo 3b.oo 4b.oo sb.oo 0 U) 344 o to 0 0.0 0 20.4s0 .o =.00 0.00m. =4.66 0 to X 0 II C,0,00 ib.oo 2b.o0 30.00o qb.oo Sb.oo 0 (degree) Figure 6.8: Channel: comparison of higher-order nonlinear approximation (solid line), higher-order linear approximation (fine dot line), lowest-order nonlinear approximation (dot line), and laboratory data (X). Arrows indicate the positions of the bank of channel. Test No.6; T = O.49sec., A0 = 0.0085m. 63 0 U) I2.2 0 01 C) col =3.44 I~rL 0 r=4.66 I~oI 0D U) x 0X To 10.00 20.00 30.00 40b.00 50.00 0 (degree) Figure 6.9: Channel: comparison of higher-order nonlinear approximation (solid line), higher-order linear approximation (fine dot line), lowest-order nonlinear approximation (dot line), and laboratory data (x). Arrows indicate the positions of the bank of channel. Test No.7; T = O.49sec., A0 = 0.0128m. 64 0 .CD 2'.2o 2'~o 3oo '~o so CD to 07 . x .oo 10.00 2b.oo 30.oo 40.00 s0.oo 0 U) M ~-3.44 JAI A0 UD ~ --.. x ---- ---....---.. 0.00 10.oo 20.00 30.00 40.00 so.oo e (degree) Figure 6.10: Channel: comparison of higher-order nonlinear approximation (solid line), higher-order linear approximation (fine dot line), lowest-order nonlinear approximation (dot line), and laboratory data (x). Arrows indicate the positions of the bank of channel. Test No.8; T = 0.74sec., A0 = O.0105m. 65 Chapter 7 SUMMARY AND CONCLUSIONS The objective of this study was to develop the parabolic approximation of the mild-slope equation and the Helmholtz equation in a polar coordinate system for application to diffraction or combined refraction-diffraction problems for the wave field inside a harbor. A general coordinate transformation of the reduced wave equation was introduced, and the parabolic approximation of the mild-slope equation in the alternate coordinate system was obtained using WKB approach. Due to the existence of cross-derivative terms resulting from the transformation, the resulting parabolic equation assumes an inconvenient form. A restriction of the transformation to the conformal case reduces the complexity of the approximation to the simple form found in the Cartesian case. The parabolic approximations of the mild-slope equation and the Helmholtz equation for the transmitted wave field in a polar coordinate system were derived using a splitting method. A lowest-order and a higher-order approximation were obtained. Wave nonlinearity and bottom damping effects were added in the models as a heuristic correction drawn from previous results. The solution to the reduced form of the parabolic approximation for the radially symmetric case was related to the large kr asymptote of the Hankel function, and the applicable domain was restricted to that in which the asymptotic form of the Hankel function was valid. The Crank-Nicolson technique was used to solve the approximations numerically. Numerical results indicated that the bottom damping effects were negligible for the experimental conditions we examined, and the wave nonlinearity caused reduction of amplitude near the centerline of the breakwater entrance and increased amplitude near the breakwaters, due to enhanced diffrraction effects. The higher-order model was found to increase the rate of amplitude reduction due to diffraction at small r in comparison to the lowest-order model. Laboratory experiments were conducted to verify the parabolic approximations. The comparison between the parabolic models and the laboratory data indicated that the higherorder model predicted the wave amplitude variation well, especially when the effect of wave nonlinearity was included in the model. The lowest-order model was least accurate in the region where the wave propagation direction had large difference from the assumed one. This trend was apparent in the vicinity of the harbor entrance and over the bank of the submerged channel. The results obtained by using the higher-order model were seriously contaminated by 66 large amplitude modulations which were too long to be sufficiently damped by the filtering process. The numerical results for very small kro, where the asymptotic form of the Hankel function is not obtained, lose physical meaning; this effect is accentuated in the higher-order model, where the discrepancy between the near field form of the governing equation and the parabolic model seem to enhance the generation of the spurious lateral Fourier components in the higher-order model. The numerical solutions were also compared with a model and laboratory data from Isobe (1987). The comparison indicated the difficulty in applying the lowest-order model in a polar coordinate system to the problem because of the difference between the real wave propagation direction and the assumed one, while Isobe's model in a ray-front coordinate system was in more reasonable agreement with the laboratory data. The higher-order models, however, compared well with Isobe's data and showed improved prediction when wave nonlinearity was included in the model. Problems with the higher-order approximation resulting from the generation of spurious lateral modes (Kirby, 1986), coupled with the fact that the higher-order approximation provides only an incremental increase in angular validity of the parabolic method (see Chapter 1) have led recently to the development of methods based on Fourier decomposition which have no explicit angular restriction. This approach has been detailed for the case of Cartesian coordinates and a flat bottom by Dalrymple and Kirby (1988) and is presently being investigated in the context of the conformal coordinate transformation for application to harbor problems. 67 Bibliography [1] Berkhoff, J. C. W., Computation of combined refraction-diffraction, Proc. 13th Coastal Eng. Conf., 471-490, 1972. [2] Berkhoff, J. C. W., N. Booij, and A. C. Radder, Verification of numerical wave propagation models for simple harmonic linear waves, Coastal Eng., 6, 255-279, 1982. [3] Booij, N., Gravity waves on water with nonuniform depth and current, Rep. 81-1, Dep. of Civ. Eng., Delft Univ. of Technol., Delft, 1981. [4] Dalrymple, R. A., J. T. Kirby, and P. A. Hwang, Wave diffraction due to areas of energy dissipation, J. Waterway Port Coastal Ocean Div. Am. Soc. Civ. Eng., 110, 67-79, 1984. [5] Dalrymple, R. A. and J. T. Kirby, Models for very wide-angle water waves and wave diffraction, J. Fluid Mech., 192, 1988. [6] Dingemans, M. W.,Verification of numerical wave propagation models with field measurements: CREDIZ verification Haringvliet, Rep. W488, part 1, Delft Hydraul. Lab., Delft, 1983. [7] Isobe, M., A parabolic refraction-diffraction equation in the ray-front coordinate system, Proc. 20th Coastal Eng. Conf., 306-317, 1987. [8] Isobe, M., H. Nishimura, and K. Horikawa, Theoretical considerations on perturbation solutions for waves on permanent type, Bull. Fac. Eng., Yokohama Nat. Univ. 31, 29-57, 1982. [9] Kirby, J. T., Higher-order approximations in the parabolic equation method for water waves, J. Geophys. Res. 91, 933-952, 1986. [10] Kirby, J. T., and R. A. Dalrymple, A parabolic equation for the combined refractiondiffraction of Stokes waves by mildly varying topography, J. Fluid Mech., 136, 453-466, 1983. [11] Kirby, J. T., and R. A. Dalrymple, Verification of a parabolic equation for propagation of weakly-nonlinear waves, Coastal Eng., 8, 219-232, 1984. [12] Kirby, J.T. and H. Kaku, Parabolic approximations for water waves in conformal coordinate sytems, in preparation. [13] Liu, P. L.-F., Viscous effects on evolution of Stokes waves, J. Waterway Port Coastal Ocean Engrg., 112, 55-63, 1986. 68 [14] Liu, P. L.-F., and C. C. Mei, Water motion on a beach in the presence of a breakwater, 1, Waves, J. Geophys. Res. 81, 3079-3084, 1976. (15] Liu, P. L.-F., and T. -K. Tsay, Refraction-diffraction model for weakly nonlinear water waves, J. Fluid Mech. 141, 265-274, 1984. [161 Lozano, C., and P. L.-F. Liu, Refraction-diffraction model for linear surface water waves, J. Fluid Mech., 101, 705-720, 1980. [17] Mei, C. C., and E. 0. Tuck, Forward scattering by long thin bodies, SIAM J. Applied Math., 39, 178-191, 1980. [181 Radder, A. C., On the parabolic equation method for water wave propagation, J. Fluid Mech., 95, 159-176, 1979. [191 Smith, R., and R. Sprinks, Scattering of surface waves by a conical island, J. Fluid Mach., 72, 373-384, 1975. [20] Tsay, T.-K., and P. L.-F. Liu, Numerical solution of water-wave refraction and diffraction problems in the parabolic approximation, J. Geophys. Res., 87, 7932-7940, 1982. [211 Whalin, R. W., The limit of applicability of linear wave refraction theory in a convergence zone, R.R.H-71-3, U.S. Army Corps of Engrs., WES, Vicksburg, 1971. [22] Yue, D. K-P., and C. C. Mei, Forward diffraction of Stokes waves by a thin wedge, J. Fluid. Mech., 99, 33-52, 1980. 69 |

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