Genişletilmiş Yumuşak Eğim Eşitlikleri İçin Sonlu Farklar Yaklaşımı

Bu çalışmada dalgaların ilerlerken geçirdikleri değişimler irdelenmiştir. Hazırlanan sayısal modelde hızlı değişen topografyalarda geçerli olan genişletilmiş yumuşak eğim eşitlikleri çözülmüştür. Bu eşitlikler sapmayı, kırınımı, sığlaşmayı, yansımayı, liman rezonansını, yüksek dereceden taban etkilerini, dalga kırılması ve taban sürtünmesine bağlı sönümleyici terimleri içermektedir. Doğrusal olmayan dalga hızı ve grup hızı, daha hassas sonuçlar elde edebilmek için çözüme dâhil edilmiştir. Mac Cormack ve Noktasal Gauss Seidel yöntemleri bu yeni yaklaşımda bir arada kullanılmıştır. Sayısal model, yarı sığlaşma alanına [1] ve kıyıya paralel dalgakıran [2] fiziksel deneylerine uygulanmış, literatürdeki sayısal model ve fiziksel deney sonuçları ile karşılaştırılmıştır. Elde edilen sonuçlar, sayısal modelin düzensiz topograflarda dalga ilerlemesini başarıyla benzeştirdiğini ortaya koymuştur.

Anahtar Kelimeler:

Sapma, Kırınım, Sayısal Model, Genişletilmiş Yumuşak Eğim Eşitliği, Sonlu Farklar Metodu

A FINITE DIFFERENCE APPROACH FOR EXTENDED MILD SLOPE EQUATION

In this study, the numerical model for the determination of transformations of waves while propagating has been presented. This numerical model was developed to solve the extended mild slope equation that is applicable to the rapidly varying topographies. It includes the effects of wave refraction, diffraction, shoaling, reflection, harbor resonance, higher order bottom configurations; dissipative terms due to wave breaking and bottom friction. Nonlinear wave celerity and group velocity were introduced in the solution to obtain results that are more accurate. Mac Cormack Method and Point Gauss Seidel Method were applied together in the proposed new solution approach. The numerical model was tested on the semicircular shoaling area [1] and shoreparallel breakwater [2]. The comparison of the numerical model in the current study and the physical experiments that are present in the literature shows the reliability of the model for wave transformations and dissipations over uneven bottoms.

Keywords:

Refraction, Diffraction, Numerical Model, Extended Mild Slope Equation, Finite Difference Method,

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Whalin, R.W., ‘The Limit of Application of Linear Wave Refraction Theory in Convergence Zone’, U.S. Army Corps of Engineers Waterways Experiment Station, Vicksburg, Report No. H-71, 329-351, 1971
Watanabe, A., Maruyama, K., ‘Numerical Modeling of Nearshore Wave Field Under Combined Refraction, Diffraction and Breaking’, Coastal Engineering in Japan, Cilt 29, 19-39, 1986.
Zhao, H., Song., Z., Xu, F., Li, R., ‘An extended time-dependent numerical model of the mild-slope equation with weakly nonlinear amplitude dispersion’, Acta Oceanologica, Cilt 29, No 2, 5-13, 2010.
Biesel, F., ‘Study of wave progression in water of gradually varying depth’, Gravity Waves, US National Bureau of Standards Circular 521, 243- 253, 1952.
Berkhoff, J. C. W., ‘Computation of Combined Refraction-Diffraction’, 13th International Conference on Coastal Engineering, ASCE, Cilt 1, 472-490, 1972.
Luke, J.C., ‘A Variational Principle for a Fluid with a Free Surface’, Journal of Fluid Mechanics, Cilt 27, 395-397,1967.
Liu, P.L.F., ‘Wave- current interactions on a slowly varying topography’, Journal of Geophysical Research, Cilt 88, No 7, 745-747, 1983.
Hsu, T.- W., Lin, T.- Y., Wen, C.C., Ou, S.- H., ‘A complementary mild- slope equation derived using higher order depth function for waves obliquely propagating on sloping bottom’, Physics of Fluids, Cilt18, No 087106, 2006.
Kirby, J.T., ‘A note on linear surface wave- current interaction over slowly varying topography’, Journal of Geophysical Research, Cilt 89, No C1, 745- 747, 1984.
Chamberlain, P.G., Porter, D., ‘The Modified Mild-Slope Equation’, Journal of Fluid Mechanics, Cilt 291, 393-407, 1995.
Maa, J.P.- Y., Hsu, T.-W., Lee, D.-Y., ‘The RIDE Model: an Enhanced Computer Program for Wave Transformation’, Ocean Engineering, Cilt 29, 1441-1458, 2002.
Radder, A.C., ‘On the Parabolic Equation Method for Water-Wave Propagation’, Journal of Fluid Mechanics, Cilt 95, 159-176, 1979.
Kirby, J.T., Dalrymple R.A., User’s Manual, Combined Refraction/ Diffraction Model, REF/DIF 1, Ver 2.3., Center for Applied Coastal Research, Dept. of Civil Engineering, Univ. of Delaware, Newark, No DE 19716, 1991.
Maa, J.P.-Y., Wang D.W.-C., ‘Wave Transformation near Virginia coast: the 1991 Halloween Northeaster’, Journal of Coastal Research, Cilt 11, No 4, 1258-1271, 1995.
Copeland, G.J.M., ‘A Practical Alternative to the Mild- Slope Wave Equation’, Coastal Engineering, Cilt 9, 125-149, 1985.
Madsen, P. A., Larsen, J., ‘An Efficient Finite- Difference Approach to the Mild- Slope Equation’, Coastal Engineering, Cilt 11, 329-351, 1987.
Panchang V.G., Pearce B.R., Wei G., Cushman –Roisin, B., ‘Solution of the Mild- Slope Wave Equation by iteration’, Applied Ocean Research, Cilt 13, No 4, 187-199, 1991.
Maa, J.P.- Y., Hwung, H-H, Hsu, T.-W., ‘A simple Wave Transformation Model, RDE-PBCG for harbor planning’. 3rd International Conference on Hydrodynamics, 407-412, 1998.
Song, ZY, Zhang, HG, Kong, J., Li, RJ, Zhang, W., ‘An efficient numerical model of hyperbolic mild- slope equation’, 26th International Conference on Offshore Mechanics and Arctic Engineering, Cilt 5, 253-258, 2007.
Bellotti, G., Cecioni, C., De Girolamo, P., ‘Simulation of small- amplitude frequency- dispersive transient waves by means of the mild slope equation’, Coastal Engineering, Cilt 55, No 6, 447- 458, 2008.
Tong, F-F., Shen, Y-M, Tang, J., Cui, L., ‘Water wave simulation in curvilinear coordinates using a time dependent mild slope equation’, Journal of Hydrodynamics, Cilt 22, No 6, 796-803, 2010.
Li, B.,Anastasiou, K., ‘Efficient Elliptic Solvers for the Mild- Slope Equation using the Multigrid Technique’, Coastal Engineering, Cilt 16, 245-266, 1992.
Walker, H.F., ‘Implementation of the GMRES Method Using Householder Transformations. SIAM’, Journal Sci. Statist. Comput. Cilt 9, 152-163, 1988.
Liu, SX, Sun, B, Sun, ZB, Li, JX, ‘Self- adaptive FEM numerical modeling of the mild- slope equation’, Applied Mathematical Modeling, Cilt 32, No 12, 2775-2791, 2008.
Maa, J.P.-Y., Maa, M.- H., Li, C., He, Q., ‘Using the Gaussian Elimination Method for large banded Matrix Equations’, Special Scientific Report No. 135, Virginia Institute of Marine Science, Gloucester Point, Va 23062, 1997.
Massel, S.R., ‘Extended Refraction- Diffraction Equation for Surface Waves’ Coastal Engineering, Cilt 23, 227-242, 1993.
Suh, K.D., Lee, C., Park, W.S., ‘Time- Dependent Equations for Wave Propagation on Rapidly Varying Topography’, Coastal Engineering, Cilt 32, 91-117, 1997.
Hsu, T.W., Wen C.C., ‘On Radiation Boundary Conditions and Wave Transformation across the Surf Zone’, China Ocean Engineering, Cilt 15, No 3, 395-406, 2001.
Isobe, M., ‘A Parabolic Equation Model for Transformation of Irregular Waves due to Refraction, Diffraction and Breaking’, Coastal Engineering in Japan, Cilt 30, No 1, 33-47, 1987.
Dally, W.R., Dean, R.G., Dalrymple, R.A., ‘Wave Height Variation across Beaches of Arbitrary Profile’, Journal of Geophysical Research, Cilt 90, No C6, 11917-11927, 1985.
Jonsson, I.G., Carlsen, N.A., ‘Experimental and Theoretical Investigations in an Oscillatory Turbulent Boundary Layers’, Journal of Hydraulic Research, Cilt 14, No 1, 45-60, 1975.
Kirby, J.T., Dalrymple, R.A., ‘An Approximate Model for Nonlinear Dispersion in Monochromatic Wave Propagation Models’, Coastal Engineering, Cilt 9, No 6, 545-561, 1986.
Behrendt Behrendt, L., A Finite Element Model for Water Wave Diffraction including Boundary Absorption and Bottom Friction, Series Paper 37, Institute of Hydrodynamics and Hydraulic Engineering, Technical University of Denmark, 1985.
Dingemans, M.W., Water Wave Propagation over Uneven Bottoms. Part 1, Linear Wave Propagation, World Scientific, Singapur, 2000.
Chen, Y, Yang, B.D., Tang, L.W., Ou, SH., Hsu, R.C., ‘Transformations of progressive waves propagating obliquely on gentle slope’, Journal of Waterway, Port, Coastal and Ocean Engineering, Cilt 130, No 4, 162- 169, 2004.
Isaacson, M., Qu, S., ‘Waves in a Harbour with Partially Reflecting Boundaries’, Coastal Engineering, Cilt 14, 193-214, 1990.
Kaya, B., Ülke, A., ‘Differential Quadrate Method for Flood Routing Using Diffusion Wave Model’, Journal of the Faculty of Engineering and Architecture of Gazi University, Cilt 27, No 2, 313-322, 2012.
İnan. A., Balas, L., ‘A Nonlinear Wave Propagation Model’, WSEAS Transactions on Mathematics, Cilt 5, No 7, 806-810, 2006.
Liu, P.L-F., Tsay, T.K., ‘Refraction-Diffraction Model for Weakly Nonlinear Water Waves’, Journal of Fluid Mechanics , Cilt 141, 265-274, 1984.
Madsen, P.A., Sorensen, O.R., ‘A new Form of the Boussinesq Equations with Improved Linear Dispersion Characteristics Part2. A Slowly Varying Bathymetry’, Coastal Engineering, Cilt 18, 183-204, 1993.