Scientia Iranica, Vol. 15, No. 3, pp 315{3 c Sharif University of Technology, June 008 Research Note Free Water Surface Oscillations in a Closed Rectangular Basin with Internal Barriers A.R. Kairi-Samani and B. Ataie-Ashtiani 1 Thee enclosed asin has certain natural frequencies of seiche, depending on the geometry of the water oundaries and the athymetry of water depths. Therefore, the variation in the water surface at a point ecomes irregular, as caused y the comination of several natural frequencies, which may e considered as the superposition of sinusoidal frequency components of dierent amplitude. This paper is mainly concerned with the motion of an incompressile irrotational uid in a closed rectangular asin with internal impervious arriers. An analytical solution is presented for predicting the characteristic of generated waves in these types of asin. The equations of free water surface oscillations and its oundary conditions are reduced to a system of linear equations, which is solved y applying the small amplitude water wave theory. The ow potential, wave amplitude, ow patterns and the natural period of waves generated in the asin with impervious internal arriers are found, ased on the asin geometry. It is shown that the natural period of the asin is strongly dependent on the location of the arriers and the size of the arrier opening. INTRODUCTION Standing waves are progressive waves reected ack on themselves and appear as alternating etween troughs and crests at a xed position. They occur in ocean asins, partly enclosed ays and seas and in estuaries. When a standing wave occurs, the surface alternately rises at one end and falls at the other end of the container (Figure 1). If dierent-sized containers are treated the same way, the period of oscillation increases as the length or depth of the container increase. Standing waves in natural asins are called seiches. Seiches can e generated y tectonic movements or winds. The free oscillations are characteristic of the system and are independent of the exciting force, except for the initial magnitude. The restoring force is provided y gravity, which returns the uid surface to its horizontal equilirium position after it is displaced y wind or pressure variations. In lakes and harours of small to medium size, the characteristic frequencies of free oscillations of the water surface (seiches) (Fig- ure 1), depend only on the asin geometry. Like water sloshing in a athtu, seiches are tide-like rises and drops in the coastal water levels of great lakes caused y prolonged strong winds that push the water toward one side of the lake, causing the water level to rise on the downwind side of the lake and to drop on the upwind side. When the wind stops, the water sloshes ack and forth, with the near-shore water level rising and falling in decreasingly small amounts on oth sides of the lake until it reaches equilirium. They occur commonly in enclosed or partially enclosed asins and are usually the result of a sudden change, or a series of intermittent-periodic changes, in atmospheric pressure or wind velocity. The period of oscillation of a seiche depends on the causative force that sets the water asin in motion and the natural or free oscillating period of the asin. Seiches can inict damage. If the natural period *. Corresponding Author, Department of Civil Engineering, Isfahan University of Technology, P.O. Box 84156, Isfahan, Iran. E-mail: akairicc.iut.ac.ir 1. Department of Civil Engineering, Sharif University of Technology, P.O. Box 11155-9313, Tehran, Iran. Figure 1. Formation of seiche.
316 A.R. Kairi-Samani and B. Ataie-Ashtiani of a moored ship matches that of a seiche, then, considerale motion will result in the moored ship. Seiches could damage structures along the coastline and create large vertical accelerations for oshore structures, such as oats, arges and oating piers. Shoreline ooding may e caused y storm surges or seiches, often occurring simultaneously with high waves. A seiche is the free oscillation of the water in a closed or semi-enclosed asin at its natural period. Seiches are frequently oserved in harours, lakes, ays and in almost any distinct asin of moderate size. They may e caused y the passage of a pressure system over the asin or y the uild-up and susequent relaxation of a wind set-up in the asin. Following initiation of the seiche, the water sloshes ack and forth until the oscillation is damped out y friction. Seiches are not apparent in the main ocean asins, proaly ecause there is no force suciently coordinated over the ocean to set a seiche in motion. If the natural period, or seiche period, is close to the period of one of the tidal species, the constituents of that species (diurnal or semidiurnal) will e amplied y resonance more than those of other species. The constituent closest to the seiche period will e amplied most of all, ut the response is still a forced oscillation, whereas a seiche is a free oscillation. A variety of seiche periods may appear in the same water level record, ecause the main ody of water may oscillate longitudinally or laterally at dierent periods. It may also oscillate, oth in the open and closed mode, if the open end is somewhat restricted, and ays and harours o the main ody of water may oscillate locally at their particular seiche periods. Seiches generally have halflives of only a few periods, ut may e frequently regenerated. The largest amplitude seiches are usually found in shallow odies of water of large horizontal extent, proaly ecause the initiating wind set-up can e greater under these conditions. Seiching is the formation of standing waves in a water ody, due to wave formation and susequent reections from the ends. These waves may e incited y earthquake motions, impulsive winds over the surface, or due to wave motions entering the asin. The various modes of seiching correspond to the natural frequency response of the water ody. There are an innite numer of seiching modes possile, from the lowest (mode 1) to innity. Realistically, the lower modes proaly occur in nature, as frictional damping aects the higher modes preferentially (higher frequency). If the rectangular asin has signicant width as well as length, oth horizontal dimensions aect the natural period given in [1]: T nm p hg n a + m 1 ; (1) where T nm is the natural free oscillation period, n and m are the modes of oscillation in longitudinal and lateral coordinates, respectively and a and are the length and width of the rectangular asin. Forces of such wave motions have een attriuted to tsunamis, surges or seiches, instaility of wind, eddy trains caused y strong currents owing across the harour entrance, and surf-eat caused y long swell. owever, the most studied are harour oscillations caused y incident waves, which have typical periods of a few minutes. Due to strong winds or long wave energy, the water ody of a harour exhiits oscillatory resonant motions, which can damage moored ships and cause navigation hazards. A numer of theoretical and numerical investigations of such resonant oscillations have een carried out, ut most of them were limited to harours with a constant depth connected to open sea. The free oscillation in closed rectangular and circular asins was analyzed y []. These solutions claried the natural periods and modes of free surface oscillations related to these special congurations. McNown [3] studied the forced oscillation in a circular harour connected to the open sea through a narrow mouth. Since the radiation eect was ruled out, the results showed a harour resonance, as it does in a closed asin. The open-sea was important in allowing for the loss of energy radiated from a harour (see [4,5]). The study of wave motion and its characteristics on a physical model of a small-oat harour was undertaken in [6]. A weakly non-linear long internal wave in a closed asin was modelled y [7]. The free oscillation of water in a lake with an elliptic oundary was studied y [8]. Ishiguro [9] developed an analytical model for the oscillations of water in a ay or lake, using an electronic network and an electric analogue computer. The study of harour resonance has een extended to take into account the eect of ottom friction y [10] and wave nonlinearity [11,1]. Ippen [1] studied free oscillation in a simple two-dimensional closed asin. As an approach to an aritrary-shaped harour with a constant depth, a numerical model, using the oundary integral element method, was developed y [13,14]. For application to real depth-varying harours, a hyrid nite element model is provided in [15] and a nite dierence model is developed in [16]. A study of irregular wave incidence was undertaken y [17,18], viscous dissipation y [19] and porous reakwater y [0]. Lee and Park [1] developed a model for the prediction of harour resonance, using the nite dierence approach. Although the application of numerical models can e used for handling any complexities in this regard, the development of an analytical solution for practical cases would e very useful for the parametric study of the free oscillation phenomenon. In this work, an analytical solution for the calculation of free water surface oscillations
Free Water Surface Oscillations 317 in a rectangular asin with internal arriers will e presented. This analytical solution could e used to verify the accuracy of numerical models and calirate experimental models, applicale in cases such as the construction of a causeway in lakes and an expansion plan for harours. The water ow is considered as an ideal and irrotational ow. Therefore, the Laplace equation governs the velocity potential function of the ow domain. The free surface oundary condition is linearized to formulate a linear set of equations for solving the small amplitude water wave in the rectangular asin. The ow potential, wave amplitude and the natural period of waves generated in the asin with impervious internal arriers are found, ased on the asin geometry. These parameters are presented for variations in asin and arrier geometries. FREE OSCILLATION IN CLOSED BASINS WIT INTERNAL BARRIERS Figure illustrates a closed rectangular asin with the width of, length of a and a constant depth of h, with two internal arriers of length u and d. The asin is divided into two semi-closed asins and it is assumed that the thickness of the arrier is small, in comparison with the other dimensions. There is an opening etween the arriers that connects the suasins. To determine the natural periods of free water surface oscillations in this asin, an ideal ow eld is considered. Therefore, the governing equation for the velocity potential function is the Laplace equation (Equation ). Also, it is assumed that the amplitude of the water surface waves is small and, therefore, a linearized form of DFSBC can e applied. r 0: () The linearized kinematics and dynamic oundary conditions of the free surface for a left semi-closed asin can e written as Equations 3-1 and 3-: 1 g t on z 0; (3-1) z on z (x; y; t): (3-) t The ottom oundary condition is as follows: w 0 on z h: (3-3) z The oundary conditions at the vicinity of arriers are as follows: u 0 on x 0 for 0 y ; (3-4) x u x 0 on x a l for 0 y d ; (3-5) u x S l on xa l for d y d +c; (3-6) u v x 0 on xa l for d + c y ; (3-7) y 0 on y 0; for 0 x a l: (3-8) In a right semi-closed asin, the linearized oundary conditions can e written as follows: 1 g t on z 0; (4-1) z on z (x; y; t): (4-) t The ottom oundary condition is as follows: w 0 on z h: (4-3) z The oundary conditions in the vicinity of the arriers are as follows: u 0 on x 0 for 0 y ; (4-4) x u x 0 on x a l for 0 y d ; (4-5) u x S r on xa l for d y d + c; (4-6) u x 0 on x a l for d + c y ; (4-7) Figure. Schematic three-dimensional closed asin with internal arriers. v y 0 on y 0; for a l x a: (4-8)
318 A.R. Kairi-Samani and B. Ataie-Ashtiani To solve the dierential equation, the method of the separation of variales is applied, hence: X(x):Y (y):z(z):t (t): (5) Therefore: 1 g (t) X(x):Y (y):z(z):dt : (6) dt By sustituting and dividing the domain in the y direction to three su-domains, the solution is: a) for 0 y d ) d y d + c g cosh k(h + z) cosh kh a l cos my cos nx a l sin t; (7) cos my cos t; (8) Velocities S l and S r are equal and are evaluated y imposing the condition that the wave velocity at x a l for a left semi-closed asin is the same as the wave velocity for a right semi-closed asin, at this point. ence: cos my c) d + c y g cosh k(h + z) cosh kh a sin t; (9) cos nx a my cos cos t; (10) The oundary conditions of this case are the same as those presented for Case a. By applying the oundary conditions for the right asin and dividing the domain in the y direction to three su-domains, as elow, the results will e given as: a) 0 y d tan na a r g cosh k(h + z) cosh kh a r +! sin nx cos my sin t; (11) a r tan na a r ) d y d + c c) d + c y a r +! sin nx cos my cos tm; (1) a r g cosh k(h + z) cosh kh a cos my cos nx a sin t; (13) my cos cos t; (14) The oundary conditions of this case are the same as those presented for Case a. For motion in oth x and y directions, at a closed asin, which is divided into two semi-closed asins, the continuity for small amplitude water waves results from equating the change in ow in the two directions to the change in storage in the control volume (Figure 3). h( x + y ) + 0: (15) t Sustitution and simplication of results, for left and right asins, yields to: T s T cl n m 1 p + hg a for d y d + c; ( " n p + hg a l m Figure 3. Selected control volume. #) 1 otherwise; (16)
Free Water Surface Oscillations 319 n m 1 T s p + hg a T cr RESULTS for d y d + c; ( " n p + hg a r m #) 1 otherwise: (17) In this section, the results of the analytical solution presented aove, for a closed rectangular asin with internal arriers, are discussed. Figure 4 shows the variation of the normalized water level,, versus xa for various values of a l a. As shown, the wavelength decreases as the ratio of a l a increases. The results are the same as those for a simple closed asin without arriers under limiting conditions of a l a equals 0 and 1. Figure 5 illustrates the variation of the normalized water level,, versus y for various Figure 4. Variation of normalized water level versus xa for various values of a l a. values of c. The wavelength decreases as the values of c and u (or d ) increase. The opening size, c, has a signicant inuence on the normalized water level () at various positions in the asin. For the limiting case of c 1, the results are the same as those in the case of a simple asin. Also, for the case of c 0, when there is no opening etween arriers, the results are the same as those for a simple half asin. Increasing the value of u decreases the wavelength and its natural period, except under limiting conditions. Figure 6 illustrates the variation of the nondimensional parameter, a n, dened as a n (ug)a, versus xa for dierent a l a and mode m n 1. From this gure and the parameter, a n, dened here, the horizontal velocity at the x coordinate is otained, knowing the dimensions of the asin. As shown in this gure, the signicant eect of a l a on the parameter, a n, can e seen. Also, the results are symmetrical (the results of a l a and 1 a l a are the same). The amplitude of variation of the velocity component, u, in an x direction, decreases as the ratio, a l a. increases. As shown, the increase in the value of a l a causes a decrease in a n and u. Figure 7 shows the variation in the maximum values of a n versus a l a. As shown, the increase in the value of a l a causes a decrease in a n and u. Figure 8 shows the variation of the nondimensional parameter, n, dened as n (vg), versus y, for dierent modes. From this gure and the parameter, n, dened here, the horizontal velocity at the y coordinate is otained, knowing the dimensions of the asin. From this gure, the signicance of the modes of oscillation on parameter n can e seen. As seen, an increase in the mode numer causes an increase in the amplitude of the horizontal velocity (v). As seen, an increase in the mode numer causes an increase in the amplitude of the horizontal velocity Figure 5. Variation of normalized water level versus y for various values of c, when u d. Figure 6. Variation of parameter ua(g) y xa for dierent a l a.
30 A.R. Kairi-Samani and B. Ataie-Ashtiani Figure 7. Variation of maximum values of parameter ua(g) versus a l a. Figure 8. Variation of parameter n v(g) versus y for dierent modes. Figure 10. Normalized natural period of coupled asin to simple asin versus a l a or a r a for dierent a ratios and m n 1. causes an increase in the natural period of the coupled asin. When the asin ecomes square, the natural period ecomes the greatest. Figure 11 illustrates thep variation of the dimensionless parameter, S n T c hg, versus al, for dierent modes. From this gure, the natural modes y varying the position of the arrier. As seen, an increase in the value of a l, decreases the non-dimensional parameter, dened aove. Figure 1 shows the normalized water level contours for dierent modes of a simple closed asin and a closed asin with an internal arrier, where ala 0:5 and c 0:, and a simple closed asin with an internal continual arrier, where ala 0:5 for dierent modes. These gures illustrate the dierences of the ow pattern among these three cases. From these gures, the signicant inuence of internal arriers on the ow map, the wavelength and, more importantly, the natural frequency of free oscillation in the asin are shown. Figure 9. Variation of the maximum values of parameter n v(g) versus dierent modes. (v). This result clearly can e seen in Figure 9. In this gure, the variation of the maximum values of n, versus mode numer, m, is presented. Figure 10 shows the normalized natural period, T cl T s (or T cr T s ), versus a l a (or a r a) for dierent values of a. As seen, an increase in the value of a Figure 11. Non-dimensional parameter T c(hg) 0:5 () variations y a l.
Free Water Surface Oscillations 31 Figure 1. The normalized water level contours for closed simple asin and closed asin with internal arrier; c 0: and a l a r. SUMMARY An analytical solution for free surface oscillations in a rectangular asin with internal arriers was presented. The water ow was considered as an ideal ow. Therefore, the Laplace equation governs the velocity potential function of the ow domain. The free surface oundary condition was linearized to formulate a linear set of equation for solving the small amplitude water wave in the rectangular asin. The ow potential, wave amplitude and the natural period of waves generated in the asin with impervious internal arriers were found, ased on the asin geometry. It was shown that arrier geometry signicantly inuences the ow map, wavelength and natural frequency of free water oscillation in the asin. NOMENCLATURE T n T wave height natural free oscillation period the largest period T cl ; T cr T s S n S l ; S r X(x); Y (y) natural period for left and right semi asins, respectively natural period of simple closed asin non-dimensional p parameter, dened as S n T c hg velocities in x direction at the station of the arrier opening for left and right semi asins, respectively functions of x and y which demonstrate ow eld or amplitude functions Z(z); T (t) functions of z and t, which demonstrate ow eld or amplitude functions a; dimensions of simple closed asin a l ; a r dimensions of semi asins in x direction dened as a n (ug)a a n a n max u ; d n n max the maximum of a n the size of up and down arriers, respectively dened as n (vg) the maximum of n
3 A.R. Kairi-Samani and B. Ataie-Ashtiani c the opening size g gravitational acceleration h water depth k wave numer l length of closed asin along the axis m; n integers show the oscillation mode t time u; v velocity components in x and y directions x; y; z coordinates REFERENCES water level (amplitude) wave frequency velocity potential function 1. Ippen, A.T., Estuary and Coastal ydrodynamics, McGraw-ill Inc., USA (198).. Lam, S.., ydrodynamics, University of Camridge, 6th Ed. (193). 3. McNown, J.C. \Waves transmission through porous structures", Journal of Waterway arour and Coastal Engrg. Division., ASCE, 100(3), pp 169-188 (195). 4. Ippen, A.T. and Goda, Y. \Wave induced oscillations in harours: The solution of a rectangular harour connected to the open sea", Report No. 59, ydrodynamics La., MIT, Camridge (1963). 5. Miles, J.W. and Munk, W.. \aror Paradox", Journal of Waterways, arors and Coastal Engrg. Division., ASCE, 87(3), pp 111-130 (1961). 6. Bottin, R.R. \Physical modelling of small-oat harours: Design experience, lessons learned, and modelling guidelines", Technical Report (ADA58700), U.S. Army Engineer Waterways Experiment Station, Vicksurg, MS, USA (199). 7. orn, D.A., Imerger, J., Ivey, G.N. and Redekopp, L.G. \A weakly nonlinear model of long internal waves in closed asins", Journal of Fluid Mechanics, Camridge University Press, 467, pp 69-87 (00). 8. Iida, F. \On the free oscillation of water in a lake of elliptic oundary", Journal of the Oceanographical Society of Japan, 1(3), pp 103-108 (1965). 9. Ishiguro, S. \An analytical method for the oscillations of water in a ay or lake, using an electronic network and an electric analogue computer", Journal of the Oceanographical Society of Japan, 11(4) pp 191-197 (1955). 10. Kostense, J.K., Meijer, K.L., Dinemans, M.W., Mynett, A.E. and Van Den Bosch, P. \Wave energy dissipation in aritrarily shaped harors of variale depth", 0th ICCE, pp 436-437 (1986). 11. Lapelletier, T.G. and Raichlen, F. \aror oscillations induced y nonlinear transient long waves", Journal of Waterway, Port, Coast and Ocean Engrg., 113(4), pp 381-400 (1987). 1. Zhou, C.P. and Cheung, Y.K. and Lee, J..W. \Response in harour due to incidence of second order low-frequency waves", Wave Motion, 13, pp 167-184 (1991). 13. wang, L.S. and Tuck, E.O. \On the oscillation of harors of aritrary shape", J. Fluid Mech., 4, pp 447-464 (1970). 14. Lee, J.J. \Wave-induced oscillations in harors of aritrary shape", J. Fluid Mech., 45, pp 375-394 (1971). 15. Mei, C.C. and Chen,.S. \yrid element method for water waves", nd Annual Symp. of Waterways, arors and Coastal Engrg. Div., ASCE, 1, pp 63-81 (1975). 16. Raichlen, F. and Naheer, E. \Wave induced oscillations of harors with variale depth", Proc. 15th ICCE., ASCE, pp 3536-3556 (1976). 17. Goda, Y., Random Seas and Design of Maritime Structure, University of Tokyo Press, Tokyo, Japan (1985). 18. Ouellet, Y. and Theriault, T. \Wave grouping eect in irregular wave agitation in harour", Jour. Wtrway., Port, Coast and Ocean Engrg., 115(3), pp 363-383 (1989). 19. Gemer, M. \Modeling dissipation in haror resonance", Coastal Engrg., 10, pp 11-5 (1986). 0. Yu, X. and Chwang, A.T. \Wave-induced oscillation in haror with porous reakwaters", Jour. Wtrway., Port, Coast and Ocean Engrg., 10(), pp 15-144 (1994). 1. Lee, J.L. and Park, C.S. \Prediction of haror resonance y the nite dierence approach", Korean Society of Coastal and Ocean Eng., Asou University, Suwon, Korea (1998).