Lineer olmayan gemi dalgalarının ışın teorisi (ray theory) ile incelenmesi
A Ray theory approach to nonlinear ship waves at low froude numbers
- Tez No: 21817
- Danışmanlar: PROF. DR. ALİ İHSAN ALDOĞAN
- Tez Türü: Doktora
- Konular: Gemi Mühendisliği, Marine Engineering
- Anahtar Kelimeler: Belirtilmemiş.
- Yıl: 1992
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: 180
Özet
ÖZET Sunulan bu çalışmanın temel amacı, sakin suda yavaş ilerleyen bir geminin çıkardığı dalgaların ışın yöntemi ile incelenmesi ve bu yöntemle bazı teorik su hattı form larının dalga paternlerinin bulunmasıdır. Bu çalışmada; giriş ve çıkış açıları eşit olan su hattı formlarının yanısıra, giriş ve çıkış açıları farklı olan su hattı formlarının ışın yolları hesaplanmış ve dalga paternleri elde edilmiştir. Bunun için viskoz ol mayan, sıkıştı rılamaz ve irrotasyonel bir akışkan ele alınmış ve akışkanın hareketsiz olarak yüzen gemiye doğru U hızı ile aktığı, koordinat sisteminin gemiye bağlı ol duğu ve gemi ile birlikte hareket ettiği varsayılmıştır. Çalışmanın birinci ve ikinci bölümünde gemi dalga pa- terninin bulunması üzerine yapılan çalışmalar ele alın mıştır. Bölüm 3'de problemin tam formulasyonu için lineer ol mayan serbest su yüzeyi koşulu ele alınmış ve toplam dal ga potansiyeli çift-gövde hız potansiyeli 4>, ile,pertürbe a dalga hız potansiyeli
Özet (Çeviri)
SUMMARY A RAY THEORY APPROACH TO NONLINEAR SHIP WAVES AT LOW FROUDE NUMBERS The first systematic analytical study of shipwaves was initiated by William Froude C1810-1879D. The main purpose for the study of ship waves was the prediction of the wave resistance of a ship. However, in the days when Froude was investigating the aforementioned phenome non, there had not been a theory which he could use to deduce the wave resistance of the ship from the measurement of the wave pattern. The first analytical investigation of the waves generated by a disturbance travelling at a uniform speed has been made by W. Thomson (1886). Following these studies, interest was awakened in determining the power supplied to the wave system by the ship through the evaluation of wave pattern characteristics, and in the problem of predicting the wave pattern of the ships by the means of model tests as well as by an analytical theory. The studies on this subject are still a challenging area of hydrodynamics both for the theoretical hydrodynamicist and the ship designer who needs to know the wavemaking resistance of the ship and who desires to minimise it. This thesis is devoted to the development of a method of computation from which reliable estimations for the wave pattern of full -form ships moving at low speed in an otherwise calm water can be obtained. In order to obtain the wave pattern of full ship forms travelling at a uniform forward speed with a low Froude number, a geometric-optical approximation called the“ray-method”is used. The so-called ray method has originated from the geometrical theory of diffraction proposed by Keller (1953,1962) which has proved to be a useful tool in electromagnetic theory, acoustics and the study of surface gravity waves (Shen, Meyer & Keller, 1968). The ray theory applied to a ships system of waves is an asymptotic theory, valid for low Froude numbers Fn=U/V gL, where U and L are the speed and length of the ship respectively and g is the acceleration due to xigravity. The advantage of this theory is its ability to predict the wave pattern of non-thin or full hull forms. The basic assumption made in the ray theory is that the waves generated by the ship are short when compared to the scale over which the flow changes, and so the waves may be assumed to have locally the same dispersive relation as in undisturbed water. The ray theory has been described by Keller in June 1974 at the Tenth Symposium on Naval Hydrodynamics, and the actual application to ship theory has begun with Inui and Kajitani (1977), who assumed that the waves were generated as in linear theory and applied the ray theory to calculate the bending of the rays. Eggers has analysed the validity of ship wave ray theories at and near the ships surface by two alternative approaches one of which is based on the conventional free surface condition of slow ship theory supplemented by surface tension considerations following Maruo (1985) and the other based on a modified free surface condition derived also by Eggers (1981, 1989). Hermans and Brandsma have investigated a class of bi -circular forms with varying angles of entrance and have determined the wave pattern by means of the ray method and used those methods to compute the wave resistance (1989). The same approach has been followed by other authors such as Yim (1980, 1981, 1983), Chung (1984), Tulin (1984), Maruo & Ikehata (1985, 1986) and Hermans (1989), who have applied the ray theory with various implications related to the wavemaking of the ships. In this thesis the concept of rays of ship waves is described primarily as a ray theory regarding a ship trevelling at a low Froude number on the surface of an otherwise calm water which is similar to the theory of geometrical optics. The ray paths involved are the directions of the energy propagation of ship waves which are the same as the optical tracing of the waves generated by the ship. The ray theory presents two separate relations: one of them is the dispersion relation for the local wave phase and the other is the transport equation for the local wave amplitude. Assuming that the fluid is inviscid, incompressible and i rrotational, the coordinate system is fixed to the ship and the incoming velocity field is (U,0,0); the wave fronts are propagating along the wave vector k with the same phase velocity. So, the phase function S has the following properties on the free surface z=0: xiiand from the i rrotationality: V XT* = 0 one can obtain the so called Eikonal function as: k = CVS)2 which gives the wave-front aberration in wavelength units. This is also called the optical difference or aberration function. Keller has used the Doppler-shif ted frequencies as in geometrical optics in order to represent the aberration of wavefront by means of the dispersion relation. Thus, the Eikonal function can be derived as: (VS)2= (VS.Vö )4 where y plane. The rays are the characteristic curves of this dispersion relation and can be regarded as the paths along which the waves travel. The phase function can then be calculated by solving ah ordinary differential equation along those rays and the amplitude function can be derived by expanding the wave potential into a series of the descending power of wave number. The initial value of the amplitude is determined by an excitation coefficient E, which depends on the shape of the hull near the source. Hence, after the calculation only for once, of the quantities such as the location of raypaths, the phase function and the amplitude function along the rays, wave patterns for different Froude numbers are calculated with the help of the same data. The exact formulation of the problem was achieved by imposing the nonlinear free-surface condition. The total velocity potential around the ship, §Kx,y,z) was assumed to be a superposition of the double-body velocity potential (x,y,z>. For the case of low Froude numbers, the free- surface condition can be linearised around the so-called double body solution by the application of the matched asymptotic expansion to the exact problem. XiiiThe free-surface condition derived by the linearisation of the nonlinear condition around the double-body solution may be used as a low Froude number condition. It can be used to obtain an approximate solution to the nonlinear waves around a slow ship. The wave potential which has to satisfy the given boundary conditions is obtained by the so-called multiple scale method. By the application of the ray theory to the same lens shaped cylindirical hulls whose waterlines consist of two circular arcs with equal bow and stern angles (Figure 3.4). the ray paths can be investigated. In addition to the lens shaped forms, the ray paths for the forms having waterlines with unequal bow and stern angles are also obtained. The ray paths for those hull forms have been computed and plotted in Chapter 5 in figures 5.5 through 5.21. The results presented in Chapter 5 have indicated that two different surface wave systems are generated from source points such as the bow and the stern. The waves that travel along those rays are called the bow and stern wave systems respectively. Since the velocity field of the double-body flow is used for the application of the ray method to the problem, the rays of the transverse bow wave system do not enter the flow. Consequently, the main contributions are calculated as the diverging bow-wave system, diverging stern-wave system and transverse stern-wave system. As can be observed from the results for the diverging bow-wave systems in figures 5.5 to 5.9; the width of the wave patterns at any streamwise ordinate increase with the increasing bow angle ft. The ray paths near the o waterline are curved in a similar manner to the waterline and far away from the hull, they tend to approach a straigth line form asymptotically. The ray paths of diverging and transverse stern-wave system are given in figures 5.10 to 5.21. The figures indicate that the width of the wave pattern decreases for the increasing values of ft. The curvature of the ray paths of stern-wave system are less than those of the bow-wave system and can be approximated by straight lines at a relatively short distance from the stagnation point at the stern; as is in the linearized theory. The ray paths of the diverging bow-wave system for the hull shape with equal bow and stern angles ft - n/8 D have been obtained via the method mentioned in this study and has been compared with the results obtained by Huang & Eggers C 2-93 and Hermans & Brandsma E SO ] (Figure xiv5.23). It can be observed that when the bow angle of the hull increases, the results require further attention: As shown in figure 5.22; for the case of large bow angles such as ft =IV8, some of the ray paths are found to t> penetrate into the body, some are found to be reflected from the hull and some cross the other ray paths. Since such behaviour does not have any physical meaning, these results can be eliminated while calculating the wave pattern (Figure 5.22-5.23) The wave patterns of the hull forms consisting of equal and unequal bow and stern angles can also be evaluated by utilising the ray paths calculated as mentioned above. Since the waterlines of the ship forms exhibit a symmetry with respect to the centerline plane (y - 0); the computations and plots are restricted to the region where y > 0. The total free surface elevation is calculated by £(x,y)= Ç (x,y) + Ç (x,y) and the results for his total ci w wave elevation are presented for different bow angles. Some realistic wave patterns for different bow angles have been evaluated at the same value of the wave number and are presented in figures (5.24,) to (5.43,) for different Froude numbers. It also can be observed that the wave patterns are strongly affected by the changes.in the velocity field of the double body flow, especially in the amidships and quartering areas near the hull. This influence becomes more pronounced for the higher values of the bow angle and the Froude number, since the deviations are larger from the parallel flow. Therefore, in the amidships and quartering areas of the hull, the diverging bow wave system is not prominant. However, at the far field the wave pattern is more obvious. For the increasing values of Froude number for the same bow angle, an increase both in the wavelength and in the amplitude can be seen. A difference between the results presented here and those of the linearised theory is also observable in the amidships and quartering areas, and no contributions from the transverse bow-wave system are observed. In addition to the results obtained by Hermans & Brandsma and Eggers, results for hull forms with unequal bow and stern angles are obtained in this research. One of these is a theoretical hull form with ft, =n/30, ft, =n/16 and the other b k hulls are two Inuid forms C-201 and M-21, and a Seikan- series railway Ferry model presented in CI 3 and [39 3 by Inui. The evaluation of the raypaths and the wave XVpatterns of these forms, the near field wave profiles of the models have been obtained and are presented in figures (5.44,5.45,5.46) where they are compared with the experimental and analytical results given in CI]. It can be seen that the near field wave profiles obtained in this thesis by the ray theory are a better approximation to those observed by the experiments than the wave profiles computed by the linearised theory. Another important result is that the calculation of the ray paths is independent of the value of the wave number. Therefore, for a given ship geometry, merely by the calculation of quantities like the location of ray paths, the phase function and the variation of the amplitude function along the rays only once; the wave patterns and the wave resistance for various Froude numbers can be calculated with the help of the same data. XVI
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