Gemi dalga direncinin kaynak-panel yöntemiyle sayısal hesabı

Abstract

Gemilerin, sakin suda seyrederken oluşturdukları dalgalar ile dalga dirençleri arasında çok yakın ve karmaşık bir ilişki vardır. Literatürde, sözkonusu dalga direnci prob lemini inceleyen değişik teoriler vardır. Uzun gemi (thin- ship) teorisi, nar in-gemi (s lender-body) teorisi, düşük-hız (low-speed) teorisi bunlardandır. Burada, Dawson' un formülasyonuna dayanan, düşük-hız (low- speed) yaklaşımı kullanılarak gemilerin dalga dirençleri ni hesaplayan sayısal bir algoritma geliştirilmiştir. Bu sayısal yöntemde, cismin yüzeyi (tekne yüzeyi) ve serbest su yüzeyinin belli bir kısmı dörtgen elemanlarda (panel lerle) temsil edilmektedir. Daha sonra, bu paneller üze - rinde, tekne yüzeyindeki kinematik koşulu ve serbest su yüzeyindeki lineerleştirilmiş sınır koşulunu sağlayacak biçimde kaynak dağılımı gerçekleştirilir, ve buna göre kay nak şiddetleri bulunur. Bu çalışmada geliştirilen algoritma, tekne (cisim) belli, sabit bir hızda sakin suda ilerleyen hız alanını, basınç dağılımını, dalga direncini ve dalga de formasyonunu bul maktadır. Düşük Froude sayılı, dolgun gemiler için prog - ram pratik olarak çok iyi bir yaklaşım sağlamaktadır.

Ship waves and wave resistance are inter-related subjects of great importance, because of the economics and perfor mance of vessels of all sizes and types. From the contem porary viewpoint one should note environmental aspects as well, namel the rate at which energy is consumed to propel ships and the undesirable effects caused directly by shipgenerated waves on other vessels, coastal struc - tures and shorelines. Scientists and engineers have followed a variety of ways to deal with the problems in that field, hoping that a continuous effort might ultimately enable satisfactory predictions of wave resistance and optimizations of ship wave-making characteristics. In the last decade of the nineteenth century, J.H. Michell (1863-1940) presented his remarkable paper on the wave resistance problem, of a thin ship with linearized boundary conditions. Michell derived an analytic expression for the wave resistance of a ship moving in a calm, inviscid fluid. In much the same way that Froude's ideas have dominated the field of the ship model testing, Michell' s results have dominated much of the subsequent analytical study of wave resistance. The ensuing developments in this field during the twen - tieth century were led by Thomas Havelock. The evaluation generalization, and experimental comparison of Michell* s wave resistance integral has occupied the time and re sources not only of Havelock, but of an impressive number of other scientists and engineers, with practical returns to the ship builder and designers. In literature, there are several classical theories for investigating the wave resistance of ships like slender- body theory, thinship theory, low-speed theory. In this study a numerical algorithm is developed by utilizing low-speed theory. In general, the conventional Froude number based on length can be reduced to the status of a small parameter by letting the velocity diminish, -vHi - keeping the geometrical length scales of the ship fixed. This leads to a "low-speed (or slow-ship) " problem which, intuitively, involve small disturbances of the fluid and hence, might be governed by linear boundary conditions on the free surface. This is a less restric tive problem than that of the thin ship, since the geo metry of the hull need not be slender (Ships can be made geometrically slender by increasing the length, while keeping the beam and draft fixed. If the velocity is also fixed, and corresponding wavelength comparable to the beam and draft, the result is a slender-body problem involving short wavelengths which is analogous to the high-frequency strip theory of seakeeping. In this situation the Froude number based on the beam and draft will be of order one, whereas the conventional Froude number based on length will be small. It is re- fered to this as the "thin-ship" approximation). It is generally assumed that the limiting case of the slow- ship approximation is the 'double model* or 'rigid - free surface' problem, of the streaming flow past the ship hull with a homogeneous boundary condition 4>z" 0 on the plane z = 0. Waves will occur in a thin Boundary layer close to this plane and will in some sense be a singular perturbation of the double model flow. On the other hand, John L.Hess and A.M.O. Smith describ ed a general method, for calculating, with the aid of electronic computer, the incompressible potential flow arbitrary, non-lifting, three dimensional bodies, more than twenty years ago. In general, the method of solu tion of potential flow (including free surface boundary condition) can be divided into two parts. One of them, is thaf of a source density distribution on the surface of the body and solves for the distribution necessary to make the normal component of fluid- velocity zero on the boundary. Plane quadrilateral surface elements are used to approximate the body surface, and the integral equation for the source density is replaced by a set of linear algebraic equations for the values of the source density on the quadrilateral elements. When this set of equations has been solved, the flow velocity both on and off the body surface is calculated. After these calculations, second part of the method is that a local portion of the undisturbed free surface are also geometrically represented by quadrilateral panels, and it is assumed that there is a simple source density distribution on the quadrilateral panels of a local portion of the undisturbed free surface. The free sur face condition is linearized as recommended by Dawson. The source density is determined so that the boundary conditions both the body surface and a local portion of the undisturbed free surface on the panels are approxi mately satisfied. Upstream waves are prevented by the use of a one-sided, upstream, finite difference operator for the free surface condition. Now, we can describe the flow of an inviscid fluid past a fixed ship. The coordinate system is right-hand and rectangular with the z-axis directed opposite to the force of gravity, the xy-plane coincides with the un disturbed free surface. A uniform stream is coming from X = -co with the speed U. We assume that the fluid is inviscid and incompressible and the flow is irrotati- onal. Furthermore we neglect surface tension. Then a steady state flow can be described by a total velocity potential <|>(x,y,z), which satisfies the Laplace equa tion: v* Cx,y,z!> = O On the free surface z = Ç (x,y), where C is the wave elevation, the velocity potential needs to satisfy the dynamic and kinematic boundary conditions ; on z = Ç Cx,y) 4>C+d>c-=o J »xx ~y y » z ¦* Where g is the gravitational acceleration constant. By combining both dynamic and kinematic conditions on the free surface, the above equations reduce to free surface equation : g <|> + v\j>. V Cg C7 = O on S, z < Ç Cx,y) where n C n, n, n 3 is the unit outward normal vector x y * at a point of S. -X Finally, it is also required that the function 4> appro aches the uniform stream potential at infinity and that are no wauves upstream from the ship. Symbolically the radiation condition : r U x + oC - 3 \ U x + OC F 3 oC - 3 x < O I L x > O as r = Cx2+ y2+ z2)*'2 ? oo o Then the wave resistance can be computed by s where the fluid pressure p is given by the Bernoulli equation P - "g C CV$>* - l£ ] - pgz and where p is the density of water. The foregoing boundary value problem given in Laplace equation and free surface equation through radiation condition is the exact formulation for a steady wave resistance problem. It is difficult to solve this exact formulation since the free surface condition is nonlinear arid the location of the free surface is not known a priori: To solve the exact nonlinear problem, a fairly general approach can be based on the concept of systematic per turbation. It is convenient to express the total velo - city potential <|> as the sum of two potential functions $ and , and into free surface equation, the latter can be -XI- linearized about the double-model solution o by neglect ing the non-linear terms of ^» and assuming that the free surface equation holds on z » 0 instead of on the free surface, that is ; % E $ C*2 + *2D + ft CS2 + »23 + 2» C» 0; + «*;C 2 x x yx yx yy xxx y y * + 2*yC*x*x + V;\ + *x C*x + *y 3x + *; C$x + $y \ 3 g 0; = o For any function F it can be verified that xx y y l l Where the subcript 1 denotes differentiation along a streamline of double-model potential on the symmetry plane z = 0. Thus the free surface condition becomes s C$ c» *+ 2* 4>\ 3, + 4,' C«2 + ft2 } + 4>' c$2 + S2 :> ] Gil I I I *x x yx y x y y g 4>' = o Z Replacing - * to get "tVt + g ?. = 2 $i.u The differentiation with respect to 1 is performed according to four point, upstream, finite difference operator. In our present study, a numerical algorithm, parallel to that of Dawson, and a computer program depending on this algorithm are developed. The output of computer program includes the velocity field, pressure distribution, plus the wave resistance and wave elevation of the free surface. The results of -xu- computer program are given for a sphere the Wighey hull and a conventional form, of Series-60. The numerical re sults, as depicted from the figures are in good agreement with the experimental results. Numerical problems usually occur when the Froude number is either too high or too low. It is turned out that good results considerably depend on the representation of the hull and the free surface by panels. The results obtained, also, imply that the choice of the panels for the representation of the hull and the free surface should depend on the Froude number to be evalua ted. Smaller panels should be used for low Froude num - bers, while a large free surface portion is required for the high Froude numbers. It is wise to vary the panel arrangement for different Froude numbers. According to the numerical experimentas of computer program for the sphere, the following restrictions are suggested. 1. The paneled region of the free surface must be about % 50 of the body length wide. 2. The paneled region of the free surface need be ex tended clown-stream about % 50 body length. 3. The paneled region of the free surface must be ex tended about % 50 of the body length upstream from the body. 4. Smaller panels should be used near the bow and aft of the body. 5. Much smaller panels should be used neart the inter section of the body and the free surface than over the down of the body. 6. At least eight panels should be used per wave length. It is showed that the above restrictions are almost true for the Wigley hull and the Series-60. As a result we can say that : 1. The present method developed from Hess-Smith- Dawson theory is efficient for evaluating the flow field, wave pattern and wave resistance for practical ship forms, especially for low Froude numbers. -xm- 2. The calculated results depend on to a certain extent on the discretization of the hull and the free sur - face. Similar panel arrangements should therefore be used if relative merits of differents competing ship designs are to be judged. 3. The accuracy of. the calculation for low Froude numbers may be improved by increasing the number of panels on the free surface. In order to enable the calculation for the high Froude numbers, a larger portion of the free surface should be panelled. Possibly, also non-linear terms in the boundary con ditions have to be taken into account.