Hidrofoillerde derinlik ve kat-kat etkisinin incelenmesi

Abstract

Ill ÖZET Bu çalışmada, tek sıralı kat-kat halinde bulunan hidrofoillerin hid rodinamik karakteristikleri incelenmiştir. Hidrofoil tipi olarak, özellik le Ayaklı Teknelerde kullanılan profiller esas alınarak, problem önce son suz derinlik hali için çözülmüş ve daha sonra sınırlı derinlik hali ve bu haldeki basınç değişimi incelenmiştir. Konuya çözüm getirecek matematik modelleme yanında, deneysel mödelle- me de yapılmış ve ayrıca analitik sonuçların bilgisayar yardımı ile nüme rik uygulaması yapılmıştır. ` İlk bölümde, tek hidrofoil ile kat-kat hidrofoil etrafındaki akım olaylarında varolan farklılıklar, kat-kat hidrofoillerin endüstrideki uy gulamaları, buradaki çalışmanın daha önceki çalışmalardan olan farkları, kat-kat akımında karşılaşılan problem tipleri ve bu problemlerin matema tik çözüm yöntemleri, konu ile ilgili çalışmalar yapmış araştırıcılar be lirtilmektedir. İkinci bölümde, önce tek bir hidrofoilin etrafındaki akıma ait bili- aen elemânter çözümler kısaca tekrarlanmakta ve böylece kat-kat akımı için gerekli matematik alt yapı verilmektedir. Daha sonra, problemin çö zümüne esas olacak matematik modelleme yapılarak hidrpfoillef, `duble+ girdap` şeklinde tekillikler ile temsil edilerek bir genel akımın içine et hücum açısında yerleştirilmektedir. Koordinat sistemi olarak, biri sa bit ve diğeri hareketli iki eksen takımı seçilmiş ve analitik ifadeler pratikliği nedeniyle hareketli eksen takımında yazılmıştır-.. '?. '`*??' Serbest su yüzeyinden yeterli derinlikte sabit bir U,» hızı ve a hücum açısı altında ilerleyen verilen bir hidroforl kat-katına ait direkt problem, akımın ' iki -boyutlu,, daimi, potansiyel, akışkanın ise ideal (viskoz olmayan) ve sıkış tınlamaz olduğu kabulleri altında çözülmüştür. Yukarıdaki kabuller altında, toplam akım, genel akım + duble dağılımı akımı + girdap dağılımı akımı şeklinde süperpozisyon ile 'elde edilmiştir. Problemin sınır şartı ise, hidrofoillerin çıkış uçlarında hızların sonlu olması, diğer adıyla Kutta, şartıdır. `Kompleks potansiyel fonksiyonu, potansiyel ve âkım fonksiyonları, kompleks hız alanı, basınç aıanı ile ilgili tüm analitik ifadeler, boyut- suzlaştı rılmış z,. kompleks değişkeni cinsinden yazılmıştır.IV Matematik modeli oluşturan dairesel silindir kat-katına ait sıfır akım çizgileri (cisim sınırı) ve durma noktaları incelenmiş, Bernoulli prensibi ile hız alanından basınç alanı ve Blasius teoremi ile de kuv vetler sistemi hesaplanmıştır. Dairesel silindir kat-katına ait tüm çözümler verildikten sonra, hidrofoil kat-katını oluşturan konform transformasyona geçilmiştir. S ÎT Z(z) = z +m.cof ( - z ) TT S şeklindeki bu dönüşümün başarılı olabilmesi için gerekli olan sınır şar tı çıkarılmış ve bunun ayni zamanda Kutta sınır şartına karşı geldiği belirtilmiştir. Ayrica, sonsuz açıklık oranı diğer bir deyimle izole profil hali, s = oo için yeniden elde edilerek, teorinin sağlaması yapılmıştır. Dönüşümü yapılan kat -kat hidrofoiller etrafındaki hız alanı, basınç alanı ve kuvvetler sistemi hesaplanmış ve hücum açısı ile sonsuzdaki genel akım hızının, konform dönüşümde korunduğu belirtilmiştir. Matematik model lemenin verildiği bu bölüm, derinlik ve kat-kat et kisinin incelenmesi ve nümerik uygulamalar ile son bulmuştur. Üçüncü bölümde, problemin deneysel yoldan çözümü için, bir deneysel mode 11 eme yapılarak tamamen orijinal bir deney modelinin dizaynı detaylı' hesaplamaları verilmeksizin özet halinde sunulmaktadır..Alışılagelmişin dışına çıkılarak, model; çekme tankında denenebilecek şekilde geliştiril miştir. Buna, başlangıçta t. T. Ü. Gemi Araştırma Merkezi'ndeki Sirkülas yon Kanalı şartlarına uyacak şekilde genel ölçüleri seçilmiş modelin de nenmesinde, daha sonra ortaya çıkan bazı teknik zorluklar neden olmuştur. Simetrik segmental kesitli ve arka arkaya üç kanat alınarak, orta daki kanadın kat-kat akımında çalıştığı düşünülmüştür. Pek tabii ki.la- borâtuvar imkanları oranında kanat sayısının 5, 7, v.s. gibi arttırıl ması mümkündür. Kanat uzunlukları, yan oranı 5 alınarak oldukça uzun tu tulmuştur. Kanatlar heriki yandan alın levhaları ile desteklenerek bir çeşit Katamaran (ikiz tekne) oluşturulmuştur. Kat-katı temsil eden orta kanadın orta kesitinde, sırtda 11 ve yüzde 5 olmak üzere toplam lfr nok taya basıncı duyan pirizler yerleştirilmiştir. Kuvvetler sistemi ölçmeleri ise, yeni geliştirilen strain-gage tipi duyucular ile çalışan Uç-bileşenli bir dinamometre ile yapılmıştır. Dinamometreler, orta kanadın heriki yanında olup, konstrüksiyonları ba sınç iletim hortumlarının çıkışına engel olmayacak şekilde düzenlenmiş tir. Dinamometreler yardımı ile strain-gage' lerden alınan elektrik sin yalleri, bir dijital strain-indikatörde kayıt ettirilerek, daha öncedenyapılmış kalibrasyon deneyi sonuçlarına göre, kuvvet yada moment olarak elde edilebilmektedir. Deneyde kullanılan modelin kanat ve yân levhaları malzeme olarak ağaçtan yapıldıklarından, uzun süreli ve geniş bir data verecek çalış-' ma.dan kaçınılmış ve sadece modelin gerçeklenmesi ile yetiniltniştir. Şüphesiz, daha üstün laboratuvar ve maddi olanaklar ile, uzun süreli olarak suya dayanabilecek malzemeler seçerek, buradaki modeli geliştir mek mümkündür. Dördüncü ve son bölümde ise, çalışmadan elde edilen analitik, nü merik ve deneysel sonuçlar ayrı ayrı belirtilmiştir.

VI SUMMARY AN INVESTIGATION OF DEPTH AND CASCADE EFFECT ON HYDROFOILS In this study, the inviscid hydrodynamic characteristics of hydrofoils in a single row cascade are examined. For a profile which is commonly employed in hydrofoil boats, the characteristics are first determined for the case of infinite depth and then for that of finite depth. Mathematical and experimental models of the problem are presented and also the numerical application of analytical results is made through a digital computer. In the first chapter, the difference between the flows around single and cascade hydrofoils is described, industrial applications of cascade hydrofoils are mentioned, the novelty of the present study is stated, various types of problems in cascade flow and techniques for their mathematical solution are briefly described, and the pertinent literature to the subject is reviewed. In the second chapter, first all existing elemantary solutions for the flow around a single hydrofoil are recorded, and hence the mathematical background needed for the cascade flow is provided. Then, the mathematical model is described for the solution of problem. Thus, hydrofoils are represented by singularities in the form of doublet + vortex, and they are immersed with a non-zero angle of attack, a, within a uniform flow. In the fundamental system of coordinates, one moving and one fixed system of coordinates are chosen, and analytical expressions are derived with respect to the moving system of coordinates for ease of computation. The results can be also given with respect to the faxed system of coordinates through- the usual coordinate transformation. Herein, the direct problem for the cascade hydrofoil moving with a constant velocity Um at a sufficient depth from the free surface of water and with an angle of attack a is solved for the case when theVII flow is taken to be two-dimensional, steady and potential, and the fluid to be inviscid and incompressible. Under the above assumptions, the analytical functions governing^ the flow are : ? ?. (i) Uniform Flow, Y (z ) ^ u (z ~ z ) ' Al a °° a a, o U '? IT (ii) Doublet Distribution Flow, X`(z ) = - cot T - (z - z )1 2 a ` u a a, o J 2tt s r '?'.. it ' ' (iii) Vortex Distribution Flow, X (z`) ` ?i la s^-n [- ` (zn~ z )3 3 a. 2tt s a'° The total flow is obtained by superposing these flows. The boundary condition of the problem is that the velocities of flow are to be finite at the trailing edges of hydrofoils, that is, the Kutta condition. The complex variable appearing in the complex function X(za) which yields the uniform flow is non-dimensionalized, the potential and flow functions and the expressions which produce the complex velocity are obtained in non-dimensional forms. Then, the zero-streamlines (body boundary) and stagnation points of the flow are investigated, the pressure field is obtained from the velocity field by means of the Bernoulli equation, the/Solutions for the force system are found through the Blasius theorem. Thus, all the solutions of mathematical model, that is, circular cylinder cascade, are obtained, and then the conformal mapping which transforms circular cylinder cascade to cascade hydrofoil is given. To apply successfully this mapping in the form, s ît. Z(z) = z --Im.cot ( - z ) 7T S the necessary boundary conditions are established.., It is also stated that these conditions correspond to the Kutta condition. In addition, the special case for the infinite gap-ratio, that iş to say, isolated profile, is solved again so as to satisfy the results of the theory.VIII The velocity, field, the pressure field and the force system for the transformed flow are computed and it is shown that the angle of attack and the velocity of uniform flow at infinity are conserved during the conformal mapping. Further in this chapter which includes the mathematical modelling, the effects of depth and cascade are determined. The pressure coefficient at finite depth is found in terms of the atmospheric pressure and the Froude number for depth as, '2 C` ~ CL + CD + `Ph. Poo Patm. Fr2 h where Cp is the pressure coefficient at infinite depth. To. investigate the cascade effect, the variation of the pressure distribution is studied with respect to varying wing clearance s. The cascade effect increases with decreasing s-values, and also, in the case of infinite (s) the result for a isolated profile is obviously obtained. In numerical application, a computer program is written for the calculation of the pressure distribution around the cascade hydrofoils, and then the program is applied to two distinct, symmetrical hollow and Joukowsky profiles. In the calculation of pressure distribution, the angle of attack a and the wing clearance s are varied in a systematic manner and all the graphs are plotted through a digital electronic computer. In chapter three, the experimental modelling of the problem is described, and a completely original design of the test model is briefly presented without going into the detail of computation. The model is developed, in an unconventional way, in a form- which can be tested in a towing tank. As is well-known, this type of experiments are generally carried out in circulating water channels due to practical considerations. In circulation channels the body is fixed and the fluid is mobile, and accordingly, it is possible to extend the duration of experiments as long as it is deemed desirable; that is to say, there exists a case of a tank with infinite length.IX The cross-section of the tank and the water velocities are, however, limited due to the energy considerations. Whereas in the towing tank, the velocities can be varied by changing the towing carriage speed' and the duration of experiments- is limited due the finite length of tank. At the outset, the dimensions of the test models are chosen here in accordance with those of the circulating water channel at the Shipbuilding Research Center,. Istanbul Technical University. Nevertheless, due to the technical -problems arising later in the experiments, all the experiments are carried out in the towing tank. Symmetric segmental type of cross-sections is chosen for wings, and three wings in a row are taken and the middle wing is considered to represent the cascade. The number of wings can be, of course, increased to 5, 7, 9 and so on within the capacity of laboratory. The length of wings, with an aspect ratio of 5, is chosen to be rather long. Within the water, struts and alike which can disturb the characteristics of flow are not used as the wing carrying elements. The wings are supported by fronts plates at their both sides, and hence a certain type of catamaran is constructed. The measurements of pressure is realized in a manometric way by placing pressure elements at the sixteen points of the middle section of the main wing; the eleven points are chosen at its back and the remaining five points at its face. The radii of pressure elements on the main wing are taken to be 2* mm. On the other hand, the measurement of the force system is realized by means of a dynamometer with three-components working with a strain- gage type transducers. The dynamometers are placed at both the edges of the wing, and they are designed such that they don't block the way öf pressure communication elements. Each dynamometer measures half of the resistance and lifting force of the wing and half of its torsional moment. The electric signals taken from the strain-gages are read in terms of e (ym/tn. ) in a digital strain-indicator and printed down. Using the earlier results of calibration, the required force and moments are then determined.The model used in experiments is completely original from the point of view of testing in the towing tank, and the dynamometers have original design as well. Further, wing and lateral plates are made of wood, and hence these elements are only employed for the realization of model and not used to obtain detailed data. In modern laboratories, the model can be, of course, further developed by using certain materials instead of wood for long-time experiments. In the fourth chapter, both the theoretical and experimental results obtained in this study are summarized. The theoretical results are : (.) For a system of cascade hydrofoils, the circular cylinder cascade can be used as a mathematical model in the superposed flow of `the uniform flow + the flow of doublet distribution + the flow of vortex distribution`. (,) A new conformal mapping function is derived so as to transform the hydrofoil cascade into the circular cylinder cascade. ?(.??) The solution for infinite number of hydrofoil cascade is obtained by varying certain parameters, (.) A relationship is obtained in terms of the Froude number for depth so as to determine the pressure at the finite depth by using that at the infinite depth. The numerical results are : (.) In a constant wing gap-ratio, there is no significant variation in the pressure distribution of the wing face with respect to the varying angles of attack, whereas at the wing leading edge, the value of C decreases (through its absolute value increases) with respect to the increasing angles of attack and this variation occurs in reverse 'sense at the wing trailing edge. (.) The difference between the curves corresponding to the values of constant angles of attack decreases with respect to the increasing values of gap-ratio.XI (.) When the gap-ratio values are varied in a constant angle of attack, the maximum effect of cascade takes place between the values s = ir and s = 2ir, this effect vanishes with the increasing values of gap and the pressure distribution of a single profile is obtained at the infinite gap-ratio. The results of experiments are : (.) The model of experiments can be pulled ât various velocities, depths, gap - ratios of wing and angles of attack in the towing tank. Thus, one can examine the effects of the Froude and Reynolds number, the Froude number for depth, gap - ratio, and angle of attack in the towing tank. ?''..... (. ) The design of cascade hydrofoil is prepared so that both the force system and the pressure distribution can be measured during the experiments. (.) It- is concluded that the calibration experiments are carried out. within the conditions of environment. (.) All the types of force systems which. can be reduced to two coplanar components of bending and the component of torsion can be measured by means of the developed dynamometer.. (.) The model where forces and moments are to be measured, should be beared by a rod construction suitable to the dynamometer. (.) By means of the Tleveloped dynamometer, all types of experiments which involve hydrofoils, rudder and under water bodies can be carried out in the cavitation tunnel, the circulating water channel and the towing tank.