Uçakların girdap kafes yöntemiyle aerodinamik analizi
Aerodynamic analysis of aircrafts by vortex lattice method
Yazar: EMRE ŞAHİN
Danışman: PROF. DR. MAHMUT ADİL YÜKSELEN
İstanbul Teknik Üniversitesi / Fen Bilimleri Enstitüsü / Uçak ve Uzay Mühendisliği Ana Bilim Dalı
Derece: Yüksek Lisans, 2013
Konu: Uçak Mühendisliği = Aircraft Engineering

Özet:
Türkiyenin Savunma Sanayisinde son yıllarda görülen önemli gelişmeler, 2023 hedefleri doğrultusunda Havacılık Sanayisine de yansımıştır. Bu çerçevede IHA, Başlangıç Eğitim Uçağı Hürkuş ve geleceğin yerli savaş ve eğitim uçağına ilişkin TX/FX projesi gibi çalışmalar gündeme oturmuştur.
Bu gibi uçak geliştirme çalışmalarının kavramsal ve başlangıç tasarım aşamalarında basit ama hızlı aerodinamik araçlarına ihtiyaç duyulmaktadır. Aerodinamik analiz çalışmalarında, günümüzde her ne kadar çok kabiliyetli CFD yöntemleri uygulamak mümkün gibi gözükse de bu gibi yöntemlerin karmaşık uçak geometrileri etrafındaki akımlar için uygulanmasında çözüm süreleri günler hatta aylara kadar çıkarak çok büyük kullanım zorluğu içermektedir. Ayrıca bu gibi hesaplamalar için çok büyük bilgisayar hızları ve hafızaları gerekmekte olup süper bilgisayarlara ihtiyaç duyulmaktadır. Kavramsal ve başlangıç tasarım aşamaları boyunca optimizasyon amacıyla uçak geometrisinin sıklıkla değişeceği de hesaba katılırsa CFD çalışmalarının maliyetinin ne kadar büyük olacağı ortaya çıkar. Bu nedenlerle uçak kavramsal ve başlangıç tasarımı aşamaları için olabildiğince basit, yaklaşık, ama hızlı aerodinamik analiz araçlarının kullanılması tercih edilir. CFD çalışmaları ise daha ziyade ayrıntılı tasarım ve geliştirme aşamalarında tercih edilir.
Uçak ve benzeri hava araçlarının kavramsal ve başlangıç tasarım çalışmalarındaki aerodinamik analiz ihtiyacına yönelik basit yöntemler arasında en ön sırayı panel yöntemleri almaktadır. Bu tip yöntemlerden önemli birisi de Girdap Kafes Yöntemi olup geçmiş yıllarda çok kullanılan bu yöntem günümüzde de halen kullanılmaktadır.
Bu tez çalışmasında uçak ve benzeri vasıtalar gibi çok sayıda taşıyıcı elemana (kanat, kuyruk, winglet vb) sahip taiyıcı yüzey sistemlerine ses altı hızlarda ve akım ayrılması görülmeyen hücum açılarında etkiyen aerodinamik kuvvetleri hesaplayabilecek bir yazılım geliştirilmesi amaçlanmıştır. Yazılımın trapez üst görünümlü, ok açılı, dihedral açılı çok sayıda taşıyıcı yüzeyi modellemesi hedeflenmiştir. 
Girdap Kafes Yönteminde her bir taşıyıcı yüzey açıklık ve veter doğrultusunda küçük panellere bölünerek her bir panel üzerine birer atnalı girdabı yerleştirilmektedir. Atnalı girdabının ön kolu panelin aerodinamik taşımasını modelleyen ve genellikle ?bağlı girdap? olarak adlandırılan taşıyıcı bir girdap olup panelin çeyrek-veter çizgisi üzerinde yer almaktadır. Atnalı girdabının yan kolları firar kenarına kadar kanat simetri düzlemine paralel olarak yüzeyi izleyip firar kenarından itibaren serbest akım doğrultusunda sonsuza uzanmaktadır. Atnalı girdaplarının şiddetleri her bir panelin üç-çeyrek veter çizgisinin ortasında yer alan bir kontrol noktasında yüzeye dik akım hızının olmayacağı şeklindeki yüzey sınır koşulu uygulanarak elde edilen bir lineer denklem takımı çözülerek elde edilmektedir. Her bir panele etkiyen aerodinamik kuvvet bu panelin bağlı girdap orta noktasında bileşke akım hızları hesaplanıp Joukowsky taşıma kanunu uygulanarak elde edilmektedir. Sistemdeki tüm panellere etkiyen kuvvetler toplanarak her bir taşıyıcı yüzeye ve sistemin tamamına etkiyen taşıma ve indüklenmiş sürükleme kuvvetleriyle bunlara ilişkin aerodinamik katsayılar elde edilmektedir.
Yöntem MatLab için programlanmış olup, formülasyonun ve programın doğruluğu çeşitli örnek çalışmalarla test edilmiştir. Tek kanat ve çift kanat üzerinde yapılan test çalışmaları simetrik kesitli kanatlarda taşıma katsayısı için yeterince doğru sonuçlar elde edebilmek için açıklık doğrultusunda en az 20 civarında panel alınması gerektiğini, veter doğrultusundaki panel sayısının taşıma için önemli olmadığını göstermiştir. Bu panel sayıları indüklenmiş sürükleme içinde uygun gözükmektedir. Ancak kambur kesitli kanatlar için veter boyunca panel sayısının önemli olduğu ve en az 10 civarında panel alınması gerektiğini ortaya koymuştur. Ayrıca VLM yöntemi sonuçları tek kanat halinde PLL yöntemi sonuçlarıyla karşılaştırılmış olup büyük açıklık oranlarında sonuçların yakın olduğu, açıklık oranı azaldıkça arada farklıkların ortaya çıktığı görülmüştür.

Abstract:
In recent years there are important improvements in the defense industry, and these important improvements also affected Aeronautical Industry in the direction of 2023 goals. In this context, projects like UAV, preliminary training aircraft Hürkuş and future fighter and training aircraft TX/FX have been started to consider.
For these kinds of aircraft improvement works, at the stage of preliminary and conceptual design, simple and fast aerodynamics tools have been needed. Although, it looks like possible to apply very talented CFD methods , but the application times of these methods for complex aircraft geometries can be more than days or months and because of this reason using these methods has very big application difficulties. Additionally for this kind of calculations very big computer speed and memories are necessary and supercomputers have been needed. At the conceptual and preliminary design stage if it is also considered that the geometry of the aircraft will changed many times for the optimization purpose, it can be understood that how big the cost of using CFD. For this reason, for the conceptual and preliminary aircraft design stages, very simple, approximate but fast aerodynamic analysis tools have been chosen. CFD methods are rather chosen for detailed design and improvement stages.
Panel methods take place in the front row among the simple aerodynamic analysis methods for the preliminary and conceptual design studies of aircraft and other air vehicles similar to aircrafts. One of the most important one of these methods is Vortex Lattice Method and it is also widely used in the past and it is still used in nowadays. Nowadays, about this topic, many package programs have been developed in order to use in the conceptual and preliminary design stage. One of these programs as mentioned in the literature summary is Tornado program. With these programs it is possible to calculate aerodynamics forces and moment coefficients at subsonic and supersonic speeds. These programs can also work at linear flow and nonlinear flow regime which is seen after the flow separation. It is possible to simulate aircrafts and air vehicles similar to aircrafts which has many lifting components like wing, tail, and winglet. These lifting components can have tapered planform, sweep angle, and dihedral angle, and twist angle.
In this thesis, it is aimed to develop a computer program that can calculate aerodynamics forces at subsonic and linear flow conditions for aircrafts and air vehicles similar to aircrafts which has many lifting components like wing, tail, winglet and vs. It is aimed to simulate many lifting surfaces which can have tapered planform, sweep angle, and dihedral angle. So, our program is not capable of simulating twisted wings and nonlinear, supersonic flow conditions in contrast to other programs. With our program it is possible to simulate interaction of lifting surfaces like wing-canard and wing-tail. It is not only possible to simulate the interaction of lifting surfaces but it is also possible to simulate the interaction of airplanes which fly close to each other. In the literature these kinds of studies have been done to increase the fuel efficiency of the airplanes.
The idea of the VLM starts with PLL theory. The idea of modeling flow filed with the Vortex lines first came with the PLL. In the PLL infinite numbers of horseshoe elements are used. The forward arms of the horseshoe elements are placed on the quarter chord line. The side arms of the horseshoe elements start from the quarter chord line and go to infinity. In the PLL method the distribution of the vortex along the span direction is the unknown the problem. The distribution of the vortex along the span direction is represented by a Fourier series. By writing these equations on every cross section of the wing an equation system can be written and the coefficients of the Fourier series can be solved. From these coefficients lift and induced drag coefficients can be found.
The PLL method is improved as Numerical Lifting Line Method. In the Numerical Lifting Line Method instead of using infinite number of vortex elements, finite number of vortex elements is used. The wing is divided into panels and along the chord direction only one panel is placed but in the span direction as many panels as wanted can be placed. The horse shoe elements have been put side by side along the span direction. Similar to PLL the forward arm of the horseshoe element has been placed to quarter chord line and the side arms of the horseshoe elements has been sent to infinity. The unknown of the problem is again the strength of the vortex elements. The formulation of the Numerical Lifting Line method is similar to the VLM and it can be found in the following parts of the thesis.
In both PLL and Numeric Lifting Line method along the chord direction there is only one horseshoe element and the distribution of the lift and induced drag force is not taking into account in these methods. These models also do not give any information about the pitching moment. In order to solve these problems VLM was developed. In VLM every single lifting surface has been divided into small panels in the span and chord direction and a vortex lattice has been placed on every panel. The aerodynamic lifting of the panel has been modeled with the forward arm of the vortex lattice and this forward arm has generally been called ?bound vortex? and it has been placed to the quarter chord line of the panel. Side arms of the vortex lattice follow the surface in parallel to the wing symmetry until the trailing edge and after the trailing edge it goes to infinity in the direction of free stream. The strength of the vortex has been calculated by solving a linear equation system which has been obtained by applying a boundary condition which says that there is no normal velocity component at a control point which is at the middle of the three quarter chord line of every panel. In order to create linear equation system an influence coefficient matrix has to be formed. Every single element of this matrix represents the induced velocity at a control point which is created by another horseshoe element by assuming every single horseshoe element has unit strength. In order to calculate induced velocity which is created by horseshoe element Bot-Savart law which is also used in electromagnetic theory has been used. In order to calculate the induced velocity by the Bot-Savart law every little part of the horseshoe element is calculated and then they are summed. After finding the strength of the horseshoe elements again by using the Biot-Savart law the velocity field and the induced velocities at the middle of the bond vortexes can be calculated. The resultant velocity at the middle of the bound vortex is the sum of free stream velocity and the induced velocity which is created by summing all of the single horseshoe elements. The aerodynamic forces for every panel have been obtained by using the resultant velocity at the middle of bound vortex and using Joukowsky lifting law. However these aerodynamic forces have to be transformed in the direction of the free stream to calculate the lift and induced drag coefficient. By summing forces on every panel in the system, the lifting and induced drag forces and related aerodynamic coefficients for every lifting surface and for the whole system can be obtained. 
The method has been programmed for the Matlab© and the accuracy of the formulation and the program have been tested for various examples. In order to use the program the geometry input for the every lifting surface has to be entered. This geometry input includes number of panels in the chord and span direction, wing root cord length, wing tip cord length, wing span, dihedral angle, sweep angle, incidence angle according to body frame, wing camber, and x,y,z position of the wing according to body frame. After entering these parameters, the program calculates lift and induced drag coefficient for any complex lifting surface planforms. The test work which has been done on the single and double wing planforms has shown that for symmetric cross-section wings in order to obtain lift coefficient accurately at least 20 panels has to be taken in the span direction, and it is also show that the number of panels in the chord direction is not important for the lift. This panel numbers also looks appropriate for the induced drag coefficient. However for cambered cross-sectioned panels it has been shown that the number of panels in the chord direction is important and at least around 10 panels have to be taken. Also VLM method results has been compared with PLL method results and it has been shown that at high aspect ratio the two results are close and at low aspect ratio the difference between results is getting bigger. 
For testing the correctness of the program another test case has been done on the F16 fighter airplane. For this study the geometry of the F16 can not be found exactly. However, by using the reference figure an approximate geometry has been obtained. The geometry of the airplane has been simplified to be used in the program and additionally to the wings, the fuselage is also modeled as a wing. No information about the camber of the wings has been found so the wings has been approximated as flat. In order to model the curved surfaces in the fuselage, many low aspect ratio wings have been used. This study has shown that complex F16 geometry which compose of totally 18 wings, which are 9 of them at the left side and 9 of them at the right side, can be modeled and solved. The simulation results show similarity with the literature data. It is considered that the difference between the simulation results and literature data is caused by the insufficient information and approximations which are made about the geometry. This study has also show that the program can not only solve single wing and biplane wing configurations but also can solve complex planforms which composed of many lifting surfaces.