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Rüzgar tüneli kollektöründe üç boyutlu akımın sonlu farklar yöntemiyle analizi

Rüzgar tüneli kollektöründe üç boyutlu akımın sonlu farklar yöntemiyle analizi

##### Dosyalar

##### Tarih

1992

##### Yazarlar

Sansar, Mehmet

##### Süreli Yayın başlığı

##### Süreli Yayın ISSN

##### Cilt Başlığı

##### Yayınevi

Fen Bilimleri Enstitüsü

##### Özet

Bu çalışmada dikdörtgen kesitli biz sesaltı rüzgar tüneli kollektöründe potansiyel akım yaklaşımı kullanılarak üç boyutlu akım alanının analizi problemi ele alınmıştır. Akım alanını temsil eden Laplace denklemi Sonla Faiklar. Yöntemi kul lanılarak ayıklaştırılmıştır. Sonlu fark ağmın oluşturulması aşamasında kol- lektörün eğrisel geometrili yüzeylerinde sınır şartlarının kolay tatbik edilmesine imkan sağlayan bir genelleştirilmiş dönüşüm kullanılmıştır. Bu dönüşüm sonunda koüektördeki akım alanı hesap uzayında dikdörtgenler prizması şeklini almak tadır. Belirli bir kollektöıü temsil eden sonla fark ağında düğüm sayısına bağlı olarak denklem sayışmış çokluğa bir iteratH yöntemi zorunla kılmıştır. Dolayısıyla bn çalışmada cebrik denklem sistemi, bir noktasal iteratif yöntem olan Gauss-Seİdel Yöntemiyle çözülmüştür. Bu yöntemde yakınsamayı hızlandırmak için de Successive Over Relaxation (SOB.) tekniği kullanılmıştır. Çıkardan bn Sonlu Farklar formülasyonu bir bilgisayar programı halinde geliştirilmiş ve doğruluğunu test amacıyla birtakım uygulamalar -ve mukayeseler yapılmış ayrıca programda çözüm yöntemi olarak kullanılan Gauss-SeideP de SOR etkisi incelenmiştir.

All engineering is a blend of theory and empiricism. The exact mix de pends on the type of engineering and the particular problem. Almost all theories must be simplified to allow mathematical solution to be obtained. The validity of the basic theory and the simplifications most always be confirmed by test. In aeronautics, a Sight test ideal, but it is enormously expensive to en gineer and baud a fall-scale airplane to find out if the design is correct. We must have substantial experimental verification of the safety and efficiency of the design before we fly. Furthermore il we wish to test any variations of a design it is almost impossible to do such tests full-scale. Additionally in the uncertain atmosphere it is difficult to find perfectly smooth air to allow steady-state flight conditions to be set up for a test. For all these reasons wind tunnels invented and it has remained the foremost instrument of aeronautical engineering. Wind tunnels usually have a contracting duct fitted upstream of their working section. This is one of the most important component of a wind tun nel which serves to improve flow uniformity and steadyness and to reduce the turbulence level in the working (test) section. VI Over the year considerable interest has been shown by many authors in methods of design for low speed wind tunnel contractions. However,until recently, most of these works is concerned with two dimensional or axisymmetrical flows and there doesn't appear to have been so many work published on the design of contractions with rectangular cross-sections. Such contractions would occur, of course, in tunnels where the working section has a rectangular cross-section and in many case this may be of high aspect ratio (ratio of height to width) when it is required to investigate flow over bridges, buildings etc. In addition, a tunnel which has a contraction of rectangular cross-section usually has the advantage of being simpler and less expensive to construct than a tunnel which has a contraction of circular or octagonal cross-section. The present study is concerned with the analysis of three-dimensional, steady, incompressible potential flow in the wind tunnel contractions of rectangu lar cross-section. Here it has been assumed that the contraction joins a parallel- sided settling section of rectangular cross-section to a parallel-sided working sec tion with a same cross-section of different aspect ratio. Thus the rectangular cross-section changes its size and aspect ratio continuously down the length of the contraction with, of course, the centre of each section lying on the axis of the tunnel. Hence there is four-fold symmetry and this simplifies the numerical aspect of the problem by investigating only one-fourth of the flowfield. The method is based on a finite difference approximation to Laplace equation. The numerical problem can be split into two parts. Firstly choosing a lattice at whose nodes the solution is to be determined and secondly choosing an algorithm to solve the sets of equations. A lattice with equal spacings in all three directions is attractive in that all internal nodes would have identical finite difference equations. Unfortunately the curved stream body, where the solution is of great interest, would not intersect va the lattice at the regular nodes. To overcome this piobiem a generalized transfor mation is used that transform the flowfield of the contraction into a rectangular prism in computational space. By applying a generalized transformation to the Laplace equation in physical space we get another form of this equation in computational space as below: 8^6 cPS & &4> &>6 d2d> The metrics included in coefficients of the latter form of the governing equation cart be readily determined if analytical expressions are available for the transformation. In this study a numerical method is used to generate the required trans formation. In this method the transformation is constructed by assigning points in the physical space along constant computational space coordinate lines and numerically computing the metrics by using finite difference equations. This method has the advantage of permitting assignment of grid points in the physical space -where desired. The disadvantage is that all metrics must be determined using numerical techniques. vxn The boundary conditions on the contraction walls can be expressed as "velocity vector is tangent to the wall surface.'* At the entrance and exit planes uniform parallel flow condition satis fied by adding constant area sections upstream and downstream to simulate the settling section and the working section respectively. The total number of equations to describe a particular contraction may be so high that it is necessary to use an economical method of solution. Hence, Gauss-Seidel iterative scheme was chosen. This is one of the most efficient and useful point-iterative procedure for large systems of equations. Another advan tage of using an iterative scheme such as Gauss-Seidel appears when working through a series of contraction shapes for which just one geometrical parameter is varied. In this case the solution for one contraction shape will provide an ex cellent starting point for another slightly different shape. So it is possible to get the solution with a minimum of computation. Additionally, Successive Over Relaxation (SOR) technique is used to ac celerate the convergence of the Gauss-Seidel procedure. For simple lattice shapes a value for SOR coefficient which gives maximum convergence may be obtained analytically. In generally, however, for more complex cases it is not possible to determine the optimal value in advance. In these cases, some numerical experi mentation should be helpful in identifying useful values for SOR coefficient. After much experimentation a value of 1.9999 was found to be most generally suitable in the present problem. A computer code has been written in FORTRAN language for the method which is described above and computations were made on the IBM 4381 machine of I. T. Ü. The program was run for the grid of 27x9x9 points by using double precision and it takes 3 minutes and 31.53 seconds of CPU time. IX Heie numerical differentiation was used to obtain the magnitudes of tîıe velocity components. In addition to principal finite difference programme (MSL3D), farther programmes calculated the velocity distribution on the center- line of the collector walls. Contours of pressure and velocity potential distribution which are superimposed on the sides of the contraction may also be plotted as shown ia Fig.3.6 and Fig.Z.7. Since the ünite difference method computes the velocity potential at internal points it is also possible to examine the velocity distribution over any plane normal to the contraction axis. For the graphical presentation of constant pressure and velocity potential lines over the surface of collector and velocity vector distribution over the sym metry planes and any planes normal to the collector axis additional computer codes have been written in QUICK BASIC language. Two comparisons were made between the present study and the other two studies. One of them was a 2-dimensional study of A. Yükselen [16] based on Complex Panel Method (CPM). The other one was a 3-dimensionaI comparative parametric design study of Y. Su [13]. In the latter study body-fitted coordinates used that transform the flowfieid of the contraction of rectangular cross-section into a unit cube in computational space. For the first comparison a three-dimensional contraction of rectangu lar cross-section for which contraction take place in only one plane was chosen. The computations were made by using the contraction wall coordinates of I.T.Û. 80cm x 110cm sabsonic ciosed-circait wind tunnel. For the second comparison a seven-parameter family contours used to define the walls of 3-dimensional contrac tion of rectangular cross-section. The velocity distribution along the contraction wall was investigated and the results of both studies against the present study were plotted on Fiğ.ZA.b and Fig.3.5.b. In the both comparisons satisfactory results were obtained.

All engineering is a blend of theory and empiricism. The exact mix de pends on the type of engineering and the particular problem. Almost all theories must be simplified to allow mathematical solution to be obtained. The validity of the basic theory and the simplifications most always be confirmed by test. In aeronautics, a Sight test ideal, but it is enormously expensive to en gineer and baud a fall-scale airplane to find out if the design is correct. We must have substantial experimental verification of the safety and efficiency of the design before we fly. Furthermore il we wish to test any variations of a design it is almost impossible to do such tests full-scale. Additionally in the uncertain atmosphere it is difficult to find perfectly smooth air to allow steady-state flight conditions to be set up for a test. For all these reasons wind tunnels invented and it has remained the foremost instrument of aeronautical engineering. Wind tunnels usually have a contracting duct fitted upstream of their working section. This is one of the most important component of a wind tun nel which serves to improve flow uniformity and steadyness and to reduce the turbulence level in the working (test) section. VI Over the year considerable interest has been shown by many authors in methods of design for low speed wind tunnel contractions. However,until recently, most of these works is concerned with two dimensional or axisymmetrical flows and there doesn't appear to have been so many work published on the design of contractions with rectangular cross-sections. Such contractions would occur, of course, in tunnels where the working section has a rectangular cross-section and in many case this may be of high aspect ratio (ratio of height to width) when it is required to investigate flow over bridges, buildings etc. In addition, a tunnel which has a contraction of rectangular cross-section usually has the advantage of being simpler and less expensive to construct than a tunnel which has a contraction of circular or octagonal cross-section. The present study is concerned with the analysis of three-dimensional, steady, incompressible potential flow in the wind tunnel contractions of rectangu lar cross-section. Here it has been assumed that the contraction joins a parallel- sided settling section of rectangular cross-section to a parallel-sided working sec tion with a same cross-section of different aspect ratio. Thus the rectangular cross-section changes its size and aspect ratio continuously down the length of the contraction with, of course, the centre of each section lying on the axis of the tunnel. Hence there is four-fold symmetry and this simplifies the numerical aspect of the problem by investigating only one-fourth of the flowfield. The method is based on a finite difference approximation to Laplace equation. The numerical problem can be split into two parts. Firstly choosing a lattice at whose nodes the solution is to be determined and secondly choosing an algorithm to solve the sets of equations. A lattice with equal spacings in all three directions is attractive in that all internal nodes would have identical finite difference equations. Unfortunately the curved stream body, where the solution is of great interest, would not intersect va the lattice at the regular nodes. To overcome this piobiem a generalized transfor mation is used that transform the flowfield of the contraction into a rectangular prism in computational space. By applying a generalized transformation to the Laplace equation in physical space we get another form of this equation in computational space as below: 8^6 cPS & &4> &>6 d2d> The metrics included in coefficients of the latter form of the governing equation cart be readily determined if analytical expressions are available for the transformation. In this study a numerical method is used to generate the required trans formation. In this method the transformation is constructed by assigning points in the physical space along constant computational space coordinate lines and numerically computing the metrics by using finite difference equations. This method has the advantage of permitting assignment of grid points in the physical space -where desired. The disadvantage is that all metrics must be determined using numerical techniques. vxn The boundary conditions on the contraction walls can be expressed as "velocity vector is tangent to the wall surface.'* At the entrance and exit planes uniform parallel flow condition satis fied by adding constant area sections upstream and downstream to simulate the settling section and the working section respectively. The total number of equations to describe a particular contraction may be so high that it is necessary to use an economical method of solution. Hence, Gauss-Seidel iterative scheme was chosen. This is one of the most efficient and useful point-iterative procedure for large systems of equations. Another advan tage of using an iterative scheme such as Gauss-Seidel appears when working through a series of contraction shapes for which just one geometrical parameter is varied. In this case the solution for one contraction shape will provide an ex cellent starting point for another slightly different shape. So it is possible to get the solution with a minimum of computation. Additionally, Successive Over Relaxation (SOR) technique is used to ac celerate the convergence of the Gauss-Seidel procedure. For simple lattice shapes a value for SOR coefficient which gives maximum convergence may be obtained analytically. In generally, however, for more complex cases it is not possible to determine the optimal value in advance. In these cases, some numerical experi mentation should be helpful in identifying useful values for SOR coefficient. After much experimentation a value of 1.9999 was found to be most generally suitable in the present problem. A computer code has been written in FORTRAN language for the method which is described above and computations were made on the IBM 4381 machine of I. T. Ü. The program was run for the grid of 27x9x9 points by using double precision and it takes 3 minutes and 31.53 seconds of CPU time. IX Heie numerical differentiation was used to obtain the magnitudes of tîıe velocity components. In addition to principal finite difference programme (MSL3D), farther programmes calculated the velocity distribution on the center- line of the collector walls. Contours of pressure and velocity potential distribution which are superimposed on the sides of the contraction may also be plotted as shown ia Fig.3.6 and Fig.Z.7. Since the ünite difference method computes the velocity potential at internal points it is also possible to examine the velocity distribution over any plane normal to the contraction axis. For the graphical presentation of constant pressure and velocity potential lines over the surface of collector and velocity vector distribution over the sym metry planes and any planes normal to the collector axis additional computer codes have been written in QUICK BASIC language. Two comparisons were made between the present study and the other two studies. One of them was a 2-dimensional study of A. Yükselen [16] based on Complex Panel Method (CPM). The other one was a 3-dimensionaI comparative parametric design study of Y. Su [13]. In the latter study body-fitted coordinates used that transform the flowfieid of the contraction of rectangular cross-section into a unit cube in computational space. For the first comparison a three-dimensional contraction of rectangu lar cross-section for which contraction take place in only one plane was chosen. The computations were made by using the contraction wall coordinates of I.T.Û. 80cm x 110cm sabsonic ciosed-circait wind tunnel. For the second comparison a seven-parameter family contours used to define the walls of 3-dimensional contrac tion of rectangular cross-section. The velocity distribution along the contraction wall was investigated and the results of both studies against the present study were plotted on Fiğ.ZA.b and Fig.3.5.b. In the both comparisons satisfactory results were obtained.

##### Açıklama

Tez (Yüksek Lisans) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 1992

##### Anahtar kelimeler

uçak mühendisliği,
kollektörler,
rüzgar tüneli kollektörü,
sonlu farklar yöntemi,
üç boyutlu akış,
Aircraft Engineering,
Collectors,
Wind tunnel collector,
Finite differences method,
Three dimensional flow