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Alan Daraltma Yöntemi Ve Sonlu Elemanlar Yöntemi İle Uygulanmasi

Alan Daraltma Yöntemi Ve Sonlu Elemanlar Yöntemi İle Uygulanmasi

##### Dosyalar

##### Tarih

1996

##### Yazarlar

Can, Kamertap

##### Süreli Yayın başlığı

##### Süreli Yayın ISSN

##### Cilt Başlığı

##### Yayınevi

Fen Bilimleri Enstitüsü

Institute of Science and Technology

Institute of Science and Technology

##### Özet

Bu tezde, üç boyutlu, viskoz, zamana bağlı akış problemleri bir Parçalı Adım - Galerkin formülasyonuna dayalı sonlu eleman yöntemi kullanılarak Navier-Stokes denklemlerinin çözülmesi ile analiz edilmiştir. Son yıllarda hesaplama yöntemi ve bilgisayar donanımındaki gelişmeler, araştırmacıların ilgisinin bu yönde toplanmasını sağlamıştır. Karmaşık akım problemlerinin analizini ve sayısal çözümünü mümümkün kılan çeşitli güçlü metodlar bu sayede geliştirilmiştir. Özellikle akışkanlar mekaniği ile uğraşan birçok bilim adamı ve mühendis için daha iyi bir analizde daha fazla nokta kullanımı ihtiyacı her zaman büyük sorun olmuştur. Bu çalışmada bilgisayar bellek kapasitesinin sorun olduğu haller için kullanılmak üzere bir alan daraltma yöntemi sunulmuştur. Hesaplama zamanını azaltmak ve aym nokta sayısı ile daha dar bir bölgeyi daha detaylı analiz edebilmek için kullanılan alan daraltma yöntemi öncelikle küre etrafındaki akışa uygulanarak test edilmiş ve sonuçların verilen referans değerleri ile uyumlu olduğu görülmüştür. Son olarak yöntem, bir bina etrafındaki akış için uygulanmıştır.

In the past decade the progress in computer hardware as well as in numerical algorithms has enabled attempts to be made towards the analysis and numerical solution of highly complex flow problems. Many methods for numerical simulation of fluid flows have been proposed and Computational Fluid Dynamics (CFD) has become a practical design tool. However, problems remain with regard to computational efficiency, accuracy and turbulence modeling. The most difficult problem maybe mesh generation for geometrically complicated domains [1]. Finite Differences and Finite Element methods are the most popular methods of Computational Fluid Dynamics. Although finite differences method is the most used and most easy method of numerical methods, as important problems are encountered during grid generation for complex geometries, it is loosing its popularity in the recent years and leaving its place gradually to finite element methods. Contrary to the finite difference methods, finite element methods are able to solve the problems efficiently and accurately around complex geometries with advanced grid generation techniques. Recently, solution of the Navier-Stokes equations with finite element methods have become one of the most popular research subjects. The most important elements that make the solution difficult are the elliptical and non-linear character of the Navier- Stokes equations. Analytical solutions to Navier-Stokes equations around simple geometries are known, but when complex geometries are concerned numerical methods have to be used. The finite element is an approximate method of solving differential equations of boundary and initial value problems in engineering and mathematical physics. Main idea of this method is to divide a continuos domain into so called finite elements of VII several convenient shapes such as triangles, quadrilaterals, bricks, rings, etc. Choosing suitable points called nodes at the corners and within the elements, the variable in the differential equation is written as a linear combination of appropriately selected interpolation functions and the values of the variable or its derivatives specified at these nodes. After the development of the finite element method the applications to the fluid dynamics or aerodynamics emphasize the effectiveness of the method. [2] In the finite element method we may use either the variational principles or weighted residuals through approximations. In finite element applications to fluid dynamics, the Galerkin Method is often considered the most convenient tool for formulating finite element models since it requires no variational principles and no higher order interpolation functions. The Finite Element Method, with its favorable features such as easy handling of complex geometries, simple solution algorithms and elegant mathematical theory, has become a powerful tool in analysis of complex flow which involve high Reynolds numbers and strong separation. In time dependent problems, numerous researchers have utilized schemes employing a fractional step approach. In this study, the time-dependent, incompressible, viscous flow is considered. The governing equations are the Navier-Stokes equations and the continuity equation. The governing equations are solved by a new version of the fractional step method proposed by Mizukami and Tsuchiya [3] which is essentially based on Chorin's fractional step method. In Chorin's method, the essential boundary conditions for the normal component of the velocity must be embodied into the discretized continuity equation which leads to great difficulties. In the present method to eliminate the above complexity a potential function with a single degree of freedom at each node is introduced and a Poisson equation for the potential is directly discretized, in which the essential boundary condition for the normal component of the velocity is treated as the natural boundary condition for the potential which still preserves the elliptic character of the physical problem. This method is also applicable to steady state flow problems [4]. VIII The Galerkin finite element method is used for the spatial discretization of the equations. 8-node parametric brick elements with trilinear interpolation functions for velocity and potential and piece wise constants for pressure are used. The pressure field at each time level is obtained from an auxiliary potential function with the solution of a Poisson's equation, where an Element by Element (EBE) iteration procedure with preconditioned conjugate gradiant (PCG) is employed. There is a restriction on the time step size in order to obtain a stable solution. Naturally, for computational efficiency, large time steps are desirable for transient-type problems. However, to obtain accurate transient solutions, reasonably small time steps must be selected. This trade-off provides a reasonable balance between accuracy and stability for the study under consideration. With adjustment for the three dimensional brick elements, the stability requirement on a time step At is At < (2 / R)(h~2 + r2 +r2)+ \u\h~x + Mr ' + Mr' (1) where Re is the flow Reynolds number and h, k and / are the smallest lengths in the x,y,z directions, respectively, of a brick element. Finally u,v,w are the velocity components whose absolute values play a role in the formula. [5] The method is more effective in three-dimensional problems than two-dimensional one since it only the poisson equation for the scalar potential which must be solved in either case. Nevertheless, analysis of high Reynolns number flow requires fine meshes, thus reducing considerably the maximum allowable time step, due to the well-known stability problem of an explicit method [4]. Viscous flow involves high gradients flow variables near solid body surfaces. Therefore, an accurate analysis demands very fine grids about the surface. As a result, the maximum allowable time step is considerably reduced, due to the well-known stability problem of an explicit method. Obviously, double precision computation is also must. However, when the researcher is limited with the available computer power, the resulting excessive memory and computing time requirement become a major problem. One way to circumvent this problem is to cluster grid points near the surface and have much larger grid cells away from the body. However unfortunately, this can not be done as desired since the the size ratio of the adjacent grid cells must IX not be greater than a certain small value if one to obtain a physically meaningful solution. In the present work, a reduced boundaries procedure which saves both from the computation time and the storage requirement for external steady flow is proposed. In this procedure, the solution is first obtained for relatively coarse and uniform grid which is not able to give accurately enough details of the flow near the body. The infinity boundary conditions are imposed along the outer boundary, as usual. Having the solution converged to the desired accuracy, the size of the domain reduced, by a factor of three, or in other words, the boundaries of the domain moved closer to the body. However, the number of grid points is kept constant. Thus, a finer near body grid results. The required conditions in and along the boundaries of the new domain are imposed from the previous coarser grid solution. The moving of boundaries can be repeated until the desired fineness near the surface is obtained. Since the first grid consist of a coarse grid of relatively larger compared to afine grid's, thus reducing the convergence time considerably. [4] The method is first tested and calibrated with investigating the flow past a sphere. For a Re = 100 the flow field results are obtained. Before applying reduced boundaries method, computations are performed on a fine grid and the results are compared with literature This solution is used as a reference for the solutions obtained from reduced boundaries method. The grid consist of 56x41 points (circumferential x radial). The grid is clustered near the body surface. Computational domain extends to about 10 radii of the sphere. The time step is taken as At = 0.005. After 1600 time steps a certain convergence is reached. Then reduced boundaries procedure is applied. The whole domain extends about 10 radii from the origin of the sphere. The whole domain in the starting grid is 56 x 25 points (circumferential x radial). Applying three times the boundary reduction was sufficient to obtain good results for surface pressure and skin friction coefficient distributions. A value of 0.01 for dt is selected. After 1000 time steps a certain convergence is reached. At this time the boundaries moved closer to the surface. The second grid extends about 4 radii from sphere center. The necessary conditions are obtained from readily obtained solution previous coarser grid solution. The conditions at the outer boundary are kept constant during each solution. This time a value of dt=0.005 is selected. After 1500 time steps again a certain convergence is reached. At the next step the domain extends to about 2.21 radii and dt=0.001 After 3319 time steps a certain converge is reached. As clearly seen the skin friction coefficient moves to its correct distribution as the domain is reduced towards a finer grid with fixed number of points. Finally the procedure is applied to the flow around a building. The building is a rectangular geometry standing on an infinite floor. The procedure used for the sphere is also used here. During reduced boundaries procedure, because of grid structure, less node have to be used on the external surface for the flow problem around a building. However, the reduced boundaries method requires the node number of the external surface to be constant. To overcome this problem, linear surface interpolation is employed on the external surface. For all solutions, pressure coefficient, friction coefficient and converge history are compared with the reference solutions and literature.

In the past decade the progress in computer hardware as well as in numerical algorithms has enabled attempts to be made towards the analysis and numerical solution of highly complex flow problems. Many methods for numerical simulation of fluid flows have been proposed and Computational Fluid Dynamics (CFD) has become a practical design tool. However, problems remain with regard to computational efficiency, accuracy and turbulence modeling. The most difficult problem maybe mesh generation for geometrically complicated domains [1]. Finite Differences and Finite Element methods are the most popular methods of Computational Fluid Dynamics. Although finite differences method is the most used and most easy method of numerical methods, as important problems are encountered during grid generation for complex geometries, it is loosing its popularity in the recent years and leaving its place gradually to finite element methods. Contrary to the finite difference methods, finite element methods are able to solve the problems efficiently and accurately around complex geometries with advanced grid generation techniques. Recently, solution of the Navier-Stokes equations with finite element methods have become one of the most popular research subjects. The most important elements that make the solution difficult are the elliptical and non-linear character of the Navier- Stokes equations. Analytical solutions to Navier-Stokes equations around simple geometries are known, but when complex geometries are concerned numerical methods have to be used. The finite element is an approximate method of solving differential equations of boundary and initial value problems in engineering and mathematical physics. Main idea of this method is to divide a continuos domain into so called finite elements of VII several convenient shapes such as triangles, quadrilaterals, bricks, rings, etc. Choosing suitable points called nodes at the corners and within the elements, the variable in the differential equation is written as a linear combination of appropriately selected interpolation functions and the values of the variable or its derivatives specified at these nodes. After the development of the finite element method the applications to the fluid dynamics or aerodynamics emphasize the effectiveness of the method. [2] In the finite element method we may use either the variational principles or weighted residuals through approximations. In finite element applications to fluid dynamics, the Galerkin Method is often considered the most convenient tool for formulating finite element models since it requires no variational principles and no higher order interpolation functions. The Finite Element Method, with its favorable features such as easy handling of complex geometries, simple solution algorithms and elegant mathematical theory, has become a powerful tool in analysis of complex flow which involve high Reynolds numbers and strong separation. In time dependent problems, numerous researchers have utilized schemes employing a fractional step approach. In this study, the time-dependent, incompressible, viscous flow is considered. The governing equations are the Navier-Stokes equations and the continuity equation. The governing equations are solved by a new version of the fractional step method proposed by Mizukami and Tsuchiya [3] which is essentially based on Chorin's fractional step method. In Chorin's method, the essential boundary conditions for the normal component of the velocity must be embodied into the discretized continuity equation which leads to great difficulties. In the present method to eliminate the above complexity a potential function with a single degree of freedom at each node is introduced and a Poisson equation for the potential is directly discretized, in which the essential boundary condition for the normal component of the velocity is treated as the natural boundary condition for the potential which still preserves the elliptic character of the physical problem. This method is also applicable to steady state flow problems [4]. VIII The Galerkin finite element method is used for the spatial discretization of the equations. 8-node parametric brick elements with trilinear interpolation functions for velocity and potential and piece wise constants for pressure are used. The pressure field at each time level is obtained from an auxiliary potential function with the solution of a Poisson's equation, where an Element by Element (EBE) iteration procedure with preconditioned conjugate gradiant (PCG) is employed. There is a restriction on the time step size in order to obtain a stable solution. Naturally, for computational efficiency, large time steps are desirable for transient-type problems. However, to obtain accurate transient solutions, reasonably small time steps must be selected. This trade-off provides a reasonable balance between accuracy and stability for the study under consideration. With adjustment for the three dimensional brick elements, the stability requirement on a time step At is At < (2 / R)(h~2 + r2 +r2)+ \u\h~x + Mr ' + Mr' (1) where Re is the flow Reynolds number and h, k and / are the smallest lengths in the x,y,z directions, respectively, of a brick element. Finally u,v,w are the velocity components whose absolute values play a role in the formula. [5] The method is more effective in three-dimensional problems than two-dimensional one since it only the poisson equation for the scalar potential which must be solved in either case. Nevertheless, analysis of high Reynolns number flow requires fine meshes, thus reducing considerably the maximum allowable time step, due to the well-known stability problem of an explicit method [4]. Viscous flow involves high gradients flow variables near solid body surfaces. Therefore, an accurate analysis demands very fine grids about the surface. As a result, the maximum allowable time step is considerably reduced, due to the well-known stability problem of an explicit method. Obviously, double precision computation is also must. However, when the researcher is limited with the available computer power, the resulting excessive memory and computing time requirement become a major problem. One way to circumvent this problem is to cluster grid points near the surface and have much larger grid cells away from the body. However unfortunately, this can not be done as desired since the the size ratio of the adjacent grid cells must IX not be greater than a certain small value if one to obtain a physically meaningful solution. In the present work, a reduced boundaries procedure which saves both from the computation time and the storage requirement for external steady flow is proposed. In this procedure, the solution is first obtained for relatively coarse and uniform grid which is not able to give accurately enough details of the flow near the body. The infinity boundary conditions are imposed along the outer boundary, as usual. Having the solution converged to the desired accuracy, the size of the domain reduced, by a factor of three, or in other words, the boundaries of the domain moved closer to the body. However, the number of grid points is kept constant. Thus, a finer near body grid results. The required conditions in and along the boundaries of the new domain are imposed from the previous coarser grid solution. The moving of boundaries can be repeated until the desired fineness near the surface is obtained. Since the first grid consist of a coarse grid of relatively larger compared to afine grid's, thus reducing the convergence time considerably. [4] The method is first tested and calibrated with investigating the flow past a sphere. For a Re = 100 the flow field results are obtained. Before applying reduced boundaries method, computations are performed on a fine grid and the results are compared with literature This solution is used as a reference for the solutions obtained from reduced boundaries method. The grid consist of 56x41 points (circumferential x radial). The grid is clustered near the body surface. Computational domain extends to about 10 radii of the sphere. The time step is taken as At = 0.005. After 1600 time steps a certain convergence is reached. Then reduced boundaries procedure is applied. The whole domain extends about 10 radii from the origin of the sphere. The whole domain in the starting grid is 56 x 25 points (circumferential x radial). Applying three times the boundary reduction was sufficient to obtain good results for surface pressure and skin friction coefficient distributions. A value of 0.01 for dt is selected. After 1000 time steps a certain convergence is reached. At this time the boundaries moved closer to the surface. The second grid extends about 4 radii from sphere center. The necessary conditions are obtained from readily obtained solution previous coarser grid solution. The conditions at the outer boundary are kept constant during each solution. This time a value of dt=0.005 is selected. After 1500 time steps again a certain convergence is reached. At the next step the domain extends to about 2.21 radii and dt=0.001 After 3319 time steps a certain converge is reached. As clearly seen the skin friction coefficient moves to its correct distribution as the domain is reduced towards a finer grid with fixed number of points. Finally the procedure is applied to the flow around a building. The building is a rectangular geometry standing on an infinite floor. The procedure used for the sphere is also used here. During reduced boundaries procedure, because of grid structure, less node have to be used on the external surface for the flow problem around a building. However, the reduced boundaries method requires the node number of the external surface to be constant. To overcome this problem, linear surface interpolation is employed on the external surface. For all solutions, pressure coefficient, friction coefficient and converge history are compared with the reference solutions and literature.

##### Açıklama

Tez (Yüksek Lisans) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 1996

Thesis (M.Sc.) -- İstanbul Technical University, Institute of Science and Technology, 1996

Thesis (M.Sc.) -- İstanbul Technical University, Institute of Science and Technology, 1996

##### Anahtar kelimeler

Akış denklemleri,
Navier-Stokes denklemleri,
Sonlu elemanlar yöntemi,
Flow equations,
Navier-Stokes equations,
Finite element method