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Student Number	962201015
Author	Chiau-Yu Huang(黃巧育)
Author's Email Address	coolradon86@hotmail.com
Statistics	This thesis had been viewed 537 times. Download 189 times.
Department	Mathematics
Year	2008
Semester	2
Degree	Master
Type of Document	Master's Thesis
Language	English
Title	Bifurcation Analysis of Incompressible Sudden Expansion Flows Using Parallel Computing
Date of Defense	2009-05-21
Page Count	60
Keyword	pseudo transient continuation parallel computing domain decomposition. incompressible flow Bifurcation
Abstract	In fluid dynamics, bifurcations phenomena, which provide the modes of transitions and instability when some physical parameter such as the Reynolds number is varied, are commonly observed. The aim of this thesis is to study numerically some parallel pseudo-transient continuation algorithm for detecting the critical points of symmetry-breaking bifurcation in sudden expansion flows. For this purpose, the resulting nonlinear partial differential algebraic equations (PDAE's) are obtained by employing a stabilized finite element method for unsteady incompressible Navier-Stokes equations as the spatial discretization. One of classical approaches for examining the stability of a stationary solution to the PDAE's is to apply the so-called pseudo-transient continuation, which can be interpreted as the context of a method-of-line (MOL) approach, beginning with some perturbed stationary solution to PDAE's and then to investigate its time-dependent response to see whether the solution returns back to original state or not after a certain time step.   In current study, after employing unconditionally stable backward Euler's method as a time integrator, at each time step, the resulting nonlinear system is solved by a fully parallel Newton-Krylov-Schwarz algorithm, where inexact Newton with backtracking as a nonlinear solver and an additive Schwarz preconditioned Krylov subspace type method such as GMRES is used to solve the corresponding Jacobian systems. While the time accuracy is not our concerns, the adaptability of time step size is a key ingredient for the success of the algorithm to speed up the time-marching process.   Our numerical results obtained from a parallel machine shows that our parallel pseudo-transient continuation algorithm is very robust and efficient and also confirmed qualitatively the bifurcation with the numerical and numerical results found by other researchers. Furthermore, it is interesting to note that imperfect pitchfork bifurcations were observed especially for the case with a small expansion ratio, in which the happening of bifurcation points is delayed due to asymmetric unstructured meshes used for the numerical simulation.
Table of Content	Tables . . . .. . . . . . . . . . . . . . . . . . . viii Figures . . . . . . . . . . . . . . . . . . . . . . . .x 1　Introduction . . . . . . . . . . . . . . . . . . . .1 2　A review of bifurcation ｔheory . . . . . . . . . . 4 　2.1 Problem statement . . . . . . . . . . . . . . . .4 　2.2 Bifurcation analysis . . . . . . . . . . . . . . 4 　2.3 Pitchfork bifurcations . . . . . . . . . . . . . 7 3 Applications to the 2D incompressible sudden expansion flows . . . . 10 　3.1 2D sudden expansion flows . . . . . . . . . . .10 　3.2 Navier-Stokes equations and their semi-discrete formulation . . . . 12 　3.3 Numerical methods . . . . . . . . . . . . . . . 15 　　3.3.1 Pseudo-transient Newton-Krylov-Schwarz method $Psi$NKS . . . . 16 　　3.3.2 Generalized eigenvalue problem . . . . . . 18 4 Numerical results . . . . . . . . . . . . . . . . . 20 　4.1 Numerical experiment setup and grid tsting. . . 20 　4.2 $Psi$NKS algorithmic parameter tuning. . . . . 26 　4.3 Bifurcation predictions.　. . . . . . . . . . . 32 5 Conclusions . . . . . . . . . . . . . . . . . . . . 43 Bibliography. . . . . . . . . . . . . . . . . . . . . 44
Reference	[1] ParaView homepage. http://www.paraview.org. [2] Online CUBIT user’s manual, 2006. http://cubit.sandia.gov/documentation.html. [3] N. Alleborn, K. Nandakumar, H. Raszillier, and F. Durst. Further contributions on the two-dimensional flow in a sudden expansion. Journal of Fluid Mechanics, 330:169–188, 1997. [4] S. Balay, K. Buschelman, W.D. Gropp, D. Kaushik, M.G. Knepley, L.C. McInnes, B.F. Smith, and H. Zhang. PETSc Web page, 2009. http://www.mcs.anl.gov/petsc. [5] F. Battaglia, S.J. Tavener, A.K. Kulkarni, and C.L. Merkle. Bifurcation of low Reynolds number flows in symmetric channels. AIAA Journal, 35:99–105, 1997. [6] W. Cherdron, F. Durst, and J.H. Whitelaw. Asymmetric flows and instabilities in symmetric ducts with sudden expansions. Journal of Fluid Mechanics, 84:13–31, 1978. [7] K.A. Cliffe, T.J. Garratt, and A. Spence. Iterative methods for the detection of Hopf bifurcations in finite element discretisation of incompressible flow problems. SIAM Journal on Scientific Computing, 4:337–356, 1992. [8] K.A. Cliffe, T.J. Garratt, and A. Spence. Eigenvalues of the discretized Navier- Stokes equation with application to the detection of Hopf bifurcations. Advances in Computational Mathematics, 1:337–356, 1993. [9] K.A. Cliffe, T.J. Garratt, and A. Spence. A modified Cayley transform for the discretized Navier-Stokes equations. Applications of Mathematics, 38:281–288, 1993. [10] T.S. Coffey, C.T. Kelley, and D.E. Keyes. Pseudo-transient continuation and differential-algebraic equations. SIAM Journal on Scientific Computing, 25:553– 569, 2004. [11] V.F. De Almeida and J.J. Derby. Construction of solution curves for large twodimensional problems of steady-state flows of incompressible fluids. SIAM Journal on Scientific Computing, 22:285–311, 2000. [12] D. Drikakis. Bifurcation phenomena in incompressible sudden expansion flows. Physics of Fluids, 9:76–87, 1997. [13] F. Durst, A. Melling, and J.H. Whitelaw. Low Reynolds number flow over a plane symmetric sudden expansion. Journal of Fluid Mechanics, 64:111–128, 1974. [14] F. Durst, J.C.F. Pereira, and C. Tropea. The plane symmetric sudden-expansion flow at low Reynolds numbers. Journal of Fluid Mechanics, 248:567–581, 1993. [15] R.M. Fearn, T. Mullin, and K.A. Cliffe. Nonlinear flow phenomena in a symmetric sudden expansion. Journal of Fluid Mechanics, 211:595–608, 1990. [16] L.P. Franca and S.L. Frey. Stabilized finite element methods. II: The incompressible Navier-Stokes equations. Computer Methods in Applied Mechanics and Engineering, 99:209–233, 1992. [17] L.P. Franca, S.L. Frey, and T.J.R. Hughes. Stabilized finite element methods. I: Application to the advective-diffusive model. Computer Methods in Applied Mechanics and Engineering, 95:253–276, 1992. [18] T.J. Garratt, G. Moore, and A. Spence. Two methods for the numerical detection of Hopf bifurcations. Bifurcation and Chaos: Analysis, Algorithms, Applications, 97:129–134, 1991. [19] T.J. Garratt, G. Moore, and A. Spence. A generalized Cayley transform for the numerical detection of Hopf bifurcations in large systems. Contributions in Numerical Mathematics, 2:177–195, 1993. [20] V. Girault and P.A. Raviart. Finite Element Approximation of the Navier-Stokes Equations. Springer, 1979. [21] V. Girault and P.A. Raviart. Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms. Springer, 1986. [22] R. Glowinski. Numerical Methods for Nonlinear Variational Problems. Springer- Verlag, 1984. [23] I.G. Graham, A. Spence, and E. Vainikko. Parallel iterative methods for Navier- Stokes equations and application to stability assessment. Concurrency and Computation: Practice and Experience, 15:1151–1168, 2003. [24] P.M. Gresho, D.K. Gartling, J.R. Torczynski, K.A. Cliffe, K.H. Winters, T.J. Garratt, A. Spence, and J.W. Goodrich. Is the steady viscous incompressible twodimensional flow over a backward-facing step at Re= 800 stable? International Journal for Numerical Methods in Fluids., 17:501–541, 1993. [25] P.M. Gresho and R.L. Sani. Incompressible Flow and the Finite Element Method. Volume 2: Incompressible Flow and Finite Element. John Wiley and Sons, 1998. [26] W.D. Gropp, D.K. Kaushik, D.E. Keyes, and B.F. Smith. High-performance parallel implicit CFD. Parallel Computing, 27:337–362, 2001. [27] M.D. Gunzburger and J.S. Peterson. Predictor and steplength selection in continuation methods for the Navier-Stokes equations. Computers and Mathematics with Applications, 22:73–81, 1991. [28] S.K. Hannani, M. Stanislas, and P. Dupont. Incompressible Navier-Stokes computations with SUPG and GLS formulations - a comparison study. Computer Methods in Applied Mechanics and Engineering, 124:153–170, 1995. [29] T. Hawa and Z. Rusak. The dynamics of a laminar flow in a symmetric channel with a sudden expansion. Journal of Fluid Mechanics, 436:283–320, 2001. [30] V. Hernandez, J.E. Roman, and V. Vidal. SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems. ACM Transactions on Mathematical Software, 31:351–362, 2005. [31] D.J. Higham. Trust region algorithms and timestep selection. SIAM Journal on Numerical Analysis, 37:194–210, 2000. [32] H. Jiang and P.A. Forsyth. Robust linear and nonlinear strategies for solution of the transonic Euler equations. Computers and Fluids, 24:753–770, 1995. [33] M. Kadja and G. Bergeles. Numerical investigation of bifurcation phenomena occurring in flows through planar sudden expansions. Acta Mechanica, 153:47–61, 2002. [34] G. Karypis. METIS homepage. http://cubit.sandia.gov/documentation.html. [35] C.T. Kelley and D.E. Keyes. Convergence analysis of pseudo-transient continuation. SIAM Journal of Numerical Analysis, 35:508–523, 1998. [36] R.B. Lehoucq and A.G. Salinger. Large-scale eigenvalue calculations for stability analysis of steady flows on massively parallel computers. International Journal for Numerical Methods in Fluids, 36:309–327, 2001. [37] R.B. Lehoucq and J.A. Scott. Implicitly Restarted Arnoldi Methods and Eigenvalues of the Discretized Navier Stokes Equations. Rutherford Appleton Laboratory, 1997. [38] A. Masud and T.J.R. Hughes. A space-time Galerkin/least-squares finite element formulation of the Navier-Stokes equations for moving domain problems. Computer Methods in Applied Mechanics and Engineering, 146:91–126, 1997. [39] S. Mishra and K. Jayaraman. Asymmetric flows in planar symmetric channels with large expansion ratio. International Journal for Numerical Methods in Fluids, 38:945–962, 2002. [40] J. Mizushima and Y. Shiotani. Structural instability of the bifurcation diagram for two-dimensional flow in a channel with a sudden expansion. Journal of Fluid Mechanics, 420:131–145, 2000. [41] M. Morzy´nski, K. Afanasiev, and F. Thiele. Solution of the eigenvalue problems resulting from global non-parallel flow stability analysis. Computer Methods in Applied Mechanics and Engineering, 169:161–176, 1999. [42] T. Mullin and C. Blohm. Bifurcation phenomena in a Taylor-Couette flow with asymmetric boundary conditions. Physics of Fluids, 13:136, 2001. [43] D. Peric and S. Slijepcevic. Computational modelling of viscoplastic fluids based on a stabilised finite element method. Engineering Computations, 18:577–591, 2001. [44] J.B. Perot. An analysis of the fractional step method. Journal of Computational Physics, 108:51–58, 1993. [45] O. Pironneau. Finite Element Methods for Fluids. John Wiley and Sons, 1989. [46] J. Sanchez, F. Marques, and J.M. Lopez. A continuation and bifurcation technique for Navier-Stokes flows. Journal of Computational Physics, 180:78–98, 2002. [47] M. Shapira, D. Degani, and D. Weihs. Stability and existence of multiple solutions for viscous flow in suddenly enlarged channels. Computers and Fluids, 18:239–258, 1990. [48] C.Y. Soong, P.Y. Tzeng, and C.D. Hsieh. Numerical investigation of flow structure and bifurcation phenomena of confined plane twin-jet flows. Physics of Fluids, 10:2910, 1998. [49] S.H. Strogatz. Nonlinear Dynamics and Chaos. Addison-Wesley Reading, 1994. [50] R. Temam. Navier-Stokes Equations: Theory and Numerical Analysis. American Mathematical Society, 2001. [51] T.E. Tezduyar. Stabilized finite element formulations for incompressible flow computations. Advances in Applied Mechanics, 28:1–44, 1991. [52] V. Thom´ee. Galerkin Finite Element Methods for Parabolic Problems. Springer, 1997. [53] G. Tiesinga. Multi-level ILU preconditioners and continuation methods in fluid dynamics. PhD thesis, University of Groningen, 2000. [54] E.M. Wahba. Iterative solvers and inflow boundary conditions for plane sudden expansion flows. Applied Mathematical Modelling, 31:2553–2563, 2007. [55] O.C. Zienkiewicz and R.L. Taylor. The Finite Element Method. Butterworth- Heinemann, 2000.
Advisor	Feng-Nan Hwang(黃楓南)
Files	962201015.pdf approve immediately
Date of Submission	2009-06-17

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