Numerical methods for viscous fluid flows in sectors, cones, and domains with corners

Numerical Methods for Viscous Fluid Flows in Sectors, Cones, and Domains with Corners Alexander V. Shapeev (M.Mech., Novosibirsk State Univ., Russia) A thesis submitted for the degree of PhD in Science Department of Mathematics, National University of Singapore Acknowledgments I would like to thank my supervisor, Prof. Ping Lin, for his guidance, sharing valuable ideas, discussions of the present work, and for all the support he offered throughout my studies in National University of Singapore. I would also like to thank Prof. Vladislav V. Pukhnachev for bringing my attention to the problems considered in my thesis and for his constant attention to my work. i Contents Acknowledgments i Contents ii Summary vi List of Figures viii List of Tables xiii Introduction and Literature Review 1.1 Overview of Viscous Flows in Sectors and Domains with Corners . . . . . . 1.1.1 Jeffery-Hamel Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Moffatt Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2.1 Flows in Infinite Sectors . . . . . . . . . . . . . . . . . . . . 1.1.2.2 Flows in Finite Domains with Corners . . . . . . . . . . . . Overview of Viscous Flows in Cones . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Flows due to a Source or a Sink at the Apex of a Cone . . . . . . . . 1.2.2 Moffatt-type Eddies in Cones . . . . . . . . . . . . . . . . . . . . . . 11 1.3 Analysis of Existing Results and the Proposed Approach . . . . . . . . . . . 12 1.4 Overview of Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . 14 1.5 Purpose and Value of the Work . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.6 Organization of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.7 Notations and Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.2 ii iii CONTENTS 1.7.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.7.2 Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 The Numerical Method for Flows in Sectors 2.1 2.2 2.3 2.4 21 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.1.1 General Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.1.2 The Navier-Stokes Equations and the Boundary Conditions . . . . . 22 2.1.3 The Navier-Stokes Equations in Terms of the Stream Function . . . 24 2.1.4 The Self-Similar Flows . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.1.5 Nondimensionalization . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.1.6 The Final Form of the Boundary-Value Problem . . . . . . . . . . . 28 2.1.7 Properties of the Problem of Self-Similar Flow . . . . . . . . . . . . 28 The Computational Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.2.1 Linearization and Transfer of Boundary Conditions . . . . . . . . . . 30 2.2.2 Spectral Discretization in the Spanwise Direction . . . . . . . . . . . 32 2.2.3 Finite Difference Discretization in the Radial Direction . . . . . . . 34 2.2.4 Solution of the Linear System of Algebraic Equations . . . . . . . . 35 Computation of Self-Similar Flows with a Source or a Sink . . . . . . . . . 36 2.3.1 Results on Stokes Flows, Different Initial and Boundary Conditions 36 2.3.2 Results on Navier-Stokes Flows with a Source . . . . . . . . . . . . . 48 2.3.3 Results on Navier-Stokes Flows with a Sink . . . . . . . . . . . . . . 51 2.3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Computation of Self-Similar Flows with Zero Net Flow Rate . . . . . . . . . 57 2.4.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 2.4.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 The Numerical Method for Steady Flows with Corners 64 3.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.2 The Computational Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.2.1 69 Discretization in the Main Subdomain . . . . . . . . . . . . . . . . . iv CONTENTS 3.3 3.2.2 Discretization in the Near-Corner Subdomains . . . . . . . . . . . . 70 3.2.3 Discretization in the Corner Subdomains . . . . . . . . . . . . . . . 72 Results of Computations and Discussion . . . . . . . . . . . . . . . . . . . . 74 3.3.1 The Lid-Driven Cavity Problem . . . . . . . . . . . . . . . . . . . . 75 3.3.2 Corner Subdomain Shrinking Factor . . . . . . . . . . . . . . . . . . 82 3.3.3 The Backward-Facing Step Problem . . . . . . . . . . . . . . . . . . 83 The Numerical Method for Flows in Cones 4.1 4.2 4.3 The Self-Similar Problem Formulation . . . . . . . . . . . . . . . . . . . . . 94 4.1.1 General Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.1.2 The Navier-Stokes Equations and the Boundary Conditions . . . . . 95 4.1.3 The Navier-Stokes Equations in Terms of the Stream Function . . . 97 4.1.4 The Self-Similar Flows . . . . . . . . . . . . . . . . . . . . . . . . . . 98 4.1.5 Nondimensionalization . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.1.6 The Final Form of the Boundary-Value Problem . . . . . . . . . . . 102 The Steady Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . 103 4.2.1 Steady Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.2.2 Nondimensionalization . . . . . . . . . . . . . . . . . . . . . . . . . . 104 4.2.3 The Final Form of the Boundary-Value Problem . . . . . . . . . . . 105 Analysis of Self-Similar and Steady Flows . . . . . . . . . . . . . . . . . . . 106 4.3.1 The Self-Similar Problem . . . . . . . . . . . . . . . . . . . . . . . . 107 4.3.2 Steady Flows Far from the Apex . . . . . . . . . . . . . . . . . . . . 110 4.3.2.1 Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . 110 4.3.2.2 Navier-Stokes Equations . . . . . . . . . . . . . . . . . . . 115 Steady Flows near the Apex . . . . . . . . . . . . . . . . . . . . . . . 118 The Computational Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 4.4.1 Linearization and Transfer of Boundary Conditions . . . . . . . . . . 120 4.4.2 Spectral Discretization in the Spanwise Direction . . . . . . . . . . . 121 4.4.3 Finite Difference Discretization in the Radial Direction 123 4.3.3 4.4 94 . . . . . . . v CONTENTS 4.5 4.4.4 Solution of the Linear System of Algebraic Equations . . . . . . . . 124 4.4.5 The Computational Method for Steady Flows . . . . . . . . . . . . . 125 Computation of Steady Flows in Cones with a Source or a Sink . . . . . . . 126 4.5.1 Steady Stokes Flows in Cones due to a Source or a Sink . . . . . . . 127 4.5.2 Steady Navier-Stokes Flows in Cones due to a Source or a Sink . . . 129 4.5.2.1 Flows due to a Sink . . . . . . . . . . . . . . . . . . . . . . 129 4.5.2.2 Dependence of Computed Flows on Discretization Parameters136 4.5.2.3 Flows due to a Source . . . . . . . . . . . . . . . . . . . . . 139 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 Computation of Self-Similar Flows in Cones with a Source or a Sink . . . . 144 4.6.1 Self-Similar Navier-Stokes Flows with a Sink at the Apex . . . . . . 144 4.6.1.1 Flows with Zero Initial Conditions . . . . . . . . . . . . . . 145 4.6.1.2 Flows with Nonzero Initial Conditions . . . . . . . . . . . . 150 4.6.2 Self-Similar Navier-Stokes Flows with a Source at the Apex . . . . . 153 4.6.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 4.5.3 4.6 4.7 Computation of Self-Similar Navier-Stokes Flows with Zero Net Flow Rate 157 4.7.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 4.7.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 Conclusion 165 Summary This thesis deals with the three fluid dynamics problems: the problems of viscous fluid flow in infinite sectors, in finite 2D domains with corners, and in infinite cones. First, unsteady flows in sectors are considered. The initial flow regime is assumed to be radial, which leads to self-similarity of the flows. Two essentially different types of flows are considered: unsteady flows with a sink or a source at the corner, and unsteady flows evolving from an initial regime in a cone with zero net flow rate. An efficient method is proposed to compute such flows. The examples of flows are computed. The efficiency of the method is confirmed on the basis of numerical experiments. The ideas of the method of computation of flows in sectors are used in the problem of flow in domains with corners. The problem is approached by a high-order finite element method with exponential mesh refinement near the corners, coupled with analytical asymptotics of the flow near the corners. Such approach allows one to compute position and intensity of the eddies near the corners in addition to the other main features of the flow. The method is tested on the problem of lid-driven cavity flow as well as on the problem of backwardfacing step flow. The results of computations of the lid-driven cavity problem show that the proposed method computes the central eddy with accuracy comparable to the best of existing methods and is more accurate for computing the corner eddies than the existing methods. The results also indicate that the relative error of finding the eddies’ intensity and position decreases uniformly for all the eddies as the mesh is refined (i.e. the relative error in computation of different eddies does not depend on their size). Last, steady flows and self-similar flows in infinite cones are considered. The problem of steady flow in cones is approached by analytical and numerical means. The results of vi SUMMARY vii asymptotic analysis and the numerical results agree with each other. Previously, there has been no complete understanding of behaviour of flows in cones with wide opening angles (wider than a half-space). In the present work, flows in cones with large opening angles are consistently described. Self-similar flows in cones are also computed and analyzed. The computational method is tested and its efficiency is confirmed. List of Figures 1.1 Illustration: a flow in a sector . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The flow regimes I, II1 , II2 , IV2 , and V2 (respectively, in left-to-right order), notations of [22]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Illustration: streamlines of a flow with Moffatt eddies . . . . . . . . . . . . 1.4 Illustration: a flow in a cone . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The grid in ζ axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.2 The prescribed stream function (2.37) . . . . . . . . . . . . . . . . . . . . . 37 2.3 The prescribed stream function (2.38) . . . . . . . . . . . . . . . . . . . . . 37 2.4 The flow with initial regime (2.37) for Re = and α = 30◦ . . . . . . . . . . 39 2.5 The flow with initial regime (2.37) for Re = and α = 90◦ . . . . . . . . . . 40 2.6 The flow with initial regime (2.37) for Re = and α = 115◦ . . . . . . . . . 41 2.7 The flow with initial regime (2.37) for Re = and α = 135◦ . . . . . . . . . 42 2.8 The flow with initial regime (2.38) for Re = and α = 30◦ . . . . . . . . . . 43 2.9 The flow with initial regime (2.38) for Re = and α = 90◦ . . . . . . . . . . 44 2.10 The flow with initial regime (2.38) for Re = and α = 115◦ . . . . . . . . . 44 2.11 The flow with initial regime (2.38) for Re = and α = 135◦ . . . . . . . . . 44 2.12 The flow with conditions (2.39) for Re = and α = 30◦ . . . . . . . . . . . 45 2.13 The flow with conditions (2.39) for Re = and α = 90◦ . . . . . . . . . . . 45 2.14 The flow with conditions (2.39) for Re = and α = 115◦ . . . . . . . . . . . 46 2.15 The flow with conditions (2.39) for Re = and α = 135◦ . . . . . . . . . . . 46 2.16 The flow with conditions (2.40) for Re = and α = 30◦ . . . . . . . . . . . 46 viii LIST OF FIGURES ix 2.17 The flow with conditions (2.40) for Re = and α = 90◦ . . . . . . . . . . . 47 2.18 The flow with conditions (2.40) for Re = and α = 115◦ . . . . . . . . . . . 47 2.19 The flow with conditions (2.40) for Re = and α = 135◦ . . . . . . . . . . . 47 2.20 The flow with a source with conditions (2.40) for Re = and α = 30◦ . . . 49 2.21 The flow with a source for Re = 9.5 and α = 30◦ . Additional (dashed) streamlines are ϕ = ±1.03 and ϕ = ±1.015. . . . . . . . . . . . . . . . . . . 50 2.22 The flow with a sink with initial regime (2.37) for Re = 100 and α = 30◦ . . 51 2.23 The flow with a sink with initial regime (2.37) for Re = 100 and α = 90◦ . . 52 2.24 The flow with a sink with initial regime (2.37) for Re = 100 and α = 115◦ . 53 2.25 The flow with a sink with initial regime (2.37) for Re = 100 and α = 135◦ . 54 2.26 Numerical resolution of the boundary layer . . . . . . . . . . . . . . . . . . 55 2.27 The self-similar Flow with zero net flow rate for Re = and α = 30◦ . . . . 58 2.28 The self-similar Flow with zero net flow rate for Re = 15 and α = 30◦ . . . 59 2.29 The self-similar Flow with zero net flow rate for Re = and α = 45◦ . . . . 60 2.30 The self-similar Flow with zero net flow rate for Re = and α = 60◦ . . . . 60 2.31 The self-similar Flow with zero net flow rate for Re = and α = 75◦ . . . . 61 2.32 The self-similar Flow with zero net flow rate for Re = and α = 90◦ . . . . 61 2.33 The self-similar Flow with zero net flow rate for Re = and α = 135◦ . . . 61 3.1 A domain decomposition near the corner . . . . . . . . . . . . . . . . . . . . 68 3.2 Argyris elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.3 Trapezia splitting of the near-corner subdomain . . . . . . . . . . . . . . . . 71 3.4 A triangular mesh of the near-corner subdomain . . . . . . . . . . . . . . . 71 3.5 A basis function near the edge A1 A2 (1st function, mesh M0) . . . . . . . . 73 3.6 A basis function near the edge A1 A2 (2nd function, mesh M0) . . . . . . . . 73 3.7 A basis function near the edge A1 A2 (1st function, mesh M1) . . . . . . . . 73 3.8 A basis function near the edge A1 A2 (2nd function, mesh M1) . . . . . . . . 77 3.9 Main subdomain mesh examples for the lid-driven cavity problem (meshes M0 and M1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 CHAPTER 4. THE NUMERICAL METHOD FOR FLOWS IN CONES 159 Ψ 2.0 1.5 1.0 0.5 10 Ζ (a) r−1 -normalized axial velocity (b) streamlines (c) streamlines (d) streamlines (e) streamlines Figure 4.37: The self-similar flow with zero net flow rate for Re = and α = 30◦ with initial conditions (4.89) CHAPTER 4. THE NUMERICAL METHOD FOR FLOWS IN CONES 160 Ψ 0.5 10 Ζ 1.0 1.5 2.0 (a) r−1 -normalized axial velocity (c) streamlines (b) streamlines (d) streamlines Figure 4.38: The self-similar flow with zero net flow rate for Re = and α = 30◦ with initial conditions (4.90) CHAPTER 4. THE NUMERICAL METHOD FOR FLOWS IN CONES 161 Ψ 2.0 1.5 1.0 0.5 10 Ζ (a) r−1 -normalized axial velocity (c) streamlines (b) streamlines (d) streamlines Figure 4.39: The self-similar flow with zero net flow rate for Re = and α = 45◦ with initial conditions (4.89) with Moffatt-type eddies. The transition zone has the reversed flow near the sidewalls of the cone (figure 4.37, graph (c)). The results of computations for wider angles (α = 45◦ , 60◦ , 75◦ ) for Re = are presented in figures 4.39–4.41. The flows differ by the size ratio and shape of eddies (in accordance with [65]), however, the results are essentially the same as for α = 30◦ . The angle α = 75◦ is close to the critical angle of 80.9◦ found by Wakiya [65] for which the eddies disappear. Hence the eddy size ratio is so huge that the consecutive eddies cannot be seen on the same graph. The flows for even wider angles (α > 80.9◦ ) monotonically decay as the apex is approached and thus not have the corner eddies. The size and intensity ratios were found from the numerical solution and compared with those found by Wakiya [65] analytically. The computations were made with Ns = and hζ = 0.01 (where hζ is a step size of the grid in ζ axis). The results of comparison are presented in Table 4.2. The “computed ratios” were taken as ratios between 4th and 5th eddy in the computations. The size and intensity ratios found from numerical solution in CHAPTER 4. THE NUMERICAL METHOD FOR FLOWS IN CONES 162 Ψ 2.0 1.5 1.0 0.5 10 (a) r−1 -normalized axial velocity (c) streamlines Ζ (b) streamlines (d) streamlines Figure 4.40: The self-similar flow with zero net flow rate for Re = and α = 60◦ with initial conditions (4.89) CHAPTER 4. THE NUMERICAL METHOD FOR FLOWS IN CONES 163 Ψ 2.0 1.5 1.0 0.5 10 (a) r−1 -normalized axial velocity (c) streamlines Ζ (b) streamlines (d) streamlines Figure 4.41: The self-similar flow with zero net flow rate for Re = and α = 75◦ with initial conditions (4.89) CHAPTER 4. THE NUMERICAL METHOD FOR FLOWS IN CONES α 30◦ 45◦ 60◦ 75◦ 164 Table 4.2: Intensity and size ratios of eddies theoretical [65] observed size ratio intensity ratio size ratio intensity ratio 3.32632 53849.9 3.33015 54284.8 7.40679 266990 7.41364 268280 26.3870 7.59639 · 106 26.416 7.63238 · 106 1327.87 3.51892 · 1012 1333.15 3.57354 · 1012 the current study are in a good agreement with the theoretical values. 4.7.2 Discussion The results on 3D self-similar flows with zero net flow rate are similar to those for 2D. The eddies occur whenever a cone angle is smaller than the critical angle. The flows at large time t tend to the quasi-steady regime with quasi-steady Moffatt-type eddies. Particularly, the eddies’ intensity and size ratios in self-similar flows are the same as those in steady flows. When the Reynolds number is nonzero, the transition between the initial flow regime and the quasi-steady regime may be different from the transition for Re = 0. Particularly, in some cases, the transition zone may have a separation of flow near the cone sidewalls. A remarkably good agreement of the theoretical and computational results indicates efficiency of the present method. The present method, similarly to the 2D case, can be used for computing the eddies in domains with conical vertices. Chapter Conclusion Viscous fluid flows in infinite sectors, finite 2D domains with corners, and infinite cones were considered. First, flows in sectors were considered. The problem of steady flow in sectors has been studied extensively in the literature, therefore we focused mainly on the unsteady self-similar formulation. The self-similar formulation allows one to set any radial flow regime at t = and study its evolution till the flow equilibrates to the steady (or quasi-steady) regime. An efficient method was proposed to compute such flows. The method is based on the combined spectral-finite difference discretization. Flows with a sink, a source, and zero net flow rate were computed. It was found that self-similar flows are unstable for the same values of parameters (α, Re) as steady Jeffery-Hamel flows are. This indicates that when the Jeffery-Hamel flow is unstable, the actual flow is not self-similar, which implies a nonuniqueness of the actual unsteady solution. The accuracy of the method was demonstrated based on numerical experiments. The ideas of this method were used for the other two problems, namely for the problem of flow in domains with corners and for the problem of axisymmetric flow in infinite cones. Second, a method for computing the infinite sequence of corner eddies in viscous fluid flows in domains with corners was proposed. The method is based on Argyris finite element discretization for the stream function formulation of the Navier-Stokes equations, exponential mesh refinement near corners, and matching the numerical solution with the asymptotics of the flow near corners. The method was applied to two benchmark prob- 165 CHAPTER 5. CONCLUSION 166 lems: the lid-driven cavity problem and the backward-facing step problem. The results of computations demonstrate high accuracy of the present method, show that the method can accurately compute the infinite series of eddies, and indicate that the relative error of finding eddies’ intensity and position decreases uniformly as the mesh is refined (i.e. the error of finding intensity and position of different eddies does not depend on their size). The comparison of the present results with the results available in the literature shows that the present method produces solutions of the same or better accuracy than the existing methods. Last, axisymmetric flows in infinite cones were considered. There has been few works on the problem of steady axisymmetric flow in cones, and some of the issues in this problem have not been resolved. Hence we first considered the steady formulation of the problem. We analyzed steady axisymmetric flows by analytic and numerical means. The results of asymptotic and numerical analysis agree with each other. The present results indicate that the flow far from the apex is radial for α < 120◦ and non-radial for α ≥ 120◦ , which is different from the results of the previous works made by Bond [8] and Ackerberg [1], who asserted that the flow ceases to be radial beyond α = 90◦ . The reasons that led previous researchers to these assertions were discussed. Then, the self-similar formulation was considered. Flows with a sink, a source, and zero net flow rate were computed. The results were interpreted and compared with the results of 2D self-similar flows. The proposed method for computation of flows in cones is similar to the method for computing flows in sectors. Based on numerical experiments, the accuracy of the method was assessed. It was confirmed that the method adequately computes the boundary layer flow near the apex and is very accurate at moderate and large distances from the apex. The approach proposed in the present work to study self-similar flows in sectors and cones can also be used to study evolution and equilibration of other types of flows, which have been studied only for the steady case. Particularly, it will be interesting to study evolution of flows in the following two cases. First, it will be interesting to study evolution of 3D flows in cones caused by a non-axisymmetric initial flow. This problem cannot be reduced to a 2D problem and will require an application of spectral discretization in both CHAPTER 5. CONCLUSION 167 transversal directions. Second, it will be interesting to study evolution of flows caused by a point source of momentum [39, 56, 60]. The method of computation of steady viscous fluid flows in domains with corners can also be applied to similar problems, including flows in 3D domains with conical vertices, and more complicated models of fluid like liquid crystal model, viscoelastic models, magnetohydrodynamics, etc. It would also be interesting to compare this method and the methods with adaptive mesh refinement near corners. Bibliography [1] R. Ackerberg, The viscous incompressible flow inside a cone, J. Fluid Mech., 21 (1965), pp. 47–81. [2] L. D. Akulenko, D. V. Georgievskii, and S. A. Kumakshev, Solutions of the Jeffery-Hamel problem regularly extendable in the Reynolds number, Izv. Ross. Akad. Nauk Mekh. Zhidk. Gaza, 39 (2004), pp. 15–32. [3] , Numerical-analytical investigation of multimodal solutions of the Jeffery-Hamel problem for a convergent channel, Izv. Ross. Akad. Nauk Mekh. Zhidk. Gaza, 40 (2005), pp. 49–60. [4] L. D. Akulenko, D. V. Georgievskii, S. A. Kumakshev, and S. V. Nesterov, New solutions and hydrodynamical effects in the Jeffery-Hamel problem, Russ. J. Math. Phys., 12 (2005), pp. 269–287. [5] L. D. Akulenko and S. A. Kumakshev, Bifurcation of a main steady-state viscous fluid flow in a plane divergent channel, Izv. Ross. Akad. Nauk Mekh. Zhidk. Gaza, 40 (2005), pp. 25–36. [6] W. H. H. Banks, P. G. Drazin, and M. B. Zaturska, On perturbations of JefferyHamel flow, J. Fluid Mech., 186 (1988), pp. 559–581. [7] E. Barragy and G. F. Carey, Stream function-vorticity driven cavity solution using p finite elements, Computers & Fluids, 26 (1997), pp. 453–468. [8] W. N. Bond, Viscous flow through wide angle cones, Phil. Mag., 50 (1925), pp. 1058– 1066. 168 BIBLIOGRAPHY 169 [9] O. Botella and R. Peyret, Benchmark spectral results on the lid-driven cavity flow, Computers & Fluids, 27 (1998), pp. 421–433. [10] M. Branicki and H. K. Moffatt, Evolving eddy structures in oscillatory Stokes flows in domains with sharp corners, J. Fluid Mech., 551 (2006), pp. 63–92. [11] S. N. Brown and K. Stewartson, On similarity solutions of the boundary-layer equations with algebraic decay, J. Fluid Mech., 23 (1965), pp. 673–687. [12] C. H. Bruneau and C. Jouron, An efficient scheme for solving steady incompressible Navier-Stokes equations, Journal Of Computational Physics, 89 (1990), pp. 389–413. [13] Z. Chen, Finite Element Methods and Their Applications, Springer, New York, 2005. [14] W. M. Collins and S. C. R. Dennis, Viscous eddies near a 90◦ and a 45◦ corner in flow through a curved tube of triangular cross-section, J. Fluid Mech., 76 (1976), pp. 417–432. [15] T. A. Davis, Algorithm 832: Umfpack v4.3—an unsymmetric-pattern multifrontal method, ACM Trans. Math. Softw., 30 (2004), pp. 196–199. [16] , A column pre-ordering strategy for the unsymmetric-pattern multifrontal method, ACM Trans. Math. Softw., 30 (2004), pp. 165–195. [17] W. R. Dean, Note on the divergent flow of fluid, Phil. Mag., 18 (1934), pp. 759–777. [18] W. R. Dean and P. E. Montagnon, On the steady motion of viscous liquid in a corner, Proc. Cambridge Phil. Soc., 45 (1949), pp. 389–394. [19] S. C. R. Dennis, W. H. H. Banks, P. G. Drazin, and M. B. Zaturska, Flow along a diverging channel, J. Fluid Mech., 336 (1997), pp. 183–202. [20] P. M. Eagles, The stability of a family of Jeffery-Hamel solutions for divergent channel flow, J. Fluid Mech., 24 (1966), pp. 191–207. BIBLIOGRAPHY 170 [21] E. Erturk, T. C. Corke, and C. Gokcol, Numerical solutions of 2-D steady incompressible driven cavity flow at high Reynolds numbers, International Journal For Numerical Methods In Fluids, 48 (2005), pp. 747–774. [22] L. E. Fraenkel, Laminar flow in symmetrical channels with slightly curved walls. i. on the Jeffery-Hamel solutions for flow between plane walls, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 267 (1962), pp. 119– 138. [23] , Laminar flow in symmetrical channels with slightly curved walls. ii. an asymptotic series for the stream function, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 272 (1963), pp. 406–428. [24] U. Ghia, K. N. Ghia, and C. T. Shin, High-Re solutions for incompressible-flow using the Navier Stokes equations and a multigrid method, Journal Of Computational Physics, 48 (1982), pp. 387–411. [25] M. Goldshtik, F. Hussain, and V. Shtern, Symmetry-breaking in vortex-source and Jeffery-Hamel flows, J. Fluid Mech., 232 (1991), pp. 521–566. [26] S. Goldstein, On backward boundary layers and flow in converging passages, J. Fluid Mech., 21 (1965), pp. 33–45. ¨ rtler, A new series for the calculation of steady laminar. Boundary layer flows, [27] H. Go J. Math. Mech., (1957), pp. 1–66. [28] O. Goyon, High-Reynolds number solutions of Navier-Stokes equations using incremental unknowns, Computer Methods In Applied Mechanics And Engineering, 130 (1996), pp. 319–335. [29] M. M. Grigoriev and A. V. Fafurin, A boundary element method for steady viscous fluid flow using penalty function formulation, International Journal For Numerical Methods In Fluids, 25 (1997), pp. 907–929. BIBLIOGRAPHY 171 [30] M. Hamadiche, J. Scott, and D. Jeandel, Temporal stability of Jeffery-Hamel flow, J. Fluid Mech., 268 (1994), pp. 71–88. [31] G. Hamel, Spiralformige bewegungen zaher flussigkeiten, Jahresbericht Deutch. Math. Verein., 25 (1916), pp. 34–60. [32] J. F. Harrison, The pressure in a viscous liquid moving through a channel with diverging boundaries, Proc. Cambridge Phil. Soc., 19 (1919), pp. 307–312. [33] H. Hasimoto and O. Sano, Stokeslets and eddies in creeping flow, in Annual review of fluid mechanics, Vol. 12, Annual Reviews, Palo Alto, Calif., 1980, pp. 335–363. [34] T. Hawa and Z. Rusak, Numerical-asymptotic expansion matching for computing a viscous flow around a sharp expansion corner, Theoretical And Computational Fluid Dynamics, 15 (2002), pp. 265–281. [35] G. B. Jeffery, The two-dimensional steady motion of a viscous fluid, Phil. Mag., 29 (1915), pp. 455–465. [36] R. R. Kerswell, O. R. Tutty, and P. G. Drazin, Steady nonlinear waves in diverging channel flow, J. Fluid Mech., 501 (2004), pp. 231–250. [37] M.-U. Kim, Slow viscous flow due to the motion of a sphere on the axis of a circular cone, Journal of the Physical Society of Japan, 47 (1979), pp. 1670–1675. [38] H. K. Kuiken, On boundary-layers in fluid-mechanics that decay algebraically along stretches of wall that are not vanishingly small, Ima Journal Of Applied Mathematics, 27 (1981), pp. 387–405. [39] L. D. Landau, On exact solution of the Navier-Stokes equations, Dokl. Akad. Nauk SSSR, 43 (1944), pp. 299–301. [40] P. A. Libby and T. Liu, Further solutions of the Falkner-Skan equation, AIAA Journal, (1967), pp. 1040–1042Roos. BIBLIOGRAPHY 172 [41] C. H. Liu and D. D. Joseph, Stokes flow in conical trenches, SIAM Journal on Applied Mathematics, 34 (1978), pp. 286–296. [42] A. Malek and T. N. Phillips, Pseudospectral collocation methods for fourth-order differential equations, IMA J. Numer. Anal., 15 (1995), pp. 523–553. [43] V. S. Malyuga, Viscous eddies in a circular cone, J. Fluid Mech., 522 (2005), pp. 101– 116. [44] A. McAlpine and P. G. Drazin, On the spatio-temporal development of small perturbations of Jeffery-Hamel flows, Fluid Dynamics Research, 22 (1998), pp. 123–138. [45] H. K. Moffatt, Viscous and resistive eddies near a sharp corner, J. Fluid Mech., 18 (1964), pp. 1–18. [46] H. K. Moffatt, Viscous eddies near a sharp corner, Arch. Mech. Stos., 16 (1964), pp. 365–372. [47] H. K. Moffatt and V. Mak, Corner singularities in three-dimensional Stokes flow, in IUTAM Symp. on Non-Linear Singularities in Deformation and Flow, Kluwer, 1999, p. 21. [48] H. Nishida and N. Satofuka, Higher-order solutions of square driven cavity flow using a variable-order multigrid method, International Journal For Numerical Methods In Engineering, 34 (1992), pp. 637–653. [49] B. Oskam and A. E. P. Veldman, Branching of the Falkner-Skan solutions for λ < 0, J. Engrg. Math., 16 (1982), pp. 295–308. [50] V. Pukhnachev, Symmetries in the Navier-Stokes equations, in Abstracts of VIII International Conference “Navier-Stokes Equations and Related Topics”, St. Petersburg, Russia, September 2002, pp. 62–63. [51] V. Putkaradze and P. Vorobieff, Instabilities, bifurcations, and multiple solutions in expanding channel flows, Physical Review Letters, 97 (2006), p. 144502. BIBLIOGRAPHY 173 [52] L. Rayleigh, Steady motion in a corner of a viscous fluid, Sci. Pap., (1920), pp. 18– 21. [53] L. Rivkind and V. Solonnikov, Jeffery-Hamel asymptotics for steady state NavierStokes flow in domains with sector-like outlets to infinity, J. Math. Fluid Mech, (2000), pp. 324–352. [54] L. Rosenhead, The steady two-dimensional radial flow of viscous fluid between two inclined plane walls, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 175 (1940), pp. 436–467. [55] O. Sano and H. Hasimoto, Three-dimensional Moffatt-type eddies due to a Stokeslet in a corner, J. Phys. Soc. Japan, 48 (1980), pp. 1763–1768. [56] H. Schlichting, Laminare strahlausbreitung, Z. Angew. Math. Mech, 13 (1933), pp. 260–263. [57] P. N. Shankar, On Stokes flow in a semi-infinite wedge, J. Fluid Mech., 422 (2000), pp. 69–90. [58] , Moffatt eddies in the cone, J. Fluid Mech., 539 (2005), pp. 113–135. [59] I. J. Sobey and P. G. Drazin, Bifurcations of two-dimensional channel flows, J. Fluid Mech., 171 (1986), pp. 263–287. [60] H. B. Squire, Some viscous fluid flow problems. 1. Jet emerging from a hole in a plane wall, Phil. Mag., 43 (1952), pp. 942–945. [61] S. R. Stow, P. W. Duck, and R. E. Hewitt, Three-dimensional extensions to Jeffery-Hamel flow, Fluid Dynamics Research, 29 (2001), pp. 25–46. [62] S. Taneda, Visualization of separating Stokes flows, J. Phys. Soc. Japan, 46 (1979), pp. 1935–1942. [63] O. R. Tutty, Nonlinear development of flow in channels with non-parallel walls, J. Fluid Mech., 326 (1996), pp. 265–284. BIBLIOGRAPHY 174 [64] A. E. P. Veldman and A. I. van de Vooren, On a generalized Falkner-Skan equation, J. Math. Anal. Appl., 75 (1980), pp. 102–111. [65] S. Wakiya, Axisymmetric flow of a viscous fluid near the vertex of a body, J. Fluid Mech., 78 (1976), pp. 737–747. [...]... necessary for studying evolution of flows in sectors and cones In the next section we are going to review the main findings for the problem of viscous fluid flow in infinite sectors and other domains with corners In section 1.2 we will give a review of research on flows in cones In section 1.3 we will analyze the existing results on flows in sectors and cones and propose a new approach to study these flows In section... numerical simulation of flows in sectors and cones 1.1 Overview of Viscous Flows in Sectors and Domains with Corners Fluid flows in sectors and cones have a wide range of applications including mechanical engineering, aerospace and water flow in rivers and canals Such flows occur whenever there is a plane corner or a conic apex in the flow domain, or when the domain has sector-like or conic outlets to in nity... were studied in an unsteady periodic formulation 3D generalizations include 3D flows in sectors (between 2 planes) [47, 55, 57], and flows in cones [37, 43, 58, 65] The later series of works will be reviewed in more detail in the next section where we will discuss the literature regarding viscous fluid flows in domains with corners 1.1.2.2 Flows in Finite Domains with Corners The property of Moffatt’s flows... occurring first, Dennis et al [19] found a pitchfork bifurcation occurring first, McAlpine and Drazin [44] reported a subcritical Hopf bifurcation occurring first, Kerswell, Tutty, and Drazin [36] predicted a subcritical pitchfork bifurcation The later work [36], numerically predicting steady flows periodic in space, was in a good agreement with the work of Tutty [63] on computing flows in expanding channels,... approached in the Stokes approximation and in the boundary layer approximation) Also, the method designed for computation of steady Moffatt eddies in domains with corners shows higher performance than the existing methods, and can be used or generalized to accurately compute other types of flows in domains with corners 1.6 Organization of the Thesis The thesis is organized as follows First, the findings on... corner, with a transition zone merging the two asymptotics together This section provides a review of research on flows in cones, mainly focusing on these two formulations and on the case of axisymmetric flows Flows with a source or a sink at the apex of a cone will be discussed in subsection 1.2.1, and flows with Moffatt-type eddies in the cone will be reviewed in subsection 1.2.2 CHAPTER 1 INTRODUCTION AND. .. discretization for the other direction There are two major versions of spectral methods: Galerkin (or Petrov-Galerkin) methods and collocation methods These two types of methods differ in the way the discretized system of algebraic equations is derived from the original differential equation In Galerkin methods, the original equation is multiplied by the basis functions and integrated over the domain In the... the in nite Moffatt’s eddy sequence were computed (maximum four corner eddies [7, 21] for certain Reynolds numbers) In addition, the accuracy of finding intensity and position of the smaller eddies was less than the accuracy for the larger eddies Another example of flows in domains with corners frequently considered in the literature is the backward-facing step flow problem 1.2 Overview of Viscous Flows in. .. of particular interest are: a symmetric purely in owing flow (denoted as I in figure 1.2), a symmetric purely outflowing flow (denoted as “I or II1 ” in figure 1.2), a symmetric flow with two zones CHAPTER 1 INTRODUCTION AND LITERATURE REVIEW 4 of in ow near to the sector sidewalls (denoted II2 in figure 1.2), and two antisymmetric flows with one in owing zone near a sidewall (denoted IV1 and V1 in figure 1.2)... self-similar flows in sectors with zero net flow rate) 4 To design an efficient method of computing steady flows with Moffatt eddies in domains with corners The method should allow one to accurately compute position and intensity of the eddies near the corners in addition to the other main features of the flow 5 To study axisymmetric flows in cones with nonzero net flow rate Steady flows were analyzed first, since their . in sectors and cones. 1.1 Overview of Viscous Flows in Sectors and Domains with Corners Fluid flows in sectors and cones have a wide range of applications including mechanical engineering, aer ospace and. flow with a sink with initial regime (2.37) for Re = 100 and α = 30 ◦ . . 51 2.23 The flow with a sink with initial regime (2.37) for Re = 100 and α = 90 ◦ . . 52 2.24 The flow with a sink with initial. Numerical Methods for Viscous Fluid Flows in Sectors, Cones, and Domains with Corners Alexander V. Shapeev (M.Mech., Novosibirsk State Univ., Russia) A thesis submitted for the degree