
NUMERICAL SIMULATION OF 2D INCOMPERSSIBLE NAVIER-STOKES FLOW DRIVEN BY ROTLETS IN A UNIT DISK A thesis submitted to Kent State University in partial fulfillment of the requirements for the degree of Master of Science by MANSOOR A. ALSULAMI August, 2017 Thesis written by MANSOOR A. ALSULAMI B.S., King Abdulaziz University, 2008 M.S., Kent State University, 2017 Approved by Xiaoyu Zheng , Advisor Andrew Tonge , Chair, Department of Mathematical Sciences James L. Blank , Dean, College of Arts and Sciences TABLE OF CONTENTS LIST OF TABLES . v LIST OF FIGURES . vi Acknowledgement . vii 1 Introduction . 1 2 Mathematical Models . 4 2.1 Incompressible Navier-Stokes Equations . 4 2.2 Stream Function and Velocity Formulation of NSEs in 2D . 5 2.2.1 Vorticity Formulation in 2D . 5 2.2.2 Poisson Equation for Stream Function . 6 2.3 Boundary Conditions . 6 2.4 Nondimensionlization of NSEs . 7 2.5 Decomposition of Solutions . 8 3 Numerical Method . 11 3.1 Spatial Discretization and Fourier Spectral Method . 11 3.2 Time Integration . 14 3.2.1 Initialization . 15 3.2.2 Vorticity Equation . 15 3.2.3 Fourier Filtering for Explicit Time Stepping . 16 3.2.4 Poisson Equation . 17 3.2.5 Vorticity at Boundary and Velocity Field . 18 iii 4 Numerical Results . 20 4.1 Convergence Rate . 20 4.2 Numerical Experiments . 23 5 Conclusion . 29 BIBLIOGRAPHY . 30 iv LIST OF TABLES 1 The errors in! ~ for Re =1............................ 22 2 The error in ~ for Re =1 ............................ 22 3 The error inu ~ for Re =1 ............................ 22 4 The error inv ~ for Re =1 ............................ 23 v LIST OF FIGURES 1 discretization of the domain where N =8.................... 12 2 Contour plots for stream function and vorticity at c = 0:2, Re = 0:1, N = 64. Top row: singular part of the solution; Middle row: combined solution; Bottom row: regular part of the solution. 24 3 Contour plots for stream function and vorticity at c = 0:5, Re = 0:1, N = 64. 25 4 Contour plots for stream function and vorticity at c = 0:7238, Re = 0:1, N = 64. 26 5 Contour plots for stream function and vorticity at c = 0:5, N = 64, Re = 1. 27 6 Contour plots for stream function and vorticity at c = 0:5, N = 64, Re = 10. 28 7 Normalized traction for different values of Re, for c = 0:5, N = 64. 28 vi Acknowledgement I would like to express my deep appreciation to my supervisor professor Xiaoyu Zheng for her support, help and encouragement. She has been always supportive, and I am grateful to her. I wish to express my sincere gratitude to the committee members, Professor Gartland and Professor Jing Le. I would like to thank them for their time that they have given me. I would like to take this opportunity to record my thanks to the Department of Mathematics at Kent State University for giving me the chance to study and gain my Master's degree. I would like to thank my father and my mother who have supported me to study abroad. Special thanks to my great family members who have been with me all the time. vii CHAPTER 1 Introduction In many active systems, individual self-propellers in a fluid generate the translational propulsion [1]. Recently, there are increased interest in active rotors. Individual particles rotate due to either internal torque or external torque. In experiments, particle rotations can be driven by chemicals [2], optical tweezers or light [3], magnetic [4] or electric fields [5]. The interaction between the individual rotors and the surrounding environment can lead to intriguing collective motion in the population. In this thesis, we study the simple steady, two dimensional flow problem when viscous fluid contained in a disk is disturbed by a fixed rotlet inside the disk. The line rotlet is simply the limiting case of the fluid generated by a rotating cylinder with radius b, angular velocity !. As the radius b ! 0, and !b2 is kept as a constant, one obtains the flow driven by a line rotlet. The constant !b2, usually denoted as Γ; which is known as strength of the rotlet [8]. For a free line rotlet in 2D infinite medium, the stream function is given by = Γ log R; R = j~x − ~x0j; (1.0.1) where R is the distance to the line rotlet. The stream function not only represents a Stokes flow, but also the exact solution of Navier-Stokes equations with an appropriate pressure field p. The velocity field and vorticity field are given by Γ ~u = θ;^ (1.0.2) R ~! = 0: (1.0.3) 1 The stream lines form concentric circles around the rotlet. The velocity is tangential to the circles, which blows up near the rotlet and decays as 1=R as one moves away from the rotlet. The vorticity is zero everywhere. The line rotlet exerts a torque per unit length with magnitude of 4πΓµ to the surrounding fluid whose dynamic viscosity is µ. When solid boundaries present, the imposed boundary conditions will alter the velocity field and break the symmetry of the flow, and might cause flow separation and eddies for- mation in the flow field [8]. A separation occurs either on the boundary or in the interior of the domain. A separation point on the boundary is where one or more streamlines meet the boundary. An interior separation point is where two or more streamlines meet. The streamlines which intersect either type of the separation point are called dividing stream lines. If P is a separation point on a smooth boundary, then the vorticity, or equivalently, the tangential stress vanishes at P . If Q is an interior separation point, then the velocity vanishes at Q. There are two kinds of eddies : attached eddy and free eddy [8]. An attached eddy is an open, simply connected region of flow, containing closed stream lines, and no flow singularities or rigid boundaries, and bounded by one or more rigid boundaries and one or more dividing streamlines. A free eddy differs from the attached eddy in that a free eddy is bounded by one or more dividing streamlines. In the Stokes limit, this problem was solved analytically by Ranger [6] . He showed that if a line rotlet is contained in a fixed disk, an attached eddy is formed when the rotlet is within a certain distance to the boundary of the disk. If a line rotlet is inside a rotating disk, then free eddies can be found for certain relative positions of the rotlet. In this thesis, we are interested in the problem where the inertial term cannot be ne- glected, and the goal is to examine the effect of the nonlinear convection term on the flow 2 field. The thesis is organized as follows. In Chapter 2, we introduce the mathematical mod- els which govern the motion of the fluid induced by a line rotlet in 2D disk. In Chapter 3, a combination of finite difference and Fourier collocation methods is developed to solve the equations. In Chapter 4, we demonstrate the numerical results with convergence rate and various numerical experiments. We conclude in Chapter 5. 3 CHAPTER 2 Mathematical Models In this chapter, we present the mathematical models that govern the dynamics of the flow field driven by a line rotlet in a 2D disk. We first write down the Navier-Stokes equations in stream function-vorticity formulation with appropriate boundary conditions. We then nondimensionalize the equations, and finally decompose the solutions into two parts, with a known singular part and a regular part, which needs to be solved. 2.1 Incompressible Navier-Stokes Equations We consider the Navier-Stokes equations (NSEs) in two dimensions. The general form of the incompressible NSEs is given by @~u p + (~u · r)~u = −∇ + νr2~u; on Ω; (2.1.1) @t ρ r · ~u = 0; on Ω: (2.1.2) where, ~u(~x;t) is the velocity field of the fluid, p(~x;t) is the pressure field, ρ is density of the fluid, and ν = µ/ρ is the kinematic viscosity of the fluid. The first equation is the consequence of the conservation of momentum, and the second equation describes the incompressibility of the fluid. We are going to consider the simplest Dirichlet boundary condition in this thesis, which is given by ~u = ~0 on @Ω: (2.1.3) In 2D, we can rewrite the NSEs in terms of the stream function-vorticity formulation. The advantage of the stream function-vorticity formulation is that we can eliminate the pressure term, which in general is difficult to handle both analytically and numerically. 4 2.2 Stream Function and Velocity Formulation of NSEs in 2D In this section, we provide the details on deriving the stream function-vorcitity formula- tion of the Navier-Stokes equations in 2D. 2.2.1 Vorticity Formulation in 2D By taking the curl of Eq. (2.1.1), we have @~u p r × + r × ((~u · r)~u) = −∇ × r + r × (νr2~u): (2.2.1) @t ρ Since the pressure p/ρ is a scalar function, so r × r(p/ρ) = 0, and using the identity 1 r × ((~u · r)~u) = r × ( 2 r(~u · ~u) − ~u × (r × ~u)), we get @ 1 r × ~u + r × r(~u · ~u) − ~u × (r × ~u) = νr2(r × ~u): (2.2.2) @t 2 The vorticity vector, ~!, is defined at a point as ~! = r × ~u; (2.2.3) which in 2D can be reduced to @u @u ~! = !z^ = 0; 0; y − x : (2.2.4) @x @y Once we substitute the expression of ~! in Eq. (2.2.3) in Eq.
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