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  • Delft University of TechnologyFaculty of Electrical Engineering, Mathematics and Computer Science

    Delft Institute of Applied Mathematics

    Fast iterative methods for solving theincompressible Navier-Stokes equations

    A thesis submitted to theDelft Institute of Applied Mathematicsin partial fulfillment of the requirements

    for the degree

    MASTER OF SCIENCEin

    APPLIED MATHEMATICS

    by

    CARLOS ECHEVERRIA SERURDelft, the Netherlands

    February 2013

    Copyright c 2013 by Carlos Echeverra Serur. All rights reserved.

  • MSc THESIS APPLIED MATHEMATICS

    Fast iterative methods for solving the incompressible Navier-Stokesequations

    CARLOS ECHEVERRIA SERUR

    Delft University of Technology

    Daily supervisor Responsible professor

    Prof. Dr. Ir. C. Vuik Prof. Dr. Ir. C. Vuik

    Other thesis committee members

    Dr. Ir. M. B. van Gijzen . . .

    . . . . . .

    February 2013 Delft, the Netherlands

  • Contents

    1 Introduction 1

    2 Discretization and linearization of the N-S equations 22.1 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 Finite element selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

    3 Projection techniques, Krylov subspace methods, and Preconditioners 73.1 Projection techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2 Krylov subspace methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

    3.2.1 Using Arnoldis Method for solving Linear Systems . . . . . . . . . 103.3 Preconditioners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

    4 Block-type preconditioners for the incompressible N-S equations 174.1 Block preconditioners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.2 Block preconditioners based on approximate commutators . . . . . . . . . 19

    4.2.1 Pressure convection-diffusion preconditioner . . . . . . . . . . . . . 194.2.2 Least squares commutator preconditioner . . . . . . . . . . . . . . 20

    4.3 Augmented Lagrangian approach . . . . . . . . . . . . . . . . . . . . . . . 214.4 SIMPLE-type preconditioners . . . . . . . . . . . . . . . . . . . . . . . . . 22

    4.4.1 SIMPLER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.4.2 hSIMPLER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.4.3 MSIMPLER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

    5 Test problems 265.1 2D Poiseuille Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.2 Driven Cavity Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.3 Backward facing step Flow . . . . . . . . . . . . . . . . . . . . . . . . . . 28

    6 Research questions 30

    iv

  • v

  • 1 Introduction

    The numerical solution of the incompressible Navier-Stokes (N-S) equations is an areaof much importance in contemporary scientific research. Except for some simple cases,the analytical solution of the (N-S) equations is impossible. Therefore, in order to solvethese equations, it is necessary to apply numerical techniques. Due to the nonlinear char-acter of the behavior of fluids, the solution of the (N-S) system of equations requires asuitable linearization of the discrete formulation of the original system. The resultinglinear system of equations gives rise to a so-called saddle-point problem. The efficientsolution of this linear system of equations is of primary interest due to the fact that mostof the computing time and memory of a computational implementation is consumed bythe solution of this system of equations. In the following report we adopt an iterativeapproach to solving this linear system, mainly by the use of a Krylov subspace methodcombined with a preconditioned linear system of equations. In particular, the type ofpreconditioners studied are known as Block-Preconditioners in literature.

    In Section 2, the Navier Stokes equations are introduced and discretized via the FiniteElement Method. The resulting algebraic system is then linearized via the Newton orPicard methods. Lastly, a brief comment on finite element selection is given. In Section3, the main concepts of linear algebra are presented. The theory behind Krylov subspacemethods is discussed and the general theory of Preconditioners is presented. In Section 4,specific Block-type preconditioners for the Navier Stokes equations are studied includingpreconditioners based on approximate commutators and SIMPLE-type preconditioners.In Section 5, a set of problems (benchmarks) that make use of the theory presented in theprevious sections is presented. The two-dimensional Poiseuille flow as well as the drivencavity flow and the backward facing step problems are shown. The problems are studiedby the use of the Matlab toolbox IFISS 1 (Incompressible Flow Iterative Solution Soft-ware) developed in the University of Manchester. Finally, in Section 6, the direction ofresearch for the master thesis project is given.

    1http://www.manchester.ac.uk/ifiss

    1

  • 2 Discretization and linearization of the N-S equations

    Partial differential equations in general, or the governing equations in fluid dynamics inparticular, are classified into three categories: (1) elliptic, (2) parabolic, and (3) hyper-bolic. The physical situations these types of equations represent can be illustrated bythe flow velocity relative to the speed of sound. Consider that the flow velocity u is thevelocity of a body moving in a fluid at rest. The movement of this body disturbs thefluid particles ahead of the body, setting off the propagation velocity equal to the speedof sound c. The ratio of these two competing speeds is defined as Mach number:

    M =u

    c

    For subsonic speed, M < 1, as time t increases, the body moves a distance, ut, which isalways shorter than the distance at of the sound wave. If, on the other hand, the bodytravels at the speed of sound, M = 1, then the observer does not hear the body ap-proaching him prior to the arrival of the body, as these two actions are simultaneous. Forsupersonic speed, M > 1, the velocity of the body is faster than the speed of sound. Thegoverning equations for subsonic flow, transonic flow, and supersonic flow are classifiedas elliptic, parabolic, and hyperbolic, respectively.

    2.1 Discretization

    In continuum mechanics, incompressible flow refers to a flow in which the material densityis constant within an infinitesimal volume that moves with the velocity of the fluid. TheMach number can be used to determine if a flow can be treated as an incompressibleflow. If M 1 and the flow is quasi-steady and isothermal, compressibility effects willbe small and a simplified incompressible flow model can be used. The incompressible flowof a Newtonian fluid is governed by the behavior defined by the set of equations

    2u + u u +p = f in (1)

    u = 0 in (2)

    Equation (1) represents the conservation of momentum, while equation (2) representsthe incompressibility condition, or mass conservation. The boundary value problem thatis considered is the system composed of equations (1) and (2) posed on a two or threedimensional domain , together with boundary conditions on = D N given by

    u = w on D, u

    n np = s on N . (3)

    The weak formulation of the Navier-Stokes equations is given by:

    (2u) v +

    (u u) v

    p v) =

    f v (4)

    ( u) q = 0 (5)

    2

  • where v and q are test functions in velocity and pressure space, respectively. After apply-ing the Gauss divergence theorem and substitution of the boundary conditions, we obtainthe equivalent problem:

    Find u H1E() and p L2() such that

    uv +

    (u u) v

    p( v) =

    N

    s v (6)

    q ( u) = 0 (7)

    where H1E denotes the Sobolev space of functions whose generalized derivatives are inL2(). The symbol denotes the outer product. The discrete version of equations (6)and (7) is formulated as:

    Given the finite dimensional subspaces Xh0 H1E0 ,Xh H1E and Mh L2(), find

    uh XhE and ph Mh such that:

    uh vh +

    (uh uh) vh

    ph( vh) =

    N

    s vh for allvh Xh0 (8)

    qh ( uh) = 0 for all qh Mh (9)

    Following the steps of the Galerkin method we define two types of basis functions, i(x)for the pressure and i(x) for the velocity. So the approximation for uh and ph is definedas

    ph =

    npj=1

    pjj(x), np is the number of pressure unknowns (10)

    and

    uh =

    nu2j=1

    u1jj1(x) + u2jj2(x) =

    nuj=1

    ujj(x) (11)

    where nu is the number of velocity unknowns, uj is defined by uj = u1j , for j = 1, ..nu2 ,

    uj+nu2

    = u2j , for j = 1, ..nu2 and j in the same way. If we make the substitution

    v = i(x), q = i(x), we get the standard Galerkin formulation:Find ph and uh, such that

    uh i +

    (uh uh) i

    ph( i) =

    N

    s i for alli = 1, ..nu, (12)

    i ( uh) = 0 for all i = 1, ...np. (13)

    3

  • This system of equations can be represented in matrix form as

    Adu+N(u) +BT p = f, (14)

    Bu = g, (15)

    where u denotes the vector of unknowns u1i and u2i, and p denotes the vector of unknownspi. The term Adu is the discretization of the viscous term and N(u) is the discretizationof the nonlinear convective term, Bu denotes the discretization of the negative divergenceof u and BT p is the discretization of the gradient of p. The right-hand side vectors fand g contain all contributions of the source term, the boundary integral as well as thecontribution of the prescribed boundary conditions.

    2.2 Linearization

    As we can see, the Navier Stokes equations are nonlinear because of th