ISTANBUL TECHNICAL UNIVERSITY FACULTY OF AERONAUTICS AND ASTRONAUTICS

GRADUATION PROJECT

JULY, 2020

A VERTEX BASED VELOCITY

FORMULATION FOR THE INCOMPRESSIBLE

NAVIER-STOKES EQUATIONS

Thesis Advisor: Prof. Dr. Mehmet ŞAHİN

Atakan OKAN

Department of Astronautical Engineering

Atakan OKAN

GRADUATION PROJECT

JULY, 2020

A VERTEX BASED VELOCITY

FORMULATION FOR THE INCOMPRESSIBLE

NAVIER-STOKES EQUATIONS

Thesis Advisor: Prof. Dr. Mehmet ŞAHİN

Atakan OKAN

(ID: 110150114)

Department of Astronautical Engineering

ISTANBUL TECHNICAL UNIVERSITY FACULTY OF AERONAUTICS AND ASTRONAUTICS

Thesis Advisor : Prof. Dr. Mehmet ŞAHİN ..............................

İstanbul Technical University

Jury Members : Prof. Dr. Mehmet ŞAHİN .............................

İstanbul Technical University

Asst. Prof. Dr. Ayşe Gül GÜNGÖR ..............................

İstanbul Technical University

Asst. Prof. Dr. Bayram ÇELİK ..............................

İstanbul Technical University

Atakan OKAN, student of ITU Faculty of Aeronautics and Astronautics student ID

110150114, successfully defended the graduation project entitled “A Vertex Based

Velocity Formulation For The Incompressible Navier-Stokes Equations”, which

he/she prepared after fulfilling the requirements specified in the associated

legislations, before the jury whose signatures are below.

Date of Submission : 13 July 2020

Date of Defense : 22 July 2020

iv

To my family and the scientific community of the world,

v

FOREWORD

Purpose of this thesis is to make a numerical discretization of incompressible

Navier-Stokes equations on a uniform, structured and collocated grid, practice some

basic and advanced concepts in partial differential equations, numerical methods and

linear algebra in the context of computational fluid dynamics, gain experience in

scientific computation practices and post-processing software.

I would like to thank Prof. Dr. Mehmet Şahin for his excellent support during

the preparation of this thesis as he answered all my questions on any day of the week,

troubleshooted my programming related issues and last but not the least, gave me a

“relatively simple, but challangeing enough” subject in the context of CFD which

helped me get familiar with many advanced topics and learn a great deal. I will always

remember this thesis as the first steps of my future CFD career in engineering.

I also would like to thank my family as they were always supportive of me and hoping

for my mental well-being while preparing my thesis during this unfortunate and

stressful times of our lives due to COVID-19. Therefore, I also salute all the essential

workers and health personnel of our world for their incredible selfless sacrifices they

had to perform for the world’s sake.

July 2020

Atakan OKAN

vi

vii

TABLE OF CONTENTS

Page

FOREWORD .................................................................................................v

TABLE OF CONTENTS ............................................................................ vii ABBREVIATIONS .................................................................................... viii

LIST OF TABLES ....................................................................................... ix LIST OF FIGURES .......................................................................................x

SUMMARY .................................................................................................. xi

ÖZET ......................................................................................................... xiii

1. INTRODUCTION......................................................................................1 2. GOVERNING EQUATIONS AND ASSUMPTIONS ..............................2

2.1 Non-dimensionalization of Navier-Stokes Equations ............................. 2 2.2 Stokes Flow ........................................................................................... 4

3. NUMERICAL TREATMENT OF THE EQUATIONS ...........................5 3.1 Grids ..................................................................................................... 5

3.2 Discretization of Momentum Equations ................................................. 6 3.3 Discretization of Continuity Equation .................................................... 9

4. SOLUTION OF SYSTEM OF EQUATIONS ......................................... 13 4.1 Linear Systems .....................................................................................13

4.2 Solution Methods .................................................................................15

5. NUMERICAL APPLICATION TO A BENCHMARK PROBLEM .... 16

5.1 Plane Pouseuille Flow ..........................................................................16 5.2 Programming of Numerical Algorithm .................................................17

5.2 Post-processing of Solution ..................................................................19

6. VALIDATION OF RESULTS AND CONCLUSION ............................ 23 6.1 Closed-form solution of plane Pouiseuille flow ....................................23 6.2 Conclusion ...........................................................................................25

REFERENCES ............................................................................................ 26 APPENDICES .............................................................................................. 28

APPENDIX A ............................................................................................28

CURRICULUM VITAE .............................................................................. 33

viii

ABBREVIATIONS

App : Appendix

BC : Boundary condition

CFD : Computational Fluid Dynamics

Eq : Equation

FDM : Finite Difference Method

Fig : Figure

FVM : Finite Volume Method

GMRES : Generalized Minimum Residual

KSP : Krylov subspace

MPI : Message Parsing Interface

PDE : Partial Differential Equation

PETSc : Portable Extensible Toolkit for Scientific Computation

RHS : Right hand side

ix

LIST OF TABLES

Page

Table 2.1 : Non-dimensionalization for flow variables. ............................................3

Table 5.1: Iteration parameters for solution on different grids…………….……... 19

x

LIST OF FIGURES

Page

Figure 3.1 : Simple 5x5 grid. ................................................................................... 5

Figure 3.2 : Grid for discretization of viscous terms. ................................................ 6

Figure 3.3 : Grid for discretization of dp/dx term.. ........................................................ 7

Figure 3.4 : Grid for discretization of dp/dx term. ......................................................... 7

Figure 3.5 : Grid for discretization of dp/dy term.. ........................................................ 8

Figure 3.6 : Grid for discretization of dp/dy term. ......................................................... 8 Figure 3.7 : Control volumes near the corner node.. .................................................... 10

Figure 3.8 : Discretization of Eq.3.14 for node-1. ....................................................... 11

Figure 5.1 : Physical domain representation of plane Pouseuille flow. ................... 16

Figure 5.2 : Solution for u component of velocity for a 11x11 grid.. ...................... 19

Figure 5.3 : Solution for v component of velocity for a 11x11 grid... ..................... 20

Figure 5.4 : Solution for pressure field on a 11x11 grid.......................................... 21

Figure 5.5 : Solution for pressure field on a 21x21 grid.......................................... 21

Figure 5.6 : Solution for u velocities on a 21x21 grid. ............................................ 22

Figure 5.7 : Solution for v velocities on a 21x21 grid. ............................................ 22

Figure A.1 : Declaration of variables and matrix/vector creation.… ....................... 28

Figure A.2 : Programming of x and y momentum equations.….............................. 29

Figure A.3 : Programming of continuity equation.… ............................................. 30

Figure A.4 : Solution step of algorithm.… ............................................................. 31

Figure A.5 : Printing the solution in Tecplot format.… .......................................... 32

xi

A VERTEX BASED VELOCITY FORMULATION FOR THE

INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

SUMMARY

Advances in computing power has made CFD a favorable complement to the

theoretical and experimental work in fluid dynamics. proper application of CFD

theory to fluid flow problems require solid knowledge of numerical analysis and

linear algebra. Analysis of viscous incompressible flows are tricky since Navier-

Stokes equations govertn the flow and they are coupled, non-linear set of partial

differential equations where the velocity and pressure is “decoupled” and has to be

treated specially. In this thesis, a subset of viscous incompressible Navier-Stokes

equations are numerically investigated with “Re<<1” assumption. Although the flow

is considered viscous, only the frictional dissipation phenomenon is taken into

account. After writing in non-dimensional form, Stokes equations are a linear set of

equation system where each equation is elliptic and of Poisson type. Stokes flow

governed by the Stokes equations is encountered in many applications of

engineering, medicine and life sciences research. Due to the characteristic of elliptic

partial differential equations, a “marching solution” typical of parabolic and

hyperbolic partial differential equations encountered in some CFD problems is not

possible and a “simultaneous solution” has to be obtained at once for every point in

the domain of Stokes flow. In order to create a well-posed problem, proper boundary

conditions need to be provided. For numerical solution of PDEs, these continuous

equations have to be transformed by numerical approximations. In this thesis, finite

difference and finite volume approaches are used.

Momentum equations for Stokes flow are discretized using finite difference scheme.

Viscous force terms of momentum equations are discretized around inner nodes

using a second-order centered difference formula with a five-point discretization

stencil. For the pressure gradient terms, a hybrid approach is adopted where the inner

nodes that are closest to the boundary use a second order centered finite difference

scheme and rest of the inner nodes use a fourth-order centered finite difference

scheme. This is done in order to eliminate a potential checkerboard distribution

caused by odd-even decoupling of pressure field. As implied by the title of the thesis,

here also a collocated grid approach is preferred with ui,j, vi,j and pi,j all being

calculated at the same node or cell vertex for simplicity. These discretizations are of

fully implicit form since an explicit discretization of a steady system does not make

sense numerically as the time term does not exist. For discretization of continuity

equation, a finite volume approach 4 cells bounded by 4 nodes are utilized where the

Green’s theorem is also used for obtaining line integrals. By means of linear

interpolation of velocity variables u and v at the middle of each four cell’s edges

using the variables at grid nodes, equations are discretized for each node at the outer

corners of inner 4 cells.

As a result of discretization of momentum and continuity equations for the Stokes

flow, a linear system of algebraic equations is formed which can be shown with a

matrix-vector product notation. If “n” is taken as the number of nodes in both x and y

direction for a uniform square grid, A is the (3n2) x (3n2) coefficient matrix obtained

from discretized equations, x is the (3n2) x 1 solution vector that contains all

xii

unknown variables in a chosen order and b is the (3n2) x 1 right-hand-side vector that

contains boundary conditions. A is square, non-symmetric, singular, sparse matrix

that can also be shown in block form which corresponds to the generalized saddle

point problem encountered in Stokes flow problems in CFD. For solution of

algebraic equation systems, direct and iterative methods are used. In this thesis, due

to the singular nature of matrix, a direct method is not viable.

Steady and laminar plane Poiseuille flow is investigated as a benchmark problem.

For the laminar flow considered, no-slip boundary conditions were applied on the top

and bottom surfaces. Inlet and outlet boundary conditions are a parabolic u velocity

profile where u=u(y) and v=0. No pressure B.C. was provided. All BCs are Dirichet

type.

The implementation of the numerical algorithm was done with the FORTRAN

programming language while utilizing PETSc data structures and routines package

for the numerical solution of PDEs. As an iterative solver for non-symmetric linear

systems and considering it can converge even in some singular cases, GMRES

method is chosen for the solution. No preconditioning is applied. Results of

numerical algorithm are visualized with Tecplot 360 post-processing software.

Results are validated with the existing analytical solution for plane Poiseuille flow

given by Hagen-Pouiseuille equation. Therefore, it is verified that the algorithm

developed is stable and converges within a tolerance in finite number of iterations.

xiii

SIKIŞTIRILABİLİR NAVIER-STOKES DENKLEMLERİ İÇİN VERTEKS

ODAKLI BİR HIZ FORMÜLASYONU

ÖZET

Hesaplama gücündeki ilerlemeler hesaplamalı akışkanlar dinamiğini teorik ve

deneysel çalışma için uygun bir tamamlayıcı haline getirmiştir. HAD teorisinin akış

problemlerine düzgün bir şekilde uygulanması, sayısal analiz ve doğrusal cebir

konusunda sağlam bilgi gerektirir. Özellikle viskoz sıkıştırılamaz akışların analizi

zordur çünkü Navier-Stokes denklemleri akışı yönetir ve bunlar, hız ve basıncın

bağlantısız olduğu ve özel olarak ele alınması gereken, doğrusal olmayan kısmi

diferansiyel denklemler kümesidir. Bu tezde, viskoz sıkıştırılamaz Navier-Stokes

denklemlerinin bir alt kümesi “Re << 1” varsayımı ile sayısal olarak incelenmiştir.

Akış viskoz olarak kabul edilmesine rağmen, sadece sürtünme kökenli dissipatif

etkiler dikkate alınmıştır. Boyutsuz formda yazıldıktan sonra Stokes denklemleri, her

denklemin eliptik ve Poisson tipi olduğu doğrusal bir denklem sistemi kümesi haline

gelir. Stokes denklemleri tarafından yönetilen Stokes akışına mühendislik, tıp ve

yaşam bilimleri araştırmalarının birçok uygulamasında rastlanmaktadır. Eliptik kısmi

diferansiyel denklemlerin karakteristiği nedeniyle, bazı HAD problemlerinde

karşılaşılan parabolik ve hiperbolik kısmi diferansiyel denklemlerin tipik bir

“adımlama çözümü” mümkün değildir ve Stokes akışının her noktası için bir defada

“eşzamanlı bir çözüm” elde edilmesi gerekmektedir. İyi konumlanmış bir problem

yaratmak için uygun sınır koşullarının sağlanması gerekir. Kısmi diferansiyel

denklemlerin sayısal çözümü için, bu sürekli denklemlerin sayısal yaklaşımlarla

ayrıklaştırılması gerekir. Bu tezde sonlu farklar ve sonlu hacim yaklaşımları

kullanılmıştır.

Stokes akışı için momentum denklemleri sonlu farklar şeması kullanılarak ayrıştırılır.

Momentum denklemlerinin viskoz kuvvet terimleri beş noktalı ayrıklaştırma

şablonuna sahip ikinci dereceden merkezi bir fark formülü kullanılarak iç noktalar

etrafında ayrıklaştırılır. Basınç gradyanı terimleri için, sınıra en yakın iç noktaların

ikinci derece merkezli sonlu fark şeması kullandıkları ve iç noktaların geri kalanının

dördüncü dereceden merkezli sonlu fark şeması kullandığı hibrit bir yaklaşım

benimsenmiştir. Bu, basınç alanının tek-çift ayrışmasının neden olduğu potansiyel bir

“dama tahtası” dağılımını ortadan kaldırmak için yapılır. Tezin başlığında da

belirtildiği gibi, burada aynı zamanda daha basit olduğundan, akış değişkenlerinin

hücre verteks noktasında hesaplandığı “kolokasyonlu” da denen bir ızgara tercih

seçilmiştir. Daimi bir sistemin açık çözüm yöntemi ile ayrıştırılması, zaman terimi

olmadığı için sayısal olarak mantıklı olmadığından, bu ayrıklaştırma “örtük” şekilde

yapılmıştır. Süreklilik denkleminin ayrıklaştırılması için, 4 noktayla sınırlandırılmış

4 hücrenin kullanıldığı sonlu hacimler yaklaşımı kullanılmıştır. Green teoremi

kullanılarak da ayrıklaştırmanın yapıldığı çizgi integralleri elde edilmiştir. Izgara

noktalarındaki değişkenleri kullanarak, içteki her dört hücrenin de kenarlarının

ortasındaki u ve v hız değişkenlerinin doğrusal enterpolasyonu ile denklemler

ayrıklaştırılır.

Stokes akışı için momentum ve süreklilik denklemlerinin ayrıklaştırılmasının bir

sonucu olarak, bir matris-vektör çarpımı ile gösterilebilen doğrusal bir cebirsel

denklem sistemi oluşturulur. Düzgün bir kare ızgara için hem x hem de y'deki nokta

xiv

sayısı olarak “n” alınırsa A, ayrıklaştırılmış denklemlerden elde edilen (3n2) x (3n2)

katsayı matrisidir, x ise bilinmeyen tüm değişkenleri içeren (3n2) x 1 çözüm

vektörüdür ve b, sınır koşullarını içeren (3n2) x 1 sağ taraf vektörüdür. A matrisi

kare, simetrik olmayan, tekil, seyrek bir matristir. Aynı zamanda Stokes akış

problemlerinde karşılaşılan genel “eyer noktası problemi”ne karşılık gelen blok

formda da gösterilebilir. Cebirsel denklem sistemlerinin çözümü için doğrudan ve

iteratif yöntemler kullanılır. Bu tezde, matrisin tekil yapısı nedeniyle, doğrudan bir

yöntem kullanılamamaktadır.

Daimi ve laminer düzlem Poiseuille akışı bir kıyaslama problemi olarak

incelenmiştir. Ele alınan laminer akış için, üst ve alt yüzeylere “kaymazlık“sınır

koşulları uygulanmıştır. Giriş ve çıkış sınır koşulları, u = u (y) ve v = 0 olan bir

parabolik u hız profilidir. Basınç içen sınır koşulu verilmemiştir. Tüm sınır koşulları

Dirichet tipindedir.

Sayısal algoritmanın pratik uygulaması, kısmi diferansiyel denklemlerin sayısal

çözümü için PETSc veri yapıları ve algoritmaları paketi kullanılarak ve FORTRAN

programlama dili ile yapıldı. Simetrik olmayan doğrusal sistemler için iteratif bir

çözücü, bazı tekil durumlarda bile çözüme yakınsadığı göz önüne alınıp GMRES

yöntemi olarak seçilmiştir. Hiçbir "ön koşullandırıcı” uygulanmamıştır. Sayısal

algoritmanın sonuçları Tecplot 360 post-proses yazılımı ile görselleştirilmiştir.

Sonuçlar Hagen-Pouiseuille denklemi ile verilen düzlem Poiseuille akışı için analitik

çözüm ile doğrulanmıştır. Bu nedenle, geliştirilen algoritmanın kararlı olduğu ve

sınırlı sayıda iterasyonda bir tolerans dahilinde sonuca yakınsadığı görülmektedir.

1

1. INTRODUCTION

Owing to the advance in computing power since 1960s for numerical

calculations on computers, computational fluid dynamics (CFD) has almost become a

discipline of its own along the fluid dynamics and aerodynamics branches of physics.

It has become a favorable complement to the common issues in fluid dynamics and

aerodynamics research when either analytical solutions either do not exist or difficult

to obtain to a problem or the practical wind/waterr tunnel experiments are costly.

However, proper application of CFD theory to fluid flow problems require significant

effort in obtaining sufficient knowledge of numerical analysis and linear algebra

concepts. CFD is by no means equal to “clicking buttons and obtaining colorful

results” by commercial/open-source software programs. This is evident in the analysis

of viscous incompressible flows where the compressibility effects are neglected and

energy equation is not necessary to solve the velocity and pressure field. This

seemingly “innocent” type of flow is governed by the Navier-Stokes equations which

are coupled, non-linear set of partial differential equations where the velocity and

pressure is “decoupled” and usually has to be treated specially by means of choosing

special higher-order discretization methods that result in stable divergence-free

solutions [7] or by using pressure correction methods that utilize the momentum

equations where the pressure field satisfies the continuity equation as found in the CFD

literature [6].

This thesis also tries to obtain a stable and divergence-free numerical solution

to the steady incompresible Navier-Stokes equations on a uniform, square and

collocated grid. Thesis is outlined as follows: in 2nd section, Navier-Stokes are non-

dimensionalized and Stokes flow assumption is made. In 3rd section, momentum and

continuity equations are discretized on collocated grids. In 4th section, solution

methods of the discretized set of algebraic equations and matrix properties are

discussed. In 5th section, plane Poiseuille flow is considered as a practical numerical

application case and programming of numerical algorithm is described. Finally, in the

6th section numerical solutions are validated with closed-form analytical solutions.

2

2. GOVERNING EQUATIONS AND ASSUMPTIONS

2.1. Non-dimensionalization of Navier-Stokes Equations

Governing equations for a viscous, incompressible flow are the Navier-Stokes

equations for momentum conservation and continuity equation for mass conservation.

Assuming a steady flow of a Newtonian fluid in the absence of body forces, the

equations in 2D can be written in Cartesian form as follows:

𝜌 ( 𝑢𝜕𝑢

𝜕𝑥+ 𝑣

𝜕𝑢

𝜕𝑦 ) = −

𝜕𝑝

𝜕𝑥+ 𝜇 (

𝜕2𝑢

𝜕𝑥2 + 𝜕2𝑢

𝜕𝑦2 ) (2.1)

𝜌 ( 𝑢𝜕𝑣

𝜕𝑥+ 𝑣

𝜕𝑣

𝜕𝑦 ) = −

𝜕𝑝

𝜕𝑦+ 𝜇 (

𝜕2𝑣

𝜕𝑥2 + 𝜕2𝑣

𝜕𝑦2 ) (2.2)

𝜕𝑢

𝜕𝑥+

𝜕𝑣

𝜕𝑦= 0 (2.3)

where u (m/s) and v (m/s) are the x and y components of velocity vector u = ui + vj, 𝜌

(𝑘𝑔

𝑚3 ) is the density, p (𝑁

𝑚2 ) is the pressure and 𝜇 (𝑘𝑔

𝑚.𝑠 ) is the dynamic viscosity with

a constant value (no temperature dependence). Due to this assumption, the energy

equation is decoupled from the above set of equations and it is not required for velocity

and pressure field solutions. Eqs 2.1 to 2.3 are a system of coupled partial differential

equations whose “general” analytical solution hasn’t been found yet.

Here, the non-linear momentum conservation equations (2.1-2.2) are

essentially “force balance” equations where the convective terms on the left-hand side

are inertial terms (acceleration), whereas the pressure and viscous terms on the right-

hand side denote the force terms. Although the flow is considered viscous, only the

frictional dissipation is taken into account. Other dissipative phenomena such as

thermal conduction or mass diffusion are assumed to be non-existent. Furthermore, for

this thesis flows where viscous effects are much more strong compared to inertial

forces (Re<<1 or creeping flows) are considered as a special case.

3

Based on this, a non-dimensionalization procedure can be applied to the

equations above to get rid of the need for dealing with consistent units during

programming of numerical analysis. For flows where viscous effects are dominant, the

appropriate characteristic scaling factors [1] for non-dimensionalization are shown in

Table 2.1.

Table 2.1: Non-dimensionalization for flow variables.

Characteristic scale Dimensionless form of variables

Velocity U (free-stream) u* = u/U , v* = v/U

Length L (characteristic length) x* = x/L, y* = y/L

Pressure 𝜇𝑈

𝐿 (for Re<<1 flows) p* =

𝑝𝐿

𝜇𝑈

After substituting the characteristic scales into the above equations, the intermediate

step for x-momentum equation is given by Eq. 2.4:

( 𝜌𝑈2

𝐿 ) ( 𝑢∗ 𝜕𝑢∗

𝜕𝑥∗+ 𝑣∗ 𝜕𝑢∗

𝜕𝑦∗ ) = (

𝜇𝑈

𝐿2 ) (− 𝜕𝑝∗

𝜕𝑥∗) + (

𝜇𝑈

𝐿2 ) ( 𝜕2𝑢∗

𝜕𝑥∗2 + 𝜕2𝑢∗

𝜕𝑦∗2 ) (2.4)

Following through by grouping terms similarly for y-momentum equation and

continuity equation, result is the following non-dimensional versions of the Navier–

Stokes and continuity equations in Cartesian form:

𝑅𝑒 ( 𝑢∗ 𝜕𝑢∗

𝜕𝑥∗ + 𝑣∗ 𝜕𝑢∗

𝜕𝑦∗ ) = − 𝜕𝑝∗

𝜕𝑥∗ + ( 𝜕2𝑢∗

𝜕𝑥∗2 + 𝜕2𝑢∗

𝜕𝑦∗2 ) (2.5)

𝑅𝑒 ( 𝑢∗ 𝜕𝑣∗

𝜕𝑥∗ + 𝑣∗ 𝜕𝑣∗

𝜕𝑦∗ ) = − 𝜕𝑝∗

𝜕𝑦∗ + ( 𝜕2𝑣∗

𝜕𝑥∗2 + 𝜕2𝑣∗

𝜕𝑦∗2 ) (2.6)

𝜕𝑢∗

𝜕𝑥∗ +𝜕𝑣∗

𝜕𝑦∗ = 0 (2.7)

where the dimensionless Reynolds number denoted by Re is calculated as:

𝑅𝑒 =𝜌𝑈𝐿

𝜇 (2.8)

4

For flows where viscous effects are dominant, taking the limit Re → 0 in Eqs. 2.5 and

2.6. leads to the non-dimensional steady Stokes equations written below.

− ( 𝜕2𝑢∗

𝜕𝑥∗2 + 𝜕2𝑢∗

𝜕𝑦∗2 ) + 𝜕𝑝∗

𝜕𝑥∗ = 0 (2.9)

− ( 𝜕2𝑣∗

𝜕𝑥∗2 + 𝜕2𝑣∗

𝜕𝑦∗2 ) + 𝜕𝑝∗

𝜕𝑦∗ = 0 (2.10)

In contrast to the non-linear nature of full Navier-Stokes equations, Stokes

equations are in linear form due to the fact that convective terms which are the source

of non-linearity are neglected in the limit Re 0. This provides a great simplification

of the discretization of equations. In addition, according to the quasi-linear partial

differential equation classification method [2], Eqs 2.9-2.10 are of elliptic type, in the

form of the Poisson equation.

2.2. Stokes Flow

Stokes flow or creeping flow governed by the Stokes equations is encountered

in many applications of engineering, medicine and life sciences research. Medical

applications [1], swimming of microorganisms [3], microfluidic device flows, viscous

polymer flows, lava flow and lubrication [4] are noteworthy examples.

It should be noted that due to the characteristic of the governing elliptic partial

differential equations, any disturbance in the Stokes flow domain is felt by all other

points immediately which implies that the solution which is sought is also influenced

by the entire boundary of the flow domain. This is significant since in this case, a

“marching solution” typical of parabolic and hyperbolic partial differential equations

encountered in some CFD problems is not possible and a “simultaneous solution” has

to be obtained at once for every point in the domain [5,6]. Therefore, in order to create

a well-posed problem, proper boundary conditions need to be provided and these can

be either of Dirichet, Neumann or mixed type [6]. Finally, since the steady Stokes flow

is of concern, this is a simple equilibrium boundary value problem - not an initial value

problem - and initial conditions are not required.

5

3. NUMERICAL TREATMENT OF THE EQUATIONS

3.1. Grids

For numerical solution of partial differential equations on computers, these

continuous equations have to be transformed into the discrete realm of computers by

numerical approximations. This can be accomplished by widely used methods such as

finite difference, finite volume, finite element methods and spectral methods. In this

thesis, finite difference and finite volume approaches are used.

For discretization of PDEs, a computational grid which consists of a finite number

of nodes is needed to represent the physical flow domain. Depending on the nature of

equation, grid nodes that lie on the same line (either vertically or horizontally) can be

thought of indicating a layer of either spatial direction (x, y) or time (n). Each unknown

or known variable can be assigned to a distinct node after discretization. In the case of

steady Stokes flow, since there is no concept of time evolution, all nodes indicate the

spatial coordinates of flow variables such as velocity and pressure on the discrete

domain. As shown in Fig. 3.1. for a simple square flow domain, while the outer blue

nodes indicate the known boundary conditions of the physical flow domain, inner

orange nodes are used for housing the unknown variables to be solved.

Figure 3.1: Simple 5x5 grid.

6

The choice of grid is not limited to the orthogonal and structured grid shown in

Fig. 3.1. For flow domains with involving more complex geometries, unstructured

grids provide flexibility and avoid Cartesian grids’ limitations encountered with

curvilinear boundaries [6]. In addition, stretched grids are preferred where the

gradients of certain flow variables are large in very small distances (i.e. boundary layer

flows) and non-uniform spacing is needed in certain regions of the computational flow

domain for finer solution [5]. This is accomplished by coordinate transformation where

the irregular physical coordinate system is mapped into a computational Cartesian

coordinate system [6].

3.2. Discretization of Momentum Equations

Momentum equations for Stokes flow are discretized using finite difference

scheme. Eqs. 2.8 and 2.9 are repeated below. The goal is to discretize the PDEs such

that the flow variables u(x,y), v(x,y) and p(x,y) are represented by their discrete and

approximate counterparts ui,j , vi,j and pi,j at each node and create a system of

algebraic equations to be solved simultaneously.

− ( 𝜕2𝑢∗

𝜕𝑥∗2 + 𝜕2𝑢∗

𝜕𝑦∗2 ) + 𝜕𝑝∗

𝜕𝑥∗ = 0 (3.1)

− ( 𝜕2𝑣∗

𝜕𝑥∗2 + 𝜕2𝑣∗

𝜕𝑦∗2 ) + 𝜕𝑝∗

𝜕𝑦∗ = 0 (3.2)

Figure 3.2: Grid for discretization of viscous terms.

Δx

Δy i,j

i+1,j

i,j-1

i,j+1

i-1,j

7

As shown by the commonly used five-point finite difference stencil on Fig.

3.2., viscous force terms of x and y momentum equations are discretized around

inner nodes using a second-order centered difference formula as in Eq. 3.3 and 3.4.

Star (*) notation for non-dimensional variables are dropped for easier reading.

− ( 𝝏𝟐𝒖

𝝏𝒙𝟐 + 𝝏𝟐𝒖

𝝏𝒚𝟐 ) = − (𝒖𝒊+𝟏,𝒋 − 𝟐𝒖𝒊,𝒋 + 𝒖𝒊−𝟏,𝒋

∆𝒙𝟐 +𝒖𝒊,𝒋+𝟏 − 𝟐𝒖𝒊,𝒋 + 𝒖𝒊,𝒋−𝟏

∆𝒚𝟐) + 𝑶(∆𝒙𝟐, ∆𝒚𝟐) (3.3)

− ( 𝝏𝟐𝒗

𝝏𝒙𝟐 + 𝝏𝟐𝒗

𝝏𝒚𝟐 ) = − (𝒗𝒊+𝟏,𝒋 − 𝟐𝒗𝒊,𝒋 + 𝒗𝒊−𝟏,𝒋

∆𝒙𝟐 +𝒗𝒊,𝒋+𝟏 − 𝟐𝒗𝒊,𝒋 + 𝒗𝒊,𝒋−𝟏

∆𝒚𝟐) + 𝑶(∆𝒙𝟐, ∆𝒚𝟐) (3.4)

Taking advantage of the uniform grid (Δx = Δy), Eqs 3.3-3.4 can be further

simplified by combining terms as follows where O(Δx2) is order of approximation.

− ( 𝝏𝟐𝒖

𝝏𝒙𝟐 + 𝝏𝟐𝒖

𝝏𝒚𝟐 ) = − (𝒖𝒊+𝟏,𝒋 + 𝒖𝒊,𝒋+𝟏− 𝟒𝒖𝒊,𝒋 + 𝒖𝒊,𝒋−𝟏+ 𝒖𝒊−𝟏,𝒋

∆𝒙𝟐) + 𝑶(∆𝒙𝟐) (3.5)

− ( 𝝏𝟐𝒗

𝝏𝒙𝟐 + 𝝏𝟐𝒗

𝝏𝒚𝟐 ) = − (𝒗𝒊+𝟏,𝒋 + 𝒗𝒊,𝒋+𝟏− 𝟒𝒗𝒊,𝒋 + 𝒗𝒊,𝒋−𝟏+ 𝒗𝒊−𝟏,𝒋

∆𝒙𝟐) + 𝑶(∆𝒙𝟐) (3.6)

For the pressure gradient term discretization, a hybrid approach is adopted where the

inner nodes that are closest to the boundary use a second order centered finite

difference scheme as in Eqs. 3.9. and 3.10 and rest of the inner nodes use a fourth-

order centered finite difference [7] as in Eqs. 3.7. and 3.8. As shown in Figs. 3.3. to

3.6 for a 7x7 uniform grid in the discretization of pressure gradient terms, nodes in

red are where higher order difference scheme is centered at and the green nodes are

where the lower order scheme is centered at.

8

Red and green nodes are not used in the difference schemes themselves. The pink

shades denote the area that includes all the nodes for centering the schemes at based

on whether the nodes are close to boundary or not.

𝝏𝒑

𝝏𝒙=

( −𝒑𝒊+𝟐,𝒋 + 𝟖 𝒑𝒊+𝟏,𝒋 − 𝟖 𝒑𝒊−𝟏,𝒋 + 𝒑𝒊−𝟐,𝒋 )

𝟏𝟐(∆𝒙)+ 𝑶(∆𝒙𝟒) (3.7)

𝝏𝒑

𝝏𝒚=

(−𝒑𝒊,𝒋+𝟐 + 𝟖 𝒑𝒊,𝒋+𝟏− 𝟖 𝒑𝒊,𝒋−𝟏+ 𝒑 𝒊,𝒋−𝟐)

𝟏𝟐(∆𝒚)+ 𝑶(∆𝒚𝟒) (3.8)

𝝏𝒑

𝝏𝒙=

( 𝒑𝒊+𝟏,𝒋 − 𝒑𝒊−𝟏,𝒋 )

𝟐(∆𝒙)+ 𝑶(∆𝒙𝟐) (3.9)

𝝏𝒑

𝝏𝒚=

( 𝒑𝒊,𝒋+𝟏 − 𝒑𝒊,𝒋−𝟏 )

𝟐(∆𝒚)+ 𝑶(∆𝒚𝟐) (3.10)

9

Reason for using a fourth-order scheme for some nodes in pressure terms is to

eliminate a potential checkerboard distribution caused by odd-even decoupling of

pressure field when both continuity and pressure gradient in the momentum equation

are discretized with second-order centered differences [7,8]. One way of organizing

variables in the grid is placing all variables at the same node.

As implied by the title of the thesis, here also this collocated grid approach is

preferred with ui,j, vi,j and pi,j all being calculated at the same node or cell vertex for

simplicity, although it may lead to nonsensical pressure and velocity oscillations in

incompressible flows [6]. Using a staggered grid where pressures could be placed at

the middle of cell edges and velocities at the vertices could help, however it is more

difficult to program [2]. Finally, these discretizations are of fully implicit form since

an explicit discretization of a steady system does not make sense numerically as the

time term does not exist.

3.3. Discretization of Continuity Equation

Non-dimensional continuity equation which is repeated below (without the star

notation) is discretized with finite volume approach.

𝝏𝒖

𝝏𝒙+

𝝏𝒗

𝝏𝒚= 𝟎 (3.11)

In order to utilize the cell partition approach near the corner node of a zoomed

portion of a uniform grid as shown in Fig. 3.7, Green’s theorem was used as follows.

Green’s theorem relates a line integral around a closed curve C and a double integral

(the convention of positive orientation is anti-clockwise as in Fig. 3.7), Green’s

theorem is given by Eq. 3.12. over the region S bounded by that curve. In the case

10

there are functions P and Q with continous partial derivatives over that region and

the curve is positively oriented [9]

∮ 𝑷(𝒙, 𝒚)𝒅𝒙 + 𝑸(𝒙, 𝒚)𝒅𝒚 = ∬ ( 𝝏𝑸

𝝏𝒙𝑺

𝑪−

𝝏𝑷

𝝏𝒚 ) 𝒅𝒙𝒅𝒚 (3.12)

Figure 3.7: Control volumes near the corner node.

Variables u and v in continuity equation are assumed to have continuous

partial derivatives and can be used in place of P and Q functions since the right hand

of Green’s theorem is similar to the form of Eq. 3.11. It can be cast into the form of

the left hand side of Eq. 3.12 by multiplying Eq. 3.11 with “dxdy” in both sides and

taking the surface integral. Since the surface integral is a definite integral over each

cell, the right hand side “0” is unaffected and we can write:

∮ −𝒗𝒅𝒙 + 𝒖𝒅𝒚 = ∬ ( 𝝏𝒖

𝝏𝒙𝑺

𝑪−

𝝏(−𝒗)

𝝏𝒚 ) 𝒅𝒙𝒅𝒚 = 𝟎 (3.13)

∮ 𝒖𝒅𝒚 − ∮ 𝒗𝒅𝒙 𝑪

𝑪

= 𝟎 (3.14)

Therefore, by means of linear interpolation of velocity variables u and v at the middle

of each four cell’s edges (shown in Fig. 3.8 as “thick” red arrows by utilizing

interpolated values at white nodes) using the variables at grid nodes as shown by the

Node - 1

(i=1, j=1)

Cell - 3

Cell - 2 Cell - 1

Cell - 4 Node - 3

(i=2, j=2)

Node - 4

(i=1, j=2)

Node - 2

(i=2, j=1)

11

thinner arrows in Fig. 3.8., Eq. 3.14 can be discretized using finite volume approach

as follows. (For easier reading, here “ij” index notation is dropped and node based

notation is used. i.e. u1,1 = u1 and u2,2 = u3 etc.)

Figure 3.8: Discretization of Eq.3.14 for cell-1.

For node-1 based on Fig. 3.6. and Eq. 3. 14 using anti-clockwise positive

orientation for the line integrals over the cell edges:

1

2[

𝑢1+𝑢2

2+

𝑢1+𝑢2 + 𝑢3+𝑢4

4 ] ∆𝑦 –

1

2[ 𝑢1 +

𝑢1+𝑢4

2 ] ∆𝑦

− 1

2[ 𝑣1 +

𝑣1+𝑣2

2 ] ∆𝑥 +

1

2[

𝑣1+𝑣4

2+

𝑣1+𝑣2 + 𝑣3+𝑣4

4 ] ∆𝑥 = 0 (3.15)

Cell-1

v1

u1 u2

u4 u3

v3 v4

v2

12

Rearranging terms in Eq. 3.15 gives the discretized continuity equation for Node-1:

Node-1: 𝑢1

2(

∆𝑦

4− ∆𝑦) +

𝑢2

2(

∆𝑦

2+

∆𝑦

4) +

𝑢3

2(

∆𝑦

4) +

𝑢4

2(

∆𝑦

4−

∆𝑦

2) +

𝑣1

2(−∆𝑥 +

∆𝑥

4 ) +

𝑣2

2(−

∆𝑥

2+

∆𝑥

4) +

𝑣3

2(

∆𝑥

4) +

𝑣4

2(

∆𝑥

2+

∆𝑥

4) = 0 (3.16)

Using similar linear interpolation approach with their respective cells for nodes 2,3

and 4 and skipping intermediate steps, the resultant discretized continuity equations

are shown in Eqs. 3.17 to 3.19.

Node-2 : 𝑢1

2(−

∆𝑦

2−

∆𝑦

4) +

𝑢2

2(∆𝑦 −

∆𝑦

4) +

𝑢3

2(

∆𝑦

2−

∆𝑦

4) +

𝑢4

2(−

∆𝑦

4) +

𝑣1

2(−

∆𝑥

2+

∆𝑥

4 ) +

𝑣2

2(−∆𝑥 +

∆𝑥

4) +

𝑣3

2(

∆𝑥

2+

∆𝑥

4) +

𝑣4

2(

∆𝑥

4) = 0 (3.17)

Node-3: 𝑢1

2(−

∆𝑦

4) +

𝑢2

2(

∆𝑦

2−

∆𝑦

4) +

𝑢3

2(∆𝑦 −

∆𝑦

4) +

𝑢4

2(−

∆𝑦

2−

∆𝑦

4) +

𝑣1

2(−

∆𝑥

4 ) +

𝑣2

2(−

∆𝑥

4−

∆𝑥

2) +

𝑣3

2(−

∆𝑥

4+ ∆𝑥) +

𝑣4

2(−

∆𝑥

4+

∆𝑥

2)= 0 (3.18)

Node-4: 𝑢1

2(

∆𝑦

4−

∆𝑦

2) +

𝑢2

2(

∆𝑦

4) +

𝑢3

2(

∆𝑦

4+

∆𝑦

2) +

𝑢4

2(

∆𝑦

4− ∆𝑦) +

𝑣1

2(−

∆𝑥

2−

∆𝑥

4) +

𝑣2

2(−

∆𝑥

4) +

𝑣3

2(−

∆𝑥

4+

∆𝑥

2) +

𝑣4

2(−

∆𝑥

4+ ∆𝑥)=0 (3.19)

13

4. SOLUTION OF SYSTEM OF EQUATIONS

4.1. Linear Systems

As a result of discretization of momentum and continuity equations for the Stokes

flow, a linear system of algebraic equations is formed which can be shown with a

matrix-vector product notation as in Eq. 4.1.

[𝑨] ∗ 𝒙 = 𝒃 (4.1)

If “n” is taken as the number of nodes in both x and y direction for a uniform square

grid, in Eq. 4.1 [A] is the (3n2) x (3n2) coefficient matrix obtained from discretized

equations, x is the (3n2) x 1 solution vector that contains all unknown variables ui,j , vi,j

and pi,j in the desired order and b is the (3n2) x 1 right-hand-side vector that contains

boundary conditions applied to the discretized equations.

In this case, vector x also contains variables on boundary nodes although they

are already known. Since a uniform square grid which involves the boundary nodes is

used for the discretization of equations, [A] is also a square matrix. For clarity, [A] in

Eq. 4.1 can be shown as a collection of square sub-matrices in the block matrix form

as in Eq. 4.2:

[𝑨𝟏𝟏 𝟎 𝑨𝟏𝟑

𝟎 𝑨𝟐𝟐 𝑨𝟐𝟑

𝑨𝟑𝟏 𝑨𝟑𝟐 𝟎] [

𝒖𝒗𝒑

] = [𝒃𝒄𝒃𝒄𝟎

] (4.2)

14

where A11 and A22 contain the coefficients of viscous terms in discretized momentum

equations, A13 and A23 contain the pressure term coefficients, A31 and A32 contain

continuity equation coefficients. This form also fits the generalized saddle point

problem classification usually encountered in the solution of linearized Navier-Stokes

equations [12] of which Stokes equation is a subset.

In addition, [A] is non-symmetric for reasons such as different choices of

numerical schemes for momentum and continuity equations (FDM vs FVM),

discretization of various terms in the momentum equations with changing order of

approximation (4th vs 2nd order in pressure terms) and finally the involvement of

boundary nodes during the discretization instead of using only the inner nodes.

Furthermore, [A] is a singular matrix due to the fact that during the discretization of

momentum equations, neither viscous nor pressure terms involve the corner nodes (in

contrast to the finite volume approach used in the continuity equation) which results

in 4 columns of matrix [A] being all zeros – hence the determinant of [A] being 0 -

due to lack of contribution from the 4 corner nodes.

Another property of [A] is that – as commonly encountered in many CFD

problems – it is a sparse matrix with very few non-zero elements compared to its size.

This can be easily seen as during the discretization with finite difference for

momentum equations, viscous terms use a 5-point stencil and pressure terms use a 4-

point stencil for inner nodes (at maximum). This means that for each row of the

coefficient matrix [A] resulting from discretization of Stokes flow, number of nonzeros

can be no more than “9”, compared to the row length of [A] which is 3n2. Considering

the fact that “n” has to be at least 5 in order to use a 4th order difference scheme for

inner nodes, minimum row length of [A] is “75” which shows how sparsely distributed

the nonzero elements are. As will be utilized during the next chapter, efficient

strategies of storage schemes are utilized on many CFD solvers which take advantage

of the sparse nature of the matrices, since storing large amount of zeros is considered

unnecessary especially as the matrix sizes get larger [10].

15

4.2. Solution Methods

Two basic approaches for solving Eq. 4.1 are direct and iterative methods.

Although the direct methods give an exact solution up to the machine epsilon in finite

number of steps [6], common methods like Gauss elimination can be computationally

very expensive as the matrix sizes get larger (unless the matrix has a special band-

diagonal structure that can be efficiently utilized by methods such as Thomas

algorithm) and they are prone to rounding-error accumulations (unless pivoting

methods are used) [11]. In this thesis, due to the singular nature of matrix, a direct

method is not viable (unless a pressure fixing method is applied to the bottom-right

“zero” block of coefficient matrix [A] to get rid of singularity). On the other hand,

iterative methods rely on providing an initial guess as an approximate solution, try to

converge to an approximate solution of equations iteratively until the convergence

criterion is met for a given tolerance. They tend to use the sparsity of matrices more

efficiently, reducing computational costs as well [6].

16

5. NUMERICAL APPLICATION TO A BENCHMARK PROBLEM

5.1. Plane Poiseuille Flow

For this thesis, numerical application of the discretized equations was made on a

CFD benchmark problem commonly used for testing new solution algorithms due to

its simplicity: laminar steady plane Poiseuille flow. This flow describes a pressure-

driven steady, incompressible flow of a Newtonian fluid between two infinitely-long

flat plates. Physical domain with appropriate boundary conditions for the application

of numerical analysis to the problem is shown in Fig. 5.1.

Figure 5.1: Physical domain representation of plane Pouseuille flow.

Although the plane Pouiseuille flow equations which will be derived in the next

chapter is for flow between infinitely-flat plates, for practical analysis purposes a finite

domain -which can be thought of as a small portion of the infinitely long hypothetical

domain- needs to be used. A unit square domain is preferred for simplicity, where

length and height is donated by L and H, respectively, as in Fig. 5.1. As mentioned in

the previous sections, an elliptic equation requires proper B.Cs on all boundary points.

H = 1

L = 1

No-slip B.C.:

u = v = 0

v=0 v=0 Parabolic

inlet B.C.

for “u”

Parabolic

outlet B.C.

for “u”

No-slip B.C.:

u = v = 0

y=H

y=0

17

For the laminar flow considered, no-slip boundary conditions were applied on the

top and bottom surfaces. In addition, as will be verified in next chapter, a laminar

steady plane Poiseuille flow has a parabolic velocity profile and essentially a one-

dimensional flow. Therefore, the choice for the inlet and outlet boundary condition are

not no-slip BCs since domain has no rigid boundaries, instead they are special: a

parabolic u velocity profile where u=u(y) and v=0. No pressure B.C. was provided. It

should be noted that all the BCs that were provided are Dirichet type.

5.2. Programming of Numerical Algorithm

The implementation of the numerical algorithm was done with the FORTRAN

programming language while utilizing PETSc data structures and routines package

developed by Argonne National Laboratory for the numerical solution of PDEs. The

package is used especially for large-scale applications that run on parallel processes

for efficiency [13]. It provides matrix and vector assembly routines and linear system

solvers based on the iterative Krylov subspace methods. The full source-code for

numerical application to plane Poiseuille flow problem which is provided in

Appendix-A is briefly described below.

In App.-A, lines 1-25 in Fig. A-1 shows the declaration of variables for sizes and

names of coefficient matrix and right hand side vector, index variables for looping

over discretized momentum and continuity equations, and variables for certain

PETSc objects that handle matrix/vector assembly routines and Krylov subspace

solvers, and include files for all these objects.

In lines 27-50 after custom initialization of PETSc and message parsing interface

MPI (although a parallel process is not used), vectors, matrices and the numerical

domain are sized. Then, a sparse matrix assembly routine is called to create the

square coefficient matrix [A] which utilizes the efficient storage schemes

(compressed sparse raw format [13]) mentioned before and finally the vectors for

right-hand-side and eigenvalue calculations are created.

18

In Fig. A-2, lines 53-132 show the programming-wise very similar x and y

momentum conservation equations for steady Stokes flow, applied to the laminar

plane Pouiseuille flow. Firstly, the boundary nodes are handled in lines 53-67 by

applying the parabolic velocity profile BCs at the inlet and outlet nodes and the no-

slip BCs at the top and bottom nodes (including the corner nodes of numerical

domain). It is of critical importance that a great attention should be payed while

handling the indices of looping since the node numbering indices of numerical

domain and programming indices for matrix elements do not correspond to each

other. For instance, the top-left corner (0,0) index of [A] matrix in computer

programming sense is the same as the bottom-left corner of the numerical domain

where imin =1 and jmin = 1.

After boundary nodes, the inner nodes where the equations are discretized are

handled. Programming of Eqs. 3.5-3.6 for Laplace equation of viscous terms are

shown in lines 107-112 and Eqs. 3.7 and 3.9 for pressure gradient term of x-

momentum equation are shown in lines 114-132. Details of y-momentum equation

are omitted due to its similarity to the programming of x-momentum equation.

Biggest difference to note is the lack of a need for velocity profile BC in y-

momentum equation due to the all-zero BCs for v velocities.

In Fig. A-3, lines 134-189 show the programming of continuity equation with

finite volume method discretization. Although rest of the nodes are not shown in this

thesis for brevity, Eqs. 3.15-3.19 that describe the discretization for bottom-left node

are shown in lines 134-152. Since no pressure boundary conditions are used for the

problem, corresponding 0 elements of the RHS of continuity equation do not need to

be explicitly set to 0 in PETSc and thus deliberately omitted in programming.

In Fig. A-4, lines 191-207 show the custom calls for assembly of coefficient

matrix and right-hand-side vector after the elements corresponding to the discretized

equations are assigned in Figs. A-2 & A-3. Then, in lines 209-228, linear system

solver is chosen from KSP methods in PETSc. As an iterative solver for non-

symmetric linear systems and considering it can converge even in some singular

cases [15], GMRES method [14] is chosen for the solution. No preconditioning is

applied as it is usually preferred in the solution of linear systems for accelerating the

convergence rate [8,10].

19

Lines 230-247 show the solution is obtained to the Eq. 4.1 using [A] and RHS

vector. Terms related to iterative methods such as iteration number where upper limit

is set as 5000 and residual norm which is used for convergence criteria based on the

tolerance set above as 10E-10 are also calculated. Finally, eigenvalues are computed

for the [A] and it is confirmed that matrix in singular -as mentioned before- due to

the zero eigenvalues appearing in the solution. In lines 249-261, after the system is

solved, the solution vector “RH1” is cast into another vector for writing into files and

all the variables, vectors, matrices and objects used for the solution are destroyed.

Finally, the solution has to be converted into “.dat” format in a special way used by

Tecplot 360 post-processing software described in [16].

5.3. Post-processing of Solution

Figure 5.2: Solution for u component of velocity for a 11x11 grid.

20

Using Tecplot 360, solutions obtained for the non-dimensional flow variables

u, v and p (dropping the * notation) on a 11x11 and a finer 21x21 grid are shown

in Figs. 5.2 to 5.7. Although all plots are contour plots, u-velocity plots in Figs.

5.2 and 5.7 are combined with vector plots to show the parabolic u-velocity profile

better. One thing to consider is the corner nodes of the grid which shows abrupt

changes in pressure values since discretization do not involve those nodes, hence

their values are disconnected from the rest of pressure field due to lack of

communication. Table 5.1 shows the iteration and convergence related parameters

for the solution. As expected, a finer grid takes longer for the algorithm to converge

to a solution. Results shown in the plots are verified in the next chapter.

Table 5.1: Iteration parameters for solution on different grids.

Grid size Maximum set

tolerance

Maximum set

iteration

number

Residual norm

of solution

Iteration number

at convergence

11 x 11 1E-10 5000 9.4742E-011 493

21 x 21 1E-10 5000 1.15422E-011 1747

Figure 5.3: Solution for v component of velocity for a 11x11 grid.

21

Figure 5.4: Solution for pressure field on a 11x11 grid.

Figure 5.5: Solution for pressure field on a 21x21 grid.

22

Figure 5.6: Solution for u velocities on a 21x21 grid.

Figure 5.7: Solution for v velocities on a 21x21 grid.

23

6. VALIDATION OF RESULTS AND CONCLUSION

6.1. Closed-form solution of plane Poiseuille flow

The original set of Navier-Stokes equations are a coupled set of non-linear partial

differential equations and there exists only a few special cases where a closed-form

analytical expression for flow variables can be obtained [17]. This usually occurs due

to geometric properties of the flow domain which allow certain terms to become 0 in

the equations, therefore lend the system to a simpler treatment beyond numerical

discretization and linear algebra solution methods which the current thesis tries to

implement. The originally derived non-dimensional governing equations of steady

Stokes flow in 2D (Eqs. 2.8-2.9) are repeated below.

0 = − 𝜕𝑝∗

𝜕𝑥∗ + ( 𝜕2𝑢∗

𝜕𝑥∗2 + 𝜕2𝑢∗

𝜕𝑦∗2 ) (6.1)

0 = − 𝜕𝑝∗

𝜕𝑦∗ + ( 𝜕2𝑣∗

𝜕𝑥∗2 + 𝜕2𝑣∗

𝜕𝑦∗2 ) (6.2)

𝜕𝑢∗

𝜕𝑥∗ +𝜕𝑣∗

𝜕𝑦∗ = 0 (6.3)

Plane Poiseuille flow is also one of those special cases where due to the the fact that it

is a flow between “infinitely-long” parallel plates, the velocity component u and v do

not change in the x-direction (the flow direction). Therefore, for a plane Poiseuille

flow all the partial derivative terms of velocity components with respect to x become

zero. After cancelling terms and dropping the star notation, Eqs. 6.1-6.3 reduce to:

𝜕𝑝

𝜕𝑥= (

𝜕2𝑢

𝜕𝑦2 ) (6.4)

0 = − 𝜕𝑝

𝜕𝑦 (6.5)

0 = 𝜕𝑣

𝜕𝑦 (6.5)

24

Eq. 6.5 shows that in the y-direction, the pressure is constant which verifies

that the pressure field plots in Figs. 5.4 - 5.5. Eq. 6.4 shows that pressure gradient in x

direction is also a constant since both sides of the equation depends on different

variables. Also, Eq 6.6 shows that, in combination with the no-slip conditions, v=0

everywhere in domain and verifies our approximate results in v-plots.

If Eq 6.4 is integrated twice from both sides, result is given by Eq. 6.6.

𝑢 =1

2(

𝜕𝑝

𝜕𝑥 ) 𝑦2 + 𝐶1𝑦 + 𝐶2 (6.6)

Applying BCs as u(0) = 0 and u(H) = 0 based on physical domain given in Fig. 5.1 for

plane Poiseuille flow to find C1 and C2 , Eq. 6.6 after grouping terms becomes:

𝑢(𝑦) = − ( 𝜕𝑝

𝜕𝑥 )

[ 𝑦(𝐻−𝑦) ]

2 (6.7)

It can be seen that for y=0 and y=H for the bottom and top plates respectively, u

becomes zero as found in results of velocity plots. In addition, pressure gradient term

is denoted by a constant in Eq. 6.8 where pressure gradient is negative for a pressure

driven flow in the positive x-direction. Therefore, Eq. 6.7 becomes after setting H=1

and L=1 for the unit square domain:

∆𝑃

𝐿= −

𝜕𝑝

𝜕𝑥 (6.8)

𝑢(𝑦) =∆𝑃

2 𝑦(1 − 𝑦) (6.9)

Clearly, this is a parabolic velocity profile which we used as a basis for our BCs and

at the center of flow channel, Eq. 6.7 becomes:

𝑢(0.5) =∆𝑃

8 (6.10)

Therefore, the velocity profile plots are also verified by this result which satisfies the

pressure difference of ∆𝑃 = 2 and u(0.5) = 0.25 shown in the Tecplot figures before.

25

6.2.Conclusion

In this thesis, steady incompressible Navier-Stokes equations with the assumption

of Re<<1 (Stokes flow) were numerically evaluated for plane Poiseuille flow on a

uniform unit square grid with no-slip BCs at the walls and parabolic velocity profile

BCs at the inlet and exit. Discretization of momentum equations used finite difference

scheme, while continuity equation used finite volume approach. Results were obtained

with FORTRAN by using PETSc package and GMRES method, visualized with

Tecplot 360 and verified with a closed-form analytical expression.

26

REFERENCES

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[2] Hoffman, J. D. (2001). Numerical methods for engineers and scientists. New York: CRC

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[9] “Chapter 5 : Line and surface integrals“ (n.d.)

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[13] Balay S, Buschelman K, Eijkhout V, Gropp WD, Kaushik D, Knepley MG, McInnes LC,

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[15] P. N. Brown & H. F. Walker, GMRES on (nearly) singular systems, SIAM J. Matrix

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[16] Tecplot 360 User Manual: Release-1 (2013)

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[17] Anderson, J. D. (2017) Fundamentals of Aerodynamics (6th Ed.). New York, McGraw-

Hill.

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APPENDIX-A: FORTRAN code for numerical solution of plane Pouiseuille flow

Figure A-1: Declaration of variables and matrix/vector creation.

29

Figure A-2: Programming of x and y momentum equations.

30

Figure A-3: Programming of continuity equation.

31

Figure A-4: Solution step of algorithm.

32

Figure A-5: Printing the solution in Tecplot format.

33

CURRICULUM VITAE