A Study of Solving Navier-Stokes Equations with a Finite Volume Method Based on Polygonal Unstructured Grids and the Computation

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A STUDY OF SOLVING NAVIER-STOKES EQUATIONS WITH A FINITE VOLUME METHOD BASED ON POLYGONAL UNSTRUCTURED GRIDS AND THE COMPUTATIONAL ANALYSIS OF GROUND VEHICLE AERO- DYNAMICS

JIANNAN TAN, B.S.

A Dissertation

MECHNICAL ENGINEERING

Submitted to the Graduate Faculty of Texas Tech University in Partial Fulfillment of the Requirements for the Degree of

DOCTOR OF PHILOSOPHY

Approved

Siva Parameswaran, Ph.D. Chair of the Committee

Jingzhou Yang, Ph.D.

Zhaoming He, Ph.D.

Sukalyan Bhattacharya, Ph.D.

Xinzhong Chen, Ph.D.

Fred Hartmeister, Dean of the Graduate School

December, 2010

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ACKNOWLEDGMENTS Although the following dissertation is an individual work, I could never complete it without the help from a lot of people. Firstly I’d like to express my sincere gratitude to my advisor, Dr. Siva Parameswaran for his consistent guidance and encouragement in the past five years. His knowledge, expertise and enthusiasm on CFD have always been the driving force throughout my graduate research career at Texas Tech Univer- sity. I want to thank my committee professors, Dr. Jingzhou Yang, Dr. Zhaoming He, Dr. Sukalyan Bhattacharya and Dr. Xinzhong Chen for their kind support and help in the past few years.

My thanks also extend to my lab mates: Dongdae Lee, Zixi Chen, Tengxiao Liu, Di- vya R. and Suranga Dharmarathne. Working with them has always been a pleasure, which makes my research progressive and my life cheerful and colorful.

Finally, my very special thanks go to my family, especially my wife, Ou Zhang. This piece of work would never exist without her supervision and support on a daily basis.

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TABLE OF CONTENTS ACKNOWLEDGMENTS………………………………………………………………………... ii

ABSTRACT……………………………………………………………………………………….. v

LIST OF TABLES……………………………………………………………………………….. vi

LIST OF FIGURES……………………………………………………………………………... vii

1. INTRODUCTION ...... 1

1.1 The Development of Computational Fluid Mechanics ...... 1 1.2 Objectives ...... 2 1.3 Contents of Dissertation ...... 3

2. THE DISCRETIZATION OF N-S EQUATIONS ...... 4

2.1 Derivation and Description of the Navier-Stokes (N-S) Equations ...... 4 2.2 N-S Equations under Cartesian Coordinating System in A 2-D Case ...... 5 2.3 Finite Volume Method ...... 5 2.4 The Discretization of 2-D N-S Equations ...... 6 2.4.1 Gauss’s Theorem in a 2-D Polygonal Cell Case ...... 6 2.4.2 Discretization of 2-D N-S Equations by casting Gauss’s Theorem ...... 7 2.4.3 Important Variables of the Discretized N-S Equations ...... 8

3. GRID STRUCTURE AND GRID GENERATION ...... 9

3.1 Structured and Unstructured Grids/Mesh ...... 9 3.2 Triangular, Quadrilateral and Polygonal Grids/Mesh ...... 10 3.3 Cartesian Grids ...... 11 3.4 Grid Generation ...... 12 3.5 The Data Structure of the Mesh ...... 12 3.6 An Example: The Data Structure of a 2-D Mesh File from ANSYS® FLUENT® ...... 14 3.6.1 Point Information Block ...... 14 3.6.2 Face Information Block ...... 16 3.6.3 Cell Information Block ...... 18 3.6.4 User Defined Information Block ...... 18 3.6.5 The Internal Code and the Corresponding Boundary Condition ...... 19 3.6.6 The Demonstrated Data structured of the Mesh in Figure 3.5 ...... 19

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4. SOLVING THE DISCRETIZED N-S EQUATIONS ...... 21

4.1 The Goals of Solving the N-S Equations ...... 21 4.2 The Positions of the Flow Variables ...... 21 4.3 The Iterative Method ...... 22 4.4 Handling Diffusive Term, Convective Term and Pressure Term ...... 23 4.4.1 The Diffusive Term ...... 24 4.4.2 The Pressure Term and Least Square Method ...... 25 4.4.3 Convective Term and Momentum Interpolation for Face Velocity ...... 27 4.4.4 A New Momentum Interpolation Method ...... 27 4.5 Conclusion...... 39

5. CFD VALIDATION ...... 40

5.1 Internal Flow in a Straight Channel ...... 40 5.2 Internal Flow in a Z-Pipe ...... 42 5.3 Internal Flow In a Jet ...... 45 5.4 Laminar Flow Past a Square Cylinder ...... 49 5.5 Conclusion...... 54

6. EFFECTS OF CROSS WIND ON SPORT UTILITY VEHICLES (SUV): A

COMPUTATIONAL STUDY...... 56

6.1 INTRODUCTION ...... 56 6.2 Aerodynamic Forces and coefficients...... 57 6.2.1 Aerodynamic forces and coefficients ...... 57 6.2.2 Method of determining yaw angle and the resultant wind velocity ...... 58 6.3 CAD models and cfd simulation ...... 59 6.3.1 Vehicle CAD model ...... 59 6.3.2 The wind tunnel geometry and boundary configuration ...... 59 6.3.3 Blockage ratio ...... 60 6.3.4 Method of dealing with different angles of attack ...... 61 6.3.5 Mesh generation and simulation ...... 62 6.4 Results and discussion ...... 63 6.5 Concluding remarks ...... 65

REFERENCES ...... 68

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ABSTRACT Navier-Stokes (N-S) equations describe the motion of fluid flow in the nature and they are called the governing equations of fluid flows. Solving Navier-Stokes equations is of great interest to the scientists and researchers. Due to the high nonlinearity, achieving the analytical solutions for the N-S equations is extremely difficult, if not impossi- ble. Thus, people have to switch to numerical solutions with putting on certain restric- tions on the N-S equations. This leads to the development of Computational Fluid Dy- namics (CFD).

This dissertation contains two major sections. The first section is about theoretical study of CFD. We go through the whole process that a CFD analysis normally re- quires: generating mesh, setting boundary conditions and achieving numerical solutions of N-S equations, and post-processing to achieve flow field plots. An in-house 2- D CFD code based on unstructured polygonal mesh is presented, in which a new momentum interpolation method is developed and implemented to calculate the flow flux on the cell faces. The 2-D code is also validated by comparing the numerical results with widely-known analytical results, if available, or by benchmarking with the results produced by commercial CFD software packages. The second section of this dissertation is about one of the applications of CFD in modern auto industry – ground vehicle aerodynamics. The cross wind effect on a sport utility vehicle (SUV) is studied and analyzed using CFD methods and compared with available wind tunnel experimental results. The first section contributes to the philosophy of mechanical physics and the second section aims to fulfill the purpose of engineering.

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LIST OF TABLES Table 3.1 Table of Internal Code ...... 19

Table 4.1 Momentum equations coefficients of cell 2 ...... 30

Table 4.2 Momentum equations coefficients of cell 6 ...... 31

Table 4.3 Momentum equations coefficients of cell 8 ...... 31

Table 4.4 Momentum equations coefficients of cell 4 ...... 32

Table 4.5 Momentum equations coefficients of cell 5 ...... 32

Table 4.6 Velocity and Pressure of Cell 5 after Each Iteration Step ...... 38

Table 6.1 Velocity inlet under different angles of attack ...... 62

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LIST OF FIGURES 2.1 A typical polygonal cell ...... 6 3.1 Structured and unstructured grids ...... 10 3.2 Quadrilateral, triangular and hexagonal grids ...... 11 3.3 Cartesian Grids...... 11 3.4 Grid point, face and volume ...... 13 3.5 A meshed rectangular box ...... 14 3.6 The point information block ...... 15 3.7 The face information block...... 17 3.8 Interpretation of face data structure ...... 18 3.9 The cell information block ...... 18 3.10 User defined information block ...... 18 3.11 Mesh in Figure 3.5 demonstrated with point, face and cell IDs ...... 20 4.1 Staggered grids...... 21 4.2 Center-stored velocity and pressure ...... 22 4.3 Definition sketch ...... 24 4.4 Least square method ...... 26 4.5 Initial velocity field and pressure distribution ...... 28 4.6 Local face ID arrangements ...... 29 4.7 Face velocity distribution of cell 5 after momentum interpolation...... 35 4.8 Pressure v.s. iteration step (Left) and mass imbalance v.s. iteration step ...... 38 5.1 Mesh of straight channel ...... 40 5.2 Flow velocity plot of my 2D code ...... 41 5.3 Flow velocity plot of FLUENT ...... 41 5.4 Mesh of Z-pipe ...... 42 5.5 Flow velocity plot of my 2D code ...... 43 5.6 Flow velocity plot of FLUENT ...... 43 5.7 Flow streamline plot of FLUENT ...... 44 5.8 Reattachment length comparison ...... 45

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5.9 Mesh of a jet ...... 46 5.10 Flow velocity plot of my 2D code ...... 47 5.11 Flow velocity plot of FLUENT...... 47 5.12 Flow streamline plot of FLUENT ...... 48 5.13 Reattachment length comparison ...... 49 5.14 Mesh of fluid domain ...... 50 5.15 Flow velocity plot of my 2D code ...... 52 5.16 Flow velocity plot of FLUENT...... 52 5.17 Flow streamline plot of FLUENT ...... 53 5.18 Symmetric vortices length comparison ...... 54 6.1 Three aerodynamic forces and coefficients ...... 57 6.2 Method of determining the angle of attack ...... 58 6.3 Method of determining the resultant wind velocity ...... 59 6.4 SUV CAD model ...... 59 6.5 Computation domain configurations ...... 60 6.6(a) Wind tunnel dimensions ...... 61 6.6(b) List of blockage ratio ...... 61 6.7 Mesh around the vehicle ...... 62 6.8 Drag coefficients v.s. angles of attack...... 64 6.9 Lift coefficients v.s. angles of attack ...... 64 6.10 Side force coefficients v.s. angles of attack ...... 65 6.11 Cd error v.s. blockage ratio...... 65

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CHAPTER 1

INTRODUCTION

1.1 The Development of Computational Fluid Mechanics The fundamental basis of computational fluid dynamics is Navier-Stokes equations which were set up in 19th century. However, due to the high non-linearity and strongly coupled pressure and velocity terms in the equations, it was extremely difficult to achieve an analytical solution in mathematics at that time. In 1960s, along with the development of modern computer technology, people started to realize that solving the

Navier-Stokes equations in a numerical way has become feasible with the help of de- cent digital computers. And so comes the new era of computational fluid dynamics

(CFD) technology.

Although it's difficult to tell who is exactly the one performing the earliest CFD calculation, a few shining attempts were noticed and recorded. Lewis Fry Richardson's weather forecasting is one of them. He proposed a method for weather forecasting by solving a set of differential equations by dividing the physical domain into many small cells which is very similar as today's finite difference method. Although his calculation results turned out to be inaccurate, his contributions were worthy compliments.

In 1933, A. Thom presented a paper "The Flow Past Circular Cylinders at Low

Speeds" [1], which is considered the earliest academic paper of computational fluids.

From 1958 to 1968, the fluid dynamics group from Los Alamos group in United States devoted lots of manpower and computer resources to develop advanced CFD technologies and they achieved many significant achievements which include: Particle-In-

Cell (PIC) method, Fluid-In-Cell (FLIC) method, vorticity and stream function method, Mark-And-Cell (MAC) method, Implicit Continuous Fluid Eulerian (ICE) me-

Texas Tech University, Jiannan Tan, December 2010 thod, SOLA and reactive flow CFD codes, Lagrangian-Incompressible (LINC) and

Arbitrary-Lagrangian-Eulerian (ALE) method, Particle-And-Force (PAF) and Free-

Lagrangian methods and the famous k-epsilon turbulence model. Many of these methods are still being widely used today.

In 1970s, Suhas V. Patankar and Dudley Brian Spalding proposed the Semi-Implicit

Method of Pressure-Linked Equations, normally referred as SIMPLE algorithm, which decouples the pressure and velocity in Navier-Stokes equations and solve the N-S equations numerically in an iterative manner. SIMPLE algorithm is truly a milestone in the CFD history. It shows the possibility to solve complex N-S equations using power- ful digital computers.

Nowadays, CFD has been recognized as a standard process as part of Computer-Aided

Engineering (CAE) spectrum in many industries and used extensively in product design and product development.

1.2 Objectives There are numerous commercial CFD codes available on the market. They are user- friendly, robust and convenient. It’s easy to perform a CFD simulation by using a commercial CFD code. However, the encapsulation of the commercial codes makes them blind to the users. User has no access to the detailed codes and cannot debug the program, which limits its function in the teaching and research activities because researchers always need full control of the codes and want to keep track of the change of every variable.

The major goals of developing this open-source CFD code based on a popular scientif- ic programming language, MATLAB, are: 1) test the new momentum interpolation method proposed in Chapter 4 and 2) provide an open-source CFD code to the De-

Texas Tech University, Jiannan Tan, December 2010 partment of Mechanical Engineering at Texas Tech University to fill in a blank for the

CFD teaching purpose.

The major goals of studying the cross wind effect on a SUV by using a commercial

CFD code are: 1) showing an example of how CFD is utilized in automobile industry and 2) giving some ideas about the accuracy of the CFD simulation compared with experimental data.

1.3 Contents of Dissertation The following chapter contains a review of N-S equations and Finite Volume Method and explains how to discretize the N-S equations. Chapter 3 explains the mesh structure in details. Chapter 4 shows the way to solve the discretized N-S equations using

SIMPLE method. Chapter 5 demonstrates the validation of the code in several cases.

Chapter 6 contains a CFD study of ground vehicle dynamics using a commercial code.

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CHAPTER 2

THE DISCRETIZATION OF N-S EQUATIONS

2.1 Derivation and Description of the Navier-Stokes (N-S) Equations As well known to the public, the general form of the equations which describe the fluid motion can be written as [2] [3]:

U  () U  U   p    T  f , (2.1) t In the above equation, U is the fluid velocity vector, p is the flow pressure, T is the stress tensor and f is considered body forces caused by the outside environment. Body forces can be various and a common example of the body forces is the gravity of the flow. In some certain circumstances the body forces can be too large to be ignored, for example, electrified flow in an electric field. However, in most cases, gravity is the only body force and its influence to the fluid motion is so small that normally it will be taken out of consideration. Under the assumptions of Newtonian, incompressible flow and constant viscosity, those cases like sound wave and shock wave are kept out of consideration, which makes the situation more reasonablely suitable to the discussion in this paper. Furthermore, the transient term is taken out of the equation. Thus, a simplified steady-state momentum equation for incompressible Newtonian flow is derived as:

()UU   P   2 U , (2.2) where μ is the constant viscosity of the fluid.

With the same assumptions, the continuity equation for steady-state, incompressible

Newtonian flow can be written as:

 (U )  0 (2.3)

Texas Tech University, Jiannan Tan, December 2010 where ρ is the constant density of the fluid.

Equation 2.2 and 2.3 are named Navier-Stokes equations in memory of Claude-Louis

Navier and George Gabriel Stokes.

2.2 N-S Equations under Cartesian Coordinating System in A 2-D Case The u-momentum equation:

P  ()Uu     2 u (2.4) x The v-momentum equation:

P ()Uv     2 v (2.5) y

The continuity equation:

()()u   v   0 (2.6) x  y

2.3 Finite Volume Method The finite volume method (FVM) is a method of discretizing partial differential equations to algebraic equations [4][5]. For the finite volume method, the usual approach is to divide the physical space into many small sub-domains which are called control volumes or “cells”. The shape of cells can be arbitrary while triangular cell and rectangular cell are the most popular two types. The partial differential equations are recast on these cells and approximated by the nodal values or central values of the control volumes. The advantage of performing finite volume method is straight as is in FVM’s definition. People can solve the approximated algebraic equations instead of the origi- nal partial differential equations, which makes life much easier and still keeps considerable levels of control of the accuracy of the solution.

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2.4 The Discretization of 2-D N-S Equations

2.4.1 Gauss’s Theorem in a 2-D Polygonal Cell Case Gauss’s Theorem, or in other words Divergence Theorem, is the basic method utilized to discretize the N-S equations. It can be mathematically stated as:

()() F  dV    F  n  dS (2.7) VS From equation 2.7, we have a very interesting observation that Gauss’s Theorem turns a volume integral of the divergence of a vector F to a surface integral of the same vector. Theoretically, the volume can be in arbitrary shape so definitely the volume can be polygonal. In this case, equation 2.7 can be recast as:

()()() F  dV  F  n  dS  F  n  S (2.8)     i i VS i where ni represents the outward normal vector of the face i of the volume and ΔSi stands for the surface area of the face i. Figure 2.1 below demonstrates a typical polygonal cell which is commonly seen.

Figure 2.1 A typical polygonal cell

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2.4.2 Discretization of 2-D N-S Equations by casting Gauss’s Theorem By doing volume integration to equation 2.4, 2.5 and 2.6 and casting Gauss’s Theorem we obtain: p ()()U  u dV     dV    2 u dV VVVx

 p  ()()()U  ni u S i face   V cell    u n i  S i face (2.9) ix i

p ()()U  v dV     dV     2 v dV VVVy

 p  ()()()U  ni v S i face   V cell    v n i  S i face (2.10) iy i  (U )  dV  0 V  (U  ni  S i ) face  0 (2.11) i

Upon observation, we find out that the term ρU niΔSi equals the mass flux of the fluid flow through the face i. Let’s use Fi to represent the mass flux. Equation 2.9, 2.10 and 2.11 can be given as:

p  ()()()Fi u face    V cell     u  n i   S i face (2.12) ix i

p  ()()()Fi v face    V cell     v  n i   S i face (2.13) iy i

(Fi ) face  0 (2.14) i

Where Fi =ρU niΔSi.is the outward mass flux going through the face. If the flow goes from the inside to the outside, the value of Fi is positive. Otherwise, Fi is negative. The terms on the left-hand side of equation 2.12 and 2.13, Fiu and Fiv together are called

Texas Tech University, Jiannan Tan, December 2010 convective term because they represent the fluid convection. The two terms on the right-hand side are pressure term and diffusive terms, separately. The ways to calculate convective term, diffusive term and pressure term will be explained in detail in next chapter, after the grid structure is well established.

2.4.3 Important Variables of the Discretized N-S Equations From equation 2.12, 2.13 and 2.14, we can find the important variables which need to be calculated unavoidably. They are:

1) Mass flux of the flow through the face, Fi =ρU niΔSi;

2) Flow velocity U on the face, or its components in x and y directions, u and v;

3) The area of the face, ΔSi.;

4) The normal vector of the face, ni;

5) The spatial gradients of the velocity components on the face;

6) The volume of the cell, ΔVi;

7) The spatial pressure gradients of the cells in x and y directions.

These variables are so important that they are not only essential to the discretized N-S equations, but also critical to building the grid structure and deciding where to put the physical parameters such as velocity and pressure components that will be calculated and/or interpolated on the finite volume grids.

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CHAPTER 3

GRID STRUCTURE AND GRID GENERATION

This chapter will have a brief review of modern popular builds of modern grids, or called mesh, which have been widely accepted and successfully applied with examples of several business CFD software packages. A few grid generation principles will be introduced, but expanded research and discussion are excluded in this paper. In the end of this chapter, one grids structure is selected and the corresponding data structure is developed and introduced, as a necessary preparation work for developing an N-S equations solver.

3.1 Structured and Unstructured Grids/Mesh Based on the connectivity, all grids/mesh can be divided into two different groups, structured girds and unstructured grids. The biggest difference of them is that every single cell of a structured grids system can be uniquely identified within an I-J-K system (or I-J system for 2-D grids) while unstructured grids cannot do so or do not do so.

The connectivity rule restricts 2-D structured grids to be quadrilateral and 3-D structured grids to be hexahedral. As an example, Figure 3.1 gives a comparison of structured grids and unstructured grids:

Figure 3.1 Structured and unstructured grids

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For structured girds, every cell has the same number of neighbor cells; every face is shared by the same number of cells; every point is shared by the same number of lines and cells.

For Unstructured grids, there are many different kinds. Theoretically, unstructured grid can be arbitrarily polygonal, so it doesn’t necessarily need to satisfy the rules for structured grids.

In some rare cases, structured grids and unstructured grids are mixed together. This kind of mesh is called hybrid grids or mixed grids.

3.2 Triangular, Quadrilateral and Polygonal Grids/Mesh According to the shape of the grids, there are three majors groups of grids: triangular girds, quadrilateral grids and polygonal grids. The most popular two types of grids are triangular grids (or tetrahedral grids in 3-D) and quadrilateral grids (or hexahedral grids in 3-D). Sometimes these two types of grids are mixed together in order to better capture the physical boundary of the geometry. In recent years, along with the development of the meshing technology, polygonal grids in 2-D and polyhedral grids in 3-D are becoming popular. A good example is the hexagonal grids used in STAR-CD, which is a commercial CFD software package from its vendor, CD- Adapco. Figure

3.2 gives a demonstration of these 3 types of grids:

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Figure 3.2 Quadrilateral, triangular and hexagonal grids

3.3 Cartesian Grids There is a special type of grids which is named Cartesian Grids. It can be classified as a unique type of unstructured quadrilateral grids with a special connectivity, as shown in Figure 3.3:

Figure 3.3 Cartesian Grids

Cartesian girds use square elements consistently and allow a cell face to be shared by

3 or more cells at the same time. The mesh generation process is easy, fast and robust, although the total number of cells is significantly increased, especially in those areas

Texas Tech University, Jiannan Tan, December 2010 which lie close to the surface of the solid geometry because a large number of cells are needed to capture the solid geometry accurately in order to reduce the model facetiza- tion.

3.4 Grid Generation Grid generation can be defined as a process of dividing a physical domain into many small sub-domains which can be called ‘cells’ or ‘elements’ [6]. Nowadays, mesh generation has become a very interesting research topic due to its wide application and large influence on those technologies which involve numerical methods and computational solutions, such as CFD analysis, FEA analysis, CAD design, etc. [7].

For structured grids, algebraic method and elliptic method are the two most popular methods for grid generation. For unstructured grids, Delaunay method and Octree method are used most [8].

3.5 The Data Structure of the Mesh Data structure refers how the mesh information is stored.

A grid has 3 basic geometric composites: points, faces and volumes, as shown in Fig- ure 3.4:

Figure 3.4 Grid point, face and volume

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Points describe the corners of the cell, and faces describe the connectivity of the points.

When points and faces are fixed, volume is already fixed. In other words, the information of points and faces can be independent, and the information of volume depends on points and faces. Hence, the information of both points and faces must be stored in the data structure.

For the information of point, every point in the whole mesh needs a unique ID number.

So does the face and cell. At the end of Chapter 2, we mentioned that the normal vector of cell face needs to be identified, which also makes the cell face directional. Be- sides, the cell face needs to be issued a boundary property, such as solid wall, interior interface inside flow, porous media, etc.. So far we conclude that a mesh data should include at least the following 3 basic information:

1) Point information. Accurate description of point position and unique ID number for every point in the mesh.

2) Face information. Description of the points which compose the face and the direction of the face.

3) Cell information. Unique ID number for every cell. Description of the faces which compose the cell.

Besides, the data structure of grids may include extra information which can be further utilized in the flow solver, just for convenience. For example, the property of the faces, such as wall, interior, mass flow inlet or outlet, pressure outlet, porous media, etc..

These kinds of information are totally optional but not necessary. They can be in- cluded in the girds information, or can be specified by users in the flow solver before the simulation is started.

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3.6 An Example: The Data Structure of a 2-D Mesh File from ANSYS® FLUENT® ANSYS® FLUENT® (Referred as FLUENT for convenience) is a popular commercial

CFD software package widely accepted by both academic and industrial fields. Al- though some general concepts about mesh structure is introduced in the user’s manual of the software, the data structure of the mesh file is never revealed officially. Howev- er, the mesh file (*.msh) is stored in such a format that it can be opened in text format and it is very readable. In the following section, the data structure of FLUENT mesh file is taken as an example and explained in details. Figure 3.5 gives a rectangular box meshed with triangular grids:

Figure 3.5 A meshed rectangular box

And boundary conditions are clearly marked in Figure 3.5. We can see that the mesh file has 4 major data blocks: point information block, face information block, cell information block and user defined information block.

3.6.1 Point Information Block In this block, the total number of points is recorded. Every point is given a unique ID number, and its Cartesian coordinates are stored as the description of the point’s spatial position, as shown in Figure 3.6:

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Figure 3.6 The point information block

FLUENT uses hexadecimal system to store the IDs of the points. Let’s take a look at the first line. It is a summary of all the points in the mesh. ‘1’ means the point ID starts from 1. Due to the hexadecimal system, ‘11’ means the last point’s ID is 17, which tells the fact that there are 17 points in total in the whole mesh. The second line is a summary of a points group. ‘1’ means this group starts with point ID 1, and ‘11’ means the last point’s ID in this group is 17. Since the number of points in this group is equal to the number of total points in the whole mesh as stated in the first line, it’s very clear that in this mesh, there are only one point group. In some other cases, points in the mesh are divided into several different groups, and the number of points in each group is not necessarily equal to each other. However, you can expect that when the numbers of points of all groups are summed together, the total number has to be equal to the points number stated in the first line which summarizes the point information of the whole mesh.

The coordinates of each point are listed as an N by 2 matrix in a format of (x, y). The point ID is not clearly listed, but it is equal to the line number. For example, the first

Texas Tech University, Jiannan Tan, December 2010 point with an ID of 1 has the coordinates of (0, 0); the second point with an ID of 2 has the coordinates of (4, 0).

3.6.2 Face Information Block In this block, the total number of faces is recorded. Every face is assigned a unique face ID. The IDs of those points which compose the face are stored. The IDs of the cells which share the face are stored, too, as shown in Figure 3.7.

Line 1 is a notation which claims the beginning of face information block. Line 2 summarizes the number of faces in the whole mesh. As we can see in Figure 3.7, the face ID starts from ‘1’ and ends with ‘24’, which means there are 2x16+4=36 faces in total in the mesh. Line 3 summarizes the information of face group 1. ‘3’ is an internal code referring the boundary condition of this face group. ‘1’ means the face ID in this group starts from 1, ‘2’ means the face ID in this group ends with ‘2’.

So far we can conclude that there are only 2 faces in this face group. And then the detailed information of the faces is listed in line 4 and line 5. Let’s take line 4 for example, it says: 2 2 7 6 0. The first ‘2’ means this face is composed of 2 points. The next

‘2’ and ‘7’ stand for the IDs of the starting point and the ending point, separately, which also defines the direction of this face. The following ‘7’ and ‘0’ are the cell IDs on the left-hand side and right-hand side of this face, as shown in Figure 3.8. A cell with a cell ID of ‘0’ is a void cell which is considered out of fluid flow.

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Figure 3.7 The face information block

Figure 3.8 Interpretation of face data structure

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3.6.3 Cell Information Block Since face information block has fixed the IDs for cells already, no more detailed cell information is needed. FLUENT just keeps a few lines to state the total number of cells in the mesh as a summary, as shown in Figure 3.9.

Figure 3.9 The cell information block

The cell ID starts from ‘1’ and ends with ‘14’, which means there are 20 cells in total.

And there is only one cell group, which implies all cells are in the same property.

3.6.4 User Defined Information Block User defined information block stores the boundary names input by the user, as shown in Figure 3.10:

Figure 3.10 User defined information block

The first line is just a notation. Let’s take a look at the second line. ‘2’ is the internal code referring the boundary condition of the face group. ‘fluid’ is the internal name of this boundary condition, and the next ‘fluid’ is the user defined or system-default name to call this face group.

This block is not essential to the flow solver and can be ignored.

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3.6.5 The Internal Code and the Corresponding Boundary Condition As we discussed in 3.6.2, all the faces in the same face group have the same boundary condition and it’s referred by an internal code in the summary of the face group information line. These referred codes can be found in User Defined Information Block.

For convenience, a table is provided here:

Table 3.1 Table of Internal Code

Boundary Code Boundary Code Wall 3 Porous-jump E Axis 25 Pressure-far-field 9 Exhaust-Fan 5 Pressure-inlet 4 Fan E Pressure-outlet 5 Inlet-Vent 4 Radiator E Intake-fan 4 Recirculation-inlet 4 Interface 18 Recirculation- 5 outlet Interior 2 Symmetry 7 Internal F Velocity-inlet A Mass-flow-inlet 14 Outlet-vent 5 Outflow 24 Periodic

From this table we find something really interesting that some boundary conditions with different names actually have the same internal boundary condition code. For example, ‘Inlet-Vent’, ‘Intake-fan’, ‘Pressure-inlet’ and ‘Recirculation-inlet’ are sharing the same code ‘4’. This probably implies the flow solver FLUENT would treat these four boundary conditions in the same way.

3.6.6 The Demonstrated Data structured of the Mesh in Figure 3.5 After analyzing the data structure of the mesh file which generates the grids in Figure

3.5, we can put on all the point IDs, face IDs and cell IDs on the mesh to achieve a clearer overview of the whole mesh data structure. It is demonstrated in Figure 3.11.

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Face IDs and cell IDs are highlighted with green and red, separately while point IDs are highlighted with white.

Figure 3.11 Mesh in Figure 3.5 demonstrated with point, face and cell IDs

In Section 2.3.3 of Chapter 2, we proposed 7 important variables which are necessary to a flow solver of N-S equations. They are: 1) mass flux of the flow through faces, 2) flow velocity vectors on faces, 3) the area of faces, 4) the normal vectors of faces, 5) the spatial gradients of the velocity components on faces, 6) the volumes of cells and 7) the spatial pressure gradients of cells in both x and y directions. From Figure 3.11, it’s clear that under the proposed mesh data structure, we have achieved 3), 4), 6) out of the 7 necessary variables. These 3 variables are determined by the geometry of the mesh and will never be changed by other physical variables of the flow field, such as flow velocity, pressure, temperature, etc..

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CHAPTER 4

SOLVING THE DISCRETIZED N-S EQUATIONS

After setting up the grids, the next step is to solve the N-S equations. Before going in- to any details, we need to do some preparations.

4.1 The Goals of Solving the N-S Equations Why are we solving the N-S equations? What are our goals? These questions need to be answered clearly. The flow velocity field and the pressure distribution are what we are interested in and they are our goals. The N-S equations govern the physics of the fluid flow, that’s the reason we are solving these equations.

4.2 The Positions of the Flow Variables Our goals are flow velocity and pressure. And here directly comes the next question:

Where to put the flow velocity and pressure on the grids?

In structured grids, the “staggered” mesh is used, and pressure is stored in the center of the cell while velocity components are stored in “staggered” faces as shown in Figure

4.1.

Figure 4.1 Staggered grids

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In unstructured grids, the velocity and pressure are both stored in the center of each cell, as shown in Figure 4.2. And the velocity and pressure on the cell face can be interpolated by the cell central values of the two neighbor cells.

Figure 4.2 Center-stored velocity and pressure

4.3 The Iterative Method In former discussion, we proposed 7 necessary variables for solving the N-S equations.

In Sections 3.6.6, we already achieve 3 out of the 7 and 4 variables are left behind.

They are: 1) mass flux of the flow through faces, 2) flow velocity vectors on faces, 5) the spatial gradients of the velocity components on faces, 7) the spatial pressure gradients of cells in both x and y directions.

We find that these 4 variables need to be derived from the cell-center based velocity and pressure. This is a huge dilemma: the cell-center based flow velocity and pressure are our goals of solving the N-S equations meanwhile we need them to derive the 4 variables to start solving the N-S equations. The only way out is the iterative method.

We first assign an arbitrary velocity and pressure filed to the cell centers, calculate the

4 variables and solve the N-S equations to get a new velocity and pressure field. Then

Texas Tech University, Jiannan Tan, December 2010 we correct the velocity and pressure somehow to make them closer to the accurate values at the end of the iteration and use the corrected values to start a new iteration level, so on and so forth until a satisfactorily accurate velocity and pressure filed is achieved. Here we will perform the famous SIMPLE method [9].

4.4 Handling Diffusive Term, Convective Term and Pressure Term As proposed in Section 2.3.2, the N-S equations are listed as:

p  ()()()Fi u face    V cell     u  n i   S i face (2.12) ix i

p  ()()()Fi v face    V cell     v  n i   S i face (2.13) iy i

(Fi ) face  0 (2.14) i Equation 2.12 and 2.13 are also called momentum equations are equation 2.14 is called continuity equation. The left-hand side of momentum equations is convective term. The right-hand side is pressure term and diffusive term, separately.

Figure 4.3 is a sketch of two neighbor cells that we will cast our equations on. We will take it as an example to demonstrate how to calculate convective, diffusive and pressure terms.

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eη P n

y eξ A

x a

Figure 4.3 Definition sketch

Cell P and Cell A shares a face. The direction of the face is from point a to point b, and its normal vector is n. ξ-η system is a curvilinear coordinate system and eξ, eη are the unit vectors along ξ and η direction. They can be expressed as follows:

y x y y x  x nˆ  iˆ   ˆ j b a iˆ  b a ˆ j (4.1)    |ab | | ab |

x x y  y x  x y  y APAPAPAPˆ ˆ ˆ ˆ eˆ  i  j  i  j (4.2)    |PA | | PA |

x y x x y  y ˆ  ˆb a ˆ b a ˆ eˆ  i  j  i  j (4.3)    |ab | | ab |

4.4.1 The Diffusive Term Diffusive term is expressed as follows:

  ( u  ni  S i ) face =  [ D ( u E  u P )+ S D  cross, u ] face (4.4) i i

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  ( v  ni  S i ) face =  [ D ( v E  v P )+ S D  cross, v ] face (4.5) i i Where:

 nˆ  nˆ D  | ab | (4.6)  nˆ  eˆ

ˆ ˆ e e  ub u a SD cross, u    | ab |  (4.7) nˆ e ˆ 

ˆ ˆ e e  vb v a SD cross, v    | ab |  (4.8) nˆ e ˆ  So far, the momentum equations 2.12 and 2.13 can be recast as:

p aPPEEP u ()() a  u    V (4.9) nb x

p aPPEEP v ()() a  v    V (4.10) nb y

Where:

aE D e  MIN(0, F e ) (4.11) aP() a E  F e (4.12) nb

4.4.2 The Pressure Term and Least Square Method The traditional way of calculate pressure term is to perform Gauss Theorem [10] on it and transform it into a discrete form so that the momentum equation 4.9 and 4.10 can be written as: aP u P ()() a E  u E   P   y face (4.13) nb nb aP v P ()() a E  v E   P   x face (4.14) nb nb

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Please note that P in equation 4.13 and 4.14 are the pressure on cell face, which need to be interpolated by the cell-center pressure PA and PP.

Here we try to avoid the pressure interpolation and introduce another method to calculate the pressure term: Least Square Method [11]. With this method, we can directly achieve the spatial pressure gradients in both x and y directions.

Figure 4.4 Least square method

nd Figure 4.4 show a cell P and its neighbor cells, A1, A2, A3, …, Ai. According to the 2 order Taylor expansion, we have:

p  p P P ()()()()  x  x   y  y (4.15) APPAPPAPx  y where (xA, yA),( xP, yP) are the coordinates of the centers of cell A and cell P.

Repeat equation 4.15 for all the neighbor cells of cell P, we have:

xAPAPA1 x y 1  y   p 1   p    xAPAPA2 x y 2  y p 2  x    xAPAPAP3 x y 3  y     p 2   p  X  p   p (4.16)  p        y     cell   xAi x P y Ai  y P   p Ai 

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X is an over-determined i by 2 matrix. The spatial pressure gradient in the center of cell P can be determined by:

TT1 p ()() X X  X  p (4.17)

Least Square Method has three distinct advantages:

1) Feasible for the grids with arbitrary shapes;

2) Matrix X is determined by the mesh geometry. It only needs to be calculated once.

3) Avoiding to interpolate face pressure.

4.4.3 Convective Term and Momentum Interpolation for Face Velocity Convective term is a little more complex than the diffusive term and pressure term.

From equation 2.12 and 2.13 we find that the convective term on a cell face is the dot product of the mass flow flux and the flow velocity component on this cell face. Al- though it seems that the mass flow flux can be directly calculated with the velocity components, that is not the case here. In order to suppress the potential ‘checker board’ oscillatory pressure distribution which cannot be detected due to the finite difference method, momentum interpolations must be taken to calculate the mass flow flux. Meanwhile, the velocity component can be treated as a normal scalar like temperature and be approximated by upwind scheme or similar schemes.

4.4.4 A New Momentum Interpolation Method Here we use a simple example to demonstrate the new momentum method, which is different from the traditional Rhie-Chow method [12]. In Figure 4.5, we have 9 cells.

Cell 5 residents in the center of the computational domain, and other 8 cells surround

Texas Tech University, Jiannan Tan, December 2010 cell 5. Only cell 5 has 0 initial velocity and 0 initial pressure. Other cells have constant velocity components u=1, v=1 and constant pressure P=1 in the center of each cell.

Figure 4.5 Initial velocity field and pressure distribution

We are going to apply iterative method and the new momentum interpolation method.

You will see how the velocity and pressure of cell 5 approach to the value of 1 while iterations advance.

Assume every cell numbers its faces locally in a clockwise manner, as shown in Fig- ure 4.6. Under this system, the 2nd face of cell P and the 4th face of cell E are actually the same face shared by cell P and cell E.

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Figure 4.6 Local face ID arrangements

Let’s go back to Figure 4.5 and think about what initial values we can calculate for equation 4.9 – 4.12 to get the iteration started.

Firstly, we can calculate the pressure term using the least square method. According to equation 4.16, we can calculate the spatial pressure gradient for cell 5: cell2 0 1 p   1    p         cell6  1 0 x   1    x   0    0     (4.18) cell8 0  1 p   1    p   0         4 1 0 y   1    ycell  cell5 cell 5

And by following the same method, we can also calculate the spatial pressure gradient of cell 2, cell 6, cell 8 and cell 4, which are the neighbor cells of cell 5: cell2 N   0 1 p  0    p         0  cell2 E  1 0x 0  x       0     1  (4.19) cell5 0  1 p    1    p           2  2 W 1 0y cell2  0    ycell  cell 2

Texas Tech University, Jiannan Tan, December 2010 cell6 N   0 1 p  0    p         1  cell6 E  1 0x 0  x       0     2  (4.20) cell6 S  0  1 p   0    p           0  5 1 0y cell5   1    ycell  cell 6

cell5 0 1 p   1    p         0  cell8 E  1 0x 0  x       0     1  (4.21) cell8 S  0  1 p   0    p           2  8 1 0y cell5  0    ycellW  cell 8 cell4 N   0 1 p  0    p         1  cell5 1 0x  1  x        0     2  (4.22) cell4 S  0  1 p   0    p           0  4 1 0y cell5  0    ycellW  cell 5

Secondly, we need to assemble the momentum equations for cell 2, cell 6, cell 8, cell 4 and cell 5.

For cell 2, we obtain Table 4.1:

Table 4.1 Momentum equations coefficients of cell 2

Local Face ID Flux through face aE Contribution to ap 1 1 1 +2 2 1 1 +2 3 -1/2 3/2 +1 4 -1 2 +1

Therefore, the discretised momentum equations for cell 2 are:

3 p 6u  u  u   u  2  u  (  V ) (4.23) 2 2N 32 5 1x  cell 2

3 p 6v  v  v   v  2  v  (  V ) (4.24) 2 2N 32 5 1y  cell 2

For cell 6, we obtain Table 4.2:

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Table 4.2 Momentum equations coefficients of cell 6

Local Face ID Flux through face aE Contribution to ap 1 1 1 +2 2 1 1 +2 3 -1 2 +1 4 -1/2 3/2 +1

Therefore, the discretised momentum equations for cell 6 are:

3 p 6u  u  u  2  u   u  (  V ) (4.25) 6 3 6E 92 5x  cell 6

3 p 6u  v  v  2  v   v  (  V ) (4.26) 6 3 6E 92 5y  cell 6

Here u6E stands for the east face u-velocity of cell 3.

For cell 8, we obtain Table 4.3.

Table 4.3 Momentum equations coefficients of cell 8

Local Face ID Flux through face aE Contribution to ap 1 1/2 1 +3/2 2 1 1 +2 3 -1 2 +1 4 -1 2 +1 Therefore, the discretised momentum equations for cell 8 are:

11 p u  u  u 2  u  2  u  (  V ) (4.27) 2 8 5 9 8S 7x  cell 8

11 p v  v  v 2  v  2  v  (  V ) (4.28) 2 8 5 9 8S 7y  cell 8

Here u8S stands for the south face u-velocity of cell 8.

For cell 4, we obtain Table 4.4:

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Table 4.4 Momentum equations coefficients of cell 4

Local Face ID Flux through face aE Contribution to ap 1 1 1 +2 2 1/2 1 +3/2 3 -1 2 +1 4 -1 2 +1

Therefore, the discretised momentum equations for cell 4 are:

11 p u  u  u 2  u  2  u  (  V ) (4.29) 2 4 1 5 7 4Wx  cell 4

11 p v  v  v 2  v  2  v  (  V ) (4.30) 2 4 1 5 7 4Wy  cell 4

Here u4W stands for the south face u-velocity of cell 4.

So far we already have the velocity and pressure distribution for all cells. However, these values are just ‘guessed values’ which are used to get the iteration started. Ac- cording to the SIMPLE method, we need to solve the momentum equations for cell 5 and get the corrected velocity of cell 5 which satisfies the momentum equation of cell

5. For cell 5, we obtain Table 4.5:

Table 4.5 Momentum equations coefficients of cell 5

Local Face ID Flux through face aE Contribution to ap 1 1/2 1 +3/2 2 1/2 1 +3/2 3 -1/2 3/2 +1 4 -1/2 3/2 +1

Therefore, the discretised momentum equations for cell 5 are:

3 3 p 5u  u  u   u   u  (  V ) (4.31) 5 2 62 8 2 4x  cell 5

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3 3 p 5v  v  v   v   v  (  V ) (4.32) 5 2 62 8 2 4y  cell 5

Let’s expand the pressure terms in the above 2 equations and we obtain:

3 3 5u  u  u   u   u  (0  1)  5 (4.33) 5 2 62 8 2 4 3 3 5v  v  v   v   v  (0  1)  5 (4.34) 5 2 62 8 2 4

And we get: u5=1, v5=1.

This is the velocity that satisfies the discretized momentum equation.

Thirdly, we will introduce our new momentum interpolation method that will be used to calculate the face velocity of cell 5.

Take a look at equation 4.9 and 4.10. The u-momentum equation of cell P can be written as:

p au u  H u ()  V (4.35) PPPPx 

And in the same way, the u-momentum equation of cell E can be written as:

p au u  H u ()  V (4.36) EEEEx 

Instead of doing simple average, let’s interpolate the face velocity using the right-hand side of the whole momentum equation:

u u uPEPE u1 H H  1 1  p 1  p  ue u  u   u ()()  V P  u    V E  (4.37) 2 2aPEPE a  2  a x a  x 

In the similar way, we are able to write down the expression of face velocity component ve:

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v v vPEPE v1 H H  1 1  p 1  p  ve v  v   v ()()  V P  v    V E  (4.38) 2 2aPEPE a  2  a y a  y 

The following page is going to show how the face velocities of cell 5 are calculated using our new momentum interpolation method:

For the local face 1 of cell 5, or in other words, the interface between cell 2 and cell 5:

3 3 3  u u   u   u u  u   u 2  u 12 62 8 2 4 2N 3 2 5 1  1 1p 1  p  u5,1    ()()  Vcell 5     V cell 2  2a5 a 2  2  a 5 x a 2  x    (4.39) 1 11 1 1 1 23 (1  )  (  0   0)  2 12 2 5 6 24

3 3 3  v v   v   v v  v   v 2  v 12 62 8 2 4 2N 3 2 5 1  1 1p 1  p  v5,1    ()()  Vcell 5     V cell 2  2a5 a 2  2  a 5 y a 2  y    (4.40) 1 11 1 1 1 1 11 (1  )  (  0   )  2 12 2 5 6 2 12 For the local face 2 of cell 5, or in other words, the interface between cell 6 and cell 5:

3 3 3 u u 2  u   u u  u   u   u 13 6E 92 5 2 6 2 8 2 4 1 1p 1  p  u5,2()()()     Vcell 5     V cell 6  2a6 a 5 2  a 5 x a 6  x  (4.41) 1 11 1 1 1 1 11 (  1)  (  0   )  2 12 2 5 6 2 12

3 3 3 v v 2  v   v v  v   v   v 13 6E 92 5 2 6 2 8 2 4 1 1p 1  p  v5,2()()()     Vcell 5     V cell 6  (4.42) 2a6 a 5 2  a 5 y a 6  y  1 11 1 1 1 23 (  1)  (  0   0)  2 12 2 5 6 24

For the local face 3 of cell 5, or in other words, the interface between cell 8 and cell 5:

3 3 u2 u 6   u 8   u 4   1u5 u 9 2  u 8S  2  u 7 2 2 1 1p 1  p u5,3 ()()()     Vcell 5     V cell 8  2a8 a 5 2  a 5 x a 8  x  (4.43) 1 6 1 1 1 23 (  1)  (  0   0)  211 2 5 11 22 2 2

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3 3 v2 v 6   v 8   v 4   1v5 v 9 2  v 8S  2  v 7 2 2 1 1p 1  p v5,3 ()()()     Vcell 5     V cell 8  2a8 a 5 2  a 5 y a 8  y  (4.44) 1 6 1 1 1 1 12 (  1)  (  0   )  211 2 5 11 2 11 2 2 For the local face 4 of cell 5, or in other words, the interface between cell 4 and cell 5:

3 3 u2 u 6   u 8   u 4   1u1 u 5 2  u 7  2  u 4W 2 2 1 1p 1  p u5,4()()()     Vcell 5     V cell 4  2a4 a 5 2  a 5 x a 4  x  (4.55) 1 6 1 1 1 1 12 (  1)  (  0   )  211 2 5 11 2 11 2 2

3 3 v2 v 6   v 8   v 4   1v1 v 5 2  v 7  2  v 4W 2 2 1 1p 1  p v5,4()()()     Vcell 5     V cell 4  2a4 a 5 2  a 5 y a 4  y  (4.56) 1 6 1 1 1 23 (  1)  (  0   0)  211 2 5 11 22 2 2 A clear view of the face velocities of cell 5 after the momentum interpolation is provided in Figure 4.7:

Figure 4.7 Face velocity distribution of cell 5 after momentum interpolation

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The next step is to build the pressure-correction equation from continuity equation.

The ‘correct’ velocity can be regarded as the sum of two parts, the ‘current’ velocity that is obtain by solving discretized momentum equations, and the correction parts that are directly obtained from pressure corrections:

*'*' ue u e  u e, v e  v e  v e (4.57) Where:

''''''y  x ue(u ) e  ( P P  P A ), v e   ( v ) e  ( P P  P A ) (4.58) aPP a

Thus, the discretised continuity equation can be cast as:

**'' [(ue y e  v e   x e )  ( u e   y e  v e   x e )]  0 (4.59) Or:

y2 x 2   ''   [(u  v )e  (P P  P E )] e  Inbalance . of . Mass . flux . flowing . into . the . cell nb. faces . of . cell .5 aPP a where:

1 1 1 ( )  ( )  (1   )  ( ) aue a u P a u E PPE (4.60) 1 1 1 (v )e  ( v ) P  (1   )  ( v ) E aPPE a a

For cell 2, cell 6, cell 8 and cell 4, the pressure is always constant and there is no need to do pressure or velocity correction. The only cell we need to worry about is cell 5.

Apply equation 4.59 and 4.60 for cell 5, we can achieve the pressure correction value for cell 5:

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111 111 111 111 1[02 ( )(1) 2 ( )](PP '  0)1[1 2 ( )0 2 ( )]( '  0) 265 2655 265 265 5 111111 111111 1[02 (  )1 2 (  )](PP ' 0)1[(1) 2 (  )0 2 (  )]( ' 0) 21125 211255 21125 21125 5 11 11 12 12     12 12 11 11

P' 0.4656 5 (4.61) Thus, the corrected pressure for cell 5 is:

PPP*'  0  0.4656  0.4656 5 5 5 (4.62)

So far we have finished the 1st iteration. And the results are:

u5 1 v5 1 P  0.4656 5 (4.63) And these values in equation 4.63 will be used as initial values to continue the iterative procedure. After 23 iteration steps, the convergence is reached. The pressure of cell 5 hits 0.9990, which can be considered no different with the ambient pressure of its all other neighbor cells. And the momentum equations are also satisfied at the same time because the mass imbalance into cell 5 drops to 0, as shown in Figure 4.8:

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Figure 4.8 Pressure v.s. iteration step (Left) and mass imbalance v.s. iteration step

A list of velocity and pressure of cell 5 after each iteration step is given in Table 4.6 for quick reference:

Table 4.6 Velocity and Pressure of Cell 5 after Each Iteration Step

Iterations Step Pressure U V 1 0.4656 1 1 2 0.5992 1 1 3 0.6994 1 1 4 0.7745 1 1 5 0.8309 1 1 6 0.8732 1 1 7 0.9049 1 1 8 0.9287 1 1 9 0.9465 1 1 10 0.9599 1 1 11 0.9699 1 1 12 0.9774 1 1 13 0.9831 1 1 14 0.9873 1 1 15 0.9905 1 1 16 0.9929 1 1 17 0.9946 1 1

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18 0.9960 1 1 19 0.9970 1 1 20 0.9977 1 1 21 0.9983 1 1 22 0.9987 1 1 23 0.9990 1 1 24 0.9993 1 1

4.5 Conclusion In this chapter, we include the detailed information on handling different terms in the discretized N-S equations and solving the equations using iterative method. A new momentum interpolation method is introduced and verified with a simple example.

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CHAPTER 5

CFD VALIDATION

CFD Validation [13] is defined as: The process of determining the degree to which a model is an accurate representation of the real world from the perspective of the in- tended uses of the model. Validation examines if the conceptual models, computational models as implemented into the CFD code and computational simulation agree with real world observations. There are 3 major ways to validate a CFD code : 1) compare the CFD numerical results with analytical solution; 2) compare the CFD simulation results with experimental results, and 3) benchmarking: compare the results of a CFD code to another CFD code which has been validated and accepted.

In this chapter, the validation will be a case-based process. Four cases are presented, including internal flow in a straight channel, internal flow in a Z-pipe, internal jet flow and external flow near a submerged cube.

5.1 Internal Flow in a Straight Channel A 4x1 straight channel is given and the inner space is divided into 400 square finite volume cells with a 0.1x0.1 size for each cell, as demonstrated in Figure 5.1. Flow is going to the +x direction. The walls are y=0 and y=1. Inlet is x=0 and outlet is x=4.

Figure 5.1 Mesh of straight channel

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We assume the inlet flow velocity = 1 m/s and the Reynolds number of the flow is 1.

The flow velocity distribution from my 2D code is shown in Figure 5.2:

Figure 5.2 Flow velocity plot of my 2D code

And the result from FLUENT is shown in Figure 5.3:

Figure 5.3 Flow velocity plot of FLUENT

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Compare Figure 5.2 and Figure 5.3, we can tell that the results are nearly identical.

The maximum velocity is 1.46 m/s in Figure 5.2 and 1.47 m/s in Figure 5.3. The velocity distributions are also in perfect match.

5.2 Internal Flow in a Z-Pipe A Z-pipe is given and meshed with 600 square finite volume cells with a 0.2x0.2 size for each cell, as demonstrated in Figure 5.4:

Figure 5.4 Mesh of Z-pipe

Flow is going to the +x direction. The internal flow domain is bounded by 6 wall boundaries. Inlet is x=-2 and out let is x=6.

In this case the inlet velocity = 1m/s and the flow’s Reynolds number =10. The reason of making the Reynolds number 10 but not 1 is that under Re=10 the small vortex in the upper right corner becomes visible while under Re=1 it cannot be seen.

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The flow velocity distribution from my 2D code is shown in Figure 5.5 and the result from FLUENT is shown in Figure 5.6.

Figure 5.5 Flow velocity plot of my 2D code

Figure 5.6 Flow velocity plot of FLUENT

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Compare Figure 5.5 and Figure 5.6, we can tell that the results are very close. The maximum velocity is 1.44 m/s in Figure 5.5 and also 1.44 m/s in Figure 5.6. Due to the existence of vortex in this case, the reattachment length and vortex flow pattern can be important criteria to judge whether the results from my 2D code and FLUENT are in good match or not. Since the color of the background of Figure 5.6 is dark and the flow velocity of the vortex is small and colored with blue, a flow streamline plot is introduced here to recognize the vortex flow pattern, as shown in Figure 5.7:

Figure 5.7 Flow streamline plot of FLUENT

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Figure 5.8 Reattachment length comparison

The reattachment length of vortex A is 1.6 in my 2D code compared with 1.59 in

FLUENT, and the reattachment length of vortex B is 0.6 in both cases, as shown in

Figure 5.8. This quantitative comparison of reattachment length indicates that a very good match of flow patterns is achieved between two codes.

5.3 Internal Flow In a Jet This case can also be called internal flow with sudden expansions. In Figure 5.9, a straight channel is connected with another wider, expanded channel. The flow is fac- ing a 90 degree expansion angle at the connection point, and the flow behavior near the expansion region is of interest of research. In this case, we will compare the flow velocity distribution and the vortex reattachment length as we did in last case.

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Figure 5.9 Mesh of a jet

As shown in Figure 5.9, an internal jet is given and meshed with 900 square finite volume cells with a 0.2x0.2 size for each cell. Flow is going to the +x direction. The internal flow domain is bounded by 6 wall boundaries. Inlet is x=0 and out let is x=8. In this case the inlet velocity = 1m/s and the flow’s Reynolds number =10.

The flow velocity distribution from my 2D code is shown in Figure 5.10 and the result from FLUENT is shown in Figure 5.11.

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Figure 5.10 Flow velocity plot of my 2D code

Figure 5.11 Flow velocity plot of FLUENT

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Compare Figure 5.10 and Figure 5.11, we find that the maximum velocity is 1.35 m/s in Figure 5.10 and also 1.33 m/s in Figure 5.11. The streamline plot (or called particle trace plot) is given in Figure 5.12 as a demonstration of the vortex flow patterns:

Figure 5.12 Flow streamline plot of FLUENT

Again here as the important criteria, we have to compare the vortex reattachment length of these two cases, as shown in Figure 5.13:

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Figure 5.13 Reattachment length comparison

The reattachment length is 2.6 in my 2D code compared with 2.59 in FLUENT.

Another quantitative good match of flow patterns is achieved.

5.4 Laminar Flow Past a Square Cylinder A square cylinder is submerged in a fluid domain. The fluid domain is meshed with

2450 square finite volume cells with a 0.2x0.2 size for each cell, as demonstrated in

Figure 5.14:

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Figure 5.14 Mesh of fluid domain

Flow is going to the +x direction. Inlet of fluid domain is x=0 and outlet is x=11. The solid cube area is bounded by 4 wall boundaries: x=4, x=5, y=4 and y=5. The inlet velocity = 1m/s and the flow’s Reynolds number =20.

This case has been studied by many researchers. And in this case, the Reynolds number is very sensitive and important. It is widely accepted that at very small Reynolds numbers (Re<1), the flow near the square cylinder’s surface is creeping flow which is dominated by viscous force and hence no flow separation will be observed behind the cylinder. When Reynolds number increases (1

Recrit, two steady, symmetric vortices can be observed behind the cylinder. The length of the vortices increases while Reynolds number increases. After Reynolds number

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exceeds a critical value Recrit, the famous Karman Vortex Street will be observed, and the vortices will no longer be steady. This critical value of Reynolds number is still under discussion, but a widely accepted range of this value is between 54 and 90

[14][15].

Furthermore, research shows that when 4.4< ReD

[16] between the length of the symmetric vortices Lr and the characteristic length of the square cylinder D:

LDr/ 0.05  Re D (5.1)

ReD is defined as:

UDUMAX MAX ReD   Re  (5.2)  U INLET

In this case, we set the Reynolds number to be 20 which makes it far away from the critical value to prevent the existence of unsteady vortices. And Re=20 is big enough to make the symmetric vortices behind the cylinder visible.

The flow velocity distribution from my 2D code is shown in Figure 5.15 and the result from FLUENT is shown in Figure 5.16. The maximum velocity is 1.42 m/s and 1.43 m/s, separately.

We can see that the maximum velocity in around 1.43 m/s. Therefore, the ReD number is actually 28.6 according to equation 5.2 for this case. From equation 5.1, we can eas- ily derive the predicted the length of the symmetric vortices behind the square cylinder:

LDr0.05  Re D   0.05  28.6  1  1.43 (5.3)

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Figure 5.15 Flow velocity plot of my 2D code

Figure 5.16 Flow velocity plot of FLUENT

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In order to compare the vortex reattachment length, the streamline plot is given in Fig- ure 5.17:

Figure 5.17 Flow streamline plot of FLUENT

Let’s compare the length of the symmetric vortices:

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Figure 5.18 Symmetric vortices length comparison

From Figure 5.18, my 2D code gives Lr =1.0 while FLUENT gives Lr =0.95, compared with the predicted value Lr =1.43 from empirical equation 5.3.

We conclude that, even my 2D code agrees with FLUENT quantitatively on the maximum velocity and qualitatively on the flow distribution, it gives a closer prediction of the symmetric vortices length behind the square cylinder.

5.5 Conclusion In this chapter, we validate the 2D code through 4 different cases by benchmarking the simulation results with a popular and validated CFD commercial code, ANSYS FLU-

ENT. And in case 4, we also compare the result with the predicted value which comes from an empirical equation based on experiments. The well-matched results prove the

Texas Tech University, Jiannan Tan, December 2010 validity of my 2D code and the success of applying the new momentum interpolation method in predicting the behavior of two-dimensional incompressible Newtonian steady-state laminar flow.

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CHAPTER 6

EFFECTS OF CROSS WIND ON SPORT UTILITY VEHICLES (SUV): A COMPUTATIONAL STUDY

6.1 INTRODUCTION Ground vehicles, especially lightweight vehicles like sedans and small SUVs, are sub- jected to various irregular road conditions as well as to different wind conditions such as cross winds. A cross wind has significant effects on vehicles so that it not only af- fects the fuel efficiency and straight line stability but can also cause accidents at ex- posed locations where the wind effects are magnified by landscape features

[17](Sigbjörnsson et al 1998). In modern industry practice, the traditional wind tunnel experiment remains as the most reliable and trustworthy way of doing research on cross wind effects. The pioneering work was carried out in 1960s [18](Beauvais et al

1967). In 2004, Charuvisit et al [19] conducted a scale model in a wind tunnel experiment to evaluate the characteristics of the aerodynamic forces acting on a vehicle. C. J.

Baker [20], at the same time, carried out full-scale and wind tunnel experiments to measure the cross wind forces and moments on trains. While these real experiments yield accurate results, they also entail a considerable amount of equipmentations, labor and time. On the other hand, Computational Fluid Dynamics (CFD) offers less expen- sive and more flexible solutions [21]. Diedrichs [22] used CFD models to exemplify the cross wind stability of high speed trains while Bettle [23] developed a CFD model of a tractor-trailer truck on a bridge to examine the relationship between wind speed and the aerodynamic forces.

Although various studies have been done on the cross wind effect on vehicles and trains, in only very rare cases a full scale, true shape vehicle model was applied and

Texas Tech University, Jiannan Tan, December 2010 the influence of the wind tunnel dimension on the simulation results was discussed. In this paper, a 1:1 full scale CFD model of a small production SUV was utilized. Results of the CFD simulations with different wind tunnel test section dimensions, under different yaw angles were generated and were compared with the wind tunnel experimental results.

6.2 Aerodynamic Forces and coefficients

6.2.1 Aerodynamic forces and coefficients The three aerodynamic forces - drag, lift and side forces are demonstrated (Figure 6.1).

The mid-point of the vehicle wheelbase at ground level on the vehicle centerline is taken as the reference point for these forces.

Figure 6.1 Three aerodynamic forces and coefficients

Drag coefficient (CD), lift coefficient (CL) and side force coefficient (CS) are defined as:

DLS CCC,,,   (6.1) DLS1 1 1 u2 A  u 2 A  u 2 A 2VVV 2 2

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in which D, L, S are drag, lift and side forces, separately. AV is the frontal area of the vehicle at a 0o angle of attack.

6.2.2 Method of determining yaw angle and the resultant wind velocity The relative velocity of wind to vehicle U(w,v), airspeed relative to ground U(w,g) and ground to vehicle U(g,v) yield the following formula:

U(,)(,)(,) w v U w g  U g v (6.2)

The angle of attack in yaw is defined to be the angle between the resultant wind velocity U(w,v) and the relative velocity of ground to vehicle U(g,v) (Figure 6.2).

Assume the angle of attack is given and the wind velocity to ground, U(w,g) is large enough. For each fixed magnitude of U(g,v) and U(w,g), two solutions are available for the resultant wind velocity to vehicle, U(w,v) (Figure 6.3). The first solution of

U(w,v) tells that the incoming wind provides a drag force on the vehicle. The second solution of U(w,v) suggests that the resultant wind seen by the vehicle is reduced by the wind which is termed a tailwind. In this paper, the first solution is always chosen as the resultant wind velocity.

Figure 6.2 Method of determining the angle of attack

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Figure 6.3 Method of determining the resultant wind velocity

6.3 CAD models and cfd simulation

6.3.1 Vehicle CAD model This is a full scale 1:1 Sport Utility Vehicle (SUV) CAD model provided by the manufacturer (Figure 6.4). Reasonable simplifications and geometry manipulations were performed before generating surface mesh.

6.3.2 The wind tunnel geometry and boundary configuration The vehicle CAD model is put into a large rectangular computational domain which is considered as the virtual wind tunnel (Figure 6.5). Proper boundary configurations are implemented.

Figure 6.4 SUV CAD model

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Figure 6.5 Computation domain configurations

6.3.3 Blockage ratio The frontal area of the vehicle and wind tunnel test section are two critical parameters to the success of the simulation because the space of wind tunnel is limited and wall boundary effects are strong. Blockage ratio is introduced as a variable to quantify the impact of the wall boundary effect, which is defined as:

A B  V , (6.3) AW in which B is blockage ratio, Av is the frontal area of vehicle and Aw is the frontal area of the wind tunnel test section.

In order to observe the variation of simulation results with different magnitudes of blockage ratio, four wind tunnel cases were tested (Figure 6.6(a)). The dimension of the vehicle is 4413 mm in length, 2007 mm in width and 1508 mm in height. Dimen- sions of wind tunnel can be calculated accordingly (Figure 6.6(b)). All numbers of dimensional ratio are stated in the format of a:b:c.

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Figure 6.6(a) Wind tunnel dimensions

Figure 6.6(b) List of blockage ratio

6.3.4 Method of dealing with different angles of attack In order to create different angles of attack, the inlet wind velocity was modified (Ta- ble 6.1). In this study, the assumed vehicle driving speed U(v,g) was 80mph or 35.763 m/s and the maximum angle of attack to be achieved was 60 degree. Thus, the mini- mum magnitude of wind velocity U(w,g) which is required to guarantee the 60 degree angle of attack is 30.972 m/s. These two velocities were fixed for all cases. 61

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Table 6.1 Velocity inlet under different angles of attack

Angle of attack (degree) U(w,v) (m/s) x-component of U(w,v) (m/s) y-component of U(w,v) (m/s) 0 66.8 66.8 0 5 66.442 66.189 5.791 10 65.563 64.567 11.385 15 64.101 61.917 16.591 30 56.260 48.723 28.130 45 43.170 30.526 30.526 60 17.882 8.941 15.486

6.3.5 Mesh generation and simulation Due to the complex surface geometry and detailed accessories (including head lights, mirrors, gas tank, engine block, etc.) of the CAD model, a rectangular ‘core space’ was made near the vehicle surface and tetrahedral mesh with small volumes (1 cm3 –

10 cm3) were applied to enhance the accuracy of calculation. The mesh of this part always remained unchanged. Those cells outside this ‘core space’ area were allowed to be bigger with a maximum volume of 10 m3 to reduce the total amount of volume cells in order to save resource and time (Figure 6.7). Commercial software ANSYS

ICEM CFD was used in the progress of mesh generation. ANSYS FLUENT was used for CFD simulation.

Figure 6.7 Mesh around the vehicle

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6.4 Results and discussion Simulations were performed for 4 cases (Figure 6.6(b)) and for each case 7 different angles of attack (0o, 5o, 10o, 15o, 30o, 45o and 60o) were tested. The standard k-e turbulence model was applied. Plots of CD, CL and CS (Figures 6.8, 6.9 and 6.10) were provided and compared with available experimental data with angles of attack of 0o, 2.5o,

5o, 10o, 15o and 20o.

A general match can be observed between the curves of computational data and experimental results. However, the difference is noticeable, especially in the CL plot (Fig- ure 6.9). This could be caused by (1) the geometric differences between the production vehicle and the CFD model which could possibly be over-simplified, (2) the different resultant airspeeds used for the CFD simulation (various numbers according to yaw angles) and the wind tunnel experiments (fixed 68 mph or 30.4 m/s), (3) the potential errors of the wind tunnel experimental data, (4) the mesh quality and (5) the resolution of the CFD software.

According to the official data provided by the manufacturer, the CD of the SUV is 0.38

o at a 0 yaw angle. Thus, the error of CD is 42.1% for blockage B=8.4%, 23.7% for B=5%, 10.5% for B=2.8% and 5.3% for B=0.5%. These numbers tell that (1) simulation results tend to give larger numbers than the experimental data and (2) the simulation error is decreasing when blockage ratio is reduced (Figure 6.11). To achieve satis- factory results of less than 10% error, we suggest that a blockage ratio of no more than

5% need to be applied not only for real wind tunnel tests [24] but also for CFD wind tunnel simulation.

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Figure 6.8 Drag coefficients v.s. angles of attack

Figure 6.9 Lift coefficients v.s. angles of attack

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Figure 6.10 Side force coefficients v.s. angles of attack

Figure 6.11 Cd error v.s. blockage ratio

6.5 Concluding remarks This study was aimed at the cross-wind effects on a ground vehicle with different angles of attack. The influence of blockage ratio was also discussed. After better agree- ment is achieved by polishing the CAD model and refining the mesh, the computational tool will be applied to study the cross-wind effect of vehicles with trailers.

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