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APPLIED COMPUTATIONAL

FLUID DYNAMICS

Edited by Hyoung Woo Oh

Applied Computational Fluid Dynamics

Edited by Hyoung Woo Oh

Published by InTech

Janeza Trdine 9, 51000 Rijeka, Croatia

Copyright © 2012 InTech

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Publishing Process Manager Jana Sertic

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First published March, 2012

Printed in Croatia

A free online edition of this book is available at www.intechopen.com

Additional hard copies can be obtained from orders@intechweb.org

Applied Computational Fluid Dynamics, Edited by Hyoung Woo Oh

p. cm.

ISBN 978-953-51-0271-7

Contents

Preface IX

Chapter 1

Computational Fluid Dynamics 1

Victor Udoewa and Vinod Kumar

Chapter 2

A Computational Fluid Dynamics Model of

Flow and Settling in Sedimentation Tanks 19

Ali Hadi Ghawi and Jozef Kriš

Chapter 3

Advances in Computational Fluid Dynamics

Applied to the Greenhouse Environment 35

Jorge Flores-Velázquez, Guillermo De la Torre-Gea,

Enrique Rico-García, Irineo L. López-Cruz

and Abraham Rojano-Aguilar

Chapter 4

Fluid Dynamics Analysis of

a Space Vehicle Entering the Mars Atmosphere 63

Antonio Viviani and Giuseppe Pezzella

Chapter 5

Air Movement Within Enclosed Road-Objects

with Contra-Traffica CFD-Investigation 73

M. Muhasilovic, A. Mededovic,

E. Gacanin, K. Ciahotny and V. Koza

Chapter 6

Computational Fluid Dynamics (CFD) and Discrete

Element Method (DEM) Applied to Centrifuges 97

Xiana Romani Fernandez, Lars Egmont Spelter

and Hermann Nirschl

Chapter 7

CFD and Thermography Techniques

Applied in Cooling Systems Designs 135

Samuel Santos Borges and Cassiano Antunes Cezario

Chapter 8

Computational Fluid Dynamics (CFD)

Modeling of Photochemical Reactors 155

Masroor Mohajerani, Mehrab Mehrvar and Farhad Ein-Mozaffari

VI

Contents

Chapter 9

Aerodynamic Design of the Vertical Takeoff Hopper

Concept of Future Launchers Preparatory Programme 177

Giuseppe Pezzella

Chapter 10

Fluid-Structure Interaction of a Radial Turbine 201

Zied Driss, Sarhan Karray, Hedi Kchaou and Mohamed Salah Abid

Chapter 11

Industrial Sprays: Experimental

Characterization and Numerical Modeling 223

Avinash Khopkar, Michael D. Cloeter and Quan Yuan

Chapter 12

Multidimensional Design

of Hydraulic Components and Systems 249

Massimo Milani, Luca Montorsi and Fabrizio Paltrinieri

Chapter 13

Study of an Individual Air-Conditioning

Energy-Saving Equipment 277

Nguyen Anh Tuan, Wu-Chieh Wu and K-David Huang

Chapter 14

Unsteady Differentiation of Aerodynamic

Coefficients: Methodology and Application 295

Carlo Necci and Nicola Ceresola

Chapter 15

Use of Fluid Dynamic Simulation to

Improve the Design of Spouted Beds 321

Rodrigo Béttega, Ronaldo Guimarães Corrêa

and José Teixeira Freire

Preface

This book is intended for use as a reference text to help advanced scientists and research engineers solve numerous and diverse fluid flow problems using

computational fluid dynamics (CFD). Each chapter has been contributed by the

practiced experts in the field of fluid dynamics. This material covers a wide spectrum

of CFD applications involving flow modeling in sedimentation tank, greenhouse

environment, hypersonic aerodynamics, computational scheme, cooling systems

design, photochemical reaction engineering, atmospheric reentry problem, fluid‐

structure interaction (FSI), atomization, hydraulic component design, air conditioning

system, and industrial applications of CFD.

Dr. Hyoung Woo Oh

Department of Mechanical Engineering,

Chungju National University,

Chungju,

Korea

1

Computational Fluid Dynamics

Victor Udoewa1 and Vinod Kumar2

1George Washington University,

USAID Development Engineer,

AAAS Science & Technology Policy Fellow, AAAS, 2009-2011,

2Mechanical Engineering,

University of Texas at El Paso,

USA

1. Introduction

Computational Fluid Dynamics (CFD) is the emerging field of fluid mechanics in which

fluid flow problems are solved and analyzed using computational methods and numerical

algorithms. In fluid mechanics, there are generally three routes of work in the field, three ways to conduct experiments. The first category is theoretical, or analytical, fluid mechanics.

Theoretical fluid mechanics includes theorizing, manipulating and solving equations with

pen and paper. The Navier-Stokes equation governing incompressible fluid flow is an

example of theoretical fluid mechanics. Secondly, many engineers and physicist work in the

area of experimental fluid mechanics. Experimental fluid mechanics involves conducting

actual physical experiments and studying the flow and the effect of various disturbances,

shapes, and stimuli on the flow. Examples include waves generated by pools, air flow

studies in actual wind tunnels, flow through physical pipes, etc. Lastly, a growing number

of engineers, mathematicians, computer scientists, and physicists work in the area of

computational fluid dynamics (CFD). In CFD, you may still run an experiment of waves

across water, an airplane in a wind tunnel, or flow through pipes, but now it is done through the computer Instead of actual, physical, 3D objects. A computer model is created,

and computer programmers code the equations representing the physical laws that govern

the flow of the molecules of fluid. Then the flow results (such as velocity and pressure) are output into files that can be visualized through pictures or animation so that you see the

result just as you do with physical experiments.

In cases where an analytical, or theoretical, solution exists, CFD simulations and the

mathematical models, which are coded in the computer program, are corroborated by

comparison to the exact solutions. This comparative check is called validation. CFD is not

yet to a point where solutions to problems are used without corroboration by existing,

known, analytical or exact solutions when available. Validation is not to be confused with

verification, however; validation is a check to make sure that the implemented, coded model

accurately represents the conceptual, mathematical description and the solution intended to

be modeled.

Still, there are many times when there is no analytical solution. In these cases, one often uses a computational approach. In such cases without a known solution, CFD is used to

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Applied Computational Fluid Dynamics

approximate a solution. Most often, CFD is used when a computational solution is faster,

cheaper, or more convenient. Convenience may be due to time or safety or another reason.

If we wanted to create a database of information about 3D flow around a cylinder for different types of fluids at varying speeds, CFD is quite helpful. It would take quite a long time to change the fluid in our flow container and clean the container for every type of fluid we desired. It would also take some time to change the speed of the flow. In this case, it’s much faster to simulate it computationally. Then, anytime we wanted to change the speed of

the flow, we simply would change one number in a computer input file. Or if we wanted to

change viscosity and density for the fluid (representing a different fluid) we would just

change the corresponding values in a computer input file. In this case, CFD is faster.

Now imagine if you were doing space defense work for a government, and you were asked

to do fluid dynamics simulation of the combustion dynamics during an explosion or when a

space shuttle launched (1). It would take immeasurable amounts of money to do test

launches over and over as you studied the combustion dynamics of space shuttle thruster

ignition. And it would take large amounts of money to test explosive devices, especially

considering the damage they cause. In these cases, CFD is, again, quite helpful. The only

costs in CFD are the time of creating a computer model, choosing the right mathematical

model, coding it, and the power and computer time required to solve the equations. But it is plain to see that CFD is cheaper.

What if you were hired to help design material for the outfits of swimming athletes? Your

company gave you the job of studying sharks and their agile ability to swim and maneuver

through the water. So you start by trying to study the fluid dynamics around the shark skin

(2). How convenient is it to locate sharks and place them in some type of testing container

where you have probes and measuring devices located? How convenient is it to place

probes on the body of the shark itself? How safe is it to work with the sharks in that manner? No, it’s better to create a computer model of a shark and get the information for the shape, design, feel, and density of its skin and to use this information to run simulations. It is clear that CFD is more convenient in this situation. Sometimes a CFD simulation can be all three—safer, cheaper, and more convenient.

Imagine a situation in which two paratroopers, jumping from both side doors of a military

cargo aircraft, always crash into each other down below (3). In order to analyze the fluid

dynamics of the problem to see what air flow forces are affecting the paratrooper paths, you would need to perform test jumps with paratroopers. However, that is potentially injurious

and not safe. You would also have to rent the plane, pay for the rental by hour, hire the test pilot, and pay for all the equipment for the jump. That is expensive. Lastly, the organization of the use of the military aircraft and personnel and equipment takes many months, and it

can take from 6 to 12 months to plan the test. In this case a CFD experiment is more convenient: faster, cheaper, and safer.

Usually solving CFD problems involves three stages. First there is the pre-processing stage.

In this stage, the geometric boundaries of the problem are defined. In 3D, a volume is

created (in 2D, an area) over which the equation will be solved. This volume is broken into

smaller units or cells creating a mesh (though there are meshless methods for computing

CFD problems). This may be uniform or non-uniform. Along with constitutive equations,

the particular equations are chosen for the problem in order to properly physically model

the flow. These equations may be manipulated depending on the mathematical method

being used. Boundary conditions are prescribed along the boundary. For time dependent

problems, initial conditions are prescribed.

Computational Fluid Dynamics

3

Secondly, the problem is solved numerically. At the least, we usually solve for velocity and pressure, but the list of unknowns can be longer depending on the mathematical model

(equation) chosen for the physical situation. Other unknowns may include temperature,

energy, and density. The numerical solution is usually computed iteratively for steady-state solutions. For time-dependent problems, a step in time is taken, and the equation is

numerically solved again, eventually producing a solution for every time step.

The final step in the CFD process is post-processing. In this stage, the solution is analyzed usually with the help of visualization and possibly animation for dynamic, or time-dependent, problems. It is in this stage that CFD results are usually compared to any

previous experimental results or known analytical, or theoretical, solutions. This

comparison is usually called validation. Today, confidence in CFD is growing, but we have

not arrived at the point of trusting CFD solutions without validation. Even if a particular

model is validated, we still corroborate the results of a simulation with experimental or

analytical results.

Because of the hybrid nature of CFD, advances in CFD are usually made in three areas:

computational and applied mathematics, mechanical/chemical engineering, and computer

science/electrical engineering. Some researchers work on new theoretical, mathematical

models creating new discretization methods (ways to discretize the problem in order to

numerically solve it over the discrete units or cells), or turbulence models. They might

publish in applied mathematics or computational mathematics journals. Others work on

computer architecture (such as different types of supercomputers or computer clusters),

coding techniques such as parallel programming, or speeding up the computational

processes through faster mesh generation and mesh reordering. They might publish their

results in electrical engineering or computer science journals. Lastly others might use CFD

to concentrate on new insights in the engineering aspect of the problem such as the

mechanics of bird flight or sharkskin-inspired speedo design for less water resistance, or

resistance to blood flow inherent to certain veins. They might publish their results in

engineering journals next to experimental or theoretical engineering results. Their focus is on the application more so than the math or computer science.

2. Pre-processing

Pre-processing refers to the work that must be done prior to the actual computational

experiment or simulation. This work can be reduced to four general areas: geometry

definition, volume division, model choice and definition, and boundary condition

definition. For the purposes of this article, we will ascribe the work of coding and the choice of computational implementation to the processing stage called simulation. Sometimes

researchers refer to mesh generation as pre-processing in general because a mesh is

generated when the volume is divided.

2.1 Geometry definition

The first step is to define the computational domain of the problem. The purpose of this

definition is to confine the problem to a finite space and limit the computation. It is true that a plane flying in the air has some residual effects on air flow patterns 1,000 miles away, but because of the negligible nature of those effects, we are relatively safe in looking at the effect of the plane on air flow within a reasonable vicinity of the plane thereby limiting our

4

Applied Computational Fluid Dynamics

computational work and making the problem finite. It would be quite a task to compute the

effect of the airplane on the air flow at a certain height around the entire earth.

To limit the flow in this way usually requires defining a domain inside which we will

compute the flow, outside of which we will not compute the flow. Usually the geometry of

the domain is chosen to be a box of some sort, usually a rectangular prism in 3D or a

rectangle in 2D. However, any closed shape may be chosen as long as the shape closes off

an inside computational domain from an outside space in which computations will not

take place.

A closed space does not imply, however, an empty domain box. For example imagine that

we are simulating intravenous blood flow (4) (5) around a cancerous growth. The vascular

domain is modeled by a 3D cylindrical prism (our domain box), but we still have an object

inside. In this case, our object is a semi-spherical cancerous growth on the surface of a wall of the vein. Traditionally, our domain must be totally closed, so the surface of the domain

goes from the wall of the vein, joins the surface of the cancerous growth, and continues on

the other side rejoining the wall of the vein creating a closed 3D space that does not go

under the tumor but continues over the surface of it. Likewise, if we were calculating flow

around a sphere (6), the domain box would be the outer half of the domain surface. The

inner half of the surface would be the sphere inside the domain box. Just as in the cancer

example, we are not calculating the flow inside the sphere, just as we were not calculating

the blood flow in the tumor. But the tumor and sphere form part of the boundaries of the

domain helping to close off the computational space in which we are interested in the

velocity and pressure of the fluid. Remember, domain boxes may contain objects inside

which no flow is calculated, but whose surface forms part of the surface of the domain

helping to limit the computational space and better define where the fluid flows.

2.2 Volume division

The second step is volume division or mesh generation. Why we must divide the volume

is not obvious until one remembers that it is easier to solve a flow problem over a smaller

area or volume than a larger one. So dividing the volume into smaller units transforms the

large problem over the entire domain into a large number of smaller problems over

smaller sub-domains. However, the real reason we divide the volume is because we seek

to find, for example, the velocity and pressure of the fluid at various points throughout

the domain volume or area. In order to do this, we fill the inner domain volume or area

with nodes—points at which we will calculate, in this instance, velocity and pressure.

Once we have filled the inner volume or area with nodes, we connect the nodes with

edges (and sides in 3D) creating smaller sub-volume or sub-area elements. For example, if

our computational box is a rectangular prism and we fill it with nodes, we can connect the

nodes to create quadrilaterals or tetrahedrals. If the domain is a rectangle and we fill the rectangle with nodes, we can connect all the nodes to create small rectangles or triangles.

This network of rectangles/triangles and quadrilaterals/tetrahedrals creates a mesh of

nodes; a mesh has been generated.

Remembering that the purpose of volume or area division is to create more manageable sub-

volumes or sub-areas, it behooves us to evenly space out the distribution of nodes. If we do not, we may find that there are large spaces (volumes or areas) with no nodes. This is problematic because it means some of the sub-volume or sub-area elements will still be

large; though our goal is to make them small.

Computational Fluid Dynamics

5

Our second goal is to make each sub-element evenly shaped. In 2D, evenness in both

directions means that 2D rectangles tend towards squares or that our triangles are equilateral.

In 3D, if evenness is desired in all 3 directions, our quadrilaterals tend toward cubes and the tetrahedrals tend to be equilateral. Oblong and unevenly shaped sub-elements also create

spaces (areas or volumes) with fewer nodes than parts of the domain with evenly shaped sub-

elements. In these cases, our second goal serves a similar purpose as our first goal: to divide the domain into smaller, more manageable sub-domains with evenly distributed nodes.

An important aspect of mesh generation is choosing the appropriate size, or refinement,

of the mesh sub-elements, such that important aspects of the flow are properly resolved.

The general rule is that no fluid particle should advance through multiple sub-elements,

cells, or units in one time step. Therefore our third goal is to increase the refinement (or the number of nodes) in areas or sub-volumes of increased fluid velocity or vorticity or

any interesting fluid flow phenomena that you would like to capture computationally.

This will allow us to visualize it later.

So far, the realm of geometry definition and mesh generation fall into the computer science

side of CFD. When generating the mesh by defining a geometry and dividing that geometric

area or volume, one must decide if one will use uniform sub-elements or non-uniform sub-

elements. Definitely in parts of the domain with special flow requiring increased refinement, the elements in those parts will not match the refinement of elements elsewhere. But in the

general flow one can still choose a uniform, structured mesh or a non-uniform, unstructured

mesh. The ability to create non-uniform meshes is important in CFD because of the physical

nature of fluids to occupy and fill any void left unoccupied. When dealing with complex geometries and small nooks, crannies, and crevices of an automobile or a model of a city

block, it helps to have unstructured, non-uniform meshes that allow for the modeler to

create the best shapes to fit the 2D or 3D space (7).

Likewise, the division of the volume or area in mesh generation requires the modeler to

choose between quadrilaterals and rectangular prisms or triangles and tetrahedrals (there

are other choices of shapes, as well, such as wedges and pyramids in 3D). Generally,

quadrilaterals and rectangular prisms have a more accurate solution than triangles and

triangular prisms, but there are ways to increase the accuracy of the latter. Because of the non-uniform and sharp geometries found in fluid problems by nature of fluids, triangles

and triangular prisms work better geometrically for CFD applications. Normally, CFD

researchers will utilize triangles and triangular prisms and then increase the number of

interpolation points inside these elements so that no accuracy is lost. Interpolation points are points inside an element at which the solution is calculated. From these interpolation points, we can approximate the solution at any location inside an element.

All of the choices in dividing the volume and discretizing the mesh have the potential to

introduce errors. Such errors, due to bad distribution of nodes or parts of the domain

where the refinement is too low, are called discretization errors. These are errors that

would disappear if we appropriately divided the volume or area or appropriately

discretized the mesh.

2.3 Physical model definition

The first two steps dealt with the computer science side of CFD and there are many CFD

engineers who work on geometric mesh discretization and mesh partitioning methods. Step

three deals with the computational and applied mathematical side of CFD—choosing the

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Applied Computational Fluid Dynamics

appropriate mathematical model. When preparing to model a certain fluid flow situation,

one must decide which equation accurately describes the fluid flow one wishes to simulate.

If no equation currently exists, the CFD engineer must do work in the theoretical side of

CFD and formulate a new equation or a more specific equation for his or her specific fluid

flow situation. If equations do exist, the CFD engineer must simply choose the correct

equation for the fluid flow. This is not a trivial step as sometimes the same situation may

require different mathematical models at different velocity regimes or different temperature regimes, for instance. So the specific parameters of the flow must be looked at in detail—

velocity, viscosity, density, pressure, etc.—so that the correct equation is chosen.

A good example from fluid mechanics is the Navier-Stokes equation which is the basic or

fundamental equation for fluid dynamics. If you remove viscosity from the equation, the

Navier-Stokes equations become the Euler equations. Since all fluids have some amount of

viscosity this approximation is important in flows in which the viscosity is negligible (8)

such as sonic flows. A plane flying at sonic speeds will have air sliding past it, relatively, as if it had no viscosity. So the use of the Navier-Stokes equations also depends on the velocity of the flow, or more accurately the Reynolds number which governs the ratio of the

kinematic forces to the inertial forces. You can still go further: if you remove vorticity from the Euler equations, you arrive at the full potential equations. The point in this illustration is that choosing the correct mathematical model is important, sometimes difficult, and always

specific to the flow situation.

The choice of the mathematical model affects the unknown values you will compute. Some

CFD simulations are really computational fluidothermodynamics because temperature and

energy are calculated as well (9). For compressible flows, density is an unknown value and

we would seek to solve for this value of density in the simulation. So the choice of

mathematical modeling affects what unknowns we will compute. More accurately, the

unknowns we want to compute in a given situation (along with other details about the flow

situation) may help positively affect our choice of a mathematical model or the need to

formulate a new one.

Remember that errors can be introduced at this step as well. If a inappropriate or poorly approximating governing equation is chosen, this affects the final solution. If a governing

solution is chosen or formulated for which no analytical solution or experimental solution

exists, we lose the opportunity for validation to reduce errors. Any simplification in the

model or any untrue assumptions the mathematical model uses introduces errors as well.

All of these types of errors can be classified as physical approximation errors because they deal with the physical, mathematical model (not the geometric model).

2.4 Boundary condition definition

After a mathematical model is chosen to model the physical phenomenon, usually boundary

conditions must be chosen. This is the fourth step of the pre-processing stage, and this step falls on the applied math side of CFD. Usually we deal with boundary-value problems which require values to be assigned along the boundary of the domain of the problem in

order to solve the problem throughout the 2D or 3D space.

For instance, in some problems, one may specify the value of the unknown on the boundary.

Imagine prescribing the value of the velocity of the fluid on the boundary. Such a boundary

condition is called a Dirichlet or a direct boundary condition because you are setting the

Computational Fluid Dynamics

7

value of the unknown. If one specifies the value of a derivative of the unknown, the

boundary condition is called a Neumann or natural boundary condition.

For example in hydrodynamic flow around a submerged rock with moss, a CFD researcher

would usually place the boundary condition of free-stream velocity on the entrance side and

exit side of the domain box assuming that the entrance side is sufficiently upstream from the rock so as to still be undisturbed, and the exit side is sufficiently downstream that the flow conditions have returned to free-stream conditions. The prescription of free-stream velocity would be Dirichlet or direct boundary conditions. The same researcher might assume stress

(a derivative of velocity) to be zero in the direction perpendicular to the side surfaces of the domain box. When she prescribes stress in that direction she is setting Neumann or natural

boundary conditions in the direction perpendicular to the side surfaces of the domain box.

Such boundaries are spatial boundaries. For time-dependent problems, there are temporal

boundaries in a sense. Time dependent problems require an initial condition, prescribed

values for the unknowns set at the temporal start of the simulation. In the same

hydrodynamic example, let us say we want to simulate the flow when a rock, half the size of

the stationary, mossy rock, was thrown into the river passing next to the mossy rock and

hitting the riverbed. To start the simulation we need to have the steady flow of the river

around the mossy, stationary rock without the 2nd rock thrown in. Once we have computed

this flow, we can use the values of velocity and pressure from this flow as initial conditions for a simulation of a moving 2nd rock that is falling to the bottom of the river. From there the simulation will march in time and use the flow results from the previous time step as initial conditions for the next time step.

3. Simulation

The second phase of CFD work is the actual simulation or the “processing” work once the

pre-processing work is completed. However, there are still some pre-simulation decisions to

be made. CFD work is done through computers which not only decreases the time it takes to

perform calculations, but also increases the amount of calculations that can be done in a

given time period. As computing power has increased over the years, CFD has been used to

solve larger and larger problems.

Large problems, however, were traditionally reserved for supercomputers. Supercomputers

are large computers made up of multiple computers or CPUs. A desktop or laptop computer

could only handle so many calculations due to hard drive limitations on different types of

computer memory. As computers in general become more advanced, not only has the

memory capacity of supercomputers increased, but so has the memory of desktops and

laptops increased. This has created a cycle where problems solved by supercomputers today

are solved by desktop computers and laptop computers tomorrow. And the problems

solved by desktop and laptop computers today were only solved by supercomputers

yesterday. For example, historically, a simulation of flow past an automobile was done on

supercomputers (10) (11). Today one can create a model of an automobile and run a flow

simulation of air flow past the automobile with one desktop or laptop machine. This

example is one of many indicative of this ever-improving cycle.

Besides memory the other limitation on computing ability in today’s world is clock speed.

CFD workers are dependent upon computer scientist researchers to continue to increase the

clock speed of microprocessors. In general, the faster computers become, the faster is the

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Applied Computational Fluid Dynamics

speed CFD scientists and engineers can compute solutions to problems. And as the speed of

computations increases, the time to do computations decreases, and CFD scientists and

engineers can compute larger problems (as long as they have the memory capacity for the

calculations and storage for the solution). Currently, as this article goes to press, the fastest machine in the world is the K computer which computes at 10 Petaflops (12). Flops are

floating point operations per second, and the prefix peta means 10^15. Therefore, the K

computer can compute 10^16 floating point operations per second.

Supercomputers come in varying shapes and sizes. A small supercomputer may have 36-100

CPUs. A larger supercomputer could have 50,000 different nodes, or CPUs. For example, the

K computer has 68,544 CPUs, each with 8 cores (octo-core) for a total of 548,352 cores (12).

As well, today we have small supercomputing clusters, in which different CPUs are linked

together to act as a supercomputer. Often one will find Linux clusters arranged in this way.

Each node of a cluster can actually contain multiple processors itself acting as a single

computer. A computer or node with 2 processors is called a dual core machine or node. A

computer or node with 4 processors is called a quad-core machine or node. Processors in a

multicore machine or cluster can use memory in different ways. Some use a shared memory

architecture. In this case, all processors can access all the memory because it is completely shared between all processors. Some may have distributed memory where each processor or

node has access only to its own memory. Finally there are hybrid machines like the K

machine. Each of the 68,544 CPUs has its own distributed memory. But inside each CPUs

memory is a system of 8 cores that share memory.

Historically supercomputers used vector-based architecture, but this created a niche market

since codes for such machines could not simply be run on non-vector based machines like a

desktop computer. Today laptop computers are very similar to supercomputers because

many supercomputers use bus-based architecture which is a modified architecture allowing

a desktop computer to run more than one processor like a quad core Linux machine (13).

Because most CFD work is done on clusters or supercomputers, CFD programmers often

learn parallel programming, a type of computer programming with instructions or

directives for communication and transmission of information between processors/cores or

nodes on a supercomputer or cluster. Parallel programming is especially important because

the purpose of supercomputing is to divide the large problem into smaller pieces given to

each node to process. However, in order to solve the larger problem, the nodes must

communicate especially and specifically about border regions of the partitioned mesh.

Once the mesh unit and resolution are chosen in the pre-processing stage, the mesh is

partitioned and a piece of the mesh is given to each processor or node. However, sometimes

researchers will re-order the mesh for large problems. Inefficient mesh partitioning

contributes increasing costs of calculations and time for larger and larger problems. To

facilitate calculations, mesh re-ordering schemes seek to minimize communication.

Communication is minimized most when each processor or node manages a contiguous

portion of the mesh. In this case, each processor only shares geometric nodes on the borders of mesh portions with processors that work on neighboring portions of the mesh. Imagine

the opposite situation in which mesh elements are randomly distributed. A processor might

have to communicate with 8 other processors if 8 of the bordering elements lie on 8 distinct processors! In CFD work, communication can take more than 50% of the computational time

depending on the specific problem and its size. So it is very important to minimize this as

much as possible, to leave more computational work for actual mathematical computations.

Computational Fluid Dynamics

9

Examples of mesh partitioning and re-ordering methods include MATLAB’s MESHPART

(14), METIS (15), and PARMETIS (16) (17). Ordering the mesh so that each CPU has access

to cells or units that are connected to each other is important. We also re-order and partition to maintain proper load balance so that no CPU has more work than any other.

All of these choices—mesh unit geometry, mesh resolution, partitioning, and order can

affect the ability of each processor to solve the resulting algebraic system of equations.

Therefore, researchers work on the parallelism of such computer codes. There are many

parallel programming language directives such as OpenMP (18), and MPI (19) and

languages such as Manticore (20) and NESL (21).

After the mesh partitioning and re-ordering scheme is chosen, the next step a CFD engineer

takes is choosing a discretization method. The mesh partitioning lies mostly in the realm of computer science, but this next step of discretization lies in the area of computational and applied mathematics. The mathematical model and governing equation has been chosen,

but the CFD scientists or engineer must choose how to discretize the solution of this model

over the entire domain.

3.1 Numerical discretization methods

In computational and applied mathematics, there are different numerical discretization

methods (22). Three popular methods are finite difference, finite volume, and finite volume

methods. Each of these classes of methods contains many variations usually specific to an

application area. Other methods include boundary element methods, higher-resolution

methods, and meshless methods like spectral methods.

3.2 Finite difference, finite volume, and finite element methods

The finite difference method is probably the oldest of the main three (23). It lends itself quite well to orderly and structured geometries. It is not used as commonly as the finite element

method or finite volume method, probably due to the geometric limitations on applications.

Still, there are modern finite difference codes that employ overlapping grids and embedded

boundaries allowing the use of the finite difference method for difficult or irregular

geometries. However, it is the easiest method to code and is often taught first in courses that teach numerical discretization methods.

The finite volume method is a method in which the governing partial differential equation is solved over smaller finite control volumes (24). Since the governing equations are cast in a conservative manner over each control volume, the fluxes across the volumes are conserved

In terms of tests, applications, validation, and literature, the most robust of all methods is the finite element method (25). The finite element method is a type of residual method in

which a residual equation is weighted and integrated over the domain. Since the domain is

broken into many elements, this integration actually takes places over each element in the

mesh. The finite element method requires more memory than the finite volume method but

is also more stable than the finite volume method.

There are other methods as well. Boundary elements methods include methods in which the

boundary is meshed into separate sub-elements (26) (27). In 3D, a boundary element method

domain would be a 2D surface. In 2D, a boundary element method domain would be

represented as a 1D surface or edge. There are also immersed boundary methods to deal

with situation in which elastic structures interact with fluid flows (28).

There are numerous other discretization methods. Each of the above discretization methods

can be used with functions of varying order. Every increase in the order of the functions

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Applied Computational Fluid Dynamics

used in the discretization method (from linear to quadratic, quadratic to cubic, from cubic to quartic, etc.) carries increased computational costs since higher order functions have more

terms and more coefficients. In fact, higher order functions require more sample points or

interpolation points to properly resolve them. Accuracy comes at a price. Still, such accuracy is sometimes warranted in cases where there are sharp gradients or shocks in velocity,

pressure, density, or temperature, for instance. In these cases, some of the previous

discretization schemes fail and introduce what we call “spurious oscillations” which are not actually physical but a byproduct of computational approximation. In these cases, CFD

engineers may choose to use higher-order discretization methods or shock capturing

methods such as Total Variation Diminishing (TVD) schemes (29), Essentially Non-

Oscillating (ENO) schemes (30), and the Piecewise Parabolic Method (PPM) (31).

Spurious oscillations are an example of discretization errors mentioned earlier. They are also called numerical errors because they are not physical. Specifically, such numerical error

resulting in spurious oscillations is often called dispersive error or dispersion. Truncation error is the type of numerical error resulting from the difference between the partial

differential equation and the finite equation that we actually code.

Additionally, with CFD we can experience a third error—computer error. For example,

for one calculation or one floating point operation, computer roundoff is usually

negligible. However, when doing repeated calculations over and over in simulations

dealing with billions of nodes at which we solve for multiple unknowns at each node,

computer roundoff error can build.

3.3 Turbulence models

Another area of CFD modeling is turbulence modeling. Turbulence modeling is a difficult

yet still potentially fruitful area of CFD research because of the computational difficulties it poses. Turbulence is a complicated phenomenon that occurs over a wide range of time

scales and length scales. This is where the trouble lies making it impossible to fully

resolve turbulent phenomena when using most approaches. Because of the wide ranges of

time periods over which turbulent periodicity manifests and the wide range of length

scales on which turbulence acts, most CFD engineers and scientist choose to resolve a

certain length scale and time scale range of turbulence and model the rest. In this case

resolving turbulence means that we actually compute and calculate turbulent quantities

like vorticity and velocities and pressures in turbulent regions. Modeling turbulence

means that we add an expression into our equation, the effect of which is approximately

to create the effect of fully resolved turbulence on our unknown values such as velocity

and pressure.

As one increases the range of time scales and length scales over which the turbulence is

resolved, one must increase the refinement of the simulation both in length (refinement of

the mesh) and in time (temporal refinement—the size of the time steps). This increased