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Computational Fluid Dynamics (CFD)

IntroductionProfessor Salvador Vargas Daz

Universidad LibreUniversidad San Buenaventura

Salvador.vargas@unilibrebog.edu.cosvargas@usb.bog.edu.co

mailto:Salvador.vargas@unilibrebog.edu.comailto:svargas@usb.bog.edu.comailto:svargas@usb.bog.edu.comailto:Salvador.vargas@unilibrebog.edu.co

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Recommended texts

Anderson, JD: Computational Fluid Dynamicswell written text excellent introduction

uses finite difference approach

Versteeg, HK and Malalasekera, W: An Introductionto CFD

good finite volume intro

Content a little out-of-date

Ferziger, JH and Peric, MExcellent treatise on finite volume method for the

mathematics and fluids expert

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What is CFD?

CFD is the simulation of fluidsengineering systems using modeling(mathematical physical problemformulation) and numerical methods(discretization methods, solvers,numerical parameters, and gridgenerations, etc.)

Historically only Analytical Fluid Dynamics(AFD) and Experimental Fluid Dynamics

(EFD). CFD made possible by the advent of

digital computer and advancing withimprovements of computer resources

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Industrial problems (Analysis and Design)

1. Simulation-based design instead of build & test

More cost effective and more rapid than EFD

CFD provides high-fidelity database for diagnosing flow field

2. Simulation of physical fluid phenomena that are difficult for EFD Full scale simulations (e.g., ships and airplanes)

Environmental effects (wind, weather, etc.)

Hazards (e.g., explosions, radiation, pollution)

Physics (e.g., planetary boundary layer, stellar evolution)

Academic problems The goal is to understand the physical aspect of the process

Why use CFD?

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Where is used?

Aerospace

Automotive

Biomedical

Chemical

Processing HVAC

Hydraulics

Marine

Oil & Gas

Power Generation

Sports

Aerospace

Automotive

Biomedical

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Where is used?

Aerospace

Automotive

Biomedical

Chemical

Processing HVAC

Hydraulics

Marine

Oil & Gas

Power Generation

Sports

6

HVAC

Chemical Processing

Hydraulics

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Marine

Oil & Gas

Sports

Power Generation

Where is used?

Aerospace

Automotive

Biomedical

Chemical

Processing HVAC

Hydraulics

Marine

Oil & Gas

Power Generation

Sports

7

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How does a CFD code work?

Preprocessor create geometry

mesh volume

Processor solve a system of equations

approximation to subset or superset of Navier-Stokesequations

Post-processor Vector plots, contour plots, integrated values (eg total

pressure)

Colour For Directors

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Using CFD to solve a problem

1. Divide the fluid volume (surface) up intomanageable chunks (gridding)

2. Simplify the equations to be solved3. Set boundary conditions

4. Initialise the other grid values

5. Step through the grid ensuring that thesesimplified equations are satisfied at thegrid points and nearest neighbours

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Discretising equations

What are we solving?

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Components of the N-S equations

Need to know

values of each variable (eg u) at each point

values of the first derivative

values of cross-derivatives

values of second derivatives

..and more

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Forward approximation to value ofthe 1st derivative of u in space

u

xi-1 i+1i

dx

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Rearward approximation to value ofthe 1st derivative of u in space

u

xi-1 i+1i

dx

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Central approximation to value ofthe 1st derivative of u in space

u

xi-1 i+1i

dx

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Approximations to values of the 1stderivative of u in space

u

xi-1 i+1i

dx

rearward

forward

central

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1st & 2nd Order Finite Difference

x

uu

x

uii

id

1

x

uu

x

uii

id2

11

1st order forward difference

2nd

order central difference

x

uu

x

uii

id

1

1st order rearward difference

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Discretising equations

(Anderson)

.......62

3

,

3

32

,

2

2

,

,,1

x

x

ux

x

ux

x

uuu

jijiji

jiji

ddd

i,j i+1,ji-1,j

i+1,j+1i,j+1i-1,j+1

i-1,j-1i,j-1 i+1,j-1

The value of the variable, u, at the grid point i+1,j can beapproximated by a Taylor expansion:

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1st & 2nd Order Finite Difference

)(,,1

,

xO

x

uu

x

u jiji

ji

d

d

2,1,1

,

)(2

xOx

uu

x

u jiji

ji

dd

From the previous equation, we can find expressions

for the derivatives:

1st order forward

difference

2nd order central

difference

2,1,,1

,

2

2

)(22 xO

xuuu

xu jijiji

ji

dd

2nd order central

difference

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Practical consequences of

discretisationErrors arise from spacing of grid needs to be

small enough to represent the key aspects of

the flowErrors arise from the order of the equations

1st order should generally not be used

Only 2nd order solutions are acceptable forjournal publication

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Testing solution

Flow over a backward-facing step

Flow expands and leaves a

recirculating vortex behind the step

Solve to 2nd order and maintainlaminar flow

How long does the domain have

to be to ensure that the solution

is validUpstream?

Downstream?

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Testing solution

Start with a coarse grid

Solve the problem

Double the grid density Compare with the first solution

If the values have not changed significantly, it

is likely that the solution is grid-independent If the values have changed significantly,

continue until they stop changing

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