# Mathematical Equations of CFD

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22 Φεβ 2014 (πριν από 7 χρόνια και 6 μήνες)

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Fluids Review

TRN
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1998
-
004

Mathematical Equations of CFD

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Fluids Review

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1998
-
004

Outline

Introduction

Navier
-
Stokes equations

Turbulence modeling

Incompressible Navier
-
Stokes equations

Buoyancy
-
driven flows

Euler equations

Discrete phase modeling

Multiple species modeling

Combustion modeling

Summary

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Fluids Review

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1998
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Introduction

In CFD we wish to solve mathematical equations which govern fluid
flow, heat transfer, and related phenomena for a given physical
problem.

What equations are used in CFD?

Navier
-
Stokes equations

most general

can handle wide range of physics

Incompressible Navier
-
Stokes equations

assumes density is constant

energy equation is decoupled from continuity and momentum if properties
are constant

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Introduction (2)

Euler equations

neglect all viscous terms

reasonable approximation for high speed flows (thin boundary layers)

can use boundary layer equations to determine viscous effects

Other equations and models

Thermodynamics relations and equations of state

Turbulence modeling equations

Discrete phase equations for particles

Multiple species modeling

Chemical reaction equations (finite rate, PDF)

We will examine these equations in this lecture

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1998
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004

Navier
-
Stokes Equations

0

V
t

z
y
x
B
y
w
z
v
y
z
u
x
w
x
V
z
w
z
z
p
z
w
w
y
w
v
x
w
u
t
w
B
y
w
z
v
z
x
v
y
u
x
V
y
v
y
y
p
z
v
w
y
v
v
x
v
u
t
v
B
x
w
z
u
z
x
v
y
u
y
V
x
u
x
x
p
z
u
w
y
u
v
x
u
u
t
u

3
2
2
3
2
2
3
2
2
Conservation of Mass

Conservation of Momentum

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1998
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004

Navier
-
Stokes Equations (2)

Conservation of Energy

g
v
Q
Q
V
p
T
k
z
E
w
y
E
v
x
E
u
t
E

Equation of State

)
,
(
T
P

T
C
C
T
k
k
T
p
p

)
(

Property Relations

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1998
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Navier
-
Stokes Equations (3)

Navier
-
Stokes equations provide the most general model of single
-
phase fluid flow/heat transfer phenomena.

Five equations for five unknowns:
,

p, u, v, w.

Most costly to use because it contains the most terms.

Requires a
turbulence model

in order to solve turbulent flows for
practical engineering geometries.

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Turbulence Modeling

Turbulence is a state of flow characterized by chaotic, tangled fluid
motion.

Turbulence is an inherently

phenomenon.

The Navier
-
Stokes equations can be used to predict turbulent flows
but…

the time and space scales of turbulence are very tiny as compared to the
flow domain!

scale of smallest turbulent eddies are about a thousand times smaller than
the scale of the flow domain.

if 10 points are needed to resolve a turbulent eddy, then about 100,000
points are need to resolve just one cubic centimeter of space!

solving unsteady flows with large numbers of grid points is a time
-

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1998
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Turbulence Modeling (2)

Conclusion
: Direct simulation of turbulence using the Navier
-
Stokes
equations is impractical at the present time.

Q: How do we deal with turbulence in CFD?

A: Turbulence Modeling

Time
-
average the Navier
-
Stokes equations to remove the high
-
frequency
unsteady component of the turbulent fluid motion.

Model the “extra” terms resulting from the time
-
averaging process using
empirically
-
based
turbulence models.

The topic of turbulence modeling will be dealt with in a subsequent
lecture.

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Incompressible Navier
-
Stokes Equations

z
y
x
B
w
z
p
w
V
t
w
B
v
y
p
v
V
t
v
B
u
x
p
u
V
t
u

2
2
2

0

V

Conservation of Mass

Conservation of Momentum

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Incompressible Navier
-
Stokes Equations (2)

Simplied form of the Navier
-
Stokes equations which assume

incompressible flow

constant properties

For
isothermal flows
, we have four unknowns: p, u, v, w.

Energy equation is decoupled from the flow equations in this case.

Can be solved separately from the flow equations.

Can be used for flows of liquids and gases at low Mach number.

Still require a turbulence model for turbulent flows.

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1998
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Buoyancy
-
Driven Flows

A useful model of buoyancy
-
driven (natural convection) flows
employs the incompressible Navier
-
Stokes equations with the
following body force term added to the y momentum equation:

This is known as the
Boussinesq model
.

It assumes that the temperature variations are only significant in the
buoyancy term in the momentum equation (density is essentially
constant).

0
0
0
)
(
T
T
g
B
y

= thermal expansion coefficient

o
T
o

= reference density and temperature

g = gravitational acceleration (assumed pointing in
-
y direction)

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Euler Equations

Neglecting all viscous terms in the Navier
-
Stokes equations yields the
Euler equations
:

0
0

p
E
V
t
E
B
z
p
w
V
t
w
B
y
p
v
V
t
v
B
x
p
u
V
t
u
V
t
z
y
x

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1998
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Euler Equations (2)

No transport properties (viscosity or thermal conductivity) are needed.

Momentum and energy equations are greatly simplified.

But we still have five unknowns:
,

p, u, v, w.

The Euler equations provide a reasonable model of compressible fluid
flows at high speeds (where viscous effects are confined to narrow
zones near wall boundaries).

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Discrete Phase Modeling

We can simulate secondary phases in the flows (either liquid or solid)
using a
discrete phase model.

This model is applicable to relatively low particle volume fractions
(< %10
-
12 by volume)

Model individual particles by constructing a force balance on the
moving particle

Particle path

Drag Force

Body Force

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1998
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Discrete Phase Modeling (2)

Assuming the particle is spherical (diameter D), its trajectory is
governed by

p
p
p
p
D
p
F
g
V
V
C
D
dt
V
d

24
Re
18
2
particle
on

acting

forces

F
number

Reynolds

relative
Re
density

particle
on
accelerati

nal
gravitatio
g
t
coefficien

drag
C
diameter

particle
D
velocity
particle
p
D

p
p
V

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Fluids Review

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1998
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004

Discrete Phase Modeling (3)

Can incorporate other effects in discrete phase model

droplet vaporization

droplet boiling

particle heating/cooling and combustion

devolatilization

Applications of discrete phase modeling

sprays

coal and liquid fuel combustion

particle laden flows (sand particles in an air stream)

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Fluids Review

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Multiple Species Modeling

If more than one species is present in the flow, we must solve species
conservation equations of the following form

Species can be inert or reacting

Has many applications (combustion modeling, fluid mixing, etc.).

i
i
i
i
i
S
R
J
m
V
t
m



mass

of

sources
other
S
reactions

chemical
by
epletion
creation/d

mass

species

of
flux
diffusion
J

species

of
fraction

mass

i
i
i
i
R
i
i
m

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1998
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004

Combustion modeling

If chemical reactions are occurring, we can predict the
creation/depletion of species mass and the associated energy transfers
using a combustion model.

Some common models include

Finite rate kinetics model

applicable to non
-
premixed, partially, and premixed combustion

relatively simple and intuitive and is widely used

requires knowledge of reaction mechanisms, rate constants (introduces
uncertainty)

PDF model

solves transport equation for mixture fraction of fuel/oxidizer system

rigorously accounts for turbulence
-
chemistry interactions

can only include single fuel/single oxidizer

not applicable to premixed systems

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Summary

General purpose solvers (such as those marketed by Fluent Inc.) solve
the Navier
-
Stokes equations.

Simplified forms of the governing equations can be employed in a
general purpose solver by simply removing appropriate terms

Example: The Euler equations can be used in a general purpose solver by
simply zeroing out the viscous terms in the Navier
-
Stokes equations

Other equations can be solved to supplement the Navier
-
Stokes
equations (discrete phase model, multiple species, combustion, etc.).

Factors determining which equation form to use:

Modeling

-

are the simpler forms appropriate for the physical situation?

Cost

-

Euler equations are much cheaper to solver than the Navier
-
Stoke
equations

Time

-

Simpler flow models can be solved much more rapidly than more
complex ones.