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Fluids Review
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1998

004
Mathematical Equations of CFD
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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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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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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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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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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
unsteady
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

consuming task
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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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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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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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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
additional
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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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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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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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.
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