Mathematical Equations of CFD

rangebeaverMechanics

Feb 22, 2014 (7 years and 8 months ago)

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Fluent Inc.
2/22/2014

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1

Fluids Review

TRN
-
1998
-
004

Mathematical Equations of CFD

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Fluent Inc.
2/22/2014

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2

Fluids Review

TRN
-
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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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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Navier
-
Stokes Equations

0






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

Conservation of Momentum

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1998
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Navier
-
Stokes Equations (2)

Conservation of Energy



g
v
Q
Q
V
p
T
k
z
E
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y
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x
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Equation of State

)
,
(
T
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T
C
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T
k
k
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)
(


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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1998
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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

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2
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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).



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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
:

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0






























































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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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Discrete Phase Modeling (2)


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



p
p
p
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D
p
F
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24
Re
18
2
particle
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acting

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additional
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Reynolds

relative
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density

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nal
gravitatio
g
t
coefficien

drag
C
diameter

particle
D
velocity
particle
p
D










p
p
V

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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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1998
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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.).









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










mass

of

sources
other
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reactions

chemical
by
epletion
creation/d

mass

species

of
flux
diffusion
J

species

of
fraction

mass




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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.