Fluids Review
TRN-1998-004
Mathematical Equations of CFD
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© Fluent Inc. 3/12/2016
Fluids Review
TRN-1998-004
Outline
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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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© Fluent Inc. 3/12/2016
Fluids Review
TRN-1998-004
Introduction
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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?
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Navier-Stokes equations
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most general
can handle wide range of physics
Incompressible Navier-Stokes equations
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assumes density is constant
energy equation is decoupled from continuity and momentum if properties
are constant
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© Fluent Inc. 3/12/2016
Fluids Review
TRN-1998-004
Introduction (2)
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Euler equations
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Other equations and models
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neglect all viscous terms
reasonable approximation for high speed flows (thin boundary layers)
can use boundary layer equations to determine viscous effects
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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© Fluent Inc. 3/12/2016
Fluids Review
TRN-1998-004
Navier-Stokes Equations
Conservation of Mass


   V  0
t
Conservation of Momentum
     u v     u w 
 u
u
u
u 
p    u 2


   u  v  w        2   V               Bx
x
y
z 
x x   x 3
 y   y x  z   z x 
 t
     u v     v w 
 v
v
v
v 
p    v 2
  B y
 u  v  w        2    V            

t

x

y

z

y

y

y
3

x

y

x

z

z

y





 
 
 
 
     w u     v w 
 w
w
w
w 
p    w 2
  Bz
u
v
 w        2
   V     
      

t

x

y

z

z

z

z
3

x

x

z

y

z

y


 
 



 
 
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© Fluent Inc. 3/12/2016
Fluids Review
TRN-1998-004
Navier-Stokes Equations (2)
Conservation of Energy
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 E
E
E
E 
   u
v
 w     kT     pV  Q v  Q g
x
y
z 
 t
Equation of State
   ( P, T )
Property Relations
   (T )
k  k T 
C p  C p T 
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© Fluent Inc. 3/12/2016
Fluids Review
TRN-1998-004
Navier-Stokes Equations (3)
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Navier-Stokes equations provide the most general model of singlephase 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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© Fluent Inc. 3/12/2016
Fluids Review
TRN-1998-004
Turbulence Modeling
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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…
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the time and space scales of turbulence are very tiny as compared to the
flow domain!
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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 timeconsuming task
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© Fluent Inc. 3/12/2016
Fluids Review
TRN-1998-004
Turbulence Modeling (2)
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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
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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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© Fluent Inc. 3/12/2016
Fluids Review
TRN-1998-004
Incompressible Navier-Stokes Equations
Conservation of Mass

 V  0
Conservation of Momentum
p
 u 

   V   u      2 u  B x
x
 t

p
 v 

   V   v      2 v  B y
y
 t
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p
 w 

   V   w      2 w  B z
z
 t

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© Fluent Inc. 3/12/2016
Fluids Review
TRN-1998-004
Incompressible Navier-Stokes Equations (2)
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Simplied form of the Navier-Stokes equations which assume
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For isothermal flows, we have four unknowns: p, u, v, w.
Energy equation is decoupled from the flow equations in this case.
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incompressible flow
constant properties
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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© Fluent Inc. 3/12/2016
Fluids Review
TRN-1998-004
Buoyancy-Driven Flows
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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:
By  (    0 ) g   0  T  T0 
 = thermal expansion coefficient
oTo = reference density and temperature
g = gravitational acceleration (assumed pointing in -y direction)
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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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© Fluent Inc. 3/12/2016
Fluids Review
TRN-1998-004
Euler Equations
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Neglecting all viscous terms in the Navier-Stokes equations yields the
Euler equations:


   V  0
t

u
p
   Vu    Bx
t
x

v
p
   Vv    B y
t
y

w
p
   Vw    Bz
t
z

E
p
   V  E    0
t


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© Fluent Inc. 3/12/2016
Fluids Review
TRN-1998-004
Euler Equations (2)
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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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© Fluent Inc. 3/12/2016
Fluids Review
TRN-1998-004
Discrete Phase Modeling
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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
Drag Force
Particle path
Body Force
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© Fluent Inc. 3/12/2016
Fluids Review
TRN-1998-004
Discrete Phase Modeling (2)
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Assuming the particle is spherical (diameter D), its trajectory is
governed by
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dV p
 18  CD Re   
 p   

 
 Fp
 V  Vp  g
2 
dt  D  24 
p


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V p  particle velocity
D  particle diameter
C D  drag coefficien t
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g  gravitatio nal accelerati on
 p  particle density
Re  relative Reynolds number
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Fp  additional forces acting on particle
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© Fluent Inc. 3/12/2016
Fluids Review
TRN-1998-004
Discrete Phase Modeling (3)
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Can incorporate other effects in discrete phase model
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droplet vaporization
droplet boiling
particle heating/cooling and combustion
devolatilization
Applications of discrete phase modeling
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sprays
coal and liquid fuel combustion
particle laden flows (sand particles in an air stream)
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© Fluent Inc. 3/12/2016
Fluids Review
TRN-1998-004
Multiple Species Modeling
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If more than one species is present in the flow, we must solve species
conservation equations of the following form
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mi
   Vmi    J i  Ri  Si
t
mi  mass fraction of species i
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Ji  diffusion flux of species i
Ri  mass creation/d epletion by chemical reactions
Si  other sources of mass
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Species can be inert or reacting
Has many applications (combustion modeling, fluid mixing, etc.).
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© Fluent Inc. 3/12/2016
Fluids Review
TRN-1998-004
Combustion modeling
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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
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Finite rate kinetics model
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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
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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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© Fluent Inc. 3/12/2016
Fluids Review
TRN-1998-004
Summary
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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
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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:
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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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© Fluent Inc. 3/12/2016