Introduction to Fluid
Dynamics & Applications
Nitin Jain
COMP 259
Class Presentation
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Motivation
• Starting point for all simulation
and animation that involves fluid
flow : indispensable for physically
based modeling
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Overview
• Understanding & Deriving the NavierStokes equations
• Solving the Navier-Stokes equations
- Basic approach
- Boundary conditions
• Applications
- Lid Driven Cavity flow
- Smoke
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So, Let’s begin the journey…
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Foundations
• Physical Quantities
-
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u
P
d
g
:
:
:
:
The velocity field
Pressure
Density
Gravity & other external forces
The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Foundations
Operators

- gradient
div
2
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u ,∂
u 
u   ∂
x
∂
y
 ∂
- divergence


div u    u  u
- Laplacian
  2u
u
2
x
x
 u
2

u
2 
y
y 2
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Navier-Stokes Equations

 u  0



 1
du
2
 (u  )u  p   u  f
dt

convection
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pressure viscosity
external
forces
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Navier-Stokes Equations
• In two
dimensions …
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Boundary Conditions
•
•
•
•
•
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No-slip condition
Free-slip condition
Inflow condition
Outflow condition
Periodic Boundary Condition
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Derivation
• Transport Theorem
• Conservation of Mass
• Conservation of Momentum
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Transport Theorem

 (c , t )

c
0
t


  
d


f ( x , t )dx    f  div( fu )( x , t )dx


 t t
dt t


  
x   (c , t )
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  
 
u ( x , t )   (c , t )
t
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Conservation of Mass




mass    ( x ,0)dx    ( x , t )dx
0
t
;  is density
Transport theorem


  
d



(
x
,
t
)
d
x



div(

u
)
(
x
,
t
)
d
x
0




 t t
dt t




  div( u )  0
t
 is constant
for incompressible
fluids
Integrand vanishes

div u  0
Continuity equation
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Conservation of Momentum
momentum  
t

  
 ( x , t )u ( x , t )dx
change in momentum   acting forces
  

body forces :   ( x , t ) f ( x , t )dx
t
 
surface forces :  σ ( x , t )n ds
 t

σ : stress tensor n : normal
  

  

 
d

(
x
,
t
)
u
(
x
,
t
)
d
x


(
x
,
t
)
f
(
x
,
t
)
d
x

σ
(
x
, t )n ds



t
 t
dt t
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Conservation of Momentum
  

  

 
d
 ( x , t )u ( x , t )dx    ( x , t ) f ( x , t )dx   σ( x , t )nds


t
 t
dt t
Transport theorem
Divergence
theorem






d
( u )  (u  )( u )  ( u ) div u  g  div σ  0
dt
…



 1
du
2
 (u  )u  p   u  f
dt

Momentum equation
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Dynamic Similarity of flows
• Dimensionless quantity =
dimensional quantity/ reference
quantity with same physical units
• The reference quantities used should
possess certain properties :
- They should be constant for the problem
- They should be known in advance
- They should be characteristic for the problem
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Dimensionless Variables
In a wind tunnel, L = length of the obstacle in the
flow, u’,P’and p’ are the upstream velocity,
pressure and density.
•
•
•
•
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x’’ = x/L
t’’ = u’t/L
u’’ = u/u’
P’’ = (P-P’) / (p’u’2)
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Reynolds Number and Froude
number
• These numbers describe the properties of
the flow
- Re = p’u’L/μ
- Fr = u’/sqrt(L*magnitude(g))
• The Reynolds number represents the
relative magnitude of the inertial and
viscous forces
• The Froude number relates the velocity of
object to velocity of fluid waves.
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Let us now see
how to solve the
Navier Stokes
equations…
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Discretization
• Refers to passing from a continuous problem to
one considered at only a finite number of points.
• Reduces differential equations to a system of
algebraic equations.
• The solution is found only at a finite number of
points in the domain.
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Discretization
• Forward difference
- Derivative at grid point i = ((function value at i+1) –
(function value at i)) / (grid spacing)
• Backward difference
- Derivative at grid point i = ((function value at i) – (function
value at i-1)) / (grid spacing)
• Central difference
- Derivative at grid point i = ((function value at i+1) –
(function value at i-1)) / (2*grid spacing)
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Pros and Cons
• With forward and backward differencing, the order
of approximation is first order. With central
differencing, it is second order.
• For convection-diffusion problems, stability
problems occur with central differencing when the
grid spacing is too coarse. Maintaining stability
hence requires restricting the grid spacing to be
very small.
• A possible compromise is to use a weighted
average of both discretizations :
- μ*forward difference + (1- μ)*backward difference
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Solving the equations
Basic Approach
1. Create a tentative velocity field.
a. Finite differences
b. Semi-Lagrangian method (Stable Fluids
[Stam 1999])
2. Ensure that the velocity field is
divergence free:
a. Adjust pressure and update velocities
b. Projection method
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Staggered Grid
pi , j 1
vi , j  1
2
ui  1 , j
pi 1, j
2
ui  1 , j
2
pi , j
vi , j  1
2
pi , j 1
pi 1, j
The staggered grid provides velocities
immediately at cell boundaries, is
convenient for finite differences,
and avoids oscillations.
Consider problem of a 2D fluid at rest
with no external forces. The
continuous solution is:
u0
v0
p  constant
On a discretized non-staggered grid
you can have:
ui , j  0
vi , j  0
pi , j  P1 for i  j even, P2 for i  j odd
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The algorithm
• Perform time discretization of
momentum equations. Discretize in a
manner explicit in the velocities and
implicit in the pressure.
• Evaluate continuity equation at time
t(n) + 1, to yield a poisson equation
for pressure. Solve for the pressure.
• Use the pressure values to compute
new velocities.
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Solving for pressure
  ( n1) ~ ( n )
du u
u
1

  p ( n1)
dt
t

Discretize in time
 ( n 1) ~ ( n ) t
u
 u  p ( n1)
Rearrange terms


t
  u ( n 1)    u~ ( n )   2 p ( n1)  0

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Satisfy continuity eq.
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Solving for pressure
We end up with the Poisson equation for pressure.
 p
2
( n 1)


t
  u~ ( n )
This is another sparse linear system. These types of
equations can be solved using iterative methods.
Use pressures to update final velocities.
 ( n 1) ~ ( n ) t
u
 u  p ( n1)

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No-slip condition
• The velocities vanish at the
boundaries to satisfy the no-slip
condition.
u0,j = 0,
vi,o = 0,
ui-max,j = 0,
vi,j-max = 0
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Free Slip Condition
• The velocity component normal to the
boundary should vanish along with
the derivative of the velocity
component tangent to the boundary.
u0,j = 0, vi,o = 0,
ui-max,j = 0, vi,j-max = 0,
v0,j = v1,j , vi-max + 1,j = vi-max ,j ,
ui,0 = ui,1 , ui,j-max+1 = ui,j-max
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Outflow Condition
• The normal derivatives of both
velocity components are set to zero at
the boundary
u0,j = u1,j , ui-max,j = ui-max -1 ,j ,
vi-max+1,j= vi-max,j, v0,j = v1,j,
ui,0 = ui,1 , ui,j-max+1 = ui,j-max
vi,0 = v1,j , vi,j-max = vi,j-max-1 ,
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Inflow Conditions
• The velocities are explicitly given
• We impose this for velocities normal
to the boundary by directly fixing the
values on the boundary line
• For the velocity components
tangential to the boundary, we
achieve this by averaging the values
on either side of the boundary.
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Periodic boundary conditions
• For periodic boundary conditions
in the x-direction, the boundary
values on the left and the right
boundaries coincide.
u0,j = ui-max-1,j , ui-max,j = u1,j
v0,j = vi-max-1,j , vi-max,j = v1,j
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Stability condition
Limits on time step
• CFL conditions – don’t move more
than a single cell in one time step
umax t  x,
vmax t  y
• Diffusion term
 1
1 
2t   2  2 
y 
 x
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1
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Satisfying the Continuity Eq.
The tentative velocity field is not necessarily
divergence free and thus does not satisfy
the continuity equation.
Three methods for satisfying the continuity
equation:
1.
2.
3.
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Explicitly satisfy the continuity equation by iteratively
adjusting the pressures and velocities in each cell.
Find a pressure correction term that will make the
velocity field divergence free.
Project the velocities onto their divergence free part.
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Explicitly Enforcing u=0
Since we have not yet added the pressure term, we can
use pressure to ensure that the velocities are
divergence free.
u>0 increased pressure and subsequent outflux
u<0 decreased pressure and subsequent influx
Relaxation algorithm
1. Correct the pressure in a cell
2. Update velocities
3. Repeat for all cells until each has u<ε
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Let’s move on
to some
applications
now…
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Lid Driven Cavity Flow
• We simulate a driven cavity flow in a
square domain.
• The physical configuration consists of
a square container filled with a fluid.
• The lid of the container moves at a
given, constant velocity, thereby
setting the fluid in motion.
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Boundary Conditions
• No slip conditions on all four
segments of the boundary, except the
upper boundary.
• Vertical velocity at the upper
boundary is zero.
• The horizontal velocity is set equal to
the given lid velocity.
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Lid Driven Cavity Flow
• Demo time :
http://www.math.ubordeaux.fr/Math_Appli/Fenetre
2/videotheque/DNS/Welcome.ht
ml
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Smoke (Fedkiw, Stam, Jensen
1999)
• Uses Euler equations instead of
the Navier Stokes equations
since
- These are more appropriate for gas
models
- Are less computationally expensive
• Uses “vorticity confinement” to
reduce numerical dissipation
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Euler Equations
• Incompressible Euler Equations
• Solve the equation in two steps
- Compute intermediate velocity without
the pressure term
- Forve the velocity field to be
incompressible
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Euler Equations
• Solve the second equation over a
timestep, ignoring the pressure term :
• Compute the pressure from the following
poisson equation
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Euler equations
• The intermediate velocity is then
made incompressible by subtracting
the gradient of the pressure from it.
• The temperature and density of the
smoke are then calculated by simply
advecting along the velocity field.
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Vorticity Confinement
• Smoke usually consists of velocity
field having a significant amount of
rotational and turbulence
components.
• Numerical dissipation damps out
these phenomena, and so they are
added back by a technique called
vorticity confinement
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Vorticity Confinement
• Vorticity is the
curl of the
velocity field.
• First, calculate
normalized
vorticity location
vectors.
• Then add
vorticity terms at
these locations.
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Some Videos…
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