Presenter: Jonathon Nooner
• Suppose that we have an f(x,y,t):
fi,nj --- n is treated as a time index, i and j are treated as spatial indices
fi,nj
¶f
=
¶x
fi,nj
¶f
=
¶y
n
- fi-1,
j
Dx
- fi,nj-1
Dy
n
n
n
fi+1,
¶2 f
j - 2 fi, j + fi-1, j
=
¶x 2
Dx 2
fi,nj+1 - 2 fi,nj + fi,nj-1
¶2 f
=
2
¶y
Dy 2
n+1
n
f
f
¶f
i, j
= i, j
¶t
Dt
• The Navier Stokes Equations represent the momentum of a fluid. A
fluid is anything that flows, it includes: air, water, oil, glass (over very
long time frames).
• Common applications would include simulations of asteroid collisions,
airflow over an air foil,, plastic printing, etc.
• In 3 dimensions there are 3 equations to represent momentum in each
dimensions, a continuity equation and an energy equation are included.

Elastic Navier-Stokes equations*:
¶ra ¶ (ra u j )
+
=0
¶t
¶x j
¶ (ui ) ¶ (ui u j )
ui ui 1 ¶p 1 ¶
¶ui
+
= - fk ´ ui - gd i3 + ( (ra K i, j ))
¶t
¶x j
r ra ¶x i ra ¶x j
¶x j
¶ (qV )
¶ (qV ) 1 ¶
¶qV
qV
+ ui
=
(rak ij
)+
¶t
¶x i ra ¶x j
¶x j C p TV
N
dQn
å
n =1 dt
* Taken From: Jacobson, M. Z., “Fundamentals of Atmospheric Modeling”,
Second Edition, 2005. Ch. 3, 4.

Anelastic Navier-Stokes equations*:
¶ (r u j )
=0
¶x j
¶ ( r ui ) ¶ (r ui u j )
¶p r q 'g p'
¶
¶ui ¶u j
+
=+(
- )d i3 +
[m (
)]
¶t
¶x j
¶x i
q
H
¶x j ¶x j ¶x i
¶ ( r q ) ¶ (r qu j ) q
¶ui ¶u j ¶ui ¶
¶T
+
=
[m (
+
)
(k
)]
¶t
¶x j
c p T ¶x j ¶x i ¶x j ¶x j ¶x j
* Taken From: Lund, T. S., and D. C. Fritts, DC (2012), Numerical
simulation of gravity wave breaking in the lower thermosphere, J of
Geophysical Research. Vol 117. D21105.
• Such problems are quite difficult to solve.
• Five Equations, Nonlinear, Computationally Expensive.
• Has dimensions 60 x 60 x 100 km in x, y z respectively.
• 300 x 300 x 500 mesh points = 45 million points.
• Computations like this are done on supercomputers; as an example,
JANUS, which has 16416 total cores, and a maximum of 184 TFLOPS
(x10^12 Floating Point Operations) available. **
** Data from: https://www.rc.colorado.edu/resources/janus
• What does the analytical solution for this problem look like? No one
knows if an analytical solution exists. A millennial prize exists for
whoever can find one.
• We start small and build up.
• The smallest equation that maintains the nonlinear characteristics of the
Navier Stokes equations is the Burger’s equation.
*
¶u
¶u ¶ 2u
+ au = b 2
¶t
¶x ¶x
• On a digital computer, the domain will need to be split into discrete
pieces. Analog computers do exist that can solve continuum equations
natively by using operational amplifiers, but they are *significantly*
harder to use, and not nearly as flexible as their digital kin.
• For simplicity, we’ll begin with one of the easiest methods: Finite
Difference – Forward Time Centered Space (Diffusion) Backward Space
(Advection) Explicit
*
• For an explicit representation of this equation, we solve for the state of
the next timestep.
n+1
n
• We are describing a continuous system using a discrete domain. Based
on the rate that the velocity information is changing, you might think that
there is a limit to how coarsely one can represent a continuum using a
discrete domain… and you would be right!
Set Initial
Conditions
Calculate Timestep
based on CFL
Condition
Enforce Boundary
Conditions
Increment
Timestep
Calculate grid
velocities for the
next Timestep
no
Meet End
Condition?
yes
End
IC : u( x,0) = 10e -( x -15 )
BCs :
u(0, t ) = 4
u(200, t ) = 4
2
+ 4
IC : u( x, 0) = BCs :
u(0, t ) = 4
u(200, t ) = 2
2
1
x + 100e -( x -15 )
+ 4
100
IC : u( x, 0) = 100e -( x -15 )
BCs :
u(0, t ) = 4
u(200, t ) = 4
2
+ 4
¶u
¶u
¶u
¶ 2 u ¶ 2v
+u
+v
= n(
+
)
¶t
¶x
¶y
¶x 2
¶y 2
¶v
¶v
¶v
¶ 2 u ¶ 2v
+u
+v
= n(
+
)
¶t
¶x
¶y
¶x 2
¶y 2
• Note that it is not necessary for the viscosity to be the same in both
directions.
• No continuity equation yet, so conservation of mass per flow area is
not necessarily obeyed.
¶u ¶u ¶u
¶ 2 u ¶ 2v
+ u + v = n( 2 + 2 )
¶t
¶x ¶y
¶x ¶y
¶v ¶v ¶v
¶ 2 u ¶ 2v
+ u + v = n( 2 + 2 )
¶t
¶x ¶y
¶x ¶y
*
h(x+dx)
h(x)
u+du
u
p=po
z
A
x
B
x
x+dx
dm = rhdx
Mass Flowrate:
¶m
¶ (uh)
= ru(x)h(x) - ru(x + dx)h(x + dx) = - r
dx
¶t
¶x
1D Continuity Equation:
2D Continuity Equation:
¶h ¶ (uh)
+
=0
¶t
¶x
¶h ¶ (uh) ¶ (vh)
+
+
=0
¶t
¶x
¶y
Gravitational Potential:
Material Derivative:
Momentum Equations:
Shallow Water
Equations:
d ¶
¶
¶
¶
= +u +v +w
dt ¶t
¶x ¶y
¶z
¶u
¶u
¶u
+u
+v
=
¶t
¶x
¶y
¶v
¶v
¶v
+u
+v
=
¶t
¶x
¶y
¶h
¶x
¶h
å Fy = - g ¶y
åF
¶h
¶ (uh)
¶ (vh)
+
+
=0
¶t
¶x
¶y
¶u
¶u
¶u
¶h
+u
+v
= -g
¶t
¶x
¶y
¶x
¶v
¶v
¶v
¶h
+u
+v
= -g
¶t
¶x
¶y
¶y
x
= -g