Computational Methods for the Euler Equations
Before discussing the Euler Equations and computational methods for them, let’s
look at what we’ve learned so far:
Method
Assumptions/Flow type
2D panel
2D, Incompressible, Irrotational Inviscid
Vortex lattice
3D, Incompressible, Irrotational Inviscid, Small
disturbance
Potential method
3D, Subsonic compressible, Irrotational, Inviscid,
PrandtlGlauert
Small disturbance
Euler CFD
3D, Compressible (no
∞
M
limit), Rotational,
Shocks, Inviscid
The only major effect missing after this week will be viscousrelated effects.
2D Euler Equations in Integral Form
Consider an arbitrary area (i.e. a fixed control volume) through which flows a
compressible inviscid flow:
≡
n
v
outward pointing normal (unit length)
≡
dS
elemental (differential) surface length
j
dx
i
dy
dS
n
v
v
v
−
=
Note: Path around surface is taken so that interior of control volume is on left.
y
x
c
dS
C
δ
n
v
dS
n
v
dy
dx
dS
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View Full DocumentComputational Methods for the Euler Equations
16.100
2002
2
Conservation of Mass
C
in
mass
of
dA
dt
d
change
of
rate
dA
C
in
Mass
C
of
out
flow
mass
of
rate
C
in
mass
of
change
of
rate
C
C
∫∫
∫∫
=
⇒
=
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
ρ
0
where
fluid
of
destiny
≡
Now, the rate of mass flowing out of
C
:
Mass flow out of
∫
=
•
=
C
u
dS
n
u
C
δ
v
v
v
velocity vector
Conservation of xmomentum
Recall that: total rate of change momentum
∑
=
forces
For xmomentum this gives:
∑
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
C
of
out
momflow
x
of
rate
C
in
momentum
x
of
change
of
rate
Forces in xdirection
∫
∫∫
∑
=
•
+
C
C
dS
n
u
u
udA
dt
d
v
v
Forces in xdirection
Now, looking closer at xforces, for an inviscid compressible flow we only have
pressure (ignoring gravity).
Recall pressure acts normal to the surface
∫
•
−
=
∑
⇒
C
dS
i
n
p
x
in
s
Force
v
v
∫
∫∫
=
•
+
C
C
dS
n
u
dA
dt
d
0
v
v
⇒
dS
dS
n
p
v
−
Into surface
Normal to surface
Gives xdirection
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 Fall '03
 willcox
 Dynamics, Fluid Dynamics, Energy, Kinetic Energy, Aerodynamics, Euler equations, Computational Methods

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