0
State the assumptions made in the development of Euler's equations and trace the origin of these assumptions?
For incompressible flow the inviscid 1D Euler equations decouple to:
ρt
+ uρx
=
0
px
=0
ut
+
ρ
et
+ uex
=
0
The 3D Euler equations are given by
ρw
ρv
ρu
ρ
ρuw
ρuv
ρu2 + p
ρu
ρv
+ ρuv
+ ρv
2 + p + ρvw
=
0
ρw2 + p
ρvw
ρuw
ρw
(E + p)w
z
(E + p)v
y
(E + p)u
x
E t
where ρ is the density, u =
(u, v, w) are the velocities, E is the total energy per unit volume and p is the
pressure. The total energy is the sum of the internal energy and the kinetic energy.
(
)
1
2
E =ρ
e+
u
2
where e is the internal energy per unit mass. The assumption of incompressiblity
gives
Show that in 3D the inviscid Euler equations with the assumption of incompressible flow decouple to:
The mass conservation equation takes the form:
=
ρe
+ ρ(u2 + v 2 + w2 )/2
∇ · u =
ux
+ vy
+ wz
=
0,
ρt
+ u · ∇ρ
=
0
px
=0
ut
+ u · ∇u
+
ρ
py
vt
+ u · ∇v
+
=0
ρ
pz
wt + u · ∇w
+
=0
ρ
et
+ u · ∇e
=
0
0 =
ρt
+ ∇ · (ρu)
=
ρt
+ ρ∇
· u + u · ∇ρ
=
ρt
+ u · ∇ρ
=
0 .
The momentum equation along the x-axis
can be condensed into
0 =
(ρu)t
+ (ρu2 )x + (ρuv)y
+ (ρuw)z
+ px
=
ρut
+ uρt
+ ρuux
+ u(ρu)x
+ ρvuy
+ u(ρv)y
+ ρwuz
+ u(ρw)z
+ px
=
ρut
+ ρuux
+ ρvuy
+ ρwuz
+ px
+ (ρt
+ (ρv)y
+ (ρu)x
+ (ρw)z
)
=
ρut
+ ρu
· ∇u
+ px
+ (ρt
+ ∇ · (ρu))
⇒ ut
+ u · ∇u
+
px
=0.
ρ
1
A similar argument reveals that the y- and z-axis
momentum equations reduce to their appropriate equations,
giving (in vector form):
∇p
(3)
⇒ ut
+ (u · ∇)u
+
=0.
ρ
Finally, The energy equation can be manipulated in the following way:
0 =
Et
+ ∇ · [(E + p)u]
=
Et
+ ∇ · (Eu) + ∇ · (pu)
=
Et
+ E∇ · u + u · ∇E + p∇
· u + u · ∇p
(
)
(
)
(
)
1
1
1
=
ρ e + u · u + ρt
e + u · u + u · ∇ ρe
+ ρ u · u + u · ∇p
2
2
2
t
(
)
(
)
1
1
=
ρet
+ ρu
· ut
+ ρt
e + u · u + u · ∇(ρe)
+ u · ∇ ρ u · u + u · ∇p
2
2
(
)
(
)
1
1
=
ρet
+ ρu
· ut
+ ρt
e + u · u + ρu
· ∇e
+ eu
· ∇ρ
+
u · u u · ∇ρ
+ ρu
· (u · ∇u)
+ u · ∇p
2
2
)
(
)
(
∇p
1
=
ρet
+ ρu
· ∇e
+ e + u · u (ρt
+ u · ∇ρ)
+ ρu
· ut
+ u · ∇u
+
2
ρ
⇒ et
+ u · ∇e
=
0 .
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study the origin and development of the universe
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