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Euler equations
• These equations are conveniently expressed in
integral form:
Φ=fluid domain
Vt ( ) = control volume
St ( ) = control surface
n = unit normal
v( x, t ) = fluid velocity (particles)
Φ
n
S
Vt ()
w
v
w( x, t ) = arbitrary velocity (mesh)
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dM d
≡ ρ dV = ρ( w −v)
n dS
dt dt
∫ ∫ i
V() t
S()
t
Euler equations (2)
transport
dQ d
≡ vdV vw ( v)•
ndS pdV gdV
dt dt
∫ ρ = ∫ ρ − − ∫ ∇ + ∫ ρ
V() t
S () t
V () t
V () t
pressure
body force
dE d
transport
≡ edV ew ( v)•
ndS pv ndS g vdV
dt dt
∫ ρ =
∫ ρ
− − ∫ i + ∫ ρ i
V() t S() t S() t V()
t
transport
pressure
body force
(Mass)
(Momentum)
(Energy)
M = mass of control volume
Q = momentum of control vol.
E = energy of control vol.
ρ = fluid density
p = pressure
g = gravity
e = total specific energy
∇=gradient operator
i = scalar product
Plus suitable equation of state: p = p( ρ, i)
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