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Euler equations (3)
• For a compressible fluid, the total specific energy is:
1 2
e= i+ v ( i = internal specific energy)
2
• Instead of the total energy equation, one may use the internal
energy form:
dI d
≡ idV iw ( v)•
ndS p vdV
dt dt
∫ ρ = ∫ ρ − − ∫ ∇i
V() t S () t V () t
pressure
transport
I = internal energy of control vol.
(Internal energy)
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Finite Element formulation
• Discretize via linear elements with velocities at nodes
• ρ and i (hence p) are uniform over each element
ρ,,
i p
v
w
ρ,,
i p
v
w
• Integral forms of mass and energy conservation may be used
directly, provided Vt () represents the current element volume
• Conservation of momentum is more complex
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