Compressible Viscous Equations
Also known as the compressible Navier-Stokes equations:
Mass:
∂
∂ρ
(ρv j ) = 0
+
∂t ∂x j
Momentum:
∂ ( ρν i ) ∂
∂p
∂
ρν iν j ) = −
+
+
(
(τ ij ) , i = 1, 2,3
∂t
∂x j
∂xi ∂x j
Energy:
∂
1
∂
(ρ e + ρ v2 ) +
∂t
∂x j
2
 ∂ν i
τ ij = µ 

 ∂x j
∂Τ
,
q = k
∂x j
+
1 2 
∂
∂
∂

+
(
ρ
e
ρ
v
)
v
p
ν
τ
ν
( q j )
=
−
+
+
(
)
(
)
j
j
ij
i


2
∂x j
∂x j
∂x j
∂ν j 
∂ν k
 + δ ij λ
∂xi 
∂xk
e = e ( p, Τ ) ← state relationship (ideal gas )
Incompressible Viscous Equations
In this case, we assume ρ = const.
Mass:
∂ν j
∂x j
Momentum: ρ
=0
∂ν i
∂
(ν iν j ) = − ∂p + ∂
+ρ
∂t
∂x j
∂xi ∂x j
  ∂ν i ∂ν j
+
 µ 
  ∂x j ∂xi




Usually, µ = µ (Τ ) . Often, when temperature variations are small, µ =const. is assumed.





∂ν j 
∂   ∂ν i ∂ν j  
∂ν i

⇒
+
=
+
µ
µ
 

 ∂x j ∂x j ∂x j ∂xi 
∂x j   ∂x j ∂xi  


∂  ∂ν j  


 =0
∂xi  ∂x j 


∂ν i
=µ
∂x j ∂x j
Usual form of momentum for incompressible flow:
ρ
∂vi
∂vi
∂p
∂
+µ
(v i v j ) = −
+ρ
∂xi
∂x j
∂x j ∂x j
∂t
In this case, the energy equation is not needed to find vi & p .
Incompressible Inviscid Equations
In this case we assume that the effects of viscous stresses are small compared to
acceleration and pressure forces:
Mass:
∂ν j
=0
∂x j
Momentum: ρ
∂ν i
∂
∂p
ν iν j ) = −
+ρ
(
∂t
∂x j
∂xi
These are known as the incompressible Euler equations.
Incompressible Potential Flow
K
In potential flow, we assume the flow is irrotational (i.e. ∇ × V = 0 ). This allows the
velocity to be written as the gradient of a scalar potential:
νi =
Mass:
∂φ
,
∂xi
∂φ j
∂x j ∂x j
φ = Potential
=0
∂ 2φ
∂ 2φ
∂ 2φ
= 0
+
+
Also written out as:
∂z 2
∂y 2
∂x 2
or
∇2φ = 0, ∇2 ≡ Laplacian
This is a single equation for a single unknown φ . It is the same for steady and unsteady
flows.
What happens to momentum?
16.100
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