It is importatnt

1. Dr Patrick Geoghegan
Book: H. Versteeg and W. Malalasekera An Introduction to
Computational Fluid Dynamics: The Finite Volume Method
FEA/CFD for
Biomedical
Engineering
Week 10: CFD

2. Last Weeks Summary

3. • 3 major equations – all derived from first principles:
– Continuity equation (mass is conserved)
– Momentum equations (F = ma, Newton’s second law)
– Energy equation (energy is conserved)

4. Continuity
• mass gain = inflow - outflow
( ) 0
=
⋅
∇
+
∂
∂
u
ρ
ρ
t
A simple check on the quality of a CFD solution is that continuity is
satisfied
Where there is no mass accumulation mass flow out should equal
mass flow in!
y
δ
z
δ
P (point)
S
T
E
x
x
u
u δ
∂
∂
−
2
1
x
x
u
u δ
∂
∂
+
2
1

5. • From Newton’s 2nd law:
“The rate of change of momentum of a fluid particle is equal to the
sum of forces on the particle”
Momentum
S
T
E
z
y
p δ
δ
y
x
z
z
zx
zx δ
δ
δ
τ
τ 





∂
∂
+
y
z
x
z
y
x
x
p
p δ
δ
δ 





∂
∂
+
y
x
zx δ
δ
τ
z
x
yx δ
δ
τ
z
x
y
y
yx
yx δ
δ
δ
τ
τ 







∂
∂
+
z
y
xx δ
δ
σ
z
y
x
x
xx
xx δ
δ
δ
σ
σ 





∂
∂
+
Surface Forces on fluid element moving with the flow

6. • The fluid element was moving with the flow: this is the non-
conservation form of the Cauchy equation
• similar equations can be derived in y and z directions
x
zx
yx
xx
f
z
y
x
x
p
Dt
Du
ρ
τ
τ
σ
ρ +
∂
∂
+
∂
∂
+
∂
∂
+
∂
∂
−
=
total
force
body
force
surface forces

7. • in the x direction
• in the y direction
• in the z direction
Cauchy in 3-D
( ) ( ) y
yz
yy
xy
f
z
y
x
y
p
v
t
v
ρ
τ
σ
τ
ρ
ρ
ρ +
∂
∂
+
∂
∂
+
∂
∂
+
∂
∂
−
=
⋅
∇
+
∂
∂
u
( ) ( ) z
zz
yz
xz
f
z
y
x
z
p
w
t
w
ρ
σ
τ
τ
ρ
ρ
ρ +
∂
∂
+
∂
∂
+
∂
∂
+
∂
∂
−
=
⋅
∇
+
∂
∂
u
( ) ( ) x
zx
yx
xx
f
z
y
x
x
p
u
t
u
ρ
τ
τ
σ
ρ
ρ
ρ +
∂
∂
+
∂
∂
+
∂
∂
+
∂
∂
−
=
⋅
∇
+
∂
∂
u

8. • Shear stresses
• direct stresses
Stokes
x
u
xx
∂
∂
+
⋅
∇
= µ
λ
σ 2
u
y
v
yy
∂
∂
+
⋅
∇
= µ
λ
σ 2
u
z
w
zz
∂
∂
+
⋅
∇
= µ
λ
σ 2
u






∂
∂
+
∂
∂
=
=
y
u
x
v
xy
yx
µ
τ
τ






∂
∂
+
∂
∂
=
=
x
w
z
u
zx
xz
µ
τ
τ






∂
∂
+
∂
∂
=
=
z
v
y
w
zy
yz
µ
τ
τ
µ
λ
3
2
=

9. • In x-direction (conservation form):
• Similar in y and z directions
• With ‘source’ 2nd order terms collected
Navier-Stokes Equations
( ) ( )
x
f
x
w
z
u
z
y
u
x
v
y
x
u
x
x
p
u
t
u
ρ
µ
µ
µ
λ
ρ
ρ
+












∂
∂
+
∂
∂
∂
∂
+












∂
∂
+
∂
∂
∂
∂
+






∂
∂
+
⋅
∇
∂
∂
+
∂
∂
−
=
⋅
∇
+
∂
∂
2
u
u
( ) ( ) ( ) Mx
S
u
x
p
u
t
u
+
∇
+
∂
∂
−
=
⋅
∇
+
∂
∂
grad
µ
ρ
ρ
u
Smx is all of the sources of x-momentum per unit volume per unit time
Including: pressure effects, body forces (gravity) and shear components from other directions of flow.

10. • Conservation of Mass
(Continuity Equation)
• Conservation of Momentum
• Conservation of internal
energy
• State Equations
Governing Equations for Viscous Compressible 3D Fluid
Flow
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
+ 𝛻𝛻 � 𝜌𝜌𝐮𝐮 = 0
𝜕𝜕 𝜌𝜌𝜌
𝜕𝜕𝜕𝜕
+ 𝑑𝑑𝑑𝑑𝑑𝑑 𝜌𝜌𝜌𝜌𝐮𝐮 = −
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
+ 𝑑𝑑𝑑𝑑𝑑𝑑 𝜇𝜇𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 + 𝑆𝑆𝑀𝑀𝑀𝑀
𝜕𝜕 𝜌𝜌𝜌𝜌
𝜕𝜕𝜕𝜕
+ 𝑑𝑑𝑑𝑑𝑑𝑑 𝜌𝜌𝜌𝜌𝐮𝐮 = −
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
+ 𝑑𝑑𝑑𝑑𝑑𝑑 𝜇𝜇𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 + 𝑆𝑆𝑀𝑀𝑀𝑀
𝜕𝜕 𝜌𝜌𝜌𝜌
𝜕𝜕𝜕𝜕
+ 𝑑𝑑𝑑𝑑𝑑𝑑 𝜌𝜌𝜌𝜌𝐮𝐮 = −
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
+ 𝑑𝑑𝑑𝑑𝑑𝑑 𝜇𝜇𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 + 𝑆𝑆𝑀𝑀𝑀𝑀
𝜕𝜕 𝜌𝜌𝜌𝜌
𝜕𝜕𝜕𝜕
+ 𝑑𝑑𝑑𝑑𝑑𝑑 𝜌𝜌𝜌𝜌𝐮𝐮 = −𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝐮𝐮 + 𝑑𝑑𝑑𝑑𝑑𝑑 𝑘𝑘𝑘𝑘𝑘𝑘𝑘𝑘𝑘𝑘𝑘𝑘 + 𝑆𝑆𝐸𝐸
𝑝𝑝 = 𝑝𝑝 𝜌𝜌, 𝑇𝑇 𝑖𝑖 = 𝑖𝑖 𝜌𝜌, 𝑇𝑇
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
+ 𝑑𝑑𝑑𝑑𝑑𝑑 𝜌𝜌𝐮𝐮 = 0
or
We wont derive this here

11. • These equations are similar because they all give information about
how quantities such as mass, momentum, and energy are transported
through the flow domain Transport Equations
• We can generalise the various transport equations into the form
The General Transport Equation
𝜕𝜕 𝜌𝜌𝛷𝛷
𝜕𝜕𝜕𝜕
+ 𝑑𝑑𝑑𝑑𝑑𝑑 𝜌𝜌𝜌𝜌𝐮𝐮 = 𝑑𝑑𝑑𝑑𝑑𝑑 𝛤𝛤
𝛷𝛷𝜇𝜇𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔Φ + 𝑆𝑆Φ

12. Φ=1 for conservation of mass
Φ =u for conservation of x-momentum
Φ =v for conservation of y-momentum
Φ =w for conservation of z-momentum
Φ =i for conservation of internal energy
• Any terms that are not in the common form have been hidden within the
source terms
• For example Γ diffusive term represents μ for the momentum equations
The General Transport Equation
𝜕𝜕 𝜌𝜌𝛷𝛷
𝜕𝜕𝜕𝜕
+ 𝑑𝑑𝑑𝑑𝑑𝑑 𝜌𝜌𝜌𝜌𝐮𝐮 = 𝑑𝑑𝑑𝑑𝑑𝑑 𝛤𝛤
𝛷𝛷𝜇𝜇𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔Φ + 𝑆𝑆Φ