wpi pi computational fluid dynamics i
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PPPPIIIIWWWW Computational Fluid Dynamics I
Reviewof
Fluid Dynamics
Instructor: Hong G. ImUniversity of Michigan
Fall 2001
PPPPIIIIWWWW Computational Fluid Dynamics IOutline
Outline
Basic relations for continuum fluid mechanicsReynolds transport theoremDivergence theorem
Derivation of equations governing fluid flowConservation of massConservation of momentumConservation of energyConstitutive relations
PPPPIIIIWWWW Computational Fluid Dynamics IBasic Relations
),( txφ
∫∫∫ ⋅+∂∂=
CSCVV
dSdVt
dVDtD
sys
)( nuφφφ
Reynolds Transport Theorem
Rate of changein system
For any vector or scalar function that represents a flow property
Rate of changein control volume
Flux through control surface
zw
yv
xu
ttDtD
∂∂+
∂∂+
∂∂+
∂∂=∇⋅+
∂∂= u
Material derivative
PPPPIIIIWWWW Computational Fluid Dynamics I
+ flux out
Basic Relations
At t=t CV
)(tφ
At t=t+dtCV
=CV
)( dtt +φ− flux in
PPPPIIIIWWWW Computational Fluid Dynamics I
The Divergence (Gauss) Theorem:Conversion of surface integral to volume integral
dVdSCVCS∫∫ ⋅∇=⋅ )()( φnφ
Basic Relations
PPPPIIIIWWWW Computational Fluid Dynamics I
Reynolds Transport Theorem:
Basic Relations
: System integral to control volume integral
∫∫∫ ⋅+∂∂=
CSCVV
dSdVt
dVDtD
sys
)( nuφφφ
∫
⋅∇+∂∂=
CV
dVt
)( uφφ
PPPPIIIIWWWW Computational Fluid Dynamics IConservation Equations
),(),( tt xx ρφ =
Conservation of Mass (Continuity)
If RTT yields
Since the equation holds for arbitrary control volume,
dVt
dVDtD
CVVsys∫∫
⋅∇+∂∂== )(0 uρρρ
0)( =⋅∇+∂∂ uρρt
PPPPIIIIWWWW Computational Fluid Dynamics IConservation Equations
Or, in Lagrangian form,
Continuity Equation:
uuu ⋅∇+∇⋅+∂∂=⋅∇+
∂∂ ρρρρρ
tt)(
0=⋅∇+= uρρDtD
0or0)( =⋅∇+=⋅∇+∂∂ uu ρρρρ
DtD
t
PPPPIIIIWWWW Computational Fluid Dynamics IConservation Equations
Conservation of MomentumIf RTT yields),(),( tt xuxφ ρ=
dVt
dVDtD
CVVsys∫∫
⋅∇+∂∂= )( uuuu ρρρ
∫∫ +⋅=CVCS
dVdS fnT ρ)(
[ ]∫ +⋅∇=CV
dVfT ρ
Surface force (stress) Body force (gravity)
Divergence theorem
PPPPIIIIWWWW Computational Fluid Dynamics IConservation Equations
Conservation of Momentum
fTuuu ρρρ +⋅∇=⋅∇+∂∂ )(t
Alternatively, subtracting
⋅∇+∂∂⋅−⋅∇+
∂∂ )()( uuuuu ρρρρ
tt
)continuity(⋅u
fTuuu ρρρρρ +⋅∇==∇⋅+∂∂=
DtD
t)(
fTu ρρ +⋅∇=DtD
PPPPIIIIWWWW Computational Fluid Dynamics IConservation Equations
Constitutive Relation – Stress Tensor
DUuT µµκ 2])32([ +⋅∇−+−= p
In tensor notation
Bulk viscosity = 0 (Stokes assumption)
∂∂
+∂∂+
∂∂−+−=
i
j
j
i
k
kijijij x
uxu
xupT µδµκδ )
32(
Unit tensor
001010100
[ ]T)()(21 uuD ∇+∇= (Deformation tensor)
PPPPIIIIWWWW Computational Fluid Dynamics IConservation Equations
Conservation of Momentum (Final):
[ ] fDuu ρµµκρ +⋅∇+
⋅∇−∇+−∇= 2)
32(p
DtD
[ ]T)()(21 uuD ∇+∇=
PPPPIIIIWWWW Computational Fluid Dynamics IConservation Equations
Conservation of Energy
From RTT, for
⋅+= uuxxφ
21),(),( tet ρ
dVeet
dVeDtD
CVVsys∫∫
⋅+⋅∇+
⋅+
∂∂=
⋅+ )
21()
21(
21 uuuuuuu ρρρ
dVdSdSCVCSCS∫∫∫ ⋅+⋅⋅+⋅−= fuTnunq ρ)()(
[ ]dVCV∫ ⋅+⋅∇⋅+⋅∇−= fuTuq ρ)(
Heat flux Work by stress
Body force work
Divergence theorem
PPPPIIIIWWWW Computational Fluid Dynamics IConservation Equations
fuTuquuuuu ⋅+⋅⋅∇+⋅−∇=⋅+⋅∇+
⋅+
∂∂ ρρρ )()
21()
21( ee
t
Total Energy Equation
0)( =⋅∇+∂∂ uρρt
fuTuquuuuu ⋅+⋅⋅∇+⋅−∇=⋅+∇⋅+
⋅+
∂∂ ρρρ )()
21(
21 ee
t
+⋅∇=∇⋅+
∂∂⋅ fTuuuu ρρρt
][ u:TT)(uT)(u ∇+⋅∇⋅=⋅⋅∇
)(: uTqu ∇+⋅−∇=∇⋅+∂∂ ete ρρ
PPPPIIIIWWWW Computational Fluid Dynamics IConservation Equations
Constitutive Relations
)(:)(: uUuT ∇−=∇ p
Tk∇−=q (Fourier’s Law)
),(),,( ρρ eTTepp == (Equation of State)
If viscous heating is neglected,
)()( uuu ⋅∇−=∇⋅+⋅−∇= ppp
)( uqu ⋅∇−⋅−∇=∇⋅+∂∂ pete ρρ
PPPPIIIIWWWW Computational Fluid Dynamics I
In Convective (Nonconservative) Form
Conservation Equations - Summary
0=⋅∇+ uρρDtD
fTu ρρ +⋅∇=DtD
)(: uTq ∇+⋅−∇=DtDeρ
PPPPIIIIWWWW Computational Fluid Dynamics I
In Conservative Form
Conservation Equations - Summary
0)( =⋅∇+∂∂ uρρt
fuuTu ρρρ +−⋅∇=∂∂ )(t
+⋅−⋅+⋅−∇=
⋅+
∂∂ qTuuuuuu )
21(
21 ee
tρρ
Discretized equations can satisfy the conservation properties more easily
PPPPIIIIWWWW Computational Fluid Dynamics I
In Integral Form (for fixed CV)
Conservation Equations - Summary
0=⋅+∂∂
∫∫CSCV
dSdVt
nuρρ
Useful in Finite Volume Methods
[ ]∫∫∫ ⋅−⋅+=∂∂
SCCVCV
dSdVdVt
)( nuunTfu ρρρ
∫
+⋅+⋅−⋅⋅+
SC
dSqe uuunTnu )21()( ρ
∫∫ ⋅=⋅+∂∂
CVCV
dVdVet
fuuu ρρ )21(
PPPPIIIIWWWW Computational Fluid Dynamics I
Compressible Inviscid Flows
Conservation Equations – Special Cases
0=∂∂+
∂∂+
∂∂+
∂∂
zyxtGFEU
=
Ewvu
ρρρρρ
U
+
+=
upEuwuv
puu
)(
2
ρρρρρ
E
+
+=
vpEvw
pvuvv
)(
2
ρρρρρ
F
++
=
wpEpvw
vwuww
)(
2
ρρρρρ
G
TchTceRTp pv === ,,ρ
ReTepRcv
)1(,)1(,1
−=−=−
= γργγor
PPPPIIIIWWWW Computational Fluid Dynamics I
Incompressible Flows
Conservation Equations – Special Cases
0=DtDρ
0=⋅∇ u
Continuity equation reduces to
0=⋅∇+ uρρDtD
(Divergence-free)
Momentum equation with constant viscosity, κ=0
fuuuu +∇+∇−=∇⋅+∂∂ 2ν
ρp
t
ρµν /= Kinematic viscosity
PPPPIIIIWWWW Computational Fluid Dynamics I
Equation for Pressure
Conservation Equations – Special Cases
Taking divergence of the momentum equation
+∇+∇−=∇⋅+
∂∂⋅∇ fuuuu 2ν
ρp
t
Poisson’s equation
fuuuu ⋅∇+⋅∇∇+∇−=∇⋅⋅∇+⋅∇∂∂ )()()( 2
2
νρp
t
fuu ⋅∇+∇⋅⋅∇−=∇ ρρ )(2 p
which replaces the incompressibility condition.
(To be continued…)
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