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Computational Fluid Dynamics
http://www.nd.edu/~gtryggva/CFD-Course/!
The
Equations
Governing
Fluid Motion!
Computational Fluid Dynamics
Outline!
Derivation of the equations governing fluid flow in
integral form!
!Conservation of Mass!
!Conservation of Momentum!
!Conservation of Energy!
Differential form!
Summary!
Incompressible flows!
Inviscid compressible flows!
Grétar Tryggvason!
Spring 2011!
Computational Fluid Dynamics
Computational Fluid Dynamics
Conservation of Mass!
In general, mass can be added or removed:!
Conservation!
of!
Mass!
time t+∆t!
time t!
The conservation law must be stated as:!
Final!
Original!
Mass! Mass!
=!
+!
-!
Mass!
Mass!
Added! Removed!
Computational Fluid Dynamics
Conservation of Mass!
Rate of
Net influx
increase of =!
of mass!
mass!
We now apply this statement to
an arbitrary control volume in an
arbitrary flow field!
Computational Fluid Dynamics
Conservation of Mass!
Control
volume V!
The rate of
change of
total mass in
the control
volume is
given by:!
Control
surface S!
∂
ρ dv
∂t ∫V
Rate of
change!
Total
mass!
Computational Fluid Dynamics
Conservation of Mass!
Control
volume V!
Computational Fluid Dynamics
Conservation of Mass!
Select a small rectangle outside the boundary
such that during time ∆t it flows into the CV:!
Control
surface S!
U∆t!
Density ρ!
Velocity Ux!
∆A!
time t! time t+∆t!
To find the mass flux through!
the control surface, letʼs examine!
a small part of the surface where
the velocity is normal to the surface!
where Ux = -u·n!
Computational Fluid Dynamics
Conservation of Mass!
The sign of the normal component of the velocity
determines whether the fluid flows in or out of the
control volume. We will take the outflow to be positive:!
u⋅n<0
Inflow!
The net in-flow through the boundary of the control
volume is therefore:!
Negative since
this is net inflow!
Integral over
the boundary!
Mass flow
through the
boundary
∆A during ∆t
is: !
ρU x ΔtΔA
∫
− ρu⋅ n ds
S
Mass flow normal
to boundary!
Computational Fluid Dynamics
Computational Fluid Dynamics
Conservation of Mass!
Conservation of Mass Equation in Integral Form!
∂
ρ dv = − ∫S ρu ⋅n ds
∂t ∫V
Rate of change
of mass!
Computational Fluid Dynamics
Conservation of momentum!
Momentum per volume: !
Conservation!
of!
momentum!
Net inflow of mass!
ρu
Momentum in control volume: !
∫
V
ρu dv
Rate of
Net influx of
+! Body +! Surface
increase of =!
momentum!
forces!
forces!
momentum!
Computational Fluid Dynamics
Rate of change of momentum!
Computational Fluid Dynamics
In/out flow of momentum!
Control
volume V!
Control
surface S!
The rate of
change of
momentum
in the control
volume is
given by:!
Select a small rectangle outside the boundary
such that during time ∆t it flows into the CV:!
Density ρ!
Ux∆t!
Velocity u!
∆A!
time t! time t+∆t!
∂
ρ u dv
∂t ∫V
Rate of
change!
( ρu )U x ΔtΔA
Ux"
Ux = -u·n so the net influx of momentum
per unit time (divided by ∆t ) is:!
Total
momentum!
Computational Fluid Dynamics
Body force!
Body forces,
such as
gravity act on
the fluid in the
control
volume.!
V
Control
volume V!
The viscous
force is given
by the dot
product of the
normal with
the stress
tensor T!
f!
∫
ρ f dv
Control
surface S!
ds
T•n!
∫ T ⋅ n dS
S
Total body force!
Total stress force!
Computational Fluid Dynamics
Viscous flux of momentum!
Stress Tensor!
T = (− p + λ ∇ ⋅ u )I + 2 µ D
Deformation Tensor!
1
D = (∇u + ∇uT )
2
Whose components are:!
1 ⎛ ∂u ∂u ⎞
Dij = ⎜ i + j ⎟
2 ⎝ ∂x j ∂xi ⎠
− ∫ ρuu ⋅n ds
Computational Fluid Dynamics
Viscous force!
Control
volume V!
dv
Momentum
flow through
the boundary
∆A during ∆t is:!
Computational Fluid Dynamics
Pressure force!
For
incompressible
flows!
∇⋅u=0
And the stress
tensor is!
T = − pI + 2µD
Control
volume V!
The pressure
produces a
normal force
on the control
surface. The
total force is:!
Control
surface S!
ds
pn!
∫
S
pn dS
Total pressure force!
Computational Fluid Dynamics
Conservation of Momentum!
Computational Fluid Dynamics
Gathering the terms!
∂
ρu dv = − ∫ ρuu ⋅ n ds − ∫ Tn ds + ∫ ρ f dv
∂t ∫
Conservation!
of!
energy!
T = − pI + 2µD
Substitute for the stress tensor!
∂
ρu dv = − ∫ ρuu ⋅ n ds − ∫ pn ds + ∫ 2 µD ⋅ n ds + ∫ ρ f dv
∂t ∫
Net inflow of
Rate of
momentum!
change of
momentum!
Total
pressure!
Total
viscous
force!
Total
body
force!
Computational Fluid Dynamics
Conservation of energy!
Computational Fluid Dynamics
The energy equation in integral form!
Net inflow of
kinetic+internal
energy!
Rate of change of
kinetic+internal
energy!
∂
1 ⎞
1 ⎞
⎛
⎛
ρ ⎜ e + u 2 ⎟ dv = − ∫ ρ ⎜ e + u 2 ⎟ u ⋅ n ds
⎝
∂t ∫ ⎝
2 ⎠
2 ⎠
+ ∫ u ⋅ f dv + 
∫ n ⋅ (uT)ds − ∫ n ⋅ q ds
V
Net work
done by
the stress
tensor!
Work
done by
body
forces!
Differential Form!
of!
the Governing Equations!
Net heat
flow!
Computational Fluid Dynamics
Differential form!
Computational Fluid Dynamics
Differential form!
Start with the integral form of the mass conservation equation!
The Divergence or Gauss Theorem can be used to
convert surface integrals to volume integrals!
∫ ∇ ⋅a dv = ∫ a ⋅n ds
V
S
∂
ρ dv = − ∫S ρu ⋅n ds
∂t ∫V
Using Gaussʼs theorem!
∫ ρu ⋅ n ds =∫ ∇ ⋅( ρu)dv
S
V
The mass conservation equations becomes!
∂
ρ dv = − ∫V ∇ ⋅( ρu)dv
∂t ∫V
Computational Fluid Dynamics
Differential form!
Computational Fluid Dynamics
Differential form!
Rearranging!
∂
ρ dv + ∫V ∇ ⋅ (ρu)dv = 0
∂t ∫V
∫
Since the control volume is fixed, the derivative can be
moved under the integral sign!
∫
V
∂ρ
dv + ∫V ∇ ⋅( ρu)dv = 0
∂t
V
⎛ ∂ρ
+ ∇ ⋅ (ρu)⎞ dv = 0
⎝ ∂t
⎠
This equation must hold for ANY control volume, no
matte what shape and size. Therefore the integrand
must be equal to zero!
∂ρ
+ ∇ ⋅( ρu) = 0
∂t
Rearranging!
∫
V
⎛ ∂ρ
+ ∇ ⋅ (ρu)⎞ dv = 0
⎝ ∂t
⎠
Computational Fluid Dynamics
Differential form!
The differential form of the momentum equation is
derived in the same way:!
Start with!
Expanding the divergence term:!
∇ ⋅ (ρu) = u ⋅ ∇ρ + ρ∇ ⋅ u
The mass conservation equation becomes:!
∂ρ
∂ρ
+ ∇ ⋅ (ρu) =
+ u ⋅ ∇ ρ + ρ∇ ⋅ u
∂t
∂t
or!
where!
Dρ
+ ρ∇ ⋅ u = 0
Dt
D() ∂ ()
=
+ u ⋅ ∇()
Dt ∂t
Computational Fluid Dynamics
Differential form!
∂
∂t
Rewrite as:!
∂
∂t
∫ ρu dv = ∫ ρ f dv + ∫ ∇ ⋅(T − ρuu )dv
To get:!
∂ρu
= ρ f + ∇ ⋅(T − ρuu)
∂t
Convective
derivative!
Computational Fluid Dynamics
Differential form!
Using the mass conservation equation the
advection part can be rewritten:!
∂ρu
+ ∇ ⋅ ρuu =
∂t
∂ρ
∂u
u
+ u∇ ⋅ uρ + ρ
+ ρu ⋅ ∇u =
∂t
∂t
⎛ ∂ρ
⎞
Du
Du
u ⎜ + ∇ ⋅ uρ ⎟ + ρ
=ρ
⎝ ∂t
⎠
Dt
Dt
=0, by mass
conservation!
∫ ρu dv = ∫ ρ f dv + ∫ (nT − ρu(u ⋅ n))ds
Computational Fluid Dynamics
Differential form!
The momentum equation equation!
∂ρu
= ρ f + ∇ ⋅ (T − ρuu)
∂t
Can therefore be rewritten as!
ρ
where!
ρ
Du
= ρ f + ∇⋅ T
Dt
⎛ ∂u
⎞
Du
= ρ⎜ + u ⋅ ∇u ⎟
⎝ ∂t
⎠
Dt
Computational Fluid Dynamics
Differential form!
Computational Fluid Dynamics
Differential form!
The “mechanical energy equation” is obtained by taking
the dot product of the momentum equation and the
velocity:!
The energy equation equation can
be converted to a differential form in
the same way. It is usually simplified
by subtracting the “mechanical
energy” !
⎛ ∂u
u⋅ ρ
+ ρ u ⋅ ∇u = ρ f + ∇ ⋅ T⎞
⎝ ∂t
⎠
The result is:!
ρ
⎛ u2 ⎞
∂ ⎛⎜ u 2 ⎞⎟
= − ρ u ⋅ ∇ ⎜ ⎟ + ρ u ⋅ fb + u ⋅ (∇ ⋅ T)
∂t ⎝ 2 ⎠
⎝2⎠
The “mechanical energy equation”!
Computational Fluid Dynamics
Differential form!
Computational Fluid Dynamics
Differential form!
The final result is!
Subtract the “mechanical energy equation” from the
energy equation!
ρ
The final result is!
ρ
De
− T⋅ ∇u + ∇ ⋅ q = 0
Dt
where!
Using the constitutive relation presented earlier for
the stress tensor and Fourieʼs law for the heat
conduction:!
De
+ p∇ ⋅ u = Φ + ∇ ⋅ k∇T
Dt
Φ = λ(∇ ⋅ u) 2 + 2µD⋅ D
is the dissipation function and is the rate at which
work is converted into heat.!
T = (− p + λ ∇ ⋅ u )I + 2 µ D
Generally we also need:!
p = p(e, ρ); T = T (e, ρ);
q = −k∇T
and equations for ! µ,
Computational Fluid Dynamics
λ, k
Computational Fluid Dynamics
Integral Form!
Summary!
of!
governing equations!
∂
∂t
∫
V
ρ dv = − ∫ ρu ⋅ n ds
S
∂
ρu dv = ∫ ρ f dv + ∫ (nT − ρu(u ⋅ n))ds
∂t ∫
∂
1
1
ρ(e + u 2 )dv = ∫ u ⋅ ρ f dv + ∫ n ⋅ (uT − ρ(e + u 2 ) − q)ds
∂t ∫
2
2
Computational Fluid Dynamics
Computational Fluid Dynamics
Conservative Form!
Convective Form!
∂ρ
+ ∇ ⋅ ( ρu) = 0
∂t
Dρ
+ ρ∇ ⋅ u = 0
Dt
∂ρu
= ρ f + ∇ ⋅ (T − ρuu)
∂t
∂
1
ρ(e + u 2 )=
∂t
2
ρ
Du
= ρ f + ∇ ⋅T
Dt
ρ
De
= T∇ ⋅ u − ∇ ⋅q
Dt
Special Cases!
Compressible inviscid flows!
Incompressible flows!
!Stokes flow!
Not in this course!!
!Potential flows!
⎛
⎞
1
∇ ⋅ ⎜ ρ (e + u 2 )u − uT + q ⎟
⎝
⎠
2
Computational Fluid Dynamics
Computational Fluid Dynamics
Inviscid, compressible flows!
µ = 0 and λ = 0
T = (− p + λ ∇ ⋅ u )I + 2 µ D
Inviscid compressible
flows!
T = − pI
Dρ
+ ρ∇ ⋅ u = 0
Dt
Du
ρ
= ρ f − ∇p
Dt
De
ρ
= − p∇ ⋅ u
Dt
Computational Fluid Dynamics
Computational Fluid Dynamics
∂ U ∂ E ∂ F ∂G
+
+
+
=0
∂t ∂ x ∂ y ∂ z
⎛
⎜
⎜
⎜
U=⎜
⎜
⎜
⎜
⎜
⎝
ρ
ρu
ρv
ρw
ρE
⎛
⎞
⎜
⎟
⎜
⎟
⎜
⎟
⎟ E=⎜
⎜
⎟
⎜
⎟
⎜
⎟
⎜
⎟
⎝
⎠
ρu
ρu 2 + p
ρuv
ρuw
( ρE + p ) u
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
⎛
⎜
⎜
⎜
F=⎜
⎜
⎜
⎜
⎜
⎝
ρv
ρuv
ρv 2 + p
ρvw
( ρE + p ) v
⎞
⎛
⎟
⎜
⎟
⎜
⎟
⎜
⎟ G=⎜
⎟
⎜
⎟
⎜
⎟
⎜
⎟
⎜
⎠
⎝
p = ρRT; e = c v T; h = c p T; γ = c p /c v
cv =
R
(γ −1)e
; p = (γ − 1)ρe; T =
;
γ −1
R
⎞
⎟
⎟
⎟
⎟
ρvw
⎟
ρw 2 + p ⎟⎟
( ρE + p ) u ⎟⎠
ρu
ρuw
Incompressible flow!
Computational Fluid Dynamics
Incompressible flows:!
Dρ
+ ρ∇ ⋅ u = 0
Dt
Computational Fluid Dynamics
Objectives:!
The 2D Navier-Stokes Equations for incompressible, !
homogeneous flow:!
Dρ
=0
Dt
∇⋅u=0
Navier-Stokes equations (conservation of momentum)!
∂
ρu = −∇⋅ ρuu- ∇P + ρ fb + ∇ ⋅ µ (∇u + ∇T u)
∂t
For constant viscosity!
∂
ρ u + ρu ⋅ ∇u = -∇P + ρ fb + µ∇ 2u
∂t
∂u
∂u
∂u
1 ∂P µ ⎛⎜ ∂ 2 u ∂ 2 u ⎞
+u +v = −
+
+
∂t
∂x
∂y
ρ ∂x ρ ⎝ ∂x 2 ∂y 2 ⎠
∂v
∂v
∂v
1 ∂P µ ⎛⎜ ∂ 2 v ∂ 2 v ⎞
+u +v = −
+
+
∂t
∂x
∂y
ρ ∂y ρ ⎝ ∂x 2 ∂y 2 ⎠
∂u ∂v
+
=0
∂x ∂y
Computational Fluid Dynamics
Computational Fluid Dynamics
Pressure and Advection!
Incompressibility (conservation of mass)!
∫ u ⋅ n ds = 0
Advection:!
!Newtonʼs law of motion: In the absence of forces,
a fluid particle will move in a straight line!
S
Navier-Stokes equations (conservation of momentum)!
∂
udV = − ∫ uu⋅ ndS + ν ∫ ∇u ⋅ ndS − 1ρ ∫ pn x dS
S
S
S
∂t ∫V
∂
vdV = − ∫Svu ⋅ndS + ν ∫S∇v ⋅ndS − ρ1 ∫S pny dS
∂t ∫V
The role of pressure:!
Needed to accelerate/decelerate a fluid particle:
Easy to use if viscous forces are small!
Needed to prevent accumulation/depletion of fluid
particles: Use if there are strong viscous forces!
Computational Fluid Dynamics
Computational Fluid Dynamics
Pressure!
Pressure!
The pressure opposes local accumulation of fluid. For
compressible flow, the pressure increases if the
density increases. For incompressible flows, the
pressure takes on whatever value necessary to
prevent local accumulation:!
Increasing pressure slows the fluid down and
decreasing pressure accelerates it!
Pressure
increases!
High Pressure!
∇⋅u < 0
Low Pressure!
∇⋅u > 0
Slowing down!
Pressure
decreases!
Speeding up!
Computational Fluid Dynamics
Computational Fluid Dynamics
Viscosity!
Diffusion of fluid momentum is the result of friction
between fluid particles moving at uneven speed. The
velocity of fluid particles initially moving with different
velocities will gradually become the same. Due to
friction, more and more of the fluid next to a solid wall
will move with the wall velocity.!
Computational Fluid Dynamics
Nondimensional Numbers!
∂u
∂u
∂u
1 ∂P µ ⎛⎜ ∂ 2 u ∂ 2 u ⎞
+u +v = −
+
+
∂t
∂x
∂y
ρ ∂x ρ ⎝ ∂x 2 ∂y 2 ⎠
u,v
x, y ˜ Ut
u˜,v˜ =
; x˜, y˜ =
; t=
U
L
L
U 2 ⎜⎛ ∂u˜
∂u˜
∂u˜ ⎞
1 ∂P U µ ⎛⎜ ∂ 2 u˜ ∂ 2 u˜ ⎞⎟
+ u˜ + v˜
=−
+
+
L ⎝ ∂˜t
∂x˜
∂y˜ ⎠
Lρ ∂x˜ L2 ρ ⎝ ∂x˜ 2 ∂y˜ 2 ⎠
∂u˜
∂u˜
∂u˜
L 1 ∂P L U µ ⎛⎜ ∂ 2 u˜ ∂ 2 u˜ ⎞⎟
+ u˜ + v˜ = − 2
+
+
∂˜t
∂x˜
∂y˜
U L ρ ∂x˜ U 2 L2 ρ ⎝ ∂x˜ 2 ∂y˜ 2 ⎠
Computational Fluid Dynamics
Nondimensional Numbers!
The momentum equations are:!
∂u˜
∂u˜
∂u˜
∂P˜ 1 ⎛⎜ ∂ 2 u˜ ∂ 2 u˜ ⎞⎟
+ u˜ + v˜ = −
+
+
∂˜t
∂x˜
∂y˜
∂x˜ Re ⎝ ∂x˜ 2 ∂y˜ 2 ⎠
∂v˜
∂v˜
∂v˜
∂P˜ 1 ⎛⎜ ∂ 2 v˜ ∂ 2v˜ ⎞⎟
+ u˜ + v˜ = −
+
+
∂˜t
∂x˜
∂y˜
∂y˜ Re ⎝ ∂x˜ 2 ∂y˜ 2 ⎠
The continuity equation is unchanged!
∂ u ∂ v
+
=0
∂ x ∂ y
Nondimensional
Numbers—the Reynolds
number!
Computational Fluid Dynamics
Nondimensional Numbers!
∂u˜
∂u˜
∂u˜
L 1 ∂P L U µ ⎛⎜ ∂ 2 u˜ ∂ 2 u˜ ⎞⎟
+ u˜ + v˜ = − 2
+
+
˜
∂t
∂x˜
∂y˜
U L ρ ∂x˜ U 2 L2 ρ ⎝ ∂x˜ 2 ∂y˜ 2 ⎠
L 1 ∂P
1 ∂P ∂P˜
=
=
2
U L ρ ∂x˜ ρU 2 ∂x˜ ∂x˜
L U µ
µ
1
=
=
2 2
U L ρ ULρ Re
where!
where!
P
P˜ =
ρU 2
Re =
ρUL
µ
Computational Fluid Dynamics
Outline!
Derivation of the equations governing fluid flow in
integral form!
!Conservation of Mass!
!Conservation of Momentum!
!Conservation of Energy!
Differential form!
Summary!
Incompressible flows!
Inviscid compressible flows!