basic equations in fluid dynamics_rev
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Basic Equations in Fluid DynamicsTRANSCRIPT
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Fluid DynamicsFluid DynamicsME 5313 / AE 5313
Basic Equations in Fluid Dynamics
Instructor: Dr. Albert Y. TongDepartment of Mechanical and Aerospace EngineeringDepartment of Mechanical and Aerospace Engineering
The University of Texas at Arlington
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Vector Operations in OrthogonalVector Operations in Orthogonal Coordinate System
Vector operations in orthogonal coordinate systems
Scalar function ),,( 321 xxx
Vector function
U it t i
),,( 321 aaaa
Unit vectors in =
Metric scale factors
321 ,, xxx 321 ,, eee
hhhMetric scale factors 321 ,, hhh
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Vector Operations in OrthogonalVector Operations in Orthogonal Coordinate System
(d l) t
(del) operator:
31 2
1 1 2 2 3 3
ee eh x h x h x
1 1 2 2 3 3h x h x h x
Basic Equations in Fluid Dynamics 3
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Vector Operations in OrthogonalVector Operations in Orthogonal Coordinate System
Cartesian Coordinates (x, y, z)
e1= i e2= j e3= k
h1= 1 h2=1 h3= 1
x x x y
x1= x x2= y x3= z
i j kx y z
Basic Equations in Fluid Dynamics 4
y
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Vector Operations in OrthogonalVector Operations in Orthogonal Coordinate System
Cylindrical coordinates (r,,z)r ze1= e2= e3=
h 1 h h 1h1= 1 h2= r h3= 1
x = r x3= zx2=
ˆˆ ˆ
x1= r x3= zx2=
r zr r z
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Vector Operations in OrthogonalVector Operations in Orthogonal Coordinate System
Spherical coordinates (r, , )
r e1= e2= e3=
h 1 h r h r sin h1= 1 h2= r h3= r sin
x1= r x2= x3=
ˆ ˆ
1 2 x3=
rr r r sin
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Vector Operations in OrthogonalVector Operations in Orthogonal Coordinate System
Gradient
The gradient of or grad is defined asThe gradient of or grad is defined as
grad 321
he
he
he
g
332211 xhxhxh
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Vector Operations in OrthogonalVector Operations in Orthogonal Coordinate System
In Cartesian Coordinates:
kji
z
ky
jx
i
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Vector Operations in OrthogonalVector Operations in Orthogonal Coordinate System
Consider a simple 1-D case:= T = T (x)
zTk
yTj
xTiT
zyx
dTidx
i
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Vector Operations in OrthogonalVector Operations in Orthogonal Coordinate System
Consider a surface (x, y, z) = C
0
dzz
dyy
dxx
d zyx
along = C
0)( rdd
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Vector Operations in OrthogonalVector Operations in Orthogonal Coordinate System
⇒ rd⇒ rd
Therefore, is normal to the f
Csurface C
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Vector Operations in OrthogonalVector Operations in Orthogonal Coordinate System
DivergenceThe divergence of is defined asa
1 2 3 1
1 2 3 1
1 ( )a h h ah h h x
3 1 2 1 2 3( ) ( )h h a h h a
3 1 2 1 2 32 3
( ) ( )x x
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Vector Operations in OrthogonalVector Operations in Orthogonal Coordinate System
Cartesian Coordinates:
31 2 aa aa
e.g.x y z
ua u v wux y z
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Vector Operations in OrthogonalVector Operations in Orthogonal Coordinate System
Potential flow:
u
)()()()(zzyyxx
2
2
2
2
2
22
(Laplacian Operator)
222 zyx 2
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Vector Operations in OrthogonalVector Operations in Orthogonal Coordinate System
Cylindrical coordinates:
2a1 21 3
a1a (ra ) (ra )r r z
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Vector Operations in OrthogonalVector Operations in Orthogonal Coordinate System
Spherical Coordinates:
212
1 ( sin )a r a
12 ( )
sinr r
2 3( sin ) ( )r a ra
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Vector Operations in OrthogonalVector Operations in Orthogonal Coordinate System
CurlaThe curl of is defined by
1 1 2 2 3 3h e h e h e1
1 2 3 1 2 3
1Curl a ah h h x x x
1 1 2 2 3 3h a h a h a
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Vector Operations in OrthogonalVector Operations in Orthogonal Coordinate System
Cartesian Coordinates:
i j k
ax y z
x y z
x y za a a
y
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Vector Operations in OrthogonalVector Operations in Orthogonal Coordinate System
aa ( )yz
aaa iy z
( ) ( )yx xzaa aaj k
z x x y
(Vorticity)u
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Vector Operations in OrthogonalVector Operations in Orthogonal Coordinate System
Velocity field: ( , , )V u v w
V
Deformation field
V
Volume dilatation
V
Rotation
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Vector Operations in OrthogonalVector Operations in Orthogonal Coordinate System
Incompressible0V
Incompressible (constant density)
0V
0V
Irrotational0V Irrotational
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Integral Theorems
Gauss’ Theorem
a ndS adV
S V
a ndS adV It is also known as the divergence theorem.
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Integral Theorems
In Cartesian coordinate system (x,y,z)
x x y y z za n a n a n dS S
yx zaa a dV
yx z
V
a a dVx y z
V
x y zn n , n , n
where
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Integral Theorems
Stokes’ Theorem
( )a dl a n dS
C S
It relates a line integralto a surface integral
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Vector Identities
(i) 0 (i)
(ii)
0
a a a (ii)
(iii)
a a a
a a a
(iii)
(iv)
a a a
a 0
(iv) a 0
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Vector Identities
(v) 1
(v)
( i)
a a a a a a2
2(vi) 2a a a
(vii) a b b a a b
(viii) a b b a a b a b b a
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Eulerian and Langrangian Coordinates
Eulerian coordinates:Eulerian coordinates:
Open system (control volume)p y ( )
Lagrangian coordinates:Closed system (control mass)Closed system (control mass)
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Eulerian Coordinate
Fixed region in space
i.e.x,y,z,t are independent
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Lagrangian Coordinate
F iFocus attention on a particular particle as it movesit moves.
i.e. x,y,z,t are no longer independentindependent
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Material Derivatives
In Eulerian coordinates:T (temperature) is a function of x,y,z,and ti.e. T = T(x,y,z,t)
D T TD T TD t t
D t t
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Material Derivatives
In Lagrangian coordinates:T = T(x,t)
DT T t T DT T t T xDt t t x t
DT T T u Dt t x
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Material Derivatives
In a 3D case, T = T(x,y,z,t)
DT T t T x T y T z
Dt t t x t y t z tDT T T T T
DT T T T Tu v w
Dt t x y z
y
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Material Derivatives
In vector form:
DT T u TDt t
Dt tu iu jv kw where
i j kx y z
u u v wx y z
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Material Derivatives
In tensor form:
kDT T Tu
kk
uDt t x
uk xkk = 1 u xk 2k = 2 v yk = 3 w z
Basic Equations in Fluid Dynamics 34
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Reynolds’ Transport Theorem
D V(t) V
D dV u dVDt t
= any fluid properties(mass)
(mass)(momentum)
u
(energy)e
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Reynolds’ Transport Theorem
Proof:
D 1(t)dV lim (t t)dV (t)dV t 0
V(t) V(t t) V(t)
(t)dV lim (t t)dV (t)dVDt t
1 By adding and subtracting V(t)
1 t t dVt
Basic Equations in Fluid Dynamics 36
( )
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Reynolds’ Transport Theorem
Then
D 1(t)dV lim (t t)dV (t t)dVD
t 0
V(t) V(t t) V(t)
( ) ( ) ( )Dt t
1lim (t t)dV (t)dV t 0
V(t) V(t)
(t t)dV (t)dVt
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Reynolds’ Transport Theorem
Second limit term =V(t)
dVt
First limit term =t 0
( ) ( )
1lim (t t)dV]t
V(t t) V(t)
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Reynolds’ Transport Theorem
S t t
S t t
dS
S t
dS u n
dV u ndS t
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Reynolds’ Transport Theorem
1 =
t 0S(t)
1lim (t t)u ndS tt
( )
= =
S(t)
(t)u ndS
V(t)
(t)u dV
S(t) V(t)
Basic Equations in Fluid Dynamics 40
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Reynolds’ Transport Theorem
Sub. back into the original equation gives
D (t)dV u dVDt t
In tensor notation, it becomes
V(t) V(t)
kk
D (t)dV u dVDt t x
k
V(t) V(t)Dt t x
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Conservation of Mass
Physical Law: matter can neither be created nor destroyeddestroyed
take
D take
D dV 0Dt
V
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Conservation of Mass
Using R T T
kk
D dV u dV 0Dt t x
k
V VDt t x
ku 0t x
kt x
u v w0
or (C ti it E ti )
0t x y z
or (Continuity Equation)
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Conservation of Mass
Special Case: i) uniform constant density
constant
u v w0
t x y z
u v wu v w 0t
t x x y y z z
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Conservation of Mass
, , 0
u v w 0 0
x y z
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Conservation of Mass
u v w 0x y z
x y z
u 0
or u 0 or
Basic Equations in Fluid Dynamics 46
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Conservation of Mass
Special Case: ii) incompressible stratified flowalong a streamline but not uniform throughoutc
D 0D 0Dt
The continuity equation can be written as
0 kk
u 0t x
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Conservation of Mass
ku => kk
k k
uu 0t x x
k
k
uD 0Dt x
Dwhere 0Dt
k
ku0
k
k
u0
x
or u 0
Basic Equations in Fluid Dynamics 48
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Conservation of Momentum
Physical Law: The directional rate of change of t t t l fmomentum = net external force.
tF ma
Two types of force
netF ma
(i) Body force, , e.g. gravityf
(ii) Surface force, , e.g. pressure, shear stressP
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Conservation of Momentum
Rate of Change of MomentumD udVDt
E t l F
V(t)Dt
PdS f dV
External Force
S V
PdS f dV
D udV PdS f dV
V S V
udV PdS f dVDt
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Conservation of Momentum
Basic Equations in Fluid Dynamics 51
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Conservation of Momentum
Stress in the x1 direction = 11 n1 + 21 n2 + 31 n3
Stress in the x2 direction = 12 n1 + 22 n2 + 32 n3
Stress in the x3 direction = 13 n1 + 23 n2 + 33 n3
j ij iP n
D => j ij i jV S V
D u dV n dS f dVDt
V S V
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Conservation of Momentum
Using Reynolds’ Transport Theorem
j j j kD u dV u u u dV
j j j k
kV V
Dt t x
Using Gauss’ Theorem
ij ij
ij ii
S V
n dS dVx
Basic Equations in Fluid Dynamics 53
S V
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Conservation of Momentum
It yields
ijj j k j
k iu u u f dV 0
t x x
Since V is arbitrary, the integrand must vanish
k iV
ijj j k ju u u f
j j k j
k it x x
Basic Equations in Fluid Dynamics 54
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Conservation of Momentum
On the other hand
jj j
uu u
t t t
and
t t t
jkj k j k
uuu u u ux x x
k k kx x x
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Conservation of Momentum
=> j j ijkj k j
u uuu u ft t x x x
k k it t x x x
u But from continuity equation, k
k
u 0t x
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Conservation of Momentum
u u j j ijk j
k i
u uu f
t x x
k it x x
D j ijj
i
Duf
Dt x
iDt x
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Conservation of Energy
First Law of Thermodynamics for a closed system(1)dE dKE Q W
d E KE) d E KE)Q W
dt
E = internal energyKE = kinetic energyKE = kinetic energy Q = heat transfer to the system
Basic Equations in Fluid Dynamics 58
W = work done by the system
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Conservation of Energy
L.H.S. :D 1e u u dVDt 2
R.H.S. : (i)
V(t)Dt 2
Q q n ds R.H.S. : (i)
(ii)s
Q q n ds W u p ds u f dV (ii)
s V
W u p ds u f dV Sign convention:Sign convention:W is positive if work is done by the systemQ is positive if heat is transferred into the system
Basic Equations in Fluid Dynamics 59
system
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Conservation of Energy
D 1 V s V s
D 1e u u dV u PdS u f dV q ndSDt 2
(2)
Using the Reynolds’ Transport Theorem
V s V s
Using the Reynolds Transport Theorem
D 1 1 1dV dV
(3)kk
V V
e u u dV e u u e u u u dVDt 2 t 2 x 2
(3)
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Conservation of Energy
Using Gauss’ Theoremj
jV V
qq n dS q dV dV
x
(4)
js V V
and
j js s
u P dS u P dS
s s
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Conservation of Energy
But j ij iP n j j
j ij i j ij iu P dS u n dS (u )n dS j ij i j ij i
s s s
( ) j ij
iV
u dVx
(5)
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Conservation of Energy
Substituting (3), (4), and (5) into (2) gives:
1 1 dV j j j j k
kV
e u u e u u u dVt 2 x 2
jj ij j j
qu u f dV
x x
j j j j
i jV
x x
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Conservation of Energy
since V is arbitrary, the integrand must vanish
j j j j kk
1 1e u u e u u ut 2 x 2 kt 2 x 2
jqu u f
jj ij j j
i ju u f
x x (6)
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Conservation of Energy
Mechanical Energy part can be removedL H S f (6)L.H.S. of (6):
11 e 1 1
1st term: j j j j j je u u e u u u ut 2 t t t 2 2 t
2nd term: kj j k k
k k k
u1 ee u u u e ux 2 x x
kj j k j j
u1 1u u u u u2 x x 2
Basic Equations in Fluid Dynamics 65
k k2 x x 2
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Conservation of Energy
From Continuity Equation:
kk
u 0t x
k
k
ux t
2nd term becomes:k
k j j k j j
k k
e 1 1e u u u u u ut x 2 t x 2
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Conservation of Energy
L.H.S. of (6)
k j j k j je e 1 1u u u u u ut x t 2 x 2 k kt x t 2 x 2
u u j jk j k j
k k
u ue eu u u ut x t x
(7)
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Conservation of Energy
R.H.S. of (6):
jj ij j j
i j
qu u f
x x
i jx x
ij j jj ij j j
u qu u f
j ij j j
i i ju u f
x x x
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Conservation of Energy
After rearranging, (6) becomes: j j ij j j
k j j k j j j ijk k i i j
u u u qe eu u u u u u ft x t x x x x
j
j j ij j j iju u u u
According to momentum equation:
j j ij j j ijj j k j j j j k j
k i k i
u u u uu u u u u f u u f 0
t x x t x x
j jk ij
k i j
u qe eut x x x
Basic Equations in Fluid Dynamics 69
j
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Remarks
Number of Equation:Continuity 1Momentum 3Momentum 3Energy 1------------------------Total 5
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Remarks
Number of unknowns:e 1uj 3uj 3qj 3σ 9σij 9ρ 1------------------Total 17
Basic Equations in Fluid Dynamics 71
Total 17
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Remarks
Introducing the constitutive equations:1) Fourier’s Law of Heat Conduction2) Newtonian Fluid2) Newtonian Fluid
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Remarks
Fourier’s Law of Heat Conduction
q k T
Newtonian Fluid
jk iij ij ij
uu up
e to a u d
ij ij ijk j i
px x x
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Deformation of Fluid Element
An infinitesimal element of fluid at time t=0 and time t=t
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Deformation of Fluid Element
1) Translation2) Rigid body rotation3) Distortion3) Distortion4) Volumetric dilatation
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Deformation of Fluid Element
∂vδx δt δy
∂x δx. δt δy
C Dx
∂vv
∂v∂x δx v δt δα tan 1
δx
δy∂v∂x δxδt
δα~δyδx
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Deformation of Fluid Element
v vt x
u
similarlyuy
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Deformation of Fluid Element
Rate of rotation (clockwise)
1 1 u v( )2 2 y x
2 2 y x
Rate of shearu v
Rate of shear
y x
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Rate of Deformation Tensor
In a 2-D case
or1 1
1 2ij
2 2
u ux x
eu u
u ux yv v
which can be broken down to1 2x x x y
ij
u v u vu 0 00y x y x1 1xe
ijev v u u v2 20 0 ( ) 0y x y y x
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Rate of Deformation Tensor
where
u 0x
v
u v0y x1
v u2
Volumedilatation Shear
v0y
v u2 0x y
u v0y x1 Rotation
u v2 ( ) 0y x
Rotation
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Rate of Deformation Tensor
In a 3-D case 31 2 11
2 1 3 11
32 2 1 2ij
uu u uu 00 0x x x xx
uu u u u1e 0 0 02
j
2 1 2 3 2
3 3 31 2
3 1 3 2 3
x 2 x x x xu u uu u0 0 0x x x x x
32 1 1
1 2 1 3
uu u u0
x x x x
32 1 2
1 2 2 3
3 31 2
uu u u1 02 x x x x
u uu u0
Basic Equations in Fluid Dynamics 81
1 3 2 30
x x x x
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Constitutive Equations
(i) Stress-strain rate relationship for Isotropic Newtonian floNewtonian flo
jk iij ij ij
uu up
h 0 i j
ij ij ijk j i
px x x
where ij = 0 i ≠ jij = 1 i = j
It is called “Kronecker delta”
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Constitutive Equations
juu u jk i
ij ij ijk j i
uu upx x x
= dynamic viscosity = second viscosity coefficient = dynamic viscosity, = second viscosity coefficient(empirical parameters)
p = thermodynamic pressure
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Constitutive Equations
Sometimes it is written as
whereij ij ijp
jk iij ij
uu ux x x
It is called the viscous stress tensor
j jk j ix x x
It is called the viscous stress tensor
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Constitutive Equations
(ii) Fourier’s Law
q k T
iTq k
or ii
q kx
or
k : thermal conductivity
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Navier-Stokes Equations
Recall Conservation of Momentum:
j j ijk j
k i
u uu f
t x x
(1)k it x x
Using constitutive relation for ij
ij jk iij ij
uu up
ij ij
i i k j ip
x x x x x
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Navier-Stokes Equations
R.H.S. (1st term):
iji
px
Among these onl the one in hich i j is non e o
1 2 31 2 3
j j jp p px x x
Among these, only the one in which i=j is nonzero
pp iji j
px x
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Navier-Stokes Equations
R.H.S. (2nd term):
k kij
u u
ij
i k j kx x x x
Therefore Eq (1) becomes:Therefore, Eq. (1) becomes:
(2)(2)
(Momentum equation)
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Navier-Stokes Equations(i) Incompressible and constant viscosity
0ku 0
kx
j ji iu uu u
i j i i j i ix x x x x x x
2 2j ji u uu
j ji
j i i i i ix x x x x x
Therefore, Eq. (2) becomes:
(3)2
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Navier-Stokes Equations
(ii) Incompressible and inviscid
u u p
Viscous terms vanish, Eq. (2) becomes:
j jk j
k j
u u pu ft x x
(4)
Known as Euler’s Equation
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Energy Equations
j jk ij
u qe eut
(1)jk i jt x x x
Applying constitutive relation for Newtonian Fluids
j j jk iij ij ij
u u uu up
ij ij ij
i k j i i
px x x x x
u u u uu u j j j jk iij ij
i k i j i i
u u u uu upx x x x x x
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Energy Equations
2j j jk k iu u uu u u
j j jk k iij
i k k j i i
px x x x x x
u j k
iji k
u upx x
2j jk i u uu u
wherej jk i
k j i ix x x x
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Governing Equations for Newtonian Fluids
Conservation Equations:
0kut x
kt x
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Governing Equations for Newtonian Fluids
Total number of unknowns = 7e u (3) T p, e, ui(3), T, p
Total number of equations = 5
2 more equations are added:equation of state: p = p( T)equation of state: p = p( ,T)
e.g. p = RTcaloric equation of state: e = e (p, T)
caloric equation of state: e e (p, T)e.g. de = Cv dT
=> All 7 unknowns now can be solved.
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Flow Kinematics
The kinematic relations for a fluid areconcerned only with the space timegeometry of the motion. They areg y yindependent of the dynamics and thethermodynamics of the continuum, andy ,are based on the continuity equation.
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Flow Kinematics
StreamlineStreamlines are lines whose tangents are everywhere parallel to the velocity vector.
For 2-D flows
dy vdx u
u v
dx uu
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Flow Kinematics
If the velocity field is known as a function of x and y (and t if the flow is unsteady) this equation can be(and t if the flow is unsteady), this equation can be integrated to gain the equations of the streamlines.
u xi y j
,u x v y
dy v ydy v ydx u x
dy dx 0dy dxy x
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Flow Kinematics
By integrating both sides
ln ln lny x c xy c
The particular
xy c
streamline that passes (1,1)
1xy c
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Flow Kinematics
PathlinesA pathline is the line traced out by a given particle as it flows from one point to another
Mathematically,
we have
dx 1t 2t 3t 4t( , )i
i idx u x tdt
0t
1t
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Flow Kinematics
StreaklinesA streakline consists of all particles in a flow that have previously passed through a common point
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Flow Kinematics
Work Example
Consider a 2-D plane flow:
11 ( )
1 1x xv u
t t
1 1t t
2 2 ( )v x v y
3 0 ( 0)v w
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Flow Kinematics
For streamlines
/(1 )dy v ydx u x t
/(1 )dx u x t
2 2 2dx v x
1 1 1 /(1 )dx v x t
dy dx (1 )dy dx ty x
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Flow Kinematics
ln (1 ) ln lny t x c ( )y
(1 )ln ln ty c x y (1 )ty cx
( 0)y cx t 2 ( 1)y cx t
Basic Equations in Fluid Dynamics 103
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Flow Kinematics
2y y 2y cxy cxy y
xx
Basic Equations in Fluid Dynamics 104
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Flow Kinematics
For pathlines
1 11 1
dx xvdt t
22 2
dx v xdt
1dt t
1 1(1 )x a t
dt
2 2tx a e 1 1( ) 2 2
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Flow Kinematics
dx x1dt t
d
dy ydt
Combining x1 and x2 gives
1 1 1( ) /2 2
x a ax a e
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Circulation and Vorticity
Definitions:
C
u dl
(i)
u (ii)
ki ijk
u i ijk
jx
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Circulation and Vorticity
ijkif any i,j,k are the same
= if i,j,k is an odd permutation0-1
if i,j,k is an even permutation+1
Basic Equations in Fluid Dynamics 108
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Circulation and Vorticity
u dl u ndA ndA
C A A
u dl u ndA ndA
0 0
( ) 0u
is divergence free (solenoidal)
Basic Equations in Fluid Dynamics 109
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Kinematics of Vortex Tubes
A vortex line is a line whose tangents are everywhere ll l t th ti it t A t t b iparallel to the vorticity vector. A vortex tube is a
region whose side walls are made up of the vortex lineslines
vortex line vortex tube
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Kinematics of Vortex Tubes
Consider unwrapping a vortex tube
vortex line
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Kinematics of Vortex Tubes
ABCDA u dl u dl u dl u dl u dl
ABCDAABCDA A B B C C D D A
u dl u dl u dl u dl u dl
note that u dl u dl
B C D A
dl dl
ABCDAA B C D
u dl u dl
Therefore,
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Kinematics of Vortex Tubes
From Stokes’ theorem
( )u dl u nds
ABCDA Area
Area
nds
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Kinematics of Vortex Tubes
Since vortex lines are tangential to the vortex tube0
0u dl u dl u dl
0n 0ABCDA and
0ABCDA A B C D
u dl u dl u dl
A B C D
u dl u dl
A B D C
u dl u dl
both in the clockwise direction
Basic Equations in Fluid Dynamics 114
A B D C
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Kinematics of Vortex Tubes
However,
and1A B
u dl
2D C
u dl
A B D C
u dl u dl
The circulation is constant over any closed contour
1 2 1 2
u dl u dl or
The circulation is constant over any closed contour about a vortex tube
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Kinematics of Vortex Tubes
u dl ndA
1 1
1
1 1 1
C A
u dl ndA
A
A1 A2
C1 C2
1 1 1A
Similarly, 2 2 2A 2 2 2
1 1 2 2A A and
Analogous to 1 1 2 2V A V A for a stream tube
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Conservative Force Fields
W F dr
C
If W is independent of the path is said to beF
If W is independent of the path, is said to be conservative.
F
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Conservative Force Fields
If W is path independent, then must be an F dr
exact differential and can be written as a gradient
of a scalar function.
F
22.
12 1
1
F
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Conservative Force Fields
dr d
i j kx y z
y
dr i dx j dy k dz
dr dx dy dz dx y z
x y z
Basic Equations in Fluid Dynamics 119
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Kelvin’s Theorem
The vorticity of each fluid particle will bed if th f ll i i tpreserved if the following requirements are
satisfied.
(i) conservative body force field
(ii) inviscid fluid(ii) inviscid fluid(iii) = constant or P = P()
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Kelvin’s Theorem
Proof:Equation of motion,
j j jk iu u uu upu f
k jk j j k i j i
u ft x x x x x x x
F i i id fl id d
jDu P G
For inviscid fluid, and are zero
j
j jDt x x
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Kelvin’s Theorem
By definition,
j jD D u dxDt Dt Dt Dt
( )j jj j
Du D dxdx u
Dt Dt
j jDt Dt
1jDu P Gd d d 1jj j j
j j
P Gdx dx dxDt x x
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Kelvin’s Theorem
( ) ( )j jj j
D dx Dxu u d
j jx xu d u
j ju u dDt Dt
j kk
u d ut x
x jx
0jxt
( )D d
jk j
k
xu u
x
and
( )jj j j
D dxu u du
Dt
Basic Equations in Fluid Dynamics 123
Dt
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Kelvin’s Theorem
Therefore, 1D P G
1
j j j jj j
D P Gdx dx u duDt x x
1dP 1 ( )2 j j
dP dG d u u
0dG 1 ( ) 02 j jd u u and
D dPDt
Basic Equations in Fluid Dynamics 124
Dt
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Kelvin’s Theorem
(i) if = constant1D 1 ( ) 0D dP
Dt
(ii) if P = P() (Barotropic)dP = P’() d
( ) ( ) 0D P d f dDt
0DDt
Basic Equations in Fluid Dynamics 125
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Bernoulli Equation
Consider (i) inviscid fluid(ii) conservative force field(ii) conservative force field
Equation of Motion
j jk
k j j
u u P Gut x x x
f G
where
k j j
( )u u u P G
( )
t
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Bernoulli EquationRecall vector identity,
1
2
a a a a a a
1u u u u u u
2u u u u u u
1 ( )2
u u u ( )
2
1 1u u u u P G
2u u u P G
t
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Bernoulli Equation
1 dPP Note:
1 1dl P dl P
dl P dl P
1 dPdP
1 dPd d dl 1 dPd dP dl
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Bernoulli Equation
1u dP 1
2u dPu u u Gt
12
u dPu u G ut
2t
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Bernoulli Equation
(i) Steady Rotational Flow:
1 ( )2
dPu u u G u u
( ) 0u u
since
1 02
dPu u u G
2
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Bernoulli Equation
Recall ( ) ( ) ( )D uDt t
1( )
2dPu u G
Dt t 2 ( ) 0
for steady flow
1 dP
0t
y
12
dPu u G B
B: Bernoulli constant
along a streamline
B: Bernoulli constant
B can vary from streamline to streamline
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Bernoulli Equation
(ii) Steady Irrotational Flow
1 02
dPu u G
2 1 dPu u G B
2u u G B
B is constant through out the entire flow field
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Bernoulli Equation
(iii) Unsteady Irrotational Flow
1 02
u dPu u Gt
0
0
(irrotational flow)
0u
( ) 0 Recall: ( )
u
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Bernoulli Equation
1( ) 0dPu u G ( ) 0
2u u G
t
1 dP 1 ) 02
dPu u Gt
1 ( )2
dPu u G B tt
B(t) is constant with respect to time through out the flow field
Basic Equations in Fluid Dynamics 134
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Vorticity Equation
Consider: constant
f 0
constant
f 0
N-S Equation:
21( )u u u P ut
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Vorticity Equation
1( )u u u u u ( )
2u u u u u
1 P1 ( )PP
212
u Pu u u ut
2t
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Vorticity Equation
Take the curl on both sides:
12
u u u ut
2p u
2( )
2( )u
t
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Vorticity Equation
where
( ) ( ) ( ) ( ) ( )u u u u u
( ) ( ) ( ) ( )u u
2( ) ( )u u
t
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Vorticity Equation
2-Dimensional plane flow:
x yu (u ,u ,0)
z(0,0, )
0
and
0 2( )u
D
( )ut
2 2( )z zz z z
Dut Dt
or
Basic Equations in Fluid Dynamics 139