6 7 irrotational flow
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Transcript of 6 7 irrotational flow
Irrotational Flow(Potential Flow)
• Bernoulli Equation
• Velocity Potential
• Two-Dimensional, Irrotational, Incompressible Flow
• Elementary Plane Flows
• Superposition of Elementary Plane Flows
Irrotational Flow
Fluid elements moving in the flow field do not undergo any rotation
0 ,0 V
0
y
u
xx
w
z
u
zy
w
0111
rzrr V
rr
rV
rr
V
z
V
z
VV
r
Rectangular coordinate
Cylindrical coordinate
Bernoulli Equation
If the flow is irrotational,
VVkgp
ˆ1
VVVVVV
2
1= 0
2
2
1
2
1ˆ1VVVkgp
rd
dzkdyjdxird ˆˆˆ
zk
yj
xi
ˆˆˆ
2
2
1Vdgdz
dp
0
2
2
gz
Vpd
constant2
2
gzVp valid between any two points
in an irrotational flow
0 V
Example: Flow field with tangential motion
- Forced Vortex rfVVr and 0
Forced Vortex (rigid body rotation)
rrfV
rzz
V
rr
rV
rV
11
2
1
2
1
rrr
r
r 2
2
11
2
1 2
a
b rarb
arbr
arbrba rdrdr
r
ppp 2
02
222
ba rr
Example: Flow field with tangential motion
- Free Vortex rfVVr and 0
Free Vortex (irrotational vortex)
r
r
r
CrfV a
2
rzz
V
rr
rV
rV
11
2
1
2
1
01
2
1
r
C
r
a
b rarb
222 abba VVpp
22
2 11
2 ab rr
C
then If 2 arC 02
2222
babba rrrpp
Velocity Potential
If the flow is irrotational, a potential function, , can be formulated to represent the velocity field.
From the vector identity, 0
The velocity field of an irrotational flow can be defined by a potential function so that
V
zw
yxu
zV
rV
rV zr
1
Stream Function and Velocity Potential
For a two-dimensional, incompressible, irrotational flow, the velocity field can be expressed in terms of both and .
xy
u
yx
u
According to the irrotationality condition,
0
y
u
x0
2
2
2
2
yx
According to the continuity equation,
0
yx
u0
2
2
2
2
yx
Laplace’s equation
Solution of Laplace’s equation represents a possible 2-D, incompressible, irrotational flow field.
Slope of Velocity Potential Line
Along a line of constant , d=0 and
0
dyy
dxx
d
The slope of a line of constant is given by
u
y
x
dx
dy
uuy
x
dx
dy
1
u
u
dx
dy
dx
dy
The slope of a line of constant is given by
Lines of constant and constant are orthogonal.
Example: Determine the Velocity Potential Line of a Flow The flow streamline function is .2axy
yyxxy
u
xz2 022
axy
ayx
ayx
axy
u 2 2
The flow is irrotational.
In term of velocity potential, the velocity components are
ayy
axx
u 2 2
222 2 2 caxxgax
dx
dguxgayay
y
122 2 2 cayyfay
dy
dfyfaxax
xu
caxay 22
Elementary Plane Flows
Uniform Flow
3c2c1c
01c2c3c
3k 2k 1k 01k 2k 3k
U
x
y
x
y
Uu
0
Uy
Uy
u
0
x
Ux
Ux
u
0
y
=0 around any closed curve
Inclined uniform flow
3k2k
1k 01k 2k 3k
x
y cosVu
sinV
xVyV sincos
cosVy
u
sinVx
xVyV cossin
cosVx
u
sinVy
V
x
y
0
3c
2c1c
1c2c
3c
=0 around any closed curve
Source Flow
r
qVr
2
0V
2
q
rq
ln2
x
y
3c2c
1c
04c
5c6c
7c
1k2k
x
y
Origin is singular point
q: volume flow rate per unit depth
=0 around any closed curve
Sink Flow
r
qVr
2
0V
2
q
rq
ln2
x
y
3c2c
1c
04c
5c6c
7c
1k2k
x
y
Origin is singular point
q: volume flow rate per unit depth
=0 around any closed curve
Irrotational Vortex
x
y 3c2c
1c
4c
3k
2k
1k
04k
5k
6k
7k
x
y
r
KV
2
0rV
2
K
rK
ln2
Origin is singular point
K: strength of the vortex
=K around any closed curve enclosing origin
=0 around any closed curve not enclosing origin
Doublet
3c
2c
1c
0
1c
2c
3c
2k
1k 1k
2k
x
y
x
y
sin2r
V
cos2r
Vr r
cos
r
sin
Origin is singular point
: strength of the doublet
=0 around any closed curve
Superposition of Elementary Plane Flows
and satisfy Laplace’s equation.
• Laplace’s equation is linear and homogeneous.
• Solutions of Laplace’s equation may be added together (superposed) to develop more complex flow patterns.
• If 1 and 2 satisfy Laplace’s equation, so does 3 = 1 + 2 .
• Any streamline contour can be imagined to represent a solid surface (there is no flow across a streamline).
Source + Uniform Flow
sin22flow uniformsource Urq
Uyq
cosln2
ln2flow uniformsource Urr
qUxr
q
x
y
U
Stagnation point
Flow past a half body
Solid surface formedby two streamlines
Source + Sink Flow
2121sinksource 222
qqq
1
221sinksource ln
2ln
2ln
2 r
rqr
qr
q
Source and sink with equalstrength, origin separated 2a apart
Source + Sink+Uniform Flow
sin222 2121flow uniformsource Urq
Uyqq
cosln2
ln2
ln2 1
221flow uniformsource Ur
r
rqUxr
qr
q
x
y
U
Stagnation point
(Flow past a Rankine body)Solid surface formedby two streamlines
Stagnation point
Doublet+Uniform Flow
2flow uniformdoublet 1sinsinsinsin
r
UUrUr
rUy
r
2flow uniformdoublet 1coscoscoscos
r
UUrUr
rUx
r
U
Stagnation point
Flow past a cylinderSolid surface formedby two streamlines
Stagnation point
aU
a
Doublet+Vortex+Uniform Flow
rK
r
aUrUrr
K
rln
21sinsinln
2
sin2
2
flow uniformvortexdoublet
21coscos
2
cos2
2
flow uniformvortexdoublet
K
r
aUrUr
K
r
a
Ua
Flow past a cylinderwith circulation
UaUK 44
Sink+Vortex
rKq
ln22vortexsink
2
ln2vortexsink
Kr
q
Vortex Pair1
221vortex2vortex1 ln
2ln
2ln
2 r
rKr
Kr
K
1221vortex2vortex1 222
KKK
Example: Flow past a cylinder
2flow uniformdoublet 1sinr
UUr
2flow uniformdoublet 1cosr
UUr
U
Stagnation point (a,)
Stagnation point (a,0)
a Ua
2
2
2
2
1cos1cosr
aU
r
aUr
rrVr
2
2
2
2
1sin1cos11
r
aU
r
aUr
rrV
Example: Flow past a cylinder- pressure distribution on the surface
U
a
Ua
2
2
1cosr
aUVr
2
2
1sinr
aUV
p
gzV
pgzU
p 22
22
22222 sin4UVVV arrar
= 0
2222222 sin41
2
1sin4
2
1
2
1UUUVUpp
2
2
sin41
21
U
pp
Pressure distribution on the surface of the cylinder
2
21
U
pp
2
2
sin41
21
U
pp
Pressure drag force acting on the cylinder
U
p
ApdFd
a
2
22
12 U
VUpp
2
0
22pressure cossin41
2
1dabUpF
Drag
2
0
322
0
22
0 sin3
4
2
1sin
2
1sin abUabUabp
0
2
0pressure coscos dpabpdAFADrag
Lift force acting on the cylinder
U
p
ApdFd
a
2
0pressure sinsin dpabpdAFALift
2
0
22pressure sinsin41
2
1dabUpF
Drag
2
0
32
2
0
22
0 cos43
cos4
2
1cos
2
1cos abUabUabp
0
Example: Flow past a cylinder with circulation
rK
r
aUr ln
21sin
2
2
flow uniformvortexdoublet
21cos
2
2
flow uniformvortexdoublet
K
r
aUr
a
Ua
UaUK 44
2
2
1cosr
aU
rVr
r
K
r
aU
rV
21sin
12
2
Stagnation points
a
KUV ar
2sin2
4
sin and at 0 1-
Ua
KarV
Flow past a cylinder with circulation- Surface pressure distribution
a
Ua
UaUK 44
a
KUV ar
2sin2
2
222
2sin2
a
KUVVV arrar
01cos2
2
a
aUV arr
2
22
12 U
VUpp
22
2
2
2sin
2sin4
Ua
K
Ua
K
U
V
22
2
2sin
2sin41
2 Ua
K
Ua
KUpp
Pressure distribution on the surface of the cylinder
2
21
U
pp
aUK
aUK 2
aUK 3
22
2
2sin
2sin41
2 Ua
K
Ua
KUpp
Pressure drag force acting on the cylinder
p
2
22
12 U
VUpp
2
0
222
pressure cos2
sin2
sin412
1dab
aU
K
aU
KUpF
Drag
ApdFd
2
0pressure coscos dpabpdAFADrag
a
2
0
22 cossin412
1dabUp
2
0
22 cos
2sin
2
2
1dab
Ua
K
Ua
KU
0
0sin22
sin22
0
22
abUa
K
Ua
Kab
Lift force acting on the cylinder
p
2
22
12 U
VUpp
2
0
222
pressure sin2
sin2
sin412
1dab
aU
K
aU
KUpF
Lift
ApdFd
2
0pressure sinsin dpabpdAFALift
a
2
0
22 sinsin412
1dabUp
2
0
22 sin
2sin
2
2
1dab
Ua
K
Ua
KU
0
bU
KUb
U
KUab
Ua
K
Ua
Kab 2
22
2
2cos
24
sin
2
2 222
0
22
Circulation around on the cylinder
p
2
0
2
0
2
0ˆˆˆˆ ˆˆˆ eerdVeerdVerdeVeV rrrrr
sdV
a
2
0 2sin2 rd
Ur
KU
0
r
KUV
2
sin2
2
2
1cosr
aUVr
KK
Ur
2
0
20 2
cos2
U
KU
b
FLift 2
2
1 2
UUK
Home work: 6.51, 6.66, 6.76, 6.93, 6.96, 6.98