# Irrotational Flow (Chapter 10)

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```					Irrotational Flow (Chapter 10)

Natalie Carroll, Ph.D.
YDAE & ABE

1
Irrotational Flow of an Ideal Fluid
Studied to examine:
1.   Flow around corners
2.   Flow over weirs
3.   Flow through constructions
4.   Flow around airfoils

5.   Flow of water through the earth
• Flow under a dam
• Regional aquifers

2
Irrotational Flow
Assumptions:
1.    Ideal Flow
• No friction between fluid and surface
• No flow perpendicular to the boundary

2. Irrotational Flow
• No rotation or distortion of fluid particles occurs
during movement

3
Flow of an Ideal Fluid

2-D Flow Formulations:
1. Stream Function y

 2y  2y
 2 0
x  2
y

2.   Velocity Potential Function f

 2f  2f
 2 0
x 2
y

4
• G.D.E.s are identical

Dx  Dy  1

G Q0

• B.C.s are different

• Lines of constant y are ^ to lines of constant f

5
Streamline Formulation
• Lines of constant y are Streamlines

• Streamlines are tangent to the velocity:
No flow ^ to a streamline

• Volume Flow Rate between any pair of streamlines
Qij  y i y j
• Velocity Components
y                     y
Vx               Vy  
y                x
6
Streamline Formulation
Assumptions:
– Motion of fluid does not penetrate into surrounding
body or separate from the surface of the body leaving
empty spaces.

– No flow ^ to fixed boundary
\ no velocity ^ to boundary

– Fixed boundaries and lines of symmetry || to flow are
streamlines

7
Example: Flow Around a Cylinder

8
Streamline Formulation
• Flow around a cylinder example:
– 2 lines of symmetry

• Volume flow rate assuming unit thickness:
Qtop ,bottom   5cm / sec 12cm 1cm   60cm / sec
3

• B.C.s:
– Horizontal axis and cylinder boundary (symmetry)
– Top boundary

9
Streamline Formulation

Boundary Conditions:               1
Q
2

Line of Symmetry
10
Streamline Formulation
Boundary Conditions:
Line of Symmetry
^ to flow

No flow ^ streamlines
Vy = 0
11
Streamline Formulation
Left and Right Boundaries:

– Left boundary: Vx is uniform 5 cm/sec (applied as y)
\                       y
Vy        0
x
– Right boundary: Line of symmetry
streamlines must be symmetrical about edge
\                       y
Vy      0
x

12
Derivative B.C.s Review – Chr. 9

13
Derivative B.C.s Review – Chr. 9

• Theoretical boundary condition:
f
Dx      Mf  S
x
f
 Dx    S
x

• Actual boundary condition
f                  S=0
0
x                  M=0

14
Example Mesh: Flow Around a Cylinder
Length – long enough for streamlines to become parallel

15
Corrections to Table 10.1

• Node 8  y = 10.8
• Node 30  x = 9.64
• Node 32  y = 10.9

16
Stream Function (Streamlines) (cm2/s)

17
Streamlines (showing symmetry)

18
Velocity Vx (cm/s)

19
Velocity -Vy (cm/s)

20
Stream Function Review
 2y  2y
 2 0
x  2
y

[KD]        [KG]       [KM]       {fQ}   {fS}
Dx=Dy=1      G=0        M=0        Q=0    S=0

21
Velocity Potential Formulation

• Velocity for an irrotational flow can be written as
the gradient of a scalar potential function.

• Velocity Components
f                     f
Vx                    Vy 
x                     y

22
Velocity Potential Formulation
• Model
– 2 Lines of Symmetry

• B.C.s:
– No velocity ^ flow
• Top Boundary and Horizontal Line of Symmetry:
f
Vy     0
y

• Cylinder:          f
0   (velocity normal to surface)
n
23
Potential Formulation

24
Velocity Potential Formulation

• Left Boundary:
f                M=0
Vx        5cm / sec
x                S = -5

• Right Boundary:
– Streamlines are ^ to line of symmetry
– Potential lines are ^ to streamlines
\ must be potential line

25
Potential Formulation
Arbitrary Value

26
Potential Function (cm2/s)

27
Potential Function (symmetry)

28
Velocity Vx (cm/s)

29
Velocity Vy (cm/s)

30
Plot of Stream and Potential Functions

31
Velocity Potential Review
 2f  2f
 2 0
x 2
y

[KD]       [KG]      [KM]       {fQ}   {fS}
Dx=Dy=1     G=0       M=0        Q=0    S=-Vx

32
B.C.s Irregular Shape
Must include downstream region
until uniform velocity is attained

33
Example: Flow Around a Corner

8 cm

14 cm

Vx = 5 cm/s
6 cm

18 cm

34
Example: Flow Around a Corner
Y formulation         f formulation

35
Groundwater Flow

• G.D.E. Seepage of groundwater under a dam

 2f    2f
Dx 2  Dy 2  0
x     y

• G.D.E. Drawdown at a well removing water from
an aquifer
 2f    2f
Dx 2  Dy 2  Q  0
x     y

36
Seepage Under a Dam
G.D.E.:
 2f    2f                        G=0
Dx 2  Dy 2  0
x     y                          Q=0

f is piezometric head (m) measured from bottom of confined
aquifer

Dx and Dy are the coefficients of permeability (m/day)

37
Seepage Under a Dam

• Boundary Conditions:
– Known values (f) beneath the water
– Zero seepage conditions on other boundaries

• Fluid Velocity Components (Darcy’s Law)
f                 f
Vx   Dx          Vy   Dy
x                 y
38
Seepage Under a Dam

Dx, Dy

Impermeable Layer
39
Seepage Under a Dam

f=a                     f=b

40
Seepage Under a Dam

f=a                     f=b

f
0
n

f
0
n                 41
Seepage Under a Dam

f=a                        f=b

f
f            0
f                    0      n
0              n
n

f
0
n
42
Example: Problem 10.7

1m
30 m

50 m

100 m

Impermeable

43

44

45

46

47
Hydraulic Flux (m/day)

48
Seepage Under a Dam Review
 2f    2f
Dx 2  Dy 2  0
x     y

[KD]     [KG]        [KM]         {fQ}           {fS}
Dx, Dy   G=0         M=0          Q=0            S=0

f
Vx   Dx
x

f
Vy   Dy
y

49
Drawdown at a Well

G.D.E.:
 2f    2f
Dx 2  Dy 2  Q  0
x     y

f is piezometric head (m) measured from bottom of confined
aquifer
Dx and Dy are coefficients of permeability (m/day)
Q represents the well (a point sink)

50
Drawdown at a Well

• Boundary Conditions:
– Known values on all or a part of the boundary
– Seepage of water into aquifer along boundary

 f              f       
  Dx    cos  Dy    sin    S
    x           y       

• Fluid Velocity Components (Darcy’s Law)
f                  f
Vx   Dx           Vy   Dy
x                  y
51
Example: Regional Aquifer

52
Example: Regional Aquifer

53
Example: Regional Aquifer Mesh
Pump Location

54
Example: Regional Aquifer Mesh

55

56

57
Velocity Vx (m/day)

58
Velocity Vy (m/day)

59
Velocity Vmax (m/day)

60
Drawdown at a Well Review
 2f    2f
Dx 2  Dy 2  Q  0
x     y

[KD]     [KG]          [KM]           {fQ}           {fS}
Dx, Dy   G=0          M=0         Q=Q            S = 0,S

f
Vx   Dx
x
f
Vy   Dy
y

61

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