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


• Adaptable to 3-D problems

                                                        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
Piezometric Head (m)




                       44
Hydraulic Gradient in x-direction (m/m)




                                          45
Hydraulic Gradient in y-direction (m/m)




                                          46
Hydraulic Gradient (m/m)




                           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
Example: Piezometric Head Contours




                                     56
Piezometric Head (m)




                       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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