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									            Point Velocity Measurements




                   P M V Subbarao
                      Professor
          Mechanical Engineering Department


Velocity Distribution is a Fundamental Symptom
         of Solid –Fluid Interactions….
               Point Velocity Measurement


• Pitot Probe Anemometry : Potential Flow Theory .
• Thermal Anemometry : Newton’s Law of Cooling.
• Laser Anemometry: Doppler Theory.
                POTENTIAL FLOW THEORY
• Ideal flow past any unknown object can be represented as a complex
  potential.
• In particular we define the complex potential

                          W    i

   In the complex (Argand-Gauss) plane every point is associated
   with a complex number
                                           i
                       z  x  iy  re
In general we can then write

                W   z   i z   f z 
   Now, if the function f is analytic, this implies that it is also
   differentiable, meaning that the limit

                 df       f z  z   f z 
                     lim
                 dz z 0         z
                          dW
                               u  iv
                           dz
so that the derivative of the complex potential W in the complex z plane
gives the complex conjugate of the velocity.
Thus, knowledge of the complex potential as a complex function of z
leads to the velocity field through a simple derivative.
   ELEMENTARY IRROTATIONAL PLANE
               FLOWS

• The uniform flow
• The source and the sink
                     THE UNIFORM FLOW
The first and simplest example is that of a uniform flow with velocity U
directed along the x axis.

   In this case the complex potential is


       W    i  Uz


and the streamlines are all parallel to the velocity direction (which is the
                                  x axis).
Equi-potential lines are obviously parallel to the y axis.
                      THE SOURCE OR SINK
   • source (or sink), the complex potential of which is

                                 m
                    W    i     ln z
                                 2
• This is a pure radial flow, in which all the streamlines converge at
  the origin, where there is a singularity due to the fact that continuity
  can not be satisfied.
• At the origin there is a source, m > 0 or sink, m < 0 of fluid.
• Traversing any closed line that does not include the origin, the mass
  flux (and then the discharge) is always zero.
• On the contrary, following any closed line that includes the origin
  the discharge is always nonzero and equal to m.
                                    Iso  lines


                                     Iso  lines



The flow field is uniquely determined upon deriving the complex
potential W with respect to z.

                           m
              W    i     ln z
                           2
      Stream and source: Rankine half-body
• It is the superposition of a uniform stream of constant speed U
  and a source of strength m.




2D Rankine half-body:

                              m
            W    i  Uz     ln z
                              2
                        m
      W    i  Uz     ln z
                        2
                          i
     Z  x  iy  re
           x  r cos 
           y  r sin 
                 y
          tan  
                 1

                x
                 i
W    i  Ure 
                   m
                   2
                           
                      ln re i
                         ln r  
                      m            m
    i  Ux  iUy                  i
                      2           2
                                   m 
                   ln r   iUy 
                m
    i  Ux                        
                2                 2 

                    m 
              Uy   
                    2 

Shape of Zero Value Stream line
                           m 
            0  Ur sin     
                           2 
2D Rankine half-body:



                  y




                               m     1  y  
                      0  Uy 
                                  tan   
                               2         x 
                                          
3D Rankine half-body:
                                           m
                          Ux 
                                   4 x 2  y 2  z 2
        Pitot Probe Anemometry : Henri Pitot in 1732
• Theory
• A constant-density fluid flowing steadily without friction
  through the simple device.
• No heat being added and no shaft work being produced by the
  fluid.
• A simple expression can be developed to describe this flow:




                u12                   u2
                                         2
         p1            gz1  p2           gz2
                 2                      2
   Apply Bernoulli’s equation along the central streamline from a
   point upstream where the velocity is u1 and the pressure p1 to the
   stagnation point of the blunt body where the velocity is zero, u2
   = 0. Also z1 = z2.
                                          1    2

This increase in pressure which bring
the fluid to rest is called the dynamic
pressure.
                2
            u
  p2  p1      1
             2
                               Dynamic pressure =

           2                    or converting this to head
  p1 u    p2
         1
  g 2 g g                        Dynamic head =
   The total pressure is know as the stagnation pressure (or total
   pressure)
                                                2
                                            u
          Stagnation pressure =   p2  p1      1
                                             2
                                                        2
                                               p1 u
 or in terms of head Stagnation head =                 1
                                               g 2 g

The blunt body stopping the fluid does not have to be a solid.
It could be a static column of fluid.
Two piezometers, one as normal and one as a Pitot tube within the
pipe can be used in an arrangement to measure velocity of flow.
Using the above theory, we have the equation for p2 ,

   p2  p1 
             u12             u  2gh2  h1 
              2             We now have an expression for
         2                  velocity obtained from two pressure
p1 u    p2
       1                  measurements and the application of
g 2 g g                   the Bernoulli equation.
                     Pitot Static Tube
• The necessity of two piezometers
  and thus two readings make this
  arrangement is a little awkward.
• Connecting the piezometers to a
  manometer would simplify things
  but there are still two tubes.
• The Pitot static tube combines the
  tubes and they can then be easily
  connected to a manometer.
• A Pitot static tube is shown below.
• The holes on the side of the tube
  connect to one side of a
  manometer and register the static
  head, (h1), while the central hole
  is connected to the other side of
  the manometer to register, as          A Pitot-static tube
  before, the stagnation head (h2).
   Consider the pressures on the level of the centre line of the
   Pitot tube and using the theory of the manometer,


        p A  p2  gX


    pB  p1  g  X  h    man gh

                   p A  pB
                                                                       X

p2  gX  p1  g  X  h    man gh
                                                                   h
                                                          B            A


                                              2
                                          u
  We know that      p2  pstag    p1       1
                                           2
                                               2
          p1  gh man     p1  
                                       u       1
                                        2


                       2 gh man   
              u1 
                               
The Pitot/Pitot-static tubes give velocities at points in the flow.
It does not give the overall discharge of the stream, which is
often what is wanted.
It also has the drawback that it is liable to block easily,
particularly if there is significant debris in the flow.
                Compressible-Flow Pitot Tube
For an ideal compressible flow coming to rest from finite
velocity:
                           2
                        V
                     h     h0
                         2
    For perfect gas :              2
                               V
                        c pT      c pT0
                                2
                 2
                  c p T0  T 
              V
               2
This process of ideal compressible flow coming to rest
is regarded as isentropic process.

          Tds  dh  vdp  0
 For perfect gas :    c p dT  vdp

                              RT
For ideal gas :      c p dT     dp
                               p
   dT dp
       
 1 T   p

            
p0  T0   1
   
p T 
                Compressible-Flow Pitot Tube
Subsonic pitot tube :
A pitot tube in subsonic flow measures the local total pressure po
together with a measurement of the static pressure p
                                      
                                          
    2
                               p0     
                                           
       c p T0  T   c pT    1
  V
   2                          p 
                                          
                                           
                         

     p0     V 2   1
                                              M 
                                                   V
        1      
     p  2c pT                                    RT
                 
                                                      1
                                                             
   p0    1 2   1                    2   p0       

      1    M                    M 
                                     2
                                                           1
   p       2                            1  p            
                                                              
The pitot-static combination therefore constitutes a Mach meter .

With M2 known, we can then also determine the dynamic
pressure.                             1
                                           
                   V 2
                           2  p0   
            M2                        1
                  p       1  p        
                                          
The velocity can be determined
from




                         1
                               
         2p  p0  
  V2                       1
          1  p          
                               
                            Supersonic pitot tube
• A pitot probe in a supersonic stream will have a bow shock
  ahead of it.
• This complicates the flow measurement, since the bow shock
  will cause a drop in the total pressure, from po1 to po2 , the latter
  being sensed by the pitot port.
• It’s useful to note that the shock will also cause a drop in o, but
  ho will not change.
• The pressures and Mach number immediately behind the shock
  are related by

                                           
                p02    1 2   1
                    1    M2 
                p2       2    
                       Normal Shock Relations
                                                           
                                   p02    1 2   1
                                       1    M2 
                                   p2       2    


                                             (  1)M1  2
                                                       2
                                           
                                       2
• Mach number relation            M2
                                             2M1    1
                                                  2
• Static pressure jump relation

                                   p2 2M    1
                                                   2
                                                 1
                                   p1     1
Mach Number Range

								
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