Variation of Flow Parameters in Time & Space - 2 | Fluid Mechanics for Mechanical Engineering PDF Download

Variation of Flow Parameters in Time and Space

Streamlines

      Definition: Streamlines are the Geometrical representation of the of the flow velocity.  

      Description:

•  In the Eulerian method, the velocity vector is defined as a function of time and space coordinates.

•  If for a fixed instant of time, a space curve is drawn so that it is tangent everywhere to the velocityvector, then this curve is called a Streamline. 
   

          Therefore, the Eulerian method gives a series of instantaneous streamlines of the state of motion (Fig. 7.2a).
                      
              
                Fig 7.2a    Streamlines
 

Alternative Definition:

A streamline at any instant can be defined as an imaginary curve or line in the flow field so that the tangent to the curve at any point represents the direction of the instantaneous velocity at that point.

       Comments:

• In an unsteady flow where the velocity vector changes with time, the pattern of streamlines also changes from instant to instant.

• In a steady flow, the orientation or the pattern of streamlines will be fixed.

From the above definition of streamline, it can be written as

            (7.3)   


Description of the terms:

        1.  is the length of an infinitesimal line segment along a streamline at a point .

        2.is the instantaneous velocity vector.

The above expression therefore represents the differential equation of a streamline. In a cartesian coordinate-system, representing

the above equation ( Equation 7.3 ) may be simplified as

Stream tube:

A bundle of neighboring streamlines may be imagined to form a passage through which the fluid flows. This passage is known as a stream-tube.

      Fig 7.2b    Stream Tube

        

           Properties of Stream tube:

       1. The stream-tube is bounded on all sides by streamlines.

       2. Fluid velocity does not exist across a streamline, no fluid may enter or leave a stream-tube except through its ends.

       3. The entire flow in a flow field may be imagined to be composed of flows through stream-tubes arranged in some arbitrary positions.

Path Lines

        Definition:  A path line is the trajectory of a fluid particle of fixed identity as defined by Eq. (6.1).


                       Fig 7.3    Path lines

A family of path lines represents the trajectories of different particles, say, P₁, P ₂, P₃, etc. (Fig. 7.3).

       Differences between Path Line and Stream Line

Note: In a steady flow path lines are identical to streamlines  as the Eulerian and Lagrangian versions become the same

 

Streak Lines

Definition: A streak line is the locus of the temporary locations of all particles that have passed though a fixed point in the flow field at any instant of time.

      Features of a Streak Line:

• While a path line refers to the identity of a fluid particle, a streak line is specified by a fixed point in the flow field.

• It is of particular interest in experimental flow visualization.

• Example:  If dye is injected into a liquid at a fixed point in the flow field, then at a later time t, the dye will indicate the end points of the path lines of particles which have passed through the injection point.

• The equation of a streak line at time t can be derived by the Lagrangian method.
  
  If a fluid particle      passes through a fixed point    in course of  time t, then the Lagrangian method of description gives the equation
  
          (7.5)
  
  Solving for  ,
  
                (7.6)
  
  If the positions  of the particles which have passed through the fixed point  are determined, then a streak line can be drawn through these points
  
  Equation: The equation of the streak line at a time t is given by

      (7.7)

  Substituting Eq. (7.5) into Eq. (7.6) we get the final form of equation of the streak line,
           (7.8)
  
  
  Difference between Streak Line and Path Line
  
  Fig 7.4    Description of a Streak line

  
  Above diagram can be described by the following points:

  Describing a Path Line:

  a)  Assume P be a fixed point in space through which particles of different identities pass at different times.

  b) In an unsteady flow, the velocity vector at P will change with time and hence the particles arriving at P at different times will traverse

       different paths like PAQ,  PBR and PCS which represent the path lines of the particle.

  Describing a Streak Line:

  a) Let at any instant these particles arrive at points Q, R and S.

  b) Q, R and S represent the end points of the trajectories of these three particles at the instant.

  c) The curve joining the points S, R, Q and the fixed point P will define the streak line at that instant.

   d) The fixed point P will also lie on the line, since at any instant, there will be always a particle of some identity at that point.

            
  Above points show the differences.

     Similarities:

      a) For a steady flow, the velocity vector at any point is invariant with time

      b) The path lines of the particles with different identities passing through P at different times will not differ

      c) The path line would coincide with one another in a single curve which will indicate the streak line too.

        Conclusion: Therefore, in a steady flow, the path lines, streak lines and streamlines are identical.