Variation of Flow Parameters in Time & Space - 1 | Fluid Mechanics for Mechanical Engineering PDF Download

Variation of Flow Parameters in Time and Space
Hydrodynamic parameters like pressure and density along with flow velocity may vary from one point to another and also from one instant to another at a fixed point.

According to type of variations, categorizing the flow:

Steady and Unsteady Flow

• Steady Flow
  A steady flow is defined as a flow in which the various hydrodynamic parameters and fluid properties at any point do not change with time.

          In Eulerian approach, a steady flow is described as,


 

 Implications:

1. Velocity and acceleration are functions of space coordinates only.
    
2. In a steady flow, the hydrodynamic parameters may vary with location, but the spatial distribution of these parameters remain invariant with time.

        In the Lagrangian approach,

1. Time is inherent in describing the trajectory of any particle.
    
2. In steady flow, the velocities of all particles passing through any fixed point at different times will be same.
    

3. Describing  velocity as a function of time for a given particle will show the velocities at different points through which the particle has passed providing the information of velocity as a function of spatial location as described by Eulerian method. Therefore, the Eulerian and Lagrangian approaches of describing fluid motion become identical under this situation. 

• Unsteady Flow
  An unsteady Flow is defined as a flow in which the hydrodynamic parameters and fluid properties changes with time. 

Uniform and Non-uniform Flows

• Uniform Flow

The flow is defined as uniform flow when in the flow field the velocity and other hydrodynamic parameters do not change from point to point at any instant of time.

For a uniform flow, the velocity is a function of time only, which can be expressed in Eulerian description as
           

 Implication:

1. For a uniform flow, there will be no spatial distribution of hydrodynamic and other parameters.
    
2. Any hydrodynamic parameter will have a unique value in the entire field, irrespective of whether it 

             changes with time − unsteady uniform flow   OR

             does not change with time − steady uniform flow.

Thus ,steadiness of flow and uniformity of flow does not necessarily go together.

•  Non-Uniform Flow

           When the velocity and other hydrodynamic parameters changes from one point to another the flow is defined as non-uniform.

 Important points:

•  For a uniform flow, there will be no spatial distribution of hydrodynamic and other parameters.
• Any hydrodynamic parameter will have a unique value in the entire field, irrespective of whether it 

             changes with time − unsteady uniform flow   OR

             does not change with time − steady uniform flow.

• Thus ,steadiness of flow and uniformity of flow does not necessarily go together.

•  Non-Uniform Flow

           When the velocity and other hydrodynamic parameters changes from one point to another the flow is defined as non-uniform.

           Important points:

           1. For a non-uniform flow, the changes with position may be found either in the direction of flow or in directions perpendicular to it.

           2.Non-uniformity in a direction perpendicular to the flow is always encountered near solid boundaries past which the fluid flows.

Reason: All fluids possess viscosity which reduces the relative velocity (of the fluid w.r.t. to the wall) to zero at a solid boundary. This is known as no-slip condition.

 Four possible combinations

Material Derivative and Acceleration

• Let the position of a particle at any instant t in a flow field be given by the space coordinates (x, y, z) with respect to a rectangular cartesian frame of reference.

• The velocity components u, v, w of the particle along x, y and z directions respectively can then be written in Eulerian form as
  
  u = u (x, y, z, t)
   v = v (x, y, z, t) 
   w = w (x, y, z, t)

• After an infinitesimal time interval t , let the particle move to a new position given by the coordinates (x + Δx, y +Δy , z + Δz).

• Its velocity components at this new position be u + Δu, v + Δv and w +Δw.

• Expression of velocity components in the Taylor's series form:

The increment in space coordinates can be written as -

Substituting the values of    in above equations, we have
 

•   In the limit      the equation becomes

Material Derivation and Acceleration...contd. from previous slide

• The above equations tell that the operator for total differential with respect to time, D/Dt in a convective field is related to the partial differential as:
  
  
  
  Explanation of equation 7.2 :

• The total differential D/Dt is known as the material or substantial derivative with respect to time.

• The first term  in the right hand side of is known as temporal or local derivative which expresses the rate of change with time, at a fixed position.

• The last three terms in the right hand side of  are together known as convective derivative which represents the time rate of change due to change in position in the field.
   

  Explanation of equation 7.1 (a, b, c):

• The terms in the left hand sides of Eqs (7.1a) to (7.1c) are defined as x, y and z components of substantial or material acceleration.

•  The first terms in the right hand sides of Eqs (7.1a) to (7.1c) represent the respective local or temporalaccelerations, while the other terms are   convective accelerations.
  
  Thus we can write,
  
  (Material or substantial acceleration) = (temporal or local acceleration) + (convective acceleration)

  Important points:

• In a steady flow, the temporal acceleration is zero, since the velocity at any point is invariant with time.
   
• In a uniform flow, on the other hand, the convective acceleration is zero, since the velocity components are not the functions of space coordinates.
   
• In a steady and uniform flow, both the temporal and convective acceleration vanish and hence there exists no material acceleration.

             Existence of the components of acceleration for different types of flow is shown in the table below.
           

• In vector form, Components of Acceleration in Cylindrical Polar Coordinate System ( r, q , z )
   
  Fig 7.1 Velocity Components in a cylindrical Polar Coordinate System
   
• In a cylindrical polar coordinate system (Fig. 7.1 ), the components of acceleration in r, θ and z directions can be written as
  
   

  Explanation of the additional terms appearing in the above equation:   

• The term    appears due to an inward radial acceleration arising from a change in the direction of V_θ ( velocity component in the azimuthal direction ) with θ as shown in Fig. 7.1. This is known as centripetal acceleration.

• The term  V_rV_θ/r represents a component of acceleration in azimuthal direction caused by a change in the direction V_r of with θ

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.