One, Two & Three Dimensional Flows | Fluid Mechanics for Mechanical Engineering PDF Download

One, Two and Three Dimensional Flows
 

• Fluid flow is three-dimensional in nature. 
   
• This means that the flow parameters like velocity, pressure and so on vary in all the three coordinate directions.

     Sometimes simplification is made in the analysis of different fluid flow problems by:

• Selecting the appropriate coordinate directions so that appreciable variation of the hydro dynamic parameters take place in only two directions or even in only one.

        One-dimensional flow

•  All the flow parameters may be expressed as functions of time and one space coordinate only.
•  The single space coordinate is usually the distance measured along the centre-line (not necessarily straight)  in which the fluid is flowing.
•  Example: the flow in a pipe is considered one-dimensional when variations of pressure and velocity occur along the length of the pipe, but any variation over the cross-section is assumed negligible.
•  In reality, flow is never one-dimensional because viscosity causes the velocity to decrease to zero at the solid boundaries.
•  If however, the non uniformity of the actual flow is not too great, valuable results may often be obtained from a "one dimensional analysis".
•  The average values of the flow parameters at any given section (perpendicular to the flow) are assumed to be applied to the entire flow at that section.

        Two-dimensional flow

•  All the flow parameters are functions of time and two space coordinates (say x and y).
   
•  No variation in z direction.
   
• The same streamline patterns are found in all planes perpendicular to z direction at any instant.

        Three dimensional flow

• The hydrodynamic parameters are functions of three space coordinates and time.
   

Translation of a Fluid Element
The movement of a fluid element in space has three distinct features simultaneously.

• Translation
• Rate of deformation
• Rotation. 

Figure 7.4 shows the picture of a pure translation in absence of rotation and deformation of a fluid element in a two-dimensional flow described by a rectangular cartesian coordinate system.

       In absence of deformation and rotation,

      a)  There will be no change in the length of the sides of the fluid element.

      b) There will be no change in the included angles made by the sides of the fluid element.

      c) The sides are displaced in parallel direction.

This is possible when the flow velocities u (the x component velocity) and v (the y component velocity) are neither a function of x nor of y, i.e., the flow field is totally uniform.

Fig 8.1  Fluid Element in pure translation

If a component of flow velocity becomes the function of only one space coordinate along which that velocity component is defined.

        For example,

• if  u = u(x) and v = v(y), the fluid element ABCD  suffers a change in its linear dimensions along with translation
• there is no change in the included angle by the sides as shown in Fig. 7.5.

            
Fig 8.2  Fluid Element in Translation with Continuous Linear Deformation

         
The relative displacement of point B with respect to point A per unit time in x direction is

Similarly, the relative displacement of D with respect to A per unit time in y direction is

Translation with Linear Deformations

         Observations from the figure:

       Since u is not a function of y and v is not a function of x

• All points on the linear element AD move with same velocity in the x direction.
   
• All points on the linear element AB move with the same velocity in y direction.
   
• Hence the sides move parallel from their initial position without changing the included angle.

       This situation is referred to as translation with linear deformation.

       Strain rate:

The changes in lengths along the coordinate axes per unit time per unit original lengths are defined as thecomponents of linear deformation or strain rate in the respective directions.

Therefore, linear strain rate component in the x direction

and, linear strain rate component in y direction

Rate of Deformation in the Fluid Element
Let us consider both the velocity component u and v are functions of x and y, i.e.,

u = u(x,y)
 v = v(x,y)
Figure 8.3 represent the above condition

Observations from the figure:
Point B has a relative displacement in y direction with respect to the point A.        

• Point D has a relative displacement in x direction with respect to point A.
• The included angle between AB and AD changes.
• The fluid element suffers a continuous angular deformation along with the linear deformations in course of its motion.

Rate of Angular deformation:
The rate of angular deformation is defined as the rate of change of angle between the linear segments AB and AD which were initially perpendicular to each other.

Fig 8.3   Fluid element in translation with simultaneous linear and angular deformation rates

From the above figure rate of angular deformation,
     (8.1)

From the geometry
     
           (8.2a)


   (8.2b)

Hence,

    (8.3)

Finally  

       (8.4)

 

Rotation

           Figure 8.3 represent the situation of rotation

          Observations from the figure:

• The transverse displacement of B with respect to A and the lateral displacement of D with respect to A (Fig. 8.3) can be considered as the rotations of the linear segments AB and AD about A.
   
• This brings the concept of rotation in a flow field.

         Definition of rotation at a point:        

The rotation at a point is defined as the arithmetic mean of the angular velocities of two perpendicular linear segments meeting at that point.

Example: The angular velocities of AB and AD about A are

  respectively.

 

Considering the anticlockwise direction as positive, the rotation at A can be written as,
    (8.5a)

or

   (8.5b)
 

The suffix z in ω represents the rotation about z-axis.

When u = u (x, y) and v = v (x, y) the rotation and angular deformation of a fluid element exist simultaneously. 

Special case : Situation of pure Rotation 

          Observation:

• The linear segments AB and AD move with the same angular velocity (both in magnitude and direction).
• The included angle between them remains the same and no angular deformation takes place. This situation is known as pure rotation. 

Vorticity

Definition: The vorticity Ω in its simplest form is defined as a vector which is equal to two times the rotation vector

    (8.6)
 

For an irrotational flow, vorticity components are zero.
        

Vortex line:
If tangent to an imaginary line at a point lying on it is in the direction of the Vorticity vector at that point , the line is a vortex line.

The general equation of the vortex line can be written as,

    (8.6b)

In a rectangular cartesian cartesian coordinate system, it becomes

     (8.6c)

where,



Vorticity components as vectors:         

The vorticity is actually an anti symmetric tensor and its three distinct elements transform like the components of a vector in cartesian coordinates.

This is the reason for which the vorticity components can be treated as vectors.

Existence of Flow

•  A fluid  must obey the law of conservation of mass in course of its flow as it is a material body.

•  For a Velocity field to exist in a fluid continuum, the velocity components must obey the mass conservation principle.

• Velocity components which follow the mass conservation principle are said to constitute a possible fluid flow

• Velocity components violating this principle, are said to describe an impossible flow.

• The existence of a physically possible flow field is verified from the principle of conservation of mass.

      The detailed discussion on this is deferred to the next chapter along with the discussion on principles of conservation of momentum and energy.