Entry Flow in a Duct - 1 | Fluid Mechanics for Mechanical Engineering PDF Download

Entry Flow In A Duct -

• Growth of boundary layer has a remarkable influence on flow through a constant area duct or pipe. 
  Consider a flow entering a pipe with uniform velocity.
  1. The boundary layer starts growing on the wall at the entrance of the pipe.
  2. Gradually it becomes thicker in the downstream.
  3. The flow becomes fully developed when the boundary layers from the wall meet at the axis of the pipe.
• The velocity profile is nearly rectangular at the entrance and it gradually changes to a parabolic profile at the fully developed region. 
• Before the boundary layers from the periphery meet at the axis, there prevails a core region which is uninfluenced by viscosity. 
• Since the volume-flow must be same for every section and the boundary-layer thickness increases in the flow direction, the inviscid core accelerates, and there is a corresponding fall in pressure. 
• Entrance length : It can be shown that for laminar incompressible flows, the velocity profile approaches the parabolic profile through a distance L_e from the entry of the pipe. This is known as entrance length and  is given by

For a Reynolds number of 2000, this distance,  the entrance length is about 100 pipe-diameters. For turbulent flows, the entrance region is shorter, since the turbulent boundary layer grows faster.

• At the entrance region, 

1. The velocity gradient is steeper at the wall, causing a higher value of shear stress as compared to a developed flow.

2. Momentum flux across any section  is higher than that typically at the inlet due to the change in shape of the velocity profile. 

3. Arising out of these, an additional pressure drop is brought about at the entrance region as compared to the pressure drop in the fully developed region.

Control Of Boundary Layer Separation -

• The total drag on a body is attributed to form drag and skin friction drag. In some flow configurations, the contribution of form drag becomes significant. 
• In order to reduce the form drag, the boundary layer separation should be prevented or delayed so that better pressure recovery takes place and the form drag is reduced considerably. There are some popular methods for this purpose which are stated as follows. 
  1. By giving the profile of the body a streamlined shape( as shown in Fig. 31.2). 
     1. This has an elongated shape in the rear part to reduce the magnitude of the pressure gradient.
     2. The optimum contour for a streamlined body is the one for which the wake zone is very narrow and the form drag is minimum.

The injection of fluid through porous wall can also control the boundary layer separation. This is generally accomplished by blowing high energy fluid particles tangentially from the location where separation would have taken place otherwise. This is shown in Fig. 31.3. 

1.  The injection of fluid promotes turbulence
2. This increases skin friction. But the form drag is reduced considerably due to suppression of flow separation
3. The reduction in form drag is quite significant and increase in skin friction drag can be ignored.
   

Mechanisms of Boundary Layer Transition

• One of the interesting problems in fluid mechanics is the physical mechanism of transition from laminar to turbulent flow. The problem evolves about the generation of both steady and unsteady vorticity near a body, its subsequent molecular diffusion, its kinematic and dynamic convection and redistribution downstream, and the resulting feedback on the velocity and pressure fields near the body. We can perhaps realise the complexity of the transition problem by examining the behaviour of a real flow past a cylinder.
  
  Fig.31.4 (a) shows the flow past a cylinder for a very low Reynolds number  . The flow smoothly divides and reunites around the cylinder. 

• At a Reynolds number of about 4, the flow (boundary layer) separates in the downstream and the wake is formed by two symmetric eddies . The eddies remain steady and symmetrical but grow in size up to a Reynolds number of about 40 as shown in Fig.31.4 (b)
• At a Reynolds number above 40 , oscillation in the wake induces asymmetry and finally the wake starts shedding vortices into the stream. This situation is termed as onset of periodicity as shown inFig.31.4 (c) and the wake keeps on undulating up to a Reynolds number of 90 . 
• At a Reynolds number above 90 , the eddies are shed alternately from a top and bottom of the cylinder and the regular pattern of alternately shed clockwise and counterclockwise vortices form Von Karman vortex street as in Fig.31.4 (d).
• Periodicity is eventually induced in the flow field with the vortex-shedding phenomenon.
   
• The periodicity is characterised by the frequency of vortex shedding f 
• In non-dimensional form, the vortex shedding frequency is expressed as   known as theStrouhal number named after V. Strouhal, a German physicist who experimented with wires singing in the wind. The Strouhal number shows a slight but continuous variation with Reynolds number around a value of 0.21. The boundary layer on the cylinder surface remains laminar and separation takes placeat about 81⁰ from the forward stagnation point. 
• At about Re = 500 , multiple frequencies start showing up and the wake tends to become Chaotic. 
   
• As the Reynolds number becomes higher, the boundary layer around the cylinder tends to become turbulent. The wake, of course, shows fully turbulent characters (Fig.31.4 (e) ).
• For larger Reynolds numbers, the boundary layer becomes turbulent. A turbulent boundary layer offers greater resistance to seperation than a laminar boundary layer. As a consequence the seperation point moves downstream and the seperation angle is delayed to 110⁰ from the forward stagnation point (Fig.31.4 (f) ).

• Experimental flow visualizations past a circular cylinder are shown in Figure 31.5 (a) and (b)

Fig 31.5 (a) Flow Past a Cylinder at Re=2000 [Photograph courtesy Werle and Gallon (ONERA)]



Fig 31.5 (b) Flow Past a Cylinder at Re=10000 [Photograph courtesy
 Thomas Corke and Hasan Najib (Illinois Institute of Technology, Chicago)]

• A very interesting sequence of events begins to develop when the Reynolds number is increased beyond 40, at which point the wake behind the cylinder becomes unstable. Photographs show that the wake develops a slow oscillation in which the velocity is periodic in time and downstream distance. The amplitude of the oscillation increases downstream. The oscillating wake rolls up into two staggered rows of vortices with opposite sense of rotation.
   
• Karman investigated the phenomenon and concluded that a nonstaggered row of vortices is unstable, and a staggered row is stable only if the ratio of lateral distance between the vortices to their longitudinal distance is 0.28. Because of the similarity of the wake with footprints in a street, the staggered row of vortices behind a blue body is called a Karman Vortex Street . The vortices move downstream at a speed smaller than the upstream velocity U.
   
• In the range 40 < Re < 80, the vortex street does not interact with the pair of attached vortices. As Re is increased beyond 80 the vortex street forms closer to the cylinder, and the attached eddies themselves begin to oscillate. Finally the attached eddies periodically break off alternately from the two sides of the cylinder.
   
• While an eddy on one side is shed, that on the other side forms, resulting in an unsteady flow near the cylinder. As vortices of opposite circulations are shed off alternately from the two sides, the circulation around the cylinder changes sign, resulting in an oscillating "lift" or lateral force. If the frequency of vortex shedding is close to the natural frequency of some mode of vibration of the cylinder body, then an appreciable lateral vibration culminates.
   
• Numerical flow visualizations for the flow past a circular cylinder can be observed in Fig 31.6 and 31.7
• An understanding of the transitional flow processes will help in practical problems either by improving procedures for predicting positions or for determining methods of advancing or retarding the transition position.
• The critical value at which the transition occurs in pipe flow is  . The actual value depends upon the disturbance in flow. Some experiments have shown the critical Reynolds number to reach as high as 10,000. The precise upper bound is not known, but the lower bound appears to be . Below this value, the flow remains laminar even when subjected to strong disturbances.

• In the case of flow through a channel,  , the flow alternates randomly between laminar and partially turbulent. Near the centerline, the flow is more laminar than turbulent, whereas near the wall, the flow is more turbulent than laminar. For flow over a flat plate, turbulent regime is observed between Reynolds numbers    of 3.5 × 10⁵ and 10⁶.

Several Events Of Transition -

Transitional flow consists of several events as shown in Fig. 31.8. Let us consider the events one after another.

1. Region of instability of small wavy disturbances-  

Consider a laminar flow over a flat plate aligned with the flow direction (Fig. 31.8).

• In the presence of an adverse pressure gradient, at a high Reynolds number (water velocity approximately 9-cm/sec), two-dimensional waves appear.
• These waves are called Tollmien-Schlichting wave( In 1929, Tollmien and Schlichting predicted that the waves would form and grow in the boundary layer).
•  These waves can be made visible by a method known as tellurium method. 

2. Three-dimensional waves and vortex formation-

• Disturbances in the free stream or oscillations in the upstream boundary layer can generate wave growth, which has a variation in the span wise direction.
• This leads an initially two-dimensional wave to a three-dimensional form.
• In many such transitional flows, periodicity is observed in the span wise direction. 
• This is accompanied by the appearance of vortices whose axes lie in the direction of flow.

3. Peak-Valley development with streamwise vortices-

• As the three-dimensional wave propagates downstream, the boundary layer flow develops into a complex stream wise vortex system.
• Within this vortex system, at some spanwise location, the velocities fluctuate violently . 
• These locations are called peaks and the neighbouring locations of the peaks are valleys (Fig. 31.9).

4. Vorticity concentration and shear layer development-
   
At the spanwise locations corresponding to the peak, the instantaneous streamwise velocity profiles demonstrate the following

•  Often, an inflexion is observed on the velocity profile.
•  The inflectional profile appears and disappears once after each cycle of the basic wave.

5. Breakdown-
     
The instantaneous velocity profiles produce high shear in the outer region of the boundary layer.

• The velocity fluctuations develop from the shear layer at a higher frequency than that of the basic wave.
• These velocity fluctuations have a strong ability to amplify any slight three-dimensionality, which is already present in the flow field.
• As a result, a staggered vortex pattern evolves with the streamwise wavelength twice the wavelength of Tollmien-Schlichting wavelength .
• The span wise wavelength of these structures is about one-half of the stream wise value. 
• The high frequency fluctuations are referred as hairpin eddies.

This is known as breakdown. 

6. Turbulent-spot development-

• The hairpin-eddies travel at a speed grater than that of the basic (primary) waves. 
• As they travel downstream, eddies spread in the spanwise direction and towards the wall.
• The vortices begin a cascading breakdown into smaller vortices.
• In such a fluctuating state, intense local changes occur at random locations in the shear layer near the wall in the form of turbulent spots.
• Each spot grows almost linearly with the downstream distance.

  The creation of spots is considered as the main event of transition .