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.
  
  Figure 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 in Fig. 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  

• In non-dimensional form, the vortex shedding frequency is expressed as  known as the Strouhal 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 (Fig31.4 (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) ).

Fig. 31.4 Influence of Reynolds number on wake-zone aerodynamics

• 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.
  
• 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.