The transition from laminar to turbulent flow in a pipe

The transition from laminar to turbulent flow In a pipe 

Laminar flow ceases to be stable when a small perturbation or disturbance in the flow tends to increase in magnitude rather than decay. For flow in a pipe of circular cross-section, the critical condition occurs at a Reynolds number of about 2100. Thus although laminar flow can take place at much higher values of Reynolds number, fliat flow is no longer stable and a small disturbance to the flow will lead to the growth of the disturbance and the onset of tuihulence. Similarly, if turbulence is artificially promoted at a Reynolds number of less than 2100 the flow will ultimately revert to a laminar condition in the absence of any further disturbance. 

In connection with the transition, Ryan and Johnson have proposed z stability parameter Z. If the critical value Zc of tiiat parameter is exceeded at any point on the cross-section of the pipe, then turbulence will ensue. Based on a concept of a balance between energy supply to a perturbation and energy dissipation, it was proposed that Z could be defined as  

Z will be zero at the pipe wall Ux = 0) and at the axis (3%/9y = 0), and it will reach a maximum at some intermediate position in the cross-section. From equation 3.6, the wall shear stress R for laminar flow may be expressed in terms of the pressure gradient along the pipe (—AP//)  

The velocity distribution over the pipe cross-section is given by  

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Often it is useful to combine variables that affect physical phenomenon into dimensionless parameters. For example, the transition from laminar to turbulent flow in a pipe depends on the Reynolds number, Re = pLv/p, where p is the fluid density, I is a characteristic dimension of the pipe, v is the velocity of flow, and // is the viscosity of the fluid. Experiments show that the transition from laminar to turbulent flow occurs at the same value of Re for different fluids, flow velocities, and pipe sizes. Analyzing dimensions is made easier if we designate mass as M, length as L, time as t, and force as F. With this notation, the dimensions of the variables in Re are ML 3 for p, (L) for L, (L/t) for v, and (FL 2t) for //. Combining these it is apparent that Re = pLu/p, is dimensionless. 

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