Turbulence in a pipe and velocity profile of the flow

The flow in a pipe becomes turbulent when its velocity exceeds a threshold defined by a critical value of the Reynolds number. For a circular pipe, the following criterion is agreed upon  

The Reynolds number is defined using the area-averaged streamwise velocity and the diameter of the pipe. The onset of turbulence is related to the notion of flow instability. The Poiseuille flow does not remain steady if the Reynolds number 

Although a turbulent flow is by definition unsteady, the average profile of the flow in a pipe is defined by averaging, for example in time, the velocity measured at every point of the pipe. We will return to this approach in more detail in Chapter 8, which presents elementary notions regarding turbulence. 

The three essential notions about turbulence that shonld be kept in mind for now are as follows  

Turbulence mixes a fluid. It therefore also mixes velocities, hence a tendency to render the velocity field uniform immediately upon moving away from the walls. 

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