Mechanical Energy Balance for Turbulent Flow

Consider the control volume shown in Fig. 8.2, with inlet at point 1 and outlet at point 2. In the following, A-variable means the variable at point 2 (outlet) minus the variable at point 1 (inlet). 

For no height change gAh = 0 for no pump in the control volume = 0 for an incompressible fluid with constant density  

By continuity the mass flow rate is the same, ivj = iVj. Viscous losses are defined 

We make all variables non-dimensional by dividing them by a dimensional standard (which has to be identified in each case) and denoting the standard with a subscript s. For the velocity standard, we use an average velocity somewhere in the system  

The pressure is made non-dimensional by dividing it by twice the kinetic energy  

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