Kinetic energy balance and dissipation

This formula involves the flow rate and the volume of the sudden expansioa 

In reality, the rate of dissipation is a local physical notion that varies in both space and time. As will be seen hereinafter, turbulence dramatically increases the rate of energy dissipation, and hence its paramount importance in fluid mechanics. The rate of energy dissipation can be defined rigorously on the basis of the Navier-Stokes equations, by estabhshing the evolution equation for kinetic energy. For this derivation, it is assumed that the fluid is incompressible. A tensor notation is used (Einstein s notation) where the velocity vector is expressed in the form Ui with index i varying from 1 to 3 to designate the three space dimensions (namely u = 

While the tensor notation is corrverrierrt for srrbsequent calcrrlatiorts, it should be remembered that, in equatiorrs [2.34] artd [2.35], the terrrrs involving index j are summed for all three values ofy, i.e. in eqrration [2.35], 

There are also three equatiorrs of dyrtamics for each value of index / = 1, 2, arrd 3. Equations [2.35] are written using the symbolic notation of the stress deviator, to simplify intermediate calculatiotts. The expression of the stress terrsor for a 

By multiplying each of the equatiorrs [2.35] of dyrramics by Uj, respectively, three equations are obtained for 1=1,2, and 3. 

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