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first, for incompressible flows you have often a decoupled energy equation expressed in terms of the temperature balance.... however, E is sum of kinetic and internal energy, the BC.s for E are consequent to the setting of BC.s for velocity and temperature 
ok, what I want to know (did not mention that), can I compute the total energy equation independent from the temperature equation? And if so, can I (at the inlet), prescribe the total energy as the sum of kinetic and internal energy (based on inlet velocity and temperature) as a Dirichlet boundary condition, dropping the potential energy contribution?

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as you see from the equation, if you want to treat as a decoupled equation you need to know apriori the velocity field and pressure gradient (steady or not). That means you already know the kinetic energy which is a part of the total energy. Therefore, the ony actual unknown field is the internal energy (the temperature) 
and what is about the potential energy which also contributes to the total energy (negleted because contribution small?)?
just out of interest then, can I solve the temperature and total energy equation if I already have obtained velocity and pressure beforehand? I mean, the energy equation is the general conservation law and the temperature equation is derived from it. So can I use both or just one equation? 
the potential energy is very often disregarded in many problems ...
My opinion is that for incompressible flows you need just to compute the temperature equation, then you sum the resulting internal energy to the (known) kinetic energy and get the total energy. I think it could be interesting computing the totale energy equation only for some special coupled cases (velocity field depending also on internal energy) 
ok, i think that has clarified the topic for me, grazie mille :)

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