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Total Pressure, Total Temperature Definitions in Compressible Flows

Hello Everyone,

I do have a question regarding the definition of total temperature, total pressure and total enthalpy.

Looking in the turbulence book of Wilcox1994 (p. 180), I noticed that he defines H_t (total enthalphy) as
$\tilde{H}_t = \tilde{h} + \cfrac{1}{2} \tilde{u}_i^2 + k$

$\tilde{H}_t$ = total enthalpy
$\tilde{h}$ = static enthalpy
$\tilde{u}$ = mean velocity

= Turbulent kinetic energy

In other words the total enthalpy has a component of the turbulent kinetic energy.

First of all, could you recommend me to any paper where the reason for this definition is defined in detail?

Second, do you think that the fraction of turbulent kinetic energy would be useful energy? From my point of view, I would expect that this fraction has to dissipate after a certain while and therefore cannot be used mechanically? Does this make sense?

Thanks in advance.

In other words, the expressions for total pressure and total temperature would be (compressible definition):
$T_t = T_s + \cfrac{\tilde{c}_{i}}{2\,c_p} + \cfrac{k}{c_p}$
and
$P_t = P_s \left(\cfrac{T_t}{T_s}\right)^{\frac{\kappa}{\kappa -1}}$

Should I use this kind of total pressure/temperature definition, if I wanted to compare CFD and EXP data? What do you guys think?