Enthalpy Explained: All You Need to Know

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I can find no "rules" for its use. No-one is saying there is a law of conservation of enthalpy.

I am a layman in aerodynamics. However, this is my view on your concern. In thermodynamics, $$dH=TdS+VdP$$ So, in an adiabatic, isobaric reversible process, $dS=0, dP=0$ the enthalpy$(H)$ remains conserved.

Air is normally considered as insulator of heat. So if you consider a chunk of air as your system, all the thermodynamical changes within it are assumed to occur in the adiabatic way. In atmospheric science, adiabatic lapse rate is an example of such assumption which works well.

In addition, if some aerodynamical study is meant to pursue in a constant pressure zone, say, at a narrow range of altitude, then the study eventually becomes adiabatic, isobaric where H is constant as mentioned above. That may be the reason to introduce enthalpy.

Having said so, I don't know and couldn't realise either what kind of thermodynamical changes we see for a parcel of air, if the process is simultaneously adiabatic and isobaric. For an ideal gas, adiabaticity implies $PV^{\gamma}=$constant. If we further set $P$ constant for isobaric process, then $V$ must become constant. Hence, no work done, no heat exchange and $dU=0$. So everything stand still.

For a non-ideal gas, some meaningful changes may occur.