Lets convert some mechanical energy to heat
Your dragon has a complex lung system and diaphragms of steel. It inhales a volume of air that is transported to smaller alveoli which compress it. Then it keeps shoving air in by trapping it in compressible chambers in its lungs.
By compressing the air, you are heating it up and changing mechanical energy from the dragon's muscles to heat in the compressed air. The dragon can't hold its breath for too long, otherwise the compressed air will cool and your exhale will have the opposite effect. (It would be an ice dragon)
When the dragon exhales, all alveoli/lung chambers are opened at once. The air will expand and cool, but at the nozzle/throat and for some distance around the mouth, you will have a superheated and pressurized air stream.
Or lets convert some heat into chemical energy
Biology is typically stingy when it comes to energy. It doesn't want to waste it. As such, mechanisms to store chemical energy are very common. Humans store ATP for bursts of energy, glycogen in our muscles for short term exercise, and fats around our bellies for long term needs.
If your dragon has a mechanism of creating glycogen from the stuff he eats, then all he has to do is poop it out and there goes an enormous amount of energy.
But dragons are cool and our answer has to be cool too
Lets put them both together then and our dragon's lungs turn into a combustion chamber. Compress in some air, create a mechanism to add in some carbohydrate (or hydrocarbon if your physiology allows) with the correct mixture. Then create a spark and the whole thing will burn out of the dragon's mouth like a chemical rocket while getting rid of excess energy.
(If you want to avoid the spark, there are fuel mixtures that will spontaneously combust when compressed, like diesel)
Fun edit: If your dragon had one or more blowholes, like a dolphin, around the rib cage and, umm, the backside, that energy could be directed during flight for thrust vectoring or speed bursts. It could also assist in initial takeoff like a JATO.