This approach is based on first-principles physics alone and is not limited to vortex tubes only, but applies to moving gas in general. It shows that temperature separation in a moving gas is due only to enthalpy conservation in a moving frame of reference.
The thermal process in the vortex tube can be estimated in the following way:
The main physical phenomenon of the vortex tube is the temperature separation between the cold vortex core and the warm vortex periphery. The "vortex tube effect" is fully explained with the work equation of Euler, also known as Euler's turbine equation, which can be written in its most general vectorial form as:
where  is the total, or stagnation temperature of the rotating gas at radial position , the absolute gas velocity as observed from the stationary frame of reference is denoted with ; the angular velocity of the system is  and  is the isobaric heat capacity of the gas. This equation was published in 2012; it explains the fundamental operating principle of vortex tubes (Here's a video with animated demonstration of how this works). The search for this explanation began in 1933 when the vortex tube was discovered and continued for more than 80 years.
The above equation is valid for an adiabatic turbine passage; it clearly shows that while gas moving towards the center is getting colder, the peripheral gas in the passage is "getting faster". Therefore, vortex cooling is due to angular propulsion. The more the gas cools by reaching the center, the more rotational energy it delivers to the vortex and thus the vortex rotates even faster. This explanation stems directly from the law of energy conservation. Compressed gas at room temperature is expanded in order to gain speed through a nozzle; it then climbs the centrifugal barrier of rotation during which energy is also lost. The lost energy is delivered to the vortex, which speeds its rotation. In a vortex tube, the cylindrical surrounding wall confines the flow at periphery and thus forces conversion of kinetic into internal energy, which produces hot air at the hot exit.
Therefore, the vortex tube is a rotorless turboexpander. It consists of a rotorless radial inflow turbine (cold end, center) and a rotorless centrifugal compressor (hot end, periphery). The work output of the turbine is converted into heat by the compressor at the hot end.