To break the vortex into bits, we create a very simple flow system - the building block of a vortex, or its discrete element.
The building block of a vortex
Consider a duct with compressed gas tank mounted to the inlet:
We shall move from simple to more complex concepts. For this reason, let's first study the behavior of the above duct system when the system is stationary. Then we can move on to the rectilinear motion of this duct system. Last, we can study its rotational motion.
The duct has adiabatic walls - no heat comes in or goes out through them. The same applies to the tank. The gas in the tank is stored at some high pressure and has room (static) temperature.
How does the duct system behave when not moving?
Let the system be stationary in the stationary frame of reference F. At some moment, gas is allowed to flow through the duct and leave through the outlet. This type of gas flow in adiabatic duct is known as "Fanno flow". The following takes place:
Then the observer in the stationary frame of reference F will see total temperature difference
and static temperature difference
This means that if the gas at the exit is stopped adiabatically, it will show the same thermodynamic temperature as the tank (all friction is neglected). Therefore, no cooling is observed yet.
What is the rectilinear motion of this duct system like?
We set the system in uniform rectilinear motion; the tank is at the leading end. Let the velocity of the system be equal to c. The inertial laboratory frame of the moving duct we denote with F'. The stationary frame we denote by F. Since the moving system F' is inertial, there is no acceleration. For the observer in F' the flow through the duct takes place just as before when the system was not moving. That is, the observer in F' sees Fanno flow in the duct. See all details above for the non-moving duct.
Does the moving observer see cooling?
and static temperature difference
This means that if the gas at the exit is stopped adiabatically by the observer in F', it will show the same thermodynamic temperature as the tank (all friction is neglected). Therefore, no cooling is observed F'.
Does the stationary observer see cooling?
Yes!!! Here is why.
Then the stationary observer concludes, that the gas in the tank and at the inlet has total temperature
As we saw above, the static temperature of the gas at the outlet of the duct is
But this static temperature is the same in all frames of reference. This means that the stationary observer sees the ejected gas to have total temperature
Therefore, the observer in the stationary frame of reference F will see total temperature difference
(let c = 330 m/s, just below sonic. For air, cp=1006 J/(kg.K) --> 108.25K separation in total temperature) and static temperature difference
Where did the energy go?
The observer in the stationary frame F sees a gas parcel with high initial kinetic energy and room (static) temperature. Upon exiting the moving system, not only has the parcel stopped, it has also been cooled! This goes against intuition. Usually, when a moving parcel is forced to stop adiabatically, its static temperature goes up. The conclusion is:
The energy of the gas parcel was delivered as propulsion to the moving system!
Discretization of Vortex Tube Flow : Rotational Motion
The discrete vortex element (the duct) rotates about an axis with uniform angular velocity:
The rotation axis is perpendicular to the duct; the rotation axis is at the duct outlet. Gas from the tank flows towards the center of rotation. As before, the static temperature of the compressed gas in the tank is
What happens in the rotating frame?
The rotating observer notices that the compressed gas in the tank has ambient static temperature. The gas in the duct moves against the centrifugal force, as if "climbing up" a gravitational well. In doing so, it does work. This work can come only from the pressure energy contained in the gas. For this reason, as the gas moves towards the center, it expands, loses internal energy and becomes cold.
where v' is the "relative velocity", or the gas velocity relative to the moving observer. In a moving, inertial frame T' is conserved throughout the duct. This meant, that the relative total temperature is the same at any cross-section of uniformly moving adiabatic duct. But in the non-inertial, rotating frame this is no longer true! In rotating frame, there is another temperature that is conserved. It is called "rotary total temperature" or "rotary stagnation temperature". Total temperature is related to the total enthalpy of the gas; Rotary total temperature is related to the rotational total enthalpy of the gas, also known as rothalpy. Therefore, the temperature that is conserved in rotating frames is
the rotary total temperature; v' is the relative velocity of the gas, omega is the angular velocity of the frame and r is the radial position. We omit the derivation of Trot; it can be derived from energy conservation considerations. At this time, it will suffice to say that in a rotating reference frame, at any cross-section of the adiabatic duct, the rothalpy of the gas is conserved, e.g.
What happens in the stationary frame?
The stationary observer monitors the total and static temperatures of the rotating gas. To derive an expression for the total temperature in F, use the velocity addition formula
where V is the velocity of the gas in the stationary frame. Now express v' in the velocity addition formula,
substitute it into the conservation of rothalpy statement
cancel the squared cross product and get
in other words
Now notice the sum of the first 2 terms is the definition of total temperature in the stationary frame; also cancel the factors of "2" in the 3rd fraction and thus
This is the vectorial form of Euler’s turbine equation, as shown in the previous section. This equation shows a quantity seen in F that is conserved at any cross-section of the rotating duct! Now, all ejected gas begins its motion at the same radial position R: the duct inlet. Denote the linear speed of the inlet (the linear speed of the rotating inlet, as seen in the stationary frame F) with c,
Look again at the conservation condition above. At the outlet, r=0; at inlet, v=c; then
The observer in the stationary frame sees this total temperature difference between inlet and outlet! (let c = 330 m/s, just below sonic. For air, cp=1006 J/(kg.K) --> 108.25K separation in total temperature). What is the static temperature difference? At inlet,
Express the static temperatures at inlet/outlet through the total temperatures at inlet/outlet and take the difference:
precisely as in the case of rectilinear motion!
This means radial heat transfer is not the reason for the radial temperature separation.
The stationary observer sees
in both static and total temperature! (let c = 330 m/s, just below sonic. For air, cp=1006 J/(kg.K) --> 54.1K maximum cooling of air, not counting any expansion effects when the gas may be injected from the tank into the duct inlet, similarly to the injection done in vortex tubes, where the maximum cooling may be somewhat larger due to nozzle expansion)
Where did the energy go?
The observer in the stationary frame F sees a gas parcel with high initial kinetic energy and room temperature (at periphery). Upon exiting the rotating system, not only has the parcel stopped, it has also been cooled! This goes against intuition. Usually, when a moving parcel is forced to stop adiabatically, its static temperature goes up. The conclusion is:
The energy of the gas parcel was delivered as propulsion to the rotating system! This is ANGULAR PROPULSION.
How To Model the Temperature Separation With CFD
How to prepare the geometry
The geometry is simple - a straight duct. The effect does not depend on size, thus choose the duct to be as long as you wish. For example, it can be chosen to be 15 m long, having rectangular cross-section of 0.3 m x 0.4 m (width x height). One end of the duct is the inlet; the other - the outlet. In this case, the geometries were prepared with Gambit (currently available only with legacy license, soon to be phased out). Alternatively, ICEM or other meshers may be used to build the grid for the duct. Here is a segment of meshed duct:
How to position the duct
Make sure to position the duct outlet face so that it goes through the origin (0,0,0) and is parallel to one of the coordinate axes, e.g. z. This will be the rotation axis. The direction of rotation (clockwise vs. counterclockwise) is unimportant. The face at (0,0,0) is outlet, the face at the other duct end is inlet.
What are the CFD simulation parameters in Ansys FLUENT?
The FLUENT solver parameters are:
Ansys FLUENT simulation results:
FLUENT should predict cold temperatures very close (within 1.5 %) to their theoretical value given by the above formula.
The information contained in this site is based on the following research articles written by Jeliazko G Polihronov and collaborators: