Frequently Asked Questions

11. Fluid Dynamics – Advanced Topics

11.1 What is the source of vortex rotation?

The source of rotation in convective vortices is amplification of the absolute rotation of ambient air converging toward the vortex axis in a thin layer. The initial rotation of the ambient air can be due to convergence of air rotating at the same speed as the Earth’s surface or to convergence of an air masses rotating relative to the earth’s surface. Convergence must be limited to a thin layer to significantly increase rotation. In a convective vortex, convergence takes place in a thin layer near the earth’s surface where tangential velocity is reduced by friction because at higher elevations there is no surface friction to reduce tangential velocity and convergence is inhibited by centrifugal force.

Angular Momentum (M ) and Circulation (G) are conserved in frictionless converging flow. Angular momentum (M ) is the product of radius (r) and tangential velocity (v ) --- (M = v r). Circulation is the integral of tangential velocity and path length. For a uniform circular path and rotation as a solid body, circulation equals angular momentum multiplied by 2p --- (G = 2pvr).

v_a r_a = v_s r_s

The absolute tangential velocity (v_s) of air stationary relative to the earth surface is:

v_s = 2 p r_s sin f / N

r_s = Sqrt (r_a v_a N / 2 p sin f)

where v_a and r_a are the velocity and radius at the radius of maximum velocity; where v_s is the absolute tangential velocity of the earth’s surface at a distance r_s from the axis of rotation, where N is the number of seconds per day (= 86400), and f is the latitude. In the following discussion, the latitude will be taken to be 30° so that sin f is 0.5.

r_s = Sqrt (r_a v_a N) / p)

Table 11.1 shows the effect of circulation on radius of maximum velocity (r_a) for convective vortices with peak tangential velocities (v_a) of 50 m/s. The radius (r_s) in Table 1 is the radius at which air stationary relative to the earth’s surface would have the same absolute circulation as the air at the radius of peak velocity.


Table 11.1 – Distance (r_s) from which air stationary relative to the earth’s surface at latitude of 30° has to converge to attain a peak tangential velocity (v_a) of 50 m/s at a range of radius of peak velocity (r_a ).
Vortex peak tangential velocity	Vortex radius at peak velocity	Vorticity at r_a rotation as solid	Angular momentum	Circulation	Source air radius	Source air absolute tangential velocity	Vorticity at r_s rotation as solid	Radius & velocity ratios
v_a (m/s)	r_{a (m)}	w (1/s)	M (m²/s)	G (m2/s)	r_s (m)	v_s (m/s)	w (1/s)	r_s / r_a
50	1	100	50	314	1,170	0.04	7.27x10^-5	1170
50	10	10	500	3,140	3,700	0.13	7.27x10^-5	370
50	100	1	5,000	31,400	11,700	0.43	7.27x10^-5	117
50	1000	0.1	50,000	314,000	37,000	1.35	7.27x10^-5	37.1
50	10,000	0.01	500,000	3,140,000	117,000	4.26	7.27x10^-5	11.7
50	20,000	0.005	1,000,000	6,280,000	166,000	6.03	7.27x10^-5	8.29
50	50,000	0.002	2,500,000	15,700,000	262,000	9.53	7.27x10^-5	5.24

Air stationary relative to the earth’s surface at a radius of 262 km has an absolute tangential velocity of 9.5 m/s relative to the vortex axis. For the 50 km radius hurricane at the bottom of Table 11.1, the air has to converge from a distance of 262 km to produce an eyewall tangential velocity of 50 m/s. For the 1 m radius dust devil at the top of the table, the air has to converge from a distance of 1170 m to produce a peak tangential velocity of 50 m/s. Producing a tangential velocity of 50 m/s from air rotating at the same speed as the earth surface requires the air to converge by a factor of 5.24 in the case of the hurricane and by a factor of 1170 in the case of the dust devil.

Surface winds, which typically have velocities around 5 m/s, have little effect on absolute circulation when the converging air comes from a location where the absolute tangential velocity of the earth’s surface is more than 4 m/s (r_s > 100 km). Local wind have a significant effect on circulation when the converging air comes from a location where the absolute tangential velocity of the earth’s surface in under 1 m/s (r_s < 10 km).

The circulation of a hurricane results from convergence of air initially rotating at velocities close to the rotation velocity of the earth’s surface. Friction between the air converging towards the base of the vortex and earth’s surface reduces the circulation of the converging air. As a result the converging air would have to come from a radius somewhat higher than 262 km to have a tangential velocity of 50 m/s at a radius of 50 km. The air in the 50 km radius hurricane could have originated at a radius of 500 km and have converged by a factor of approximately 10.

The circulation of dust devils results form convergence in air masses swirling relative to the earth’s surface. The initial swirling action could be the result of the vortex forming within a random eddy or near the boundary between two air streams. Hurricanes always have cyclonic circulation. Dust devils can have cyclonic or anti‑cyclonic circulation. Tornadoes at intermediate size predominantly have cyclonic circulation but occasionally have anti-cyclonic circulation. While the rotation of the earth’s surface plays a major role in large vortices like hurricanes, the earth’s rotation has little effect on small vortices like dust devils.

(Speculative) Like in hurricane, the air in small convective vortices may converge from an initial radius 5 to 20 times the radius of maximum velocity. The air in the 100 m diameter vortex of Table 11.1 could have been air originally at a radius of 1 km rotating at a velocity of 5 m/s or air originally at a radius of 11.7 km rotating at the speed earth’s surface (0.43 m/s). The speed of rotation of the earth’s surface of 0.04 m/s at a radius of 1 km is small relative to a velocity of 5 m/s and would have negligible effect on absolute rotation. The source of rotation in the 100 m radius tornado is more likely to be air originally at a radius of 1 km with a random rotation of 5 m/s than air originally at radius of 11.7 km with a rotation of 0.43 m/s. Convergence ratios of more than 20/1 seem improbable.

In the absence of energy source the atmosphere would tend to rotate at the same speed as the earth’s surface. The vorticity (w)of the earth’s surface is low and is maximum at the pole and zero at the equator. The vorticity of air rotating as a solid body is circulation divided by enclosed area --- (w=G/pr²). The absolute vorticity of an air mass is the sum of the vorticity of the earth's surface and of the vorticty of the air relative to the earth's surface. Vorticities significantly higher than the vorticity of the earth’s surface are rare since surface friction tends to attenuate relative vorticity. Table 11.1 shows the vorticity of air rotating as solid body at the radius of maximum velocity r_a and at the radius of origin r_s. The only mechanism capable of significantly increasing vorticity is convergence in a thin layer which requires an energy source. The high tangential velocity of convective vortices is always the result of convergence in a relatively thin layer of air close to the earth’s surface. In the case of hurricane, of air initially rotating at the same speed as the earth’s surface; in the case of dust devils, of randomly rotating air masses. Upturning of streamwise (horizontal) vorticity cannot produce vertical vorticity. For information vorticity see: http://en.wikipedia.org/wiki/Vorticity

Convergence does not necessarily produce tight convective vortices; convergence can be the result of uplift over a broad area. Convergence in a thunder storm can produce rotation in large air masses such as rotating super cells. The rotation of surface air resulting from convergence in super cell enhances the rotation of surface air producing air with high rotation whose rotation can be further enhanced by convergence in a small diameter vortex. Super cells clouds are not the cause tornadoes. Convergence in a thin layer is the cause rotation in both super cells and tornadoes. Without sustained convergence the rotation of a large air mass is dampened by friction.
The diameter of convective vortices increases with circulation.. Hurricane diameter increase during their intensification period indicating that the distance from which the air converge increases as the hurricane intensifies. When the source of the rotationis the earth's rotation, the circulation of the converging air increases as the air's initial distance from the vortex axis increases.

11.2 How much energy is required to overcome friction losses?

Friction losses are much lower for laminar flow than for turbulent flow. Flow in pipes becomes turbulent when the Reynolds Number exceed 5000. The Reynolds Number in atmospheric vortex is much higher than 5000 and therefore the flow should be turbulent but centrifugal force inhibits turbulence and keeps the flow laminar. The smooth thread shape of some waterspouts shows that flow in a vortex can be laminar. Turbulence is inhibited in vortex flow because when a particle of fluid moves towards the axis of rotation its tangential velocity increases to conserve angular momentum which increases centrifugal force and pushes the particle back towards its original position.

Table 11.2 shows that friction losses at a velocity of 40 m/s in a 40 m diameter pipe friction losses are 8500 more for turbulent flow than for laminar flow. The work loss due to friction is 1200 J/kg for turbulent flow and only 0.14 J/kg for laminar flow. Thermodynamic calculations show that the work produced when air is raised can be in the order of 5000 J/kg. For convection work of 5000 J/kg, friction losses are 24% for turbulent flow and 0.003% for laminar flow. Laminar flow reduces friction losses to negligible level.

Table 11.2 Friction loss in 10 km long pipe - for constant density air like fictitious fluid (Density: r = 1.0 kg/m², Viscosity: m = 1.8x10^-5 Pa )
Diameter	Velocity	Reynolds Number	Mass flow	DP Turbulent flow	DP Laminar flow	Loss ratio	Friction loss turbulent (specific)	Friction loss laminar (specific)	Friction loss turbulent (total)	Friction loss laminar (total)
d (m)	v (m/s)	ratio	Q (kg/s)	DP_t (Pa)	DP_l (Pa)	DP_t/DP_l	w_t (J/kg)	w_l (J/kg)	W_t (J)	W_l (J)
20	20	22x10⁶	6,300	730	0.29	2500	730	0.29	4.6x10⁶	1.8x10³
20	40	44x10⁶	12,600	2700	0.58	4600	2700	0.58	33x10⁶	7.2x10³
40	10	22x10⁶	12,600	91	0.036	2500	91	0.036	1.1x10⁶	0.45x10³
40	20	44x10⁶	25,000	333	0.07	4600	333	0.07	8.4x10⁶	1.8x10³
40	40	88x10⁶	50,000	1200	0.14	8500	1200	0.14	61x10⁶	7.2x10³

For the same flow, doubling the pipe diameter reduces friction loss by a factor of by a factor of 16 for laminar flow and by a factor of 30 for turbulent flow. Friction loss are inversely proportional to the diameter to the 4th power for laminar flow and to the diameter to the 5th power for turbulent flow.

Friction losses in a vortex would be slightly higher than in laminar pipe flow because the flow path is longer and there is more shear. Friction loss in vortex flow could be 5 times the friction losses in laminar pipe flow which is still much less than in turbulent pipe flow. Friction losses in unconfined flow such as cumulus updrafts could be 5 times the friction losses in turbulent pipe flow. Entrainment further increases friction losses in cumulus flow.

The Pax Scientific vortex mixer invented by Jay Harman demonstrated that using a vortex to circulate the water in a tank requires much less less energy than alternative methods such as pumps or agitators. Jay Harman stated that he is convinced vortices could be used to control atmospheric circulation and to bring rain to arid or drought ridden regions.
Pax Scientific vortex mixer web site
Article on Jay Harman

References
http://en.wikipedia.org/wiki/Reynolds_number
http://en.wikipedia.org/wiki/Darcy_friction_factor
http://en.wikipedia.org/wiki/Moody_chart
http://en.wikipedia.org/wiki/Colebrook_equation

11.3 How is energy conserved in a vortex?

Thermodynamic analysis permits calculating the ideal work (w_i) produced when air is raised reversibly. Ideal work (w_i) is capable of producing an ideal velocity (v_i), where: w_i = v_i² / 2. Energy must be conserved irrepective of the details of the lifting process. Fig. 11.3 shows how energy is conserved in five steady-state processes. For each process, the mass flow (M) is uniform throughout the length of the pipe.

Figure 11.3: Energy conservation in adiabatic upward flow

In case A, the air is raised in a vertical pipe in an adiabatic, frictionless (inviscid), steady-state process. The ideal work (w_i) results in ideal velocity (v_i) at the pipe outlet. The kinetic energy is mainly produced as the air accelerates in the bottom of the pipe but the upward velocity also increases as the density of the rising air decreases in order to produce a uniform mass flow. The enthalpy of the air leaving the pipe (h₂) is equal to the enthalpy of the air entering the pipe (h₁) minus the increase in its potential energy (gz). The process is reversible; the entropy of the air leaving the pipe (s₂) is equal to the entropy of the air entering the pipe (s₁).

In case B, the flow in the bottom of the pipe is limited by a restriction. The kinetic energy (w_i) is produced at the restriction. The cross-sectional area of the restriction is small relative to the cross-sectional area of the pipe so that the velocity in the pipe approaches zero. The kinetic energy of the jet dissipates downstream of the restriction. The enthalpy of the air leaving the pipe is increased by the kinetic energy dissipated. The process is irreversible; the entropy of the air leaving the pipe (s₂) is greater than the entropy of the air entering the pipe (s₁).

In case C, the kinetic energy of the air leaving the restriction is captured by a turbine and leaves the system as shaft work. The enthalpy of the air leaving the pipe (h₂) is equal to the enthalpy of the air entering the pipe (h₁) minus the increase in its potential energy (gz). The process is reversible; the entropy of the air leaving the pipe is equal to the entropy of the air entering the pipe.

In case D, the cross-sectional area of the pipe increases with height so that the velocity decreases with height. The kinetic energy is produced as the air enters the pipe and is dissipated as the velocity decreases. The enthalpy of the air leaving the pipe is increased by the kinetic energy dissipated. The process is irreversible; the entropy of the air leaving the pipe is greater than the entropy of the air entering the pipe.

In case E, the restriction is replaced with a Venturi. The velocity of the air at the throat of the Venturi can be greater than ideal velocity (v_i) without violating the law of conservation of energy because the kinetic energy is converted back to heat resulting the same exit enthalpy as in the irreversible cases B and D. The enthalpy of the air leaving the pipe is increased by the kinetic energy dissipated. The process is irreversible; the entropy of the air leaving the pipe is greater than the entropy of the air entering the pipe. For a given differential pressure the velocity produced in a Venturi can be greater than the velocity produced in a restriction or nozzle because the kineitc energy is recovered.

The ideal work (w_i) reverts to heat unless it is taken out of the system with a turbine or leaves the system as exit velocity. In any case the ideal work (w_i) is lost unless taken out of the system as shaft work. Producing useful work requires a shaft to take the work out of the system.

Reference
Venturi tube Wikipedia

11.4 What role does centrifugal force play in a vortex?

Low level flow in tornadoes is characterized by an outer region where converging air rotates faster to conserve angular momentum, and by an inner region where the air rotates as a solid body and where the tangential velocity falls to zero. This flow pattern, called a combined Rankine vortex, is illustrated by the solid lines in the graph at the bottom of Fig 11.4. The effect of rotation on upward flow in the annular tube shown in Fig. 11.4 will be used to try to understand what happens isn a convective vortex. For information on Rankine vortex see: http://en.wikipedia.org/wiki/Rankine_vortex

Figure 11.4: Upward flow in an annular tube

Tangential velocity in a Rankine vortex is related to radius according to:

v = v_m (r / r_c) 0 < r < r_c

v = v_m (r_c / r) r > r_c

where v_m is the maximum tangential velocity v, and r_c is the radius of the annulus of rising air. The inward differential pressure is balanced by centrifugal force except in the boundary layer where friction is significant. In the bottom graph of Fig. 11.4, the solid curves represent the frictionless Rankine vortex and the dotted curve shows the effect of friction in reducing tangential velocity. In order to explain vortical flow is it necessary to understand the contribution of tangential velocity to outward centrifugal force. The radial pressure across a thin shell is:

jP = r v² (jr / r)

By integration, the radial pressure differential across both the inner and outer parts of the Rankine vortex are:

DP = r (v_m)²/ 2

The total radial differential pressure across the two parts of the Rankine vortex is:

DP = r (v_m)²

For a given DP, the tangential velocity required to balance radial pressure differential is independent of vortex diameter.

The total radial differential pressure is the integral of the radial pressure over all the shell elements. If the tangential velocity were constant between radius r_a and r_b, as shown by the solid line in the graph at the top of Fig. 1, the total radial pressure differential pressure would be:

DP = LN (r_b / r_a) r (v_m)²

A uniform tangential velocity between radius 10 m and 20 m would result in a total radial pressure differential of 0.69 rv². A uniform tangential velocity between radius 10 m and 100 m would result in a total radial pressure differential of 2.3 rv², 4.6 times the radial pressure differential in the outer part of a Rankine vortex. For a given peak velocity, broadening the peak increases the differential pressure produced by centrifugal force. There is a small decrease in density as the pressure decreases as one moves towards the vortex axis; for purpose of initial discussion the density will be taken to be uniform in the horizontal direction.

11.5 What role does viscous friction play in a vortex?

Friction plays an essential role in the existence of convective vortices. The slight reduction in velocity resulting from friction against the earth’s surface in convergence zone ‘A’ of Fig 11.4 reduces centrifugal force and allows the air in the layer next to the earth's surface to converge into the bottom of the annular tube. Increasing tangential velocity requires energy; the energy is produced thermodynamically by the air rising in the annular tube. Tangential velocity increases as it converges towards the bottom of the tube to conserve angular momentum.

If the inner tube were to vanish, the air in zone ‘C’ of the inner tube would be dragged by friction by the air in the annulus and rotates as a solid body except next to the earth’s surface where the air is slowed by friction against the ground. The reduction in axial pressure is greatest at a level slightly above the bottom of the tube and tends pulls air axially both upward and downward towards level ‘C’. Axial pressure reduction is greatest at level ‘C’ near the bottom of the tube because the rising air loses angular momentum by friction against the outer tube or against the surrounding air and because the vortex diameter is smaller near the bottom. Level ‘C’ acts like the rotor of a centrifugal pump. Air is pulled in the eye of the rotor both from below and from above. These two axial flows which are small compared to the flow in the annulus are added to the annulus flow.

The reduction in angular momentum by surface friction in zone ‘B’ enables some of the converging air to penetrate past the annular tube and to rise in the inner tube. In dust devils, sand entrained with the rising air gets thrown out by centrifugal force once the air is high enough for the effect of friction against the bottom surface to diminish and for the effect of drag from the rotating annulus to increase centrifugal force. The sand is thrown out in a cone which can have sides with a slope of approximately 30°. In some dust devils a sand filled dark cone is visible above a clear sand free space.

The core is dragged in solid rotation for the whole length of the vortex. The stronger axial pressure reduction near the bottom of the vortex tends to pull air downward from zone ‘F’ towards zone ‘C’. The downward pull is resisted by the buoyancy of the sinking air because the temperature of descending air increases as it is compressed. In hurricanes, the air at the centre of the eye can be very dry above the 2 km level and have a high relative humidity below the 2 km level indicating that the core air descends from high in the troposphere and that air does not descend all the way to the bottom of the vortex.

If the outer tube were to vanish, the centrifugal force produced by the air rotating in the annulus could replace the physical outer annular wall and prevent the surrounding air at higher pressure than the air in the annulus from entering the rising air stream in the annulus. The tangential velocity profile in the graph at the top of Fig. 1 shows that the peak velocity can be reduced by friction or by divergence and how the velocity profile can be broadened. The rising air is high entropy air coming from the bottom of the atmosphere while the air surrounding the vortex is low entropy air. The surrounding air stays layered; it rotates but does not rise. At high elevations the radial pressure differential is balanced partly by the centrifugal force in the annulus of rising air and partly by the centrifugal in the non rising surrounding air. The equation in the top graph shows that broadening the tangential velocity profile permits balancing radial pressure differential with lower peak tangential velocities.

The air surrounding the annulus of rising air, zone ‘D’, is entrained by friction. Angular momentum is transferred from the rising air to the surrounding air. Centrifugal force in vortex flow inhibits turbulence because when a particle of air converges, tangential velocity increases to conserve angular momentum and the increased centrifugal force tends to push the particle back where it came from. The smooth thread shape of some dust devils, waterspouts and tornadoes shows that flow in the tuba can be laminar. Friction losses are much lower for laminar flow than for turbulent flow. For example the friction loss for air flowing in a horizontal tube 100 m in diameter and 10 km long at a velocity of 50 m/s is 670 J/kg for turbulent flow and 0.03 J/kg for laminar flow. The friction coefficient for turbulent flow is 23,000 times the friction coefficient for laminar flow. Atmospheric models use a turbulent viscosity approximately 100,000 greater than the laminar the viscosity of air.

Convective vortices have been simulated with computational fluid dynamics (CFD) models. CFD models calculate the time varying properties of a flow field. The fact that CFD results are able reproduce many of the features observed in natural vortices is taken as an indication of the validity of the model. Testing the effect of a variation can require redefining problem geometry. It difficult to distinguish whether a change in the flow field is due to a change in geometry, a change in initial conditions or a change in a model parameter. Traditional fluid mechanics are still used to solve most fluid mechanics problems. A CFD model must be supplemented by traditional fluid mechanic approach. A CFD model matching observations does not necessarily provide understanding of the role of friction at various point in the flow field.The above approach based on Fig. 11.4 shows that traditional analysis method can throw some light on the role of friction.

11.6 What are the factors controlling base pressure reduction and vortex diameter?

The pressure under the annular column of rising air depends on the buoyancy of the rising air. The maximum pressure reduction under the annulus of rising air is the ideal pressure reduction (DP_i), in the table of FAQ 9.6. The air converging in zone 'A' of the Rankine vortex turns up as it reaches the annulus diameter because buoyancy of the rising air is insufficient to draw the converging air any further.

The central core of the vortex is entrained in solid body rotation by friction. The vertical flow in the central core is small and the central core plays no role in energy production. Bringing air downward in the core requires rather than produces energy. The forced rotation of the core is capable of producing a maximum axial pressure reduction (DP_a) equal to twice the annulus pressure reduction (DP_i).

The diameter of the annulus is the diameter for which the differential pressure due to centrifugal force in the outer part of the rankine vortex (DP_ro) is equal to the ideal annulus pressure reduction (DP_i). The ideal annulus pressure recution (DP_i) is determined by the thermodynamic properties of the rising air. The annulus diameter is determined by the circulation of the converging air. As shown in Table 11.1, diameter increases with circulation. For a given annulus pressure reduction annulus diameter is proportional to circulation.

The fact that observed annulus pressure reduction agrees fairly well with ideal process pressurereduction indicates that frictional losses in a convective are small irrespective of diameter.

11.7 How is an AVE vortex different from a natural convective vortex?

In natural vortices, the source of the rotation is the rotation of the air mass within which the vortex forms which may due to the earth's rotation. In the atmospheric vortex engine (AVE), the rotation is produced by having the air enter a circular arena via tangential entry duct.The circulation is the result of forcing the air to enter a circular area tangentially at a short distance from the vortex axis rather than because the air was initially stationary relative to the earth’s surface at a large distance from the vortex axis. In the natural vortex of Fig. 11.4 and in the AVE of Fig. 11.7, the kinetic energy of the air is derived from the energy produced thermodynamically as the preceding air rises in the vortex.

Figure 11.7: AVE upward flow in an annular tube.

Natural vortices are rare in spite of the fact that surface air often has sufficient heat content to produce a vortex. An AVE would have additional features to ensure that a vortex can be started, controlled and stopped. A roof with a circular opening is used to force the air to converge. The diameter of the vortex could be 10 to 30% of the roof opening. The lower level of the vortex would is surrounded by a wall to prevent the vortex form being disturbed by the wind until the vortex is well established. Heat can be injected in the center of the station to create a strong updraft to start the vortex. The air can be heated in heat exchangers located upstream of the tangential entries to increase the energy production. The size of the vortex is limited by the amount of air that can flow through the tangential entries; in addition the air flow can be limited with dampers located either in the tangential entries or upstream of the heat exchangers.

The AVE provides a controlled supply of warm rotating air whose rotation provides its own chimney. Should the upper part of the vortex be blown away by a gust of wind, the lower part of the vortex re-establishes itself with warm spinning air from below. The upper part of the vortex, which was initially filled with warm spinning air from the bottom of the atmosphere, fills with the cool non spinning air at its base loses its buoyancy and die out. If a chimney filled with warm flue gas was cut in two and the upper part moved horizontally to the side, the upper half would quickly fill from the bottom with cools air and the flow in the upper half would quickly stop.

11.9 How is the ideal work dissipated in a natural vortex? Why will withdrawing energy with turbines not reduce vortex intensity?

Looking at energy conservation in open systems helps understand what happens to the energy of convection. Fig. 11.3 panels B, C & D show that provided no work is taken out of the system and provided that the exit velocity is negligible, the enthalpy of the air leaving the top of the tube is equal to the enthalpy of the air entering the tube, minus the increase in its potential energy and plus the ideal work.
Fig. 11.3 Upward flow in a vertical tube

Engineers use the open thermodynamic system shown in Fig. 9.1 to calculate the maximum quantity of work that can be produced when a fluid is transferred from one state to another (or the minimum quantity of work required to transfer a fluid from one state to another). Ideal work can only be calculated by assuming: negligible friction loss, negligible velocity (kinetic energy), and no heat transfer (adiabatic process). By contrast, the atmospheric science and fluid dynamics try to determine the properties of the flow field. It is not readily apparent that looking at a process with negligible velocity will shed light on flow field. Engineers have learned from experience that the most effective way of understandind energy transformations is to look at rigorously defined closed or open systems. It is not uncommon to use both open and close system to gain understanding from different perspective. The fact that results of thermodynamic anaylysis are confirmed by observations on real systems has given engineers confidence in defining thermodynamic systems which are are not always obviously related to the real system. Ideal thermodymnamic systems often bear little relationship to real systems but they are necessary to understand real systems. Open complex systems can be understood by looking at simpler closed systems. Atmospheric scientist have been reluctant to consider closed systems because the atmosphere is obviously an open system.
Fig. 9.1 Open Thermodynamic System without additional heat input

Fig. 9.6 shows that the addition of heat in non adiabatic process can be accomodated in open system. Ideal system analysis requires that the velocity be considered negligible in the adiabatic system of Fig. 9.1 and the non adiabatic system of Fig. 9.6.
Fig. 9.6 Open Therodynamic System with additional heat input

The open thermodynamic system shown in the figure below will be used to examine what happens to ideal work in an atmospheric vortex. The boundaries of the open thermodynamic system are the walls of the conduit consisting of the convergence zone, the upward flow annular tube, and the divergence zone. Boundaries of thermodynamic systems can be set arbitrarily as required to facilitate analysis. The velocity of the air entering and leaving a vortex can be made negligible by setting the system boundaries a long way from the vortex axis.

Fig. 11.9 Open system of the atmospheric vortex.

The ideal work (w_i) produced when air is raised is used to produce the tangential velocity (v_t) of the spinning air at the red dot under the annular tube; but the fact that the work of convection is required to produce tangential velocity does not mean that the work of convection cannot be taken out of the system. The kinetic energy of the spinning air is restituted as the air diverges in the upper part of the vortex.

There is a physics 101experiment where spinning masses are pulled toward the axis of rotation with a string. The experiment shows that energy supplied by pulling on the string is responsible for the increase in the kinetic energy of the rotating masses and that the energy is restored by the spinning masses pulling back on the string as the external pull is reduced. The purpose of the experiment to demonstrate the principle of conservation of mechanical energy. The energy required pull the masses in is equal to the energy produced when the masses move back out. Should the string is cut instead of gradually let go the mechanical energy would not be recovered and would somehow revert to heat. A skater pulling his arms in spins faster; the mechanical energy he supplied is not restored when he lets his arms out because there is no mechanism for doing so.

The energy of the spinning air leaving the vortex is recovered as the vortex diverges and becomes available to give kinetic energy to the air entering the vortex. Therefore it should be possible to remove energy from the system with peripheral turbines without significantly attenuating vortex intensity. As explained in FAQ 11.2 friction losses in a laminar vortex are extremely low. Nonetheless unless there are not machine to remove energy from the system the mechanical energy dissipates merely because the expansion process is not constrained as illustrated by the unconstrained piston of FAQ 9.5.
Dissipation in unconstrained expansion

A similar argument applies to vertical velocity. Upward velocity increases as the air enters the annular tube but the the upward kinetic energy is recovered when the upward velocity at the upper end of the vortex. The upward motions stops when the air reaches its level of neutral buoyancy.The fact that flow in a vortex is laminar is important for energy recovery because turbulence dissipates energy.

Should removing energy from the vortex with peripheral turbines reduce vortex intensity excesively it there are other alternatives available namely: injecting air without spin directly in the center of the vortex via an opening in the center of the arena floor or using a vertical axis turbine in the center of the arena.

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