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Among the renewable energy sources, high altitude wind power is gaining increased attention for its better strength, steadiness, and coverage compared to the traditional ground-based wind power with wind turbines. However, unlike the latter, the technology for high altitude wind is still immature and the works on the field are mostly empirical. In our research, we try to set up a framework about force analysis and provide a stepping stone for other kite energy researchers and engineers to develop more efficient systems. In this paper, we analyzed and experimentally verified the effects of acting aerodynamic forces at different angles of attack ranging from 0° to 90°. We also studied the power potentials of a kite corresponding to these varying forces. The work will enable a researcher or engineer to design a more feasible and more efficient kite power system with better understanding of the kite dynamics.

Wind energy has changed from an almost negligible part of the total electricity supply of the world to a much larger, more important, and fast expanding component after recent explosive growth [

Nevertheless, most people agree that wind energy should claim a much greater share of the future energy supply [

On the other hand, wind is stronger and steadier if we move upward. Because there is less friction between the air and ground and there is less obstruction from ground topologies, wind blows faster when the altitude increases [^{3}, where ρ is the air density, A is the cross section area of the wind, and v is the wind speed. Hence, the total wind energy per unit area grows even faster than the wind speed. For this reason, the typical height of wind turbine towers increased from about 20 m in the 1980s to around 100 m today. Correspondingly, the tip of the rotor blades can reach nearly 200 m high.

However, there is a limit on the wind turbine height. The increased energy-flow through the rotor means a greater force on the tip of the tower, and correspondingly a greater load on the entire tower structure and the underground foundation. Meanwhile, the increased height makes it harder to install, repair or simply inspect the generator, gearbox, controller, and blades. The associated constructing and maintenance cost will significantly reduce the economic margin achieved by the higher tower and larger rotor.

Recently, many people started to think about ways of tapping the high altitude wind energy without building a big wind turbine. One method is to substitute the tower with a flying device. For example, Joby Energy [

Another approach is to separate the wind harnessing component and the power-generating component of the wind turbine. Then we can send the wind harnessing part to higher altitude and catch the full potential of the wind while keeping the heavier and more expensive power generating part on the ground for safety and maintenance purposes. For example, Laddermill [

While many enthusiastic groups and startup companies are working on the direction, most designs are based on trial and error methods as most new technologies at their embryonic stage. Even among the researchers, most efforts are focused on design and control [13-16]. Systematic study of kite aerodynamics is rare [1,17,18]. Many people simply borrow the results from the study of airplane wings or horizontal wind turbines [19,20], and the latter are mostly based on wing study. However, there are several fundamental differences between the two. First, the airplane wings and wind turbine blades are sturdy and fixed while the kites used by most power-generating group are soft. Second, the wings and blades are designed to generate upward lift forces to keep the airplane afloat while the kite generator is to pull the tether that is connected to the generator. The force needs to not only overcome the combined weight of the kite and tether, but also the working load from the generator, which is usually significantly greater than the former loads. Third, most wings and blades are straight while most kites are curved in shape.

In this paper, we will help to pave the road of harness-

ing high altitude wind power by studying the aerodynamics of the kite. We will review the wind profile, develop a model of a simple weightless kite, and then theoretically analyze the forces of the kite corresponding to the varying angles of attack. We will further verify ourmodel with experimental results and then give some summary and conclusions.

Due to the surface friction of the Earth, the wind speed is positively correlated to the height at the altitude we are interested. Although no single equation can capture the wind profile anytime anywhere, there are still two prevailing wind laws that are used in the field. They are the logarithm law and the power law [

The logarithm law of wind profile over the height is based on the Monin-Obukhov similarity theory [

where κ is the von-Larman constant, is the velocity scale generated by the turbulent friction τ and air density ρ of the surface wind over land or sea, H_{0} is the roughness length, and L_{m} is the length scale or the Monin-Obukhov length determined by the absolute temperature, gravity, air density, heat flux, and specific heat at the location. Both V* and L_{m} are independent of height.

According to Monin-Obukhov, we have

Then with some mathematical derivation, which we will not repeat here, we can get

where V_{g} is the ground speed at L_{m}, K_{g} is a constant that can be determined empirically. Please note that the logarithm law does not consider the Coriolis effect and is valid for flat, uniform surface with neutral atmosphere conditions [

Power law is also used widely by the wind turbine industry:

where V_{g} is the reference wind speed at the height H_{g}, α is a constant that can be determined empirically and is often taken as 1/7. This law was not developed mathematically like the logarithm law, but generally provide a close description of wind profile in many cases [1,14].

According to Thuiller and Lappe [

It is worth mentioning that due to the fluctuation of the temperature, the wind speed fluctuates accordingly since the air density ρ and the length scale L_{m} (a function of air density and temperature) both changes with temperature. Although the wind speed varies over time, its speed distribution is Rayleigh. Therefore, we can predict the average power generation of the kite at a given hour of the day. In fact, observation over time also shows that the change of wind speed can be assumed as sine curves over time [

When the kite is at horizontal position, or the zenith point, the wind is parallel to the chord plane of the airfoil. In most kite designs, each airfoil plane is connected to two bridal lines, one on leading edge and the other on trailing edge. A virtual effective tether can be considered to be attached to the chord plane through the aerodynamic center.

When fully inflated, the cross-section of a modern kite takes the shape of an airfoil. Therefore, we choose to model a flying kite as a number of flat airfoil planes linked together in parallel as shown in _{L} and the drag coefficient C_{D} of the airfoils are homogeneous throughout. When the kite is gliding in the sky like a parachute, its aerodynamic behavior is similar to that of an airplane wing.

However, there is one critical difference between a kite and an airplane wing. That is, a kite generally flies with a large angle of attack, which is often greater than the stall angle. On the contrary, the airplane seldom flies beyond this critical angle due to the possibility of losing speed. Therefore, we choose to define the aerodynamic forces slightly differently and set up a separate framework for kites. As seen in _{WF} = V_{W}cosθ as the Flying velocity, and V_{WP} = V_{W}sinθ as the Pushing velocity. Both V_{WF} and V_{WP} will generate aerodynamic forces that will affect

the flight of the kite.

Like an airplane wing, the flying velocity V_{WF} will generate both lift force and drag force. We define L_{ift} and D_{rag} as the induced lift and drag due to the flight of the airfoil in this head wind. They are perpendicular and parallel to the chord plane respectively, as shown in _{W}, the L_{ift} and D_{rag} forces can be obtained as

Meanwhile, V_{WP} will generate a force P_{ush}, which can be considered as the chord plane of the kite being pushed away. That is,

For simplicity, we can use vector summation to obtain the total aerodynamic force P_{ull} generated by the kite with the incoming wind

where P_{ush} and L_{ift} are collinear and both are perpendicular to D_{rag}.

Combining L_{ift}, D_{rag} and P_{ush}, we can project them to the horizontal and vertical directions, which are defined as F_{loat} and D_{rift} respectively. F_{loat} is the force to keep the kite floating in the sky while D_{rift} makes the kite drift away along the wind if it is free.

If we change the angle of attack, we can observe different aerodynamic behaviors of a kite, where both the absolute and relative magnitudes of each aerodynamic force will change.

1) Laminar dominant range: This is where the kite flies overhead and the angle of attack is smaller than the stall angle. The head wind, V_{WF}, is laminar and regular effect of an airfoil dominates the flight of the kite. The power carried by the transverse wind component V_{WP} is negligible compare to that carried by V_{WF}. Since the maximum lift coefficient C_{L} is generally between 0.5 ~ 1 for typical large parafoils [

2) Turbulent dominant range: When the angle of attack is sufficiently greater than the stall angle (without loss of generality, we arbitrarily choose it to be when P_{ush} = 3L_{ift}), the airflow separates from the kite surface and becomes turbulent. Hence, the induced L_{ift} and D_{rag} from the headwind become insignificant and the form drag P_{ush} of the kite due to the crosswind becomes dominant. Since the form drag coefficient C_{DL} (~2) of a kite is much greater than the lift coefficient C_{L} (0.5 ~ 1), the force generated by the kite as a whole with its chord plane facing the wind is contributing far greater to the total force than the combined induced forces generated by the individual airfoils.

3) Transition range: The transition range is where the airflow changes from laminar dominant to turbulent dominant and both effects will have comparable contribution to the flight of the kite. In this paper, we will take the liberty to define it as from stall angle to the angle when P_{ush} = 3L_{ift}.

In _{DL} ≈ 2. As shown in the figure, L_{ift} and D_{rag} forces are more evident at the laminar range. Meanwhile, P_{ush} dominates the overall aerodynamic forces at large angle of attack due to the increased V_{WP} and greater C_{DL} over C_{L} and C_{D}.

The F_{loat} increases quickly as the angle of attack increases in the laminar dominant range. As we can see from _{loat} will keep mostly flat or increase slightly in a long range after the stall angle. However, it will deteriorate quickly at the end of the turbulent zone.

In _{rag} to P_{ull} ratio of a typical kite corresponding to the change of angle of attack. Beyond the stall angle, the ratio is extremely small, generally less than 2%. Therefore the D_{rag} force and its corresponding angle deviation from overall pulling force will be negligible. That is, we can approximate that the aerodynamic force is perpendicular to the chord plane of the kite. For a weightless kite, or when the kite weight is negligible to the air force, we can further approximate that the tension or tether direction is perpendicular to the kite chord plane too. In this case, the angle of attack is equal to the angle of tether measured from the vertical line or z-axis.

By doing so, we can plot the relationship between T_{ension} and the tether position as shown in _{ension} at the directions. They are calculated as the equivalent total force coefficients The outline curve is the change of T_{ension} with angle of attack. As we can see, T_{ension} increases in general as the tether rotates from upright position to horizontal position if we assume the wind speed is constant. This assumption is valid for the most kite sports like kite surfing or kite boards.

For large-scale power-generating kites, the length of the tether is long enough (>200 m) that wind speed changes over height. Therefore, the T_{ension} force of the kite tether is not monotonically increasing. For a fixed length tether, T_{ension} force increases (in general) when the kite moves away from the zenith point and reaches a maximum at certain angle. Then T_{ension} starts to decrease due to the diminishing wind speed.

The blue falling zone is where the float force generated by the kite cannot support the combined weight of the kite and the tether. It includes both the very large and very small angles of attack. The falling zone with very large angle of attack is often considered as the real cross

wind zone. In kite sports, the kite can briefly dip into this zone when it is moving along a certain pattern such as a circle. However, it is vulnerable to the error of flight control or a suddenly weakened wind. It is recommended to avoid this zone for a stable power or force generating kite.

The black park/reel-back zone is where a kite can float by itself yet with drastically decreased tension on the tether. It can be used to park or reel back a kite. The latter case is used when a pump or Yo-Yo style motion is chosen for power generation. That is, a kite can be released in the power zone. The strong tension will pull the tether, which in turn power the generator through a gearbox. When the tether reaches its length limit, the kite can be controlled to fly into the reel-back zone. Then an electric motor will reel the kite back. The tension in this zone is significantly smaller than that in the power zone, yet the aerodynamic force still maintains the shape of the kite and keeps it from falling. This will make it more stable to control the kite than in the falling zone.

To verify the accuracy of the kite model, we conducted a wind tunnel test as described in

It is evident that the theoretical calculation in _{DL}. The result also matches the wind tunnel test data extracted from the report by Klimes and Sheldahl [

One interesting observation from the experiment is that the three models generate similar F_{loat} and D_{rift} forces provide that they have equal projected or shadow areas on Y direction. It is seemingly contradict to the general teaching of a fluid textbook, where concave half circle comes with a higher C_{DL} (~2.3) than that of a flat surface (~2). We contribute the effect to the narrow kite width compare to the overall length, i.e., the high aspect ratio of the models. For a concave half cylinder with infinite length, the incoming air stream has to go through the obstacle over the two curved ends. However, the stream can take the easier path to escape from the top and bottom parts of a short half cylinder just like passing over a flat plate.

One implication of the above observation is the calculation of total power generated by the kite. For the soft kite without backbone, its shape will become very close to a half circle under the strong wind. This will effecttively decrease more than one third () of the projected area compare to the original status when it is flat and fully expanded.

Tapping the higher altitude wind for renewable energy is gaining momentum in the recent years. The purpose of this paper is to provide some fundamental results on the force analysis of kites and serve as stepping stones for the future development of kite based power generating system.

The forces that are beneficial or useful to generate power in kites are different from those in airplanes or horizontal axial wind turbines. Therefore, we choose a different framework to describe the aerodynamics of kites. In this framework, the angle of attack of the kite can range from 0˚ to 90˚. The regular sense of lift and drag will not physically lift or hinder the flight of the kite like they do for an airplane. Therefore we defined the F_{loat}, D_{rift} and P_{ush} forces to characterize the aerodynamic forces on the kite. However, they can be easily converted to lift and drag forces in conventional sense.

The F_{loat} and D_{rift} forces vary greatly in the full range of angle of attack, as we have shown from both theoretical calculation and wind tunnel experiment. Their combined effect, or P_{ush} on the kite by the wind, grows in general (except right after the stall angle) with the climbing of angle of attack. We therefore divided the entire range of the angle of attack to Laminate, Transition and Turbulent ranges. Each range corresponds to the contribution of the resulted forces due to the head wind and transverse wind. By mapping the forces to the applications, we can further divide the flight of the kite to three zones: Power Zone, Falling Zone and Park/Reelback Zone. From this point, we can design the flight of a kite to obtain steady force, and then generate steady power or electricity.

As the next step, we are currently developing an innovative kite engine cycle based on this research. It will take advantage of the different power zones to maximize the power output of a kite from higher altitude wind.