The dynamic motion of tethered undersea kites (TUSK) is studied using numerical simulations. TUSK systems consist of a rigid winged-shaped kite moving in an ocean current. The kite is connected by tethers to a platform on the ocean surface or anchored to the seabed. Hydrodynamic forces generated by the kite are transmitted through the tethers to a generator on the platform to produce electricity. TUSK systems are being considered as an alternative to marine turbines since the kite can move at a high-speed, thereby increasing power production compared to conventional marine turbines. The two-dimensional Navier–Stokes equations are solved on a regular structured grid to resolve the ocean current flow, and a fictitious domain-immersed boundary method is used for the rigid kite. A projection method along with open multiprocessing (OpenMP) is employed to solve the flow equations. The reel-out and reel-in velocities of the two tethers are adjusted to control the kite angle of attack and the resultant hydrodynamic forces. A baseline simulation, where a high net power output was achieved during successive kite power and retraction phases, is examined in detail. The effects of different key design parameters in TUSK systems, such as the ratio of tether to current velocity, kite weight, current velocity, and the tether to kite chord length ratio, are then further studied. System power output, vorticity flow fields, tether tensions, and hydrodynamic coefficients for the kite are determined. The power output results are shown to be in good agreement with the established theoretical results for a kite moving in two dimensions.

## Introduction

Fossil fuels are presently used to satisfy the majority of global energy needs. Due to climate change concerns, researchers are studying sustainable energy technologies as alternatives to fossil fuels. Renewable energy resources that have not been adequately utilized include ocean and tidal currents. This is in part due to limitations in existing hydrokinetic technology, such as conventional fixed marine turbines. The power output of these turbines is limited by the local current speed, leading to high production costs per kilowatt hour. However, new technologies which have been proposed, such as TUSK, seek to overcome this limitation. TUSK systems consist of a rigid-winged kite moving in an ocean current. The kite is connected by tethers to a rotating spool and generator on a fixed platform on the ocean surface or anchored to the seabed. Hydrodynamic forces on the kite cause the kite to move downstream in the ocean current, thus tensioning the tethers and rotating the spool and generator to produce electricity during a power (or reel-out) phase. During a kite retraction phase, the hydrodynamic kite forces are reduced (by decreasing the kite angle of attack, for example), and the kite is reeled-in by the spool. If energy output during the power phase is larger than the energy required to retract the kite, then a net power output can be achieved. The main advantage of TUSK systems [1] is that a kite during its power phase can move across the current at high velocities of up to $Vk=(2L/3D)Vc,$ where Vk is the kite velocity, Vc is the ocean current velocity, and L/D is the kite's lift-to-drag ratio (which may conservatively reach $L/D≅6−9$).

Consequently, TUSK kites move at least 4–6 times faster than the ocean current speed, and since power output is proportional to flow velocity cubed, at least 100× more power can be harnessed compared to a fixed marine turbine with the same turbine rotor area. Thus, TUSK systems are flow speed enhancement devices, and the potential power output increase of these systems compared to marine turbines means that they may find use in tidal locations with lower mean velocities.

Airborne wind energy (AWE) systems are another renewable energy technology, which are similar to TUSK systems, except that the kites fly in air. AWE systems are being considered as an alternative to conventional wind turbines. AWE systems were first studied by Loyd [1] and re-introduced by Ockels [2]. Archer and Caldeira [3] summarized the global high-altitude wind power resource. Since then, AWE systems have been studied extensively, and the state-of-the-art has been recently summarized in Ahrens et al. [4]. Due to the higher density of water compared to air, it is expected that more power would be generated by using ocean currents compared to AWE systems.

In contrast to AWE systems, underwater kite systems have rarely been studied since they were first proposed by Landberg [5]. However, Moodley et al. [6] conducted a preliminary economic analysis on the feasibility of the Minesto Deep Green technology, which uses underwater kites with mounted turbines to extract tidal and ocean current energy, for the Agulas Current near South Africa. They found Deep Green to be more cost-effective than other hydrokinetic technologies. Lazakis et al. [7] also conducted a risk assessment of installation and maintenance activities for the Minesto Deep Green system, and they [7,8] have investigated operational and maintenance parameters that affect system cost. Minesto UK, Ltd., has advanced their research and development to the sea-trial stage, but to the best of our knowledge, it has published limited technical analyses of its system in the archival literature, although Jansson [9] developed a kite motion simulator that includes hydrodynamic force, tether, buoyancy, and added mass models to aid Deep Green technology development. HydroRun Technologies, Ltd. [10] has also been developing a system since about 2012 that uses underwater gliders during successive power (reel-out) and retraction (reel-in) phases.

Olinger and Wang published some of the first technical analyses of TUSK systems [11,12]. Initial design studies including detailed power output estimates were conducted. A six degrees-of-freedom kite–tether model was also used to simulate TUSK kite trajectories and confirm power estimates and other system performance parameters. A preliminary study of cavitation effects in TUSK systems was also conducted. Li et al. [13,14] studied the aspects of kite motion control, in underwater kite systems for the first time. Ghasemi et al. [15,16] developed a two-dimensional and three-dimensional computational tool to study a baseline simulation of TUSK systems by solving the full Navier–Stokes equations. Coiro et al. [17] studied the Generator Electrical Marino (GEM) system which is a submerged, buoyant turbine tethered to the ocean floor. However, the GEM turbine was not attached to a winged hydrokite, and so does not achieve the high-speed, cross-current motion required for power output enhancement.

The present work seeks to develop numerical simulations to model the kite and tether dynamics of a simple TUSK system. The studies [1114] cited above use TUSK system models that are differential equation based on relatively simple (linear, inviscid) models to obtain kite lift and drag forces. Nonlinear effects are captured in the present study since the full Navier–Stokes equations are solved for the ocean current flow field.

A two-dimensional computational tool is developed to capture the interaction of a TUSK kite and tether with the current flow by solving the full Navier–Stokes equations. The computational tool includes a flow solver which uses a two-step finite-volume method with open multiprocessing (OpenMP), explained in detail in Ref. [18], to solve the flow equations. A rigid kite is assumed. The numerical interaction of a solid and fluid flow is widely studied in Refs. [1925], and here, we use the fictitious domain-immersed boundary method to simulate the interaction of the kite with the flow. TUSK system kites generally fly across the current along three-dimensional paths, and our final goal is to simulate these complex motions. However, the first step toward that goal is to formulate and solve the governing equations for a two-dimensional TUSK system in this work.

This paper is organized as follows: In Sec. 2, a summary of the computational model is given including the flow solver, the fictitious domain-immersed boundary method, and the methods used to control the kite trajectory. In Sec. 3, results from a baseline simulation of the kite–tether system are presented including predicted kite hydrodynamic forces, vorticity flow fields, and power generation values. These power estimates are compared to theoretical predictions for Loyd's simple kite [1] undergoing two-dimensional motion. A parameter study is also performed in which the effect of the ratio of tether to current velocity, kite weight, current velocity, and tether length has been investigated. Results summary and directions for future work conclude this paper.

## Governing Equations

The computational model consists of a flow solver on a defined numerical domain and grid, a kite model that uses the fictitious domain-immersed boundary method, a Hooke’ s law model for tether elasticity, and a kite tether length control model. The viscous incompressible fluid flow is governed by the continuity and momentum equations
$∇·V=0$
(1)
$∂(ρV)∂t+∇·(ρVV)=−∇p+∇·τ+Fb+Ftether+Fs$
(2)

Here, $V$ denotes the velocity vector, ρ the density, p the pressure. $Fb$ represents any body forces such as gravity, $Ftether$ is the tether tension force, and $Fs$ represents the fluid–solid interaction force. The basic numerical formulation for each term of the above equations is described in more detail in Ref. [26].

The governing equations are solved with an explicit second-order predictor–corrector method on a staggered grid. A second-order essentially nonoscillatory method and a simple second-order centered difference approximation were used to discretize the advection and viscous terms, respectively. The pressure equation is solved using a semicoarsing multigrid method [27]. A flow chart of the algorithm for numerical modeling of a tethered undersea kite, based on the Navier–Stokes equations, is shown in Fig. 1.

Fig. 1
Fig. 1
Close modal
In order to track the rigid kite, the fictitious domain-immersed boundary method [28,29] is used in which the geometry of the kite is defined by a marker function varying smoothly from solid to fluid. In this method, the Navier–Stokes equations are solved in the whole domain, and the linear and angular momentum are calculated in the solid region, and used to apply a rigid body constraint on the grid points occupied by the rigid kite to update its position. A NACA0012 airfoil was used for the cross section shape of the kite, which is specified by
$ytc=5t({ 0.2969xc−0.1260(xc)−0.3516(xc)2+0.2843(xc)3−0.1015(xc)4})$
(3)
Here, yt is the half thickness at a given value of x, c is the chord length, x is the position along the chord, and yt is the airfoil half thickness [30]. To identify the solid object, a marker function, ψ, is defined such that ψ = 1 inside the airfoil equation and zero everywhere else. The marker function transits smoothly near the airfoil interface by
$ψ=0.5+0.5(y−yt)3+1.5ϵ2(y−yt)((y−yt)2+ϵ2)1.5$
(4)

where $ϵ$ can be adjusted to control the thickness of the transition zone.

Various support structure and tether attachment concepts are being considered for TUSK systems, including tether attachment to the ocean floor, floating or fixed structures at the ocean surface, or submerged buoyant structures. While each of these has its own advantages and drawbacks, in this work for simplicity we assume a fixed support structure for tether attachment at the ocean surface.

To model the tension forces in the elastic tethers that connect the kite to the fixed structure at the surface ocean, a simple Hooke's law for the tether tension is used
$Ft=k(max(xi−x0)−x0)$
(5)

The tethers are designed to remain in tension in most situations, but if the tension reduces to zero, the tethers can go slack. Tether weight and hydrodynamic forces of the tethers are not modeled. To reduce the computational cost by reducing the domain size, the fixed attachment point of the two tethers at the ocean surface is placed outside of the computational domain as shown in Fig. 2(a). A uniform, regularly structured grid that encompasses a domain size is used in the simulation as shown in Fig. 2(b).

Fig. 2
Fig. 2
Close modal

The unstretched tether lengths of the two straight-line tethers (between a kite control unit and the kite) are specified as an input control command in order to control the geometric angle of attack of the kite as shown in Fig. 3. A periodic triangular wave shape is prescribed on both tether unstretched lengths for this purpose. Figure 3 shows that the unstretched (input) has a small difference with the actual (output) tether lengths due to fairly large tether spring stiffnesses. The tether reel-out velocity is set to near optimal [1], 30% of the ocean current speed, in the baseline simulation.

Fig. 3
Fig. 3
Close modal

The amplitude, time period, and phase shift (between the two tether length curves) were systematically varied over multiple simulation runs to identify a baseline simulation case with the highest achievable net power production. Application of more complex kite control schemes [13,3135] will be studied in future work.

A grid refinement study has been carried out to ensure the independency of the numerical results with grid mesh resolution. Figure 4 illustrates the vertical position and velocity of the kite's center of mass versus time for three different mesh resolutions. The results converged for an 880 × 720 mesh which was then used in the baseline simulation. Moreover, four simulations with Reynolds number equal to 3000, 6000, 12,000, and 20,000 have been performed in which Reynolds number is calculated based on the chord length of the kite. It is shown in Fig. 5 that the effect of viscous forces is small in the baseline simulation.

Fig. 4
Fig. 4
Close modal
Fig. 5
Fig. 5
Close modal

The computational tool was initially developed by extending and modifying a three-dimensional code used to model tethered floating structures in Ref. [26]. The accuracy of the computational tool was validated in Ref. [26] by comparing simulation results with various analytical, numerical, and experimental predictions for floating bodies exposed to unsteady wave loadings.

## Results

In this section, the results for a baseline simulation of the tethered undersea kite shown in Fig. 2(a) are presented. Then, the effects of key design parameters, such as the ratio of tether velocity to currents velocity, kite weight, current velocity, and tethers length on TUSK systems performance are examined.

### Baseline Simulation.

The input parameters for the simulation are listed in Table 1. The tether spring stiffness is set to match the elasticity of Nylon 6, a potential TUSK tether materials. The phase shift for unstretched tether length (see Fig. 3) was adjusted to achieve the maximum achievable power output for the baseline simulation. Input values in Table 1 are set to model typical TUSK system parameters. The computational domain size and the resultant kite motion amplitudes are kept small in this study to limit computational run time. Instantaneous snapshots (in time) from an animation of the kite motion during a power-retraction cycle are shown in Fig. 6. Vorticity is created near the kite leading edge and is convected downstream. Larger vortices are observed during the power phase (Fig. 6(b)) compared to the retraction phase (Fig. 6(d)) as expected since the kite effective angle of attack is higher in the power phase.

Fig. 6
Fig. 6
Close modal
Table 1

Input parameters for baseline simulation

 Domain size Kite chord length Tether spring stiffness Reynolds number $Re=3×103−20×103$ Power-retraction cycle time period Phase shift (unstretched tether length) Tether velocity Ocean current velocity Number of tether springs 2 Water viscosity Water density Kite density Time step size Tether attachment point coordinates
 Domain size Kite chord length Tether spring stiffness Reynolds number $Re=3×103−20×103$ Power-retraction cycle time period Phase shift (unstretched tether length) Tether velocity Ocean current velocity Number of tether springs 2 Water viscosity Water density Kite density Time step size Tether attachment point coordinates

In Fig. 7, time records of the kite geometric $(αG)$ and effective $(αeff)$ angle of attack are presented. The velocity diagrams in Figs. 6(b) and 6(d) define the kite effective angle of attack. All the velocity vectors are with respect to an observer on the kite. The effective angle of attack is defined based on the kite chord line and local current vectors, and thus accounts for the effect of the kite motion on angle of attack. Positive geometric and effective angles of attack are required during the power phase to obtain larger hydrodynamic forces. Effective angles of attack near 0 deg during the retraction phase yield low kite hydrodynamic forces and less power consumed. By adjusting the rest length of the tethers as in Fig. 3, appropriate geometric and effective angles of attack can be achieved to maximize net power production. The effective angle of attack values during power phase approaches $40$ deg which is above the stall angle of NACA 0012 airfoil in steady flow.

Fig. 7
Fig. 7
Close modal

The trajectory of the kite center of mass for three power-retraction cycles is presented in Fig. 8. It is shown that after the first cycle, the kite travels along a fairly stable periodic path. Hydrodynamic lift and drag coefficient are shown in Fig. 9. Lift and drag forces were obtained by determining the resultant force of tethers in the vertical and horizontal directions, respectively. Kite weight and buoyancy forces were subtracted from the resultant force on the tethers to isolate the hydrodynamic lift. The lower L/D ratios observed in Fig. 9 are due to the higher effective angle ($αeff=40 deg$) above stall and the lower Reynolds number. The tether tensions during the power and retraction cycle are shown in Fig. 10. Tether tensions can help determine required tether materials for TUSK systems.

Fig. 8
Fig. 8
Close modal
Fig. 9
Fig. 9
Close modal
Fig. 10
Fig. 10
Close modal

The reel-in and reel-out velocity of the tethers and the instantaneous power are shown in Figs. 11 and 12, respectively. Power is calculated by multiplying the tether tension by the reel-out or reel-in velocity of the tethers. It is demonstrated that the average power that is produced during power phase is almost twice that of the power consumed during retraction phase, yielding a positive net power production.

Fig. 11
Fig. 11
Close modal
Fig. 12
Fig. 12
Close modal

### Parametric Study

#### Effect of Tether to Current Velocity Ratio.

In order to validate our power estimates and numerical scheme, a parametric study on input tether velocity was undertaken to allow us to compare with earlier theoretical results of Loyd [1]. Six different simulations with different $Vt/Vc$ ratios have been conducted. In these six simulations, other input values are adjusted to keep the average lift-to-drag ratio L/D of the kite at $L/D=0.6$. By carefully studying the close-up views in Fig. 13, it is observed that the average power output during the power phase peaks for $0.4≤Vt/Vc≤0.6$. However, during retraction there is increasing consumed power as $Vt/Vc$ increases from 0.4 to 0.6.

Fig. 13
Fig. 13
Close modal
Therefore, the optimal ratio should be near $Vt/Vc=0.4$. The results are compared with a theoretical power prediction for a simple moving kite in two-dimensional motion during the power phase from Loyd [1]. The power coefficient, Fs, defined by Loyd [1] is given in the following equation:
$Fs=P0.5ρV3ACL$
(6)
which is used to nondimensionalize the average power phase output, p, from the simulation (after subtracting the power created by the net buoyancy force). The average lift coefficient CL in Eq. (6) is output from the simulation and varies between $0.5≤CL≤1.8$ for different simulations. The theoretical power coefficient from Loyd [1] is given in the following equation:
$FsL=VtVc{1+1(L/D)2−(VtVc)2−VtVc(L/D)}21+1(L/D)2$
(7)

There is a good agreement between our normalized power output and Loyd's results as shown in Fig. 14, which also confirms that the maximum power is obtained at $Vt/Vc≅0.4$.

Fig. 14
Fig. 14
Close modal

#### Effect of Kite Weight on Power Output.

Another study was performed where kite to water density ratios ($ρk/ρf=2.0$, 3.0, and 4.0) were varied with other parameters set as in Table 1. The instantaneous power output during reel-out and reel-in cycles is presented in Fig. 15, which shows that when increasing the kite's density, although more power is obtained during the power phase, more power is consumed during retraction phase. Figure 16 shows the variation of power coefficient, which is defined by

Fig. 15
Fig. 15
Close modal
Fig. 16
Fig. 16
Close modal
$Cp=P0.5ρkV3c$
(8)

where ρk is the kite density, and c is the chord length of the kite. Figure 16 shows that the net generated power is independent of the kite's density in our simulation. This is expected since the generated power due to the falling kite weight (power phase) must be consumed to reel-in the kite weight to a higher elevation during the retraction phase.

#### The Effect of Ocean Current Velocity.

The ocean current velocity effect is studied by carrying out three simulations in which , and , with other input values matching the baseline simulation. The power output during reel-out and reel-in cycles and the net output power are presented in Figs. 17 and 18, respectively, and it is observed that the power output is proportional to $Vc3$, as expected.

Fig. 17
Fig. 17
Close modal
Fig. 18
Fig. 18
Close modal

#### Effect of Tether Length.

The effect of tether length on power output was also studied. For this study, the computational domain is , and the ratios of the tether to kite chord length are $Lt/c=4.5,15$, and 30, with other input values matching the baseline simulation.

Figure 19 shows the instantaneous power output during the power and retraction phases for different ratios of tether to chord length. Since the geometric and effective angle of attack are controlled to have equal values in all the three simulations, the hydrodynamic forces and consequently the power are the same for all the cases. Figure 20 shows the vorticity contours at various times for a simulation performed with $Lt/c=30$. The attachment point of the tethers is placed outside of the computational domain as shown in Fig. 20(a). Also, the trajectory of the kite center of gravity is plotted in Fig. 21 during the time. The kite requires four cycles to reach a near-periodic motion for a larger tether to chord length ratio, while in the baseline simulation, where $Lt/c=4.5$, the kite obtained the near-periodic path after the second cycle. The simulations assume a uniform current velocity with ocean depth. The effect of current velocity decreases with increasing ocean depth, and the effect of varying tether retraction velocity will be studied in future simulations.

Fig. 19
Fig. 19
Close modal
Fig. 20
Fig. 20
Close modal
Fig. 21
Fig. 21
Close modal

## Conclusion

A computational tool for simulation of the motion of a tethered undersea kite has been developed where the kite forces are transmitted via extendable tethers to produce power from an ocean current flow. A projection method with OpenMp acceleration was used to solve the two-dimensional Navier–Stokes equations. The fictitious domain-immersed boundary method in which the rigidity constraint is imposed on the solid domain by conserving the linear and angular momentum was used to capture the kite-flow field interaction. The kite geometric angle of attack is controlled by altering the unstretched length of the tethers to obtain periodic power-retraction phases. A baseline simulation and the effect of varying key parameters in TUSK systems are investigated in this study. The power generated for the two-dimensional baseline simulation matches well with theoretical results from Loyd [1]. Reynolds number independence of the results has also been established. Vorticity flow fields, tether tension, tether reel-in and reel-out velocities, and hydrodynamic coefficients for the kite are determined. The weight of the kite has little effect on the net power output in TUSK systems. The computational tool also allows simulation of longer tethers with larger tether to kite chord length ratios. Extension to a three-dimensional flow domain, to model kite cross-current motions that yield higher power output, is needed in the future.

## Acknowledgment

The authors gratefully acknowledge the support of the NSF Energy for Sustainability Program through Grant Nos. CBET-1336130 and CBET-1033812.

## Nomenclature

• A =

area of the kite

•
• as =

acceleration of the solid object

•
• c =

kite chord length

•
• CL =

lift coefficient

•
• Cp =

power coefficient

•
• $Fb$ =

body force

•
• $Fs$ =

fluid–solid interaction force

•
• $Ftether$ =

tether force

•
• Fs =

theoretical power coefficient

•
• I =

inertia tensor

•
• K =

tether spring stiffness

•
• L/D =

lift to drag ratio

•
• p =

pressure

•
• P =

power

•
• Re =

Reynolds number

•
• t =

time

•
• Td =

phase shift (unstretched tether length)

•
• us =

velocity of the solid object

•
• $V$ =

velocity vector in the computational domain

•
• Vc =

current velocity

•
• Vk =

kite velocity

•
• Vt =

tether velocity

•
• xs =

location of the solid object

•
• yt =

maximum thickness of the airfoil equation

•
• αs =

angular acceleration of the solid object

•
• $Δt$ =

time step size

•
• μ =

water viscosity

•
• ρ =

density

•
• ρk =

kite density

•
• ρw =

water density

•
• τ =

viscous tensor

•
• ψ =

marker function of the solid object

•
• ωs =

angular velocity of the solid object

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