Analysis of Influencing Factors on Lift Coefficients of Autonomous Sailboat Double Sail Propulsion System Based on Vortex Panel Method

Sail is the core part of autonomous sailboat and wing sail is a new type of sail. Wing sail generates not only propulsion but also lateral force and heeling moment. The latter two will affect the navigation status and bring resistance. Double sail can effectively reduce the center of wind pressure and heeling moment. In order to study the effect of distance between two sails, airfoil and attack angle on the total lift coefficient of double sail propulsion system, pressure coefficient distribution and lift coefficient calculation model have been established based on vortex panel method. By using the basic finite solution, the fluid dynamic forces on the two-dimensional sails are computed. The results show that, the distance in the range of 0 to 1 time chord length, when using the same airfoil in the fore and aft sail, the total lift coefficient of the double sail increases with the increase of distance, finally reaches a stable value in the range of one to three times chord length. Lift coefficients of thicker airfoils are more sensitive to the change of distance. The thicker the airfoil, the longer distance is required of the total lift coefficient toward stable. When different airfoils are adopted in fore and aft sail, the total lift coefficient increases with the increase of the thickness of aft sail. The smaller the thickness difference is, the more sensitive to the distance change the lift coefficient is. The thinner the fore sail is, the lower the influence will be on the lift coefficient of aft sail.


Introduction
Autonomous sailboat is a new kind of marine mobile observation platform at the air-sea interface, which uses wind power as driving force. Solar panels or other devices are used to generate electricity to provide power for the control system and sensors (Yu et al., 2018a). Compared with traditional marine mobile observation platforms, the marine observation of long-endurance, large range and high spatial and temporal resolution observation especially the air−sea interface marine environmental element observation such as sea surface layer meteorological data and subsurface layer marine data can be realized with low cost with the help of autonomous sailboat (Yu et al., 2018b). The collected data can serve for the research frontier and hot topics such as simulation of global climate change, ocean acidification, ocean carbon cycle and air−sea−ice interaction, etc.
Sail plays an important role in propelling autonomous sailboat, which can be divided into two kinds. One is con-ventional soft sail and the other is rigid wing sail. The structure of wing sail is different from soft sail, as wing sail looks like an airplane wing. Compared with conventional soft sail, rigid wing sail is widely used in autonomous sailboats because of its large lift−drag ratio, large lift coefficient, good shape stability and simple aerodynamics and it is also easy to automatic control.
Wing sail produces lift and drag at the same time. Lift and drag can be decomposed into propulsion and lateral force. What is more, the wind force acting on the wing sail will cause the autonomous sailboat to heel, affecting navigation status and increasing sailing resistance. Reducing the pressure center of wing sail can effectively reduce the heeling arm. One of the effective methods is to mount double sail. Compared with a single sail, double sail propulsion system provides propulsion by two sails, and the force on each sail and the height of the wind pressure center can be reduced, thus the heeling moment can be reduced. In prac-tical application, A-Tirma G2 autonomous sailboat developed by Universidad De Las Palmas De Gran Canari (Dominguez-Brito et al., 2016), Force 12 xplorer autonomous sailboat developed by Open Ocean Robotics company (Open ocean robotics, 2019), and Arrtoo autonomous sailboat co-researched by the United States Naval Academy and Aberystwyth University (Miller et al., 2014) all adopted double sail design scheme, as shown in Fig. 1.
However, reciprocal interference exists between two sails. The aft sail, in particular, is susceptible to the influence of the fore sail's wake. The smaller the distance between the fore sail and aft sail, the greater the interference will be. In order to maximize the advantages of double sail and minimize the interference, it is necessary to study the effect of the distance of double sail on the lift coefficient. At the same time, the influence of airfoil combination on the lift coefficient should also be analyzed.
In preliminary design of wing sail, it is approximated that the lift coefficient of two-dimensional airfoil is equal to that of three-dimensional sail. Airfoil aerodynamic parameters mainly include lift, drag, aerodynamic center and pressure center. Propulsion of autonomous sailboat is mainly provided by the lift of the wing sail. The lift characteristics of the sail are usually expressed by the curve of lift coefficient C l varying with the attack angle.
Various methods exist to estimate lift and pressure of a given airfoil (Millikan, 2018;Bell, 2015;Liu et al., 2012). For a two-dimensional airfoil, thin airfoil theory will lose accuracy as the airfoil thickness increases (Spreiter and Alksne, 1958). Panel method provides increased accuracy over the former and can be applied to a variety of airfoil configurations. Computational fluid dynamics method is more accurate, but more computationally expensive (Cox, 2011). Panel method can be used to calculate the pressure distribution, lift and moment characteristics of airfoil flow with arbitrary shape, thickness and attack angle. Panel method contains source panel method and vortex panel method. In this paper, we adopted the vortex panel method. By satisfying no penetration condition and Kutta condition, a linear algebraic equation with unknown strength can be solved, with which, the lift coefficients and pressure distribution can be easily predicted.
In addition to this introduction, the rest of the paper is organized as follows: Section 2 presents the numerical method for calculating aerodynamic characteristics of double sail based on vortex panel method. The influence of distance and attack angle on the total lift coefficient of fore and aft sails with the same airfoil is detailed in Section 3. The influence of airfoil combination and distance on lift coefficient when fore and aft sails use different airfoils is presented in Section 4. Finally, Section 5 concludes the paper and discusses future works.

Numerical solution of double sail based on vortex panel method
In airfoil research, the four-digit airfoil developed by the National Advisory Committee for Aeronautics (NACA) is a typical low-speed airfoil. Four-digit airfoils have larger lift coefficient and smaller drag coefficient ( Public Domain Aeronautical Software, 2018). Four-digit symmetrical airfoils are commonly used on autonomous sailboats. In actual use, the thickness at trailing edge should be completely zero. The modified airfoil equation for NACA four-digit symmetric airfoil is as follows (NACA, 2018): where, x is the coordinate in airfoil length direction, from 0 to c, in this paper c=1; z t is the coordinate in airfoil thickness direction; t is the maximum airfoil thickness (last two digits of NACA four digit airfoil). The method to analyze single airfoil's lift coefficients and pressure distribution can be seen from Katz and Plotkin (2011) and Kuethe and Chow (1998). For double wing sail, set the leading edge of fore airfoil at the origin of the coordinate. Assuming that the airfoil of fore and aft sails were mounted on the same mid-line of the deck, reflected in co- ordinates the leading edge of aft airfoil locates at x axis. L represents the distance from the trailing edge of the fore sail to the leading edge of the aft sail, as shown in Fig. 2. In order to analyze the influence of the distance between fore sail and aft sail on the lift coefficient, distance factor d between sails can be dimensionless treated as L/c, the distance/chord length ratio. The theoretical range of L is from 0 to the whole deck length. When the fore sail and aft sail at different attack angles respectively, the moment on two sails will cause the boat yawing, so it is assumed that the rotation of the two sails is synchronous, that is, they have the same attack angle.
Key differences between double airfoil panel method and single airfoil panel method are the separation of the two geometries and a twice Kutta condition. The Kutta condition is applied precisely at the trailing edge to ensure a smooth flow at the trailing edge (Rao and Yang, 2017).
From references we can rewrite the linear equation: for i<m+1: The elements in the coefficients matrix of double airfoil are calculated by the vertical coefficients and . If we calculate through Eq. (4) directly, there will be a virtual panel from the trailing edge of the fore airfoil to the trailing edge of the aft airfoil, the angle between the virtual panel and x axis is zero, so the last coefficient of the fore airfoil and the first coefficient of the aft airfoil will be miscalculated due to the virtual panel, and the virtual panel should be removed in actual calculation.
The vertical coefficients matrix is a (2m+2)×(2m+2) matrix which contains two blocks. The first m+1 rows belong to the upper block standing for the coefficients of fore airfoil, and the rest m+1 rows belong to lower block representing the coefficients of aft airfoil. In the coefficients matrix, the separation of the two geometries was manifested in matrix columns, hereof, it is most crucial to well treat the leading edge point and trailing edge point respectively of the fore and aft airfoil.
For the first m rows and the rows from m+2 to 2m+1 (the lower block), the first column , and the elements from column two to column m can be calculated by Eq. (4). For the column m+1, the elements can be calculated by Eq. (5). For column m+2, the elements can be calculated by Eq. (3), and the elements from column m+3 to column 2m+1 can be calculated by Eq. (4). At last for the (2m+2)-th column, the elements can be calculated by Eq. (5).
Kutta condition was manifested in coefficients matrix rows. For the fore airfoil, the m+1 row was related with Kutta condition, for the aft airfoil, the 2m+2 row was related with Kutta condition. To obtain a determined system, we define the first column and (m+1)-th column to be 1, the rest coefficients of m+1 row is zero. We define the (m+2)-th column and (2m+2)-th column of 2m+2 row to be 1. The rest elements of (2m+2)-th row are zero. The linear equation can be obtained as follows: . . a n 1,m+1 a n 1,m+2 a n 1,m+3 . . . a n 1,2m+2 a n 2,1 a n 2,2 . . . a n 2,m+1 a n 2,m+2 a n . . . a n 2m+1,m+1 a n 2m+1,m+2 a n 2m+1,m+3 . . . a n 2m+1,2m+2 The linear algebraic equation can be solved for the fore sail and the aft sail, and the distribution of vortex on each panel can be determined.
With the known circulation densities, the velocity at each control point can be computed, and the pressure distribution and lift coefficients can be obtained. Since there is only tangential velocity at the control point on the panel surface because of the vanishing of the normal component there, the dimensionless velocity can be obtained, which can be given by where and are tangential coefficients. The calculation methods of tangential coefficients are similar to that of the vertical coefficients discussed above. The virtual panel should also be removed. But the tangential coefficients matrix is a (2m+1)×(2m+2) matrix. Similar to the vertical coefficients matrix, the tangential coeffi-cients matrix also consists of two blocks. One block is the first m rows standing for the coefficients of fore airfoil, and the other block is the last m rows representing the coefficients of aft airfoil.
For the first m rows and the last m rows, the first column , and the elements from column two to column m can be calculated by Eq. (9). For column m+1, the elements can be calculated by Eq. (10). For column m+2, the elements can be calculated by Eq. (8), and the elements from column m+3 to column 2m+1 can be calculated by Eq. (9). At last for the (2m+2)-th column, the elements can be calculated by Eq. (10).
As mentioned above, the virtual panels between two airfoils should be removed from the tangential coefficients matrix. Actually, the vortex on the virtual panel has no influence on other panels, so the coefficient of virtual panel can be regarded as zero. The method to solve this problem is to insert a row between two blocks, and the elements of the inserted row are zero. So the tangential coefficients matrix can be obtained. Based on the above theory, and using MATLAB software programming, the pressure distribution and lift coefficients can be obtained. In order to verify the accuracy of computation program, the pressure distribution coefficient of NACA0012 airfoil at 10 degrees of attack is taken as an example and compared with Gregory and O'Reilly's data (Gregory and O'Reilly, 1970). As can be seen from Fig. 3, the results calculated by the program in this article well coincided with the contrast data. Therefore, it is believed that the calculation method and the program used in this paper are credible.

Influence of distance and attack angle on the total lift coefficient of fore and aft sails with the same airfoil
Assuming that the airfoil of fore and aft sails are the same, the lift coefficients of various airfoils at different distances and attack angles were analyzed. Eight airfoils, NACA0008, NACA0009, NACA0010, NACA0012, NACA0015, NACA0018, NACA0021 and NACA0024, were selected. The lift characteristic of airfoil is related to the flow around it. According to the attack angle, airfoil can be divided into three flow zones: attachment flow zone, stall zone and deep stall zone. The attack angle of attachment flow zone ranges from −10° to 10°. As a result of Liu's research, when n>160, the results become relatively stable and the panel method cannot account for viscous effects on the lift when the angle is larger than 15° (Liu, 2018); moreover, because the potential flow of ideal fluid is solved without calculating the influence of fluid viscosity and flow separation, there will be deviation from the actual situation at a high attack angle (Zhang, 1984). Therefore in this pa- per, the maximum attack angle is 11° and the number of n is 200.
In theory, the distance L between the fore sail and aft sail ranges from 0 to positive infinity. But the actual situations are first, the distance between the two sails is limited by the deck length and cannot be extended indefinitely; second, considering the rotating space of the sail, the fore and aft sail cannot be joined end to end. By changing the airfoil and value of the distance factor d, the lift coefficient distribution of double sail can be obtained, by summing up the lift coefficients of the fore sail and aft sail, the total lift coefficient of two sails can be obtained. By taking the distance and attack angle of two sails as variables, the variation of the total lift coefficients with the increase of distance at different attack angles is calculated, as shown in Fig. 4.
As can be seen from Fig. 4, the total lift coefficients of the fore and aft sails change sharply when distance factor d varies from 0 to 1. The change tended to be gentle after 3c distance. The total lift coefficient increased with the increase of distance no matter which airfoil was chosen, on the premise that the fore and aft sail airfoils are the same. Compared with the lift coefficient of a single sail, the average lift coefficient of double sail is lower at the same attack angle. The farther the distance between two sails is, the smaller the interference between fore and aft sails will be. The influence of distance on different airfoils is also different. As can be seen from Fig. 4, with the increase of airfoil thickness, the point where the lift coefficient tends to be stable gradually shifts to the right, which means that the interaction between thin airfoil sails is lower than that between thick airfoils at the same distance. It can also be seen from Fig. 4 that the lift coefficient can be increased by increasing the attack angle and the thickness of the airfoil.
In order to compare with a single sail, the total lift coefficient of double sail can be averaged. The lift coefficients of single sail of NACA0012 airfoil at different attack angles were calculated in MATLAB environment and meets well with classical solution (Anderson, 2010). Lift coefficients at various attack angles are shown in Table 1.
Assuming that lift coefficient of a single airfoil is C l,sin , the total lift coefficient of double airfoil is C l,tot , and 0.5C l,tot <C l,sin according to the calculation results, and the difference between the averaged coefficient and single airfoil lift coefficient is called the loss of sail interference. Compared with the lift coefficient of single airfoil the lift coefficient loss percentage is shown in Fig. 5.
From Fig. 5, it can be seen that the lift coefficient loss percentage decreases gradually with the increase of distance at different attack angles. Before the distance is c, the lift coefficient loss percentage changes dramatically, which means that with the increase of distance, the influence on the aft sail decreases rapidly. After 1c, the lift coefficient loss percentage declines with the increase of the distance and the loss of lift coefficient decreases more smoothly. When the distance is 3c, the lift coefficient loss decreases to smaller than 1.8%.

Influence of airfoil combination and distance on total lift coefficient
In the third section, the lift coefficients are analyzed with the distance and attack angle as variables when the fore and aft sail airfoils are the same. For further study, this section analyzes the different combination of the fore and aft sail airfoils. Similarly, based on the calculation program written in this article, the influence of airfoil combination and distance on lift coefficient at 9° of attack was calculated. We have tacitly assumed that the fore sail airfoil is thinner than the aft sail airfoil. The analysis results are shown in Fig. 6. The curves in Fig. 6 show the variation of the total lift coefficient corresponding to the aft sail airfoil.
From Fig. 6, it can be seen that the total lift coefficient of the two sails increases with the increase of the thickness of the aft sail airfoil, and the total lift coefficient increases to a stable state with the increase of the distance between the fore and aft sail. When the distance factor is between 0 and 1, the fore sail airfoil is fixed. The larger the thickness of the aft sail airfoil is, the faster the total lift coefficient increases.
In order to analyze the influence of the fore sail on the aft sail, the aft sail airfoil is set to NACA0024, and the attack angle is 9°. By changing the airfoil of the fore sail and the distance between two sails, the lift coefficient of the aft sail was extracted and the change curve was drawn as shown in Fig. 7. As can be seen from Fig. 7, no matter which kind of airfoil the fore sail is, with the increase of distance factor, the lift coefficient of the aft sail increases gradually, which is close to the lift coefficient of the NACA0024 airfoil at attack angle of 9°. With the increase of the thickness of the fore sail airfoil, the lift coefficient of aft sail decreases gradually, which indicates that the thicker the fore sail is, the greater the interference and influence on the aft sail will be. However, the variation trend of the total lift coefficient of the double sail propulsion system is consistent with the increase of the distance.

Conclusions
The lift coefficient changes caused by different distances between two sails, different attack angles and different airfoils were investigated.
First of all, compared with a single sail, double sail can effectively reduce the height of wind pressure center, reduce the heeling arm, and reduce the sail resistance caused  Fig. 6. Variation of lifting coefficient at different distance and airfoils at attack angle 9°. SUN Zhao-yang et al. China Ocean Eng., 2019, Vol. 33, No. 6, P. 746-752 by heeling.
Then, according to the theory of the vortex panel method, the calculation method of double sail aerodynamic coefficients was established.
Furthermore, the aerodynamic performance of the aft sail will be affected by the fore sail in the range of 1°−11°a ttack angles. With the increase of distance, the influence of the fore sail on the aft sail will gradually decrease, and the total lift coefficient of two sails will gradually rise and tend to be stable in the later period. In the double sail propulsion system, when the aft sail airfoil is fixed, the thinner the fore sail airfoil is, the smaller the interference on the aft sail will be, and the smaller the distance is required when the total lift coefficient tends to be stable.
Finally, for the NACA0012 airfoil commonly used on autonomous sailboats, the distance between fore and aft sails should be no less than the length of the chord. If the deck is long enough, three times of chord length can be adopted.
In addition to the reference lift coefficient, the lift-drag ratio should also be taken into account in the selection of airfoils for autonomous sailboat. The variation of lift coefficients of different airfoils with different distances studied in this paper can provide references for the selection of airfoils and installation of airfoils for autonomous sailboat. In the follow-up study, the aerodynamic characteristics of three-dimensional wing sail should be further explored. In addition, the calculation in this paper is based on the potential flow theory. The calculation results do not contain the correction of the viscous boundary layer, therefore they may not be accurate enough. Later, the analysis method in this paper can be combined with the boundary layer theory.