Abstract
As the helicopters in a multi-lift system fly in a close formation, there is severe aerodynamic interference between the wake of the rotors, bringing complex aeromechanics coupling. So it is necessary to investigate interference and resulting performance changes before studying performance optimization and advanced formation control. A baseline configuration of four tandem helicopters carrying a load cooperatively in a “2-lead” formation is performed to explore the interference and performance. A vortex-panel approach based on the viscous vortex particle method is employed to investigate the performances and flow fields in a steady-flight state. The steady-flight state is obtained by a hierarchical trimming method, and the vortex-panel approach is validated by wind tunnel experiments. On this basis, aerodynamic interferences and performances at different flight speeds and variant relative positions are investigated. Computational results indicate that for the baseline configuration, there exists serious interference between helicopters in the front-and-rear arrangement, especially at forward flight. At the advance ratio of 0.1, there exists a 20% thrust loss and a 15% power increase for the front rotor of the tandem helicopter behind the formation. The aerodynamic interference will be reduced significantly if the distance between the front and the rear helicopter meets any of the three conditions below: more than 3.5D(D represents the rotor diameter) in the longitudinal direction, more than 0.75D in the lateral direction, or more than 0.5D in the vertical direction.
The capability of heavy cargo transportation makes helicopters unique and invaluable in both civil and military applications, especially in situations where other aircrafts cannot easily reach the specified destinations due to operational limit
The concept of using two or more helicopters to carry a load cooperatively, known as the multi-lift system, has been proposed for several decades. Two basic configurations named pendant and spreader bar (shown in
Fig.1 Two basic configurations
Separation distances of two to four rotor diameters are expected to ensure safety for a close formation flight of manned helicopters. The multi-lift system is a typical proximity flight demonstration. Modeling and control analyses for twin-lift system were conducted in Refs.[
Related studies have shown that there exist complex and serious aerodynamic interferences in a proximity flight of rotorcrafts. Ref.[
Fig.2 Different arrangements for two helicopters
Moreover, latest research focused more on load distribution for the multi-lift system. The load distribution concept of equalizing cable tension to improve system performance for dual lift system was accepted in most studie
The aerodynamic interference between helicopters is investigated in this paper for the multi-lift system. Section 1 presents the baseline configuration of four tandem helicopters carrying a load cooperatively. The aerodynamic interference computation method based on the vortex method is introduced in Section 2. The validation for the computation method is demonstrated in Section 3. On the basis of Section 4, the aerodynamic interference between helicopters at a certain forward speed is calculated and a recommended formation is proposed. At last, Section 5 summarizes the conclusions.
The baseline configuration of the multi-lift system consists of four small tandem helicopters and a slung load with a mass of 10 kg. It forms a “2-lead” formation as shown in
Fig.3 Baseline of the multi-lift system with helicopters
Moreover, the configuration of the small tandem helicopter is summarized in
Item | Value |
---|---|
Takeoff mass / kg | 15.0 |
Inertia on x‑axis / () | 0.284 |
Inertia on y‑axis / () | 2.065 |
Inertia on z‑axis / () | 2.083 |
Radius of rotor / m | 0.9 |
Chord of rotor / m | 0.069 |
Twist angle of rotor / (°) | 0 |
Blade number | 2 |
Rotational speed of rotor / (rad∙ | 113.1 |
Position of front rotorin body axis / m | (0.582 5, 0, -0.25) |
Position of rear rotorin body axis / m | (-0.582 5, 0, -0.25) |
Position of suspend pointin body axis / m | (0,0,0) |
Viscous vortex particle method (VVPM) in Refs.[
(1) |
where , and are the spatial position, vorticity density vector and radius of the vortex particle, respectively. is the cutoff function considering the distribution of vorticity caused by the induction influence of each particle, where is the smoothing parameter.
Then consider the vortex dynamics equation (curl form of Navier-Stokes equation) as
(2) |
where is the vorticity field, the material derivative, and the Laplacian operator. The governing equations of vortex particle density and position can be obtained by solving the joint
(3) |
where is the viscous diffusion effect of vortex particles, which represents the influence of air viscosity in the whole process of vorticity transport.
The equation above is the discrete governing equation in the convection-diffusion form, with the vorticity and spatial position of vortex particles changing with time. The local velocity of vortex particles is determined by the combination of free-stream velocity and induced velocity of vorticity field. Here, the induced velocity term can be solved by the Biot-Savart’s theorem, i.e.
(4) |
where is the Biot-Savart kernel, and the vector form of Green’s function. By substituting the governing equation of vortex particle field into the above equation, the solution formula of induced velocity generated by vortex particle field can be obtained as
(5) |
The expression of the kernel function should correspond with cutoff function , and the kernel function is the Rosenhead-Moore kernel, where is the radius of the vortex core, which is determined by the resolution of the vortex particles filed as
(6) |
In this study, is solved by particle strength exchange (PSE) metho
(7) |
where and are the and particles’ volume, respectively. Since the kernel function will decay rapidly with the increase of distance, only the vortex particles close to pth particle will be considered, and the influence of all particles exceeding the set truncation distance will be ignored. Therefore, only the influence of adjacent particles will be reflected in the calculation. In this study, the cutoff distance of PSE method is consistent with the region division distance of acceleration algorithm discussed below.
It can be found from the above induced velocity calculation of vortex particles that the total contribution of N particles needs to be included in the solution of particle convection velocity term or velocity gradient term. This is similar to the classical N-body problem. For this type of problem, TreeCod
Because the interference between multiple rotors is involved, the order of magnitude of vortex particles involved in this study is large. To optimize computational efficiency to the greatest extent possible, the FMM algorithm is used as the acceleration technology. In FMM algorithm, if the interval between two vortex element regions (a divided group of vortex particles) is greater than the set truncation distance, the influence of the source region on the target region is first expanded into a multipole series. In the target region, the multipole series is transformed into a local Taylor expansion, so as the induced velocities of all vortex particles in the region can be quickly gained. FMM algorithm is used to calculate the induced velocities, velocity gradients and viscous diffusion effects of vortex particles.
The aerodynamic model of rotor blade involves the calculation of rotor thrust, blade flapping and induced velocity. The method of lifting lin
In the study, the whole blade is divided into several micro segments along the spanwise direction first, then it is represented by the middle arc surface without thickness. For the blade with sweepback and taper, the corresponding mesh lines should be modified at the front and rear edges of the blade, and the grid partition should be corresponding to the blade segments.
Fig.4 Vortex panel grid partition of a blade with sweepback
As the blade is divided into several parallel columns along the spanwise direction and several rows along the chord direction, the blade surface is composed of many spanwise and chord grids. Although the spanwise number of each blade segment is different, the chord grid number should remain the same. The middle arc surface of the blade is replaced by the vortex quadrilateral, the spanwise attached vortex of the vortex quadrilateral is located at the quarter chord line of the grid, and the chord attached vortex is along the spanwise grid. At the boundary of the grid, the trailing edge vortex of the vortex quadrilateral is located at the quarter chord of the adjacent grid in the chord direction. In the trailing edge area of the blade, the vortex lattice equivalently generates vortex particle clusters, as shown in
Fig.5 Lifting surface vortex panel distribution and vortex particles generated in trailing edge
The vorticity source generated at each trailing edge of blade segment and shed into rotor wake can be calculated by
(8) |
where is the strength of the newly generated vortex, the vector form of the blade bound circulation, and the local velocity vector of the bound vortex consist of the blade structure (including rotation and flapping) motion, free-stream wind, induced velocity and the interference velocities from other sources. In
The hybrid hierarchical optimization trimming method from Ref.[
Fig.6 Hybrid hierarchical optimization trimming method in Ref.[
The trimming method at a certain flight speed mainly includes the following steps:
Step 1 Load trim. Calculate total forces and moments from cables to trim the load.
Step 2 Using the total forces and moments, apply the optimal force allocation strategy to determine the force distribution among each cable.
Step 3 Trim the four tandem helicopters, and get related trimming variables.
Step 4 Based on the trimming variables, calculate aerodynamic interference by vortex method at the certain flight speed.
Step 5 Introduce the aerodynamic interference into the helicopter and load models.
Step 6 Retrim the load and the helicopter by improved delta trimming method, and get the new trimming variables.
Step 7 If the difference between the last two trimming results is small enough, end trimming; if not, return to Step 4.
It should be noted that the above-mentioned optimal force allocation strategy and trimming results can be found in Ref.[
To validate the computational method, the calculated thrust coefficient and sectional lift coefficients of the Caradonna‑Tung rotor in hover, downwash velocities of the scaled model rotor in forward flight, thrust and power coefficients of tandem rotors are predicted and compared with the wind-tunnel test data.
The experiment for Caradonna‑Tung rotor in hover was conducted in the Army Aeromechanics Laboratory’s hover test facilit
Fig.7 Calculation results for the Caradonna‑Tung rotor in hover
The scaled model rotor in the wind-tunnel tes
Fig.8 Wake structure for the scaled model rotor at advanced ratio of 0.15
Fig.9 Inflow velocities in longitudinal direction
Fig.10 Inflow velocities in lateral direction
The study involves the prediction and comparison of thrusts and power requirements for the tandem-rotors configuration in both hover and level forward flight states with experimental dat
Considering the symmetry arrangement of tandem rotors in hover, only the performance of the front rotor is shown in
Fig.11 Hover performance of tandem-rotors
Fig.12 Level-flight performance at CT=0.003 4
The numerical simulations of the multi-lift system are based on the steady flight states obtained by the hybrid hierarchical optimization trimming method. As shown in
Advance ratio | 0 | 0.04 | 0.06 | 0.08 | 0.1 | 0.14 | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Heli 1 | Heli 4 | Heli 1 | Heli 4 | Heli 1 | Heli 4 | Heli 1 | Heli 4 | Heli 1 | Heli 4 | Heli 1 | Heli 4 | |
/ (°) | 5.5 | 5.7 | 4.0 | 5.0 | 3.7 | 4.8 | 4.2 | 4.6 | 4.1 | 4.6 | 4.6 | 5.2 |
/ (°) | 0.0 | 0.0 | 0.4 | 0.8 | 0.9 | 1.2 | 0.7 | 1.4 | 0.9 | 1.2 | 1.0 | 1.0 |
/ (°) | 1.7 | 1.8 | 1.4 | 1.7 | 1.5 | 1.7 | 1.8 | 1.6 | 1.8 | 1.6 | 1.8 | 1.5 |
/ (°) | 0.0 | 0.0 | 0.6 | 1.0 | 1.2 | 1.5 | 1.2 | 1.7 | 1.5 | 1.7 | 1.9 | 1.9 |
/ (°) | 5.6 | 5.8 | 4.5 | 5.3 | 4.4 | 5.7 | 4.7 | 5.8 | 4.9 | 5.7 | 5.4 | 6.1 |
/ (°) | 0.0 | 0.0 | 0.4 | 0.7 | 0.8 | 1.1 | 0.7 | 1.2 | 0.7 | 1.1 | 0.7 | 0.9 |
/ (°) | 1.7 | 1.8 | 1.7 | 1.8 | 1.8 | 2.1 | 2.0 | 2.1 | 2.2 | 2.1 | 2.2 | 2.0 |
/ (°) | 0.0 | 0.0 | 0.7 | 1.0 | 1.2 | 1.5 | 1.3 | 1.8 | 1.6 | 1.8 | 2.1 | 2.1 |
(°) | 3.4 | -3.9 | 2.8 | -5.3 | 1.8 | -7.2 | 0.5 | -7.7 | -0.8 | -8.9 | -4.3 | -11.4 |
It should be noted that it is essential and necessary to get the steady-flight states before interference exploration. Without trimming, arbitrarily given attitudes of helicopters or manipulations of the rotors cannot reflect the actual flight status of the multi-lift system, and related results of interference or performance will be meaningless. Trimming results in
Based on the trimming results above, several steady-flight states are calculated first to explore the aerodynamic interferences. For example,
Fig.13 Wake structure in hovering
Fig.14 Wake structure at advance ratio of 0.1
Fig.15 Sectional velocity fields of Heli 1(right) and Heli 4(left) at different flight speeds
At the advance ratio of 0.06 shown in
In conclusion, there is almost no aerodynamic interference while hovering for the baseline configuration of four tandem helicopters carrying a load cooperatively. With the increased flight speed, the interference becomes more and more serious, especially between the helicopters in the front-rear arrangement. Besides, it can be found that interference is complex and will bring different effects on performance at different flight speeds. For seeking ways to improve performance and reduce interference meanwhile, the following sections focus on exploring the interference variations and related influence factors.
In the following discussion, Heli 1 and Heli 4 are taken as examples. The definitions of rotational directions of the front and rear rotors and longitudinal and lateral relative distances between Heli 1 and Heli 4 are shown in
Fig.16 Arrangements of Heli 1(right) and Heli 4(left)
Besides, we assume that in the following discussions, the differences of lateral, longitudinal and height relative positions have little effect on the trimming state. So, the trimming variables are fixed as those in
And in the following figures, 100% thrust of the front rotor corresponds to 95 N and that of the rear rotor to 83 N. Meanwhile, 100% power of the front rotor corresponds to 310 W and that of the rear rotor to 452 W. These values of Heli 1 are obtained by the calculation results when Heli 4 is excluded.
This section shows the aerodynamic interferences with changes in lateral relative positions from -1.5 to 1.5, where represents the rotor diameter, -1.5 represents the arrangement in which Heli 1 is located at the lateral position of negative 1.5 rotor diameters relative to that of baseline configuration, and 1.5 represents the arrangement in which Heli 1 is located at the lateral position of positive 1.5 rotor diameters relative to that of baseline configuration.
In
Fig.17 Sectional vorticity fields of Heli 1(right) and Heli 4(left) with different lateral relative positions
Fig.18 Sectional force at 0.75R of the front rotor with different lateral relative positions of Heli 1
Fig.19 Sectional force at 0.75R of the rear rotor with different lateral relative positions of Heli 1
Fig.20 Thrust variation with the change of lateral relative position
Fig.21 Power variation with the change of lateral relative position
In summary, there exist larger thrust loss and higher power consumption for Heli 1 in the baseline configuration. The thrust loss and power consumption will be reduced by adjusting the relative lateral distance between Heli 1 and Heli 4 appropriately. Furthermore, within lateral distance ranges from -1.5D to -0.75D and from 0.75D to 1.5D, the performance is even better than that of the case without interference.
Calculation of aerodynamic interference with the change of longitudinal relative position is demonstrated in this section.
Fig.22 Sectional vorticity fields of Heli 1(right) and Heli 4(left) with different longitudinal relative positions
Fig.23 Sectional force at 0.75R of the front rotor with different longitudinal relative positions of Heli 1
Fig.24 Sectional force at 0.75R of the rear rotor with different longitudinal relative positions of Heli 1
Fig.25 Thrust variation of the rear rotor with different longitudinal relative positions
Fig.26 Power variation of the rear rotor with different longitudinal relative positions
Fig.27 Thrust variation of the front rotor with different longitudinal relative positions
Fig.28 Power variation of the front rotor with different longitudinal relative positions
The sectional vorticity fields in X-Z plane with three typical relative heights are shown in
Fig.29 Sectional vorticity fields with different relative heights
Sectional forces at 0.75R of the front and rear rotors with different relative heights are shown in
Fig.30 Sectional force at 0.75R of the front rotor with different relative heights of Heli 1
Fig.31 Sectional force at 0.75R of the rear rotor with different relative heights of Heli 1
Fig.32 Thrust variation of the front rotor with different relative heights
Fig.33 Power variation of the front rotor with different relative heights
Fig.34 Thrust variation of the rear rotor with different relative heights
Fig.35 Power variation of the rear rotor with different relative heights
Based on the explorations above, at advance ratio of 0.1, the latera relative distances from -1.5D to -0.75D and 0.75D to 1.5D are beneficial to reduce interference, thrust loss and power consumption. There is almost no interference when the longitudinal distance is larger than 3.5D. Locating the rear helicopter 0.5D higher or more than the front one is favorable to improve performance. Since the interference at advance ratio of 0.1 corresponds the relatively worst case, the conclusions above are suitable for hovering and other advance ratios smaller than 0.1.
Therefore, it can be concluded that larger lateral or longitudinal distance and locating the helicopter behind at higher height are favorable from the perspective of reducing aerodynamic interference and improving performance. This means the isosceles trapezoid formation and the rectangle formation in which helicopters behind are located at higher heights or far enough in the longitudinal direction as shown in
Fig.36 Isosceles trapezoid formation
Fig.37 Rectangle formation
0.75D lateral | 3.5D longitudinal | 0.5D height | |||
---|---|---|---|---|---|
Thrust | Power | Thrust | Power | Thrust | Power |
7.9%↑ | 8.1%↓ | 6.7%↑ | 5.8%↓ | 2.9%↑ | 4.7%↓ |
The aerodynamic interference and resulted performance changes of helicopters in the multi-lift system at steady flight state are investigated by the vortex approach. A baseline configuration of four tandem helicopters carrying a load cooperatively with the “2-lead” formation is introduced to explore the interferences. The vortex approach combining the lifting surface theory and viscous vortex method is validated by related wind tunnel test data. The steady flight states are calculated based on the hybrid hierarchical trimming method. On the basis, several numerical simulations are developed and the following conclusions are obtained:
(1) There indeed exists serious interference between helicopters in front-and-rear arrangement at forward flight state. The interference is complex and resulted effects on performance are different at different advance ratios.
(2) At the advance ratio of 0.1, for the baseline configuration there exist a 20% thrust loss and 15% power increase for the front rotor of the tandem helicopter behind.
(3) At the advance ratio of 0.1, the lateral relative distances from -1.5D to -0.75D and 0.75D to 1.5D are beneficial to reduce interference, thrust loss and power consumption. There is almost no interference when the longitudinal distance is larger than 3.5D. Locating the rear helicopter higher than the front one is favorable to improve performance.
(4) The isosceles trapezoid formation and the rectangle formation in which rear helicopters are located at higher heights or far enough in the longitudinal direction are recommended to reduce interference and improve performance.
Contributions Statement
Dr. DING Zhiwei wrote the core code of arodynamics and conducted related analysis. Dr. DUAN Dengyan contribured to model components and data analysis. Mr. ZHAO Gang and Ms. XUAN Jinting contributed to the discussion and background of the study. Prof. LI Jianbo proposed the problem and provided technical support. All authors commented on the manuscript draft and approved the submission.
Conflict of Interest
The authors declare no competing Interests.
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Authors Dr. DING Zhiwei received his B.S. degree in engineering from the National Key Laboratory of Rotorcraft Aerodynamics, Nanjing University of Aeronautics and Astronautics, in 2013. In 2014, he studied for a M.S. degree at the same laboratory and switched to studying for a Ph.D degree in 2016. His primary focus is rotorcraft aerodynamic modeling and optimization design, and his research interests mainly include high-speed rotorcraft and new configuration eVTOL. [Baidu Scholar]
Prof. LI Jianbo received the B.S. and Ph.D. degrees in aerospace vehicle design from Nanjing University of Aeronautics and Astronautics (NUAA), Nanjing, China, in 2000 and 2003, respectively. From 2006 to 2008, he was an assistant professor in State Key Laboratory of Helicopter Rotor of Dynamics, NUAA and in 2009, he became a full professor. His research has focused on aerodynamics, flight dynamics and preliminary design of traditional helicopter, wind milling rotor aircraft, ducted fan aircraft and rotor/wing compound aircraft. [Baidu Scholar]