Effects of fuselage shape on helicopter aerodynamic performance in case of intermeshing-rotor interference

To study the influence of fuselage shape on the aerodynamic performance of a helicopter under the interference of an intermeshing-rotor, a numerical simulation method of flow field under the forward flight state of Synchropter is established based on the lattice Boltzmann method. Firstly, the Maryland helicopter was calculated to verify the scientific validity of the method. Then, the simplified S-97 fuselage was used as the object of analysis to analyze the aerodynamic data such as surface pressure, friction, and lift resistance of the fuselage with different head curvature, cross-section shapes, and transition segments under Synchropter interference with different attack angles in forward flight. The results show that increasing the radius of curvature of the nose can reduce the total drag of the fuselage; the fuselage with a circular cross-section shape has better aerodynamic characteristics of lift resistance, and the fuselage with rhombic cross-section shape has better transverse stability; the transition section with backward position, smaller upward dihedral angle, and smooth transition can avoid serious airflow separation and achieve the purpose of drag reduction.


Introduction
The intermeshing-rotor helicopter has the advantages of a large load, stable handling, strong wind resistance, etc.It considers the advantages of the twin rotor configuration, which is smaller than other twin rotor shape resistance and better forward flight performance.It is a helicopter configuration with good comprehensive performance.In helicopter flight, the fuselage drag significantly influences flight performance, and the fuselage drag is more than 50% of the total forward flight drag [1] .Therefore, to increase the flight speed and improve the flight performance of the intermeshing-rotor helicopter, the study of fuselage shape is very important.
Research has been conducted on the aerodynamic and flow field characteristics of intermeshing-rotor helicopters both domestically and internationally.Kenji [2] theoretically analyzed the waving motion and induced power loss of the cross-twin rotor helicopter in the hovering state, providing a theoretical basis for analyzing the aerodynamic characteristics of this configuration.Masashi et al. [3] proposed an additive and subtractive mesh method and carried out numerical simulation calculations on the K-MAX helicopter to prove the validity of the method.The effectiveness of the method is proven.Gao [4] developed a crossover unmanned helicopter based on aerodynamic research and carried out hovering tests and field test flights.Wang [5] established the aerodynamic calculation model, the fuselage aerodynamic model, and the flight dynamics model of the crossover dual-rotor helicopter.Wu et al. [6] IOP Publishing doi:10.1088/1742-6596/2764/1/012012 2 established a numerical simulation method suitable for the aerodynamic characterization of a crossed dual rotor for the problem of aerodynamic interference and calculated the rotor tension and hovering efficiency.Shen et al. [7] analyzed the aerodynamic interference and non-constant loads of a crossed dual-rotor under a hovering state by using the non-constant vortex lattice method in combination with the foliation momentum theory, which provides a specific theoretical basis for the study of complex aerodynamics of crossed twin-rotor helicopters.However, there are fewer studies on the effects of different shapes on the airframe drag and flow field under the interference of a crossed twin-rotor helicopter.
Based on CFD numerical simulation software, this paper establishes a model for calculating the flow field of a cross-twin rotor helicopter fuselage, verified by an example of a Maryland fuselage under single rotor interference.On this basis, the method is used in the numerical simulation of the sample fuselage and compared with the flow field of the changed fuselage shape to analyze the effects of the fuselage with different head curvature, cross-section shapes, and transition segments on the aerodynamic performance of the fuselage under the interference of the cross-twin-rotor.

Control equations
The Lattice Boltzmann Method (LBM) is an approach based on mesoscopic scale modeling and the Boltzmann theory of molecular dynamics of gases by discretizing the Boltzmann equation, i.e., obtaining the basic equation of LBM, the Lattice Boltzmann equation.
In Equation (1): is the velocity of the hypothetical particle in the direction of i , δx is the lattice length, δt is the discrete time step,  0 is the average time interval between the two collisions, which is also called the relaxation time.
The Wall-Adapting Local Eddy viscosity model is suitable for large eddy simulation of complex wall cases and was chosen to provide continuous local eddy viscosity and near-wall characteristics [8] .
In this model, the vortex viscosity coefficient for sublattice motion is:


is the filter size, cw is the model constant 0.325 indicating the filter width, s is the strain rate tensor at the solution scale, and G d αβ is the traceless symmetric part of the square of the velocity gradient tensor.
A generalized wall law that takes into account the effects of unfavorable and favorable pressure gradients is used to model the boundary layer: In Equation (3): U is the average velocity at a certain distance from the wall; y denotes the distance to the wall; ur is the surface friction velocity; up is the characteristic velocity of the counterpressure gradient;   is the shear stress of the turbulent wall;

d / dp
x ω is the wall pressure gradient; f 1 and f 2 are the interpolation functions given by Shih [9] .3

Arithmetic validation
The experimental data of the single-rotor helicopter are used to validate the numerical simulation method because there is less experimental data on the cross twin-rotor helicopter, and the single-rotor helicopter also has aerodynamic interference to the helicopter fuselage, so the experimental data of the single-rotor helicopter are used to validate the numerical simulation method.The Maryland fuselage was chosen for the example validation of the fuselage because of its rich experimental data, which can provide accurate data assurance.The computational domain and lattice refinement of the Maryland fuselage are shown in Figure 1, and the computational conditions are set according to the test conditions of the Maryland fuselage in the wind tunnel [10] .The surface pressure coefficient versus azimuth change curves of the fuselage corresponding to the positions of the monitoring points are taken through post-processing and compared with the experimental values proposed by Bi and Leishman [10] .The results are shown in Figure 2. The pressure coefficient versus azimuth change curves of the fuselage at the various monitoring points is in good agreement with the experimental values, which indicates that the numerical calculation method used in this paper is effective for monitoring the details of the interference between rotor and fuselage of the helicopter when the helicopter is forward-flying.

Characterization of frictional resistance of fuselage parts
The simplified Sikorsky S-97 fuselage is selected as the research object.The prototype fuselage is simplified, as shown in Figure 3.When the length of the fuselage is L, the fuselage can be divided into four parts according to the scale: the head of the fuselage (the red part), the middle part of the fuselage (the yellow part), the transition part of the fuselage (the green part), and the tail part of the fuselage (the blue part).The blade airfoil is NACA0012 airfoil; the blade is a flat blade with no torsion, the center distance between the left and right two rotors is L=0.3R, the initial crossing angle of the two rotor blades is 90°, the steering is reversed, the rotor axis angle is 24°, ignoring the influence of the blade hub, and the fuselage assembly is shown in Figure 4. Using the above numerical simulation method, the scaled-down model is used to calculate and set the parameters related to the calculation: The incoming velocity of the virtual wind tunnel is 27.78 m/s, and the fuselage head-on angles are -5°, 0°, and +5°, respectively, to calculate the airframe flow field under the interference of the crossed twin rotor in the forward flight state.
Figure 5 shows the pressure and surface friction coefficient cloud diagrams around the fuselage at different angles of approach.From the pressure cloud diagram of the symmetric section of the fuselage at a certain moment, it can be seen that the static pressure near the head of the fuselage is the largest, and with the increase of the angle of approach, the area of the largest static pressure region decreases.The pressure over the rotor is smaller than that under pressure from the fuselage surface friction coefficient cloud diagram.It can be seen that the smallest surface friction coefficient is in the small area of the head of the fuselage, and with the increase of the angle of attack, the area of this area decreases.The surface friction coefficient is the largest at the largest area in the fuselage cross-section, and with the decrease of the cross-sectional area, the surface friction coefficient decreases.angles of attack.The drag coefficients for each part of the fuselage at the three angles of approach are shown in Table 1.It can be seen that the total surface friction coefficient of the fuselage increases as the angle of approach increases, which leads to an increase in the drag force on the fuselage.In practice, because the tail will be installed at the tail of the fuselage, it is not very meaningful to analyze the tail of the fuselage alone, and the surface friction coefficient of the fuselage cross section at the maximum of the fuselage mid-section can be known from the above analysis.Therefore, the next relevant analysis is carried out with the nose curvature, fuselage cross-section, and fuselage transition section as the research objects.

Effect of nose curvature on aerodynamic performance
The smooth curvature can reduce the separation and turbulence when the air flows through the nose and reduce the generation of resistance.The nose curvature can also affect the stability of the aircraft.A IOP Publishing doi:10.1088/1742-6596/2764/1/0120126 reasonable nose curvature design can help to maintain a stable flight attitude and reduce aerodynamic disturbances during flight.Therefore, the radius of curvature of the leading edge of the nose is taken as the object of study, and its effect aerodynamic performance is analyzed under the interference of the intermeshing rotor.Figure 6 gives a schematic diagram of the three cases of nose leading edge radius of curvature, and the radius of curvature of the three cases gradually increases.Numerical simulation is carried out for them under the flight condition with a heading angle of 0°, and the relevant data are shown in Table 2. Table 2. Data related to the three cases.From Table 2, it can be seen that the total fuselage drag gradually decreases with the increase of the nose curvature radius.To further study the mechanism of the influence of the nose curvature on the total fuselage drag, the cloud diagram of the pressure distribution at the nose under different curvature radii is given in Figure 7.It can be seen that, with the increase of the nose curvature radius, the area of the most pressurized area of the nose increases significantly.However, at the same time, the area of the whole high-pressure area decreases significantly, which leads to the reduction of the differential pressure drag of the fuselage.Figure 8 shows the pressure cloud of a symmetric section of the fuselage with different curvature radii at a particular moment, from which it can be seen that the smaller the radius of curvature of the nose is, the larger the contact area between the high-pressure area of the fuselage front and the fuselage is, which ultimately leads to an increase in the fuselage drag.
The surface pressure coefficient and friction coefficient of the nose with different curvature radii at a certain moment of symmetric cross-section are obtained after post-processing, and the comparison graphs are shown in Figure 9.It can be seen that the peak value of the surface pressure coefficient and the minimum value of the surface friction coefficient appear in the most forward end of the nose.With the increase of the nose curvature radius, the larger the peak interval of the surface pressure coefficient is, which leads to the increase of the overall surface pressure.At the same time, the surface friction coefficient decreases, which ultimately leads to the reduction of the fuselage resistance.

Effect of fuselage cross-section on aerodynamic performance
From the above calculations and analysis, it can be seen that the drag in the transition section between the rear fuselage and the tail beam accounts for a more significant proportion of the total drag in the fuselage.This section will produce more aerodynamic severe separation phenomenon.Hence, this section investigates the effect of the transition section of the fuselage on the aerodynamic performance by changing the shape of the transition region from the location of the largest section underneath the fuselage to the tail beam (as shown in Figure 10).Figure 11 shows the calculated pressure coefficient distributions along the top and bottom fuselage lines at a head-on angle of 0°.The pressure on the lower surface of the fuselage decreases at the transition section.In contrast, the pressure value recovers when approaching the tail beam due to the effect of airflow separation caused by the shape change of the transition section.Compared with the lower surface, the pressure coefficient of the upper surface of the fuselage does not change drastically due to the slight change in shape, except for the immense pressure change at the nose.The comparison of the four transition forms shows that the peak pressure fluctuation caused by the small arc transition form is relatively the largest, followed by the concave form, and the smallest is the convex form.The pressure changes on the upper surface of the fuselage for the four transition forms are the same, indicating that the change in the shape of the lower surface of the fuselage has almost no effect on the pressure changes on the upper surface.sections.Through the above calculation and analysis, no matter what kind of transition form is used, the pressure on the lower surface of the fuselage will be decreased after the transition section, so when choosing the position of the transition section, under the premise of careful consideration of the weight of the fuselage, the position of the beginning of the transition section should be as far as possible back to postpone the position of the beginning of the pressure drop.At the same time, because the increase of the slope angle of the transition section will lead to an increase in resistance and will cause more 0

Figure 1 .
Figure 1.Schematic diagram of computational domain and lattice refinement.

Figure 2 .
Figure 2. Comparison of maryland airframe calculation results.The surface pressure coefficient versus azimuth change curves of the fuselage corresponding to the positions of the monitoring points are taken through post-processing and compared with the experimental values proposed by Bi and Leishman[10] .The results are shown in Figure2.The pressure coefficient versus azimuth change curves of the fuselage at the various monitoring points is in good agreement with the experimental values, which indicates that the numerical calculation method used in this paper is effective for monitoring the details of the interference between rotor and fuselage of the helicopter when the helicopter is forward-flying.

Figure 3 .
Figure 3. Schematic diagram of fuselage segment dimensions.The blade airfoil is NACA0012 airfoil; the blade is a flat blade with no torsion, the center distance between the left and right two rotors is L=0.3R, the initial crossing angle of the two rotor blades is 90°, the steering is reversed, the rotor axis angle is 24°, ignoring the influence of the blade hub, and the fuselage assembly is shown in Figure4.

Figure 4 .
Figure 4. Rotor and fuselage assembly diagram.Using the above numerical simulation method, the scaled-down model is used to calculate and set the parameters related to the calculation: The incoming velocity of the virtual wind tunnel is 27.78 m/s, and the fuselage head-on angles are -5°, 0°, and +5°, respectively, to calculate the airframe flow field under the interference of the crossed twin rotor in the forward flight state.Figure5shows the pressure and surface friction coefficient cloud diagrams around the fuselage at different angles of approach.From the pressure cloud diagram of the symmetric section of the fuselage at a certain moment, it can be seen that the static pressure near the head of the fuselage is the largest, and with the increase of the angle of approach, the area of the largest static pressure region decreases.The pressure over the rotor is smaller than that under pressure from the fuselage surface friction coefficient cloud diagram.It can be seen that the smallest surface friction coefficient is in the small area of the head of the fuselage, and with the increase of the angle of attack, the area of this area decreases.The surface friction coefficient is the largest at the largest area in the fuselage cross-section, and with the decrease of the cross-sectional area, the surface friction coefficient decreases.

Figure 5 .
Figure 5. Cloud diagram of pressure and surface friction coefficient around the fuselage at differentangles of attack.The drag coefficients for each part of the fuselage at the three angles of approach are shown in Table1.It can be seen that the total surface friction coefficient of the fuselage increases as the angle of approach increases, which leads to an increase in the drag force on the fuselage.Table1.Surface friction coefficients of fuselage parts at different angles of attack.

Figure 6 .
Figure 6.Schematic diagram of three types of nose leading edge curvature radii cases.Table2.Data related to the three cases.

Case 1 (
small curvature) Case 2 (medium curvature) Case 3 (large curvature) Radius of curvature of a

Figure 7 .
Figure 7. Pressure distribution cloud at the nose with different radii of curvature.Figure8shows the pressure cloud of a symmetric section of the fuselage with different curvature radii at a particular moment, from which it can be seen that the smaller the radius of curvature of the nose is, the larger the contact area between the high-pressure area of the fuselage front and the fuselage is, which ultimately leads to an increase in the fuselage drag.

Figure 8 .
Figure 8. Pressure clouds of symmetric sections of fuselage with different radii of curvature at a certain moment in time.

Figure 9 .
Figure 9.Comparison of pressure coefficient and friction coefficient on the surface of the head with different radii of curvature.

Figure 10 .
Figure 10.Schematic diagram of the shape of different fuselage transition segments.

Figure 11 .
Figure 11.Distribution of pressure coefficients of fuselage top and bottom lines for different transitionsections.Through the above calculation and analysis, no matter what kind of transition form is used, the pressure on the lower surface of the fuselage will be decreased after the transition section, so when choosing the position of the transition section, under the premise of careful consideration of the weight of the fuselage, the position of the beginning of the transition section should be as far as possible back to postpone the position of the beginning of the pressure drop.At the same time, because the increase of the slope angle of the transition section will lead to an increase in resistance and will cause more

Calculation results and analysis
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Table 1 .
Surface friction coefficients of fuselage parts at different angles of attack.