Study and analysis of lift to drag ratio performance of supercritical wing based on computational fluid dynamics method

As the main airfoil of large fixed-wing aircraft at present, the supercritical wing has better performance than the traditional airfoil, especially its aerodynamic performance. Therefore, it is necessary to calculate and experiment with the supercritical wing to optimize its structure. In this paper, computational fluid dynamics (CFD) technology is used to evaluate and compare the performance of a supercritical wing (half wingspan) and a flat wing (half wingspan) with similar size and the same boundary conditions. First, shrink modelling of the two airfoils is performed using the similarity principle. Then it was imported into the simulation environment to calculate the lift-drag ratio under different boundary conditions and draw a corresponding dot plot for comparison. The results show that the aerodynamic indexes of the supercritical wing are better than those of the flat wing of the same size at the cruise stage. Under the standard cruise condition (the angle of attack of the flow is 2°, the air density is 0.5259 kg/m3, and the incoming flow velocity is 800 km/h, for example), the lift-drag ratio of the supercritical wing is increased from 15.0855 to 28.1962 by 1.869 times.


Introduction
As one of the most important parts of aircraft and the main source of lift, the design of the wing has always been the focus of aeronautical engineers' efforts and improvements.With the evolution of aircraft power from piston engines to jet engines, people found that the traditional configuration of the wing could not meet the flight requirements of the new passenger aircraft in the high subsonic state.In 1967, the American aeronautical engineer R.T. Whitcomb proposed the design idea of the supercritical wing.This kind of wing not only has a better lift-drag ratio and buffeting characteristics compared with the traditional flat wing, but it also greatly reduces the phenomenon of steep wing drag under high subsonic conditions.Today, the supercritical wing is still used on a variety of advanced passenger and military aircraft, including the Boeing 787 and C17 transport aircraft in the United States, the European A350, and China's latest C919 aircraft.
There are various research methods for supercritical wing, which can be roughly divided into experimental methods, theoretical analysis methods, and numerical simulation methods, all of which have been widely used: The experimental method and theoretical analysis were widely used in the early stages of research and are still currently the most effective methods, but the experimental method has very strict requirements for the experimental environment and usually requires the construction of a wind tunnel, a water tunnel, and other complicated facilities to carry out experiments, while the theoretical analysis method is difficult to verify the details inside.With the development of computer technology, computational fluid dynamics (CFD) technology has become one of the most important research methods for aircraft design.Compared with traditional experimental methods, this method does not require the construction of large-scale research facilities and greatly reduces the cost of experiments.Taking wing design as an example, the CFD method can easily adjust various parameters of the research object without the need to customize the model for each airfoil, which reduces the research period and improves the efficiency.Compared with theoretical analysis methods, the CFD method can intuitively observe various states inside the flow field and conveniently observe the flow state at a certain point.Therefore, computational fluid dynamics has become the most important development method for major aircraft manufacturers.The research method adopted in this study is also CFD combined with three-dimensional modelling to study the aerodynamic characteristics of a supercritical wing.
For the supercritical wing and numerical simulation methods, predecessors have carried out rich and detailed research: Zhang Wu used CFD technology to simulate the aerodynamic characteristics of a large passenger aircraft, the A380, and used numerical optimization technology to design the approach of the wing [1] .Once the cross section is obtained, the optimized plan for the airfoil configuration with reverse cross section torsion is determined.Zhang Y studied and optimized flight indices such as the pressure distribution characteristics of supercritical wings by combining the optimization design platform developed by the advanced CFD method with the RANS genetic algorithm and other theories [2], and finally obtained the wing and integrated multi-terminal airfoil design scheme applicable to practical engineering, which was selected.Chen Z et al. adopted the continuous adjoint method based on N-S equations to optimize the aerodynamic shape of the airfoil with viscose conditions and finally optimized the shape of the two-dimensional airfoil, which made a great improvement compared with the original scheme [3].L.N. Sarkar and D.H. Silva designed the transonic airfoil by using the Garabedian McFadden inversion method [4], compared the test data with the experimental data, and finally achieved a great improvement in the transonic aerodynamic performance of the redesigned airfoil.Salim Koc, Hyoung-Jinh, and Nakahashi Kused a set of threedimensional unstructured Euler equations and its discrete adjoint code to calculate the overall aerodynamic disturbance by introducing the Hicks-Henne morphological function to the inner wing shape and hanger shape and analyzing the flow field and sensitivity [5] .The resistance coefficient is reduced by 16 counts, and the buffeting strength around the hanger is successfully reduced while the constraint conditions are satisfied and the lift coefficient is maintained.Yang Q used infinite interpolation theory to generate structural networks in his research [6].The Baldwin-Lomax turbulence model and lattice center finite volume method are used.The non-viscous flux terms were discretized by the AUSM+ scheme, and the viscous flux terms were discretized by the second-order central difference scheme.The flow field around the wing was analyzed and simulated, and it was found that the shape of the airfoil had a significant influence on flutter characteristics.The flutter velocity of the airfoil with a smaller relative thickness is higher.The flutter velocity of the supercritical airfoil is higher than that of the NACA00xx series airfoil when the thickness is basically the same.In the transonic range, limit cycle oscillation is common due to the nonlinear effect of aerodynamics.
It can be found that in previous studies, most of them tend to use a variety of equations to calculate and fit to obtain the flow field characteristics or aerodynamic characteristics around the wing, while few use CFD and other means for direct aerodynamic simulation.Compared with previous studies, the main research method adopted in this study is the combination of CFD aerodynamic simulation and 3D modeling.Under the premise of following the similarity principle, the outflow field of the supercritical machine's warp surface is explored by changing variables such as atmospheric density or cruising altitude, air velocity, the incoming angle of attack, and airfoil type.

Geometric model
In order to reflect the excellent aerodynamic performance of a supercritical wing, two types of wing models are established in this study for comparison, one of which is the flat and straight airfoil using the Boeing 737 midspan airfoil.The airfoil section is shown in Figure 1.The other type of airfoil is the supercritical wing, which is the main research object of this study.In order to restore the aerodynamic performance of the actual supercritical wing as much as possible, different airfoils are used in each section of the wing in this study, and the torsion Angle of each section is adjusted to make it close to the physical characteristics of the actual supercritical wing.The airfoil parameters of each section are as follows: The section at the wing root adopts the NACA0015 airfoil, with a relative thickness of 15.00%.The airfoil section is shown in Figure 3.The NASA SC(2)-0410 airfoil is adopted at the wing tip, and the relative thickness is 9.97%.The airfoil section is shown in Figure 4.The airfoil parameters are shown in Table 1.

Tip chord 1320mm
Half wingspan length 17900mm Sweep angle at the quarter string 25.0° Section distribution: root chord, spanwise 11.6%, spanwise 18.5%, spanwise 49.0%, spanwise 70.0%, spanwise 85.0%, cusp chord, and kink position.The NACA0015 airfoil is used for the section at the root chord, the NASA SC(2)-0410 is used for the section at the tip chord, and the NASA SC(2)-0712 is used for the other sections.
The parameter settings for torsion angle and airfoil chord length at each section are shown in Table 2.
Table 2. Section characteristic parameters of each chord of supercritical wing.According to the existing research results and the inherent law of the CFD calculation method, the volume of the calculated object will directly determine the number of grids occupied during calculation.The larger the research object is, the more grids will be occupied, and more powerful computing resources will be needed to carry out the calculation for a longer time.Due to the limitations of the experimental conditions, this research lacks a computer with powerful computing power.Experiments in wind tunnels or water tunnels are also carried out unconditionally.Therefore, it is necessary to scale down the model given above to a certain extent so as to reduce the number of grids occupied during calculation, reduce the memory occupied by rendering, and reduce rendering time.To ensure that the model is still useful for research after scaling, the entire scaling process should adhere to the similarity principle, which states that the solid geometry before and after scaling should be similar, the similar parameters of the flow field before and after scaling should be equal, and the boundary conditions before and after scaling should be similar.A dimensionless number is primarily a Reynolds number Re and a Newton number Ne in the choice of similarity, and its expression, such as type (1), is shown in (2).
Among them, ρ represents the density of the fluid, V represents the flow rate of the fluid, L represents the characteristic length of the entity, μ is the dynamic viscosity coefficient, and F is the force acting on the entity.
As a result, the following is calculated based on the flat wing overall size shrinkage ratio: for supercritical wing exhibition to the total length of shrink than standard, shrinkage ratio of 0.04; the previous size is 17900 mm; the shrink after that size is 17900 * 0.04 = 716 mm; the overall volume is reduced to the original 1.5625 1/4 size.The scale-volume function in UG-NX12.0 was used for uniform scaling of the original model, and the supercritical wing after scaling was shown in Figure 8.It can be found that the size of the supercritical wing with a reduced ratio is close to that of the flat wing as the reference object, which can be used for follow-up research.
When the characteristic length of the wing is greatly reduced, the parameter of air velocity cannot be changed hastily because it may lead to fundamental changes in the physical mechanism of the flow field.For example, the original subsonic flow around the wing may become supersonic flow, which will fundamentally change the nature of the research.Therefore, this study will mainly change the fluid density to ensure that the Reynolds number is basically equal, which is similar to the idea of the water tunnel experiment.
Due to the particularity of hydrodynamics research, the actual research object is not the wing itself, strictly speaking, but the flow field around it.Therefore, it is necessary to build a cuboid area around the wing and carry out the difference calculation with the wing to get the flow field area around the wing.This part is called the computational geometry area.The selection of the computational geometry area is strictly restricted: if the area is too large, a large number of grids will need to be rendered during the calculation, which will pose a great challenge to the function of the computer, and the required time will be greatly extended.If the area is too small, information such as the boundary layer around the wing may not be fully reflected, thus making the experimental results unreliable.It is generally believed that the dividing standard of the calculated geometric area is as follows: the height of the lower and lower fields on the wing should be 3 times the average thickness of the wing; the width of the left and right flow fields should be twice the wingspan length; the length of the chord length of the wing should be 1.5~2 times in front of the wing; and the length of the rear should be 2.5~3 times.According to this principle, the calculated geometric regions of the flat wing and the supercritical wing are established as follows: The calculated geometric area of the flat wing is a cuboid with length * width * height =2100mm*2100mm*360mm, and its shear section is shown in Figure 9.

Basic flow equations and key parameter concepts
Fluid motion follows the Navier-Stokes equations, as shown in equations (3)(4)(5).
momentum equation： energy equation： Among them: DV ��⃗ Dt characterization of fluid momentum changes over time, or call the inertial force.fb ���⃗ is body force - is the increase in mass at some point in space ∇ × �ρV � �⃗ � represents the mass that flows out of that point 2 � represents change of total energy of fluid micelle (including internal energy and kinetic energy) ρfb ���⃗ represents the work done by a volumetric force on a fluid micelle ρq̇ represents fluid micelle receive heat from the outside world by radiation ∇�V � �⃗ × τ ij � represents work done by surface forces (pressure and viscous forces) on fluid micelle ∇(λ∇T) represents fluid micelle receive heat from the outside world through heat conduction At the same time, the lift force and drag force caused by it also follow certain physical laws.The formula of lift drag ratio derived from this formula is as follows: 2 ，Pa S is reference area.We usually choose the wing area, m2

Grid and boundary conditions
In fact, the number of grids is related not only to the size of the calculated volume but also to the division of the grids.Generally speaking, the finer the division of the grids and the larger the number of grids, the more accurate the fluid-related indicators or results obtained by the solution will be and the more reliable the data will be.The finer the grid division, the more equations need to be solved, the more computing resources are occupied, and the more time is required for rendering and solving the grid.Therefore, after determining the computational geometry region of the computational object, the mesh density in the region cannot be blindly increased.
In addition, with the constant increase in grid density, the solutions to key variables should change constantly.However, in practice, it is found that when the grid density increases to a certain extent, the values of key variables no longer change with the increase in the number of grids, and the grid density at this time is generally referred to as the "grid independent solution," that is, the minimum grid density that makes the solutions of variables no longer change.According to this theory, when the Fluent module in AIM Discovery 2020 R2 is adopted in this study, the division density is set as the third gear (the highest is the first gear) in the setting of mesh conditions, and the calculation step size is unified to 2000 steps so as to strike a balance between accuracy and efficiency.
The CFD calculation needs to set a series of boundary conditions to limit the solution conditions for the fluid so as to obtain the results expected by researchers.Such conditions usually include the velocity inlet of computational geometry, pressure outlet, turbulence model, solution format, and so on.The software ANSYS Discovery AIM 2020 R2 adopted in this study also requires these conditions to be limited, and the settings for each condition are shown in Table 3.

Velocity inlet
The plane directly opposite the leading edge of the wing is set as the velocity inlet, namely the direction of incoming flow.

Pressure outlet
The plane opposite the trailing edge of the wing is set as the pressure outlet.

Turbulence model
The study of the default SST turbulence model using ANSYS -k - model.

Solution scheme
Second-order upwind scheme is adopted for convection terms.

Variable selection
In this study, four groups of variables are selected for numerical simulation and analysis, which are: airfoil type, incoming angle of attack, air velocity, and air density.In each experiment, the type of airfoil was determined first, followed by a group of reference variables, and then changes were made to the other two groups of variables, yielding a total of five values from each group of variables to calculate the lift drag ratio and other performance parameters in each state.Finally, the same parameters of the two airfoils were compared.

Straight wing
Using flow rate as a baseline variable, the flow velocity was set to 800 km/h, 825 km/h, 850 km/h, 875 km/h, and 900 km/h, and the angle of attack of the wing was calculated based on air density and flow.
Considering that the cruising altitude of general civil airliners is mostly between 29,000 feet (about 8839 m) and 41,000 feet (12500 m), the five air density levels studied are set at the atmospheric densities of 8000 feet, 9000 feet, 10000 feet, 11000 feet, and 12000 feet, respectively.The comparison standard of atmospheric density and pressure with altitude adopted in this study was calculated based on the standard pressure value and gravitational acceleration of sea level at 45° north latitude.The air density was calculated based on 1.293 of the standard density, and the sea level temperature was set at 15 °C without considering the influence of water vapor content on the atmospheric density.The air density of five grades was 0.5259 kg/m 3 , 0.4671 kg/m 3 , 0.4135 kg/m 3 , 0.3648 kg/m 3 , and 0.3117 kg/m 3 , according to this standard.The angle of attack of the wing's incoming flow was set at 0°, 1°, 2°, -1°, and -2°, respectively.According to these conditions, CFD numerical simulation can yield the following values: (lift drag ratio is uniformly taken to four decimal places) The lift-drag ratio of the straight wing varies with the Angle of attack of the incoming flow, as shown in Figure 11   By analyzing the lift-drag ratio dot plot of the flat wing at different angles of attack, it can be found that: 1.At each angle of attack, with the increase of inlet velocity and atmospheric density, the lift-drag ratio of the straight wing generally presents an upward trend, and with the increase of atmospheric density, the increase potential of the lift-drag ratio gradually slows down.
2. Under the same atmospheric density, the lift-drag ratio of the wing increases obviously with the increase in the angle of attack of the incoming flow.
3. Under the joint action of factors such as angle of attack and atmospheric density, the variation trend of the wing at the same speed is different in each state.At 0 degrees of attack, the lift-drag ratio of the wing at all speeds changes stably.When the angle of attack increases, the variation trend of the lift-drag ratio fluctuates at some speeds.
4. Angle of attack at incoming flow = 2, air density = 0.4671 kg/m 3 , and flow rate = 875 km/h; under the condition of a lift-to-drag ratio, this can be observed in the group data, that is, in the process of numerical simulation operation, result error caused by insufficient step operation.

Supercritical airfoil
The boundary conditions are consistent with those of the straight wing.The calculated lift-drag ratio changes with the angle of attack of the incoming flow, as shown in Figure 14 to 16.  Analysis of the dot plot of the lift-drag ratio of the supercritical wing at different angles of attack shows that: 1.Under different angles of attack, as flow speed and air density have increased, the supercritical airfoil lift-drag ratio has also increased, while the change trend is more flat, turning smaller; 2.Under the same boundary conditions, the supercritical airfoil lift-drag ratio increased significantly when compared to the straight wing; 3.Under the comprehensive action of various boundary conditions, the lift-drag ratio of a supercritical wing is stable with no obvious fluctuation and a similar trend.
4. A straight wing, supercritical wing, in boundary conditions is as follows: density = 0.3117 kg/m 3 ; coming flow angle of attack = 0; under the condition of a flow rate of 850 km/h also appearing in the group's data, it is speculated that this causes similar effects to a straight wing.
By comparing the schematic diagram of the lift-drag ratio between a supercritical wing and a straight wing under the same boundary conditions, it can be found that: 1. Compared with the straight wing, the lift/drag ratio of the supercritical wing under the same boundary conditions is significantly higher, which means that the supercritical wing can provide better lift conditions than the straight wing and can carry larger loads for flight.
2. In the process of constantly changing air density, the lift-drag ratio of the supercritical wing is more stable than that of the flat wing, and there is no big turning point and fluctuation, which means that the supercritical wing can make the transition more smoothly under the condition of changing air layer and velocity, i.e., the change of its flight performance is more predictable, the flight process is safer, and the operation performance is better.
3. All other conditions are the same, but with the change in angle of attack, the lift-drag ratio of the supercritical wing changes less than that of the flat wing, which means that the lift gain of the passenger plane equipped with the supercritical wing is actually smaller than that of the flat wing through the change in angle of attack.The aircraft using a supercritical wing can steadily rise in

Figure 1 .
Figure 1.Boeing 737 midspan airfoil cross section.The airfoil was imported into the UG-NX12.0 software for airfoil generation.The half-wingspan length of the airfoil generated was 700 mm, and the relative thickness was 12.54%.The resulting three-dimensional wing morphology is shown in Figure2.

Figure 2 .
Figure 2. Boeing 737 midspan airfoil three-dimensional flat wing model.The other type of airfoil is the supercritical wing, which is the main research object of this study.In order to restore the aerodynamic performance of the actual supercritical wing as much as possible, different airfoils are used in each section of the wing in this study, and the torsion Angle of each section is adjusted to make it close to the physical characteristics of the actual supercritical wing.The airfoil parameters of each section are as follows:The section at the wing root adopts the NACA0015 airfoil, with a relative thickness of 15.00%.The airfoil section is shown in Figure3.

Figure 3 .
Figure 3. NACA0015 airfoil section at wing root.The NASA SC(2)-0410 airfoil is adopted at the wing tip, and the relative thickness is 9.97%.The airfoil section is shown in Figure4.

Figure 4 .
Figure 4. NASA SC(2)-0410 airfoil section at the wing tip chord.The NASA SC(2)-0712 airfoil is used in the wing bend (KINK) section, with a relative thickness of 11.99%.The airfoil section is shown in Figure5.

Figure 5 .
Figure 5. NASA SC(2)-0712 Airfoil section at the wing bend.The data of the above three airfoils were edited into DAT text and imported into UG-NX12.0software to generate the supercritical airfoils used in this study.The plane configuration of the wing and its parameters are shown in Figure 6.

Figure 6 .
Figure 6.Screenshot of plane configuration and parameters of supercritical wing.The airfoil parameters are shown in Table1.Table1.Supercritical wing parameter table.

Figure 7 .
Figure 7. Supercritical wing 3D model schematic.According to the existing research results and the inherent law of the CFD calculation method, the volume of the calculated object will directly determine the number of grids occupied during calculation.The larger the research object is, the more grids will be occupied, and more powerful computing resources will be needed to carry out the calculation for a longer time.Due to the limitations of the experimental conditions, this research lacks a computer with powerful computing power.Experiments in wind tunnels or water tunnels are also carried out unconditionally.Therefore, it is necessary to scale down the model given above to a certain extent so as to reduce the number of grids occupied during calculation, reduce the memory occupied by rendering, and reduce rendering time.To ensure that the model is still useful for research after scaling, the entire scaling process

Figure 8 .
Figure 8. Schematic diagram of a supercritical wing with reduced ratio (1/25).It can be found that the size of the supercritical wing with a reduced ratio is close to that of the flat wing as the reference object, which can be used for follow-up research.When the characteristic length of the wing is greatly reduced, the parameter of air velocity cannot be changed hastily because it may lead to fundamental changes in the physical mechanism of the flow field.For example, the original subsonic flow around the wing may become supersonic flow, which will fundamentally change the nature of the research.Therefore, this study will mainly change the fluid density to ensure that the Reynolds number is basically equal, which is similar to the idea of the water tunnel experiment.Due to the particularity of hydrodynamics research, the actual research object is not the wing itself, strictly speaking, but the flow field around it.Therefore, it is necessary to build a cuboid area around the wing and carry out the difference calculation with the wing to get the flow field area around the wing.This part is called the computational geometry area.The selection of the computational geometry area is strictly restricted: if the area is too large, a large number of grids will need to be rendered during the calculation, which will pose a great challenge to the function of the computer, and the required time will be greatly extended.If the area is too small, information such as the boundary layer around the wing may not be fully reflected, thus making the experimental results unreliable.It is generally believed that the dividing standard of the calculated geometric area is as follows: the height of the lower and lower fields on the wing should be 3 times the average thickness of the wing; the

Figure 9 .
Figure 9. Computation domain diagram of flat and straight wings.The calculated geometric area of the supercritical wing is a cuboid with length * width * height =1340mm*2160mm*300mm, and its shear section is shown in Figure 10.