Research on the Performance of Slender Aircraft with Flare-Stabilized-Skirt

Abstract

In this paper, a morphing aircraft with a deployable flared skirt is proposed, and the influence of the flare skirt on the static stability of hypersonic aircraft is studied. The theoretical model of static stability of slender aircraft is established, and the position of the center of pressure can be used as a theoretical basis to measure static stability. Three-dimensional compressible Reynolds-averaged Navier–Stokes (RANS) solver and an SST k-ω turbulence model are used to analyze the aerodynamic characteristics of the aircraft and the influence effect of the flared skirt on the pressure distribution of the body, and the influence trend on the position of the pressure center is verified. At the same time, the reliability of the code and grid is verified by a classic example, and the results are in good agreement with the experimental data in the current literature. Finally, the static stability of an aircraft with flared skirts with different deployment angles is quantitatively measured by defining the stability derivative at the common point. The results show that the static stability of the aircraft with the same forebody is improved by more than 100% under different flight Mach numbers when the flared skirt deployment angle is 30° compared to 0°.

1. Introduction

Stability is the premise of a safe flight for an aircraft, and its importance is self-evident. However, maneuverability and stability are contradictory. The enhancement of maneuverability means the weakening of stability, and the problem raised by the improvement of stability is that maneuverability needs to be weakened, which needs to be weighed according to the actual situation in the design stage of the aircraft [1]. The morphing aircraft achieves a good balance between stability and maneuverability. It can change its aerodynamic shape according to the flight environment and mission profile and obtain the best aerodynamic performance and flight state [2,3,4,5,6,7,8,9,10,11].

The external flow field of a high-speed aircraft is complex and easily disturbed by various factors during flight, which causes the aircraft to deviate from the preset flight state. At this time, restoring the aircraft to a stable state through active control is necessary. Van Arnhem [12] et al. studied the influence of Tail-mounted Propellers on aerodynamic performance and static stability, and considered that the changes in propeller loading due to the airframe-induced flow field are the dominant factor in changing airframe stability and performance. Bykerk [13,14] et al. studied the influence factors of hypersonic wave riders’ lateral and longitudinal static stability during horizontal take-offs and landings by comparing numerical simulation results and experimental data. Phillips [15] and others studied the contribution of tail dihedral to the static stability of a V-tail by using lift line theory. Rokhsaz [16] considers that the scissor wing is similar to the variable sweep and oblique wing, which can enhance the stability and control of the aircraft. The aircraft’s static and dynamic stability derivatives are studied experimentally using a novel 3-DOF mechanism in the United States Air Force Academy [17]. Kohler [18] and others explored the influence of a Bio-inspired Rotating Empennage on a fighter’s static and dynamic stability and handling quality. Luckert [19] used the second-order shock expansion theory proposed by Syvertson and Dennis to study the static stability and drag characteristics of a rotating body in supersonic flow. Bryan [20] et al. verified the static stability of a series of spherically blunted 10-degree cones at different Mach numbers and angles of attack by viscosity theory and wind tunnel tests. Stetson [21] et al. conducted a comparative study on the static stability of conical and biconical aircrafts. He verified the aerodynamic changes when the front half of a 5 deg half-angle cone was replaced with a 10 deg half-angle cone. Scholars have conducted a large quantity of research on the balance between the stability and maneuverability of the aircraft. One of the attempts to combine these two requirements is to use variable geometry. Currently, the methods mainly focus on exploring morphing wings, and there is little research on a rotating body without wings.

The main innovation and contribution of this paper are as follows: the influence of a deployable flared skirt on the static stability of a hypersonic vehicle is analyzed by combining the three-dimensional compressible RANS solver and SST k-ω turbulence model. It is found that the flared skirt can significantly improve static stability without significantly reducing the flight efficiency of the aircraft.

This article is organized as follows: Section 2 presents a morphing aircraft with a deployable flared skirt, and the maximum deployment angle is 30 degrees. Section 3 establishes a theoretical model for the static stability of the morphing aircraft and proposes the basis for determining the static stability. In Section 4, the aerodynamic performance of different variants is analyzed by numerical methods, and the effects of the deployable flared skirt on static stability and flight efficiency of the aircraft are finally evaluated.

2. Geometrical Model Description

In order to explore the effect of the deployable flared skirt on the aerodynamic characteristics and stability of the aircraft, as shown in Figure 1, this paper describes a cone–column–skirt aircraft configuration. During flight, the opening angle of the flared skirt can be adjusted according to different flight profiles to adjust the position of the aircraft’s pressure center, improve the aircraft’s static stability margin, and further improve the overall flight efficiency.

In order to eliminate the multiple factors that may affect stability, the aircraft is simplified into a slender body of rotation, and the head cone is passivated with a blunt radius of 0.05 m and a body diameter of d = 0.35 m. Table 1 describes four aircraft configurations with different flare skirt opening angles, which change aerodynamic parameters and flight characteristics significantly.

3. Theoretical Modeling of the Aircraft Static Stability

Static stability means that the aerodynamic forces on the aircraft produce a moment during which the aircraft has a tendency to return to the original flight state when the factors that interfere with flight disappear. There are at least two ways of comparing static stability [22], one being the position of the center of pressure as a measure of stability. The total pressure distribution on the aircraft determines the position of the center of pressure. For aircrafts with identical forebodies, the change in the total pressure distribution can be directly attributed to the effect of the flare skirt. Another standard measure of stability for non-rotating (or slowly rotating) aircrafts is the pitch moment coefficient gradient $C_M^{\alpha }$ [1].

The force analysis schematic of flared-skirt stabilized aircraft in the body coordinate system O_bX_bY_bZ_b, and velocity coordinate system O_vX_vY_vZ_v is shown in Figure 2. The aerodynamic force on the aircraft is decomposed into three components along the velocity coordinate system, the drag X, the lift Y, and the lateral force Z, wherein the resistance passes through the center of mass and produces no moment on the center of mass. The lift force Y and lateral force Z produce a pitching moment M_z and a yawing moment M_y to the center of mass, i.e.,

(1)$\{\begin{array}{l}{M}_x=0\hfill \\ M_y=Z(\overline{x_p}-\overline{x_g})L\hfill \\ M_z=-Y(\overline{x_p}-\overline{x_g})L\hfill \end{array}$

The pitching moment M_z and yawing moment M_y can also be expressed as follows:

(2)$M_z=-c_y^{\alpha }\alpha q{S}_{ref}(\overline{x_p}-\overline{x_g})L=C_M^{\alpha }\alpha q{S}_{ref}L$

(3)$M_y=c_z^{\beta }\beta q{S}_{ref}(\overline{x_p}-\overline{x_g})L=C_M^{\beta }\beta q{S}_{ref}L$

For axisymmetric aircrafts, through coordinate transformation, we obtain the following:

(4)$c_y^{\alpha }=-c_z^{\beta }$

Then,

(5)$C_M^{\alpha }=C_M^{\beta }=-c_y^{\alpha }(\overline{x_p}-\overline{x_g})$

Taking the pitching moment in the longitudinal plane as an example, when the aircraft flies in equilibrium at the angle of attack ${\alpha }_0$, the accidental disturbance increases the angle of attack by $\mathrm{\Delta }\alpha (\mathrm{\Delta }\alpha >0)$, which causes the lift acting on the focus to increase by $\mathrm{\Delta }Y$. For a rotating aircraft with zero rudder deflection angle, the focus coincides with the pressure center, so the additional pitching moment due to the lift increment can be expressed as follows:

(6)$\mathrm{\Delta }{M}_z(\alpha )={C_M^{\alpha }|}_{\alpha ={\alpha }_0}\mathrm{\Delta }\alpha q{S}_{ref}L$

As shown in Figure 3a, when ${C_M^{\alpha }|}_{\alpha ={\alpha }_0}<0$, the perturbation of the increase in the angle of attack under the effect of aerodynamic forces produces an additional pitching moment that inhibits the increase in the angle of attack or even eliminates the increment in the angle of attack, causing the aircraft to “bow the head”, during which the angle of attack has a tendency to return to ${\alpha }_0$. In this case, the aircraft has static stability, and the larger the absolute value of $C_M^{\alpha }$, the more stable it is. As shown in Figure 3b, when ${C_M^{\alpha }|}_{\alpha ={\alpha }_0}=0$, the perturbation of the increase in the angle of attack under the effect of aerodynamic forces produces no additional pitching moment, the additional angle of attack caused by the disturbance is no longer increased and also cannot be eliminated. In this case, the aircraft is statically neutral and stable. As shown in Figure 3c, when ${C_M^{\alpha }|}_{\alpha ={\alpha }_0}>0$, the perturbation of the increase in the angle of attack under the effect of aerodynamic forces produces an additional pitching moment that exacerbates the increase in the angle of attack, causing the aircraft to “lift the head” and the angle of attack to increase further. In this case, the aircraft is statically unstable when $\mathrm{\Delta }\alpha <0$, and vice versa.

Typically, c_y is linearly related to α, so $C_M^{\alpha }$ can be replaced by $C_M^{c_y}$, i.e.,

(7)$C_M^{c_y}=\frac{\partial C_M}{\partial c_y}=\frac{C_M^{\alpha }}{c_y^{\alpha }}=-\left(\overline{x_p}-\overline{x_g}\right)$

where the difference, $\left(\overline{x_p}-\overline{x_g}\right)$, between the relative coordinates of the center of pressure and the center of mass is called the static stability margin, and the closer the center of pressure is to the bottom, the greater the static stability of the aircraft, provided that the position of the center of mass is determined.

4. Aerodynamic Performance Analysis of the Flared-Skirt Stabilized Aircrafts

4.1. Computational Model and Numerical Method

The morphing aircrafts model is an axisymmetric cone-cylinder-flare structure, and the shape of the external flow field is set as a cylinder. The boundary conditions are shown in Figure 4. Considering the imitative effect and the calculation efficiency of the model, set L_o = 3 L. At the same time, to observe the influence of the flare skirt on the flow field, the distance from the aircraft head to the inlet of the flow field is D = 10 d so that the aircraft is located in the forward position of the flow field center.

After preprocessing the model, the tetrahedral grid is used to divide the computational domain of the flow field, as shown in Figure 5a, and the grid near the aircraft is encrypted, as shown in Figure 5b. The height of the first layer of the grid near the wall has an important influence on accurately describing the flow effect of the flow field. The height of that is set to ensure Y⁺ ≈ 1. At the same time, the mesh draw ratio of the first 15 layers is kept below 1.2.

The simulated flight conditions are set as follows: flight altitude, h = 10 km; local temperature, T = 223.3 k; local pressure, p = 26,499.9 pa; ideal gas density, ρ = 0.41351 kg/m³; and dynamic viscosity, μ = 1.46 × 10⁻⁵ kg/(ms). The controlling equation is the Reynolds Averaged Navier–Stokes equation. The turbulence model adopts the format, and this model sums the models through a hybrid function, namely the model that can accurately predict the inverse pressure gradient flow near the wall when activated at the near wall, and the model that can accurately calculate the free flow when activated at the far wall.

4.2. Code Validation

Based on the results of the hypersonic wind tunnel tests of the slender body in Dr. Singh’s paper, the model parameters and experimental conditions are set to verify the example [23]. The slender experience model consists of three parts: the semi-spherical head, the 5° half cone, and the cylindrical section. The parameters are shown in Figure 6. The reference area of the model is 0.000314 m², and the reference length is 0.177 m.

The simulation conditions are set as shown in Table 2, with the flight angles of attack set to −3°, 0°, 3°, 5°, 7°, and 10°. Unstructured grids are used in the calculation domain of the flow field, and the number of grids is 3,010,704.

As shown in Figure 7, the comparison of the lift–drag ratio between the experimental data and the calculation model shows that the numerical calculation results are in good agreement with the experimental data when the flight angle of attack is not greater than 5°. When the angle of attack is greater than 5°, the numerical results are slightly larger than the experimental data.

A comparison of the pitching moment coefficients between the experimental data and the calculated model is shown in Figure 8, where the pitching moment reference point is located at 5.2 cm from the model’s base. The results obtained agree with the experimental data in the published literature for all the angle of attack conditions tested.

The comparison of the center of pressure between the experimental data and the calculation model is shown in Figure 9, and it can be seen that the position of the center of pressure of the slender body is near the cone of the model head. By normalizing the center of pressure position, the numerical calculation results are slightly higher than those obtained from the experimental tests, with an average error of 8.06%, which verifies the usability of the numerical calculation method.

4.3. Influence of Deployable Flare Skirt on Aircraft Pressure Distribution

The aerodynamic characteristics of flare skirts with different tapers are analyzed by using numerical methods. Taking Mach 5 and 0° angle of attack as an example, the cloud map of the surface pressure distribution of the aircraft is shown in Figure 10. The diagram shows that under the same inflow conditions, the compression effect of the cone–column forebody of the aircraft on air is approximately the same. The flared skirt on the bottom compresses the secondary airflow, and the compression effect increases as the flare angle of the skirt increases.

Figure 11 shows the change of the surface pressure coefficient C_p of aircrafts with different flare angles of the skirt along the axial of the aircraft body under Mach 5. The pressure coefficient curve can be divided into three stages: conical head, cylindrical body, and flared skirt. During supersonic flight, the nose of the aircraft exerts significant compression on the airflow, with the highest coefficient of pressure at the point of the blunt tip, followed by the conical section, and close is to zero for the cylindrical body. The coefficients of pressure in the cone–column precursor of the four configurations of aircrafts follow a similar trend, which is consistent with the pressure distribution demonstrated by contour. As shown in Figure 11a, the pressure coefficients of the flared skirt and the cylinder are maintained near 0 for a tail skirt spread angle of θ = 0°, and a local negative pressure is generated at the bottom of the aircraft, where the pressure coefficient is less than 0. As shown in Figure 11b–d, the pressure coefficients on the flared skirt increase as the flared skirt opening angle increases and eventually return to negative at the bottom of the aircraft.

The variation curves of the relative position of the center of pressure with the angle of attack for different configurations of aircrafts are shown in Figure 12. Figure 12 also shows that the center of pressure moves towards the bottom of the aircraft for the determining angle of attack as the flare skirt opening angle increases. The change is most significant at a 0° angle of attack, where the V4 moves the center of pressure back by 25.6% compared to the V1. As the angle of attack increases, the flared skirt’s effect on the pressure center’s backward movement diminishes. At an angle of attack of 10°, the V4 moves the center of pressure backward by 10.3% compared to the V1. At the same time, it can be seen that the effect of the angle of attack on the center of pressure decreases as the flared skirt opening angles increase.

4.4. Influence of Flared Skirts on the Static Stability of Aircrafts

The static stability of the aircraft is determined by a combination of the mass distribution, which can be predicted from the opening angle of the flared skirt, and the aerodynamic characteristics. This section attempts to isolate the contribution of aerodynamic properties to flight stability. The pitch moment coefficient gradient $C_M^{\alpha }$ is used as a measure of stability. To eliminate the influence of centroid position change of different configurations, $C_M^{\alpha }$ is normalized by Formula (8) [22]:

(8)${\overline{C}}_M^{\alpha }=C_M^{\alpha }+\left(C_L+C_D\right)\left(\overline{C}G-CG\right)$

Among them, ${\overline{C}}_M^{\alpha }$ is related to the position selection of $\overline{C}G$, and the closer $\overline{C}G$ is to the actual $CG$, the smaller the error is. Therefore, the selected position of $\overline{C}G$ is close to the average position of the centroid of all configurations, as shown in Table 3.

As shown in Figure 13a, Figure 14a, Figure 15a and Figure 16a the curves of ${\overline{C}}_M^{\alpha }$ with different configurations changing with the angle of attack under the inflow of Mach 4~Mach 7 show that the flare skirt significantly influences the aircraft’s static stability. As the angle of attack increases, the static stability of the aircraft decreases slowly, and the changing trend of different configurations is approximately similar.

The V1 to V3 aircrafts are all statically unstable within the tested angle of attack, while the V4 is statically stable at small angles of attack. For aircrafts with the same forebody, the maximum increase in static stability is 101%, 102%, 103%, and 103% for a flared skirt opening angle of 30° compared to an opening angle of 0°, respectively. Also, to better measure the comprehensive influence of the flared skirt on flight performance, the change of the lift-to-drag ratio with an angle of attack for the corresponding configuration is shown in Figure 13b, Figure 14b, Figure 15b and Figure 16b, with a maximum reduction of 17%, 11%, 7%, and 4% for a flared skirt opening angle of 30° compared to an opening angle of 0°, respectively.

5. Results and Discussion

The analysis of the experimental results leads to the following conclusions:

• (1). The deployable flared skirt has a significant effect on adjusting the pressure distribution, which can make the center of pressure of the aircraft move backward.

• (2). The influence of flared skirt opening angle θ on static stability: The aircraft is statically unstable when the opening angles are 0°, 10°, or 20°, and the aircraft is statically stable under the partial angle of attack conditions when the opening angle at 30°.

• (3). The influence of the angle of attack α on static stability: Under the same Mach number conditions, the static stability of different configurations has the similar trend with the angle of attack. The static stability of the aircraft decreases as α increases when α < 5°. The static stability of the aircraft does not change significantly as α increases when α > 5°.

• (4). The influence of Mach number on static stability: At the exact angle of attack, the greater the Ma, the stronger the static stability of the aircraft.

• (5). The deployable flared skirt can adjust the flight stability without significantly reducing the lift–drag ratio of the aircraft.

6. Conclusions

This paper investigates the flared skirt’s role in the static stability of hypersonic aircraft. Two methods are proposed to discriminate the stability of slender spinning bodies: the relative positions between the center of pressure and the center of mass, and the gradient of the pitch moment coefficient. The aerodynamic characteristics of aircrafts with different configurations are analyzed by numerical method, and the influence of the deployable flared skirt on the pressure distribution of aircrafts and the influence trend on the position of the pressure center is verified under the condition of Mach 5 inflow. At the same time, the reliability of the code is verified using classic examples. Finally, by defining the pitch moment coefficient gradient at the common point, the effect of the change in the position of the center of mass of different configurations is eliminated, and the effect of the flared skirt on the stability of the aircraft at different deployment angles in multiple flight profiles is measured quantitatively. At the same time, it is found that the deployable flared skirt will not significantly reduce the lift–drag ratio of the aircraft, indicating that the flared skirt has little influence on the control efficiency of the aircraft while adjusting the flight stability.

Author Contributions

Design and modelling, X.P.; CFD simulation, X.P. and S.L.; validation, F.Y. and Y.L.; investigation, G.C.; writing—original draft preparation, X.P.; writing—review and editing, H.Y., S.L., F.Y., Y.L. and G.C.; project administration, H.Y. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

The datasets generated or analyzed during this study are available from the corresponding author upon reasonable request.

Conflicts of Interest

The authors declare no conflict of interest.

Nomenclature

$d$	Diameter of the aircraft body (m);
$X$	Drag (N);
$Z$	Lateral forces (N), $Z=c_Z^{\beta }\beta q{S}_{ref}$;
$Y$	Lift (N), $Y=c_Y^{\alpha }\alpha q{S}_{ref}$;
$M_x$	Rolling moment (Nm);
$M_y$	Yawing moment (Nm);
$M_z$	Pitching moment (Nm);
$L$	Reference length (m);
$\overline{x_p}$	The relative coordinate of the pressure center on the axis of the body (the dimension is 1);
$\overline{x_g}$	The relative coordinate of the centroid on the axis of the body (the dimension is 1);
$c_z^{\beta }$	The derivative of the lateral force coefficient;
$c_y^{\alpha }$	The derivative of the lift coefficient;
$\beta$	Yaw angle (°);
$\alpha$	Angle of attack (°);
$q$	Incoming flow pressure (Pa), $q=\frac{1}{2}\rho v^2$;
$S_{ref}$	Reference area (m2);
$C_M^{\alpha }$	Pitch moment coefficient gradient;
$C_M^{\beta }$	Yaw moment coefficient gradient;
$C_M^{c_y}$	Derivative of the pitching moment coefficient concerning the lift coefficient;
${\overline{C}}_M^{\alpha }$	Pitch moment coefficient gradient at the common point.

Footnotes

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Figures and Tables

Figure 1. Configuration of the aircraft with the flared skirt at the bottom. (XCP is the distance between the pressure center of the aircraft and the head, XCG is the distance between the aircraft’s center of mass and the head.)

Figure 2. Schematic of force analysis of flared-skirt stabilized aircraft (yaw angle β = 0°).

Figure 3. Static stability division conditions. (a) Static steady-state when [Forumla omitted. See PDF.]; (b) Static neutral steady-state when [Forumla omitted. See PDF.]; (c) Static unsteady-state when [Forumla omitted. See PDF.].

Figure 4. Calculation domain of the aircraft’s external flow field (D and Lo represent the diameter and length of the flow field, and d and L represent the diameter and length of the aircraft).

Figure 5. Grid division of the computational domain. (a) Grid division of the flow field; (b) near-wall mesh encryption.

Figure 6. Schematic view of the hypersonic test model [23].

Figure 7. Comparison of the lift–drag ratio between the experimental data and the calculation model.

Figure 8. Comparison of the pitching moment coefficient between the experimental data and the calculation model.

Figure 9. Comparison of the center of pressure between the experimental data and the calculation model.

Figure 10. Pressure distribution of the aircraft surface at different flare angles of the skirt. (a) V1: θ = 0°; (b) V2: θ = 10°; (c) V3: θ = 20°; (d) V4: θ = 30°.

Figure 11. Variation of the aircraft surface pressure coefficient Cp along the axial under the inflow of Mach 5. (a) V1: θ = 0°; (b) V2: θ = 10°; (c) V3: θ = 20°; (d) V4: θ = 30°.

Figure 12. Relative position of the center of pressure with the angle of attack for different configurations.

Figure 13. Aerodynamic performance of the aircraft under Mach 4 inflow conditions. (a) Variation in the pitch moment coefficient gradient with the angle of attack; (b) variation in lift-to-drag ratio with the angle of attack.

Figure 14. Aerodynamic performance of the aircraft under Mach 5 inflow conditions. (a) Variation in the pitch moment coefficient gradient with the angle of attack; (b) variation in lift-to-drag ratio with the angle of attack.

Figure 15. Aerodynamic performance of the aircraft under Mach 6 inflow conditions. (a) Variation in the pitch moment coefficient gradient with the angle of attack; (b) variation in lift-to-drag ratio with the angle of attack.

Figure 16. Aerodynamic performance of the aircraft under Mach 7 inflow conditions. (a) Variation in the pitch moment coefficient gradient with the angle of attack; (b) variation in lift-to-drag ratio with the angle of attack.

Table 1

Structure parameters of the aircraft.

Configuration	Length of Head Ln/m	Length of Body Lb/m	Length of Skirt Ls/m	Deployment Angle of the Skirt θ/°
V1	3 d	11 d	d	0
V2	3 d	11 d	d	10
V3	3 d	11 d	d	20
V4	3 d	11 d	d	30

Table 2

Verification conditions [23].

Experimental Conditions	Temperature (k)	Atmospheric Density (kg/m3)	Mach Number	Viscosity Coefficient (Ns/m2)
Value	89.3	0.0371	8.2	6.161 × 10−6

Table 3

Locations of the center of mass of the aircraft.

Configuration	V1	V2	V3	V4
CG/m	2.9096	2.9081	2.9063	2.9059
$\overline{C}G$/m	2.908

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