In the present study, the governing differential equations are discretized using the finite volume method, and the k − ω shear stress transport (SST) turbulence model is employed to account for turbulent effects. The governing equations consist of the three-dimensional incompressible Reynolds-averaged Navier-Stokes (RANS) equations, in which a momentum source term is incorporated. In accordance with the assumptions adopted in this investigation, the effects of body forces and heat conduction are neglected:

In Equation (1), W represents the variable for conservation of flow field, F c represents the convection flux, F v represents the viscous flux, S represents the source item, v represents the volume of the control body element, d v represents the surface of the control body element, d s represents the area of the control body element.

The momentum source method replaces the physical blade geometry with a virtual actuator disk model. The aerodynamic forces exerted by the blades on the surrounding flow are transformed into equivalent momentum source terms, which are incorporated into the governing equations and solved in a coupled manner with the flow field. Let the Cartesian coordinate system of the computational domain be defined as O-XYZ, and the Cartesian coordinate system attached to the rotor disk be defined as O-XYZ. The transformation relationship between these two coordinate systems can be expressed as follows:

In the above formulation, β 1 s and β 1 c denote the rotor disk roll (lateral tilt) angle and pitch (longitudinal tilt) angle, respectively. When the vehicle operates in helicopter mode, both angles are equal to zero. The vector ( x c , y c , z c ) represents the coordinates of the rotor disk center in the Cartesian coordinate system of the computational domain.

Figure 1. Schematic of the rotor disk coordinate system and the blade micro-tip segment.

In the derivation of the momentum source terms, a blade element differential segment of spanwise length dr is extracted along the blade radial direction. An infinitesimal blade element located at radius r, with chord length c and azimuth angle ψ , is considered for analysis, as illustrated in Figure 1. According to Newton’s third law, the force acting on an infinitesimal blade element implies an equal and opposite force exerted by that blade element on the fluid. Therefore, the force applied to the airflow at the location of the blade element per unit time can be expressed as:

In Equation (2),where S Δ * denotes the momentum source term; N is the number of blades of the propeller; Ω / 2 π represents the angular displacement per unit time; and Δ ψ / Ω is the time required for the rotation. By transforming S Δ * into the computational-domain coordinate system ( x , y , z ) , the corresponding momentum source term ( S x , S y , S z ) can be obtained.

The rotor in helicopter mode was numerically simulated using the momentum source method. The dynamic pressure distributions were monitored at two radial stations located at 0.215R and 0.660R from the rotor center, where R denotes the rotor radius. The computed results were compared with the experimental data reported in Ref. [17]. The comparison is presented in Figure 2. As shown in the figure, the dynamic pressure distributions predicted by the momentum source method agree well with the experimental trends, thereby demonstrating the validity and reliability of the present computational approach.

Figure 2. The comparison chart of the dynamic pressure distribution beneath the rotor.

To investigate the aerodynamic interference characteristics of a tiltwing aircraft operating in helicopter mode, the geometric model of the vehicle is first established, as shown in Figure 3. The geometric configurations of the aircraft under three distinct operating modes are presented. The aircraft adopts a twin-propeller tiltwing configuration, with the propellers mounted on the tilting wing segments.

Figure 3. Geometric models of a tiltwing aircraft in three flight modes. (a) Helicopter Mode; (b) Tilting Transition Mode; (c) Fixed-Wing Mode.

This study focuses on the aerodynamic interference characteristics between the wing's tilt segment and the rotor in a tilt-rotor aircraft. To eliminate the influence of geometric parameters such as the upward sweep angle and root-taper ratio on the results, the wing is modeled using the simplest configuration-a rectangular, straight wing. In order to reduce computational complexity and emphasize the primary research focus, the aerodynamic coupling effects between the aircraft’s fuselage, horizontal tail, and the wing-rotor system are not considered in this study. Additionally, the effects of lateral-directional forces are neglected. Therefore, the computational model is simplified to include only a single wing and the rotor. The specific parameters are provided in Table 1.

Table 1. Wing modeling parameters.

Since the propeller is represented using the momentum source method, explicit geometric modeling of the propeller is not required. Instead, a locally refined mesh region is generated at the propeller location. This region is defined as a cylindrical volume with a radius of 1.35 m and a height of 0.2 m. Figure 4 presents the front view of the wing model used in the numerical simulations.

Figure 4. Wing model main view.

The entire flow field computational domain is a rectangular region with dimensions of 90 m × 90 m × 45 m. The inlet is defined as a velocity inlet, while the outlet is set as a pressure outlet. The wing root plane is designated as the symmetry plane, and the wing surface is treated as a wall. The remaining boundaries are set as symmetry planes. The distribution of these boundary conditions is illustrated in Figure 5.

A non-structured (unstructured) mesh is employed to discretize the computational domain. Mesh motion techniques are utilized to simulate the deflection process of the flaperon, and a conventional mesh remeshing trigger is implemented such that predefined mesh operations are automatically executed when the mesh quality falls below specified criteria. To more accurately resolve the near-wall flow over the wing surface, prism layer meshes are generated along the wing surface, with the height of the first layer set to 2.40668 × 10⁻⁵ m. A total of 22 prism layers are applied, with a growth ratio of 1.3. Figure 6 presents the sectional meshes of the wing and the rotor, respectively.

Figure 5. Computational domain.

Figure 6. Grid view.

A grid independence study is a critical step to ensure the accuracy and reliability of numerical simulation results. Both grid quality and grid resolution have a significant influence on computational accuracy and efficiency. To assess the effect of grid discretization on the numerical results, multiple computational grids with varying levels of refinement were generated. Numerical simulations were conducted for each grid under identical boundary conditions, followed by a comparative analysis of the results.

The computational boundary conditions were specified as follows: the freestream velocity was set to 5 m/s, the propeller rotational speed was 1500 RPM, and the flaperon deflection angle was 0˚. The thrust generated by the rotor is denoted as T. Under the influence of the rotor wake, the wing experiences both horizontal and vertical aerodynamic forces. The horizontal force acting on the wing is denoted as F_x, defined as positive along the negative direction of the body-fixed x-axis. The vertical force on the wing is represented by Fz, defined as positive along the negative direction of the body-fixed y-axis. The numerical results obtained with different grid resolutions are presented in Figure 7. As shown in the figure, the ratio Fz/T increases with increasing grid resolution and approaches convergence when the total number of grid cells exceeds 7.18 million. Considering both the accuracy of the numerical results and the computational cost, the grid containing 7.18 million cells was selected for the simulations presented in this study.

Figure 7. Grid independence verification.

The definition of the tilt angle of a tiltwing aircraft is based on the flight mode: a tilt angle of 0˚ corresponds to the fixed-wing mode, while 90˚ corresponds to the helicopter mode. In the helicopter mode, the angle between the wing chord line and the fuselage longitudinal axis is set to 4˚, so that the thrust direction generated by the rotors is aligned with the direction corresponding to the zero-lift angle of attack of the tiltwing. This configuration effectively reduces the aerodynamic interference of the rotors on the wing in the tilting segment [18]. Therefore, in the present study, the tilt angle of the wing is set to 86˚.

To investigate the aerodynamic interaction between the wing in the tilting segment and the rotors under varying inflow velocities, the freestream velocity at the inlet is prescribed as constant and set to 0 m/s, 1 m/s, and 2 m/s, respectively, while the angle of attack of the fixed-wing segment is maintained at 0˚. The rotor rotational speed is fixed at 1500 RPM, and the flaperons are kept undeflected. The detailed computational cases are summarized in Table 2.

Table 2. Computational conditions for studying the effects of different freestream velocities.

Figure 8 illustrates the variation of aerodynamic loads on the tilt-segment wing as a function of inflow velocity (where F_x denotes the force in the horizontal direction, Fz the force in the vertical direction, and V the inflow velocity). The computational results indicate that, under the influence of rotor wake, both the fixed-segment wing and the tilt-segment wing experience a vertical force Fz and a horizontal force F_x. However, the fixed-segment wing is located far from the core region of the rotor wake and is therefore subjected to relatively weak aerodynamic interference, which can be neglected. For the tilt-segment wing, a pronounced coupling with the rotor wake is observed. As shown in Figure 8(a), with increasing inflow velocity, the ratio of the horizontal force F_x to the rotor thrust T increases monotonically.

Owing to the relatively small projected area of the tilt-segment wing in the xy-plane, its blockage effect on the rotor wake is limited. As shown in Figure 8(b), the ratio of the vertical force Fz to the rotor thrust T also increases with increasing inflow velocity; however, its magnitude does not exceed 0.2% of the total rotor thrust. Consequently, this vertical force has a negligible influence on the investigation of aerodynamic interference between the wing and the rotor. Based on the

Figure 8. Variation of aerodynamic loads on the wing during the transition (tilt) phase with changing freestream conditions. (a) Variation of F_x/T with with V; (b) Variation of F_Z/T with with V.

above analysis, the subsequent sections of this chapter focus primarily on the effects of the horizontal force F_x acting on the tilt-segment wing.

As shown in Figure 9, the velocity slice contours are presented for inflow velocities of 3 m/s and 5 m/s. With the gradual increase in the inflow velocity, the overall velocity level of the rotor wake increases significantly. Influenced by the upstream freestream, the high-speed wake impinging on the right side of the tilt-transition wing is convected closer to the wing surface. The resulting asymmetric distributions of velocity and pressure gradients on the two sides of the wing generate a rearward horizontal force, F_x acting on the tilt-transition wing.

Figure 9. Velocity contour slices and corresponding velocity streamline cross-sections.

Under freestream velocities of 3 m/s and 5 m/s, unsteady numerical simulations were conducted to investigate the flaperon deflection process. The flaperon deflection angle was continuously varied from an initial 0˚ to −45˚. To examine the influence of different deflection rates on the aerodynamic interference between the wing and the rotor in the transition segment of a tilt-rotor configuration, the flaperon deflection durations were set to 0.5 s, 1 s, and 2 s, respectively. A constant angular deflection rate was maintained throughout the entire deflection process. The specific computational cases are summarized in Table 3.

Table 3. The flaperon deflection condition under unsteady calculation.

Figure 10 presents a comparison of the variations in F_x/T with respect to the flaperon deflection angle, as computed using steady and unsteady methods under an incoming flow velocity of 3 m/s, for different flaperon deflection completion times. Overall, the unsteady results exhibit trends consistent with those obtained from the steady approach, with F_x/T decreasing as the absolute value of the flaperon deflection angle increases.

When the flaperon deflection angle lies within the range of −20˚ to 0˚, the F_x/T values predicted by different computational methods show only minor discrepancies. The results follow similar trends with respect to the deflection angle and display an approximately linear variation, indicating that within the small-deflection regime, the aerodynamic response is relatively insensitive to the choice of computational method and to unsteady effects. As the flaperon deflection angle further increases and enters the range of −20˚ to −45˚, F_x/T transitions from positive to negative values, and noticeable differences begin to emerge among the results obtained using unsteady methods. In particular, for a deflection completion time of 0.5 s, F_x/T exhibits a pronounced reduction near a deflection angle of approximately −25˚, yielding values significantly lower than those predicted by the steady computation as well as those corresponding to other deflection completion times.

Figure 10. Variation of F_x/T with flaperon deflection angle under freestream velocities of 3 m/s.

Figure 11 presents a comparison of the variation of F_x/T with the flaperon deflection angle obtained using different computational approaches at an inflow velocity of 5 m/s. In terms of both overall trend and magnitude, the unsteady simulation results are in good agreement with those from the steady-state calculations. Specifically, F_x/T decreases monotonically as the absolute value of the flaperon deflection angle increases.

The F_x/T results obtained using different computational approaches exhibit noticeable quantitative discrepancies, which are primarily concentrated in the flaperon deflection angle range of −20˚ to −45˚. Within this deflection interval, the F_x/T values predicted by the steady-state computations are generally slightly higher than those obtained from the unsteady simulations. The overall variation trend is consistent with that observed at an inflow velocity of 3 m/s. Further comparison of the results corresponding to different deflection completion times indicates that, near a deflection angle of approximately −25˚, the unsteady simulation with a deflection completion time of 0.5 s yields a markedly lower F_x/T value than both the steady-state result and the unsteady results associated with other deflection completion times. This behavior is similar to that observed under the 3 m/s inflow condition. A detailed analysis focusing on this specific deflection angle is presented in the subsequent sections.

Given that the overall variation trends of F_x/T with respect to flaperon deflection angle are essentially consistent for inflow velocities of 3 m/s and 5 m/s, and considering that the unsteady aerodynamic force fluctuations are relatively smaller and the flow exhibits higher stability at an inflow velocity of 5 m/s, the 5 m/s condition is selected as a representative case for subsequent investigations of flow-field structural characteristics. Figure 12 illustrates the Q-criterion-based vortex structure distribution of the overall tilt-wing flow field, where a Q-criterion

Figure 11. Variation of F_x/T with flaperon deflection angle under freestream velocities of 5 m/s.

Figure 12. Overall vortex distribution of tilt-wing.

threshold of 1500 s⁻² is adopted to identify and characterize the dominant vortical structures.

The spiral vortex structure observed in the low-velocity region at the center of the virtual disk is essentially an equivalent representation of the hub vortex reproduced by the virtual disk model under the condition of an “absence of a physical hub.” In practical propeller or rotor systems, a concentrated axial vortex structure is generated downstream of the hub as a result of the combined effects of boundary-layer separation on the hub surface and the pronounced circumferential velocity shear within the rotating wake. The vortex core of the hub vortex is characterized by distinct low-pressure and low-velocity features and develops helically in the downstream direction. Its influence may extend to the wing surfaces on both sides of the tilt-transition section, where local attachment can occur.

When the flaperon begins to deflect, the aerodynamic characteristics of the control surface undergo significant changes. As the deflection angle increases progressively, the adverse pressure gradient on the lower surface of the flaperon intensifies, leading to boundary-layer separation and the formation of a pronounced separation vortex. During its development, this separation vortex interacts and couples with the upstream hub vortex, thereby further modifying the local flow-field structure. Figure 13 presents the velocity streamline distribution for a flaperon deflection angle of −22.5˚ with a deflection completion time of 2 s, where a distinct and finite-scale separation vortex can be clearly identified beneath the flaperon.

Figure 13. Velocity streamline plot at −22.5˚ flaperon deflection.

Different flaperon deflection rates have a pronounced influence on the evolution of the propeller hub vortex. Under different inflow conditions, the aerodynamic loads exhibit an overall consistent trend with respect to the flaperon deflection angle. In the present study, the case with an inflow velocity of 5 m/s is selected for detailed analysis. Figure 14 presents vortex structure contours on the windward side of the tilting-wing segment for flaperon deflection completion times of 0.5 s, 1 s, and 2 s, all with a deflection angle of 22.5˚. As shown in the figure, the overall vortex system becomes increasingly stable with longer deflection completion times. When the flaperon deflection completion time reaches 2 s, the separation vortex generated on the flaperon couples with the propeller hub vortex, resulting in a more coherent and stable vortex structure.

Figure 15 presents vortex-structure contours on the windward surface of the tilting wing on the leeward side for flaperon deflection completion times of 0.5 s, 1 s, and 2 s, with a constant deflection angle of −22.5˚. The propeller hub vortex initially forms a columnar helical vortex convecting downstream along the propeller axis. As it passes over the upper surface of the flaperon, the imposed flaperon deflection forces the root vortex filament to follow the increasingly curved wing surface, thereby reducing the distance between the vortex core and the wall and intensifying wall confinement effects. The abrupt increase in trailing-edge camber

Figure 14. Vortex distribution on the windward side of the tilt-wing segment under different deflection completion times.

Figure 15. Vortex distribution on the leeward side of the tilt-wing segment under different deflection completion times.

subjects the hub vortex to an adverse pressure gradient, under which it progressively destabilizes and disintegrates. In the contour plots, this process is manifested as a sudden interruption of the vortex structure accompanied by a rapid transition in the color scale. As the deflection completion time increases from 0.5 s to 2 s, the hub vortex remains more coherent and exhibits a more stable core.

Figure 16 presents contour plots of the surface pressure coefficient distribution on the leeward side of the wing during the transition phase, for flapleron deflection completion times of 0.5 s, 1 s, and 2 s, all with a deflection angle of −22.5˚. The integrity of the vortex system exhibits a strong correspondence with the surface low-pressure regions: the more stable the vortex core, the lower the resulting surface pressure. The degree of near-wall attachment of the hub vortex emerges as a key factor accounting for the discrepancies between the local pressure distribution and the horizontal force response.

Figure 16. Surface pressure coefficient contour plots of the tilt-wing segment under different deflection completion times.

Key Weaknesses:

Geometric Simplification: The model is highly simplified, utilizing a rectangular straight wing and neglecting the fuselage, horizontal tail, and lateral-directional forces. While this isolates the rotor-wing interaction, it may limit the applicability of the results to real-world aircraft configurations where these interactions are significant.