Delayed Detached-Eddy Simulations of Aerodynamic Variability During Carrier-Based Aircraft Landing with a Domain Precursor Inflow Method

Abstract

Flight tests and wind tunnel experiments face difficulties in investigating the impact of aircraft carrier air-wake on the landing process. Meanwhile, numerical methods generally exhibit low overall computational efficiency in solving such problems. To address the computational challenges posed by the disparate spatiotemporal scales of the ship air-wake and aircraft motion, a domain precursor inflow method is developed to efficiently generate unsteady inflow boundary conditions from precomputed full-domain air-wake simulations. This study investigates the aerodynamic variability of carrier-based aircraft during landing through the turbulent air-wake generated by an aircraft carrier, employing a hybrid RANS-LES methodology on dynamic unstructured overset grids. The numerical framework integrates a delayed detached-eddy simulation (DDES) model with a parallel dynamic overset grid approach, enabling high-fidelity simulations of coupled aircraft carrier interactions. Validation confirms the accuracy of the precursor inflow method in reproducing air-wake characteristics and aerodynamic loads compared to full-domain simulations. Parametric analyses of 15 distinct landing trajectories reveal significant aerodynamic variability, particularly within 250 m of the carrier, where interactions with island-generated vortices induce fluctuations in lift (up to 25%), drag (18%), and pitching moments (30%). Ground effects near the deck further amplify load variations, while lateral deviations in landing paths generate asymmetric forces and moments. The proposed methodology demonstrates computational efficiency for multi-scenario analysis, providing critical insights into aerodynamic uncertainties during carrier operations.

1. Introduction

During the landing process of carrier-based aircraft, the carrier normally sails into the prevailing wind to maximize the value of “wind over deck” [1], which would generate a highly turbulent airflow around the ship’s superstructure and over the flight deck. This turbulent flow, referred to the ship’s ‘air-wake’, can adversely affect aircraft performance, disturbing the aircraft’s flight path and requiring immediate corrective action from the pilot to compensate [2]. Under the influence of aerodynamic turbulence disturbances, aircraft landing operations within restricted areas may incur a heightened risk of catastrophic incidents due to potential deviations from stabilized glide paths [3,4]. Therefore, the landing phase of carrier-based aircraft is a critical and highly complex maneuver, characterized by trajectory deviations and stochastic entry into the ship air-wake, leading to significant variability in aerodynamic characteristics. Operating aircraft to land on the carrier through the air-wake presents significant challenges. Therefore, conducting research on the impact of wake flow on the aerodynamic characteristics of carrier-based aircraft during landing is of great significance.

Flight testing is the most direct method for conducting such research; however, due to factors such as sea conditions, weather, measurement methods, and safety, this method can only be used as the ultimate means [5]. It is thus highly desirable to have a thorough understanding of the air-wake characteristics and their effects on landing aircraft before the ship sets sail. In the design phase, wind tunnel testing is usually the most important and commonly used method for studying air-wake characteristics [6,7]. The air-wake flow around the ship can be measured in the wind tunnel tests only by using a scaled model due to the limitation of wind tunnel size. For example, a 1:100 scaled model was mounted in an NRC 2 m × 3 m low-speed wind tunnel to study the air-wake of a generic simple frigate shape (SFS2) [8], and a 1:200 scale model of the Queen Elizabeth Class aircraft carrier was submerged in a water channel to provide experimental data [9]. However, the scaling of ship models not only affects the temporal and spatial distribution of the air-wake field but also poses difficulties for the study of the landing process of carrier-based aircraft. This is because the carrier-based aircraft will be scaled down to a size where measurement equipment cannot be installed. In addition, there is usually not enough space in the wind tunnel to study the entire landing process of carrier-based aircraft. Therefore, it is difficult to conduct research on the aerodynamic characteristics of carrier-based aircraft landing in the wake under the existing wind tunnel test conditions.

With advancements in numerical calculation methods and the improvement of high-performance computing resources, there has been growing confidence in using computer modeling, with computational fluid dynamics (CFD) emerging as a viable alternative to wind tunnel testing for studying ship air-wakes [2]. The CFD method has been applied in the study of air-wake flow in general ships [10,11,12] and aircraft carriers [2,13,14,15,16]. However, the simulation of carrier-based aircraft landing using CFD involves complex, unsteady aerodynamic phenomena due to interactions between the dynamic aircraft and the ship’s air-wake. While the traditional Reynolds-averaged Navier–Stokes (RANS) method has been widely employed, it exhibits significant limitations in capturing the intricate flow physics of this process. RANS models are computationally efficient but struggle to resolve unsteady flow features critical for landing simulations, such as boundary layer separation, vortex shedding, and interactions between the aircraft wake and ship air-wake. Large eddy simulation (LES) can improve turbulent structure resolution by directly simulating large eddies [17], but its computational cost is prohibitive for full-scale simulations even with today’s high-performance computing systems [15]. The requirement for extremely fine grids and small time steps makes LES impractical for coupled simulations involving dynamic aircraft landing and ship air-wake effects.

In addition, as the length/time scales of the ship air-wake are much larger than those of the aircraft, another significant challenge in the simulating aircraft landing on a ship is the disparity in length and time scales between the ship and aircraft flow fields [18,19]. In order to capture the influence of flow structures of different scales in the air-wake flow field on the aerodynamic characteristics of carrier-based aircraft during the landing process, it is necessary to use a computational grid with a scale equivalent to that of carrier-based aircraft. This makes the grid number very large and computation too expensive, posing significant challenges for studying the variability of aerodynamic characteristics of carrier-based aircraft landing under different conditions. Therefore, a method that can reduce computational costs and guarantee the precision of the results is essential. It can also be used to investigate the uncertainties in the landing process.

To address the challenges in studying the aerodynamic variability of carrier-based aircraft during landing using CFD methods, this paper employs a DDES-based RANS-LES hybrid approach on dynamic parallel unstructured overset grids [20] and adopts a domain precursor inflow method [21] to investigate aerodynamic characteristic variations under different landing trajectory deviations. The paper is organized as follows: Section 2 presents the numerical methodology for simulating the carrier-based aircraft landing process. Section 3 describes the computational models and the proposed domain precursor inflow method. The variations in aerodynamic characteristics during the landing process are analyzed in Section 4, followed by concluding remarks in the final section.

2. CFD Methodology for Carrier-Based Aircraft Landing Simulation

2.1. The Unsteady Flow Solver

The governing equations in this paper are the unsteady Navier–Stokes (N–S) equations based on the calorically perfect gas model. To eliminate the numerical stiffness in low-Mach number regimes around the carrier, the preconditioning techniques are utilized in the unsteady flow solver [22]. A preconditioned dual-time-step algorithm for solving the unsteady compressible N–S equation in the integral form is

$$\Gamma M^{-1}{\displaystyle \frac{\partial }{\partial \mathsf{\tau }}}\underset{\mathsf{\Omega }}{∰}\mathit{W}\mathrm{d}\mathsf{\Omega }+{\displaystyle \frac{\partial }{\partial \mathrm{t}}}\underset{\mathsf{\Omega }}{∰}\mathit{W}\mathrm{d}\mathsf{\Omega }+\underset{\partial \mathsf{\Omega }}{∯}\left({\mathit{F}}_c-{\mathit{F}}_v\right)\mathrm{dS}=0,$$

(1)

where $\mathsf{\tau }$ is pseudo time, $\mathit{W}$ represents the vector of conservative variables, ${\mathit{F}}_c$ is the vector of convective fluxes, and ${\mathit{F}}_v$ is the vector of viscous fluxes, which are defined in the following:

$$\mathit{W}=\left[ \begin{array}{c}\rho \\ \rho u\\ \rho v\\ \rho w\\ \rho E\end{array}\right], {\mathit{F}}_c=\left[ \begin{array}{c}\rho V_r\\ \rho u{V}_r+p{n}_x\\ \rho v{V}_r+p{n}_y\\ \rho w{V}_r+p{n}_z\\ \rho H{V}_r+p{V}_g\end{array}\right], {\mathit{F}}_v=\left[ \begin{array}{c}0\\ {\tau }_{xx}{n}_x+{\tau }_{xy}{n}_y+{\tau }_{xz}{n}_z\\ {\tau }_{yx}{n}_x+{\tau }_{yy}{n}_y+{\tau }_{yz}{n}_z\\ {\tau }_{zx}{n}_x+{\tau }_{zy}{n}_y+{\tau }_{zz}{n}_z\\ {\mathsf{\Theta }}_x{n}_x+{\mathsf{\Theta }}_y{n}_y+{\mathsf{\Theta }}_z{n}_z\end{array}\right],$$

(2)

where $\rho$ represents the fluid density, ${\tau }_{ij}$ is viscous stress, and $u,$ $v,$ and $w$ represent the Cartesian components of velocity in the $x,$ $y,$ and $z$ directions, respectively. In the above, $E$ and $H$ are the total energy and total enthalpy per unit mass, respectively. $p$ denotes the static pressure, and $V_r$ is the contravariant velocity relative to the motion of the grid:

$$V_r=V-V_g=\left(u-u_g\right)n_x+\left(v-v_g\right)n_y+\left(w-w_g\right)n_z,$$

(3)

In Equation (3), $V_g=u_g{n}_x+v_g{n}_y+w_g{n}_z$ is the contravariant velocity at the surface, $\partial \mathsf{\Omega }$, of the control volume. In Equations (2) and (3), $n_x,$ $n_y,$ and $n_z$ denote the components of the unit outward-facing normal vector, ${\tau }_{ij}$ is the component of viscous stress, and ${\mathsf{\Theta }}_x,$ ${\mathsf{\Theta }}_y,$ and ${\mathsf{\Theta }}_z$ are the thermal fluxes.

Equation (1) includes a pseudo-time derivative which is preconditioned by the matrix $\Gamma M^{-1}$. The $\Gamma$ preconditioning matrix Γ is given by

$$\Gamma =\left[\begin{array}{ccccc}\Theta & 0& 0& 0& {\rho }_T\\ \Theta u& \rho & 0& 0& {\rho }_{T}u\\ \Theta v& 0& \rho & 0& {\rho }_{T}v\\ \Theta w& 0& 0& \rho & {\rho }_{T}w\\ \Theta H-1& \rho u& \rho v& \rho w& {\rho }_{T}H+\rho c_p\end{array}\right],$$

(4)

and the Jacobian matrix $M$ is given by

$$M={\displaystyle \frac{\partial \mathit{W}}{\partial \mathit{Q}}}=\left[\begin{array}{ccccc}{\rho }_p& 0& 0& 0& {\rho }_T\\ {\rho }_{p}u& \rho & 0& 0& {\rho }_{T}u\\ {\rho }_{p}v& 0& \rho & 0& {\rho }_{T}v\\ {\rho }_{p}w& 0& 0& \rho & {\rho }_{T}w\\ {\rho }_{p}H-1& \rho u& \rho v& \rho w& {\rho }_{T}H+\rho c_p\end{array}\right],$$

(5)

where $\mathit{Q}={\left[p,u,v,w,T\right]}^{\mathrm{T}}$ is the vector of primitive variables. The preconditioning matrix $\Gamma$ is designed to scale the eigenvalues of the system to improve convergence for all flow speed regimes. In addition, to close Equation (1), the ideal gas equation of state is employed, which is expressed as

$$p=\rho RT$$

(6)

Here, $T$ is temperature, and $R$ represents the gas constant of air.

In this work, Equation (1) is solved by using a median dual-cell vertex finite-volume method on the unstructured grid [23]. The convective fluxes ${\mathit{F}}_c$ are computed by using the Riemann scheme of HLLC/E [24]. Second-order accuracy in the space discretization is achieved by a piecewise linear reconstruction. Venkatakrishnan’s limiter [25] is used due to its superior convergence property. The viscous fluxes ${\mathit{F}}_v$ are computed by using a Galerkin finite-element approach. The physical time derivative in Equation (1) is discretized to the second order using a three-point backward-difference in time, while the implicit LU-SGS [26] scheme in pseudo-time is applied to the pseudo-time derivative.

2.2. DDES-Based RANS-LES Hybrid Method

The simulation of carrier-based aircraft landing using computational fluid dynamics (CFD) involves complex, unsteady aerodynamic phenomena due to interactions between the aircraft and ship air-wake. RANS models are computationally efficient but struggle to resolve unsteady flow features critical for landing simulations. In contrast, LES can improve turbulent structure resolution by directly simulating large eddies, but its computational cost is prohibitive for full-scale simulations. To accurately resolve turbulent structures in the unstable ship air-wake flow field with an acceptable computational cost, the DDES-model-based RANS-LES hybrid method is employed in this work. The main idea of the RANS-LES hybrid method is that the RANS turbulence length scale in the near wall region is smaller than the local grid scale, while the local grid in the region far from the wall is fine enough to support LES [27]. As a result, the RANS branch is activated near the wall, and the LES branch is activated far from the wall. This is realized by redefining the turbulence length scale in the dissipative term of the turbulence kinetic energy transport equation. In the SST-DES model [28], the turbulence length scale is defined as follows

$$L_{DES}=\min \left(L_{RANS},L_{LES}\right),$$

(7)

where $L_{RANS}=\sqrt{k}/\left(C_{\mu }\omega \right)$ is the RANS turbulence length scale, and $L_{LES}=C_{DES}\Delta$ is the LES length scale, i.e., local grid scale. The hybrid characteristic of DES is determined by comparing the RANS character length scale with the local grid scale. The LES may be activated within the RANS region where the local grid is too coarse to support LES; this can lead to an incorrect decrease in eddy viscosity and premature occurrence of unphysical flow separation. To protect the RANS region from being invaded by the LES region, DDES modified the character turbulence length scale by introducing the delay function [29]. The delay function takes the following form:

$$f_d=1-\tan \mathrm{h}\left({\left(8r_d\right)}^3\right),$$

(8)

where $r_d={\displaystyle \frac{{\nu }_t+\nu }{\sqrt{u_{ij}{u}_{ij}}{\kappa }^2{d}^2}}$ is the delay factor. In the near wall boundary layer, $f_d$ is equal to 0, while in the separated region far from wall, $f_d$ approaches 1. The RANS turbulence length scale of SST-DDES [30] is defined as

$$L_{DDES}=L_{RANS}-f_d\max \left(0,L_{RANS}-L_{LES}\right),$$

(9)

One can see that $L_{DDES}$ is promised to be $L_{RANS}$ in the boundary layer, where it is supposed to be covered by the RANS region.

Based on Menter’s SST two-equation model [31], one can obtain SST-DDES model as

$${\displaystyle \frac{\partial }{\partial t}}\left(\rho k\right)+{\displaystyle \frac{\partial }{\partial x_i}}\left(\rho u_{i}k\right)=P_k-C_{\mu }\rho k\omega F_{DDES}+{\displaystyle \frac{\partial }{\partial x_i}}\left[\left(\mu +{\sigma }_k{\mu }_t\right){\displaystyle \frac{\partial k}{\partial x_i}}\right],$$

(10)

$$\begin{array}{c}{\displaystyle \frac{\partial }{\partial t}}\left(\rho \omega \right)+{\displaystyle \frac{\partial }{\partial x_i}}\left(\rho u_i\omega \right)= \\ {\displaystyle \frac{\rho \alpha }{{\mu }_t}}{P}_k-C_{\mu }\rho {\omega }^2+{\displaystyle \frac{\partial }{\partial x_i}}\left[\left(\mu +{\sigma }_{\omega }{\mu }_t\right){\displaystyle \frac{\partial \omega }{\partial x_i}}\right]+2\left(1-F_1\right)\rho {\sigma }_{\omega 2}{\displaystyle \frac{1}{\omega }}{\displaystyle \frac{\partial k}{\partial x_i}}{\displaystyle \frac{\partial \omega }{\partial x_i}},\end{array}$$

(11)

where $F_{DDES}$ is defined as

$$F_{\mathrm{DDES}}=\max \left[{\displaystyle \frac{L_{RANS}}{L_{LES}}}{f}_d,1\right],$$

(12)

2.3. Dynamic Parallel Overset Grid Method

During landing simulation, the aircraft moves relative to the carrier, and the grid must be adjusted to follow its movement. In this work, a dynamic parallel overset unstructured grid method [20] is utilized to deal with the moving boundary of the landing aircraft.

In the present overset unstructured grid method, hole cutting, the first step in overset grid assembly, is achieved by classifying the grid nodes based on a wall-distance-based criterion. As depicted in Figure 1a, the region nearest to a body is designated as its corresponding active zone, whereas the region closer to other objects is considered to be its inactive zone. When the nodes of a component grid are situated within its inactive zone, they are categorized as inactive nodes, or “hole points”. Conversely, nodes located in the active zone are classified as active nodes and are utilized in the flow field calculations. This is illustrated in Figure 1b, where the red grid nodes represent active nodes, while the black ones indicate inactive nodes. Figure 1c displays the results of hole cutting for the overset unstructured grid system, which consists of four circles, as well as the distribution of active nodes in each component grid. In the current hole-cutting method, the wall distance of the grid node is a crucial parameter for classifying grid nodes. Once the wall distances of all grid points are calculated, the hole cutting or classification of grid nodes is also completed simultaneously. To efficiently identify the active zone for hole cutting, this paper employs a parallel calculation algorithm for global minimum wall distance. This algorithm is based on the oriented-bounding-box-based kd-tree data structure of the wall surface grid, as proposed in Ref. [20].

After classifying the grid nodes into active and inactive nodes, the interpolation nodes are defined as the two layers of inactive nodes next to the active nodes, see Figure 2a, which shows the double-layer interpolation boundaries of the overset unstructured grid systems illustrated in Figure 1. Their flow values are interpolated from the donor component grid, in which the interpolation points are located within its active zone. As the sub-grids have been split into parts, the position information of interpolation nodes is sent to the candidate processes, which have the partitions of the candidate donor grid with their oriented bounding boxes (OBBs) surrounding the interpolation nodes. Figure 2b,c illustrate this parallel strategy of searching the donor cell. The detail of the current parallel overset grid method can be find in Refs. [20,32].

When applying the current parallel overset grid approach to the simulation of aircraft landing in the ship air-wake, the sub-grids are first generated around the carrier and aircraft, respectively. Each sub-grid needs to be generated only once and just moves rigidly along with its body during the whole simulation. With the parallel dynamic overset grid method, the handling of dynamic boundaries in unsteady simulations is highly efficient.

3. Simulation Model for Aerodynamic Variability in Aircraft Landing

3.1. Profile of Aircraft Landing to Carrier

The landing approach profile for a carrier-based aircraft is illustrated in Figure 3a. The aircraft approaches the carrier on a flight path or glide slope and flies with a fixed angle of attack α with respect to the aircraft relative freestream [19]. Its pitch attitude relative to the horizon depends on the deck wind speed and the airspeed of the aircraft. The deck wind speed is the composition velocity of ship speed and wind speed. It can be regarded as a result of either the forward velocity of the ship or the wind speed blowing over a stationary ship. The airspeed of an aircraft is the relative velocity of the air flowing over the aircraft. To illustrate the calculation methods of the pitch-up attitude and approach velocity of the aircraft, a vector diagram showing the associated velocity components is presented in Figure 3b, where ${\mathit{V}}_{air}$ is the airspeed of the aircraft, ${\mathit{V}}_{\mathrm{A}}$ is the approach velocity of the aircraft, ${\mathit{V}}_{\mathrm{D}}$ is the deck wind speed, ${\theta }_{air}$ is the angle of airspeed relative to the horizon, ${\theta }_{\mathrm{A}}$ is the angle of approach velocity relative to the horizon, ${\varphi }_{air}$ is the angle between the projection of the airspeed vector on the horizontal plane and the deck wind velocity, and ${\varphi }_A$ is the angle between the projection of the approach velocity vector on the horizontal plane and the deck wind velocity. The components of airspeed ${\mathit{V}}_{air}$ can be written in terms of the magnitude of the approach speed $V_{\mathrm{A}}$ as

$$V_{airx}=V_{air}\cos {\theta }_{air}\cos {\varphi }_{air}=V_{Ax}+V_D=V_A\cos {\theta }_A\cos {\varphi }_A+V_D,$$

(13)

$$V_{airy}=V_{air}\cos {\theta }_{air}\sin {\varphi }_{air}=V_{Ay}=V_A\cos {\theta }_A\sin {\varphi }_A,$$

(14)

$$V_{airz}=V_{air}\sin {\theta }_{air}=V_{Az}=V_A\sin {\theta }_A.$$

(15)

Therefore,

$$V_{air}^2={\left(V_A\cos {\theta }_A\cos {\varphi }_A+V_D\right)}^2+{\left(V_A\cos {\theta }_A\sin {\varphi }_A\right)}^2+{\left(V_A\sin {\theta }_A\right)}^2$$

(16)

The solution of approach speed $V_{\mathrm{A}}$ can be obtained by using Equation (16). The angle of airspeed relative to the horizon ${\theta }_{air}$ and the angle between the projection of the airspeed vector on the horizontal plane and the carrier velocity ${\varphi }_{air}$ can be calculated from Equations (13)–(15) as

$${\theta }_{air}={\sin }^{-1}\left({\displaystyle \frac{V_{A}sin{\theta }_A}{V_{air}}}\right)$$

(17)

$${\varphi }_{air}={\tan }^{-1}\left({\displaystyle \frac{V_{\mathrm{A}}cos{\theta }_{\mathrm{A}}sin{\varphi }_{\mathrm{A}}}{V_{\mathrm{A}}cos{\theta }_{\mathrm{A}}cos{\varphi }_{\mathrm{A}}+V_{\mathrm{D}}}}\right)$$

(18)

The pitch-up attitude of the aircraft is $\theta =\mathsf{\alpha }-{\theta }_{air}$, and the yaw attitude of the aircraft relative to the flight patch would be $\varphi ={\theta }_A-{\theta }_{air}$.

3.2. Domain Precursor Inflow Approach

The impact of the carrier air-wake on carrier-based aircraft landing can be examined by numerically conducting a coupled solution calculation of the air-wake field and the moving aircraft flow field. However, due to the high instability, nonlinearity of the coupled flow field, and the disparate spatiotemporal scales between the carrier and aircraft, a small time step and fine grid are necessary for simulating the landing of the carrier-based aircraft in the wake environment. As a result, the computational time of conventional numerical simulation methods is generally too expensive to exploit the aerodynamic variability of carrier-based aircraft landing. According to the characteristic flow structure of the ship air-wake, the influence of the ship air-wake on the approaching aircraft is confined to the aft region of the flight deck. Therefore, to address the above-mentioned issues, this paper employs a domain precursor inflow method, which is frequently adopted to generate turbulent flow boundary conditions [21,33,34].

As illustrated in Figure 4, the schematic of the domain precursor inflow simulation approach is presented. There are two simulation domains. The first one is the entire air flow field of carrier, as shown in Figure 4a, and the other only includes the air-wake of the aft carrier region, see Figure 4b. Face $P_I$ in the entire domain and the precursor inflow boundary face in the aft air-wake domain are in the same plane. The unsteady numerical simulation of the air-wake flow field without the presence of the aircraft is first performed in the entire domain shown in Figure 4a. When the unsteady air-wake field is fully developed, the solution of the entire domain at a moment $t_0$ and the transient solution at each physical time step in face $P_I$ after $t_0$ are saved. The aft air-wake domain involved is initialized by interpolating from the solution saved in the entire domain. The air-wake field involving aircraft landing is computed in the aft domain of Figure 4b with an unsteady precursor inflow boundary condition, which can be interpolated from the time series solution saved in plane $P_I$. With the domain precursor inflow simulation approach, the simulation of carrier-based aircraft landing can be completed by the following steps:

(1)

The entire carrier air-wake domain, the aft carrier air-wake domain, and the aircraft flow field domain are determined, and the meshes in those domains are generated separately.

(2)

The unsteady flow field of in the entire carrier air-wake domain is computed by using the RANS-LES hybrid solver to a fully developed state.

(3)

The solution of the entire domain at a moment $t_0$ and the transient solution at each physical time step after $t_0$ in the same plane as the inflow of aft air-wake domain are saved.

(4)

The steady flow field in the aircraft domain is calculated, and the solution is saved as the initialized state of landing simulation.

(5)

The aft carrier air-wake field is initialized by interpolating from the solution of the entire carrier air-wake domain at $t_0$.

(6)

The mesh and solution of the aircraft domain are adjusted to the proper state according to landing simulation condition.

(7)

The aircraft landing through the carrier air-wake is simulated on the overset grids of the aft air-wake domain mesh and aircraft mesh with the unsteady precursor inflow boundary condition.

Notably, there may not necessarily be a boundary face in the entire domain mesh that is in the same plane as the inflow boundary face of the aft air-wake domain as shown in Figure 4. In this work, the solution in plane P_I is saved by recording the flow solution on the nodes of the grid cells intersecting with the plane where the precursor inflow boundary face in the aft air-wake domain is located. The unsteady precursor inflow boundary condition in the aft air-wake domain is interpolated from the solution of those nodes. The interpolation stencils used in initializing the aft air-wake field and setting the unsteady precursor inflow boundary are built by employing the same method as the overset grid method. The steady flow field around the aircraft is computed with a fixed angle of attack α. When the landing path is given, the direction of airspeed and the magnitude of approach speed can be determined. The aircraft mesh is placed at the starting point of the landing path and rotated to the pitch angle $\theta$ and the yaw angle ${\theta }_A$. Keeping the approach speed and attitude constant in the landing simulation, the aircraft can land on the carrier on the given flight path with a fixed angle of attack.

Compared with using the entire flow field around the carrier to calculate the aircraft landing, the domain precursor inflow approach method can omit the computational domain to a great extent, resulting in a large-scale reduction in the number of computational grids. Its speed of computation allows for several landings to be computed within a reasonable time.

3.3. Landing Simulation Setup

The configurations and the reference frame used in this study are illustrated in Figure 5. The complete ship model is a generic aircraft carrier model [35], with a length of 335 m and a width of 74 m. It is a “clean deck” configuration such that no aircraft or support equipment are present on the deck. The aft portion of carrier deck selected for landing simulation is shown in red in Figure 5c, and the dimensionless length, scaled by the aircraft carrier’s length, is 0.153. The carrier-based aircraft is a simplified model, with a length of 20 m, a wingspan of 12 m, and a height of 12.5 m.

The computational domain for numerical simulation of aircraft carrier wake field is selected as a rectangular region, as shown in Figure 4. According to Refs. [13,16] and considering the influence region of air-wake, the distance from the entrance to the bow of the ship is set to twice the length of the ship, the length from the exit to the stern is set to four times the length of the ship, the width of the calculation domain is set to three times the width of the ship, and the height of the calculation domain is set to ten times the height of the ship. The computational domain of aft air-wake is the aft portion of full air-wake simulation as shown in Figure 4b. The aircraft’s computational domain is a cylindrical region with a length of five times the length of the aircraft and a diameter of five times the span of the aircraft. The unstructured grids are generated in these three computational domains. The wall surfaces on the carrier and aircraft are the triangular elements. The prism elements are generated in the boundary layer near the wall, while isotropic tetrahedral cells are generated in the field away from the wall. To capture the complex flow structures in the air-wake of the carrier, the downstream regions of the carrier in the full air-wake grid and aft air-wake grid are refined locally. To verify the accuracy of interpolation between the component grids in the overset grid system, the cell size of aircraft’s field grid is the same as that of aft air-wake field grid. As a result, the grids in the full air-wake domain, aft air-wake domain, and aircraft domain have 61.15 million, 26.23 million, and 17.12 million cells, respectively. Figure 6 illustrates the details of these three grids.

Following Ref. [36], a deck wind speed of 25 m/s, along the x-direction, is used in this work. The aircraft’s airspeed during landing is maintained at 75.0 m/s and the angle of attack, α, is 6.0°. The angle ${\varnothing }_{\mathrm{A}}$ between the horizontal projection of the approach velocity vector and the deck wind velocity is fixed at 5.0° to align the landing path with the angled flight deck. The time-accurate computation of the unsteady air-wake at 25 m/s is first carried out on the grid of the entire domain using the RANS-LES hybrid method with a time step size of 0.01 s. When the computation is run for a period of 40 s, the air-wake fluctuations produced by the carrier are completely propagated through the domain. The solution in the entire domain at this moment and the solutions on the nodes of the grid cells intersecting with plane $P_I$ at the time steps within the next 40 s are saved. The aft air-wake domain is then initialized by interpolating from the solution in the entire air-wake domain at 40 s. Before performing the landing simulation, the steady computation of the aircraft flow with an angle of attack is carried out at 75 m/s. The simulations of aerodynamic variability of the carrier-based aircraft at the given landing paths are run at last on the overset grids of the aft air-wake domain and aircraft domain.

4. Results and Discussion

4.1. Validation of the Domain Precursor Inflow Approach

The CFD solver and dynamic overset grid method have been well validated, and results can be found in [20,22,37]. To elucidate the differences between the results obtained using the DDES method and the RANS method, the ship air-wake was computed under identical conditions with both methods. The vortex structures of different methods illustrated with the iso-surface of the Q-criterion [38] are shown in Figure 7, and it can be observed that the DDES method exhibits a superior capability to capture multi-scale flow structures. Nevertheless, validation for the domain precursor inflow approach is given here. The unsteady air-wake fields of the carrier from 40 s are computed on the grids of the entire domain and aft domain with the precursor inflow boundary condition, respectively. As shown in Figure 8, four probe points are selected to compare the consistency of the results obtained by the two methods. The dimensionless distances from these four probe points to the precursor inflow boundary surface, scaled by the length of carrier, are 0.0597, 0.149, 0.448, and 1.045, respectively.

The velocity fluctuation history for 40 s at four probe points calculated using the precursor inflow method is shown in Figure 9 and compared with the results of the entire domain simulation. It can be observed that the unsteady fluctuation history of the flow field at probe points 1 and 2 obtained by the two methods is completely consistent throughout the entire simulation time range. The velocity histories at probe points 3 and 4 calculated by the two methods are also consistent for more than the first ten seconds, but the results calculated by the two methods begin to show differences from 55 s and 52 s, respectively. This difference gradually increases over time. However, the frequency of fluctuations in the flow field variables over time is consistent. The instantaneous vortex structures are visualized by displaying the iso-surface of the Q-criterion at 10 s, 20 s, and 30 s, which are given in Figure 10. The entire domain simulation and aft air-wake domain simulation with precursor inflow boundary can capture the irregular vortex shedding. Both simulations have the same iso-surface of the Q-criterion in the air-wake field near the tail of the carrier. Differences in the vortex structure are observed only in the later part of the wake field after a long period of simulation. As shown in Figure 10c(i,ii), the dimensionless distance scaled by the length of the carrier from the precursor inflow boundary surface to the region where the difference in vortex structures begins to appear is about 1. It can be seen that the results in Figure 10 concur with those in Figure 9. This is mainly due to the strong unsteady and nonlinear nature of the air-wake field, which has flow field structures varying in spatial and temporal scales. As a result, a small disturbance upstream or at the previous moment can cause differences in flow field distribution over long distance or periods of time. However, this difference does not alter the spatiotemporal distribution characteristics of the air-wake field.

In fact, the differences in the wake field in the region far from the aircraft carrier will not affect the simulation of the landing process of the carrier-based aircraft. This is because when differences begin to appear in this region after a period of time, the carrier-based aircraft has already approached the tail of the aircraft carrier, where the simulation result is completely consistent with that of the entire domain simulation. This can be further validated by performing simulations of carrier-based aircraft landing with the two different air-wake field simulation methods. As illustrated in Figure 11, The aerodynamic force coefficients and moment coefficients of the landing aircraft calculated on the aft domain with the precursor inflow boundary condition are compared with those calculated on the entire domain. The calculation method of these coefficients is shown in Equations (19) and (20).

$$C_i=i/(q_{\infty }S) \left(i=L,D,Y\right)$$

(19)

$$C_j=j/\left(q_{\infty }Sc\right) \left(j=l,m,n\right)$$

(20)

where $C_i$ represents the aerodynamic force coefficient, $L$ is lift force, $D$ drag force, $Y$ lateral force, $C_j$ represents the aerodynamic moment coefficient, $l$ is rolling moment, $m$ pitching moment, $n$ yawing moment, and $S$ represents the reference area of the aircraft, which is 72 m². The reference length, $c$, is 3.5 m. The dynamic pressure $q_{\infty }$, defined by the aircraft’s airspeed, is given by $q_{\infty }=0.5\rho V_{air}^2$.

In Figure 11, the aerodynamic coefficients computed with entire air-wake domain simulation are denoted as “EFS”, in contrast to the “DPIS” coefficients. It can be seen that the two results are completely consistent with each other. The comparison of several instantaneous flow fields during the landing process is shown in Figure 12; it is difficult to see the difference between the results obtained by the two methods from the vortex structures around the carrier-based aircraft. Figure 13 illustrates the assembly results of overset grid when the carrier-based aircraft reaches two different positions during the landing. It can be seen that the hole-cutting of the overset grid is reasonable. The sub-grids of the aircraft and the carrier have comparable scales near the interpolation boundary, which can guarantee the interpolation accuracy.

From the results presented above, it is implied that using the domain precursor inflow method to predict the air-wake field and the aerodynamic characteristics of the carrier-based aircraft landing on the carrier is highly reliable. The precursor inflow method reduces computational costs by 57% compared to full-domain simulations while maintaining fidelity in the carrier air-wake flow field. In the next subsection, the aerodynamic variability due to the differences in landing trajectory and timing will be investigated on the aft air-wake domain simulation with the precursor inflow boundary condition.

4.2. Variability Analysis of Aerodynamic Loads

This paper mainly considers the aerodynamic variability of the carrier-based aircraft caused by differences in landing point, landing trajectory, and landing time. The reference origin is set at the expected landing point, and the lateral deviations of the landing point are ±10 m. The angle of approach velocity relative to the horizon, ${\theta }_{\mathrm{A}}$, would vary within the range of 4.0° to 6.0°. The longitudinal distances between the landing point and starting point vary from 500 m to 600 m, which can result in a landing time difference of over 1.0 s. To reduce the number of CFD solver runs, an optimal Latin hypercube sampling method is employed to generate 15 sample points in the spaces of the angle of approach velocity relative to the horizon, ${\theta }_{\mathrm{A}}$, lateral deviation of landing point, ${\delta }_y$, and longitudinal distance between the landing and starting points, $D_x$. The parameters of the computation samples are summarized in

Table 1. The parameters of the computation states.

ID	${\mathit{\theta }}_{\mathbf{A}}$ (°)	${\mathit{\delta }}_{\mathit{y}}$ (m)	${\mathit{D}}_{\mathit{x}}$ (m)
1	4.6000	1.3333	550.0000
2	4.8667	−2.6667	523.3333
3	5.8000	−6.6667	536.6667
4	5.9333	2.6667	576.6667
5	4.3333	8.0000	563.3333
6	5.4000	5.3333	590.0000
7	5.0000	9.3333	543.3333
8	4.2000	−5.3333	583.3333
9	5.6667	6.6667	516.6667
10	5.5333	−4.0000	556.6667
11	4.7333	−1.3333	596.6667
12	5.2667	0.0000	503.3333
13	4.0667	4.0000	530.0000
14	4.4667	−8.0000	510.0000
15	5.1333	−9.3333	570.0000

. As a result, the landing paths utilized in the computations to demonstrate the aerodynamic variability are illustrated in Figure 14.

The positions of the aircraft’s center of mass for all the landing simulations are presented in Figure 15, as a function of time. Due to different landing trajectories and starting positions, the position of the aircraft relative to the carrier is different at the same time for different landing simulations. For a better reading, the aerodynamic loads are shown as functions of the centroid coordinates of carrier-based aircraft in the following.

As the initial flow field around the aircraft is calculated by a steady simulation, there will be an adjustment process between the aircraft flow field and ship air-wake field at the beginning of the coupling calculation. The adjustment process of the coupled flow field would cause a certain change in the aerodynamic force in the initial stage of the landing simulation. Therefore, Figure 16 presents the lift coefficient $C_L$, drag coefficient $C_D$, lateral force coefficient $C_Y$, rolling moment coefficient $C_l$, pitching moment coefficient $C_m$, and yawing moment coefficient $C_n$ as functions of longitudinal coordination of the aircraft from X = 480 m, and Figure 17 provides the standard deviation of these coefficient values as a function of longitudinal distance to illustrate the uncertainty associated with these coefficients. In the subsequent landing process, the aerodynamic forces and moments have different variabilities along the longitudinal position, and the standard deviation of different coefficients is relatively small. Prior to 400 m from the landing point, across all carrier landing trajectories, the aerodynamic loads experienced by the carrier-based aircraft vary minimally with its longitudinal position relative to the carrier. As illustrated in Figure 18, the aircraft is located above the vortex structures of the air-wake field within this range, and the streamline at the location of the aircraft is approximately a straight line. From the simulated flow field of the landing process shown in Figure 19a, it can be further observed that there is no mutual interference between the aircraft flow field and the aircraft carrier wake flow field within this position range. Analysis of the air-wake flow field illustrated in Figure 18 and Figure 19 indicates that the aircraft has not yet entered the carrier’s wake flow field at this stage.

At about 375 m, all aerodynamic coefficients show some fluctuations, as can be seen from Figure 16 and Figure 17, and the standard deviation of the coefficients also increases at this position. From Figure 18 and Figure 19b, it can be seen that the reason for the fluctuation of aerodynamic coefficients is that from this position, the aircraft begins to enter the wake region, and the aircraft flow field and the carrier’s air-wake field begin to interfere with each other. However, the fluctuation amplitudes of the aerodynamic coefficients near this location are relatively small, and there may be some differences in the location of aerodynamic coefficient fluctuations when landing along different trajectories. Afterwards, the lift coefficient, drag coefficient, and pitch moment coefficient of the carrier-based aircraft began to decrease with the decrease in the longitudinal relative position from the aircraft carrier, while the lateral force coefficient, roll moment coefficient, and yaw moment coefficient remained very small, near zero. At this stage, the fluctuation amplitude of all aerodynamic coefficients of carrier-based aircraft is relatively small, and the different landing trajectories of the aircraft have little effect on the aerodynamic coefficients. This phenomenon arises because the wake of the aircraft carrier tends to become uniform after long-distance evolution, and its velocity loss increases with decreasing distance from the aircraft carrier, resulting in a decrease in the effective wind speed of the aircraft, which in turn leads to a decrease in lift coefficient, drag coefficient, and pitch moment coefficient.

Starting from a position of approximately 250 m, significant changes in aerodynamic coefficients began to occur along each landing trajectory, and the standard deviation of these coefficients also increases correspondingly. First of all, these aerodynamic coefficients experience significant fluctuations as they vary with the position of the carrier-based aircraft. The fluctuation amplitude of the lift coefficient (up to 25%), drag coefficient (18%), and pitch moment coefficient (30%) is most severe near the position of approximately 200 m, and the standard deviations of these coefficients also reach their maximum values at this location. From the flow field distribution shown in Figure 18 and Figure 19c, it can be seen that at this position, the carrier-based aircraft enters the wake influence zone of the carrier island, where the velocity fluctuation amplitude is large, and the frequency is high. In the following, as the aircraft approaches the carrier, the intensity of the wake vortex in the wake increases, and the velocity field fluctuates more with time and space, see Figure 18 and Figure 19d–f. Therefore, when the carrier-based aircraft flies in this area, all aerodynamic coefficients exhibit significant high-frequency fluctuations. Secondly, as the distance to the aircraft carrier decreases, the overall trend of the lift coefficient, drag coefficient, and pitch moment coefficient is to first rapidly decrease, reaching a minimum at about 50 m from the landing point, and then sharply increase, with values exceeding those of the lift coefficient, drag coefficient, and pitch moment coefficient when flying in an undisturbed flow field away from the aircraft carrier. As shown in Figure 20, the distributions of the pressure and streamline in the vertical plane where the glide slope line is located are presented for the carrier-based aircraft at the longitudinal positions of 200 m, 125 m, 50 m, and 0 m. It can be seen from Figure 20a–c that as the distance to the aircraft carrier decreases, the high-pressure distribution range on the lower surface of the carrier-based aircraft gradually decreases, causing both its lift coefficient and drag coefficient to decrease simultaneously. But when the carrier-based aircraft enter the deck, due to ground effect, the lift coefficient and drag coefficient of the carrier-based aircraft increase sharply, which can amplify load variations by 12–15%, see Figure 20d. At last, due to the significant vertical and lateral velocity distribution in the wake field of this area, the aircraft will experience large asymmetric aerodynamic loads such as lateral forces, rolling moments, and yaw moments, see Figure 16c,d,f, and the standard deviations of these coefficients also consistently maintain relatively high values, which can be found in Figure 17c,d,f. For example, the fluctuation amplitude of the lateral force coefficient can reach the same level as the drag coefficient. Meanwhile, the instantaneous vortex structure is illustrated in Figure 21 with the iso-surface of the Q-criterion around the carrier-based aircraft at the longitudinal positions of 200 m, 125 m, 50 m, and 0 m.

It should be noted that although the aerodynamic load changes of carrier-based aircraft on different landing trajectories are similar, there are certain differences in the locations where strong pulsations occur. This is mainly because the carrier’s air-wake flow field changes in space and time. When carrier-based aircraft enter the wake field with different trajectories, the wake structure they experience is different, so the change history of aerodynamic loads experienced by carrier-based aircraft will be different. In addition, when entering the wake flow field along the same trajectory at different times, the wake flow field experienced by the aircraft is also different, which can also cause different changes in the aerodynamic load history of the carrier-based aircraft.

In this subsection, the aerodynamic variability due to the differences in landing trajectory has been investigated on the aft air-wake domain simulation with a precursor inflow boundary condition. The aerodynamic force coefficient and moment coefficient are presented as functions of longitudinal coordination. Significant aerodynamic variability within 250 m of the carrier, due to distinct landing trajectories, is revealed. The results suggest that the proposed method is efficient and can be used for initial investigations of carrier-based aircraft landing on the carrier before flight testing.

5. Conclusions

To address the computational challenges posed by the disparate spatiotemporal scales of the ship air-wake and aircraft motion, this work investigates the aerodynamic variability during carrier-based aircraft landing based on an SST-DDES-based RANS-LES hybrid method with dynamic parallel unstructured overset grids. A domain precursor inflow method is developed to efficiently generate unsteady inflow boundary conditions from precomputed full-domain air-wake simulations. The main conclusions are summarized as follows:

(1)

The domain precursor inflow method can significantly reduce computational costs compared to full-domain simulations while maintaining fidelity in air-wake simulation.

(2)

Parametric analyses of 15 distinct landing trajectories reveal significant aerodynamic variability, particularly within 250 m of the carrier, where interactions with island-generated vortices induce fluctuations in lift (up to 25%), drag (18%), and pitching moments (30%).

(3)

Ground effects near the deck further amplify load variations by 12–15%, while lateral deviations in landing paths generate asymmetric forces and moments.

The results indicate that the proposed methodology exhibits remarkable computational efficiency in analyzing aerodynamic variability, providing critical insights into aerodynamic uncertainties during carrier-based aircraft landing. Future work will extend the methodology to include atmospheric turbulence effects and pilot-in-loop control strategies; the impact of aircraft or support equipment on the deck on the air-wake will also be discussed.