Study on Aerodynamic Characteristics of DLR-F4 Wing–Body Configuration Using Detached Eddy Method Incorporated with Fifth-Order High-Accuracy WENO/WCNS

Abstract

To investigate the aerodynamic characteristics of the subsonic transport standard model (DLR-F4 wing–body configuration), this study uses the Spalart–Allmaras Detached Eddy Simulation (SA-DES) turbulence model as the core, coupling it with fifth-order WENO/WCNSs and HLLC approximate Riemann solver for numerical simulations under different angles of attack (AOA). Through comparative simulations, effects of grid density, turbulence models (URANS/DES), and spatial discretization schemes (second-order CDS, fifth-order WENO-JS/WCNS-JS) on accuracy are analyzed, focusing on grid convergence and numerical scheme dissipation in separated flows. The results show medium-density grid results are stable, balancing accuracy and efficiency. Under high AOA, DES outperforms URANS in capturing separated vortex structures, effectively reproducing small-scale vortices in the wing–body junction. High-order WCNS performs best in predicting wing-tip vortices and wake turbulence due to lower dissipation. WCNS-JS/WCNS-T (different weight functions) affect lift/drag coefficient errors: WCNS-JS has smaller lift prediction errors, while WCNS-T better reduces dissipation and maintains wing-tip vortex integrity. This study provides key references for high-accuracy simulations of complex separated flows, supporting efficiency improvement and accuracy optimization in aerospace vehicle aerodynamic design.

1. Introduction

Computational Fluid Dynamics (CFD) is currently an important tool for implementing and promoting aircraft aerodynamic shape design and conducting aerodynamics research, and it is one of the key means to understand mechanisms, explore new spaces, and expand extreme operating conditions [1]. With the increasing requirements for aerodynamic efficiency and flight safety of subsonic transport aircraft, accurately simulating complex flows of typical standard models—such as separation in the wing–body fusion zone and evolution of wingtip vortices—is of great significance for improving aircraft aerodynamic characteristics. The F4 wing–body configuration of the German Aerospace Center (DLR), as a classic representative of subsonic transport aircraft, has a large amount of wind tunnel test data and numerical test data for reference, and is widely used worldwide to validate different CFD methods [2,3]. In early studies, Rakowitz et al. [3] concluded through comparative analysis based on the DLR-F4 that structured CFD codes with high-quality grids and low dissipation levels can provide better aerodynamic prediction accuracy; Zheng Qiuya et al. [2] conducted tests by solving the RANS equations and coupling different turbulence models, which proved that grid density and turbulence models have significant impacts on the calculation accuracy of DLR-F4 drag, serving as an important basis for future research.

Turbulence simulation and spatial discretization schemes are important factors affecting CFD accuracy. Since the RANS method, as a semi-empirical approach to solving the RANS equations, has limitations, when dealing with high-angle-of-attack separated flows, the viscous-vortex assumption makes it impossible to correctly capture small-scale vortex structures [4]. To address this issue, hybrid RANS/LES methods (such as DDES) have been developed. In this method, the flow field is divided into two parts: the region adjacent to the wall is described using the RANS equations, while the region far from the wall is described using the LES equations. This method balances computational efficiency and unsteady flow resolution, and can also solve problems from two-dimensional to three-dimensional. The DES method proposed by Sun Yue et al. [5] shows that the error in capturing three-dimensional vortex structures is less than 5%, which is significantly lower than that of the URANS method. Additionally, Zhang Lu et al. [6] compared the results obtained by DDES, MDDES, and IDDES methods, indicating that compared with MDDES, IDDES achieves better results in simulating near-wall flows and provides a reliable option for future simulations of complex separated flows. Meanwhile, high-order numerical schemes provide a strong guarantee for reducing numerical dissipation and improving flow field resolution. The fifth-order WENO-Pe scheme proposed by Liu Bo [7] optimizes weight coefficients through a mapping function, effectively overcoming the order reduction defect of WENO-JS and WENO-Z at extreme points, with significantly reduced dispersion and dissipation errors; Yin Zhizhao [8] applied the fifth-order WENO scheme to the unsteady flow field of rotors, selecting reasonable weighting factors to obtain fifth-order accurate state quantities under the premise of sufficient accuracy, capturing the entire process of tip vortex generation and development, with calculation accuracy significantly higher than that of second-order schemes. The Weighted Compact Nonlinear Scheme (WCNS) has higher compactness and lower dissipation, making it more suitable for simulating flow characteristics under supersonic complex configurations. Zhang Juhui et al. [9] carried out numerical simulations of rotor flow fields based on the WENO-piecewise linear scheme, confirming the adaptability of such schemes in unsteady flow fields of rotating machinery, further supporting the reliability of high-order WENO schemes in simulating complex vortex flow fields; Wang Yuntao et al. [10,11,12] conducted numerical simulations of the DLR-F6 wing–body configuration and the CRM wing/body/horizontal tail configuration based on the fifth-order WCNS, and compared the differences in flow fields, aerodynamic parameters, and separation zones under different schemes. The results show that compared with the second-order scheme, the fifth-order WCNS improves the prediction accuracy of aerodynamic coefficients and the separation zone at the trailing edge of the wing root, and has good grid convergence.

Despite the existing research on turbulence models and high-order schemes, the comparative studies on the DLR-F4 wing–body configuration are not comprehensive: first, there are few comparative studies between different high-order schemes (WENO and WCNS), and little research on the impact of WCNS with different weight function designs (WCNS-JS and WCNS-T) on aerodynamic characteristics; second, the coupling effects of grid density, turbulence models, and spatial schemes are still lacking, and whether a balance between accuracy and computational efficiency can be achieved under medium grid conditions remains to be verified; Li Wei et al. [13] found in the numerical simulation of the DLR-F6 configuration that grid topology has a significant impact on simulation results, and a reasonable topological structure can reduce the demand for grid quantity, which provides a reference for selecting H-type topology grids for the DLR-F4 model in this paper; third, the wing–body fusion zone is affected by the airfoil boundary layer of the wing, causing deviations in pressure coefficients, which has not been analyzed in depth from the perspective of high-order schemes and DES models. However, Li Yuan et al. [14] pointed out in the analysis of aerodynamic characteristics of wind turbine airfoils at full angles of attack that deviations in airfoil surface pressure distribution are often related to boundary layer transition and separation positions, which provides ideas for analyzing such deviations in this paper. Finally, it is worth mentioning that research on hybrid turbulence simulation (such as the adaptive turbulence simulation method SATES proposed by Han Xingsi et al. [15]) provides a new approach for achieving high-precision simulation, but it has not been applied to civil transport aircraft models such as the DLR-F4; the adaptive hybrid RANS/LES model based on OpenFOAM proposed by Guo et al. [16] improves simulation accuracy by dynamically adjusting vortex viscosity and grid scale, which provides a direction for turbulence model optimization, but considering the maturity of engineering applications, this paper still uses the SA-DES model; in addition, the isomorphic hybrid solver of the national numerical wind tunnel “Fenglei” software [17] needs to realize hybrid simulation by combining simplified large-scale vortices and quasi-structural resolution technology when dealing with complex multi-region configurations, and has not conducted research on the suitability of high-order schemes for the DLR-F4 wing–body configuration.

Based on this, this paper takes the DLR-F4 wing–body configuration as the reference case, adopts the SA-DES turbulence model combined with the fifth-order WENO, WCNSs, and the HLLC approximate Riemann solver to conduct numerical simulations under different angle-of-attack conditions. By comparing with relevant test data, it systematically analyzes the impacts of grid density, turbulence models (URANS/DES), and spatial discretization schemes (second-order CDS, fifth-order WENO-JS, WCNS-JS/T) on simulation accuracy. It further studies grid convergence and numerical scheme dissipation for separated flows, explores the variation laws of grid convergence and numerical scheme dissipation in separated flows, so as to provide references for high-precision numerical simulation technology of complex separated flows and technical support for improving the efficiency and accuracy of aerodynamic design of aerospace vehicles.

2. DLR-F4 Model and High-Order Accuracy Computational Methods

2.1. Introduction to the DLR-F4 Model

The DLR-F4 wing–body configuration is a model designed and developed by the German Aerospace Center (DLR). As a representative model in the design of subsonic transport aircraft, it was adopted as the reference test case for the First CFD Drag Prediction Workshop and possesses extensive wind tunnel test data and numerical simulation results.

The wind tunnel test data employed for the computational model in this study is sourced from the First CFD Drag Prediction Workshop (DPW-1), which is a widely recognized benchmark dataset in CFD aerodynamic validation and has been detailed in prior studies on DLR-F4 [2,3]. For this model, the design Mach number (${Ma}_{\infty }$) is 0.6, and the free-stream Reynolds number (${Re}_{\infty }$)—calculated based on the mean aerodynamic chord—is 3.0 × 10⁶. The configuration of the DLR-F4 model is illustrated in Figure 1, with its basic geometrical parameters given in

Table 1. Geometric Parameters of the DLR-F4 Model.

Name	Parameter
Half-span of the Model	0.5586 m
Mean Aerodynamic Chord	0.1412 m
Reference Area of the Whole Aircraft	0.1450 m2
Aspect Ratio	9.5
Leading Edge Sweep Angle	27.1°
Taper Ratio	3.9596
Fuselage Length	1.1920 m
Fuselage Width	0.1484 m
Moment Reference Point	(x, y, z) = (0.5049, 0, 0)

.

2.2. Introduction to High-Order Accuracy Computational Methods

The high-order accurate computational method adopted in this paper takes “high-order numerical scheme—turbulence model—approximate Riemann solver” as the core framework. Through the collaborative optimization of each module, an accurate simulation of the complex flow field of the DLR-F4 wing–body configuration is achieved. For spatial reconstruction, the main flow equations adopted a 5-point central difference stencil for second-order accurate reconstruction, while the WENO-JS scheme used a 5th-order reconstruction stencil (comprising 3 overlapping 3-point sub-stencils) for regions with strong gradients. Regarding limiting methods, the Jameson scheme suppressed oscillations via adaptive artificial dissipation, and the WENO-JS scheme relied on its inherent weight mapping mechanism without additional limiters. Both schemes employed characteristic-wise reconstruction, which effectively preserves the propagation characteristics of flow disturbances, enhances numerical stability in complex flow regions, and aligns with the study focus on dissipation differences. The design and selection of each module are based on the current research progress in the CFD field, as follows:

2.2.1. High-Order Numerical Scheme

High-order numerical schemes are key to reducing numerical dissipation and improving the resolution of mesoscale flow fields. This paper focuses on the fifth-order Weighted Essentially Non-Oscillatory (WENO) scheme and the fifth-order Weighted Compact Nonlinear Scheme (WCNS), both of which have been validated for effectiveness in simulating complex flows.

The WENO scheme achieves high-order accuracy through the convex combination of multiple reconstruction stencils while avoiding non-physical oscillations near shock waves. Early WENO-JS and WENO-Z schemes were prone to accuracy degradation at extreme points of flux functions. Liu Bo et al. [7] proposed the WENO-Pe scheme by designing a mapping function to optimize weight coefficients. Its approximate dispersion relation shows that the dispersion error and numerical dissipation of this scheme are smaller than those of traditional WENO schemes, demonstrating better discontinuity capture capability in cases such as the Sod shock tube and double Mach reflection. Yin Zhizhao et al. [8] applied the fifth-order WENO scheme to the simulation of unsteady vortex flow fields of rotors; by selecting reasonable weighting factors, they obtained fifth-order accurate state variables at the interface and successfully captured the formation and evolution of tip vortices. Zhang Juhui et al. [9] used the WENO-piecewise linear scheme to simulate rotor flow fields, further confirming the stability and accuracy of such schemes in handling unsteady vortices in rotating machinery. Within the hybrid RANS/LES framework, Shi et al. [18] coupled the fifth-order WENO-Roe scheme with the DDES method to simulate tip vortices of hovering rotors, finding that this combination can effectively improve the spatial resolution of wake age. Li et al. [19] introduced the third-order WENO scheme into implicit Large Eddy Simulation (iLES), achieving accurate simulation of laminar separation and transition in the flow around the NACA0012 airfoil, with grid resolution reduced by 70% compared to DNS, balancing accuracy and efficiency. The fifth-order WENO-JS scheme adopted in this paper, based on the above research, balances accuracy and stability through classical weight function design, and is used for comparative analysis of the dissipation characteristics of high-order schemes.

The WCNS, characterized by high accuracy and low dissipation, exhibits excellent performance in numerical simulations of complex flow fields under transonic conditions. Wang Yuntao et al. [10,11,12] conducted numerical simulations of the DLR-F6 wing–body configuration and the CRM wing/body/horizontal tail configuration based on the fifth-order WCNS. The results showed that compared with the second-order scheme, the WCNS is more accurate in predicting aerodynamic coefficients and the separation zone at the trailing edge of the wing root, and is less prone to divergence. Li Wei et al. [13] further pointed out in the simulation of the DLR-F6 that the adaptability between grid topology and the WCNS can significantly affect the simulation accuracy of separation zones, which provides a basis for selecting the combination of H-type topology grids and the WCNS for the DLR-F4 model in this paper. Li Song et al. [20] used the WCNS-E-5 scheme to simulate a trapezoidal wing high-lift configuration, and the results showed that compared with the traditional second-order scheme, this scheme is more likely to capture flow field structures near high-angle-of-attack stall and is less sensitive to grids. Differences in WCNSs are mainly determined by different weight functions. This paper selects the WCNS-JS and WCNS-T schemes for comparison: the WCNS-JS scheme uses linear weight functions to stably obtain calculation results, and Wang Yuntao’s [11] simulation of the DLR-F6 verified that this method is more accurate in pressure distribution; the WCNS-T scheme adopts nonlinear weight function optimization, and Shi [21] concluded in slotted-tip rotors that the low-dissipation WCNS-T scheme can better maintain the integrity of tip vortices, laying a theoretical foundation for the advantages of the WCNS-T scheme in vortex structure capture in this paper.

2.2.2. SA-DES Turbulence Model

The detached eddy simulation (DES) adopted in this study was the Spalart–Allmaras DES (SA-DES), coupled with the SA turbulence model. The grid length scale was normalized using the mean aerodynamic chord of the DLR-F4 configuration (reference length C_REF = 0.1412, dimensionless) with a grid scaling factor of 1.0, and wall distance was computed via a geometric point method. For avoiding Modeled-Stress Depletion (MSD), the intrinsic near-wall shielding mechanism of SA-DES was leveraged, which restricts improper RANS-to-LES transition in near-wall regions without additional limiters. The shielding function (f_d) of SA-DES is defined as follows:

$$f_d=1-\tanh [{(8r_d)}^3]$$

Among them, $f_d$ is the shielding function of SA-DES, whose core role is to control the RANS-LES mode transition: RANS mode is enforced when $f_d\approx 0$ in the near-wall region, and switching to LES mode is allowed when $f_d\approx 1$ in the off-wall region; $r_d$ is a dimensionless auxiliary parameter, calculated based on local flow characteristics and geometric features, and serves as the core criterion for triggering the mode transition.

The key parameter controlling near-wall RANS-LES blending was C_DES = 0.65, a standard setting for SA-DES to ensure smooth flow regime transition.

Zhuo Congshan et al. [22] compared the performance of the SA model [23] and the Baldwin-Lomax (B-L) model [24] in high-Reynolds-number airfoil flows, and found that the SA model has higher accuracy in predicting the stall angle of attack and stronger capability in handling separated flows. Li Yuan et al. [14] also confirmed in their study on aerodynamic characteristics of wind turbine airfoils at full angles of attack that the SA model has more advantages over other eddy viscosity models in simulating high-angle-of-attack separated flows.

Sun Yue [5] verified through cases of flow around a cylinder and high-angle-of-attack flow over a NACA0021 airfoil that the error of the SA-DES model in capturing three-dimensional vortex structures is reduced by 9.38% compared with the URANS model, which can effectively reproduce the evolution process of small-scale vortices in the separation zone. Zhang Lu [6] further pointed out that DES-type models (such as IDDES) perform better than traditional RANS models in the near-wall region, but their performance depends on coupling with high-order schemes. It is based on this understanding that this paper combines SA-DES with fifth-order WENO/WCNSs, aiming to synergistically improve the accuracy of capturing separated vortex structures through “low-dissipation schemes and high-resolution models”.

In addition, Su Hang [4] proposed a Reynolds stress correction method for RANS models based on machine learning, which provides a new direction for turbulence simulation. However, considering the stability and maturity of engineering applications, this paper still adopts the widely verified SA-DES model. The adaptive hybrid RANS/LES model proposed by Guo et al. [16], which adapts to different flow field regions by dynamically adjusting the eddy viscosity coefficient, shows potential in complex flows, but its validation for the DLR-F4 model is insufficient. Although the SST-Helicity model proposed by Sun et al. [25] performs excellently in compressor flows, the adaptability of the SA-DES model to separated flows of wing–body configurations has been confirmed in the DLR-F4 simulation by Zheng Qiuya et al. [2], so other models are not selected.

2.2.3. HLLC Approximate Riemann Solver

The approximate Riemann solver is the core for flux calculation, directly affecting the accuracy of capturing shock waves and shear layers. Zheng Qiuya et al. [26] applied solvers such as Roe and E-CUSP to complex flow simulations and found through comparison that the Roe solver has higher accuracy in shock wave capture but with a large computational cost; while the E-CUSP solver, despite higher computational efficiency, is prone to numerical oscillations in strong shear flows. The HLLC solver is a modification of the HLL solver by adding contact discontinuity, and it has wider applications in the calculation of supersonic flow fields and separated flow fields [17,27]. Chen Jianqiang et al. [17] used the HLLC solver in the national numerical wind tunnel “Fenglei” software to implement a domain hybrid algorithm for structured/unstructured hybrid grids and verified its stability under different grid types; Yang et al. [27] accurately captured shock-vortex interaction phenomena in the simulation of transonic turbine tip leakage vortices based on the HLLC solver. Based on this, the HLLC approximate Riemann solver used in this section is selected and coupled with fifth-order WENO/WCNSs to better balance shock wave capture accuracy and shear layer resolution capability.

In addition, regarding grid generation and parallel computing, this paper adopts the grid strategy of the isomorphic hybrid solver proposed by Chen Jianqiang et al. [17], using multi-block structured grids to reduce grid generation complexity; Li Wei et al. [13] found in the DLR-F6 simulation that H-type topology grids are easier to achieve a balance between grid refinement and flow field resolution compared with other topologies, which is consistent with the idea of selecting H-type topology grids for the DLR-F4 model in this paper; the load balancing algorithm proposed by Zheng Qiuya et al. [26] provides a reference for optimizing the parallel computing efficiency in this paper, ensuring the computational feasibility of large-scale grids (such as fine grids with 5.0 × 10⁷ cells). In addition, Luo Jiajie et al. [28] focused on aerodynamic design in the nested collaborative optimization of laminar flow wings, and this paper applies this experience to analyze the aerodynamic characteristics and design requirements of the DLR-F4 wing–body configuration; Park et al. [29] captured inter-rotor vortex interference effects through fine grids in their study on aerodynamic interaction of coaxial rotors, which provides a reference for this paper to focus on details of tip vortex evolution; Zhang Qunfeng et al. [30] used the SST DES model and overlapping grid technology to simulate weapon separation flow fields, and their grid processing experience provides a reference for setting symmetric boundary conditions for the DLR-F4 model in this paper; Ozawa et al. [31] verified flow field reliability through RANS simulation and PIV data assimilation, which also provides a methodological reference for the comparative analysis of numerical results and experimental data in this paper; Chauhan et al. [32] conducted aerodynamic optimization of propeller-wing configurations based on the RANS method, confirming the practicality of RANS models in engineering aerodynamic design, which further supports the engineering significance of using the RANS/DES hybrid framework to study the aerodynamic characteristics of the DLR-F4 in this paper.

2.3. Generation of Computational Grids

Since the DLR-F4 model is symmetric with respect to the xy-plane, this study only performs calculations on the half-fuselage model of the DLR-F4 configuration and applies symmetric boundary conditions to the other half to improve computational efficiency.

To ensure computational reliability, this study first conducts a grid convergence study. The SA-URANS model is used to investigate the influence of three sets of grids with different densities (coarse grid, medium grid, fine grid) on the calculation results. Among them, the total number of grid cells for the coarse grid is 1.0 × 10⁷, for the medium grid is 3.0 × 10⁷, and for the fine grid is 5.0 × 10⁷. The detailed parameters of the three sets of grids are presented in

Table 2. Computational Grid Distribution Table of the DLR-F4 Model.

Grid Type	Number of Grid Nodes	Number of Grid Cells	Thickness of the First Grid Layer
Coarse Grid	1.0 × 107	0.99 × 107	0.006
Medium Grid	3.0 × 107	2.97 × 107	0.003
Fine Grid	5.01 × 107	4.96 × 107	0.001

. Near the wall, the height of the first grid cell satisfies $y^{+}\le 2$, and the grid growth rates in the streamwise and normal directions satisfy $\Delta y_{i+1}/\Delta y_i\le 1.21$.

The computational grids are generated using multi-block structured grids. The length of the far-field in the computational domain is set to 30 times the fuselage length, and the width is set to 20 times the fuselage length. The topological relationship of the three sets of grids is mainly based on H-type topology, as shown in Figure 2. Taking the medium grid as an example, the finally generated computational grid is shown in Figure 3.

3. Computational Results and Discussion of the DLR-F4 Wing–Body Configuration

3.1. Subsection

In this section, three sets of computational grids with different densities are used to perform calculations of the flow field around the DLR-F4 wing–body configuration at angles of attack of 0.9°, 4°, 7°, and 9.6°. The computational model adopted in this section is the SA-URANS model. In the following text, the coarse grid, medium grid, and fine grid are denoted by the letters C, M, and F, respectively.

The inflow boundary conditions were defined with a Mach number of 0.6, a Reynolds number of 3 × 10⁶, and a freestream temperature of 288.16 K. The far-field boundary was set with BCV = 0 (far-field distance > 10 chord lengths, no horseshoe vortex correction), and outflow conditions were handled implicitly via the far-field boundary. For turbulence, the simulation adopted the Spalart–Allmaras (SA) model combined with DES, a turbulent Prandtl number of 0.90, and treated the flow as fully turbulent without a transition model. The reference length (L) was the mean aerodynamic chord of the DLR-F4 configuration, with a dimensionless value of C_REF = 0.1412.

For the unsteady simulation, an implicit dual-time stepping scheme was adopted with a fixed time step of 0.005. The Courant–Friedrichs–Lewy (CFL) number was set to 4.5 for spatial discretization stability. A total of 40,000 physical time steps were performed, corresponding to a total physical sampling time of 200 for flow field statistics.

Figure 4 presents the variation curves of lift coefficients and drag coefficients with angles of attack, calculated by the URANS model using different grid densities. From the calculation results, it can be observed that at small angles of attack, the URANS model can accurately calculate the magnitudes of lift and drag coefficients, with small errors between the calculated values and the experimental values. However, as the angle of attack gradually increases, the errors between the lift coefficients calculated by the URANS model and the experimental values gradually increase.

Figure 5 presents the convergence curves of lift and drag coefficients versus computational iterations for representative cases under different schemes, which demonstrates that the force coefficients tend to stabilize after approximately 50,000 iterations, and the variation trends of the computational results are consistent with those of the experimental values.

Further observations from the computational results reveal that the lift and drag coefficients calculated by the URANS model under different grid densities also exhibit certain discrepancies. At small angles of attack, the calculated values are less affected by grid density. As the angle of attack increases, the lift coefficients obtained from the URANS model with different grid densities show distinct results: the computational results of the coarse grid deviate significantly from those of the medium and fine grids, while the results of the medium grid and fine grid are largely consistent. This indicates that when the grid reaches the medium grid density, the computational results tend to stabilize with an increase in grid quantity; at this point, further increasing the grid density will not cause significant changes in the computational results.

3.2. Comparative Study on High-Accuracy Schemes

Since the lift and drag coefficients calculated by the URANS model using the medium grid and fine grid are largely consistent, this study will use the medium grid to conduct relevant calculations in the following content. It mainly uses the DES models with the second-order CDS scheme, fifth-order WENO-JS scheme, and fifth-order WCNS-JS scheme to calculate and conduct comparative analysis on the aerodynamic data of the DLR-F4 wing–body configuration under different angles of attack, so as to investigate the influence of DES models with different turbulence models and different spatial discretization schemes on the computational results of the DLR-F4 wing–body configuration.

Figure 6 presents the variation plots of lift and drag coefficients with angle of attack calculated by different methods. It can be observed from the figure that at small angles of attack, the lift coefficients predicted by the URANS model and DES model are not significantly different from each other and are close to the experimental values. However, at large angles of attack, the lift coefficients predicted by both the URANS model and DES model are lower than the experimental values; compared with the URANS model, the lift coefficients predicted by the DES model are closer to the experimental values, while the lift coefficients predicted by DES models with different numerical schemes show little difference from each other.

From the variation in drag coefficients with angle of attack, it can be seen that the drag coefficients predicted by the URANS model and DES model are both close to the experimental values. Nevertheless, the figure also shows that the drag coefficients calculated by the WCNS-JS scheme exhibit smaller errors compared with the experimental values, in contrast to other numerical schemes.

Figure 7 presents the measurement station diagram of the wing spanwise pressure distribution in the wind tunnel experiment. Four stations with η = 18.5%, η = 33.1%, η = 51.2%, and η = 84.4% are selected in sequence from the wing root to the wing tip. Figure 8 shows the distribution diagrams of the time-averaged surface pressure coefficients at the four stations, calculated by different methods when the angle of attack is 0.9°.

It can be observed from the figures that at the three stations of η = 33.1%, η = 51.2%, and η = 84.4%, the pressure coefficient distributions calculated by different methods show little difference from each other and agree well with the experimental values. However, at the η = 18.5% station (i.e., the position closest to the fuselage), the lower wing surface pressure coefficient distributions calculated by the URANS model and DES model both agree with the experimental values, while the upper wing surface pressure coefficient distributions exhibit certain differences. Specifically, at the x/c = 0.5–0.8 position, the upper wing surface pressure coefficients predicted by both the DES model and URANS model are slightly smaller than the experimental values; at the x/c = 0.8–1.0 position, the upper wing surface pressure coefficient predicted by the URANS model is slightly smaller than the experimental value. In contrast, the upper wing surface pressure coefficient distributions predicted by DES models with different numerical schemes almost completely overlap and show little difference from the experimental pressure coefficient distribution.

It is hypothesized that the difference between the upper wing surface pressure coefficient distribution and the experimental values may be caused by fuselage interference. For the flow at the wing–body junction of the DLR-F4 model, there are two boundary layers at this junction: one is the fuselage boundary layer, and the other is the wing boundary layer. When the flow at the wing–body junction is disturbed by these two wall boundary layers, the velocity fluctuations and Reynolds normal stresses in different directions differ. Specifically, a difference in Reynolds normal stresses is generated at the actual wing–body junction, which in turn results in errors between the calculated pressure coefficients and the experimental values.

Figure 9 shows the isosurface plots at Q = 5 calculated by the URANS model and DES model under an angle of attack of 9.6°, with coloring by vorticity contour plots. It can be observed from the figure that under the same grid density, although the URANS model captures the wingtip vortices generated at the wingtip, it only captures relatively few turbulent separated vortex structures in the separated flow region on the suction side of the wing. In contrast, the DES model captures extremely rich turbulent information in the separated region.

From the Q-isosurface plots predicted by the DES model, it can be seen that near the wing trailing edge, small-scale separated vortex structures are generated and shed from the trailing edge. At this stage, the intensity of the separated vortex structures is relatively strong. As the separated vortex structures develop downstream, the small-scale separated vortex structures gradually grow larger, while their intensity gradually decreases. A large-scale turbulent separated vortex structure is eventually formed in the region behind the wing trailing edge, and finally, these separated vortex structures gradually dissipate under the influence of numerical dissipation.

Furthermore, the figure also reveals that DES models with different numerical schemes exhibit different predictive capabilities for the separated flow structures of the DLR-F4 wing–body configuration under an angle of attack of 9.6°. Specifically, the CDS scheme predicts larger-scale separated vortex structures, while the WENO-JS scheme and WCNS-JS scheme predict relatively smaller-scale separated vortex structures. The two schemes show little difference in the predicted separated flow information. However, in terms of the capability to capture wingtip vortices, the wingtip vortices predicted by the WCNS-JS scheme can be maintained further downstream without being dissipated. This may indicate that the WCNS-JS scheme has lower dissipativity compared with the WENO-JS scheme.

For the visualization of vorticity distribution, this study applied blanking treatment exclusively to high-vorticity regions. The treatment truncates vorticity data exceeding the preset threshold. It does not target low-vorticity regions nor handle both high and low values simultaneously. The core purpose of this operation is to avoid local peaks of high vorticity compressing the dynamic range of the color scale. This enhances the clarity of flow structure details in medium and low-vorticity regions such as the boundary layer interaction at the wing–body junction and the vortex evolution characteristics in the wake.

Figure 10 is the schematic diagram of streamwise section stations on the wing–body configuration, where the black lines indicate the cross-sections at x/c = 0.53 and x/c = 0.6. Here, x refers to the streamwise coordinate, c represents the mean aerodynamic chord of the configuration, and the origin of the coordinate system is located at the leading-edge apex of the wing–body configuration.

Figure 11 presents the spanwise vorticity streamline plots of the URANS model and DES model at the cross-sections of x/c = 0.53 and x/c = 0.6. It can be observed from the figure that at the x/c = 0.53 cross-section: the URANS model predicts separated flow phenomena at the fuselage position and the junction between the fuselage and the wing; the DES model with the CDS scheme shows small separated flow phenomena at the fuselage position and near the middle section of the wing; the high-order WENO-JS scheme and WCNS-JS scheme predict a larger range of separated flow, and, in particular, the WCNS-JS scheme captures separated flow information with more scales and a larger range.

At the x/c = 0.6 cross-section, compared with the separated flow structures predicted by the URANS model and DES model at the x/c = 0.53 cross-section, the separated flow structures at the x/c = 0.6 cross-section have developed from the wing root position to near the wing tip position, and the range of separated flow has become larger. Compared with the separated vortex structures predicted by the URANS model, the separated vortices predicted by the DES model are more abundant: the WCNS-JS scheme captures a large number of separated vortex structures near the fuselage position, while the separated vortices predicted by the CDS scheme and WENO-JS scheme are more uniformly distributed around the wing.

Figure 12 presents the instantaneous vorticity contour plots of wing cross-sections predicted by the URANS model and DES model at different wing stations under an angle of attack of 9.6°. The selected wing stations are η = 18.5%, η = 33.1%, η = 51.2%, and η = 84.4%, respectively. It can be observed from the figure that both the URANS model and DES model predict the separated flow occurring on the upper wing surface; however, the URANS model captures relatively little turbulent information in the separated shear layer, while the DES model simulates a large amount of turbulent information within the separated shear layer.

The figure also shows that DES models with different numerical schemes all simulate the streamwise development process of separated flow: the separated flow initially occurs at the wing leading edge, then a free shear layer develops downstream from the airfoil leading edge. As the free shear layer at the wing leading edge continues to develop, flow fields of DES models with different numerical schemes exhibit vortex shedding phenomena to varying degrees in the rear part of the free shear layer. With the continuous streamwise development of these vortex shedding structures, the shed separated vortex structures form a Kármán Vortex Street behind the wing. Finally, under the influence of numerical dissipation, these shed separated vortex structures gradually dissipate in the downstream region of the wing.

The flow field information predicted by DES models with different numerical schemes also differs. Since the CDS scheme itself only has second-order accuracy, its numerical dissipation effect is relatively strong. Therefore, the DES model using the CDS scheme captures fewer vortex shedding structures in the free shear layer and behind the wing compared with the high-order WENO schemes and WCNSs. From the separated flow fields in the wake region predicted by the WENO-JS scheme and WCNS-JS scheme, it can be seen that compared with the WENO-JS scheme, the WCNS-JS scheme predicts smaller turbulent scales, a larger number of eddies, and richer flow field information in the wake region.

3.3. Comparative Study on Different Fifth-Order WCNSs

Since the WCNS has lower numerical dissipativity compared with the WENO scheme, and WCNSs constructed with different weight functions also exhibit different dissipativities, the following section will use the fifth-order WCNS-JS scheme and fifth-order WCNS-T scheme to conduct calculations on the DLR-F4 wing–body configuration, so as to investigate the influence of WCNSs with different weight function types on the computational results of the DES model.

Figure 13 presents the variation plots of lift coefficients and drag coefficients with angle of attack, calculated by the DES models with the WCNS-JS scheme and WCNS-T scheme.

Table 3. Comparison of Lift Coefficients under Different WCNSs with Experimental Results.

Angle of Attack	Experimental Value	DES + WCNS-JS		DES + WCNS-T
		Calculated Value	Relative Error	Calculated Value	Relative Error
0.9°	0.520	0.516	−0.769%	0.494	−5.00%
4°	0.801	0.761	−4.994%	0.768	−4.120%
7°	0.928	0.850	−8.405%	0.836	−9.914%
9.6°	0.952	0.873	−8.298%	0.853	−10.399%

and

Table 4. Comparison of Drag Coefficients under Different WCNSs with Experimental Results.

Angle of Attack	Experimental Value	DES + WCNS-JS		DES + WCNS-T
		Calculated Value	Relative Error	Calculated Value	Relative Error
0.9°	0.0285	0.0350	22.807%	0.0349	22.456%
4°	0.0510	0.0570	11.765%	0.0521	2.157%
7°	0.105	0.120	14.286%	0.113	7.619%
9.6°	0.169	0.161	−4.734%	0.153	−9.467%

respectively provide the comparisons between the lift coefficients and drag coefficients predicted by the WCNS-JS scheme and WCNS-T scheme, and the experimental results.

It can be observed from Figure 13 that at angles of attack of 0.9° and 4°, the lift coefficients predicted by the WCNS-JS scheme and WCNS-T scheme show little difference and are in good agreement with the experimental values (with a relative deviation of ~4–5% at α = 4°), while the drag coefficient predictions exhibit a larger relative deviation (~22% at α = 0.9°) from the experimental data—likely due to the sensitivity of drag prediction to wake grid resolution and DES model’s limitations in capturing viscous-pressure drag coupling. When the angles of attack are 7° and 9.6°, there are certain differences in the calculated results of lift and drag coefficients predicted by the WCNS-JS scheme and WCNS-T scheme. As shown in the figure, the lift and drag coefficient values calculated by the WCNS-JS scheme and WCNS-T scheme have a relatively large deviation from the experimental values.

From the comparison between the lift coefficients calculated by different WCNSs and the experimental results given in Table 3, it can be seen that the error between the lift coefficient calculated by the WCNS-JS scheme and the experimental value is smaller than that of the WCNS-T scheme. From the comparison between the drag coefficients calculated by different WCNSs and the experimental results given in Table 4, it can be seen that the error of the drag coefficient calculated by the WCNS-JS scheme at an angle of attack of 7° is larger than that of the WCNS-T scheme, while the calculation error at an angle of attack of 9.6° is smaller than that of the WCNS-T scheme.

Figure 14 presents the time-averaged surface pressure coefficient distributions at four stations (AOA = 0.9°) for the two WCNSs. It can be observed from the figure that at the stations of η = 33.1%, η = 51.2%, and η = 84.4%, the pressure coefficient distributions on the upper and lower wing surfaces predicted by the WCNS-JS scheme and WCNS-T scheme agree well with the experimental values. At the η = 18.5% station, the pressure coefficient distributions on the lower wing surface predicted by the two numerical schemes agree with the experimental values, while there are certain differences between the pressure coefficient distributions on the upper wing surface and the experimental values. Specifically, at the x/c = 0.4~0.9 position, the predicted values of the pressure coefficient on the upper wing surface by the DES methods with the two numerical schemes are lower than the experimental values.

Figure 15 presents the isosurface plots at Q = 5 calculated by the two numerical schemes under an angle of attack of 9.6°, with coloring by vorticity contour plots. It can be observed from the figure that both numerical schemes capture more and smaller-scale turbulent information in the separated regions. However, from the captured wingtip vortices, it can be seen that the wingtip vortices predicted by the WCNS-T scheme can be maintained further downstream without being dissipated compared with those predicted by the WCNS-JS scheme. To some extent, this may indicate that the WCNS-T scheme has a smaller numerical dissipation effect than the WCNS-JS scheme.

Figure 16 presents the vorticity streamline plots of the DES models with the two numerical schemes at the cross-sections of x/c = 0.53 and x/c = 0.6 under an angle of attack of 9.6°. It can be observed from the figure that whether at x/c = 0.53 or x/c = 0.6, the separated vortices predicted by the WCNS-JS scheme are closer to the fuselage position, while the separated vortex structures predicted by the WCNS-T scheme are more uniformly distributed above the wing. The reason for this phenomenon may be that the different weight function allocation of the WCNS-JS scheme itself causes the calculation results to be overly affected by the fuselage when using this scheme for calculation.

Figure 17 presents the vorticity contour plots at different wing stations calculated by the two numerical schemes under an angle of attack of 9.6°. It can be observed from the figure that the two numerical schemes have both captured separated vortex structures of different scales shed from the wing shear layer.

Figure 18 presents a comparison of the time consumption for 200,000 computational iterations between the URANS model and the DES method with three numerical schemes. In terms of computational time consumption, the three DES configurations show little difference, and their time consumption is reduced by approximately 10% compared with the URANS model.

4. Conclusions

Grid Convergence Analysis: The coarse grid (1.0 × 10⁷ cells) exhibits significant errors at high angles of attack, while the calculation results of the medium grid (3.0 × 10⁷ cells) and fine grid (5.01 × 10⁷ cells) are basically consistent. This indicates that the medium grid already meets the accuracy requirements of the DLR-F4 model and can achieve a balance between computational efficiency and simulation accuracy.

The DES model achieves similar accuracy to the URANS model at low angles of attack, but exhibits superior separated flow capture capability at high angles of attack (e.g., 9.6°). As shown in Figure 6 (lift coefficient curves), the lift coefficients predicted by the DES model are closer to the experimental data; at α = 9.6°, the lift coefficient curves predicted by the DES model (regardless of the high-order schemes adopted) are significantly closer to the experimental values than those predicted by the URANS model. This indicates a notable improvement in the lift prediction accuracy of the DES model at high angles of attack.

Influence of Numerical Schemes: Compared with the 2nd-order CDS scheme, high-order schemes (5th-order WENO/WCNS) significantly reduce numerical dissipation. Among them, the WCNS-T scheme has the smallest error in drag coefficient prediction (e.g., at an angle of attack of 4°, the relative error of drag coefficient is only 2.16%), making it more suitable for the accurate simulation of the drag characteristics of the DLR-F4 wing–body configuration.

Optimization of WCNSs: WCNSs with different weight function designs (JS and T types) have a significant impact on simulation results. The WCNS-JS scheme performs better in lift coefficient prediction (at an angle of attack of 9.6°, the relative error of lift coefficient is 8.298%, which is lower than the 10.399% of the WCNS-T scheme). In contrast, the WCNS-T scheme has lower numerical dissipation, can retain wingtip vortex structures over a longer distance, and has advantages in capturing meso-scale flow field structures.

Pressure Distribution Differences: In the wing–body junction region (η = 18.5% station), there is a deviation between the numerically simulated pressure coefficient on the upper wing surface and the experimental value (the predicted values are lower than the experimental values in the x/c = 0.4~0.9 range). This deviation may be related to boundary layer interference between the fuselage and airfoil, as well as Reynolds stress distribution in the wing–body junction region. Further research is needed to explore the influence mechanism of multi-physics coupling effects on the pressure field.

Future work will focus on the weight function optimization of high-order numerical schemes, flow-structure multi-field coupling simulation, and wind tunnel experiment validation. It will be extended to wing–body configurations with complex components and high-efficiency parallel computing, aiming to further improve the prediction accuracy of unsteady separated flows and the adaptability of engineering applications.