A Comparative Study of Airfoil Stall Characteristics Based on Detached Eddy Simulation Incorporated with Weighted Essentially Non-Oscillatory Scheme and Weighted Compact Nonlinear Scheme

Abstract

In this paper, the detached eddy simulation (DES) method is used to calculate the aerodynamic characteristics of NACA0015 airfoil by combining the Riemann approximate solution HLLC (Harten–Lax–van Leer Contact) with the high-order weighted essentially non-oscillatory (WENO) scheme and the weighted compact nonlinear scheme (WCNS), respectively. By comparing the calculation results of the two different numerical schemes with the wind tunnel test results, it is found that both numerical schemes can accurately calculate the aerodynamic parameters at small angles of attack. However, in the range of near-stall angle (in the range of 10–15°), the calculation results of various numerical schemes have a certain degree of deviation. The calculation results of the fifth-order WCNS and the fifth-order WENO scheme are closer to the experimental values. The fifth-order WCNS predicts the stall angle of attack more accurately than the fifth-order WENO scheme. The calculation accuracy of the fifth-order WCNS is better than that of the fifth-order WENO scheme under the post-stall condition (where the angle of attack is greater than 15°). By comparing the vorticity contours calculated by different numerical schemes, it is found that the numerical dissipation of the fifth-order accuracy is smaller than that of the third-order accuracy, and the vortex capture ability is stronger. WCNS captures the small vortex structure that the WENO scheme does not.

1. Introduction

With the continuous development of computer technology, computational fluid dynamics (CFD) has also developed rapidly in the past decade, and has become an important means of aeronautical and space vehicle design and analysis [1]. NASA pointed out in the ‘2030 CFD Vision’ report [2] that CFD technology can provide more abundant flow information on the premise of effectively reducing risks and costs in the design process. For the unsteady turbulent separated flows, three key factors affecting the calculation accuracy of the flow simulation are the turbulence simulation method, the numerical scheme and the computational grid. Among them, the selection of turbulence simulation methods and numerical schemes has always been the focus of computational fluid dynamics.

In accordance with the accuracy of the turbulent flow simulation, the turbulence simulation methods can be classified into three categories [3]: direct numerical simulation (DNS) method, large eddy simulation (LES) method and Reynolds average Navier–Stokes (RANS) method. The DNS method is to obtain all the information of the flow field by directly solving the Navier–Stokes (N-S) equation. Although it can ensure high calculation accuracy, it requires huge computational resources in order to obtain all the information of the flow field. Spalart [4] predicted that the application of the DNS method to the numerical simulation of wing–body combination will be realized in 2080. The LES method decomposes the flow field information into large-scale fluctuations and small-scale fluctuations by setting a filter. The large-scale fluctuations are solved directly by the numerical simulation method, and the influence of small-scale fluctuations on the flow flied is obtained by the model of the sub-grid scale model assumptions. However, practical engineering problems often have a very high Reynolds number and a very thin boundary layer, which still requires a great deal of computational resources in order to use LES to simulate the flow within the thin boundary layer. At present, the RANS method is still the most commonly used method for solving aerodynamic flow problems and, in particular, practical engineering problems. The RANS method requires a small number of grids and can achieve the computational results quickly. However, the RANS method has weaker capability for simulating complex flow phenomena, and it is difficult to give detailed information of the complex flow field including separated flow and vortex shedding [5].

In order to improve the computational efficiency and ensure a certain degree of accuracy, a RNAS/LES hybrid method was proposed. Spalart [6] was the first to propose the RANS/LES method. He combined the RANS method based on the SA one-equation model with the LES method, and proposed the Spalart–Allmaras detached eddy simulation (SA-DES) method. This method uses the RANS method in the near-wall region and the LES method in the far-wall region, thus avoiding the consumption of huge computing resources of the LES method in the near-wall solution. Subsequently, Strelets [7] developed the SST-DES model based on the Shear Stress Transport (SST) two-equation model. The DES method has received great attention since its inception. It has been shown that it achieves high computational accuracy and vortex capture capability in the cylindrical flow and large separation flows. However, the DES method also has some defects [8]. One of the most typical problems is Modeled Stress Depletion (MSD), which is mainly caused by improper mesh refinement. Due to the improper refinement of the grid in the boundary layer, the interface between RANS and LES moves near the wall, and the RANS modeling area is reduced. However, the grid resolution at this time cannot support the accurate solution of the LES model, resulting in a small computational stress, that is, the MSD problem. In order to reduce the dependence of the DES method on the grid, Strelets and Spalart proposed the SST-DDES (delayed detached eddy simulation) [9] method and the SA-DDES [10] method.

In the simulation of complex flow problems, in order to obtain abundant flow field information, the core issue is to construct an excellent numerical scheme. Compared with low-order numerical schemes, high-order schemes have lower dissipation and dispersion errors, and the calculation results are more accurate. Although the computational cost is higher than that of low-order schemes, with the continuous development of computer technology, high-order schemes are more and more widely used to solve various numerical problems.

Among many high-order schemes, the WENO scheme [11] can guarantee no oscillation at the discontinuous point and has high accuracy in the smooth region, which is an ideal choice for solving the N-S equation. In order to improve the computational stability at the discontinuity points, Jiang and Shu [12] proposed a fifth-order WENO-JS scheme by introducing a sub-template smoothing factor. Later, according to the different structure of the weight function, various forms of WENO schemes have been developed, such as WENO-M, WENO-Z and so on.

WCNS is another type of high-order scheme. Based on the compact nonlinear scheme (CNS) [13], Deng and his collaborators [14,15] proposed a weighted compact nonlinear scheme by introducing the idea of nonlinear WENO interpolation. Xu Zhang [16] compared the dispersion and dissipation characteristics of the fifth-order WCNS and WENO schemes by Fourier analysis, and found that WCNS has less dissipation and dispersion than the WENO scheme. Taku Nonomura [17] used the WENO scheme and WCNS to calculate the two-dimensional vortex motion problem and the three-dimensional steady-state vortex problem. It was found that the free flow and vortex retention effect of the WENO scheme are not good in either the generalized coordinate system or the Cartesian coordinate system, and the resolution is lower than that of WCNS. Kamiya [18] used the same flux evaluation method and smoothness index to compare the resolution and robustness of the WENO scheme and WCNS. The comparison results show that the resolution of WCNS is higher than that of WENO, and the WENO scheme is more robust than WCNS. Ge Mingming [19] combined the seventh-order WCNS with the implicit large eddy simulation to study the high-speed flow noise problem of the M291 cavity, and compared it with the experimental results to confirm the reliability and efficiency of the method in the simulation of high-speed cavity noise problems.

However, at present, the comparative study of these two high-order schemes is limited to typical examples such as the shock tube problem, the two-dimensional Riemann problem and the double-Mach reflection problem, and there is a lack of comparative studies on problems such as flows around airfoils or wing–body configurations. Therefore, in this paper, the low dissipation HLLC approximate Riemann solver [20], combined with the high order WENO scheme and WCNS, is used in the simulations of separation flows around a NACA0015 airfoil. The calculation results using the two schemes are compared and analyzed.

2. Turbulence Model and Numerical Method

2.1. Turbulent Model

The equation of the S-A model used is as follows [21]:

$$\begin{array}{ll}{\displaystyle \frac{\partial \stackrel{\mathrm{\wedge }}{\nu }}{\partial \mathrm{t}}}\mathrm{+}{\mathrm{u}}_{\mathrm{j}}{\displaystyle \frac{\partial \stackrel{\wedge }{\nu }}{\partial {\mathrm{x}}_{\mathrm{j}}}}& ={\mathrm{c}}_{\mathrm{b}1}\left(1-{\mathrm{f}}_{\mathrm{t}2}\right)\stackrel{\wedge }{\mathrm{S}}\stackrel{\wedge }{\nu }-\left[{\mathrm{c}}_{\mathrm{\omega }1}{\mathrm{f}}_{\omega }-{\displaystyle \frac{{\mathrm{c}}_{\mathrm{b}1}}{{\kappa }^2}}{\mathrm{f}}_{\mathrm{t}2}\right]{\left({\displaystyle \frac{\stackrel{\wedge }{\nu }}{\mathrm{d}}}\right)}^2\\ & +{\displaystyle \frac{1}{\sigma }}\left[{\displaystyle \frac{\partial }{\partial {\mathrm{x}}_{\mathrm{j}}}}\left(\left(\mathrm{v}+\stackrel{\wedge }{\nu }\right){\displaystyle \frac{\partial \stackrel{\wedge }{\nu }}{\partial {\mathrm{x}}_{\mathrm{j}}}}\right)+{\mathrm{c}}_{\mathrm{b}2}{\displaystyle \frac{\partial \stackrel{\wedge }{\nu }}{\partial {\mathrm{x}}_{\mathrm{i}}}}{\displaystyle \frac{\partial \stackrel{\wedge }{\nu }}{\partial {\mathrm{x}}_{\mathrm{j}}}}\right]\end{array}\mathrm{,}$$

(1)

where $\mathrm{d}$ is the distance from the center point of the current grid to the wall surface, and other specific parameters can be found in Reference [21].

The SA-DES model [6], which is based on the SA model, only needs to modify the $\mathrm{d}$ in the original equation and replace the $\mathrm{d}$ in the original equation with ${\mathrm{L}}_{\mathrm{D}\mathrm{E}\mathrm{S}}$. The equation of ${\mathrm{L}}_{\mathrm{D}\mathrm{E}\mathrm{S}}$ is defined as follows, where ${\mathrm{C}}_{\mathrm{D}\mathrm{E}\mathrm{S}}$ is 0.65 and ∆ is the maximum value of the grid in three directions.

$${\mathrm{L}}_{\mathrm{D}\mathrm{E}\mathrm{S}}=\mathrm{m}\mathrm{i}\mathrm{n}\left(\mathrm{d}\mathrm{,}{\mathrm{C}}_{\mathrm{D}\mathrm{E}\mathrm{S}}\Delta \right)\mathrm{,}$$

(2)

2.2. Numerical Method

2.2.1. HLLC Scheme

The HLLC scheme is an improved method proposed by Toro et al. to solve the problem that the HLL scheme [22] will produce large numerical dissipation at the discontinuity surface. The structure of the approximate Riemann solution is four different states separated by three waves. Figure 1 shows the HLLC Riemann fan with two intermediate states. This scheme can accurately capture shock waves, contact discontinuity waves and rarefaction waves in theory [23]. Compared with the Roe scheme, the HLLC scheme does not require entropy correction and is more suitable for low-speed and transonic flow calculations [24].

2.2.2. WCNS

Based on the characteristics of the N-S equations, the diffusion term is generally discretized by the central scheme, while the convection term has a variety of discrete schemes, and its accuracy directly affects the characteristics of the numerical solution. Therefore, this paper uses a variety of high-order spatial discretization schemes to discretize the convection term, and studies the influence of different spatial discretization schemes on the calculation results.

The WENO scheme uses the idea of Newton interpolation to construct the sub-template, while WCNS uses the Lagrange interpolation to construct the sub-template. The two schemes are similar in structure, but the interpolation ideas are different. Here, the construction of the third-order and fifth-order WCNS-JS schemes is taken as an example to briefly explain the interpolation process of this type of scheme.

As shown in Figure 2, the third-order WCNS-JS scheme is to construct second-order sub-templates at three points, and then combine these second-order sub-templates to obtain the numerical flux at half-node. Now, the left value ${\mathrm{f}}_{\mathrm{L},\mathrm{i}+1/2}$ at half point ${\mathrm{x}}_{\mathrm{i}+1/2}$ is taken as an example to illustrate the interpolation process. The right value ${\mathrm{f}}_{\mathrm{R},\mathrm{i}+1/2}$ can be obtained by the symmetrical characteristics.

The left value of the third-order accuracy constructed on the global template is:

$${\mathrm{f}}_{\mathrm{i}+1/2}=-{\displaystyle \frac{1}{8}}{\mathrm{f}}_{\mathrm{i}-1}+{\displaystyle \frac{3}{4}}{\mathrm{f}}_{\mathrm{i}}+{\displaystyle \frac{3}{8}}{\mathrm{f}}_{\mathrm{i}+1},$$

(3)

The second-order interpolation on the sub-template is:

$$\left\{\begin{array}{l}{\mathrm{f}}_{\mathrm{i}+1/2,0}={\displaystyle \frac{1}{2}}\left(3{\mathrm{f}}_{\mathrm{i}}+{\mathrm{f}}_{\mathrm{i}-1}\right)\\ {\mathrm{f}}_{\mathrm{i}+1/2,1}={\displaystyle \frac{1}{2}}\left({\mathrm{f}}_{\mathrm{i}}+{\mathrm{f}}_{\mathrm{i}+1}\right)\end{array}\right.,$$

(4)

According to the third-order interpolation on the global template and the second-order interpolation on the sub-template, the linear weights can be obtained as follows:

$${\mathrm{d}}_0=1/4, {\mathrm{d}}_1=3/4$$

(5)

The smoothing factors of the third-order WCNS are:

$$\left\{\begin{array}{l}\mathrm{I}{\mathrm{S}}_0={\left({\mathrm{f}}_{\mathrm{i}}-{\mathrm{f}}_{\mathrm{i}-1}\right)}^2\\ \mathrm{I}{\mathrm{S}}_1={\left({\mathrm{f}}_{\mathrm{i}+1}-{\mathrm{f}}_{\mathrm{i}}\right)}^2\end{array}\right.,$$

(6)

The fifth-order WCNS-JS scheme is interpolated on five points. The interpolation template is shown in Figure 3. The left value of the fifth-order accuracy constructed on the global template is:

$${\mathrm{f}}_{\mathrm{i}+1/2}={\displaystyle \frac{3}{128}}{\mathrm{f}}_{\mathrm{i}-2}-{\displaystyle \frac{5}{32}}{\mathrm{f}}_{\mathrm{i}-1}+{\displaystyle \frac{45}{64}}{\mathrm{f}}_{\mathrm{i}}+{\displaystyle \frac{15}{32}}{\mathrm{f}}_{\mathrm{i}+1}-{\displaystyle \frac{5}{128}}{\mathrm{f}}_{\mathrm{i}+2},$$

(7)

The sub-template interpolation of the scheme is:

$$\left\{\begin{array}{l}{\mathrm{f}}_{\mathrm{i}+1/2,0}={\displaystyle \frac{1}{8}}\left(3{\mathrm{f}}_{\mathrm{i}-2}+10{\mathrm{f}}_{\mathrm{i}-1}+15{\mathrm{f}}_{\mathrm{i}}\right)\\ {\mathrm{f}}_{\mathrm{i}+1/2,2}={\displaystyle \frac{1}{8}}\left(-{\mathrm{f}}_{\mathrm{i}-1}+6{\mathrm{f}}_{\mathrm{i}}+3{\mathrm{f}}_{\mathrm{i}+1}\right)\\ {\mathrm{f}}_{\mathrm{i}+1/2,3}={\displaystyle \frac{1}{8}}\left(3{\mathrm{f}}_{\mathrm{i}}+6{\mathrm{f}}_{\mathrm{i}+1}-{\mathrm{f}}_{\mathrm{i}+2}\right)\end{array}\right.,$$

(8)

The linear weights are:

$${\mathrm{d}}_0=1/16, {\mathrm{d}}_1=10/16, {\mathrm{d}}_2=5/16$$

(9)

The smoothing factors are:

$$\left\{\begin{array}{l}\mathrm{I}{\mathrm{S}}_0={\displaystyle \frac{1}{4}}{\left({\mathrm{f}}_{\mathrm{i}-2}-4{\mathrm{f}}_{\mathrm{i}-1}+3{\mathrm{f}}_{\mathrm{i}}\right)}^2+{\left({\mathrm{f}}_{\mathrm{i}-2}-2{\mathrm{f}}_{\mathrm{i}-1}+{\mathrm{f}}_{\mathrm{i}}\right)}^2\\ \mathrm{I}{\mathrm{S}}_1={\displaystyle \frac{1}{4}}{\left({\mathrm{f}}_{\mathrm{i}-1}-{\mathrm{f}}_{\mathrm{i}+1}\right)}^2+{\left({\mathrm{f}}_{\mathrm{i}-1}-2{\mathrm{f}}_{\mathrm{i}}+{\mathrm{f}}_{\mathrm{i}+1}\right)}^2\\ \mathrm{I}{\mathrm{S}}_2={\displaystyle \frac{1}{4}}{\left(3{\mathrm{f}}_{\mathrm{i}}-4{\mathrm{f}}_{\mathrm{i}+1}+{\mathrm{f}}_{\mathrm{i}+2}\right)}^2+{\left({\mathrm{f}}_{\mathrm{i}}-2{\mathrm{f}}_{\mathrm{i}+1}+{\mathrm{f}}_{\mathrm{i}+2}\right)}^2\end{array}\right.,$$

(10)

The commonly used JS-type nonlinear weights [12] are:

$${\omega }_{\mathrm{k}}={\displaystyle \frac{{\mathrm{\alpha }}_{\mathrm{k}}}{{\displaystyle {\sum }_{\mathrm{k}=0}^{\mathrm{n}}{\alpha }_{\mathrm{k}}}}},$$

(11)

$${\alpha }_{\mathrm{k}}={\displaystyle \frac{{\mathrm{d}}_{\mathrm{k}}}{{\left(\mathrm{I}{\mathrm{S}}_{\mathrm{k}}+\epsilon \right)}^2}},$$

(12)

The final reconstructed nonlinear weight function is:

$${\mathrm{f}}_{\mathrm{i}+1/2}={\displaystyle {\sum }_{\mathrm{k}=0}^{\mathrm{n}}{\omega }_{\mathrm{k}}{\mathrm{f}}_{\mathrm{j}+1/2,\mathrm{k}},}$$

(13)

where ${\mathrm{d}}_{\mathrm{k}}$ is a linear weight, and ε is a small number introduced to avoid zero denominator. In this paper, ε is set to 10⁻⁶; the value of n varies with the order. When it is a third-order scheme, n = 2; when it is a fifth-order scheme, n = 3.

2.3. Time-Marching Scheme

In this paper, the fifth-order Runge–Kutta method [25] is adopted in the time-marching process, and the DDADI (Diagonal Dominant Alternating Direction Implicit) scheme [26] is used for solving the turbulence equations. Similar to LU-SGS (lower–upper symmetric Gauss–Seidel) [27], this scheme decomposes the complex matrix equations into several sub-steps, and there are also decomposition errors. Since it needs to be solved once in each discrete direction, the DDADI method can only be applied to structural grids.

3. Numerical Simulation Results and Discussion

3.1. Mesh Subdivision

The calculation program used in this paper is CHNS3D, an in-house program of Professor Zhong’s research team of Nanchang Hangkong University, and the calculation is completed on the computer cluster of Nanchang Hangkong University. The computational grid uses multi-block structural grids, and ICEM(CFD 2020 R2) software is used for grid generation. The grid division is mainly based on C-H topology. In order to ensure the accuracy of the calculation, the grids at the trailing edge where the flow separation may occur and the grids in the wake are refined. In addition, before starting the calculation of this paper, the grid influence research has been carried out. The computational grid used in this paper is a grid that can give better results.

The far field of the computational domain is set to 15c (c is the chord length of the wing), the span length is 0.2c, the number of spanwise grid points is 25, the thickness of the first layer grid is $1\times 10^{-6}$,${\mathrm{y}}^{+}\le 2$ and the total number of grids is 1.3 × 10⁶. The calculation model and grid are shown in Figure 4.

The incoming flow conditions are adopted from the LM Glasfiber [28] wind tunnel experiment. The stall angle of attack is about 14.3°, the incoming flow Mach number $\mathrm{M}\mathrm{a}=0.079$ and the Reynolds number based on the chord length is $1.6\times 10^6$. In this paper, the time step $\Delta \mathrm{t}$ is set to 0.01, and the corresponding physical time step is $3.5\times 10^{-5}\mathrm{s}$.

3.2. Analysis of Time-Average Results

Figure 5 shows the variation of time-averaged lift coefficients and drag coefficients with the angles of attack. It can be seen from Figure 5 that the lift coefficients and drag coefficients calculated by various numerical schemes are consistent with the experimental values at small angles of attack. However, with the increase in the angles of attack, different extents of separation flows appear at the airfoil surface at different angles of attack, and the calculation results gradually begin to deviate from the experimental values. In the near-stall state, the lift coefficients calculated by the third-order WENO scheme are significantly smaller than the experimental values, and the calculation errors are significantly larger than that using the third-order WCNS. There is no significant difference between the results of the fifth-order WENO scheme and the fifth-order WCNS, and the calculation errors are smaller than that of the third-order schemes. However, the calculated lift coefficients are still smaller than the experimental values, while the calculated drag coefficients are larger than the experimental values. The reason for this phenomenon may be due to the grid-induced separation caused by improper grid partition. It can be seen from Figure 5 that the experimental stall angle of attack is about 14.3°. Except for the fifth-order WCNS, the stall angle of attack calculated by other numerical schemes is larger than the experimental stall angle. In the post-stall state, all numerical schemes overestimate the lift coefficients, and the calculation error of the fifth-order WENO appears to be the largest.

Figure 6 shows the comparison between the time-averaged pressure coefficients and the experimental results at different angles of attack. It can be seen from Figure 6 that when the angle of attack is 8.13°, all four numerical schemes predict the pressure coefficient distributions on the upper and lower surfaces of the wing considerably well.

For the case of the angle of attack being 15.66°, the pressure coefficient distributions calculated by the four numerical schemes are slightly different from the experimental pressure coefficients. Among them, the suction peak predicted by the fifth-order WENO scheme is much larger compared with other schemes, and the separation point position is relatively delayed; the peak suction and separation point predicted by the third-order WENO scheme have smaller errors than other numerical schemes. The prediction accuracy of WCNS is between the fifth-order WENO scheme and the third-order WENO scheme. However, no matter which difference scheme is used, the separation point position and the pressure coefficient distribution on the upper surface of the wing cannot be accurately predicted.

It can be seen from the iteration diagram of the lift coefficient and drag coefficient with time given in Figure 7 that when the angle of attack is 15.66°, the lift coefficients calculated by the four numerical schemes fluctuate above the experimental values, while the drag coefficients fluctuate near the lower part of the experimental values. This may be due to the fact that the pressure peak at the leading edge of the wing is too high, and the separation point is too late, resulting in a small calculated lift coefficient and a high drag coefficient. Figure 8 shows the time-consuming comparison diagram of the four numerical schemes for 300,000 iterations. From the point of view of calculation time, the calculation time of the fifth-order scheme is about 23% higher than that of the third-order scheme.

For the case of the angle of attack being 20.11°, the airfoil is already in the post-stall state. It can be seen from Figure 6 that the calculation results of the fifth-order WCNS are the closest to the experimental values. At a distance of 0.2c from the trailing edge of the wing, the pressure coefficient results calculated by other numerical schemes all have a certain degree of ‘protrusion’ except that the fifth-order WCNS predicts the pressure coefficients quite well.

Figure 9 shows the iteration diagram of the lift coefficient and drag coefficient with time at an angle of attack of 20.11°, where the lift coefficients and drag coefficients calculated by the four numerical schemes fluctuate near the experimental value.

Figure 10 shows the distributions of the time-averaged friction coefficients on the upper surface of the airfoil when the angle of attack is 20.11°. When the friction becomes zero, the separation flow begins. The position is located at 0.1–0.2c from the leading edge of the wing. From Figure 6, it can also be seen that the pressure coefficients of the upper wing surface for these four numerical schemes gradually become flat at the positions around 0.1–0.2c from the leading edge of the airfoil, which indicates where the separation flow occurs. From Figure 10, it can be seen that in the middle and near the trailing edge of the wing, except for the fifth-order WCNS, other numerical schemes have the phenomenon of separation and reattachment, and the calculated friction coefficients are smaller when compared to the fifth-order WCNS.

3.3. Instantaneous Flow Field Analysis

In this section, the flow field at an angle of attack of 15.66° and 20.11° is analyzed. Figure 11 and Figure 12 give the vorticity contour and the instantaneous vorticity map of the Q criterion calculated by different numerical schemes at an angle of attack of 15.66°.

From the experimental pressure coefficient distribution diagram shown in Figure 6, it is known that the experimental separation point is about 0.3c from the leading edge of the wing. From Figure 11, it can be seen that the separation points calculated by the four numerical schemes are all at around 0.5–0.6c from the leading edge of the wing. Hence, the separation positions predicted by the four numerical schemes are all delayed. From the vorticity contour and streamline diagram, it can be seen that the vortex capture capabilities of the four numerical schemes are slightly different. Compared with the third-order WENO scheme, the third-order WCNS can capture smaller vortex shedding structures, and the fifth-order WCNS is better than the fifth-order WENO scheme.

Figure 12 is the contour map when Q = 0.01, and it is colored by a vorticity cloud chart. It can be seen from the instantaneous vorticity diagram of Q criterion that the third-order WENO scheme has the largest numerical dissipation and poor ability to capture vortices. Compared with the calculation results of third-order WENO scheme, the vortex structure of the third-order WCNS and the fifth-order WENO can be maintained to further downstream without being dissipated. The fifth-order WENO scheme is slightly better than the third-order WCNS. The fifth-order WCNS captures the most abundant and finer separation flow structure. This shows that the fifth-order WCNS has less numerical dissipation, so it is better than other schemes in terms of vortex capture capability and vortex retention characteristics.

Figure 13 and Figure 14 give the vorticity cloud diagram and Q iso-surface diagram at an angle of attack of 20.11°, respectively. From the instantaneous vorticity cloud diagram and streamline diagram, it can be seen that the four numerical schemes have separated flow phenomena at the leading edge of the wing around 0.1–0.2c, which is consistent with the separation positions obtained from the distribution of friction coefficients and pressure coefficients.

From the instantaneous vorticity diagram of the Q criterion given in Figure 14, it can be seen that under the influence of numerical dissipation, the third-order WENO scheme and the third-order WCNS only capture the large-scale flow structure in the separation region, while the fifth-order WENO scheme and the fifth-order WCNS capture more abundant flow structures, and the fifth-order WCNS captures more delicate vortex structures.

4. Conclusions

In this paper, the DES method is combined with the high-order WENO scheme and WCNS to study the separated flow of the NACA0015 airfoil for the first time using different numerical schemes. From the calculations of this paper, the following conclusions can be drawn:

Firstly, the lift coefficients and drag coefficients calculated by the DES method incorporated with all the high-order schemes, namely, third-order WCNS, third-order WENO, fifth-order WCNS and fifth-order WENO, are consistent with the experimental values at small angles of attack. However, with the increase in the angles of attack, the calculated lift coefficients by all the schemes deviate from the experimental values at certain degrees; the fifth-order WCNS and the fifth-order WENO scheme appear to be the most accurate, and the fifth-order WCNS predicts the stall angle of attack considerably well.

Secondly, by comparing the pressure coefficient distributions at different angles of attack, it is found that when the angle of attack is 8.13°, the calculation results of the four numerical schemes are in good agreement with the experimental values. For the case of the angle of attack being 20.11°, all four schemes predict the flow separation well and the fifth-order WCNS appears to be the most accurate in the prediction of the pressure distributions. However, in the case of the angle of attack being 15.66°, none of the four numerical schemes predicts the separation point well, which results in a discrepancy in the prediction of the pressure distributions.

In addition, by comparing the vorticity contours calculated by different numerical schemes, at all three angles of attack, it can be seen that the third-order WCNS can capture finer vortex structures than the third-order WENO scheme, and the fifth-order WCNS can capture finer vortex structures than the fifth-order WENO scheme; this may indicate that WCNS has smaller numerical dissipation and better vortex capture capability than the WENO scheme of the same discretization order for the separated flow simulations around an airfoil. Finally, the calculation program in this paper is also applicable to other airfoils and wing–body configurations under different Re and Ma numbers.