Evolution and Influence of Multivortex Structure on the Head of Canard-Controlled Missile at High Angle of Attack

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1. Introduction

Due to its high maneuverability and other advantages, the canard-controlled missile is widely used in vertical launching air-defense missiles. Vertical launching missiles may face the issue of uncertain roll angle ($\phi$) and a large angle of attack ($\alpha$) during the initial stage of launch. The canards of a tube-launched missile are positioned on the nose due to the constraints imposed by the launch tube on the span dimension. The random roll angle generated during the initial stage of launch induces interactions between vortices, resulting in a greater influence of the roll angle on the lateral forces. Therefore, it is significantly important to study the variations in multivortex flow field structure and aerodynamic characteristics caused by canards at different roll angles under large angles of attack.

A revolutionary body is a commonly used simplified model in research on missiles with large angles of attack. It has been confirmed in numerous studies that slender bodies produce unexpected lateral forces at sufficiently large angles of attack [1, 2]. At such large angles of attack, the vortices at the head of the slender body transition from symmetric to asymmetric as they develop downstream [3]. Micro disturbances on the head, such as surface roughness [4], manufacturing defects [5, 6], and even dust particles [7], are considered one of the causes of the natural asymmetry phenomenon. Extensive simulations and experiments have demonstrated the influence of micro disturbances on the asymmetric direction of the flow field by introducing micro perturbation [8–10] or changing the roll angle [11, 12]. However, when additional disturbances on the head are significantly stronger than surface roughness and manufacturing defects, the impact of micro disturbances on the flow field diminishes. Therefore, introducing additional disturbances on the head becomes an important means of controlling asymmetrical flow at large angles of attack. Asymmetric disturbances on the head determine the asymmetric direction of the flow field by eliminating the bistable phenomenon of asymmetric flow around the slender body [13, 14], while symmetric disturbances on the head, such as tip disturbance [15], nose vortex generators [16], and weak jets [17, 18], enhance flow field stability, allowing a stable symmetric flow field to extend to larger angles of attack. Additionally, periodic variations in disturbances, such as rotating tips perturbation [19], eliminate unexpected lateral forces through time-averaging techniques.

Compared to the vortices generated by weak disturbances, which are significantly weaker than the head vortices, the vortices produced by strong disturbance sources in missiles, such as canards, strakes, drag rings, and wings, are comparable to or even stronger than the head vortices. The introduction of strong disturbance induces interaction between vortices and interaction between vortices and the wall, further affecting the aerodynamic force of the missile. Johnson et al. [20] conducted research on wing-body combinations and found that the interaction between asymmetric vortices and wing significantly affects lateral force and roll moment and may lead to severe control issues. Kumar [21] carried out wind tunnel experiments on a slender body with a circular drag ring and found that adding a ring to the nose significantly reduces lateral force but greatly increases the roll moment. Karn et al. [22] investigated the unsteady characteristic of a slender body with circular drag rings using CFD methods and found that the ring reconstructs the flow field around the slender body and generates significant fluctuating loads behind the ring. Yuan et al. [23] conducted experiments to measure the pressure distribution and flow field structure of a slender body with strakes at $\alpha ={55}^{°}$. They found that the vortices of strakes absorbed the energy originally injected into the separated vortices of the body, resulting in a weakening of the separated vortices of the body and a decrease in lateral force. Xu and Deng [24] carried out experiments to investigate the wing rock motion of a slender body with a low-swept wing at different angles of attack. They discovered that the asymmetric vortices generated by the forebody induce the new wing vortices and make the uncommanded lateral motion of the model more complex. Furthermore, they found that this motion can be suppressed by employing a rotating nose tip, which effectively inhibits the asymmetry of the vortices. Maynes and Gebert [25] performed wind tunnel tests on the AIM-120A missile and found that the lateral force of the missile at large angles of attack is influenced by the roll angle, although this influence does not exhibit regular patterns. Wang et al. [26] obtained the fluctuation characteristics of surface pressure on a canard-controlled missile at large angles of attack through wind tunnel experiments and observed that the shedding frequency of the Karman-like vortex on the body played a dominant role in the frequency of the aerodynamic forces. The strong disturbance changes the spatial structure of the flow field through interactions between vortices and interactions between vortices and the wall. Vortex merging is a typical interaction between vortices that has a significant influence on the flow field structure. Nielsen et al. [27] used topological methods to describe the changes in vortex structure during the interaction of vortex pairing and found that the core growth model of vortex can be reduced to a lower dimension at sufficiently low Reynolds numbers. Dizes and Verga [28], through direct numerical simulations, made an intriguing finding that when the radius of vortices exceeds the critical size, which is 0.22 times the separation distance, with the diffusion of vortices, the two adjacent corotating vortices merge into a larger vortex. Meunier and Leweke [29] further analyzed the three-dimensional short-wave instability of vortices during the merging process. Dam et al. [30] determined the merging process of three vortices with equal intensity under viscous effects and provided an analytical equation for this process. Near-wall vortices alter the distribution of surface pressure and further change the lateral force. McKenna, Bross, and Rockwell [31] investigated the evolution of vortex structures in the interaction between a streamwise vortex and a wing. They found that the wall causes a decrease in the streamwise vorticity, an increase in the azimuthal vorticity, a reduction in downwash, and a decrease in axial velocity. Zhijun, Gu, and Zhao [32] implemented particle image velocimetry (PIV) experiments and found that the attached streamwise vortices extract vorticity from the boundary layer, leading to the generation of three-dimensional multivortex structures.

Current researches on missiles at large angles of attack typically focused on explaining and managing the natural asymmetry of slender revolutionary bodies, with less emphasis on the multivortex structures that arise from canard. To investigate the causes of unexpected aerodynamic changes in canard-configured missiles at low speeds and large angles of attack, this study focuses on the interaction between canard vortices and body vortices at different roll angles, examining the evolution of these multivortex structures and their effects on missile aerodynamics. This study is aimed at clarifying the mechanisms behind unexpected aerodynamic variations in canard-configured missiles at large angles of attack and informing flight control strategies in these conditions. The existing literature covers the study of aerodynamic mechanisms in simple models and the engineering applications of aerodynamics in complex models. It lacks an in-depth look at the aerodynamic mechanisms within complex models. This paper enhances the understanding of aerodynamic forces in canard-configured missiles at large angles of attack and extends the research on missile aerodynamics from basic revolutionary bodies to the more practically relevant canard configurations, offering valuable insights for engineering applications.

2. Methodology

2.1. Governing Equations

After decades of development, numerical simulation is now considered to be a mature and reliable means of analyzing the characteristics of flow, which can yield more information on the flow field than experiments. The Reynolds-averaged Navier–Stokes (RANS) equations use a turbulence model to solve for the viscous term and are closed to reduce the difficulty of obtaining the solution. The $k-\omega$ shear stress transfer (SST) turbulence model [33, 34] has been widely used in research on aerospace because of its high accuracy and speed.

The equations of continuity, momentum, and energy can be written in differential form as follows: $$\begin{array}{}\text{(1)}& \frac{\partial \rho }{\partial t}+\frac{\partial }{\partial x_i}\rho u_i=0\text{(2)}& \frac{\partial \rho u_i}{\partial t}+\frac{\partial \rho u_i{u}_j}{\partial x_j}=-\frac{\partial p}{\partial x_i}+\frac{\partial }{\partial x_j}\mu \frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}-\frac{2}{3}\frac{\partial u_i}{\partial x_j}{\delta }_{ij}+\frac{\partial }{\partial x_j}-\rho \overline{u{\prime }_{i}u{\prime }_j}\text{(3)}& \frac{\partial \rho E}{\partial t}+\nabla \cdot \stackrel{⟶}{v}\rho E+p=-\nabla \cdot \sum _j{h}_j{J}_j+S_h\end{array}$$

The Boussinesq hypothesis [35] is used to relate the Reynolds mean pressure to the mean velocity. $$\begin{array}{}\text{(4)}& -\rho \overline{u{\prime }_{i}u{\prime }_j}={\mu }_t\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}-\frac{2}{3}\rho k+{\mu }_t\frac{\partial u_i}{\partial x_j}{\delta }_{ij}\end{array}$$

The $k-\omega$ SST turbulence model is a two-equation model in which the expressions for $k$ and $\omega$ are introduced to close the equations. $$\begin{array}{}\text{(5)}& \frac{\partial \rho k}{\partial t}+\frac{\partial \rho u_{i}k}{\partial x_i}=\frac{\partial }{\partial x_j}{\Gamma }_k\frac{\partial k}{\partial x_j}+G_k-Y_k+P_k{f}_r\text{(6)}& \frac{\partial \rho \omega }{\partial t}+\frac{\partial \rho u_i\omega }{\partial x_i}=\frac{\partial }{\partial x_j}{\Gamma }_{\omega }\frac{\partial \omega }{\partial x_j}+G_{\omega }-Y_{\omega }+D_{\omega }\end{array}$$where $G_k$ represents the turbulent kinetic energy ($k$), $G_{\omega }$ is the expression of turbulent dissipation rate (ω), ${\Gamma }_k$ and ${\Gamma }_{\omega }$ represent the effective diffusion terms of $k$ and $\omega$, respectively, and $Y_k$ and $Y_{\omega }$ represent their divergence terms, respectively. $P_k$ is a production term, and $f_r$ is a correction factor. $D_{\omega }$ represents orthogonal divergence. The turbulence kinetic energy is defined as: $$\begin{array}{}\text{(7)}& G_k=-\rho \overline{u{\prime }_{i}u{\prime }_j}\frac{\partial u_j}{\partial x_i}\end{array}$$

Equation of effective diffusion term: $$\begin{array}{}\text{(8)}& {\Gamma }_k=\mu +\frac{{\mu }_t}{{\sigma }_k}\text{(9)}& {\Gamma }_{\omega }=\mu +\frac{{\mu }_t}{{\sigma }_{\omega }}\end{array}$$where ${\sigma }_k$ and ${\sigma }_{\omega }$ represent the Prandtl number of $k$ and $\omega$, respectively. Turbulence viscosity coefficient ${\mu }_t$: $$\begin{array}{}\text{(10)}& {\mu }_t=\frac{\rho k}{\omega }\frac{1}{\max 1/{\alpha }^{\ast },\Omega F_2/a_1\omega }\end{array}$$

Among them, $$\begin{array}{}\text{(11)}& a^{\ast }=a_{\infty }^{\ast }\frac{a_0^{\ast }+{\mathrm{Re}}_t/R_k}{1+{\mathrm{Re}}_t/R_k}\text{(12)}& \Omega =\sqrt{2{\Omega }_{ij}{\Omega }_{ij}}\text{(13)}& {\sigma }_k=\frac{1}{F_1/{\sigma }_{k1}+1-F_1/{\sigma }_{k2}}\text{(14)}& {\sigma }_{\omega }=\frac{1}{F_1/{\sigma }_{\omega 1}+1-F_1/{\sigma }_{\omega 2}}\end{array}$$where ${\Omega }_{ij}$ is the rate of rotation and $F_1$ and $F_2$ are defined as follows: $$\begin{array}{}\text{(15)}& F_1=\tanh {\min \max \frac{\sqrt{k}}{{\beta }^{\ast }\omega y},\frac{500\mu }{y^2\omega },\frac{4\rho k}{{\sigma }_{\omega 2}{D}_{\omega }^{+}{y}^2}}^4\text{(16)}& F_2=\tanh {\max \frac{2\sqrt{k}}{0.09\omega y},\frac{500\mu }{\rho y^2\omega }}^2\end{array}$$where $y$ is the distance to the other surface and $D_{\omega }^{+}$ is the positive direction of the orthogonal diffusion term. $$\begin{array}{}\text{(17)}& D_{\omega }^{+}=\max 2\rho \frac{1}{{\sigma }_{\omega 2}}\frac{1}{\omega }\frac{\partial k}{\partial x_j}\frac{\partial \omega }{\partial x_j},{10}^{-20}\end{array}$$

In Equation (5), $G_{\omega }$ represents $\omega$ equation $$\begin{array}{}\text{(18)}& G_{\omega }=\frac{a}{v_t}{G}_k\end{array}$$

Definition of coefficient a: $$\begin{array}{}\text{(19)}& a=\frac{a_{\infty }}{a^{\ast }}\frac{a_0+{\mathrm{Re}}_t/R_{\omega }}{1+{\mathrm{Re}}_t/R_{\omega }}\text{(20)}& a_{\infty }=F_1{a}_{\infty 1}+1-F_1{a}_{\infty 2}\text{(21)}& a_{\infty 1}=\frac{{\beta }_{i1}}{{\beta }_{\infty }^{\ast }}-\frac{K^2}{{\sigma }_{\omega 1}\sqrt{{\beta }_{\infty }^{\ast }}}\text{(22)}& a_{\infty 2}=\frac{{\beta }_{i2}}{{\beta }_{\infty }^{\ast }}-\frac{K^2}{{\sigma }_{\omega 2}\sqrt{{\beta }_{\infty }^{\ast }}}\end{array}$$

Turbulent kinetic energy divergence term: $$\begin{array}{}\text{(23)}& Y_k={\rho \beta }^{\ast }k\omega \end{array}$$

In incompressible flows, $$\begin{array}{}\text{(24)}& {\beta }^{\ast }={\beta }_{\infty }^{\ast }\text{(25)}& {\beta }_i^{\ast }={\beta }_{\infty }^{\ast }\frac{4/15+{{\mathrm{Re}}_t/R_{\beta }}^4}{1+{{\mathrm{Re}}_t/R_{\beta }}^4}\end{array}$$

Divergence in $\omega$: $$\begin{array}{}\text{(26)}& Y_{\omega }={\rho \beta \omega }^2\end{array}$$

In incompressible flows, $$\begin{array}{}\text{(27)}& \beta ={\beta }_i\text{(28)}& {\beta }_i=F_1{\beta }_{i1}+1-F_1{\beta }_{i2}\end{array}$$

Correction for orthogonal divergence: $$\begin{array}{}\text{(29)}& D_{\omega }=21-F_1{\rho \sigma }_{\omega 2}\frac{1}{\omega }\frac{\partial k}{\partial x_j}\frac{\partial \omega }{\partial x_j}\end{array}$$

The constant term in this equation is as follows: $K=0.41$, ${\beta }_{i1}=0.075$, ${\beta }_{i2}=0.0828$, ${\sigma }_{k1}=1.176$, ${\sigma }_{k2}=1.0$, ${\sigma }_{\omega 1}=2.0$, ${\sigma }_{\omega 2}=1.168$, ${\beta }_{\infty }^{\ast }=0.09$, $R_{\beta }=8$.

2.2. Validation and Verification

2.2.1. Validation of Numerical Algorithm

The k-ω SST turbulence model balances the model’s effectiveness with computational efficiency. Spalart [36] recognizes the practical value of unsteady Reynolds-averaged Navier–Stokes (URANS) equations. As depicted in Figure 1, 3D URANS simulation is capable of capturing the main vortex structures in the flow field and provides drag coefficient $C_D$ predictions that are in close agreement with the experiment. Moreover, Teso-Fz-Betoño et al. [37] compared the large eddy simulation (LES) WALE method with the URANS method using the $k-\omega$ SST turbulence model for the reattachment process in a backward-facing step flow. The two methods, while showing some differences, are largely consistent, showing similar fluid behavior dynamics. Dejoan, Jang, Leschziner [38] also contrasted LES with URANS for periodic perturbations in separated flows over a backward-facing step, demonstrating that URANS can effectively replicate the overall effect of disturbances, aligning well with experimental observations. These studies indicate that URANS can reliably capture the main flow characteristics and is suitable for studying separated flows. Although LES and DES offer greater precision in simulations, the cells required for grid independence far exceed that of URANS. The Reynolds stress model (RSM) needs to solve additional Reynolds stress equations, demands higher computational costs, and introduces more uncertainty into the model. Table 1 outlines the advantages and disadvantages of these methods. Given its maturity and stability, URANS is deemed more appropriate for engineering applications, prompting its adoption for numerical simulation in this study. Within the RANS framework, the $k-$ω turbulence model is designed for high Reynolds numbers and does not offer particular advantages in solving boundary layer flows. In contrast, the $k-\omega$ SST turbulence model, a low Reynolds number model, performs well in modeling boundary layer flows, as validated by Teso-Fz-Betoño et al. [37]. The two-equation $k-\omega$ SST turbulence model offers a balance between accuracy and computational speed, making it the choice for URANS-based numerical simulation in this paper.

[figure(s) omitted; refer to PDF]

Table 1

Comparison of methods for solving viscous terms.

Method	Advantage	Disadvantaged
LES	High precision	Requires very high precision mesh and very high computational costs
DES	Reduce the high computational cost of LES in the near-wall region while maintaining the accuracy of large-scale vortices	The parameters need to be carefully adjusted to ensure a smooth transition
RANS	Mature, stable, and low computational cost	The transient structure of turbulence cannot be captured
RSM	Provides anisotropic and nonlinear effects of turbulent pulsation	The large number of equations increases the uncertainty of the model

A validation methodology for numerical approaches that align with the techniques detailed in reference [26] was employed in this study. To verify the feasibility of the numerical algorithm for the canard-controlled missile at a large angle of attack, the three-dimensional flow around a cylinder and an airfoil under a large angle of attack were compared with references. The case in Ref. [39] was used to examine the flow around a cycle cylinder. The diameter of the cylinder was 150 mm. Symmetric boundary conditions were used on both sides. The velocity of incoming flow was 35 m/s, pressure was 0.1 MPa, temperature was 60°C, turbulence was 0.7%, and the Reynolds number was $\mathrm{Re}=3\times {10}^5$. The calculation was carried out by using 5.08 million grid cells, which were refined near the wall, the time step used for the calculations was 0.001 s, and the time averaging on the calculation results were performed. The results were compared with those under several Reynolds numbers [39–41], as shown in Figure 2(a). Even at identical Reynolds numbers, there are variations in the experimental data reported by Achenbach [39], Giedt [41], and Fage and Falker [40]. Thus, the present CFD result falls within a reasonable range when compared to the literatures. The comparison of the flow separation angle is illustrated in Figure 3, where present CFD results are compared with Achenbach’s experimental data [39]. Since the separation angle varies continuously with the Reynolds number, it is reasonable to conclude that present CFD results are consistent with the experimental trends. The results of the numerical simulations based on the $k-\omega$ SST turbulence model were found to be in good agreement with the experimental results.

[figure(s) omitted; refer to PDF]

The distribution of the coefficient of surface pressure of the NACA0021 airfoil at $\alpha ={60}^{°}$ and Reynolds number of $\mathrm{Re}=2.7\times {10}^5$ was compared with that of the reference [42]. The length of the chord was 125 mm. Symmetric boundary conditions were set on both ends of the wing. The calculations for the wings were performed by using 5.31 million grid cells, and the grid near the walls was refined. The time step used for the calculations was 0.001 s, and the time average was used. The results of the coefficient of surface pressure obtained in the numerical simulations based on the $k-\omega$ SST turbulence model were close to those of experiments [42], as shown in Figure 4(a).

[figure(s) omitted; refer to PDF]

2.2.2. Model and Verification

The model used in the calculations and experiments is shown in Figure 5(a). The canards and fins were added based on a pointed arch-shaped slender body with a slenderness ratio of 17.5. To demonstrate the multivortex interactions that may occur under the influence of large angles of attack and roll and to derive universally applicable conclusions, this study employs a common design for canard-configured missiles. These designs encompass trapezoidal canard planforms and hexagonal airfoils with the tips of leading and trailing edges removed. For canard-configured missiles launched in a tube, the canards are restricted to a location on the ogive segment, with a limitation that the canard span should not exceed the missile diameter. The chord length and axial position of the canards are determined based on the requirement of static stability. When the roll angle of the model was 0°, the form of the canard was a + shape. The diameter of the model was $D=36\text{mm}$, the deflection angle of the canard was 0°, and its span was equal to the diameter of the missile. Considering that there was a support structure in the tail in the experiment, the model also had a supporting rod with a diameter of 0.5D. Disregarding the manufacturing errors of the experimental model, the model used in the experiment is consistent with the CFD simulations model.

[figure(s) omitted; refer to PDF]

The domain of calculation is shown in Figure 5(b). The front face, 55.6D from the vertex, and the lower half of the cylinder were used as the inlet of velocity. The back face, 83.3D from the tail, and the upper half of the cylinder were used as the outlet of pressure. A Cartesian coordinate system with the origin of the vertex was established. The $x$-axis of coordinate system pointed to the tail, and the $y$-axis of coordinate system was directed straight upward. The calculations were carried out under a pressure of 100.00 kPa, a temperature of 30.0°C, and the speed of incoming wind was set to 17 m/s. The reference length was $D=0.036\text{m}$. The reference area was calculated according to the formula $S=D^2/4\pi$. At low speeds, viscous forces predominate, thus Reynolds number similarity is taken into account. The unit length Reynolds number for both experiments and simulations is $1.05\times {10}^6$. Under the conditions described in this paper, the aerodynamic forces do not exhibit strong unsteadiness; therefore, Strouhal number similarity is not considered. A variable time step was used in the calculations. The time step at the beginning of the calculation was 0.2 s and was gradually reduced as the calculation progressed. The time step was kept constant at 0.0005 s once a stable flow field had been established, and 2000 time steps after stabilization are used for statistical analysis. Calculations were performed at attack angles of 5°, 15°, 25°, 35°, 45°, and 55°, and roll angles of 0°, 15°, 30°, and 45°. The definitions of the configurations under different roll angles are shown in Table 2. Configurations A and D were symmetric, configurations B and C were asymmetric, and configuration E serves as a reference for comparison.

Table 2

The definitions of the configurations under different roll angles.

	$\mathit{\phi }={\mathbf{0}}^{°}$	$\mathit{\phi }={\mathbf{15}}^{°}$	$\mathit{\phi }={\mathbf{30}}^{°}$	$\mathit{\phi }={\mathbf{45}}^{°}$	$\mathit{\phi }={\mathbf{0}}^{°}$wingless
Configuration	A	B	C	D	E

To ensure the availability of data, it was necessary to verify mesh independence. Calculations were performed with a coarse mesh with 3.05 million cells, a medium-mesh with 6.32 million cells, and a fine mesh with 8.71 million cells, and their results were compared. In addition, the $y^{+}=1$, which was suggested by literature [43–45], was guaranteed for all three grids. And the courant number is set as $CFL=1$ [46]. Configuration C was used to verify grid independence under a velocity of flow of $v_{\infty }=17\text{m}/\text{s}$ and $\alpha ={55}^{°}$. The grid convergence index (GCI) was defined as: $$\begin{array}{}\text{(30)}& \text{GCI}=\frac{Fs}{r^n}\frac{f_2-f_1}{f_1}\end{array}$$where $f_1$ and $f_2$ are the aerodynamic coefficients of the fine mesh and the coarse mesh, respectively, $Fs$ is a constant that was set to three, $n$ is the order of the spatial splitting scheme, and $r$ is the factor of mesh fineness. $$\begin{array}{}\text{(31)}& r={\frac{N_1}{N_2}}^{1/\dim }\end{array}$$

$N_1$ and $N_2$ represent the number of cells of the fine and coarse meshes, respectively, and dim represents the dimensions of the computational domain. The results of the three grids are shown in Table 3. The mesh with 6.53 million grid cells satisfied the computational requirements and, thus, was used for subsequent calculations. The wall $y^{+}$ distribution of medium mesh is shown in Figure 3.

Table 3

Comparison of results of three kinds of cells.

Number of cells	CA	CN	CZ	Cm	GCI of CA	GCI of CN	GCI of CZ	GCI of Cm
3.05 million	0.1648	13.7501	−1.1346	−6.2406
6.53 million	0.1216	15.3227	−1.4467	−7.9860	0.8362	0.2415	4.1987	0.5143
8.71 million	0.1255	15.2706	−1.5742	−7.9132	0.0841	0.0092	0.2185	0.0248

2.2.3. Experimental Validation

To further validate the results of numerical calculations, PIV experiments are conducted to obtain the velocity distribution of the cross-section for comparison with the results obtained from numerical simulations. The PIV experiments were carried out in an open-jet return-flow wind tunnel with turbulence of lower than 1%. The experimental arrangement is shown in Figure 6. A 532 nm solid double-pulse laser with a maximum frequency of 10 Hz and a laser controller was connected through cables and cooling pipes. Following reflection by the light guide arm, the laser generator generated sheet light through refraction from a group of laser lenses. The synchronizer was, respectively, connected to the computer, the laser controller, and the CCD camera. The CCD camera and the computer were also connected through a cable and an image acquisition card. In the experiment, the particles generated by the particle generator for imaging were dispersed into the wind tunnel and diffused evenly. The computer simultaneously triggered the generation of a solid pulse laser and the exposure of the CCD camera through the synchronizer. The double-pulse laser triggered two laser pulses each time, and the interval between them was 10 μs. The CCD camera captured two frames of images of particles through two instances of exposure and transferred them to the Dantec PIV software for cross-correlation analysis. The two frames of images are divided into multiple interrogation zones. Within each interrogation zone, cross-correlation statistical analysis is used to obtain the average displacement vector of particles within the zone. This further allows for the calculation of the velocity vector field across the entire solution domain.

[figure(s) omitted; refer to PDF]

Lazar et al. [47] categorizes the uncertainties in PIV measurements into the equipment, particle dynamics, data acquisition, and image analysis and, subsequently, provides a method for quantifying the uncertainty in PIV measurements. In the context of the current PIV measurements, uncertainties are quantified using the method from the literature [47]. Since the experiment was conducted in an open-return low-speed wind tunnel without observation windows, optical distortions caused by window glass are not present and, thus, are neglected. The velocity uncertainty induced by the equipment is as follows: $$\begin{array}{}\text{(32)}& w_u=\sqrt{{\tilde{u}}^2{\frac{1}{L}{w}_l}^2+{\frac{-l}{L^2}{w}_L}^2+{\frac{l}{\lambda L}{w}_{\lambda }}^2+{\frac{-\tilde{u}}{\Delta t}\frac{l}{L}}^2{w}_{t1}^2+w_{t2}^2}\end{array}$$

In the equation, $\tilde{u}$ represents velocity in the pixel-time coordinate system, with units of pixels/s, and other parameters are as shown in Table 4.

Table 4

Parameters and uncertain values.

Category	Parameter	Description	Values	${\mathit{w}}_{\mathit{i}}$
Calibration	$$l$$	Calibration scale physical length	36 mm	1 mm
	$$L$$	Calibration scale image plane length	2048 pixels	1 pixel
	$$\lambda$$	Distance from calibration scale to lens	1 m	0.5 mm
Timing	$$t1$$	Laser pulse timing	200 ns	2 ns
	$$t2$$	Accuracy of delay generator	200 ns	2 ns

From Equation (32), the velocity uncertainty caused by the equipment is obtained as 0.075 m/s, with a relative uncertainty of 0.54%.

The uncertainty in particle dynamics primarily stems from the particle slip phenomenon caused by large velocity gradients or strong velocity fluctuations, which is predominantly observed in compressible flows. In PIV experiments, atomized paraffin oil is utilized as tracer particles, with a density of 780 kg/m³ and droplet diameters of approximately 2 to 3 μm. Given the minuscule size of the droplets, the effect of gravity is neglected. Due to the significantly higher density of paraffin oil relative to air, the forces acting on the particles can be simplified to Stokes’ drag. The particle slip velocity can be expressed as follows: $$\begin{array}{}\text{(33)}& u_{\text{slip}}=\frac{1}{18}\frac{{\rho }_P{d}_P^2}{{\mu }_f}\frac{\partial u_P}{\partial x_P}\frac{\text{d}{x}_P}{\text{d}t}+\frac{\partial u_P}{\partial y_P}\frac{\text{d}{y}_P}{\text{d}t}\end{array}$$

In the equation, ${\rho }_P$, $d_P$, and $u_P$ represent the particle density, particle diameter, and velocity, respectively, and ${\mu }_f$ denotes the dynamic viscosity coefficient of the fluid. In the experiment conducted in a low-speed wind tunnel, the maximum slip velocity in the flow field is calculated and is approximately an order of magnitude of 10⁻⁴. Therefore, the uncertainty due to particle slip phenomena can be neglected.

Sampling of each set of images can be considered as being conducted independently within the same spatial window [48]. The samples can be assumed to follow a normal distribution [49, 50]. For samples that fall within two standard deviations from the mean, the confidence level is 95%. For a sample size of $n$, with a standard deviation $\sigma$, and an average velocity $\overline{u}$, the confidence interval for velocity can be expressed as follows: $$\begin{array}{}\text{(34)}& \overline{u}\pm \frac{z_c\sigma }{\sqrt{n}}\end{array}$$

In the equation, $z_c$ is the confidence factor, and $\sigma$ also represents the inherent fluctuations in the flow at a fixed spatial position, which corresponds to the turbulence intensity. By acquiring 50 image samples in a single acquisition and performing statistical analysis, the standard deviation is 0.185 in a point away from the wall. With a 95% confidence level, the confidence factor is 1.96, resulting in an uncertainty velocity of 0.051 m/s, and the uncertainty is 0.37%. When statistical analysis is conducted on a point with a larger standard deviation of 0.5 in the flow field, the uncertainty velocity at a 95% confidence level is 0.139 m/s, and the uncertainty is 0.995%.

The uncertainty in image analysis arises from the image processing techniques and the PIV algorithms used. Factors such as particle intensity, particle density, image size, turbulence fluctuations, velocity gradients, background noise, and the size of the interrogation windows can all impact the level of uncertainty. The PIV algorithm employs bilinear interpolation on the images. To achieve a smooth flow field, noise is filtered and corrected by comparing the average values of adjacent points. In this experiment, a PIV acquisition and processing system from Dantec Inc was employed. Cross-correlation computations were performed on the 50 sets of images using interrogation zones of $16\times 16$ pixels. This smaller interrogation zone allows for more precise capture of regions with high-velocity gradients, reducing the need for extensive smoothing. Although a smaller interrogation area might lead to a higher noise level, even in regions of high-velocity gradients at supersonic speeds, the standard deviation typically remains below 5% [47].

The error propagation formula for uncertainty is given as follows [47]: $$\begin{array}{}\text{(35)}& {\epsilon }_T=\sqrt{{\epsilon }_E^2+{\epsilon }_L^2+{\epsilon }_S^2+{\epsilon }_P^2}\end{array}$$

In the equation, ${\epsilon }_T$, ${\epsilon }_E$, ${\epsilon }_L$, ${\epsilon }_S$, and ${\epsilon }_P$ represent the total uncertainty, equipment-induced uncertainty, particle dynamics uncertainty, acquisition uncertainty, and image analysis uncertainty, respectively. As it is challenging to quantify the uncertainty specifically caused by the PIV algorithm, even if the image analysis uncertainty is taken as 5%, the total uncertainty remains no greater than 5.13%.

Sources of error in PIV measurements are commonly categorized into installation and alignment errors, system component errors, flow-induced errors, and measurement errors [51, 52]. Installation and alignment errors involve model machining inaccuracies, discrepancies in mounting angles, and the precise alignment of the laser plane. System component errors encompass laser synchronization delays and image distortion. Flow-induced errors are attributed to the particle slip phenomenon, which can arise from significant velocity gradients, intense velocity fluctuations, or high streamline curvature. Within the uncertainty analysis, the influence of particle slip at low velocities has been discussed and is deemed insignificantly small, thus ignorable. Measurement errors include invalid measurements, systemic error, and random error. In the experiment, the sideslip angle was utilized as a proxy for the angle of attack. To rigorously mitigate errors, a laser level was employed in the calibration of the model and experimental apparatus, ensuring that the model’s longitudinal axis was aligned within a horizontal plane. The sideslip angle was fine-tuned with an accuracy of 0.05° using a precision angle adjustment mechanism, and a similar calibration approach with a 0.5° precision was applied to the roll angle, aided by a laser level. To address random measurement errors, the average and standard deviation of 50 sets of PIV images were computed, and the resulting curves and error bands were charted and compared with CFD results as depicted in Figure 7.

[figure(s) omitted; refer to PDF]

Configuration C was used for the PIV experiment at $\alpha ={55}^{°}$. The experiment was carried out at a wind speed of 17 m/s, ambient pressure of 99.8 kPa, air temperature of 25°C, and relative humidity of 44%. Due to limitations of space at the experimental site, sections constituted by $x/D=2$ and $x/D=8.5$ were selected to measure the average 2D velocity distribution within 5 s, and its contours were compared with those of the numerical simulations. Figure 7 shows that the contours of velocity obtained by the PIV experiment were close to those of the numerical simulation at both cross-sections, which verifies the accuracy of the numerical method.

3. Results and Discussion

3.1. The Evolution of the Multivortex Structure

At large angles of attack, the flow around a slender body typically consists of a pair of main vortices and multiple smaller secondary vortices. However, when there is a canard present on the head, it introduces additional pairs of vortices. These corotating vortices tend to merge due to viscous diffusion. This merging process occurs within a certain downstream distance of the canard, which is defined as the region of vortex interaction. The multivortex structure undergoes multiple cycles of merging until it eventually converges into a simpler structure dominated by two main vortices. Throughout this process, the reconstruction of the flow field structure from a multivortex configuration to a steady Karman-like vortex structure affects the surface pressure coefficient, the downstream flow field structure, and the overall aerodynamic force coefficients.

3.1.1. The Merging of Near-Wall Streamwise Vortices

In canard-controlled missile, there are multiple vortices present such as the tip vortices of canard. Unlike at low angles of attack where these vortices operate independently, neighboring vortices with the same rotational direction may merge due to the effects of viscous diffusion at large angles of attack. Within the region of vortex interaction, typical flow phenomena such as strong vortex entrainment of weaker vortices, two-vortex merging, and three-vortex merging occur, leading to the transformation of the flow field into relatively simpler separated flows. These vortex interactions are primarily observed in the range of $\alpha ={35}^{°}$ to $\alpha =55$°. Hence, it is crucial to analyze the characteristics of vortex merging processes occurring in this range. The $Q$ criterion is used to identify vorticity. $Q$ is the second invariant of the tensor of the velocity gradient. $$\begin{array}{}\text{(36)}& Q=0.5\times {\mathbf{\Omega }}_F^2-{\mathbf{S}}_F^2\end{array}$$where the matrix $S$ is the symmetric tensor of the velocity gradient, also known as the strain rate tensor, and ${\mathbf{S}}_F^2$ represents the square of the norm of the matrix $\mathit{S}$. Matrix $\mathbf{\Omega }$ is the antisymmetric tensor of the velocity gradient, also known as the tensor of the rate of rotation or the vorticity tensor. $$\begin{array}{}\text{(37)}& \mathbf{S}=0.5\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\text{(38)}& \mathbf{\Omega }=0.5\frac{\partial u_i}{\partial x_j}-\frac{\partial u_j}{\partial x_i}\end{array}$$

The larger the value of $Q$ was, the larger the tensor of the rate of rotation or the vorticity tensor. Therefore, the value of $Q$ also represents the intensity of the vortex. $C_P$ is the pressure coefficient, which is defined as follows: $$\begin{array}{}\text{(39)}& C_P=\frac{P-P_{\infty }}{1/2{\rho }_{\infty }{v}_{\infty }^2}\end{array}$$where $P$ is the local pressure, $P_{\infty }$ is the freestream pressure, ${\rho }_{\infty }$ is the density of the freestream flow, and $v_{\infty }$ is the freestream velocity.

In the theoretical framework proposed by Dizes and Verga [28], the merging of two parallel vortices with equal intensity begins when their relative distance, normalized by the vortex radius, reaches 0.22. However, in practical flow situations, it is rare to have two parallel vortices with exactly equal intensities. For configuration D at $\alpha ={45}^{°}$, typical vortex merging occurs between two nonparallel vortices with approximate intensities, as shown in Figure 8. The dimensionless distance ($\text{dr}$) is defined as the ratio of the distance between the two vortex cores ($d$) to a reference diameter ($D$). It is important to note that the definition of $\text{dr}$ here differs from that in Reference [28]. In Figure 8, due to the unparallel flow directions of two vortices, the decrease of $\text{dr}$ before merging approximates to a line, until reaching a position of $x=2.33D$. At this location, the dimensionless distance is $dr=0.21$. Beyond this point, the distance between the vortices undergoes nonlinear changes, ultimately reaching zero at $x=3.67D$, indicating completion of the merging process. Thus, the vortex interaction process between $x=2.33D$ and $x=3.67D$ is defined as the vortex merging process. Within the region of vortex interaction, the merging of the main vortices is the final stage. Before the merging of the main vortices, the vortex generated from the head gains angular momentum from the tip flow of the canard as it passes over the upper surface of the canard. After merging with the weaker tip vortex of the canard, they merged into an enhanced main vortex V11. The main vortex V12 originates from the separation of the lower canard and gradually dissipates due to reattachment and reseparation during development at downstream. At the beginning of the merging process between the two main vortices, the intensities of them are similar. During the merging process, V11 maintains a relatively higher intensity, and the two vortices intertwine and rotate around each other, as observed from the streamlines passing through the vortex cores. The intertwining causes the $\text{dr}$ become nonlinear. As the size of the vortex core increases due to viscous effects, the two vortices merge into a larger vortex, denoted as V1. The merged vortex V1 also receives additional angular momentum from the free shear layer, resulting in slower dissipation and dominating the downstream vortex structure. As the size of the merged vortex core expands, the $C_P$ on the symmetry plane is reduced.

[figure(s) omitted; refer to PDF]

In the region of vortices interaction, it is also possible to observe the phenomenon of a strong vortex entraining a weaker vortex and the participation of a third vortex during the merging process of two vortices, as depicted in Figure 9. In Figure 9(a), the vorticity distribution of configuration A at $\alpha ={45}^{°}$ illustrates the merging process of a strong and a weak vortex. The flow over the upper surface of the canard completely separates at large $\alpha$, disrupting the vortex generated at the vertex of the head and generating a new separation vortex, V21. The high-pressure region on the lower surface of the canard increases the velocity gradient of the boundary layer, leading to the roll-up of a new separated vortex, V22, due to the Kelvin–Helmholtz instability. The separated vortex of the canard, V21, dissipates quickly, while the separated vortex of the body, V22, receives additional angular momentum from the free shear layer. During the merging process, the stronger main vortex V22 is hardly affected by the presence of vortex V21, maintaining its trail structure. In Figure 9(c), the low-pressure region on the surface of the body is observed due to the influence of the separated vortex of the canard and the separated vortex of the body. As shown in Figure 9(b), a three-vortex-merging phenomenon occurs on configuration A at $\alpha ={35}^{°}$. The vortex generated at the head is split into two parts by the leading edge of the canard. A portion of the vortex washes over the upper surface of the canard and generates the tip vortex of the canard, V31, under the effect of the wingtip circulation, while the other part dissipates under the influence of the lower separated vortex of the canard. The lower separated vortex of the canard, V32, merges with the vortex V31 with similar intensity during the process of upwashing. During this merging process, the separated vortex of the body, V33, on the lower side separates and participates in the merging of vortices V31 and V32. The three vortices ultimately merge into a larger vortex, V3. In Figure 9(d), it is evident that the surface pressure coefficient on the body is influenced by the near-wall vortex, V32, resulting in the generation of strip-shaped low-pressure regions. The merged vortex, V3, also leads to lower surface pressure coefficients on the body. Initial symmetrical vortices maintain their symmetry after merging, while initially asymmetric vortices retain their asymmetry.

[figure(s) omitted; refer to PDF]

Figure 10 presents the variation in vorticity $Q$, turbulence intensity $I$, and turbulent kinetic energy $k$ for configuration D at $\alpha ={45}^{°}$ and configuration C at $\alpha ={35}^{°}$ during the merging of vortices. Figures 10(a) and 10(b) identify vortices through the $Q$ criterion, illustrating the changes in vortex strength during the merging process. Figures 10(c) and 10(d) depict the evolution of turbulence intensity $I$, which is the dimensionless root-mean-square velocity, defined as $I=v_{\text{rms}}/v_{\infty }$. Figures 10(e) and 10(f) show the changes in turbulent kinetic energy k throughout the merging process. As the vortices approach each other during the merging process, their interaction leads to an increase in both turbulence intensity and turbulent kinetic energy, although the increase in turbulence intensity for configuration C at $\alpha ={35}^{°}$ is not pronounced. After the vortices begin to merge, the internal chaos within the vortices intensifies, and the high-frequency fluctuations are amplified by the action of the free shear flow, resulting in an increase in turbulence intensity and turbulent kinetic energy.

[figure(s) omitted; refer to PDF]

3.1.2. Interaction Between Vortices and Walls

In the region of vortex interaction, there are various cases of interaction between vortices and the wall, such as the reattachment and reseparation of vortex, the Conda effect of curved surfaces on near-wall vortices, the interaction between vortices and canards, and the influence of vortices on surface pressure coefficients.

Figure 11 illustrates the process of reattachment and separation of a canard separation vortex. The definition of the dimensionless velocity $v_{\dim }$ is defined by $v_{\dim }=v/v_{\infty }$, where $v$ is the local velocity and $v_{\infty }$ is the freestream velocity. From the vorticity distribution in Figure 11(a), it can be observed that the vortex core undergoes significant deformation after reattachment. The contour plot of surface $C_P$ on the body in Figure 11(b) shows distinct strip-shaped low-pressure regions corresponding to the position of the canard separation vortex in Figure 11(a). Figures 11(c), 11(d), 11(e), 11(f), 11(g), and 11(h) display the distribution of $C_P$ and $v_{\dim }$ at three different cross-sections. From the cross-section of $x=1.4D$ to $x=1.6D$, after reattachment of the vortex, the pressure coefficient near the vortex core decreases, and the velocity increases, resulting in an elliptical region of low velocity near the wall. From the cross-section of $x=1.6D$ to $x=1.8D$, the vortex starts to move away from the body, and the pressure coefficient and velocity near the vortex core increase. Both pressure potential energy and kinetic energy increase simultaneously, indicating the mixing of the incoming flow with the low-energy separated flow.

[figure(s) omitted; refer to PDF]

The interaction between the vortex and canard at large $\alpha$ involves the interaction of the separated vortex from the head and the canard at the upper side. This interaction occurs in two ways: one scenario is the vortex lies exactly within the plane of the canard and is split into two weaker vortices by the leading edge of the canard, as shown in Figure 12(a); the other is the vortex washes over one side of the canard, enhancing the tip vortex of the canard, as illustrated in Figure 12(b). When the vortex of the head is split into two parts, such as in the vortex on configuration B at $\alpha ={45}^{°}$ in Figure 12(a), the vortex of the head collides with the leading edge of the canard and breaks up. The intensity of the two resulting vortices is greatly weakened, having no significant influence on the flow around the canard. When the vortex of the head washes over one side of the canard, it significantly enhances the leading-edge vortex of the canard, resulting in a pronounced region of low pressure on the upper surface of the canard. The strong leading-edge vortex is pushed towards the tip of the canard by the spanwise flow, creating a stronger tip vortex.

[figure(s) omitted; refer to PDF]

The influence of the vortex on the surface $C_P$ can be divided into the region directly affected by the vortex and the region indirectly affected by the vortex, as shown in Figure 13. To analyze the generation mechanism of these two regions, the contours of $v_{\dim }$ and $C_P$ are plotted at the cross-sections of $x=2.5D$ and $x=4.5D$, as illustrated in Figure 14. In the flow of the vortex, the closer the position to the center is, the lower the pressure is. Therefore, when the vortex core approaches the wall, the $C_P$ at that location is reduced due to the influence of the vortex. In Figure 14(b), the vortex in the upper left corner of the body leads to a decrease in the $C_P$ on the wall. The wall corresponds to the region directly affected by the vortex, marked in Figure 13, and the region directly affected by the vortex follows the trajectory of the vortex in a streak-like region. In Figure 13, there is a larger nonstreak-shaped region of low $C_P$ on the surface of the body, which is not generated by the proximity of the vortex. From Figure 14(a), it can be observed that this region is caused by a large velocity near the wall, which is driven by the pressure difference between the upstream and downstream. The high-pressure zone on the upstream does not show significant changes, so the low-pressure region on downstream largely determines the pressure difference and the pressure gradient between the upstream and downstream. From Figure 14(b), it can be seen that the pressure gradient between the upstream and downstream, determined by the low pressure of the vortex, is a major driving factor for the velocity near the wall. The velocity near the wall, in turn, is a determining factor for the surface $C_P$ in that region. Therefore, this region is defined as the region indirectly affected by the vortex. In Figure 13, the high-pressure on the lower surface of the canard and the low pressure on the separated vortex of the canard together create a large pressure gradient behind the canard, resulting in a significant velocity gradient in the boundary layer. This leads to the generation of a strong separated vortex of the body. The low pressure of the strong vortex further accelerates the flow near the wall, and the increase in velocity means an increase in the velocity gradient of the free shear layer, thereby enhancing the separated vortex of body. In Figure 13, the region indirectly affected by the vortex is larger at the cross-section of $x=2.5D$ compared to the cross-section of $x=4.5D$. Although the $C_P$ near the vortex core in Figure 14(d) is higher than the one in Figure 14(b), the separation of vortex in Figure 14(d) results in a smaller pressure gradient near the wall. The lower velocity near the wall leads to a higher $C_P$ at the cross-section of $x=4.5D$. As the size of the vortex increases during the separation process of the separated vortex of body, the region indirectly affected by the vortex also increases, as shown in Figure 13. When the separated vortex of body detaches and generates secondary vortices in the free shear layer, the region indirectly affected by the vortex becomes smaller.

[figure(s) omitted; refer to PDF]

The multiple vortices generated by canards in a canard-controlled missile create a complex flow field structure at the head. At large angles of attack, these vortices interact with each other through vortex merging and other interactions, eventually forming a structure of a flow field similar to the flow field around a slender body. In addition to the direct impact of the low-pressure vortex on the body, strong separated vortices on one side induce high velocities and low pressures in near-wall flow, resulting in a large area of low-pressure on the surface, defined as the regions indirectly affected by the vortex. The regions indirectly affected by the vortices, which are located on the side of the body and significantly affect the lateral aerodynamic forces, are larger compared to the regions directly affected by the vortices.

3.2. The Influence of the Interaction of Vortices on Flow Field and Aerodynamic Characteristics of the Missile

3.2.1. The Influence of the Head Vortices on the Downstream Structure of Vortices

The multivortex structure in the flow field of a canard-controlled missile not only affects the surface $C_P$ within the region of vortices interaction but also influences the downstream flow field structure. To study the downstream flow field structure, the vortices of four configurations at $\alpha ={35}^{°}$, 45°, and 55° were obtained, using iso-surface of $Q=30000{\text{s}}^{-2}$, and colored according to $v_{\dim }$, as shown in Figure 15. Based on the characteristics of the vortices, the lee-side vortices are divided into three regions. Region A is defined as the region of vortices interaction, where multiple vortices interact and eventually form a symmetric or asymmetric pair of dominant vortices after multiple vortex mergers. The symmetry of the dominant vortices depends on the symmetry of upstream geometry. Region B is defined as the region of asymmetric vortices. The asymmetric vortices rolling up from the free shear layer exhibit distinct alternating separation. In symmetric configurations, the separated vortices are still influenced by the symmetric dominant vortices in region B, with weaker asymmetry compared to asymmetric configurations. In asymmetric configurations, there is evident alternating shedding of separated vortices. After the shedding of the dominant vortices, secondary vortices generated by the rolling-up of the free shear layer enhance and become new dominant vortices. Region C is the region affected by the separated vortices of the fin. This region also exhibits the phenomenon of interaction between the separated vortices of the body and the separated vortices of the fin. Comparing different angles of attack in Figure 15, as the angle of attack increases, the number of lee-side vortices significantly increases, region A decreases significantly, and the velocity on the iso-surface increases while region B expands noticeably. The length and angle of the iso-surface decrease with increasing angle of attack for asymmetric configurations, indicating that the dissipation of vortex increases and downwash decreases. Comparing different roll angles, in symmetric configurations, the $v_{\dim }$ on the iso-surface for the separated vortices of canard is significantly higher than that in asymmetric configurations, and configuration D has higher $v_{\dim }$ on the iso-surface of $Q=30000{\text{s}}^{-2}$ than configuration A. Even though different roll angles induce notable differences in the vortex intensities in region B for asymmetric configurations, the number of separated vortices remains consistent.

[figure(s) omitted; refer to PDF]

To analyze the structure of downstream vortices, the configuration C at $\alpha ={55}^{°}$ was selected as a representative state for analysis, as shown in Figure 16. To distinguish the vortices on each side of the body, the vortices with the angular velocity pointing forward were marked in orange while those with the angular velocity pointing backward were marked in purple. For the convenience of analysis, the side with $z>$0 was defined as the left side, while the other side was the right side. The vortices were captured by taking cross-sectional views at different axial positions, resulting in subplots (b) and (c) which reflect the vortex intensity and structure at different sections.

[figure(s) omitted; refer to PDF]

The vortex distribution of configuration C at $\alpha ={55}^{°}$ is shown in Figure 16, where a clear alternating separation and shedding can be observed in region B. In Figure 16(b), there is a significant correlation between the surface $C_P$ and the separation of vortices. When the vortices attach, a significant low-pressure region appears on the sides. The stronger the vortex is, the larger the region of low pressure on the sides is, and the lower the pressure is. As the vortices separate, the pressure gradually recovers until the free shear layer rolls up a newly attached vortex under the influence of the Kelvin–Helmholtz instability. This process is repeated by the newly attached vortices. The low-pressure regions on the sides during this process are the regions indirectly affected by the vortices mentioned in the previous section. In Figures 16(a) and 16(b), the separated vortex V6 has a larger size of vortex core and intensity, while the separation vortex V7 on the left side is significantly weaker. The higher pressure on the lower surface of the canard located in the lower right corner is the reason for the larger velocity gradient and stronger separation vortex (V6) generated in the right boundary layer. In Figure 16(c), after the separation of V5 on the left side, the left free shear layer rolls up a new vortex V7. However, during the development of V7, the strong vortex V6 on the right side shifts towards the left side due to the Conda effect, causing the leftward displacement of V7. This leads to a decrease in the velocity gradient on the left side and a weakening of the separation vortex V7. On each side, the free shear layer rolls up new vortices after the previous vortex separates. During the attachment of the new vortices, the vortex on the other side starts to separate. This phenomenon results in the axial fluctuations of surface $C_P$. From the axial variations, this process is similar to the shedding of a Karman vortex, but it occurs in a steady state. In region C, the separated vortices of the body interact and merge with the separated vortices of the fin which have lower velocity.

Figure 17 presents a comparison of the pressure coefficient across three sections for configuration C at an angle of attack of 55°. Sections 1 and 3 are each oriented at an angle of 60° relative to Section 2. The distribution of the pressure coefficient on both sides corresponds to the surface pressure coefficient distribution depicted in Figure 18, with alternating low and high-pressure regions and a significant drop in the pressure coefficient within the vortex region. On the leeward side, the presence of a strong vortex at the initial separation is associated with a very low-pressure coefficient, which corresponds to the large velocity regions indicated by the contour plots in Figure 16.

[figure(s) omitted; refer to PDF]

The comparison of vortex distribution between configuration A and configuration E at $\alpha ={55}^{°}$ is shown in Figure 19. The main difference between the two configurations lies in the presence or absence of canards. It can be observed that configuration E exhibits a flow field structure with alternating separation and shedding similar to asymmetric configurations B and C, while configuration A does not show significant vortex alternation and does not exhibit axial fluctuation in surface $C_P$. In Figure 19(a), compared to configuration E, the head vortices in configuration A which demonstrates vortex merging have larger vortex core sizes but weaker intensity, and they are also further away from the body. In configuration A, the separated vortices of canards were elongated in the downstream direction along the vertical axis, ultimately causing the free shear layer to roll up new vortices. In Figure 19(c), configuration E maintains the symmetry of vortices in the ogive section initially but quickly develops natural asymmetry. The vortex on the left side separates first, and the vortex on the right side, which separates later, accumulates more angular momentum from the free shear layer, resulting in a stronger separation vortex. Comparing Figures 19(a) and 19(c) and 19(b) and 19(d), the distance between the vortices and wall at configuration A leads to lower velocities on the iso-surface. Lower velocities on the iso-surface imply smaller velocity gradients near the wall, which makes it less likely to generate new vortices. As a result, the flow on configuration A remains symmetric without alternating separation and maintains a longer symmetric region. On the other hand, the proximity of the separated vortices to the body leads to significant velocity gradients in the free shear layer on configuration E, making it easier to generate new vortices and induce alternating separation. Compared with the spatial distribution of asymmetric vortices at $\alpha ={55}^{°}$ in Figure 15, the number of separated vortices on the body of configuration E is consistent with that of the asymmetric configurations in region B. Additionally, the asymmetric flow on the body also results in an asymmetric flow field around the fin at configuration E.

[figure(s) omitted; refer to PDF]

For symmetric canard-controlled missiles, the merging of vortices in the region of vortices interaction eventually leads to the generation of a symmetric pair of separated vortices. The separation of these symmetric main vortices reduces the velocity gradient in the free shear layer, thereby decreasing the asymmetry in the downstream flow field structure and the asymmetry in the surface pressure distribution on the body. However, for asymmetric canard-controlled missiles, the merging of vortices in the region of vortices interaction results in the generation of a determined asymmetric flow field structure. The downstream flow field structure and the surface $C_P$ distribution on the body resemble that of configuration E, which does not have canards.

3.2.2. The Integration of the Surface Pressure Coefficient and the Coefficients of Aerodynamic Force

The interaction of vortices and the alternating separation of vortices both result in the changes of surface $C_P$, while aerodynamic force coefficients are obtained by integrating the surface pressure. Therefore, the effects of vortices are ultimately reflected in the aerodynamic forces. To investigate the distribution of the normal force and lateral force along the axial direction, 630 cross-sections are taken along the $x$-axis on the missile. For each cross-section, the pressure is integrated with the $y$-direction and $z$-direction, and the integrated results are nondimensionalized. The resulting dimensionless coefficients are defined as the sectional normal force coefficient ($C_{Sf\text{y}}$) and sectional lateral force coefficient ($C_{Sf\text{z}}$): $$\begin{array}{}\text{(40)}& C_{Sf\text{y}}=\frac{{\int }_{2\pi }^0\Delta l\cdot P\theta \cos \theta d\theta }{1/2\rho v^{2}D}\text{(41)}& C_{Sf\text{z}}=\frac{{\int }_{2\pi }^0\Delta l\cdot P\theta \sin \theta d\theta }{1/2\rho v^{2}D}\end{array}$$

In the equation, $\Delta l$ represents the distance between two adjacent nodes, and $P\theta$ is a function of pressure concerning the circumferential angle $\theta$.

A comparative analysis of the $C_{Sf\text{z}}$ and surface $C_P$ for five configurations at $\alpha ={55}^{°}$ is shown in Figure 18. In Figure 18(a), configurations B, C, and E exhibit noticeable axial fluctuations in $C_{Sf\text{z}}$. These three configurations reach the maximum absolute amplitude at the first trough, followed by a gradual decrease in amplitude. Meanwhile, configuration E shows a phase difference compared to configurations B and C. The asymmetric separated vortices from the asymmetrical canard result in the generation of asymmetrical main vortices in the region of vortices interaction. This causes the asymmetrical main vortices to separate earlier on configurations B and C, leading to the observed phase difference. Configurations A and D have much smaller fluctuations in $C_{Sf\text{z}}$ compared to the other configurations, with their values consistently close to zero. The distributions in Figures 18(b) and 18(c) correspond to the values in Figure 18(a). On the right side, configurations B and C exhibit significant low-pressure regions at the first trough, corresponding to the region indirectly affected by the vortex as shown in Figure 13. On the left side, there are high-pressure regions in a slightly elevated position, which corresponds to the high pressure due to the separation of the vortex as seen in Figure 16(c). The alternating low-pressure and high-pressure regions on the body correspond to the attached and separated states of the K-H vortices, as depicted in Figure 16(b). The alternating separation of vortices on both sides leads to the fluctuations of pressure and causes significant fluctuations in $C_{Sf\text{z}}$. The canards and fins produce pressure difference between the left and right sides due to their roll angles, resulting in negative $C_{Sf\text{z}}$. Additionally, the fins also generate positive $C_{Sf\text{z}}$ at the front due to the influence of the separated vortex of body.

The overall coefficient of lateral force ($C_Z$) is obtained by integrating the surface pressure in the z-direction, which is shown in Figure 20(a). At a small angle of attack $\alpha =5^{°}$, the $C_Z$ for all configurations are close to zero. However, significant differences in the $C_Z$ can be observed between $\alpha ={15}^{°}$ and $\alpha ={35}^{°}$. At $\alpha ={45}^{°}$ and $\alpha ={55}^{°}$, the absolute values of $C_Z$ for the asymmetrical configurations are significantly larger than those for the symmetrical configurations, indicating that the influence of roll angle on $C_Z$ dramatically decreases as the angle of attack increases. In Figure 20(a), at $\alpha ={35}^{°}$, configurations B and E have positive values of $C_Z$, while at an angle of attack $\alpha ={45}^{°}$, configurations B, C, D, and E encounter dramatically decrease in $C_Z$. So, a comparison of the distributions along the axial direction for these six states is made, as shown in Figure 20(b). For configuration B, the changes in the distribution of $C_{Sf\text{z}}$ along the missile at $\alpha ={35}^{°}$ and $\alpha =45$° are compared with the results shown in Figure 18(a). With increasing $\alpha$, the period of fluctuations in $C_{Sf\text{z}}$ along the axial direction becomes shorter, and the amplitude of $C_{Sf\text{z}}$ at the head increases. The significant decrease in $C_{Sf\text{z}}$ at the head section is an important factor leading to the decrease in $C_Z$. Additionally, the increase in $\alpha$ also accelerates the attenuation of the downstream amplitude. Configuration C, compared to configuration B, exhibits stronger separated vortices of head, resulting in lower surface $C_P$ and hence lower $C_{Sf\text{z}}$ and $C_Z$. In Figure 20(b), at $\alpha ={35}^{°}$, the leading edge of the fin is positioned where there is a significant pressure difference between the two sides, resulting in a positive $C_{Sf\text{z}}$ at the front part of the fin. This is influenced by the length of the missile [53].

[figure(s) omitted; refer to PDF]

By integrating the pressure along the normal direction of the entire missile, the variation of the coefficient of normal force ($C_N$) concerning the $\alpha$ is obtained as shown in Figure 21(a). Configuration E, which lacks canards, exhibits a significantly lower $C_N$ compared to the other configurations, while configuration D has a smaller $C_N$ than configurations A, B, and C at large angles of attack. The analysis is conducted on the values of $C_{Sf\text{y}}$ for the five configurations at $\alpha ={55}^{°}$, to explain the reasons for the differences in $C_N$. From the distribution of $C_{Sf\text{y}}$ along the axial direction, it can be observed that at all $\alpha$ and $\phi$, the $C_{Sf\text{y}}$ at canards and fin are significantly greater than those at the body, as shown in Figure 21(b). The $C_{Sf\text{y}}$ at the fin decreases as the roll angle increases within the range of 0° to 45°. This is one of the main reasons why configuration D has a significantly lower $C_N$ than the other configurations. Configuration E, which is the absence of canards, has a significantly lower $C_{Sf\text{y}}$ at the head compared to other configurations. Although the $C_{Sf\text{y}}$ on the body is larger for configuration E, it still leads to a smaller $C_N$. The symmetric configurations have slightly smaller $C_{Sf\text{y}}$ in the body compared to the asymmetric configurations. Even though the $C_{Sf\text{y}}$ at the fin for the asymmetric configurations is smaller than that for configuration A, the longer body brings the $C_N$ of the asymmetric configurations closer to that of configuration A. It is worth noting that another significant difference exists behind the canards, where the $C_{Sf\text{y}}$ decreases as the roll angle increases. This difference is caused by different pressure distributions near the root of the canards at different roll angles. And it is also one of the reasons for the variation of $C_N$.

[figure(s) omitted; refer to PDF]

4. Conclusions

In this study, the vortex interaction process at the head of a canard missile at large angle of attack was investigated, and its influence on the flow field and aerodynamic forces was analyzed. The main conclusions are as follows:

At large angles of attack, the flow field over a canard-controlled missile generates a complex multivortex structure due to the disturbance caused by the canards. The evolution of the multivortex structure significantly impacts the pressure distribution, the flow field, and the aerodynamic forces. The multivortex structure in the head evolves into a steady-state structure of flow field dominated by a pair of main vortices through vortex merging. This evolutionary process occurs within a certain range in the head.

The configurations with symmetric geometry maintain the symmetry of the flow field during the evolution process. The process of vortex merging is dominated by the separated vortices of the canard, which results in the merged main vortices further away from the body and maintaining downstream symmetry. In comparison to the “+” sharp configuration of canard missiles, the X-sharp configuration exhibits stronger main vortices after mergence in the region of vortices interaction. It maintains a longer symmetric region in the downstream flow field, demonstrating higher flow field stability.

The configurations with asymmetric geometry induce inconsistent vortex intensities and positions on both sides, leading to a strong asymmetry in the main vortices after merging. These intense vortices not only directly influence the surface pressure distribution near the vortices but also create large areas of low pressure on the sides by affecting the velocity. The asymmetric vortices in the downstream flow field cause alternating separation and shedding on the left and right sides, resulting in fluctuating sectional lateral force coefficients. As the angle of attack increases, the amplitude of the fluctuations in the sectional lateral force coefficient increases and gradually dominates the lateral force coefficient, leading to a significant decrease in the lateral force coefficient.

As a comparison, the symmetric configuration without canards exhibits a naturally asymmetric flow at large angles of attack. When the main vortex becomes asymmetric, it also induces alternating separation and shedding of vortices on both sides of the body. However, the separation process is delayed compared to the asymmetric canard configuration. This is because the asymmetric canards generate a pair of asymmetric main vortices, leading to earlier detachment of the vortices.

In conclusion, the flow field in the head of a canard-controlled missile exhibits a multivortex structure. This multivortex structure not only affects the surface pressure coefficients within the region of vortex interaction but also determines the downstream flow field structure, pressure distribution, and dominates the lateral force.

Author Contributions

Conceptualization: Kai Wei and Shaosong Chen; methodology: Kai Wei and Yihang Xu; validation: Yihang Xu and Xujian Lyu; formal analysis: Dongdong Tang; investigation: Dongdong Tang; resources: Shaosong Chen; data curation: Qing Chen; writing—original draft preparation: Kai Wei; writing—review and editing: Shaosong Chen, Qing Chen, and Kai Wei; visualization: Xujian Lyu; supervision: Shaosong Chen; project administration: Kai Wei. All authors have read and agreed to the published version of the manuscript.

Funding

The present work is financially supported by the Science and Technology on Underwater Information and Control Laboratory (No. 2021-JCJQ-LB-030-05).

Acknowledgments

The support of the Science and Technology on Underwater Information and Control Laboratory is greatly appreciated.

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