Study on the Aerothermoelastic Characteristics of a Body Flap Considering the Nozzle–Jet Interference

Abstract

A body flap/RCS-integrated configuration is often used to achieve pitch trimming and controlled flight in near space for hypersonic vehicles. Under the high temperature and pressure load induced by the expansion wave at the nozzle exit, the body flap is prone to significant structural deformation, which leads to a change in the resulting moment, even comparable to the control ability, and bring additional challenges to the control system. Based on the CFD/CTD/CSD coupling method, the aerothermoelastic effect on the aerodynamic characteristics and structural deformation of the body flap under jet interaction is systematically studied. Numerical results indicate that the pitching moment coefficients show an increasing trend for all the models, rigid, elastic and thermoelastic, while the increment significantly decreases with the increase in trajectory altitudes. With the increase in deflection angle, the pitching moment coefficients of the three models decrease nonlinearly at high altitude, and the aerothermoelastic effect significantly decreases. At a middle-lower altitudes, the pitching moment coefficient is reversed at a lager deflection angle, and the trailing edge of the body flap presents the deformation characteristics of upward bending, which makes the aerothermoelastic phenomenon degenerate into an aeroelastic problem. From the station along the chord and spanwise direction, the change in displacement increment of the thermoelastic model reflects the competitive relationship between normal stress and thermal stress imposed by jet interaction.

1. Introduction

The gas vanes are always adopted for the thrust vector control of the early rocket engines. Since the gas vanes will block the engine exhaust flow to varying degrees at different stages, this is bound to produce some drag and reduce the performance of the engine. Moreover, strict requirements for the thermal protection ability of its structure must be made for the high-temperature flow from the nozzle at the tail. In order to realize controlled flight in near space, body flaps are usually adopted and combined with the rocket jet reaction control system (RCS) of modern aircrafts [1,2], which are placed on the tail to realize pitch trimming and control (shown in Figure 1). This design scheme is the key technology of orbit control commonly used in the ascending and descending stages of hypersonic vehicles. Different flight stages can be controlled with a single approach or a composite controlling mode, such as X-34, X37-B and HOPE-X [3]. When the engine is working, the body flap will inevitably be subject to the effect of high-pressure load and high heat flux with shear effect of the boundary flow due to the high-speed expanding jet gas, resulting in an increase in the structural temperature. Under extreme force and thermal load, the body flap is prone to large thermal deformation. And the change in material properties and temperature gradient due to aerodynamic heating causes additional thermal stress in the structure of the thin body flap [4]. Therefore, the coupling between aerodynamic heating and aeroelasticity needs to be comprehensively considered. Although the force due to the aerothermoelastic effect on the jet interference is much less than the engine thrust, its force arm is significantly greater than that of the engine thrust. The pitching moment generated by the jet interference can bring considerable moment compared with that generated by the engine thrust, even comparable to the control ability, which can bring additional challenges to the aircraft control system [5,6].

The difficulty of the numerical method in solving the interference of the body flap and nozzle jet under aerothermoelastic effect is mainly due to the huge difference in the time scales of heat transfer, structural vibration and unsteady flow, which can reach more than three orders of magnitude. If the aerodynamic/thermal/structural coupling simulation is adopted in the full-time domain, the computational resource consumption will be almost unacceptable, especially for the complex configuration and structures [7]. A series of previous studies on multi-field coupling studies have shown that the change in structural temperature field is not obvious, and the change in material properties and structural thermal stress is not significant in the full-time domain [8]. The influence of structural deformation on heat transfer can be ignored. Thus, it is reasonable and efficient to solve the aeroelastic problem via freezing the structure temperature distribution [9]. Due to the multidisciplinary coupling characteristics and the increasing complexity of the configuration of modern aircrafts, the accuracy of the engineering methods used to predict aerodynamic data show more limitation. By comparison, the CFD (computational fluid dynamics) method is a more applicable approach that can increase the data accuracy of unsteady aerodynamic load and heat flux [10]. Furthermore, the modal method is widely used as a linear method for the calculation of structural deformation, which ignores the effects of geometric and material nonlinearity. It is inappropriate to use the modal method in the deformation analysis of a complex structure with time-varying structural features. Therefore, the finite element method (FEM) is adopted to study the aerothermoelastic deformation in this paper, which is more accurate to analyze structural deformation due to the coupling effect under aerodynamic load and aerothermal load [11].

At present, the jet interference flow field between the engine and body flap is mainly analyzed based on rigid-body assumption, while research related to the aerothermoelastic effect on the control surface caused by jet interference is very limited in the literature. When considering the elastic effect of the body flap of a symmetrical aircraft under the action of the wake jet, the structural deformation is always dominated by a bending mode shape, and the classical flutter instability of modal coupling does not usually occur. Since the time scale of structural vibration is much smaller than that of heat transfer, the structural temperature can be regarded as constant during each time step of calculating deformation. Therefore, the static aerothermoelastic analysis method based on CFD/CTD/CSD is used to study the aerodynamic interaction characteristics of jet flow/body flap in this paper.

2. Numerical Methodology and Validation

2.1. Numerical Methodology

The process of the static aerothermoelastic solution used in this paper is shown in Figure 2, which is based on the coupling of computational fluid dynamics (CFD)/computational thermodynamics (CTD)/computational solid dynamics (CSD) without considering the coupling relationship between structural deformation and heat transfer. When analyzing structural deformation, the static aeroelastic problem with the given structural temperature distribution at the specified time can be solved using this numerical framework. The advantage of this method is that it can quickly obtain the steady solutions of structural deformation and flow field without huge time-consuming calculation of dynamic aerothermoelasticity directly in the time domain. The steps are as follows:

• The steady flow field is solved based on CFD under the given flow field parameters;
• Taking the steady flow field as the initial field coupled with the transient heat conduction equation, the structural temperature distribution can be solved at the specified time. It is worth noting that the coupling time step is determined by the characteristic time scale of heat conduction;
• The obtained node temperature of the structure field is assigned to the finite element node as the temperature load, and the structural stiffness matrix considering the thermal effect is obtained. Taking the steady flow field as the initial field coupled with the structural static equation, the structural displacement can be solved. The surface force load at the interface is obtained using an interpolation method;
• The flow field mesh is updated according to the structural displacement using the mesh deformation method. Based on the updated flow field mesh, the steady flow field solver is recalled to obtain the aerodynamic force of the current coupling step;
• If the aerodynamic force meets the convergence criterion, the simulation is over. Otherwise, return to step 3.

The aforementioned analysis framework mainly includes four modules: CFD solver module, heat conduction solver module, structural deformation solver module and grid-treating module. A detailed description of the modules in both the mathematical and numerical method is provided next.

2.1.1. Fluid Flow Governing Equations and Numerical Method

Fluid dynamics computations are performed via the in-house solver, which is widely used in solving the flow field related to jet interference [12]. The three-dimensional unsteady Reynolds-averaged governing equation in the Cartesian coordinate system is

$$\frac{\partial \mathit{Q}}{\partial t}+\frac{\partial \mathit{F}}{\partial x}+\frac{\partial \mathit{G}}{\partial y}+\frac{\partial \mathit{H}}{\partial z}=\frac{\partial {\mathit{F}}_v}{\partial x}+\frac{\partial {\mathit{G}}_v}{\partial x}+\frac{\partial {\mathit{H}}_v}{\partial x}$$

(1)

where $\mathit{Q}={(\rho ,\rho u,\rho v,\rho w,\rho E)}^T$ is the vector of conservative variables; $\rho$ is the density of the perfect gas; u, $v$ and $w$ are the velocities in three directions; $E$ is the total energy per unit mass of gas; $\mathit{F}$, $\mathit{G}$, $\mathit{H}$ are the convection terms;${\mathit{F}}_v$, ${\mathit{G}}_v$, ${\mathit{H}}_v$ are the viscous terms expressed as follows, respectively.

$$\mathit{F}=\left\{\begin{array}{l}\rho u\\ \rho u^2+p\\ \rho uv\\ \rho uw\\ (\rho E+p)u\end{array}\right\}, \mathit{G}=\left\{\begin{array}{l}\rho v\\ \rho vu\\ \rho v^2+p\\ \rho vw\\ (\rho E+p)v\end{array}\right\}, \mathit{H}=\left\{\begin{array}{l}\rho w\\ \rho wu\\ \rho wv\\ \rho v^2+p\\ (\rho E+p)w\end{array}\right\}$$

(2)

$${\mathit{F}}_v=\left\{\begin{array}{l}0\\ {\tau }_{xx}\\ {\tau }_{xy}\\ {\tau }_{xz}\\ q_x+u{\tau }_{xx}+v{\tau }_{xy}+w{\tau }_{xz}\end{array}\right\}, {\mathit{G}}_v=\left\{\begin{array}{l}0\\ {\tau }_{xy}\\ {\tau }_{yy}\\ {\tau }_{yz}\\ q_y+u{\tau }_{yx}+v{\tau }_{yy}+w{\tau }_{yz}\end{array}\right\}, {\mathit{H}}_v=\left\{\begin{array}{l}0\\ {\tau }_{xz}\\ {\tau }_{yz}\\ {\tau }_{zz}\\ q_z+u{\tau }_{xz}+v{\tau }_{yz}+w{\tau }_{zz}\end{array}\right\}$$

(3)

$${\tau }_{ij}=\left[({\mu }_L+{\mu }_T)\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right)\right]-\frac{2}{3}({\mu }_L+{\mu }_T)\frac{\partial u_l}{\partial x_l}{\delta }_{ij};q_i=\left(\frac{{\mu }_L}{{\mathrm{Pr}}_L}+\frac{{\mu }_T}{{\mathrm{Pr}}_T}\right)\frac{\partial T}{\partial x_i}$$

(4)

where ${\tau }_{ij}$ is stress term, ${\mu }_L$ is viscous coefficient of laminar flow, ${\mu }_T$ is viscous coefficient of turbulent flow, ${\mathrm{Pr}}_L$ and ${\mathrm{Pr}}_T$ are the Prandtl number, and $T$ is the static temperature.

The governing equations are discretized using the finite volume method (FVM) based on unstructured meshes. The advection upstream splitting method (AUSM) scheme and central difference scheme are used for convection terms and viscous terms discretization with an implicit LU-SGS scheme for temporal integration. The one-equation SA turbulence model is implemented to treat the turbulent boundary layers for computations of the viscous flow. The wall boundary is defined as an isothermal wall with 300 K and no-slip condition. The stationary chamber is taken as the starting point of the nozzle with the prescribed total temperature and total pressure. The far field is defined as the boundary condition of pressure outlet with the prescribed pressure and temperature equal to that of the specified ballistic altitude.

2.1.2. Heat Transfer Governing Equation

The unsteady heat conduction equation in the Cartesian coordinate system is

$${\rho }_S{c}_{pS}\frac{\partial T_S}{\partial t}=\left({\lambda }_x\frac{{\partial }^2{T}_S}{\partial x^2}+{\lambda }_y\frac{{\partial }^2{T}_S}{\partial y^2}+{\lambda }_z\frac{{\partial }^2{T}_S}{\partial z^2}\right)+q_{\mathsf{\Gamma }}$$

(5)

where $T_S$ is the temperature of the structure, ${\lambda }_i$ is the thermal conductivity coefficient, $c_{pS}$ is the specific heat capacity, and $q_{\mathsf{\Gamma }}$ is the heat flux applied to the surface of the structure.

The governing equations are discretized using FEM with the unstructured mesh:

$$\left[\mathit{C}\right]\left\{\dot{\mathit{T}}\right\}+\left[{\lambda }_T\right]\left\{\mathit{T}\right\}=\left\{Q_T{}^{\mathrm{\Gamma }}\right\}$$

(6)

where $\left[\mathit{C}\right]$ is the global heat capacity system matrix, $\left[{\lambda }_T\right]$ is the global thermal conductivity system matrix, and $\left\{Q_T{}^{\mathrm{\Gamma }}\right\}$ is the boundary thermal load matrix.

The initial and boundary value condition are as follows.

$${\left.T_S\right|}_{t=0}=T_{wall}\left(x,y,z\right)=const, \mathrm{in} \mathrm{\Omega }$$

(7)

$${\left.q_{\mathsf{\Gamma }}\right|}_t=q\left(\mathsf{\Gamma },t\right), \mathrm{in} \mathrm{\Gamma }$$

(8)

where $\mathrm{\Omega }$ is the spatial domain and $\mathrm{\Gamma }$ is the boundary of the structure.

2.1.3. Structural Static Equation Considering Thermal Effect

The tensor form of equilibrium equation is provided as follows.

$$\frac{\partial {\sigma }_{S,ji}}{\partial x_j}+F_i=0$$

(9)

where ${\sigma }_{S,ji}$ is the stress tensor and $F_i$ is the external force load.

The tensor form of the geometric equation is provided as follows.

$${\epsilon }_{S,ij}=\frac{1}{2}\left(\frac{\partial u_{s,i}}{\partial x_j}+\frac{\partial u_{s,j}}{\partial x_i}\right)$$

(10)

where ${\epsilon }_{S,ij}$ is the strain tensor and $u_s$ is the displacement vector.

The stress–strain constitutive relation is given as Equation (11).

$${\sigma }_{S,ij}=\frac{E(T)\nu }{(1+\nu )(1-2\nu )}\theta {\delta }_{ij}+\frac{E(T)\nu }{(1+\nu )}{\epsilon }_{S,ij}-\frac{E(T)}{1-2v}\alpha (T-T_{ref}){\delta }_{ij}$$

(11)

where $E(T)$ is the Young’s modulus dependent on the temperature, $\theta$ is the volumetric strain, $\alpha$ is the coefficient of thermal expansion, and $T_{ref}$ is the reference temperature.

Saint-Venant’s deformation compatibility condition is adopted as follows.

$${\nabla }^2{\sigma }_{S,ij}+\frac{1}{1+v}\frac{\partial {\sigma }_{S,kk}}{\partial x_i\partial x_j}=-\frac{E(T)}{1-v}\alpha {\nabla }^2(T-T_{ref}){\delta }_{ij}-\frac{E(T)}{1+v}\alpha \frac{{\partial }^2(T-T_{ref})}{\partial x_i\partial x_j}$$

(12)

The governing equations are discretized using FEM as follows.

$$\left[K_s\left(T\right)\right]\left\{w\right\}=\left\{{\mathbf{Q}}_F\right\}$$

(13)

where $\left[K_{\mathbf{s}}\left(T\right)\right]$ is the global stiffness matrix with the temperature, $\left\{w\right\}$ is the displacement matrix, and $\left\{{\mathbf{Q}}_F\right\}$ is the force load matrix of the boundary nodes. The boundary value conditions are

$$\left\{\begin{array}{l}{\sigma }_{s,ij}{n}_j={\overline{f}}_i\\ u_{s,i}={\overline{u}}_{S,i} \end{array}\right., \mathrm{at} \mathrm{\Gamma }$$

(14)

where $n_j$ is the normal vector of the boundary face of a finite element, ${\overline{f}}_i$ is the surface force, and ${\overline{u}}_{S,i}$ is the given displacement.

2.1.4. Dynamic Mesh Technique and Interface Interpolation Method

The mesh deformation technology and the interface interpolation technology used in this paper are both radial basis function (RBF) methods [13,14,15,16,17]. This method is simple and easy to implement. The quantity to be interpolated at the node i is expressed in the following form:

$$w(x)={\displaystyle \sum _{i=1}^N{\alpha }_i\varphi (‖x-x_{\mathrm{i}}‖})+{\displaystyle \sum _{j=1}^{N_p}{\beta }_j{p}_j(x)}$$

(15)

where x is the coordinates of the target interface and x_i is the coordinates of the template interface. The displacement matrix $w(x)$ consists of two parts. The first term of the right hand side ${\displaystyle \sum _{i=1}^N{\alpha }_i\varphi (‖x-x_i‖})$ is the basic quantity of RBF, while the second term ${\displaystyle \sum _{j=1}^{N_p}{\beta }_j{p}_j(x)}$ is the polynomial that guarantees the uniqueness and solvability of the equation. For conditionally positive definite basis functions of the order which is no more than two, ${\displaystyle \sum _{j=1}^{N_p}{\beta }_j{p}_j(x)}$ can be represented by linear polynomials. $‖x-x_i‖$ is the Euclidean distance between the target point and the interpolation point. RBF interpolation ultimately boils down to solving a system of linear equations. When there is a large amount of data, this is a large-scale matrix inversion problem, which is prone to numerical instability. Wendland’s C2 function in the form of Equation (16) is adopted as the RBF, which is conditionally compact and positive on ${\eta }^3$, and it can yield good results for surface reconstruction.

$$\varphi (\eta )=\left\{\begin{array}{l}{\left(1-\eta \right)}^4\left(1+4\eta \right)\eta <\mathrm{R}\\ 0\eta \ge \mathrm{R}\end{array}\right.$$

(16)

where $R$ is the specified working distance and $\eta =\frac{‖x-x_i‖}{R}$.

Here, the polynomial is constructed based on a linear basis, i.e., $p_j(x)=[1,x,y,z{]}^T$. The weight coefficients ${\alpha }_i$ and ${\beta }_j$ can be solved through interpolation and orthogonal conditions, which are expressed as Equation (17).

$$\left\{\begin{array}{l}{\displaystyle \sum _{i=1}^N{\alpha }_i\varphi (‖x_l-x_i‖})+{\displaystyle \sum _{j=1}^4{\beta }_j{p}_j(x_l)=w(x)\left|{}_{x=x_l}\right.}\\ {\displaystyle \sum _{i=1}^N{\alpha }_i}={\displaystyle \sum _{i=1}^N{\alpha }_i}{x}_i={\displaystyle \sum _{i=1}^N{\alpha }_i}{y}_i={\displaystyle \sum _{i=1}^N{\alpha }_i}{z}_i=0\end{array}\right.$$

(17)

where $w(x_l)$ is the known quantity of the template interface.

Equation (15) can be written in the matrix form as follows.

$$\left[\begin{array}{l}\mathbf{\Phi }\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}P\\ P^T\hspace{0.33em}\hspace{0.33em}0\hspace{0.33em}\end{array}\right]U=W$$

(18)

where $\mathbf{\Phi }$ is the RBF matrix, P is the polynomial matrix, and U is the weight coefficient matrix. The matrices have the following forms.

$$\left\{\begin{array}{l}\mathbf{\Phi }=\left[\begin{array}{l}\varphi (‖x_1-x_i‖)\hspace{0.33em}\hspace{0.33em}\varphi (‖x_1-x_2‖)\hspace{0.33em}\cdots \hspace{0.33em}\varphi (‖x_1-x_N‖)\\ \varphi (‖x_2-x_1‖)\hspace{0.33em}\hspace{0.33em}\varphi (‖x_2-x_2‖)\hspace{0.33em}\cdots \hspace{0.33em}\varphi (‖x_2-x_N‖)\\ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}⋮\\ \varphi (‖x_N-x_1‖)\hspace{0.33em}\hspace{0.33em}\varphi (‖x_N-x_2‖)\hspace{0.33em}\cdots \hspace{0.33em}\varphi (‖x_N-x_N‖)\end{array}\right]\\ U={\left[{\alpha }_i\hspace{0.33em}\hspace{0.33em}{\beta }_1\hspace{0.33em}{\beta }_2\hspace{0.33em}{\beta }_3\hspace{0.33em}{\beta }_4\right]}^T\\ P=\left[\begin{array}{l}1\hspace{0.33em}\hspace{0.33em}{x}_1\hspace{0.33em}{y}_1\hspace{0.33em}{z}_1\hspace{0.33em}\hspace{0.33em}\\ 1\hspace{0.33em}\hspace{0.33em}{x}_2\hspace{0.33em}{y}_2\hspace{0.33em}{z}_2\\ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}⋮\\ 1\hspace{0.33em}\hspace{0.33em}{x}_N\hspace{0.33em}{y}_N\hspace{0.33em}{z}_N\end{array}\right]\end{array}\right.$$

(19)

The weight coefficient matrix U can be obtained as Equation (18) with the following expression.

$$U={\left[\begin{array}{l}\mathrm{\Phi }\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}P\\ P^T\hspace{0.33em}\hspace{0.33em}0\hspace{0.33em}\end{array}\right]}^{-1}W$$

(20)

By bringing the calculation results into Equation (15), the function values at the interpolation points can be solved.

2.2. Verification Cases

Since three-dimensional experimental data of aerothermoelasticity in hypersonic have not been found in existing research, the calculation module of transient heat transfer and static aeroelasticity are verified respectively, to confirm the rationality of the static aerothermoelasticity analysis method in this paper.

2.2.1. Verification Case of Transient Heat Transfer

The aerodynamic heating experiment of the leading edge of a cylindrical shell under shock wave interaction conducted by Wieting in the NASA Langley 8 ft high temperature wind tunnel was selected [18]. The geometric parameters, the structural parameters and the incoming flow parameters of the experimental object are shown in

Table 1. Calculation parameters of the body flap.

Geometric Parameters		Structural Parameters		Incoming Flow Parameters
chord length	609.6 mm	$\rho$	8030 kg/m3	Ma	6.47
lead edge (relative to the centroid of the aircraft)	(4000 mm, 0, 0)	Cp	502.48 J/(kg K)	$T_{\infty }$	equal to the atmospheric parameter
spanwise length	1148 mm	Tinitial	300 K	$P_{\infty }$

. The thermal conductivity is shown in Figure 3. It can be regarded as a rigid body that does not take into consideration the impact of elastic deformation on aerodynamic heat.

The case is simplified to a two-dimensional problem. Figure 4 shows the mesh stretched in the extension direction. The distribution of CFD mesh is 160 × 100, and the height of the first boundary layer is 6 × 10⁻⁶ m. The solution of the structural deformation for FEM adopts the hexahedral mesh with 20 nodes.

The comparison between the calculated and experimental data of the pressure and heat flow on the surface at the initial time is shown Figure 5. It can be seen that the numerical results of the distribution of pressure and heat flux both agree well with the experimental data. The calculated heat flux at stagnation is 678.2 kW/m², which is very close to the experimental value of 664.1 kW/m² . Figure 6 shows the comparison between the calculated and experimental data of the stagnation point temperatures. It can be seen that the calculated time-variant temperature is in good agreement with the experimental results, which shows that the numerical method can better simulate the change process of aerodynamic heating effect on the structure with time.

2.2.2. Verification Case of Staticaeroelsticity

In this paper, the aeroelastic study of the HIRENASD wing-body [19,20] is selected to evaluate the accuracy of the aeroelastic simulation of the solver. The configuration and the hybrid mesh (shown in Figure 7) were provided by the AePW Conference sponsors. This simulation is performed at a Mach number of 0.80 and a Reynolds number of 14 million, since interesting transonic flow phenomena and significant wing deformation were found in the experiment [21]. The level of the dimensionless dynamic pressure q/E is 0.47 × 10⁻⁶, where q is the dynamic pressure of the free stream and E is the elastic modulus of the structure. Fully turbulent solution is adopted for viscous flow with the SA model, and the RBF method is applied for mesh deformation.

Figure 8 shows the comparison between the calculated elastic deformations and the experimental data at the leading edge and the trailing edge. There are significant deformations at the wingtip, and the calculated deformations agree well with the experimental data, which shows the numerical simulation accuracy concerning the flow field and the interaction with the elastic structure.

3. Calculation Model and Result Analysis

3.1. Calculation Model

The research object of this paper is shown in Figure 9, which is the engine nozzle protruding from the bottom of the plane symmetry aircraft and the embedded body flap under the nozzle at the bottom. The afterbody of the aircraft is provided to limit the interference zone of the body flap and the nozzle without the whole aircraft. The main boundary conditions include the pressure outlet with the parameter in Table 1, the afterbody defined as the adiabatic wall, the plenum chamber of the nozzle defined as the pressure inlet with the parameter in

Table 2. Parameters of plenum chamber of the nozzle.

Parameter Name		Values
Plenum chamber	Total temperature	3800 K
	Total pressure	1.2 × 107 Pa
	Expansion ratio	28.0

, the nozzle defined as the adiabatic wall, and the body flap defined as the isothermal wall with an initial temperature of 300 K.

The structural configuration and the computational mesh are shown in Figure 9. Three different mesh densities for mesh convergence are provided in

Table 3. Mesh statistics of different refinements.

Mesh Refinement	Mesh Number
coarse	2.66 million
medium	4.85 million
fine	7.23 million

. The body flap adopts a solid configuration with the material of C/Sic, and the material parameters are shown in

Table 4. Structural material parameters.

Parameter Name	Values
Density	1900 kg/m3
Thermal conductivity	10.0 w/m/C
Specific heat capacity	1200 J/kg/C
Young’s modulus	change with temperature
Poisson’s ratio	0.3
Coefficient of thermal expansion	1.2 × 10−5

. The impact of the elastic deformation (including the deformation considering the thermal effect) on the aerodynamics characteristic of the body flap caused by the jet interference is studied at different trajectory altitudes with different deflection angles.

According to the inherent characteristics of the research object in this paper, the body flap will be continuously heated by the high-temperature gas of the tail jet during the flight, and accordingly, structural stiffness will be significantly weakened. The variation in Young’s modulus due to the influence of temperature on the material properties, is shown in Figure 10.

3.2. Results and Analysis

The body flap deflection angle is represented by $\delta$ in the following text. The pitching moment coefficient is defined from the perspective of aircraft handling and stability control. The moment reference point is the assumed aircraft centroid (2.4 m, 0, 0), which is 60% relative to the whole length of the assumed aircraft provided in Ref. [22], the reference length is the aircraft bottom diameter, the reference area is the upper surface area of the body flap, and the reference dynamic pressure is uniformly dependent on the incoming flow parameters listed in Table 1. The definition of the deviation angle of the body flap and the positive or negative moment is as follows: the trailing edge of the rudder deviation angle is downward, and the pitching moment is positive when the aircraft raises its head.

3.2.1. Jet Interaction under $\delta =0^{\circ }$ of Body Flap at the Altitude of 30 km

Under atmospheric environment at the altitude of 30 km, the ratio between the total pressure of the plenum chamber and the back pressure is 10,025, while the complete expansion ratio corresponding to the supersonic nozzle configuration is only 634. Thus, there exists a strong expansion wave and jet interaction between the nozzle outlet and the body flap.

Figure 11 shows the pressure contour and the velocity streamline of the symmetrical plane for the rigid model. It can be seen that the shear layer on both sides of the outer edge of the nozzle outlet shows obvious expansion, and its interaction area takes up most of the upper surface of the body flap. Figure 12a displays that a high-pressure area is concentrated in the middle of the trailing edge of the upper surface, and it gradually decreases towards the tip and the root. Furthermore, there is a low-pressure area, less than that of the far field, ranging from the midpoint of the chord to the root. Due to the three-dimensional effect of the nozzle and the installation position of the body flap, the middle of the rear edge of the body flap is subjected to the most significant aerodynamic and aerothermal loading due to the jet interaction. The strong effect of the shear layer in the expansion zone has a significant suction effect on the non-expansion zone between the nozzle and the body flap. In addition, the expanded wake affects the trailing edge of the body flap vertically, and the velocity stagnation forms a local stagnation point. Accordingly, there is a local high-pressure area above the trailing edge, and it forms a reverse pressure gradient. Thus, the area with little jet interaction on the upper surface of the body flap is a low-pressure reflux area.

Due to the aforementioned flow characteristics, coupled with the structural characteristics of the thin tip, the body flap is prone to significant elastic deformation. In addition, because the gas of the nozzle wake is of high temperature, the temperature near the upper surface of the body flap is close to the total temperature of the jet flow, as shown in Figure 12b. The aerothermal load transmitted to the upper surface of the structure is very large. Therefore, it is extremely important to study the influence of elastic deformation, taking into consideration the thermal effect on the aerodynamic characteristics of the body flap.

Comparison data in Figure 13a indicate that as the density of the mesh is refined, the spatial resolution of the solver is enhanced, and the numerical results have good qualitative consistency. In this research, the medium grid is adopted considering the balance between efficiency and accuracy.

The results in Figure 13b indicate that the peak pressure of the elastic and thermoelastic models is significantly lower than that of the rigid model, because the normal effective angle between the upper surface and the shear flow becomes smaller when the structural deformation is taken into consideration. Noticeably, the station in X direction is represented by $\overline{x}$, and the station in Z direction is represented by $\overline{z}$ in the following.

$$\begin{array}{l}\overline{x}=\frac{x-x_{leadingedge}}{L_1},\overline{z}=\frac{z}{L_2}\\ L_1=L_{chord}=x_{trailing edge}-x_{leadingedge}\\ L_2=z_{\max \mathrm{spanwise}}/2\end{array}$$

(21)

Figure 14 shows that the maximum temperature near the middle of the trailing edge can reach about 1600 K at 200 s. Figure 15 shows that the temperature of the monitor point rises due to aerodynamic heating. The convergence of the time step is performed with time intervals equal to 1 s, 2 s and 4 s. It can be seen that the results of the time intervals equal to 1 s and 2 s are very close to each other, while the results of the time interval equal to 4 s is larger during the whole time history. Therefore, 2 s is adopted as the time step in this research, considering the balance between efficiency and accuracy. Moreover, it can be indicated that the characteristic time scale of heat transfer is large. Since the structural temperature increases, the stiffness of the structure will be significantly weakened, and the structural deformation of the thermoelastic model is greater than that of the elastic model, as shown in Figure 16.

Figure 17a shows the torsion angle of the elastic and thermoelastic models at different spanwise stations in the z direction of the body flap. The comparison results indicate that the torsion angle of the thermoelastic model becomes gradually lager than that of the elastic model, from the midpoint of the chord line to the trailing edge. The maximum angle of the thermoelastic model is 7.5°, while the magnitude of the elastic model is only 4.5° at the symmetric plane ($\overline{z}=0$). Moreover, the bending deformation at $\overline{z}=0$ plane is significantly greater than that of the edge position, and the comparison results of the thermoelastic is more obvious, because the body flap is prone to larger deformation when the aerothermal heating effect is considered. In addition, the deformation is concentrated in the normal direction, which occurs because the body flap mainly deformed in the first-order bending mode. According to the deformation compatibility condition, the torsion angle of the thermoelastic model near the root will be smaller than that of the elastic model.

Figure 17b shows the torsion angle at different stations along the chord direction of the elastic and the thermoelastic models. It is shown that the torsion angle of the elastic model changes monotonically at $\overline{x}=1$, with the maximum angle being equal to −0.8°. The difference is that the deformation angle of the thermoelastic model changes non-monotonically, with the maximum angle being equal to 0.6°. The extreme points exist near the 1/4 and 3/4 spanwise station of the body flap, respectively, which demonstrate that its structural deformation characteristics are more complex than that of the elastic model. The reason for the above phenomenon is that the thermal expansion effect is considered in the thermoelastic model, and the thermal stress reduces the normal strain. In addition, the tip of the body flap is less affected by the jet interaction and the significant thermal expansion, which leads to the reverse direction of the torsion angle. The convex deformation characteristics on the structure are unique to the thermoelastic model. Moreover, the elastic effects of the two models are weakened when the location is near the root. Especially at the plane close to the root ($\overline{x}=0.14$), it is worth noting that the thermoelastic effect almost disappears and degenerates into a pure aeroelastic effect.

The aforementioned deformation characteristics of the thermoelastic model are a result of the comprehensive effect of aerodynamic and aerothermal loading. The displacements at different positions represent the competitive relationship between thermal stress and external force load. Therefore, the deformation characteristics of different stations along the flow direction at the equilibrium position are closely related to the area and the intensity of the jet interaction.

The results in

Table 5. Change rate of pitching moment when $\delta =0^{\circ }$ at 30 km.

Structure Model	Pitching Moment Coefficient	Rate of Change
Rigid	−0.651	-
Elastic	−0.504	22.58%
Thermoelastic	−0.461	29.19%

indicate that the change in aerodynamic characteristics caused by the elastic effect cannot be ignored in the flow interaction of this configuration, and the change rate of the pitching moment coefficient of the body flap exceeds 20%. When the thermal effect is taken into consideration, the influence amount is 26%, indicating that the aerothermoelastic problem caused by the jet interaction puts forward strict requirements for the evaluation of the stability and design of the control law.

3.2.2. Jet Interaction under Different Deflection Angles of the Body Flap

Figure 18 indicates that the pitching moment decreases with the increase in $\delta$. It can be seen that the difference in the pitching moment among the three models is the largest than that of the other deflection angles when $\delta$ is equal to 0°. With the increase in $\delta$, the difference in the three models decreases, and the elastic effect is very small when $\delta$ is equal to 15°. It can be predicted that when $\delta$ exceeds 15°, the effect of the jet can be ignored for the change in the aerodynamic characteristics of the body flap. Figure 19 shows the change in the peak pressure on the body flap of the three models with various $\delta$. It can be seen that the jet interaction on the body flap gradually decreases with the increase in $\delta$. This is also illustrated by the nonlinear variation in the pitching moment coefficient.

From Figure 20, it can be seen that the area of jet interaction is significantly reduced compared with $\delta$ = 0°, and the separation position of the rigid upper surface is slightly different from the that of the thermoelastic model, which is because the expansion angle of the jet flow is close to that of $\delta$ = 15°.

As shown in Figure 21, the temperature when $\delta ={15}^{\circ }$ is much lower than that of $\delta =0^{\circ }$ at 200 s, and the maximum temperature is only about 600 k. The area affected by the high-temperature gas is significantly reduced. Accordingly, the pitching moments of the three models are almost the same, and the thermoelastic effect is not significant.

Figure 22 shows that as $\delta$ increases, the torsion angle of the elastic and thermoelastic models at the symmetric plane gradually decreases. When $\delta ={15}^{\circ }$, the difference in the maximum torsion angle between the two models is 1°, indicating that the aerothermoelastic effect is significantly weakened.

Figure 23 shows that the extreme points of the thermoelastic model at 1/4 and 3/4 of the spanwise stations nearly disappear when $\delta ={15}^{\circ }$ at $\overline{x}=1$, while the thermal expansion effect at the tip of the body flap still exists, and the characteristic curve of the thermoelastic deformation is still maintained. The above phenomena show that the aerodynamic and aerothermal loading of the jet interaction significantly decrease with the increase in the deflection angle of the body flap. If the heating time is long enough, the internal temperature distribution of the structure will gradually approach the temperature of the gas close to the wall surface. Thus, the body flap will still show the deformation characteristics in the thermoelastic model.

In general, the aerothermoelastic effect still exists at the altitude of 30 km up to $\delta ={15}^{\circ }$, and its impact extent is mainly related to the relative position of the body flap and the nozzle exit.

3.2.3. Jet Interaction under Different $\delta$ of Body Flap at Different Altitudes

At the trajectory altitudes of 10 km, 20 km, 30 km and 40 km, the pressure ratios of the nozzle and the far field are 453, 2170, 10,025 and 41,793, respectively, and the expansion angle shows significant change. As shown in Figure 24, the interaction between the jet flow and the body flap is the strongest at the altitude of 40 km, while the jet hardly acts on the body directly at the altitude of 10 km, which makes the interaction area limited. This paper focuses on the study of aerodynamic interaction and elastic deformation on the body flap due to jet interaction at the altitudes of 20 km, 30 km and 40 km.

As shown in Figure 25, the point where pressure begins to increase on the upper surface of the body flap move forward as the altitude increases, indicating that the jet interaction area increases. In addition, the low-pressure area on the upper surface compared with the far-field pressure disappears with the increase in altitude. This phenomenon occurs because the jet acting on the structure with a larger expansion angle leads to a reduction in the suction effect on the non-expansion zone. The peak pressures at the three altitudes are almost equal.

Figure 26 shows the heat flux on the upper surface near the trailing edge of 30 km and 40 km at the initial time, which is of no significant difference. The temperature distribution of the structure has little difference after 200 s of aerodynamic heating. Therefore, the difference in the structural deformation is mainly determined by the pressure load between the upper and lower surfaces of the body flap.

The pitching moment coefficients of different $\delta$ and altitudes are shown in Figure 27. There is little difference in the expansion angle and the jet interaction area between the results of 30 km and 40 km at $\delta =0^{\circ }$, which is because the pressure ratios of the nozzle and the far field are large enough for this relative location of the nozzle and the body flap at these two heights. Since the pressure integral on the upper surface is almost the same, the difference in the pitching moment coefficient mainly stems from the far-field pressure integral on the lower surface. At 20 km, the area on the upper surface influenced by the jet interaction is much smaller, while the pressure integral on the lower surface is larger, making the pitching moment coefficient very different from that of 30 km. The relationship between pitch force rejection and altitude in Figure 27b indicates that as the altitude decreases, the absolute value of pitching moment difference between different models gradually decreases. Overall, at high altitudes (30 km and 40 km), the control surface efficiency is significantly higher under aerodynamic interference than at lower altitudes (20 km).

The influence of the elastic effect on the pitching moment is shown in

Table 6. The relative value of the influence of the elastic effect on the pitching moment.

Deflection Angle (∘)	20 km		30 km		40 km
0	elastic	thermoelastic	elastic	thermoelastic	elastic	thermoelastic
	25.38%	31.83%	22.52%	29.20%	21.36%	27.30%
5	17.29%	20.85%	17.45%	22.64%	15.88%	21.77%
10	−21.68%	−18.22%	10.90%	16.68%	11.25%	16.56%
15	−6.19%	−5.78%	15.52%	18.23%	4.05%	8.58%

. The control surface efficiency (CSE) of the body flap for different models is provided in

Table 7. CSE of different calculation models.

Deflection Angle (∘)	20 km			30 km			40 km
	Rigid	Elastic	Thermoelastic	Rigid	Elastic	Thermoelastic	Rigid	Elastic	Thermoelastic
5	4.76%	3.41%	3.05%	5.72%	4.06%	3.57%	5.66%	3.96%	3.62%
10	2.08%	1.71%	1.66%	3.77%	2.88%	2.70%	3.67%	2.84%	2.59%
15	0.51%	0.53%	0.53%	2.08%	1.92%	1.75%	2.54%	2.05%	1.89%

, which is defined as follows:

$$valu{e}_{CSE}=\frac{△Cm}{\Delta \delta }\times 100\%$$

(22)

In general, the CSE of the body flap will be reduced for both the thermoelastic and elastic models at 30 km and 40 km, especially for the thermoelastic model. Moreover, the trend of the relative value of the influence of the elastic effect on the pitching moment is complex, and the influence of the pitching moment and the elastic effect of the rigid body model changes nonlinearly at the same time at different altitudes and different $\delta$.

As shown in Figure 28 and Figure 29, the effective angle between the body flap and the expansion wave decreases at the altitude of 20 km with $\delta ={10}^{\circ }$, and the aerodynamic and aerothermal loads acting directly on the upper surface are very limited. Since the suction effect still exists, the pressure on most areas of the upper surface is smaller than that of the bottom surface. Thus, the reverse sign of the pitching moment coefficient shown in Table 4 and Table 5 appears. When $\delta$ increases to 15°, the pitching moment coefficient continues to increase synchronously, and the aerothermoelastic effect almost degenerates into a pure aeroelastic problem. Although the calculation results in this paper do not consider the incoming flow conditions, the conclusion that the pitching moment coefficient begins to show an inverse sign at a certain $\delta$ has a reference value. In the medium and low altitudes, the aerodynamic load under the body flap will be larger because of the increase in the inflow dynamic pressure and the back pressure. It can be inferred that the pitching moment coefficient will show an inverse sign at a smaller $\delta$.

Table 8. Equivalent deflection angles at different trajectory altitudes.

Δ (∘)	Model	${\mathit{\delta }}_{\mathit{A}\mathit{T}\mathit{E}}$(°)
		20 km	30 km	40 km
0	elastic	1.73	2.56	2.78
	thermoelastic	2.16	3.32	3.55
5	elastic	0.71	1.69	1.96
	thermoelastic	0.86	2.19	2.69
10	elastic	−0.79	0.93	1.19
	thermoelastic	−0.66	1.42	1.76

presents the equivalent deflection angle under different trajectory altitudes. The calculation formula is shown in Equation (23), where ${\delta }_{ATE}^i$ is the equivalent rudder deflection angle caused by the aerothermoelastic effect; $C{m}_{ATE}^i$ is the pitching moment coefficient of the thermoelastic model; $C{m}_{R\mathit{G}}^i$ is the pitching moment coefficient of the rigid model.

$${\delta }_{ATE}^i=\frac{C{m}_{ATE}^i-C{m}_{RG}^i}{(C{m}_{RG}^{i+1}-C{m}_{RG}^i)/5}, i=0,1,2$$

(23)

Results from Table 8 indicate that ${\delta }_{ATE}^i$ caused by the aerothermoelastic effect is the largest at the altitude of 40 km when $\delta =0^{\circ }$, which is 3.55°. And the difference in ${\delta }_{ATE}^i$ of the two models turns to be smaller as $\delta$ increases. Noticeably, it is the reverse sign of the equivalent deflection angle that makes the inverse sign of the pitching moment coefficient at the altitude of 20 km when $\delta ={15}^{\circ }$.

Figure 30, Figure 31, Figure 32 and Figure 33 show the displacement of the elastic and thermoelastic models in the y direction at different stations in the chord direction at different altitudes and $\delta$. With the increase in $\delta$, the downward bending displacement gradually decreases at the altitudes of 30 km and 40 km. When $\delta$ is less than 10°, the downward bending degree decreases with the increase in $\delta$ at the altitude of 20 km. When $\delta$ increases to 15°, the displacement is reversed.

At the altitudes of 30 km and 40 km, the maximum deformation of the body flap with different $\delta$ at the station $\overline{x}=1$ indicates that the displacement increment of the thermoelastic model is significantly greater than that of the elastic model. As the station moves towards the root, the thermoelastic effect weakens, and the displacement decreases rapidly and nonlinearly with the increase in $\delta$. When $\delta$ increases to 15°, the displacement increment of the two models at different stations is almost the same at the altitude of 20 km. The results indicate that the aerothermoelastic problem of the body flap turns into be a pure aeroelastic problem under this deflection angle. In addition, the deformation characteristics near the trailing edge ($\overline{x}=1$) of the body flap is a result of the competitive relationship between the thermal stress caused by the thermal load and the jet aerodynamic load. The area near the symmetric plane is the main action area of the jet, the external aerodynamic load is much greater than the thermal stress, and the deformation shows contraction towards the interior of the structure. When the location is near the spanwise edge ($\overline{z}=\pm 1$), the jet interaction force is small, which is realized as the thermal expansion. As the deformation is dominated by the aerodynamic load, the main deformation form is the bending mode, and there will be two extreme points of the torsion angle as described above.

As the station moves towards the root, the jet interaction rapidly decreases, and the deformation of the body flap is characterized by the thermal expansion, while the displacement increment is determined by the maximum displacement of the trailing edge, which is still downward bending. With the increase in $\delta$, the jet interaction force rapidly decreases. Compared with the smaller $\delta$, the thermal expansion effect increases at the trailing edge of the symmetric plane.

4. Conclusions

In this paper, three configurations of the rigid body model, the elastic model and the thermoelastic model are used to investigate the coupling feature with the flow field based on the CFD/CTD/CSD coupling method. The effects of jet interaction on the deformation and aerodynamic characteristics of the body flap at different deflection angles are systematically studied at different trajectory altitudes. The following conclusions are obtained:

(1) Since the pressure ratio of the high altitude is much larger than that of the middle and low altitudes, the jet interaction, as well as the thermal load and aerodynamic load, acts on the body flap in a larger area, which makes the bending deformation of the body flap more significant in the elastic model at 30 km and 40 km.

(2) As the deflection angle of the body flap increases at high altitudes, the pitching moment coefficient of the three models decreases nonlinearly, and the aerothermoelastic effect significantly decreases. Nevertheless, it is very likely to degenerate into a pure aeroelastic problem in the case of low altitudes at a large deflection angle. Furthermore, it can be judged that there must be a turning point of the deflection angle that is causing the rise of the body flap at this altitude.

(3) The deformation characteristics of the body flap section along the flow direction are very complex when the thermal effect is taken into consideration, which is the result of the combined result of the aerodynamic and thermal load. There are significant differences in the deformation characteristics of the object surface at different stations, which is caused by the different extent of jet interference in different areas of the upper surface of the body flap.