0 Introduction

The reentry vehicle with a large angle of attack and hypersonic speed flies in the chemical non-equilibrium thermal environment for a long time. The thermal environment has four distinctive characteristics, including high enthalpy, low pressure, low heat flux and chemical non-equilibrium flow, which provide favorable conditions for transiting the vehicle's thermal protection system from ablative to non-ablative type. The surface catalytic and anti-oxidation characteristics of non-ablative thermal protection materials reduce the surface aerodynamic heat load and control the appearance of the vehicle^[1-3]. Therefore, the objective and effective evaluations of catalytic and anti-oxidation properties are of utmost important in the selection of thermal protection materials. Moreover, the optimization design of thermal protection systems are based on these criterions.

At present, the rule of Three-Parameter-Simulation and the method of Stagnation-point-Simulation are generally adopted to investigate the performance of thermal protection materials. The Three-Parameter-Simulation, consisting of total enthalpy at the outer edge of boundary layer, wall pressure and heat flux of stagnation-point, suits with the heat transfer in the chemical equilibrium boundary layer^[4-13]. In order to apply Three-Parameter-Simulation to simulate the heat transfer in the chemical frozen or non-equilibrium boundary layer (the flow has reached the chemical equilibrium state at the outer edge of boundary layer), the conception of "local heat transfer simulation" (LHTS^[14]) is presented. The new three-parameters are corrected as total enthalpy, velocity gradient at the outer edge of boundary layer, and wall pressure on the stagnation-point.

However, if the chemical state is non-equilibrium at the outer edge of boundary layer, the Three-Parameter-Simulation is impossible to obtain the material's effective service performance from the ground simulation test. Hence, in order to improve the service performance of non-ablative thermal protection materials, the fourth parameter simulation on the chemical non-equilibrium degree of the outer edge of boundary layer should be introduced into the aforementioned Three-Parameter-Simulation.

The current study analyzes the main flow field parameters and the effect of the stagnation-point heat flux of hypersonic blunt body on the basis of Fay-Riddell formula^[15]and Goulard formula^[16]. Subsequently, the paper introduces the simulated environmental characteristics provided by ground high-enthalpy simulation facility and highlights main differences from real flight environment. By combining the differences between simulation and flight tests, the validity of Three- and Four-Parameter-Simulations are theoretically analyzed. Moreover, the validity of simulation parameters selection has been demonstrated by a numerical simulation method. If the ratio of dissociation and total enthalpy at the outer edge of boundary layer cannot be simulated in the high enthalpy simulation facility, the stagnation-point heat flux of spacecraft cannot be simulated by the simulation test.

1 Analysis of the influence parameters of the stagnation-point heat flux for hypersonic vehicle 1.1 Influence parameters of the stagnation-point heat flux

For an axisymmetric blunt body, its stagnation-point heat flux can be determined by using different formulae, according to the different chemical states, within the boundary layer of the flow around the body. The stagnation-point heat flux within an equilibrium boundary layer can be determined by using Fay-Riddell formula (T_w≤2000K and the wall dissociation enthalpy h_Dw=0):

where C_Hs is the heat transfer coefficient of the stagnation point, h_te is the total enthalpy at the outer edge of boundary layer, h_fw=c_pfT_w is the wall frozen enthalpy (T_w and c_pf are the wall temperature and the equivalent specific heat capacity, respectively.), h_De=∑h_i⁰C_ie is dissociation enthalpy at the outer edge of boundary layer (C_ie and h_i⁰ are the concentration at the outer edge of boundary layer and the formation enthalpy of i species, respectively.), Le_f is the frozen Lewis number.

The stagnation-point heat flux of the frozen boundary layer is related to the rate constant of vehicle's surface catalytic reaction, k_w, which can be determined by using Goulard formula:

The C_Hs is,

where the frozen Prandtl number, Pr_f, is the velocity gradient at the stagnation point, (du_e/dx)_s is the velocity gradient at the outer edge of boundary layer, ρ_e and μ_e are the mix gas density and the viscosity coefficient at the outer edge of boundary layer, respectively.

According to the Fay-Riddell formula, the K value equals to 0.763, whereas the Goulard formula results in a K value of 0.664. The Damköhler number (Dam_w) of the surface catalysis reaction can be defined as:

For the approximation of Le_f=Pr_f=1 and h_teh_fw, the formulae for stagnation-point heat flux within equilibrium and frozen boundary layers can be approximated as:

where α is the ratio of dissociation to total enthalpy at the outer edge of boundary layer.

Equation (2) indicates that C_Hs is controlled by (du_e/dx)_s, ρ_e and μ_e. Because both of the mix gas density and viscosity coefficient are functions of enthalpy, pressure and species concentration (ρ=f_ρ(p, h, C_i) and μ=f_μ(p, h, C_i)). At the chemical equilibrium outer edge of boundary layer, ρ_e and μ_e are the function of p_e, h_te. For the chemical non-equilibrium at the outer edge of boundary layer, the functions are ρ_e=f_ρ(p_e, h_te, C_ie) and μ_e=f_μ(p_e, h_te, C_ie), where p_e is pressure at the outer edge of boundary layer, which equals to the stagnation pressure, p_s.

Hence, if the outer edge of the chemidcal frozen boundary layer is chemical equilibrium, the stagnation-point heat flux is influenced by three field parameters, h_te, p_s and (du_e/dx)_s. If the outer edge of boundary layer is chemical non-equilibrium, the stagnation-point heat flux is also influenced by a fourth field parameter, α.

1.2 Coupling effect between dissociation enthalpy and catalytic reaction rate on the stagnation-point heat flux

Within the stagnation-point domain, the dissociation enthalpy is controlled by the vehicle's head radius and flight orbit. In addition, the flow characteristic time (τ_f) around the body is proportional to the vehicle's nosetip radius (R_n), given by the following expression:

As a result, under the specific flight height and velocity, τ_f decrease with R_n. However, τ_f of the ionization and dissociation reactions remain constant after normal shock. Therefore, the h_De exhibits a direct relationship with R_n. For convenience, the stagnation-point heat transfer theory for frozen boundary layer is used to analyze the coupling effect of dissociation enthalpy and the rate constant of surface catalytic reaction on the stagnation-point heat flux.

Because the maximum of α equals to 1, while Dam_w equals to 100, according to Goulard formula (equation (4b)), and the error is less than 1%. When Dam_w=0 (k_w=0), its heat flux on the non-catalytic surface can be defined as , whereas when Dam_w=100, the dissociation energy entirely releases on the material's surface and for the fully-catalytic surface, the heat flux can be defined as . According to equation (4b):

For a partially-catalytic surface, 0 < Dam_w < 100, the relationship between the heat flux of partially-and the fully-catalytic surface can be given as follows:

According to the above analysis, the stagnation-point heat flux of the limited catalytic surface is directly influenced by α and the catalytic reaction Damköhler number of the material surface, Dam_w. Fig. 1 presents the variation of stagnation-point dimensionless heat flux with α and Dam_w.

For the fully-catalytic surface, the stagnation-point heat flux is not influenced by dissociation enthalpy. However, for the partially-catalytic surface, the stagnation-point heat flux shows linear relationship with dissociation enthalpy. In conclusion, the stagnation-point heat flux decreases with the increasing of dissociation enthalpy, which implies that the surface catalytic characteristics of non-ablative material have a significant impact on the stagnation-point heat flux. Therefore, it is indispensable to consider the dissociation enthalpy in the ground simulation test study for the service performance evaluation of these materials.

In summary, the stagnation-point heat flux affected by the principal flow field factors include h_te, p_s, (du_e/dx)_s and α. The first three parameters affect the stagnation-point heat flux within the chemical frozen and non-equilibrium boundary layer whose outer edge is chemical equilibrium. If the outer edge of boundary layer is chemical non-equilibrium, the stagnation-point heat flux is directly influenced by α.

2 Difference between ground simulation and flight environment

The arc-heated wind tunnel (AHWT) and induction-heated wind tunnel (IHWT) are main facilities to perform ground simulation research to evaluate the performance of thermal protection materials. Such kind of facility utilizes arc or induction heating to heat the in-chamber gas to the pre-set enthalpy value. Then, the in-chamber high-temperature gas gets dissociated and ionized and air dissociation enthalpy accounts for 10%~70% of the total enthalpy (see Fig. 2), with the enthalpy range of 4~32 MJ/kg.

When the in-chamber high-enthalpy air passes through the supersonic nozzle, a complex three-body collision reaction between the dissociated atoms occurs in the nozzle contraction section, which gradually reduces the dissociation enthalpy in the fluid. With the increase of flow rate, the frozen point of chemical flow state occurs at some positions in the nozzle's expansion section (nozzle with a higher Mach number), resulting in the dissociation enthalpy of nozzle jet is higher than the equilibrium state.

In order to quantitatively analyze the flow characteristics in the supersonic nozzles, the flow characteristics of the supersonic nozzles in the arc-heated and induction-heated wind tunnels are calculated, respectively. Table 1 presents the parameters of the in-chamber gas and the geometric parameters of the nozzle.

Along the axis of the supersonic nozzle, the distribution of C_O and C_N in the chemical equilibrium and the non-equilibrium are shown in Fig. 3(a). Besides, the distribution of C_O and C_N in the chemical equilibrium and the non-equilibrium states, the distribution of temperature (T) and velocity (U) of jet flow, and the distribution of α and h_t along the cross-section of the nozzle are shown in Fig. 3(b), (c) and (d), respectively. The numerical simulation results exhibit that under the supersonic conditions, the jet provided by the ground simulation equipment exhibits a severe chemical non-equilibrium state.

Therefore, the ground simulation environment is able to provide the same total enthalpy, stagnation-point pressure and velocity gradation as those in the actual flight environment. The main difference between simulated flow field and flight environment is the dissociation enthalpy (because the dissociation enthalpy is zero before the shock wave in the real flight environment).

3 Selection of ground simulation parameters

To study the service performance of non-ablative thermal protection materials in a chemical non-equilibrium environment, the FPS should be applied in the ground test simulation. The Four-Parameter-Simulation includes stagnation pressure, stagnation-point velocity gradient, total enthalpy and dissociation enthalpy, and the ground test results can reproduce the aerodynamic heat load at the stagnation-point region under the flight conditions.

When the flight environment parameters (remarked by superscript F) and simulation environment parameters (remarked by superscript S) are determined, the crucial step is to determine the effective size of the test piece. The following relationships between the total enthalpy, stagnation-point pressure, stagnation-point velocity gradients, flight velocity (U_∞^F), free flow density (ρ_∞^F) and pressure (p_∞^F) and effective spacecraft nosetip radius (R_n^F) under the flight conditions should be used for this purpose.

Total enthalpy:

Stagnation-point pressure:

Stagnation-point velocity gradient:

where ε refers to the density ratio of gas "before" and "after" the shock wave, ε=ρ_∞/ρ_e. The value has the following relationship with the Mach number Ma_∞, and the ratio of specific heat capacities of the incoming flow γ_∞.

In a ground simulation environment, the jet energy is consisting of gas kinetic energy, internal energy and dissociation energy. The total enthalpy of gas flow can be determined by the following equation:

The jet stagnation-point pressure can be determined from the given expression:

The velocity gradient, within a stagnation-point domain, can be determined according to the following relationship:

According to the relationships of h_te^F=h_te^S, p_s^F=p_s^S and (du_e^F/dx)_s=(du_e^S/dx)_s, the effective dimension in the ground simulation environment can be given as following:

3.1 Validity of simulation parameters

To simulate the performance of material with a fully-catalytic surface, the Three-Parameter-Simulation (consisting of stagnation-point pressure, stagnation-point velocity gradient and total enthalpy) can provide the required data. However, for the material with partially-catalytic surface, dissociation enthalpy should be also considered.

In order to verify the validity of parameter selection, the stagnation-point heat flux of vehicle under specific flight condition is compared with the stagnation-point heat flux of test piece under the corresponding ground simulation conditions. With the help of CFD, the selected flight conditions are shown in Table 2.

According to the flow field parameters of AHWT and formulae of Three-Parameter-Simulation, the dimensions and shapes of test samples are determined and listed in Table 3, R_n^S=26.65mm for hemisphere-headed cylinder (corresponding to R_n^F=35mm) and R_n^S=35mm for flat-head cylinder (corresponding to R_n^F=200mm) and the surface temperature is T_w= 350K.

The simulation results of stagnation-point conduction heat flux (), diffusion heat flux (), and pressure are shown in Table 4~5, respectively. The simulation results indicates that, for the fully-catalytic surface, the stagnation-point heat flux is consistent under the two conditions, with a maximum deviation of heat flux (between simulation and flight environments) less than 2.01% (showned in Table 6).However, for the thoroughly non-catalytic surface, the stagnation-point heat flux is consistent for the large blunt body, but shows 30% deviation for the small blunt body (see Table 6). For the big blunt body (R_n^F=200mm), the temperature and atomic concentration distributions on the stagnation-point line near the surface are both consistent (see Fig. 4 and Table 6). This result indicats that, in flight and simulated environments, the conduction and diffusion heat flux of the stagnation-point are consistent, respectively. For the small blunt body (R_n^F=35mm), despiting the consistent conduction heat flux, the atomic concentration distribution has a larger deviation, which results in a larger difference in dissociation enthalpy (see Fig. 6) and diffusion heat flux.

3.2 Selection of simulation parameters and validity analysis for partially-catalytic surface

According to equation 6(a) and (b), the α can be determined by the non-catalytic and fully-catalytic stagnation-point heat flux:

According to equation (7) and (16):

The dimensionless stagnation-point heat flux equation is only determined by the Damköhler number of surface catalytic reaction and independent on α. Fig. 6 demonstrates the variation of with Dam_w.

According to the condition of Three-Parameter-Simulation, there are Dam_w^F=Dam_w^S and . Using equation (7) and (16), the following equation can be obtained:

It indicates that the value of is determined by Dam_w^F, α^F and α^S. If α^S=α^F(h_De^S/h_te=h_De^F/h_te), the is equal to and the stagnation-point heat flux can be simulated. Fig. 7 demonstrates the variation of with Dam_w, α^S=0.4 and θ=α^F/α^S.

According to equation (18), in the chemical frozen boundary layer, only under the condition of α^S=α^F, the simulation performed by Three-Parameter-Simulation method is valid. Therefore, for the stagnation-point heat transfer in the chemical frozen boundary layer, a reasonable simulation method should consist of total enthalpy, pressure, velocity gradient and the ratio of dissociation enthalpy to total enthalpy at the outer edge of boundary layer, that is the Four-Parameter-Simulation. In general, α^F < α^S results in that the value of stagnation-point heat flux produced by ground simulation is smaller than that in the real flight environment.

4 Conclusions

(1) When the outer edge of boundary layer is in the chemical equilibrium state, the Three-Parameter-Simulation is valid in both the ground simulation and real flight environment.

(2) When the outer edge of boundary layer is in the non-equilibrium state, the Three-Parameter-Simulation will result in heat flux reduction in both the ground simulation and real flight environment, but the Four-Parameter-Simulation adopted in the test can compensate this problem.

(3) While the dissociation enthalpy cannot be simulated, the only effective way to solve the problem of not being able to simulate the dissociation enthalpy is to determine the difference between the simulated stagnation point heat flux of partially-catalytic surface and the stagnation point heat flux in flight. The simulation results are then extrapolated to the flight environment via mutual verification between the ground simulation and the flight test.