Effect of gas blowing on the head of thermal ablation vehicle

The reentry vehicle will encounter thermal ablation, especially at the stagnation point regime. A theoretical work has been done to analyze the thermal effect of gas blowing due to thermal ablation of surface material on the head of a general hypersonic vehicle. By deriving the formulation, research takes into account the effect of gas blowing on the thermal dynamics balance, and then solves them by numerical discretization. It is found that gas blowing will increase the temperature and heat flux at the surface of stagnation point regime.

complicated [22,23] to predict. It is worth noting that, the solid material experiences a chemical reaction under high temperature of fluid flow at stagnation point regime [1,[6][7][8]. Therefore, thermal dynamics of single phase stagnation point flow is one of the crucial factors to understand the complicated mass loss process during the thermal ablation.
Research focuses our attention on the near wall of thermal dynamics of the stagnation point regime of a sphere with typical physical properties. According to the dimensionless analysis, the local Mach number in the near wall regime is approximately zero due to the local high sonic speed, so we assume that the stagnation point flow in the very near wall regime of our objective problem is incompressible. If taking incompressible stagnation point flow as a prototype model to study, there are differences. The multi-physics nature of thermal ablation makes the wall inject gas into the stagnation point regime, while the effect of gas blowing on the thermal dynamics is not clear.
Research studies this effect on the thermal dynamics of a compressible stagnation point flow in a typical hypersonic vehicle cruising condition by means of theoretical research, so as to have some qualitative understanding of the thermal ablation problem, paving the way for future study and planning.

Incompressible assumption in the near wall region
Firstly, in order not to lose generality, we consider a prototype of hypersonic incoming flow passing a sphere, with physical properties of air being that of altitude height around 20 ~ 30 km, which is typical parameter ranges where thermal ablation happens, see Fig. 1. By default, all the physical parameters in this paper use the ISO unit. In the current study, the typical incoming flow Mach number is around M 1 = 8 ~ 20, the density of fluid flow is ρ 1 = 1.8 × 10 2 kg/m 3 , the temperature is T 1 = 226.5 K, the pressure is p 1 = 1197.0 Pa, the sound speed is a 1 = 301.7 m/s, the fluid flow velocity is U 1 = 6034.0 m/s, the dynamic viscosity is μ 1 = 1.5 × 10 −5 Pa ⋅ s, and the kinematic viscosity is ν 1 = 8.2 × 10 −4 m 2 /s. The incoming flow forms a detached shock wave over the sphere which can be seen in many hypersonic vehicles; behind the shock wave, the fluid flow is compressed, its entropy increases, its thermal dynamics properties, such as pressure, density and temperature increase significantly as well, and the chemical reaction of air happens due to high temperature [16,17]. Having assumed the gas being in a perfect state, research focuses attention on the stagnation point flow region and considers neither the chemical reaction nor the electric dissociation. Therefore, research calculates the flow and thermal dynamics properties by normal shock relations [14,15], see Eqs. (1)-(6). Let's take M 1 = 20 as an example, the properties behind the normal shock wave can be calculated, with the Mach number being M 2 = 0.4, the density of fluid flow being ρ 2 = 0.1 kg/m 3 , the temperature being T 2 = 1.8 × 10 4 K, the pressure being p 2 = 5.5 × 10 4 Pa, the sound speed being a 2 = 2667.0 m/s, the fluid flow velocity being U 2 = 1006.0 m/s, the dynamic viscosity being μ 2 = 1.9 × 10 4 Pa ⋅ s, and the kinematic viscosity being v 2 = 1.8 × 10 −3 m 2 /s. In addition, it is found that, for M 1 = 8 ~ 20, the physical properties behind the shock wave are in the same order, and the variations are small.
(1) Fig. 1 The schematic of the head of a hypersonic vehicle in a typical incoming flow Secondly, the detached distance of detached shock wave can be estimated by a semiempirical relation [11][12][13], see Eq. (7), which is weakly related to chemical reactions [13,24,25]. D is the typical diameter of blunt nose hypersonic vehicle, with D = 0.2m, δ h being the distance of detached shock wave, and δ h = 13.8e − 3m estimated by Eq. (7), see (4)

Traditional incompressible dynamic and thermal dynamic equations
In the following, theoretical analysis is conducted in the Cartesian coordinate system, where coordinate (x, y) represents (R, Z), and velocity [u, v] T represents [u r , u z ] T in Fig. 1, respectively.
Keep in mind that the local Mach number is around zero M τ ~ 0, allowing us to use incompressible Hiemenz flow to analyze the thermal dynamics of objective problem.
In order to keep generality, research takes full 2D N-S equations to begin, see Eqs. (8)- (9). In order to better understand the problem, research firstly examines the standard Hiemenz solution without thermal dynamics equation.

Theoretical derivation of effect of gas blowing and suction on thermal dynamics
For the case of blowing gas from bottom wall at stagnation point regime, the governing equations are Eqs. (15)-(18): In this section, thermal dynamics balance is controlled by heat conduction and fluid convection. The gas blowing effect will alter the dynamics of stagnation point flow, so as to indirectly influence the thermal dynamics by convection effect. Eqs. (15)- (18) 21) and (23) represent the dynamics with gas blowing [31], while Eqs. (19)- (22) represent the dynamics with gas suction [31].
Set f(0) = 0, f ' (0) = 0, f '' (0) = 1.2326 to satisfy the no-slip and no-penetration wall boundary condition for standard, blowing and suction cases, the gas injection effect can be represented by dimensionless mass flux f w . Set θ(0) = 0, θ ' (0) = 0.0 for thermal dynamics boundary condition at wall. Then integrate numerically by the Runge-Kutta 4th order method, the ODE system Eq. (17) and Eq. (18) for blowing case; integrate the ODE system Eq. (21) and Eq. (22) for suction case. Figure 3 shows that, the effect of dimensionless mass flux f w on the wall normal velocity is to increase or decrease the wall normal velocity. Keep in mind that, thermal dynamics balance is controlled by heat conduction k ∂ 2 T ∂y 2 and fluid flow convection ρC p v ∂T ∂y , and the variation of wall normal velocity profile v(η) will surely alter the thermal dynamics balance. Figures 4 and 5 show the effect of dimensionless mass flux f w on the dimensionless temperature θ and dimensionless heat flux θ' of stagnation point flow. It is clear that, the blowing increases significantly the temperature and heat flux, while suction does the contrast. This means that: in reality, thermal ablation-induced gas injection will deteriorate the heat environment of hypersonic vehicle. In terms of the effect of suction, the current conclusion is purely academic. In reality, if the very hot fluid flow is sucked into the vehicle body, the heat will be accumulated, which means it is impossible to use suction to actively control the heat over the vehicle surface, although the current study suggests that suction will diminish the surface temperature and heat flux. Nevertheless,   it is valuable for basic research, which provides many options and basic knowledge for the final design.

Conclusion
Research conducted theoretical analysis on the thermal dynamics of stagnation point flow regime of a general hypersonic vehicle. The effect of gas blowing due to thermal ablation on the thermal dynamics has been studied by theoretical derivation of new ODE.
In terms of effect of gas blowing, increment of mass flux of gas blowing will deteriorate the heat environment of stagnation point flow, because the normal fluid convective effect on the heat is augmented by the wall normal mass flux. However, the current conclusion is based on the assumption: the gas injection temperature is by default equal to the wall temperature, and when it comes to the application, practical caution is needed. Similarly, the suction will alleviate the heat flux, but in practice, when such extreme high temperature gas is sucked into the vehicle head, the thermal ablation resist material reacts, and the material mechanical properties are deteriorated. Therefore, the current research results serve as a qualitative direction for basic understanding of thermal dynamics of hypersonic stagnation point flow. The detached shock wave distance from the wall δ h , and its relation with δ in terms of order are: δ h ≈ 120 × 10 −4 m, δ ≈ 2 × 10 −4 m, δ h /δ = 60.

Explanation of incompressible approximation
According to the theoretical solution of Hiemenz flow, far from the bottom wall, larger than δ, fluid parcel vertical velocity can be approximately estimated as inviscid strain In the stagnation point regime, the temperature can be estimated: T δ > 10 4 K, a δ > 2 × 10 3 m/s, V δ ≈ 16 m/s and the local sonic speed is around a δ ≈ 2000 m/s, thus M δ = V δ a δ . According to the definition, in this respect, the blue interest domain can be treated as incompressible. In detail, starting from complete compressible fluid dynamic equations, and introducing with prior knowledge M δ ≈ 0.01, the incompressible approximation can be proved.
Firstly, some important dimensional and dimensionless parameters will be defined as follows: ∞ represents the reference physical variable, and it can be the value at infinite far field, or certain specific regime that we are interested in. For example, for wall-bounded turbulence, the reference velocity is U ∞ = U τ , while for homogeneous turbulence, it is U ∞ = U λ .
Reference mass flux: Reference momentum flux: Reference energy flux: Three basic control equations: mass conservation: momentum conservation: energy conservation: 5th term: pressure gradient work − → u * · ∇ * p * and pressure work on the compressible fluid parcel p * ∇ * · − → u * 6th term: heat conduction 7th term: viscous work on the fluid parcel dimensionless energy conservation: dimensionless energy conservation·γ(γ-1)Ma 2 : Thus, it is clear that, when γ(γ − 1)Ma 2 becomes vanishingly small, certain terms can be neglected, which are the kinetic energy of fluid parcel, the pressure gradient and pressure work on the fluid parcel, and the viscous work on the fluid parcel.