# Assessment of Radiative Heating for Hypersonic Earth Reentry Using Nongray Step Models

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

^{2}and show a consistent tendency with the engineering estimation. The Planck-mean absorption coefficient is calculated to show the radiative transfer significantly occurs in the shock layer. By performing the steady simulation at each flight trajectory point, the present algorithm using a nongray step model with moderate efficiency and reasonable accuracy is promising to solve the real-time problem in engineering for predicting both convective and radiative heating to the atmospheric reentry vehicle in the future.

## 1. Introduction

## 2. Physical Models and Numerical Methods

#### 2.1. Flow Governing Equations with Thermochemical Nonequilibrium Models

_{tr}, and the vibrational, electronic, and electron energies are uniformly described by one vibrational-electronic temperature, T

_{ve}[42]. In this manner, the mass, momentum, and energy conservation equations of the hypersonic nonequilibrium flow can be expressed as follows [3]:

_{i}is the ordinate variable in the i direction, N

_{s}is the total number of the air species, ρ

_{s}and ρ are the species density and total density, u

_{i}is the flow velocity component in the i direction, p is the pressure, τ

_{ij}is the viscous stress tensor, e and e

_{ve}are the total energy and vibrational-electronic energy, h is the total enthalpy, q

_{j}and q

_{ve,j}are the total heat flux and vibrational-electronic heat flux, h

_{s}and h

_{ve,s}are the enthalpy and vibrational-electronic enthalpy of the species s, J

_{s,j}is the mass diffusion flux of the species s in the j direction, ω

_{s}is the mass production rate of the species s per unit volume, ω

_{ve}is the vibrational-electronic energy source term, and ω

_{r}is the radiative source term. The state equation of the air follows:

_{r}is the total number of chemical reactions, M

_{s}is the molecular mass per mole of species s, ν

^{f}

_{r}

_{,s}and ν

^{b}

_{r}

_{,s}are the forward and backward reaction stoichiometric coefficients of species s of the r-th reaction, k

_{f,r}and k

_{b,r}are the forward and backward reaction rate coefficients of the r-th reaction, respectively. For each chemical reaction, the forward reaction rate coefficient is calculated using the Arrhenius formula [46], while the backward reaction rate coefficient is obtained from the corresponding forward reaction rate coefficient divided by the equilibrium constant, which is computed using the temperature fitting expression [44].

_{ve}is modeled in the expression proposed by Gnoffo et al. [46], in which the translational-vibrational energy exchange part is calculated by the Landau–Teller model [47], and the relaxation time is calculated by the Millikan–White expression [48] with Park’s high-temperature correction [49]. For the present uncoupled radiation–flowfield simulation, the radiative source term ω

_{r}is neglected.

#### 2.2. Flowfield Solver

#### 2.3. Step Models for Radiation Properties

#### 2.4. Tangent Slab (TS) Approach for RTE

**x**represents the spatial position vector, ν is the radiation frequency, I

_{ν}and I

_{bν}is the spectral radiative intensity and blackbody radiative intensity at frequency ν, respectively, B

_{j}is the j-th component of unit vector

**B**in the transmission direction of I

_{ν}, and κ

_{ν}is the spectral absorption coefficient at

**x**.

_{ν}is the optical thickness at frequency ν perpendicular to the body surface (τ

_{ν}= 0 at body surface), the subscript “δ” and “w” represent the outer edge of the shock layer and the wall, respectively; t is a dummy variable of integration, and ε

_{w}and T

_{w}are the wall emissivity and temperature, respectively. E

_{n}is the integro-exponential function of order n as follows:

#### 2.5. Radiation–Flowfield Uncoupling Algorithm

_{r}in Equation (4) [22]. Assuming a two-dimensional problem is solved, the mesh has a total number of grid nodes of N

_{ξ}× N

_{η}, where N

_{ξ}is the discretized number parallel to the wall and N

_{η}is the discretized number normal to the wall. Thus, the coupling simulation should need about N

_{ξ}× N

_{η}× M × N

_{η}calculations in one radiation iteration, for which the last product factor N

_{η}is due to the numerical integration of radiative heat flux divergence and M is the number of radiation step regions.

_{ξ}× M × N

_{η}calculations for radiation solution in total. Therefore, in one iteration, the number of calculations for coupled radiation simulation is far greater with an order of magnitude of O(N

_{η}) than that of the present uncoupled algorithm. Particularly, the present scheme only needs to compute radiation one time in total, while the coupled simulation has to update the radiation in each step over the whole computational process. Even for the loosely coupled manner, it still requires a considerable computational cost, in which the radiation is updated one time after a certain number of flow iterations. Therefore, the present method is more time-efficient. Additionally, the present algorithm can provide more radiation information including the radiative heating to the whole surface of the reentry vehicle and the absorption properties distributed in the flowfield, but the engineering methods certainly cannot make it, that will be seen later in Section 3.4.

## 3. Results and Discussion

_{∞}and Ma are the flight velocity and Mach number, R

_{N}is the vehicle nose radius, ρ

_{∞}and T

_{∞}are the freestream density and temperature, and T

_{w}is the wall temperature. For each case in Table 2, the time of flow over the vehicle, R

_{N}/V

_{∞}, is of an order of magnitude around 10

^{−4}s, but the time scale of entry down in the altitude is around 0.5 s. The latter is far greater than the former, which means a steady flowfield establishes very quickly. Therefore, we can use the present algorithm to perform steady simulation for each case to fulfill the real-time prediction throughout the Fire II trajectory. Only the Fire II forebody is considered in this paper, the axisymmetric geometry and grids of which are shown in Figure 1. The computational mesh has the dimensions of 153 × 128 (axial × radial) for every case and the spacing of the first grid layer perpendicular to the wall can ensure the cell Reynolds number with an order of magnitude of one in order to predict the reliable aerodynamic heating [58]. The noncatalytic wall condition is used and the wall emissivity is uniformly set to be one.

#### 3.1. Convective Heating

_{N}is the vehicle nose radius; ρ

_{∞}and V

_{∞}are the freestream density and velocity; H

_{w}and H

_{e}are the enthalpies at the wall and the outer edge of the boundary layer, which can be evaluated by using the wall temperature T

_{w}and freestream total temperature T

_{0}, respectively. T

_{0}can be calculated by the freestream temperature T

_{∞}and Mach number Ma. R

_{N}, ρ

_{∞}, V

_{∞}, T

_{w}, T

_{∞}, and Ma are all listed in Table 2.

^{2}and increases continuously up to almost 8 MW/m

^{2}. Figure 3 shows the convective heating over the whole surface of the Fire II forebody for each trajectory point, which presents that the convective heat transfer still maintains a high level of magnitude greater than 1 MW/m

^{2}in the region outside the stagnation point for all cases. Figure 4 further compares the forebody convective heating line at t = 1636 s predicted by the present method with those obtained by DPLR and LAURA [59], the good agreement of which shows the high prediction accuracy of the present PHAROS solver again.

#### 3.2. Thermochemical Nonequilibrium Flowfield

_{tr}and vibrational-electronic temperature T

_{ve}in the Fire II flowfield throughout the trajectory from t = 1634 s to 1645 s. At t = 1634 s, there is a remarkable difference between T

_{tr}and T

_{ve}in a distance closely behind the bow shock, which suggests the translational-rotational and vibrational-electronic energy modes are highly in nonequilibrium in this region. As time goes on with the altitude and velocity down, the thermodynamic nonequilibrium tends to be weakened steadily and the total level of magnitude of temperature also decreases gradually. From t = 1637 s to 1645 s, T

_{tr}and T

_{ve}have become consistent in most areas of the shock layer. For all cases, both T

_{tr}and T

_{ve}exceed 10

^{4}K in the shock layer around the Fire II forebody, and the peak of the translational-rotational temperature even reaches up to 44,000 K at t = 1634 s. Such extreme high temperature directly leads to the harsh aerothermal environment for Earth reentry and results in significant radiative heating [19]. Figure 6 presents the number densities of the species O, N, O

_{2}, N

_{2}, NO, and e

^{−}along the stagnation line for each case. The high atomic and electron concentrations demonstrate the strong dissociation and ionization reactions of air in the reentry shock layer of Fire II. At t = 1634 s, the number density of each species changes remarkably along the stagnation line, while at the following trajectory points, such variations become more and more unnoticeable and the concentration of each species approaches a constant in most regions of the shock layer. Due to the high ionization, the number density of electrons (that is the sum of the number densities of all positive ions, such as O

^{+}, N

^{+}, and NO

^{+}) is greater than 10

^{21}m

^{−3}in most areas of the shock layer. The free electrons and ions constitute a plasma sheath around the reentry vehicle, which absorbs the radio-frequency radiation and causes the communication blackout [60].

#### 3.3. Radiative Heating

_{N}is the vehicle nose radius; ρ

_{∞}and V

_{∞}are the freestream density and velocity; a and b are empirical exponents; f is a tabulated function of freestream velocity V

_{∞}. The detailed descriptions of a, b, and f can be found in Ref. [61].

#### 3.4. Radiation Field

_{m}is the absorption coefficient for the m-th step, I

_{bν}is the spectral blackbody intensity at radiation frequency ν, σ is the Stefan–Boltzmann constant, and T is the temperature. In the present hypersonic nonequilibrium flow, κ

_{P}is calculated using the vibrational-electronic temperature [62].

_{P}is remarkable only in the shock layer, particularly nearly behind the bow shock with peak values, while κ

_{P}is very small in the freestream. It suggests that the radiation energy transfer mainly occurs in the high-temperature shock layer. The five-step model predicts the greatest κ

_{P}, the eight-step model second, and the two-step model the smallest. As time goes on with the altitude and velocity down, the total level of κ

_{P}grows gradually. Although the temperature in the shock layer decreases as shown in Figure 5 and makes a negative contribution to radiative transfer as the Fire II flight altitude descends, the air density increase significantly promotes the radiation effect in the flowfield. The results of the three-step models all support this point. Another interesting thing is that κ

_{P}predicted by the two-step model shows the Fire II shock layer is close to being optically thin with the optical thickness being an order of magnitude of 10

^{−2}, while the results of the five-step and eight-step models do not agree with this, which can only be clarified further in the future using the more detailed radiation property model, such as the line-by-line or narrowband calculations [57]. Generally, the present radiation–flow uncoupling procedure is a good selection with higher efficiency in time cost than the coupling method to provide rich radiation information on the whole hypersonic nonequilibrium flowfield of Fire II reentry that the engineering methods cannot make.

## 4. Conclusions

^{2}, which is comparable to the convective heating and even exceeds the latter. Although there are remarkable differences among the two-, five-, and eight-step models, the three models all show essentially consistent trends in predictions of radiative transfer. The uncoupling calculated radiative heating can be regarded as the upper limit in the engineering application. In the future, more efforts need to be made to clarify the level of the optical thickness for the flowfield of Earth reentry vehicles at hypervelocity above 10 km/s, which the two-step model predicts to be optically thin, while the five- and eight-step models do not agree. The present scheme can also provide more radiation information in the nonequilibrium flowfield than the previous engineering relations.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

e | total energy |

e_{ve} | vibrational-electronic energy |

h | total enthalpy |

h_{s} | enthalpy for the species s |

h_{ve,s} | vibrational-electronic enthalpy for the species s |

k_{f,r} | forward reaction rate coefficients of the r-th reaction |

k_{b,r} | backward reaction rate coefficients of the r-th reaction |

p | pressure |

q | total heat flux |

q_{ve} | vibrational-electronic heat flux |

q_{cw,stag} | convective heat flux at stagnation point |

q_{rw,stag} | radiative heat flux at stagnation point |

B_{j} | j-th component of the unit directional vector |

H | altitude |

H_{w} | enthalpy at the wall |

H_{e} | enthalpy at the outer edge of the boundary layer |

I_{ν} | spectral radiative intensity at frequency ν |

I_{bv} | blackbody radiative intensity at frequency ν |

J_{s,j} | mass diffusion flux of the species s in the j-th direction |

M_{s} | molecular mass per mole of species s |

N_{s} | total number of air species |

N_{r} | total number of chemical reactions |

R_{s} | gas constant for the species s |

R_{N} | nose radius |

T_{tr} | translational-rotational temperature |

T_{ve} | vibrational-electronic temperature |

T_{w} | wall temperature |

T_{∞} | freestream temperature |

ε_{w} | wall emissivity |

κ_{m} | absorption coefficient for the m-th spectral step |

κ_{P} | Planck-mean absorption coefficient |

κ_{ν} | absorption coefficient at frequency ν |

ν | radiation frequency |

${\nu}_{r,s}^{b}$ | stoichiometric coefficient of the species s in the r-th backward reaction |

${\nu}_{r,s}^{f}$ | stoichiometric coefficient of the species s in the r-th forward reaction |

ρ_{s} | density of the species s |

ρ_{∞} | freestream density |

τ_{ij} | viscous stress tensor |

τ_{ν} | optical thickness at frequency ν |

ω_{r} | radiative source term |

ω_{s} | mass production rate of the species s |

ω_{ve} | vibrational-electronic energy source term |

## Appendix A. Nongray Two-, Five- and Eight-Step Models

#### Appendix A.1. Two-Step Model

_{0}= 1.225 kg/m

^{3}; for the second step, the absorption coefficient is

_{bλ}is the blackbody radiative intensity expressed as:

_{B}is the Boltzmann constant, λ is the wavelength of radiation, and T is the temperature.

#### Appendix A.2. Five-Step Model

_{N}is the number density of nitrogen, and T is the temperature; am, bm, and cm are fitting coefficients depending on the general range of temperature, which can be found in Ref. [64].

#### Appendix A.3. Eight-Step Model

^{−1}) are formulated as follows:

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**Figure 2.**Convective heating at the stagnation point of Fire II predicted by the present algorithm (present), Anderson’s engineering relation (Anderson Eng.), Olynick et al., and Gupta et al.

**Figure 3.**Forebody convective heating along the Fire II surface throughout the trajectory from t = 1634 s to t = 1645 s.

**Figure 4.**Forebody convective heating at t = 1636 s predicted by the present algorithm (present), DPLR, and LAURA codes.

**Figure 5.**Temperature distribution in the Fire II flowfield (unit: 10

^{4}K): (

**a**) t = 1634 s; (

**b**) t = 1636 s; (

**c**) t = 1637 s; (

**d**) t = 1640 s; (

**e**) t = 1643 s; (

**f**) t = 1645 s.

**Figure 6.**Number density of species along the stagnation line: (

**a**) t = 1634 s; (

**b**) t = 1636 s; (

**c**) t = 1637 s; (

**d**) t = 1640 s; (

**e**) t = 1643 s; (

**f**) t = 1645 s.

**Figure 7.**Radiative heating at the stagnation point of Fire II throughout the trajectory predicted by the present algorithm with 2-step model, 5-step model, 8-step model, T & S engineering relation, Gupta et al., and Olynick et al.

**Figure 8.**Forebody radiative heating throughout the trajectory predicted by (

**a**) two-step model; (

**b**) five-step model; (

**c**) eight-step model.

**Figure 9.**Planck-mean absorption coefficient throughout the trajectory (unit: m

^{−1}): (

**a**) two-step model; (

**b**) five-step model; (

**c**) eight-step model.

Model | Step No. | Wavelength (Å) | Spectral Band |
---|---|---|---|

Two-step | 1 | 0–1100 | VUV (vacuum ultraviolet) |

2 | 1100–∞ | Visible | |

Five-step | 1 | 620–1100 | VUV continuum |

2 | 1100–1300 | VUV continuum | |

3 | 1300–1570 | VUV lines | |

4 | 1570–7870 | Visible | |

5 | 7870–9552 | IR (infrared) lines | |

Eight-step | 1 | 400–852 | VUV continuum |

2 | 852–911 | VUV continuum | |

3 | 911–1020 | VUV continuum | |

4 | 1020–1130 | VUV continuum | |

5 | 1130–1801 | Continuum + line wings | |

6 | 1130–1801 | Line “centers” | |

7 | 1801–4000 | Visible | |

8 | 4000–∞ | Visible + infrared |

Time (s) | H (km) | V_{∞} (km/s) | Ma | R_{N} (m) | ρ_{∞} (kg/m^{3}) | T_{∞} (K) | T_{w} (K) |
---|---|---|---|---|---|---|---|

1634 | 76.42 | 11.36 | 40.58 | 0.935 | 3.72 × 10^{−5} | 195 | 615 |

1636 | 71.02 | 11.31 | 38.94 | 0.935 | 8.57 × 10^{−5} | 210 | 810 |

1637 | 67.05 | 11.25 | 37.17 | 0.935 | 1.47 × 10^{−4} | 228 | 1030 |

1640 | 59.62 | 10.97 | 34.34 | 0.935 | 3.86 × 10^{−4} | 254 | 1560 |

1643 | 53.04 | 10.48 | 31.47 | 0.805 | 7.80 × 10^{−4} | 276 | 640 |

1645 | 48.37 | 9.83 | 29.05 | 0.805 | 1.32 × 10^{−3} | 285 | 1520 |

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**MDPI and ACS Style**

Yang, X.; Wang, J.; Zhou, Y.; Sun, K.
Assessment of Radiative Heating for Hypersonic Earth Reentry Using Nongray Step Models. *Aerospace* **2022**, *9*, 219.
https://doi.org/10.3390/aerospace9040219

**AMA Style**

Yang X, Wang J, Zhou Y, Sun K.
Assessment of Radiative Heating for Hypersonic Earth Reentry Using Nongray Step Models. *Aerospace*. 2022; 9(4):219.
https://doi.org/10.3390/aerospace9040219

**Chicago/Turabian Style**

Yang, Xinglian, Jingying Wang, Yue Zhou, and Ke Sun.
2022. "Assessment of Radiative Heating for Hypersonic Earth Reentry Using Nongray Step Models" *Aerospace* 9, no. 4: 219.
https://doi.org/10.3390/aerospace9040219