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During atmospheric reentry, radiative heating is one of the most important component of the total heat flux. In this paper, we investigate how the thermal radiation coming from the postshock region interacts with the spacecraft structure. A model that takes into account the radiation reflected by the surface is developed and implemented in a solid solver. A partitioned algorithm performs the coupling between the fluid and the solid thermal fields. Numerical simulation of a hollow cone head and a deployed flap region shows the effects of the radiative cooling and the significance of the surface radiation.

When a space vehicle travels through the different layers of the atmosphere in the reentry phase, its most external part is subject to extreme thermal conditions. A thermal protection system (TPS) is hence installed to insulate the vehicle’s parts or its contents from high temperatures and heat fluxes. A significant amount of the total heat flux reaching the vehicle is due to the radiation coming from the postshock hot plasma [

Numerical simulations are one of the most widely used tools to design the TPS, and a detailed mathematical model allows to reduce margins in the sizing. A higher level of accuracy is obtained by including in the physical model the thermal response of the solid to the heat fluxes, through a coupled approach. In the latter, atmosphere gases and the spacecraft structure form two distinct systems that interact through the external surface of the TPS, and through which energy exchanges occur. To enhance the thermal control, the material of the TPS is chosen such that it reflects part of the incident radiation. However, in particular geometric configurations such as cavities, the reflected radiation can reach other surfaces of the vehicle structure. This paper exposes the development and the implementation of an implicit method for the calculation of surface radiation. This solution of the structure problem is coupled to the fluid flow through a partitioned algorithm.

Some space vehicles can be equipped with side flaps that are used to steer the vehicle. The deployed configuration of these flaps presents a gap, and radiative effects in this area are of major interest [

During atmospheric reentry, the hot gas around the spacecraft transfers energy fluxes to the structure through the fluid-structure interface. The solid then reacts, with part of the heat being conducted in the deeper layers of the solid, and part being reradiated from the surface into the fluid. A continuous exchange is hence established on a transient basis, with a thermal energy equilibrium on the interface. The solution of the coupled problem is obtained using the domain decomposition approach, where each domain is described by its own model, and coupling conditions are defined on the boundary [

The flow-field solver code

The conservation of mass, momentum, and total energy is expressed as

In (

In (

To complete the system, the conservation equations have to be supplemented by the equation of state that provides pressure as a function of density and temperature, that is,

Finally,

The in-house solid solver is a code implemented in C++ based on the finite element method and solves the heat transfer problem by means of conduction within the solid, according to the following transient equation:

On the walls that are in contact with the fluid, we impose a Neumann boundary condition, that remains constant between two successive coupling times.

In the classical discretised FEM matrix formulation of (

All forms of matter emit and exchange thermal radiation with their surroundings. Radiation heat transfer can occur in the absence of matter and so also without temperature gradients.

In our work, we model the radiative heat flux taking into account the received radiation from adjacent surfaces (irradiation), and the radiation reflected and then emitted by the surface.

Given two different bodies or a single concave one, the radiative heat fluxes between the surfaces that “see” each other is modeled under two hypotheses:

the individual surfaces absorb, reflect, and emit diffusively, independently of the temperature and the spectrum;

the gas between two surfaces is treated as optically thin, that is, it does not interact with the radiation; this implies that the radiative heat exchange occurs only between surfaces.

The components of the total radiative heat flux incident on a surface are [

total irradiation towards the surface,

emission from the surface, given by the Stefan-Boltzmann law:
^{−2} K^{−4}]

reflected part of irradiation from the surface,

The sum of the last two components is called radiosity, denoted with

In the computational domain, the solid boundaries are discretised into

verify if the normal vectors to the two surfaces are intersecting (Figure

verify if no other surface

Two tests are performed to verify if two faces have a view factor

We can express (

This system is solved for

We consider a partitioned algorithm for fluid-structure interaction, that is, separate solvers are used for the fluid and the structure. This allows an independent time integration, and the coupling is obtained via a series of continuity relations at the interfaces and need to be solved iteratively.

The three elements of the solution of the coupled problem are then:

solution of the fluid equations;

solution of the heat transfer problem;

definition of the solid-fluid interface conditions.

The thermal coupling between the fluid and the structure is given by the thermal equilibrium at the interface. The Dirichlet-Neumann method is used to impose continuity of temperatures and heat fluxes through the interface

When we pass information from the fluid solver to the solid one, we impose a Neumann boundary condition on the interface

We will denote with

The physical characteristic time scales of the aerofield and the heat transfer in the structure differ by many orders of magnitude, that is,

A loose coupling, however, does not guarantee conservation of the energy at the interface [

Taking this into account, we use in this work a strong coupling, that is, we subiterate interface data exchange until the residual of the coupled problem is less than a given threshold.

The algorithm develops as follows (see Figure

pass the solid interface physical quantities

integrate the Navier-Stokes equations in time up to

pass the fluid interface physical quantities

integrate the heat equation in time up to

repeat steps (1) to (4) until convergence, then start next time step.

Partitioned coupling algorithm.

This kind of algorithm is computationally very expensive if the coupling is performed at every

Given the solvers we use, the computational domains are discretised according to the cell-centered finite volume and finite elements methods, respectively for the fluid and the solid domain. The junction is performed by matching the FE boundary nodes with the mid-points of the border sides of the FV cells.

A first evaluation of the coupling techniques is made on a simple hollow 0.5 m long cone head geometry immersed in air at high speed (Mach 10) in thermal and chemical equilibrium. The empty space inside the ogive is in vacuum conditions. We considered a very refractory material, with small values of thermal diffusivity

A significant issue in the simulations is to verify the importance of radiative heat transfer; hence, we simulate two cases: the first without and the second with surface reradiation. In the first case, the walls of the cone head are considered nonradiating, and we imposed an initial condition

For all the simulations in this section, on the fluid domain we imposed supersonic inflow and outflow conditions, respectively on the left and right boundaries. Initial conditions are summarized in Table

Cone head simulation parameters.

Fluid | Structure | ||

3200 ms^{−1} | ^{2} s^{−1} | ||

10 | 10 Wm^{−1} K^{−1} | ||

5000 Pa | 0.5 | ||

250 K | 0.5 | ||

300 K | |||

Cells | 9600 | Elements | 1654 |

Fluid domain and structure shell meshes used for the simulations of the hollow cone head.

In Figures

Temperature field within the hollow cone without (a) and with (b) radiation effects at time points

When surface reradiation is taken into account, the evolution of heat transfer in the structure changes considerably (Figure

This simulation demonstrates that radiation between solid surfaces plays a considerable role, and the obtained temperature values, are of the same order of magnitude of those encountered in real applications, with such flow fields.

We consider now a more complex geometry: a model of a flap that, when deployed, creates a small cavity. The particularity of this case is that two surfaces of the same structure “see” each other, and the medium between these faces is a gas. We want to study the phenomena that take place in the cavity, as done in [

We analyze here a cross-section of the flap. The total length of the flap is 0.4 m, and base dimensions of the cavity are 0.075 m by 0.025 m. The meshes used for these simulations are shown in Figure

Flap simulation parameters.

Fluid | Structure | ||

1700 ms^{−1} | ^{2} s^{−1} | ||

5 | 10 Wm^{−1} K^{−1} | ||

50000 Pa | 0.5 | ||

290 K | 0.5 | ||

300 K | |||

Cells | 4705 | Elements | 2283 |

Mesh for the simulation of thermal coupling in a flap region.

The external flow field is composed of a lip shock, a separation shock and a reattachment region as clearly shown within the solutions, Figures

Before making any consideration from the thermal point of view, we notice from Figure

Velocity of the flow inside the cavity at two random time points.

Temperature evolution inside and around a flap without (a) and with (b) radiation effects.

The temperature fields are shown at two instantaneous times without and with radiation respectively in Figures

Evolution of the temperature within the flap without (a) and with (b) radiation effects.

In this work, we defined a model of surface radiation and thermal interactions between atmospheric gases and the structure of a space vehicle in the phase of atmospheric entry. Because of this specific application, a special focus was made on radiation heat exchange. An efficient way of calculating the radiative heat flux and view factors has been devised and implemented. A mathematical formulation of the thermal coupling algorithm has been given. The consistent interface conditions are continuity of the temperature and heat flux within each one of the domains.

In the second part of this work, we focused on the realistic geometry of flap. The importance of radiative effects in this complex configuration has been analyzed. From the results we evince that, at such high temperatures, radiative heat transfer within the structure surfaces cannot be neglected. The coupled model shows the lowering of the temperature distributions due to radiation losses in the solid and gives a more reliable prediction of the thermal loads and their evolution. It is crucial to consider radiative heat transfer among surfaces in high-speed atmospheric entries, when even higher temperatures and heat loads than our case are attained.

The modelling here has been simplified and needs to be extended, especially for what concerns the physics and the chemistry of the flow and material properties.

This work was partially supported by the ESA-NPI Program, CCN C21872-1.