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Background of the study: To prevent damage to a re-entry vehicle, its windward side must be protected by a heat shield made from advanced thermal protection materials (TPMs). In-flight testing of new TPMs is prohibitively expensive, so the extreme heat experienced during hypersonic re-entry is simulated by exposing a TPM sample to a hot plasma jet. Inductively coupled plasma (ICP) facilities are particularly useful for this testing because they generate large, contamination-free plasma volumes without using electrodes, making them a preferred choice for evaluating thermal protection systems. Numerical simulation of ICPs plays an important role in supplementing the experiments in the facility in order to accurately characterize the ablative properties of the TPM. In aerothermodynamics, the primary quantities of interest are the surface heat flux and the time-integrated heat flux (heat load). Conventionally, a decoupled approach using the film coefficient engineering model is used to study the material response, where a fixed boundary condition obtained from CFD calculations is given to the material response code [1]. However, the most effective way to account for all the strongly interacting physics is to combine the fluid flow field and thermal response analysis into a single computational simulation. This requires a fully coupled approach where the flow solver is tightly coupled with the material response solver to accurately predict the ablative characteristics of the material being tested in the plasma facility. Hence, this work aims to develop a strongly coupled framework for accurate prediction of the response of TPM samples under the inductively coupled plasma environment.
Methodology: The dynamics of the gaseous plasmas treated in this work are governed by the species mass, global momentum and energy, and vibronic energy equations which have been implemented in a finite-volume solver HEGEL [2] which uses the PLATO [3] library for the evaluation of plasma-related quantities (e.g. thermodynamic properties, etc.). The electromagnetic field inside ICPs is governed by the set of Maxwell’s equations which are solved in a mixed finite-element solver FLUX [4]. The material response is simulated using CHyPS [5] employing a high-order discontinuous Galerkin (DG) formulation.
The governing equations for the plasma are coupled to those for the electromagnetic field via the Lorentz forces and Joule heating source terms in the momentum and energy equations, respectively. At the same time, the electric and magnetic fields within the plasma are affected by the electrical conductivity of the plasma. As a result, the two datasets (Lorentz forces and Joule heating for HEGEL, and electrical conductivity for FLUX) are to be passed at every fluid time-step to accurately capture the magneto-hydrodynamic phenomena occurring within an ICP. Similarly, HEGEL passes the heat flux, pressure, and species fluxes to CHyPS for the surface to be coupled. CHyPS in return provides the surface temperature and composition along with the blowing rates. The coupling between HEGEL and FLUX takes place in the entire volume of the domain and hence is denoted as volume coupling. The HEGEL-CHyPS coupling occurs only along the surface of the material denoted as surface coupling. The required coupling between the above-mentioned solvers is here realized using preCICE [6], an open-source coupling library for partitioned multi-physics simulations.
Results: Simulations have been conducted for the UIUC Plasmatron X facility under the Local Thermodynamic Equilibrium (LTE) assumption with a circular sample placed in an air plasma jet. The torch has two injectors: central injection consisting of 15 holes which are inclined 15 degrees down and have 24 degrees swirl angle, and sheath gas injection consisting of 72 holes which inject gas straight in the axial direction. For simplification, both injectors are assumed to be continuous annular injectors for 2D axi-symmetric simulations conducted in this work. At the end of the torch, a straight nozzle of length 129.54 mm has been used. Mass flows in the central and the sheath gas injection is kept fixed to be 0.86 g/s and 7.13 g/s respectively. The working gas is air modeled using an 11-species mixture: $\mathcal{S} = \left\{\mathrm{e}^-,\, \mathrm{N}_2, \, \mathrm{O}_2, \, \mathrm{NO},\mathrm{N},\, \mathrm{O}, \, \mathrm{N}_2^+, \, \mathrm{O}_2^+,\, \mathrm{NO}^+,\, \mathrm{N}^+,\, \mathrm{O}^+\right\}$. The operating conditions for the simulation are as follows: pressure 590 Pa, and power 300 kW. The coil frequency in the facility is fixed to be 2.1 MHz. First, an uncoupled ICP simulation (i.e. without the material solver) is performed for the above operating condition which gives a supersonic plasma jet in the chamber. This agrees with the observations from experiments where the torch becomes choked at low-pressure and high-power combinations, giving a supersonic plasma jet in the facility.
Next, a fully coupled simulation is performed where the plasma flowfield is coupled with the material response. The preliminary result presented here considers only heat conduction calculation in the material response code ignoring the pyrolysis and ablation effects to showcase the correct implementation of the coupling framework. The material is assumed to be non-porous with thermal conductivity, density, and specific heat capacity of 0.5W/mK, 180 Kg/m3, and 2000 J/KgK, respectively. For the coupled simulation, the material code CHyPS runs in a time-accurate manner for 10 s. The coupling occurs every 0.01 s i.e. CHyPS provides the surface temperature to the plasma solver HEGEL and gets back the heat flux every 0.01 s. For each coupling window, HEGEL converges to a steady state for the given temperature boundary condition. This two-way coupling ensures that the flow boundary layer in the fluid domain responds to the changes occurring in the material, thereby updating the heat flux in every coupling window. This allows for a more accurate prediction of material response. The plasma and sample temperature contours at different instances of time show the coupled dynamics of the material. At t = 0 s, the material is completely cold (300K) everywhere. As time progresses, the sample temperature increases due to the heat flux being applied by the plasma jet at the sample surface. At t = 10 s, the maximum temperature in the sample reaches around 2400K. The plasma temperature remains unchanged everywhere except for regions very close to the sample surface.
Conclusion: A high-fidelity multi-physics simulation framework for inductively coupled plasma discharges has been presented where a fluid solver, an electromagnetic solver, and a material response solver, have been coupled to provide a flexible formulation for predicting TPS material response under ICP environments. A preliminary plasma-material coupled simulation has been presented for a heat conduction problem indicating the correct implementation of the framework. Future work will include the pyrolysis and ablation effects in the material incorporating the effect of material recession in the coupled simulations via the mesh movement capability of the solvers.
Acknowldegments: The present research is funded by Vannevar Bush Faculty Fellowship OUSD(RE) Grant No: N00014-21-1-295. The work is also supported by the Center for Hypersonics and Entry Systems Studies (CHESS) at UIUC. Computations were performed on Frontera, an HPC resource provided by the Texas Advanced Computing Center (TACC) at The University of Texas at Austin, on allocation CTS20006.
References: [1] Cooper, J., Salazar, G., and Martin, A., “Numerical investigation of film coefficient approximation for chemically reacting boundary-layer flows,” J. Thermophys. Heat Transf., Vol. 37, No. 3, 2023, pp. 644–655.
[2] Munafò, A., Alberti, A., Pantano, C., Freund, J. B., and Panesi, M., “A computational model for nanosecond pulse laser-plasma interactions,” J. Comput. Phys., Vol. 406, 2020, p. 109190.
[3] Munafò, A., and Panesi, M., “Plato: a high-fidelity tool for multi-component plasmas,” AIAA Aviation 2023 Forum, 2023, p. 3490.
[4] Kumar, S., Munafò, A., Le Maout, V., Mansour, N., and Panesi, M., “Self-consistent magneto-hydrodynamic modeling of ICP discharges,” AIAA SCITECH 2022 Forum, 2022, p. 1619.
[5] Chiodi, R. M., Stephani, K. A., Panesi, M., and Bodony, D. J., “CHyPS: A high-order material response solver for ablative thermal protection systems,” AIAA SCITECH 2022 Forum, 2022, p. 1501.
[6] Bungartz, H.-J., Lindner, F., Gatzhammer, B., Mehl, M., Scheufele, K., Shukaev, A., and Uekermann, B., “preCICE–a fully parallel library for multi-physics surface coupling,” Comput. Fluids, Vol. 141, 2016, pp. 250–258.
Summary
The present work demonstrates a high-fidelity coupled framework to study thermal protection material response under inductively coupled plasma environments. The modular nature of the framework allows a tightly coupled approach to model the material response of a sample tested in ICP facilities. A preliminary plasma-material coupled simulation has been presented for a heat conduction problem indicating the correct implementation of the framework, paving the path for incorporating more complex phenomena like ablation, surface recession, etc. to accurately predict the material response during testing in ICP facilities.
