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Numerical simulation of inductively coupled plasma flows under chemical non-equilibrium
Abstract
This paper presents a detailed review of the numerical modeling of inductively coupled air plasmas under local thermodynamic equilibrium and under chemical non-equilibrium. First, the physico-chemical models are described, i.e. the thermodynamics, transport phenomena and chemical kinetics models. Particular attention is given to the correct modelling of ambipolar diffusion in multi-component chemical non-equilibrium plasmas. Then, the numerical aspects are discussed, i.e. the space discretization and iterative solution strategies. Finally, computed results are presented for the flow, temperature and chemical concentration fields in an air inductively coupled plasma torch. Calculations are performed assuming local thermodynamic equilibrium and under chemical non-equilibrium, where two different finite-rate chemistry models are used. Besides important non-equilibrium effects, we observe significant demixing of oxygen and nitrogen nuclei, which occurs due to diffusion regardless of the degree of non-equilibrium in the plasma.
... The subsonic plasma flow in the Plasmatron chamber was numerically simulated using a two-dimensional magnetohydrodynamic solver, referred to as CF-ICP in the following, which couples the Maxwell induction equations with the Navier-Stokes equations under the assumptions of Local Thermodynamic Equilibrium (LTE) and axisymmetric steady flow [48]. The code is integrated into the Computational Object-Oriented Library for Fluid Dynamics (COOLFluiD) [49] and relies on the Mutation++ library [50] to compute the thermodynamic and transport properties of an eleven-species air mixture (O 2 , N 2 , O 2 + , N 2 + , NO, NO + , O, O + , N, N + , e − ). ...
... The subsonic steady state plasma flow field in the Plasmatron chamber is numerically simulated using the in-house ICP magnetohydrodynamics solver. 27,28 This solver couples the Maxwell equations with the Navier-Stokes equations under Local Thermodynamic Equilibrium (LTE) and axisymmetric steady flow assumptions. The VKI ICP code is integrated into the Computational Object-Oriented Library for Fluid Dynamics (COOLFluiD). ...
Spacecraft entering a planetary atmosphere are surrounded by a plasma layer containing high levels of ionization. The high electron number densities cause attenuation of the electromagnetic waves emitted by the on-board antennas, leading to communication blackout for several minutes. This work presents experimental measurements of signal propagation in an ionized plasma flow. These measurements are conducted at the VKI plasma wind tunnel using conical horn antennas transmitting in the Ka-band, between 33 and 40 GHz. Clear attenuations are observed when the signal is propagating through the plasma. These vary between 5 and 15 dB depending on the testing conditions. Preliminary evidence of Faraday rotation effect caused by the plasma is also observed. Additionally, high speed imaging analysis shows significant jet fluctuations, related with the chamber pressures and power settings.
... ICP facilities are modeled using magneto-hydrodynamic (MHD) simulations [6,[13][14][15][16]. In Ref. [6] and [13], the fields are assumed to be 2D axisymmetric. ...
The objective of this work is to model the plasma jet in the plasmatron X ICP facility, located at the University of Illinois Urbana-Champaign (UIUC) using high-fidelity numerical schemes. The governing equations are solved using a finite volume based fluid solver called hegel, developed at the Center for Hypersonics and Entry Systems Studies (CHESS). The plasma is assumed to be in the state of local thermodynamic equilibrium (LTE). The convective terms are discretized using a combination of non-dissipative central skew-symmetric scheme and a dissipative upwind scheme. This combination is used to achieve appropriate amount of filtering of high frequency scales and dissipation due to the sub-grid scales, hence performing an implicit large eddy simulation (ILES). The validity of the scheme is studied by applying it to Taylor-Green vortex testcase. To replicate no-reflection boundary conditions, sponge regions are added near each boundary by adding source terms to force the flowfield to a target state. The ILES method and sponge boundary zones are applied to simulate a subsonic turbulent jet of Reynolds number 3600 and Mach number 0.9. The sponge zones can be seen to avoid reflections back into the physical domain. Using the same numerical schemes, the plasma jet in plasmatron X ICP facility is simulated. Due to very high translational temperatures in the core compared to the ambient flow (≈ 10000 K), a sharp gradient is seen in the density field across the jet shear layer. The high temperatures also cause the transport properties to vary by an order of magnitude within the physical domain. It is seen that the plasma jet dynamics were affected by these significant differences in density and viscosity between the plasma core and the ambient fluid. The cold dense ambient fluid is seen to be periodically entrained into the plasma core.
... We can reduce the dimensionality of the problem by taking into account some physical relationships from which we can recuperate some boundary layer edge parameters. The subsonic VKI Plasmatron flowfield, composed by the torch and test chamber, is numerically simulated using an axisymmetric Local Thermodynamic Equilibrium (LTE) magnetohydrodynamics solver which computes the solution of the Navier-Stokes equations coupled to the Maxwell's equations under certain assumptions (VKI ICP code [26][27][28]). Initially developed as a standalone tool, the VKI ICP code is nowadays integrated into the Computational Object-Oriented Library for Fluid Dynamics (COOLFluiD) [29]. ...
In this work, we calibrate a carbon nitridation model for a broad span of surface temperatures from existing plasma wind tunnel measurements by accounting for experimental and parametric uncertainties. A chemical non-equilibrium stagnation line model is proposed to simulate the experiments and obtain recession rates and CN densities, the measured model outputs. First, we establish the influence of the experimental boundary conditions and nitridation parameters on the simulated observations through a sensitivity analysis. Results show that such quantities are mostly affected by the efficiency of nitridation reactions at the gas-surface interface. We then perform model calibrations for each experimental condition and compare them based on the experimental data used. This allows us to check the consistency of the experimental dataset. Using only the trustworthy experimental data, we perform a calibration of Arrhenius law parameters for nitridation efficiencies considering all available experimental conditions jointly, allowing us to compute nitridation efficiencies even for surface temperatures for which there are no reliable experimental data available. The stochastic Arrhenius law agrees well with most of the data in the literature. This result constitutes the first nitridation model extracted from plasma wind tunnel experiments with accurate uncertainty estimates.
... We can reduce the dimensionality of the problem by taking into account some physical relationships from which we can recuperate some boundary layer edge parameters. The subsonic VKI Plasmatron flowfield, composed by the torch and test chamber, is numerically simulated using an axisymmetric Local Thermodynamic Equilibrium (LTE) magnetohydrodynamics solver which computes the solution of the Navier-Stokes equations coupled to the Maxwell's equations under certain assumptions (VKI ICP code [26][27][28]). Initially developed as a standalone tool, the VKI ICP code is nowadays integrated into the Computational Object-Oriented Library for Fluid Dynamics (COOLFluiD) [29]. ...
In this work, we calibrate a carbon nitridation model for a broad span of surface temperatures from existing plasma wind tunnel measurements by accounting for experimental and parametric uncertainties. A chemical non-equilibrium stagnation line model is proposed to simulate the experiments and obtain recession rates and CN densities, the measured model outputs. First, we establish the influence of the experimental boundary conditions and nitridation parameters on the simulated observations through a sensitivity analysis. Results show that such quantities are mostly affected by the efficiency of nitridation reactions at the gas-surface interface. We then perform model calibrations for each experimental condition and compare them based on the experimental data used. This allows us to check the consistency of the experimental dataset. Using only the trustworthy experimental data, we perform a calibration of Arrhenius law parameters for nitridation efficiencies considering all available experimental conditions jointly, allowing us to compute nitridation efficiencies even for surface temperatures for which there are no reliable experimental data available. The stochastic Arrhenius law agrees well with most of the data in the literature. This result constitutes the first nitridation model extracted from plasma wind tunnel experiments with accurate uncertainty estimates.
... The subsonic VKI Plasmatron flowfield, composed by the torch and test chamber, is numerically simulated using an axisymmetric LTE magnetohydrodynamics solver which computes the solution of the Navier-Stokes equations coupled to the Maxwell's equations under certain assumptions (VKI ICP code [141][142][143]). Initially developed as a standalone tool, the VKI ICP code is nowadays integrated into the Computational Object-Oriented Library for Fluid Dynamics (COOLFluiD) [144]. ...
Space vehicles entering dense planetary atmospheres must withstand extreme heating conditions to protect the astronauts and cargo from damage. Aerospace engineers rely either on catalytic or ablative materials to protect the spacecraft from the intense heat. Qualitatively, both types of materials are different in the way they cope with high temperatures and how they can dissipate large amounts of energy. Catalytic materials re-radiate most of the heat back to the surrounding gas without undergoing fundamental changes in their structure. On the other hand, ablative materials transform the thermal energy into decomposition and removal of the protection material itself. Quantitatively, their differences are related to the physico-chemical models and the amount of model data needed for the simulation of their thermal response, as well as the observations that we can obtain in testing facilities.
The investigation of gas-surface interaction phenomena relies on the development of predictive theoretical models and the capabilities of current experimental facilities. Both resources are strong assets of scientific research. However, due to the complexity of the physics and the various phenomena that need to be investigated in ground-testing facilities, both numerical and experimental processes generate data subjected to uncertainties. Nevertheless, it remains a common practice in the field of aerothermodynamics to resort to calibration and validation methods that are not apt for rigorous uncertainty treatment. Further, as the complexity and level of definition of gas-surface interaction models increase, so does the number of parameters that need estimation. To alleviate this problem, the current state-of-the-art inverse methodology is to project as many assumptions about the physics as considered plausible in order to reduce the number of parameters sought out, potentially biasing the results and substantially slowing down the progress for these aerospace systems.
This thesis investigates the process of scientific inference and its ramifications for selected gas-surface interaction experiments. Its main contributions are the improvement and re-formulation of model calibrations as statistical inverse problems with the consequent extension of current databases for catalysis and ablation. The model calibrations are posed using the Bayesian formalism where a complete characterization of the posterior probability distributions of selected parameters are computed.
The first part of the thesis presents a review of the theoretical models, experiments and numerical codes used to study catalysis and ablation in the context of the von Karman Institute’s Plasmatron wind tunnel. This part ends with a summary on the potential uncertainty sources present in both theoretical-numerical and experimental data. Subsequently, the methods used to deal with these uncertainty sources are introduced in detail.
The second part of the thesis presents the various original contributions of this thesis. For catalytic materials, an optimal likelihood framework for Bayesian calibration is proposed. This methodology offers a complete uncertainty characterization of catalytic parameters with a decrease of 20% in the standard deviation with respect to previous works. Building on this framework, a testing strategy which produces the most informative catalysis experiments to date is studied. Experiments and consequent stochastic analyses are performed, enriching existing catalysis experimental databases for ceramic matrix composites with accurate uncertainty estimations.
The last contribution deals with the re-formulation of the inference problem for nitridation reaction efficiencies of a graphite ablative material from plasma wind tunnel data. This is the first contribution in the literature where different measurements of the same flowfield are used jointly to assess their consistency and the resulting ablation parameters. An Arrhenius law is calibrated using all available data, extending the range of conditions to lower surface temperatures where no account of reliable experimental data is found. Epistemic uncertainties affecting the model definition and ablative wall conditions are gauged through various hypothesis testing studies. The final account on the nitridation reaction efficiency uncertainties is given by averaging the results obtained under the different models.
This thesis highlights the fact that the process of scientific inference can also carry deep assumptions about the nature of the problem and it can impact how researchers reach conclusions about their work. Ultimately, this thesis contributes to the early efforts of introducing accurate and rigorous uncertainty quantification techniques in atmospheric entry research. The methodologies here presented go in line with developing predictive models with estimated confidence levels.
... The subsonic VKI Plasmatron flowfield, composed by the torch and test chamber, is numerically simulated using an axisymmetric LTE magnetohydrodynamics solver which computes the solution of the Navier-Stokes equations coupled to the Maxwell's equations under certain assumptions (VKI ICP code [2,41,42]). Initially developed as a standalone tool, the VKI ICP code is nowadays integrated into the Computational Object-Oriented Library for Fluid Dynamics (COOLFluiD) [43]. ...
This work focuses on the development of a dedicated experimental methodology that allows for a better stochastic characterization of catalytic recombination parameters for reusable ceramic matrix composite materials when dealing with uncertain measurements and model parameters. As one of the critical factors affecting the performance of such materials, the contribution to the heat flux of the exothermic recombination reactions at the vehicle surface must be carefully assessed. In this work, we first use synthetic data to test whether or not the proposed experimental methodology brings any advantages in terms of uncertainty reduction on the sought out parameters compared to more traditional experimental approaches in the literature. The evaluation is done through the use of a Bayesian framework developed in a previous work with the advantage of being able to fully and objectively characterize the uncertainty on the calibrated parameters. The synthetic dataset is adapted for testing ceramic matrix composites by carefully choosing adequate auxiliary materials whose heat flux measurements have the capability of reducing the resulting uncertainty on the catalytic parameter of the thermal protection material itself when tested under the same flow conditions. We then propose a comprehensive set of real wind tunnel testing cases for which stochastic analyses are carried out. The physical model used for the estimations consists of a 1D boundary layer solver along the stagnation line in which the chemical production term included in the surface mass balance depends on the catalytic recombination efficiency. All catalytic parameters of the auxiliary and thermal protection materials are calibrated jointly with the boundary conditions of the experiments. The testing methodology confirms to be a reliable experimental approach for characterizing these materials while reducing the uncertainty on the calibrated catalytic efficiencies by more than 50 %. An account of the posteriors summary statistics is provided to enrich the current state-of-the-art experimental databases.
View Video Presentation: https://doi.org/10.2514/6.2022-3732.vid In this work, we seek to infer nitrogen ablation model parameters from different experimental measurements. First, the dependencies of the proposed stagnation line forward model are studied to establish the influence of the different parameters on the simulated counterparts of the available observations. Results show that recession rate and CN local density predictions in the boundary layer are mostly affected by the efficiency of the nitridation reactions at the gas-surface interface for the experimental conditions considered. In turn, it is expected that the available measured counterparts provide enough information to calibrate nitridation parameters. We then carry out different stochastic model calibrations and compare them on the basis of the experimental data considered. When used independently, measured recession rates and CN densities each give information about nitridation reactions. As they are both part of the same experiments, one model should be capable of explaining both observations at once. Checking for this consistency with the available data has the potential of improving the characterization of nitridation mechanisms and signaling issues with the model and/or the experiments. It is identified that the results obtained when using a particular measurement of local CN densities present inconsistencies when compared to the results obtained using the measured recession rate for the same experimental condition and under the same chosen physical model. The uncertainties of the experimental data are then included as free parameters in the inference framework with the results showing great departure of the calibrated standard deviation for the problematic condition from the rest of the dataset, suggesting that the measurement should be repeated in the future.
This work presents a dedicated plasma wind tunnel experimental methodology that significantly improves the stochastic characterization of TPS catalytic efficiencies when dealing with uncertain measurements and model parameters. We first use synthetic data to test whether the proposed experimental methodology brings any advantages in terms of uncertainty reduction. The evaluation is done using a Bayesian framework developed in a previous work with the advantage of being able to fully and objectively characterize the uncertainty on the calibrated parameters. We then propose a comprehensive set of real wind tunnel cases for which stochastic analyses are carried out. All model parameters are calibrated jointly with the boundary conditions of the experiments. The testing methodology confirms to be a reliable experimental approach able to reduce the uncertainty on the TPS catalytic efficiencies by more than 50%. An account of the posteriors statistics is provided to enrich the current state-of-the-art experimental databases.
High frequency induction (electrodeless) plasmatrons provide the wide possibilities for modeling of nonequilibrium processes of dissociated gas flow thermochemical interaction with a surface at the conditions corresponding to hypersonic vehicle’s flight in atmosphere at high altitude [1].
The Stefan-Maxwell relations for the Champan-Enskog method for solving the linearized Boltzmann equations are derived with correction for the higher approximations. The coefficients in these relationships are expressed in terms of determinants, whose elements are complete integral parentheses, which are a function of the collision integrals of particles of all the pairs and of the composition of the mixture. Fick's laws are derived for three-component (with the electron treated as an element), mixtures of gases in which an arbitrary number of equilibrium chemical reactions can occur.
A new model is presented which is well-suited for the calculation of the transport properties of ionized species in CFD simulations. A multicomponent mixture formulation and simple mixing rules are used to compute the transport coefficients of dissociated and ionized air from room temperature up to 30000 K at different pressures. Both formulations share little input data, which simplify the handling of the transport model. The calculated binary diffusion coefficients are in good agreement with the results of Levin et al. and Capitelli et al. The viscosity coefficients of the mixture differ from the computations of Gupta et al. and Murphy, especially with increasing pressure. The thermal and electrical conductivities are consistent with the results of Murphy and deviate from the calculations of Gupta et al. and Yos.
The hydrodynamic equations and transport coefficients with diffusion and heat transfer in a multicomponent partially ionized gas in the presence or absence of an external magnetic field are derived using the kinetic theory of gases. Expressions for heat and diffusion fluxes and formulae for transport coefficients have been obtained by the Chapman-Enskog and Grad methods. The latter formulae are fairly elaborated for practical calculations. The aim is to obtain mass and heat transfer equations, and also exact formulae for transport coefficients in a simpler form, convenient for the solution of hydrodynamical problems.
Preconditioned Krylov subspace techniques have had a substantial impact on the numerical solution of engineering and scientific problems. These methods can be mandatory for large problems arising from 3-dimensional models. However, although iterative methods are rapidly gaining ground, there are still a number of open questions and unresolved issues related to their use in representative applications. In this paper, we give an overview of the most commonly used techniques, and discuss open questions and current trends.
Stefan-Maxwell relations allowing for higher approximations are obtained
for the Chapman-Enskog solution to the linearized (low Knudsen numbers)
Boltzmann equation. The coefficients in these relations are expressed in
determinants incorporating complete integral brackets which are a
function of the particle collision integrals for all the pairs and of
the mixture composition. The Fick law is obtained for three-component
gas mixtures (with allowance for the electron element) in which an
arbitrary number of equilibrium chemical reactions can occur.
(For abstract see issue 21, p. 2746, Accession no. A73-40516)