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This paper aims to describe the physical and numerical modeling of a new computational fluid dynamics solver for hypersonic flows in thermo-chemical non-equilibrium. The code uses a blend of numerical techniques to ensure accuracy and robustness and to provide scalability for advanced hypersonic physics and complex three-dimensional (3D) flows.

The solver is based on an edge-based stabilized finite element method (FEM). The chemical and thermal non-equilibrium systems are loosely-coupled to provide flexibility and ease of implementation. Chemical non-equilibrium is modeled using a laminar finite-rate chemical kinetics model while a two-temperature model is used to account for thermodynamic non-equilibrium. The systems are solved implicitly in time to relax numerical stiffness. Investigations are performed on various canonical hypersonic geometries in two-dimensional and 3D.

The comparisons with numerical and experimental results demonstrate the suitability of the code for hypersonic non-equilibrium flows. Although convergence is shown to suffer to some extent from the loosely-coupled implementation, trading a fully-coupled system for a number of smaller ones improves computational time. Furthermore, the specialized numerical discretization offers a great deal of flexibility in the implementation of numerical flux functions and boundary conditions.

The FEM is often disregarded in hypersonics. This paper demonstrates that this method can be used successfully for these types of flows. The present findings will be built upon in a later paper to demonstrate the powerful numerical ability of this type of solver, particularly with respect to robustness on highly stretched unstructured anisotropic grids.

Wall-modeled large eddy simulation (LES) is a practical tool for solving wall-bounded flows with less computational cost by avoiding the explicit resolution of the near-wall region. However, its use is limited in flows that have high non-equilibrium effects like separation or transition. This study aims to present a novel methodology of using high-fidelity data and machine learning (ML) techniques to capture these non-equilibrium effects.

A precursor to this methodology has already been tested in Radhakrishnan et al. (2021) for equilibrium flows using LES of channel flow data. In the current methodology, the high-fidelity data chosen for training includes direct numerical simulation of a double diffuser that has strong non-equilibrium flow regions, and LES of a channel flow. The ultimate purpose of the model is to distinguish between equilibrium and non-equilibrium regions, and to provide the appropriate wall shear stress. The ML system used for this study is gradient-boosted regression trees.

The authors show that the model can be trained to make accurate predictions for both equilibrium and non-equilibrium boundary layers. In example, the authors find that the model is very effective for corner flows and flows that involve relaminarization, while performing rather ineffectively at recirculation regions.

Data from relaminarization regions help the model to better understand such phenomenon and to provide an appropriate boundary condition based on that. This motivates the authors to continue the research in this direction by adding more non-equilibrium phenomena to the training data to capture recirculation as well.

Non‐equilibrium hypersonic viscous flows with high enthalpy conditions have been computed with an implicit time‐dependent finite‐difference scheme. This scheme accounts for both chemical and vibrational non‐equilibrium processes in air flow around a hemispherical cylindrical body. The air was assumed to decompose into the following five species N, O, NO, N₂ and O₂ and only the two diatomic species N₂ and O₂ are taken in thermal non‐equilibrium. A range of Mach number from 14 to 18 has been investigated. The numerical results have been compared with those obtained by other workers and are in agreement with ballistic range data concerning the standoff shock distance at M = 15.3. The computed heat flux wall follows the trends of the experiments with an under prediction increasing with the Mach number. The influence of the thermal non‐equilibrium assumption on the computed standoff shock distance is investigated.

Partially‐decoupled upwind‐based total‐variation‐diminishing (TVD) finite‐difference schemes for the solution of the conservation laws governing two‐dimensional non‐equilibrium vibrationally relaxing and chemically reacting flows of thermally‐perfect gaseous mixtures are presented. In these methods, a novel partially‐decoupled flux‐difference splitting approach is adopted. The fluid conservation laws and species concentration and vibrational energy equations are decoupled by means of a frozen flow approximation. The resulting partially‐decoupled gas‐dynamic and thermodynamic subsystems are then solved alternately in a lagged manner within a time marching procedure, thereby providing explicit coupling between the two equation sets. Both time‐split semi‐implicit and factored implicit flux‐limited TVD upwind schemes are described. The semi‐implicit formulation is more appropriate for unsteady applications whereas the factored implicit form is useful for obtaining steady‐state solutions. Extensions of Roe's approximate Riemann solvers, giving the eigenvalues and eigenvectors of the fully coupled systems, are used to evaluate the numerical flux functions. Additional modifications to the Riemann solutions are also described which ensure that the approximate solutions are not aphysical. The proposed partially‐decoupled methods are shown to have several computational advantages over chemistry‐split and fully coupled techniques. Furthermore, numerical results for single, complex, and double Mach reflection flows, as well as corner‐expansion and blunt‐body flows, using a five‐species four‐temperature model for air demonstrate the capabilities of the methods.

This work aims to describe the physical and numerical modeling of a CFD solver for hypersonic flows in thermo-chemical non-equilibrium. This paper is the second of a two-part series that concerns the application of the solver introduced in Part I to adaptive unstructured meshes.

The governing equations are discretized with an edge-based stabilized finite element method (FEM). Chemical non-equilibrium is simulated using a laminar finite-rate kinetics, while a two-temperature model is used to account for thermodynamic non-equilibrium. The equations for total quantities, species and vibrational-electronic energy conservation are loosely coupled to provide flexibility and ease of implementation. To accurately perform simulations on unstructured meshes, the non-equilibrium flow solver is coupled with an edge-based anisotropic mesh optimizer driven by the solution Hessian to carry out mesh refinement, coarsening, edge swapping and node movement.

The paper shows, through comparisons with experimental and other numerical results, how FEM + anisotropic mesh optimization are the natural choice to accurately simulate hypersonic non-equilibrium flows on unstructured meshes. Three-dimensional test cases demonstrate how, for high-speed flows, shocks resolution, and not necessarily boundary layers resolution, is the main driver of solution accuracy at walls. Equally distributing the error among all elements in a suitably defined Riemannian space yields highly anisotropic grids that feature well-resolved shock waves. The resulting high level of accuracy in the computation of the enthalpy jump translates into accurate wall heat flux predictions. At the opposite end, in all cases examined, high-quality but isotropic unstructured meshes gave very poor solutions with severely inadequate heat flux distributions not even featuring expected symmetries. The paper unequivocally demonstrates that unstructured anisotropically adapted meshes are the best, and may be the only, way for accurate and cost-effective hypersonic flow solutions.

Although many hypersonic flow solvers are developed for unstructured meshes, few numerical simulations on unstructured meshes are presented in the literature. This work demonstrates that the proposed approach can be used successfully for hypersonic flows on unstructured meshes.

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.

To compute the Navier‐Stokes equations of a non‐equilibrium weakly ionized air flow. This can help to have a better description of the flow‐field and the wall heat transfer in hypersonic conditions.

The numerical approach is based on a multi block finite volume method and using a Riemann's solver based on a MUSCL‐TVD algorithm. In the flux splitting procedure the modified speed of sound, due to the electronic mode, is implemented.

A good description of the shock standoff distance, of the wall heat fluxes and of the peak of electron density number in the shock layer.

The radiative effects are not included in this paper. For the very high Mach numbers, this can modify the shock layer parameters.

The knowledge of the wall heat transfer in the re‐entry body problems.

The building of a robust numerical code in order to well describe hypersonic air flow in high Mach numbers.

A combination of highly conductive porous media and nanofluids is an efficient way for improving thermal performance of relevant applications. For precisely predicting the flow and thermal transport of nanofluids in porous media, the purpose of this paper is to explore the inter-phase coupling numerical methods.

Based on the lattice Boltzmann (LB) method, this study combines the convective flow, non-equilibrium thermal transport and phase interactions of nanofluids in porous matrix and proposes a new multi-phase LB model. The micro-scale momentum and heat interactions are especially analyzed for nanoparticles, base fluid and solid matrix. A set of three-phase LB equations for the flow/thermal coupling of base fluid, nanoparticles and solid matrix is established.

Distributions of nanoparticles, velocities for nanoparticles and the base fluid, temperatures for three phases and interaction forces are analyzed in detail. Influences of parameters on the nanofluid convection in the porous matrix are examined. Thermal resistance of nanofluid convective transport in porous structures are comprehensively discussed with the models of multi-phases. Results show that the Rayleigh number and the Darcy number have significant influences on the convective characteristics. The result with the three-phase model is mildly larger than that with the local thermal non-equilibrium model.

This paper first creates the multi-phase theoretical model for the complex coupling process of nanofluids in porous structures, which is useful for researchers and technicians in fields of thermal science and computational fluid dynamics.

A finite difference method for solving the quasi one‐dimensional non‐equilibrium hypersonic flow equations in a diverging nozzle is presented and discussed. In chemically reacting flows the system of equations to be solved is very stiff. Some reactions may be several orders of magnitude faster than others and generally, they are much faster than the convective process except for very high Ma numbers. For this reason the development of a numerical scheme whose stability is independent of the chemical reaction rates is of importance. The main advantage of this scheme is the conservation of each chemical component, the positivity of densities and vibrational energies, as well as its relative simplicity, which results in a fast computer code.

A report is given on the above Congress which was attended by approximately 550 delegates from twenty‐seven countries. The sixty papers presented embraced the subjects of aerodynamics, materials, structures, propulsion, air transport, aerospace medicine, space Hight and re‐entry; a selection of these papers are summarized below.