A finite element solver for hypersonic flows in thermo-chemical non-equilibrium, Part I

2019 ◽  
Vol 29 (7) ◽  
pp. 2352-2388
Author(s):  
Jory Seguin ◽  
Song Gao ◽  
Wagdi George Habashi ◽  
Dario Isola ◽  
Guido Baruzzi
Keyword(s):  
Finite Element ◽  
Three Dimensional ◽  
Coupled System ◽  
Hypersonic Flows ◽  
Computational Time ◽  
Content Type ◽  
Stabilized Finite Element ◽  
Loosely Coupled ◽  
Stabilized Finite Element Method ◽  
Non Equilibrium

Purpose 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. Design/methodology/approach 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. Findings 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. Originality/value 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.

Characteristic stabilized finite element method for non-stationary conduction-convection problems

2019 ◽  
Vol 30 (2) ◽  
pp. 625-658
Author(s):  
Yongshuai Wang ◽  
Md. Abdullah Al Mahbub ◽  
Haibiao Zheng
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Time Derivative ◽  
Content Type ◽  
Stabilized Finite Element ◽  
Stabilized Finite Element Method ◽  
Advection Term ◽  
The Stability ◽  
Element Method ◽  
Low Order

Purpose This paper aims to propose a characteristic stabilized finite element method for non-stationary conduction-convection problems. Design/methodology/approach To avoid difficulty caused by the trilinear term, the authors use the characteristic method to deal with the time derivative term and the advection term. The space discretization adopts the low-order triples (i.e. P1-P1-P1 and P1-P0-P1 triples). As low-order triples do not satisfy inf-sup condition, the authors use the stability technique to overcome this flaw. Findings The stability and the convergence analysis shows that the method is stable and has optimal-order error estimates. Originality/value Numerical experiments confirm the theoretical analysis and illustrate that the authors’ method is highly effective and reliable, and consumes less CPU time.

Immersed volume method for solving natural convection, conduction and radiation of a hat‐shaped disk inside a 3D enclosure

2012 ◽  
Vol 22 (6) ◽  
pp. 718-741 ◽  
Author(s):  
E. Hachem ◽  
H. Digonnet ◽  
E. Massoni ◽  
T. Coupez
Keyword(s):  
Heat Transfer ◽  
Finite Element ◽  
Thermal Properties ◽  
Natural Convection ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Content Type ◽  
Stabilized Finite Element ◽  
Level Set Function ◽  
Volume Method

PurposeThe purpose of this paper is to present an immersed volume method that accounts for solid conductive bodies (hat‐shaped disk) in calculation of time‐dependent, three‐dimensional, conjugate heat transfer and fluid flow.Design/methodology/approachThe incompressible Navier‐Stokes equations and the heat transfer equations are discretized using a stabilized finite element method. The interface of the immersed disk is defined and rendered by the zero isovalues of a level set function. This signed distance function allows turning different thermal properties of each component into homogeneous parameters and it is coupled to a direct anisotropic mesh adaptation process enhancing the interface representation. A monolithic approach is used to solve a single set of equations for both fluid and solid with different thermal properties.FindingsIn the proposed immersion technique, only a single grid for both air and solid is considered, thus, only one equation with different thermal properties is solved. The sharp discontinuity of the material properties was captured by an anisotropic refined solid‐fluid interface. The robustness of the method to compute the flow and heat transfer with large materials properties differences is demonstrated using stabilized finite element formulations. Results are assessed by comparing the predictions with the experimental data.Originality/valueThe proposed method demonstrates the capability of the model to simulate an unsteady three‐dimensional heat transfer flow of natural convection, conduction and radiation in a cubic enclosure with the presence of a conduction body. A previous knowledge of the heat transfer coefficients between the disk and the fluid is no longer required. The heat exchange at the interface is solved and dealt with naturally.

scholarly journals An Efficient Algorithm with Stabilized Finite Element Method for the Stokes Eigenvalue Problem

2017 ◽  
Vol 2017 ◽  
pp. 1-9 ◽  
Author(s):  
Zhifeng Weng ◽  
Yaoxiong Cai
Keyword(s):  
Finite Element ◽  
Eigenvalue Problem ◽  
Stokes Problem ◽  
Mixed Finite Element ◽  
Computational Time ◽  
Element Space ◽  
Stabilized Finite Element ◽  
Stabilized Finite Element Method ◽  
Stokes Eigenvalue Problem ◽  
Element Pair

This paper provides a two-space stabilized mixed finite element scheme for the Stokes eigenvalue problem based on local Gauss integration. The two-space strategy contains solving one Stokes eigenvalue problem using theP1-P1finite element pair and then solving an additional Stokes problem using theP2-P2finite element pair. The postprocessing technique which increases the order of mixed finite element space by using the same mesh can accelerate the convergence rate of the eigenpair approximations. Moreover, our method can save a large amount of computational time and the corresponding convergence analysis is given. Finally, numerical results are presented to confirm the theoretical analysis.

Finite element modeling of nonlinear reaction–diffusion–advection systems of equations

2018 ◽  
Vol 28 (11) ◽  
pp. 2688-2715 ◽  
Author(s):  
Sanjay Komala Sheshachala ◽  
Ramon Codina
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Reaction Diffusion ◽  
Diffusion Reaction ◽  
Content Type ◽  
Numerical Examples ◽  
Stabilized Finite Element ◽  
Stabilized Finite Element Method ◽  
Nonlinear Reaction ◽  
Element Method

Purpose This paper aims to present a finite element formulation to approximate systems of reaction–diffusion–advection equations, focusing on cases with nonlinear reaction. The formulation is based on the orthogonal sub-grid scale approach, with some simplifications that allow one to stabilize only the convective term, which is the source of potential instabilities. The space approximation is combined with finite difference time integration and a Newton–Raphson linearization of the reactive term. Some numerical examples show the accuracy of the resulting formulation. Applications using classical nonlinear reaction models in population dynamics are also provided, showing the robustness of the approach proposed. Design/methodology/approach A stabilized finite element method for advection–diffusion–reaction equations to the problem on nonlinear reaction is adapted. The formulation designed has been implemented in a computer code. Numerical examples are run to show the accuracy and robustness of the formulation. Findings The stabilized finite element method from which the authors depart can be adapted to problems with nonlinear reaction. The resulting method is very robust and accurate. The framework developed is applicable to several problems of interest by themselves, such as the predator–prey model. Originality/value A stabilized finite element method to problems with nonlinear reaction has been extended. Original contributions are the design of the stabilization parameters and the linearization of the problem. The application examples, apart from demonstrating the validity of the numerical model, help to get insight in the system of nonlinear equations being solved.

A finite element solver for hypersonic flows in thermo-chemical non-equilibrium, Part II

2019 ◽  
Vol 30 (2) ◽  
pp. 575-606
Author(s):  
Song Gao ◽  
Jory Seguin ◽  
Wagdi G. Habashi ◽  
Dario Isola ◽  
Guido Baruzzi
Keyword(s):  
Heat Flux ◽  
Finite Element ◽  
Hypersonic Flow ◽  
High Speed ◽  
Unstructured Meshes ◽  
Hypersonic Flows ◽  
Anisotropic Mesh ◽  
Content Type ◽  
Edge Based ◽  
Non Equilibrium

Purpose 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. Design/methodology/approach 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. Findings 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. Originality/value 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.

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