Residual‐based artificial viscosity for simulation of turbulent compressible flow using adaptive finite element methods

In this paper, we present a finite element method with a residual‐based artificial viscosity for simulation of turbulent compressible flow, with adaptive mesh refinement based on a posteriori error estimation with sensitivity information from an associated dual problem. The artificial viscosity acts as a numerical stabilization, as shock capturing, and as turbulence capturing for large eddy simulation of turbulent flow. The adaptive method resolves parts of the flow indicated by the a posteriori error estimates but leaves shocks and turbulence under‐resolved in a large eddy simulation. The method is tested for examples in 2D and 3D and is validated against experimental data. Copyright © 2012 John Wiley & Sons, Ltd.     

Efficient augmented Lagrangian‐type preconditioning for the Oseen problem using Grad‐Div stabilization

Efficient preconditioning for Oseen‐type problems is an active research topic. We present a novel approach leveraging stabilization for inf‐sup stable discretizations. The Grad‐Div stabilization shares the algebraic properties with an augmented Lagrangian‐type term. Both simplify the approximation of the Schur complement, especially in the convection‐dominated case. We exploit this for the construction of the preconditioner. Solving the discretized Oseen problem with an iterative Krylov‐type method shows that the outer iteration numbers are retained independent of mesh size, viscosity, and finite element order. Thus, the preconditioner is very competitive. Copyright © 2012 John Wiley & Sons, Ltd.     

On the Finite Increment Calculus method for stabilizing advection‐diffusion equations,analysis and computation of the stabilization parameter

In this work we are concerned with the finite increment calculus (FIC) method. The method has been developed for efficient approximation of advection‐diffusion equations with high Péclet numbers. Since the natural application of FIC is within the framework of the FEM, we consider the BVP in a weak sense on finite dimensional spaces. Here we provide a result on existence and uniqueness of the solution as well as an error analysis. Also we propose a choice of the stabilization parameter. We test the method on some troublesome 2D problems. Copyright © 2011 John Wiley & Sons, Ltd.     

Parallel domain decomposition method for finite element approximation of 3D steady state non‐Newtonian fluids

We introduce a stabilized finite element method for the 3D non‐Newtonian Navier–Stokes equations and a parallel domain decomposition method for solving the sparse system of nonlinear equations arising from the discretization. Non‐Newtonian flow problems are, generally speaking, more challenging than Newtonian flows because the nonlinearities are not only in the convection term but also in the viscosity term, which depends on the shear rate. Many good iterative methods and preconditioning techniques that work well for the Newtonian flows do not work well for the non‐Newtonian flows. We employ a Galerkin/least squares finite element method, with stabilization parameters adjusted to count the non‐Newtonian effect, to discretize the equations, and the resulting highly nonlinear system of equations is solved by a Newton–Krylov–Schwarz algorithm. In this study, we apply the proposed method to some inelastic power‐law fluid flows through the eccentric annuli with inner cylinder rotation and investigate the robustness of the method with respect to some physical parameters, including the power‐law index and the Reynolds number ratios. We then report the superlinear speedup achieved by the domain decomposition algorithm on a computer with up to 512 processors. Copyright © 2015 John Wiley & Sons, Ltd.     

A stabilized finite element method for the Stokes problem based on polynomial pressure projections

A new stabilized finite element method for the Stokes problem is presented. The method is obtained by modification of the mixed variational equation by using local L² polynomial pressure projections. Our stabilization approach is motivated by the inherent inconsistency of equal‐order approximations for the Stokes equations, which leads to an unstable mixed finite element method. Application of pressure projections in conjunction with minimization of the pressure–velocity mismatch eliminates this inconsistency and leads to a stable variational formulation. Unlike other stabilization methods, the present approach does not require specification of a stabilization parameter or calculation of higher‐order derivatives, and always leads to a symmetric linear system. The new method can be implemented at the element level and for affine families of finite elements on simplicial grids it reduces to a simple modification of the weak continuity equation. Numerical results are presented for a variety of equal‐order continuous velocity and pressure elements in two and three dimensions. Copyright © 2004 John Wiley & Sons, Ltd.     

An immersed boundary solver for inviscid compressible flows

In this paper, a simple and efficient immersed boundary (IB) method is developed for the numerical simulation of inviscid compressible Euler equations. We propose a method based on coordinate transformation to calculate the unknowns of ghost points. In the present study, the body‐grid intercept points are used to build a complete bilinear (2‐D)/trilinear (3‐D) interpolation. A third‐order weighted essentially nonoscillation scheme with a new reference smoothness indicator is proposed to improve the accuracy at the extrema and discontinuity region. The dynamic blocked structured adaptive mesh is used to enhance the computational efficiency. The parallel computation with loading balance is applied to save the computational cost for 3‐D problems. Numerical tests show that the present method has second‐order overall spatial accuracy. The double Mach reflection test indicates that the present IB method gives almost identical solution as that of the boundary‐fitted method. The accuracy of the solver is further validated by subsonic and transonic flow past NACA2012 airfoil. Finally, the present IB method with adaptive mesh is validated by simulation of transonic flow past 3‐D ONERA M6 Wing. Global agreement with experimental and other numerical results are obtained.     

The partition‐of‐unity method for linear diffusion and convection problems: accuracy,stabilization and multiscale interpretation

We investigate the effectiveness of the partition‐of‐unity method (PUM) for convection–diffusion problems. We show that for the linear diffusion equation, an exponential enrichment function based on an approximation of the analytic solution leads to improved accuracy compared to the standard finite‐element method. It is illustrated that this approach can be more efficient than using polynomial enrichment to increase the order of the scheme. We argue that the PUM enrichment, can be interpreted as a subgrid‐scale model in a multiscale framework, and that the choice of enrichment function has consequences for the stabilization properties of the method. The exponential enrichment is shown to function as a near optimal subgrid‐scale model for linear convection. Copyright © 2003 John Wiley & Sons, Ltd.     

On pressure separation algorithms (PSepA) for improving the accuracy of incompressible flow simulations

We investigate a special technique called ‘pressure separation algorithm’ (PSepA) (see Applied Mathematics and Computation 2005; 165 :275–290 for an introduction) that is able to significantly improve the accuracy of incompressible flow simulations for problems with large pressure gradients. In our numerical studies with the computational fluid dynamics package FEATFLOW ( www.featflow.de ), we mainly focus on low‐order Stokes elements with nonconforming finite element approximations for the velocity and piecewise constant pressure functions. However, preliminary numerical tests show that this advantageous behavior can also be obtained for higher‐order discretizations, for instance, with Q₂/P₁ finite elements. We analyze the application of this simple, but very efficient, algorithm to several stationary and nonstationary benchmark configurations in 2D and 3D (driven cavity and flow around obstacles), and we also demonstrate its effect to spurious velocities in multiphase flow simulations (‘static bubble’ configuration) if combined with edge‐oriented, resp., interior penalty finite element method stabilization techniques. Copyright © 2008 John Wiley & Sons, Ltd.     

Design and analysis of a Schwarz coupling method for a dimensionally heterogeneous problem

In the present work, we propose and analyse an efficient iterative coupling method for a dimensionally heterogeneous problem. We consider the case of a 2D Laplace equation with non‐symmetric boundary conditions coupled with a corresponding 1D Laplace equation. We first show how to obtain the 1D model from the 2D one by integration along one direction, by analogy with the link between shallow water equations and the Navier–Stokes system. Then we focus on the design of a Schwarz‐like iterative coupling method. We discuss the choice of boundary conditions at coupling interfaces. We prove the convergence of such algorithms and give some theoretical results related to the choice of the location of the coupling interface, and to the control of the difference between a global 2D reference solution and the 2D coupled solution. These theoretical results are illustrated numerically. Copyright © 2014 John Wiley & Sons, Ltd.     

The Multidimensional Optimal Order Detection method in the three‐dimensional case: very high‐order finite volume method for hyperbolic systems

The Multidimensional Optimal Order Detection (MOOD) method for two‐dimensional geometries has been introduced by the authors in two recent papers. We present here the extension to 3D mixed meshes composed of tetrahedra, hexahedra, pyramids, and prisms. In addition, we simplify the u2 detection process previously developed and show on a relevant set of numerical tests for both the convection equation and the Euler system that the optimal high order of accuracy is reached on smooth solutions, whereas spurious oscillations near singularities are prevented. At last, the intrinsic positivity‐preserving property of the MOOD method is confirmed in 3D, and we provide simple optimizations to reduce the computational cost such that the MOOD method is very competitive compared with existing high‐order Finite Volume methods.Copyright © 2013 John Wiley & Sons, Ltd.     

An efficient and accurate algebraic interface capturing method for unstructured grids in 2 and 3 dimensions: The THINC method with quadratic surface representation

We present in this paper an efficient and accurate volume of fluid (VOF) type scheme to compute moving interfaces on unstructured grids with arbitrary quadrilateral mesh elements in 2D and hexahedral elements in 3D. Being an extension of the multi‐dimensional tangent of hyperbola interface capturing (THINC) reconstruction proposed by the authors in Cartesian grid, an algebraic VOF scheme is devised for arbitrary quadrilateral and hexahedral elements. The interface is cell‐wisely approximated by a quadratic surface, which substantially improves the numerical accuracy. The same as the other THINC type schemes, the present method does not require the explicit geometric representation of the interface when computing numerical fluxes and thus is very computationally efficient and straightforward in implementation. The proposed scheme has been verified by benchmark tests, which reveal that this scheme is able to produce high‐quality numerical solutions of moving interfaces in unstructured grids and thus a practical method for interfacial multi‐phase flow simulations. Copyright © 2014 John Wiley & Sons, Ltd.     

On the discontinuous Galerkin method for the numerical solution of the Navier–Stokes equations

The paper deals with the use of the discontinuous Galerkin finite element method (DGFEM) for the numerical solution of viscous compressible flows. We start with a scalar convection–diffusion equation and present a discretization with the aid of the non‐symmetric variant of DGFEM with interior and boundary penalty terms. We also mention some theoretical results. Then we extend the scheme to the system of the Navier–Stokes equations and discuss the treatment of stabilization terms. Several numerical examples are presented. Copyright © 2004 John Wiley & Sons, Ltd.     

Viscous regularization of the full set of nonequilibrium‐diffusion Grey Radiation‐Hydrodynamic equations

A viscous regularization technique, based on the local entropy residual, was proposed by Delchini et al. (2015) to stabilize the nonequilibrium‐diffusion Grey Radiation‐Hydrodynamic equations using an artificial viscosity technique. This viscous regularization is modulated by the local entropy production and is consistent with the entropy minimum principle. However, Delchini et al. (2015) only based their work on the hyperbolic parts of the Grey Radiation‐Hydrodynamic equations and thus omitted the relaxation and diffusion terms present in the material energy and radiation energy equations. Here, we extend the theoretical grounds for the method and derive an entropy minimum principle for the full set of nonequilibrium‐diffusion Grey Radiation‐Hydrodynamic equations. This further strengthens the applicability of the entropy viscosity method as a stabilization technique for radiation‐hydrodynamic shock simulations. Radiative shock calculations using constant and temperature‐dependent opacities are compared against semi‐analytical reference solutions, and we present a procedure to perform spatial convergence studies of such simulations. Copyright © 2017 John Wiley & Sons, Ltd.     

A least‐square weighted residual method for level set formulation

In this paper, a least‐square weighted residual method (LSWRM) for level set (LS) formulation is introduced to achieve interface capturing in two‐dimensional (2D) and three‐dimensional (3D) problems. An LSWRM was adopted for two semi‐discretized advection and reinitialization equations of the LS formulation. The present LSWRM provided good mathematical properties such as natural numerical diffusion and the symmetry of the resulting algebraic systems for the advection and reinitialization equations. The proposed method was validated by solving some 2D and 3D benchmark problems such as those involving a rotating slotted disk, the rotation of a slotted sphere, and a time‐reversed single‐vortex flow and a deformation problem of a spherical fluid. The numerical results were compared with those obtained from essentially non‐oscillatory type formulations and particle LS methods. Further, the proposed LSWRM for the LS formulation was coupled with a splitting finite element method code to solve the incompressible Navier–Stokes equations, and then, the collapse of a 3D broken dam flow was well simulated; in the simulation, the entrapping of air and the splashing of the surge front of water were reproduced. The mass conservation of the present method was found to be satisfactory during the entire simulation. Copyright © 2011 John Wiley & Sons, Ltd.     

A stabilized finite element formulation for high‐speed inviscid compressible flows using finite calculus

This paper aims at the development of a new stabilization formulation based on the finite calculus (FIC) scheme for solving the Euler equations using the Galerkin FEM on unstructured triangular grids. The FIC method is based on expressing the balance of fluxes in a space–time domain of finite size. It is used to prevent the creation of instabilities typically present in numerical solutions due to the high convective terms and sharp gradients. Two stabilization terms, respectively called streamline term and transverse term, are added via the FIC formulation to the original conservative equations in the space–time domain. An explicit fourth‐order Runge–Kutta scheme is implemented to advance the solution in time. The presented numerical test examples for inviscid flows prove the ability of the proposed stabilization technique for providing appropriate solutions especially near shock waves. Although the derived methodology delivers precise results with a nearly coarse mesh, a mesh refinement technique is coupled to the solution process for obtaining a suitable mesh particularly in the high‐gradient zones. Copyright © 2014 John Wiley & Sons, Ltd.     

A new positive‐definite regularization of incompressible Navier–Stokes equations discretized with Q1/P0 finite element

A new regularization method is proposed for the Galerkin approximation of the incompressible Navier–Stokes equations with Q1/P0 element, by newly introducing a square‐type linear form into the variational divergence‐free constraint regularized with the global pressure jump (GPJ) method. The addition of the square‐type linear form is intended to eliminate the hydrostatic pressure mode appearing in confined flows, and to make the discretized matrix positive definite and then non‐singular without the pressure pegging trick. Effects of the free parameters for the regularization on the solutions are numerically examined with a 2‐D driven cavity flow problem. Furthermore, the convergences in the conjugate gradient iteration for the solution of the pressure Poisson equation are compared among the mixed method, the GPJ method and the present method for both leaky and non‐leaky 3‐D driven cavity flows. Finally, the non‐leaky 3‐D cavity flows at different Re numbers are solved to compare with the literature data and to demonstrate the accuracy of the proposed method. Copyright © 2003 John Wiley & Sons, Ltd.     

Flux schemes based finite volume method for internal transonic flow with condensation

This paper describes the development of numerical method for the solution of condensing steam flow in internal aerodynamics problems. The numerical method is based on the fractional step method, where the resulting set of ODEs is solved by the two‐stage Runge–Kutta method and the homogeneous set of PDEs by a finite volume method. The flow does not contain both phases (gas and liquid) in the whole domain, therefore we discuss properties of used finite volume methods in several cases of single‐phase transonic flow in a channel and a turbine cascade. We present numerical results of two‐phase flow of condensing steam in 2D nozzle achieved by several numerical methods and show the differences in results caused by numerical diffusion. Copyright © 2010 John Wiley & Sons, Ltd.     

Level‐set based flamelet approach for simulating turbulent lifted jet flame

The present study focuses on numerically investigating the flame structure, flame liftoff, and stabilization in a lifted turbulent H₂/N₂ jet flame with a vitiated coflow. To realistically represent the turbulent partially premixed nature in the flow region between nozzle exit and flame base, the level‐set approach coupled with the conserved scalar flamelet model has been applied. The unstructured‐grid level‐set approach has been developed to allow the geometric flexibility and computational efficiency for the solution of the physically and geometrically complex reacting flows. The pressure–velocity coupling is handled by the multiple pressure‐correction method. The predicted flame pattern is in good conformity with the measured one. In terms of the liftoff height, the agreement between prediction and experiment is quite good. Even if there are noticeable deviations in a certain region, the predicted profiles for the overall flame structure agree reasonably well with the experimental data. These numerical results indicate that the present level‐set‐based flamelet approach in conjunction with the unstructured‐grid finite‐volume method is capable of realistically predicting the essential features and precise structure of the turbulent‐lifted jet flame with computational efficiency. Copyright © 2008 John Wiley & Sons, Ltd.