A local domain‐free discretization method to simulate three‐dimensional compressible inviscid flows

In this paper, the domain‐free discretization method (DFD) is extended to simulate the three‐dimensional compressible inviscid flows governed by Euler equations. The discretization strategy of DFD is that the discrete form of governing equations at an interior point may involve some points outside the solution domain. The functional values at the exterior‐dependent points are updated at each time step by extrapolation along the wall normal direction in conjunction with the wall boundary conditions and the simplified momentum equation in the vicinity of the wall. Spatial discretization is achieved with the help of the finite element Galerkin approximation. The concept of ‘osculating plane’ is adopted, with which the local DFD can be easily implemented for the three‐dimensional case. Geometry‐adaptive tetrahedral mesh is employed for three‐dimensional calculations. Finally, we validate the DFD method for three‐dimensional compressible inviscid flow simulations by computing transonic flows over the ONERA M6 wing. Comparison with the reference experimental data and numerical results on boundary‐conforming grid was displayed and the results show that the present DFD results compare very well with the reference data. Copyright © 2009 John Wiley & Sons, Ltd.     

A local domain‐free discretization method for simulation of incompressible flows over moving bodies

This paper presents a local domain‐free discretization (DFD) method for the simulation of unsteady flows over moving bodies governed by the incompressible Navier–Stokes equations. The discretization strategy of DFD is that the discrete form of partial differential equations at an interior point may involve some points outside the solution domain. All the mesh points are classified as interior points, exterior dependent points and exterior independent points. The functional values at the exterior dependent points are updated at each time step by the approximate form of solution near the boundary. When the body is moving, only the status of points is changed and the mesh can stay fixed. The issue of ‘freshly cleared nodes/cells’ encountered in usual sharp interface methods does not pose any particular difficulty in the presented method. The Galerkin finite‐element approximation is used for spatial discretization, and the discrete equations are integrated in time via a dual‐time‐stepping scheme based on artificial compressibility. In order to validate the present method for moving‐boundary flow problems, two groups of flow phenomena have been simulated: (1) flows over a fixed circular cylinder, a harmonic in‐line oscillating cylinder in fluid at rest and a transversely oscillating cylinder in uniform flow; (2) flows over a pure pitching airfoil, a heaving–pitching airfoil and a deforming airfoil. The predictions show good agreement with the published numerical results or experimental data. Copyright © 2010 John Wiley & Sons, Ltd.     

Extension of domain‐free discretization method to simulate compressible flows over fixed and moving bodies

This paper is the first endeavour to present the local domain‐free discretization (DFD) method for the solution of compressible Navier–Stokes/Euler equations in conservative form. The discretization strategy of DFD is that for any complex geometry, there is no need to introduce coordinate transformation and the discrete form of governing equations at an interior point may involve some points outside the solution domain. The functional values at the exterior dependent points are updated at each time step to impose the wall boundary condition by the approximate form of solution near the boundary. Some points inside the solution domain are constructed for the approximate form of solution, and the flow variables at constructed points are evaluated by the linear interpolation on triangles. The numerical schemes used in DFD are the finite element Galerkin method for spatial discretization and the dual‐time scheme for temporal discretization. Some numerical results of compressible flows over fixed and moving bodies are presented to validate the local DFD method. Copyright © 2006 John Wiley & Sons, Ltd.     

Numerical modelling of gas–solid fluidized beds using the two‐fluid approach

This paper details an approach to modelling gas–solid fluidized beds using the two‐fluid granular temperature model. Details concerning the difficulties associated with the boundary conditions, particularly for curved boundaries, are described along with a novel means of obtaining the internal stress of the solid‐phase, in part, by solving an implicit equation. This results in a scheme that is stable even when the solid volume fraction is close to maximum packing. A transient, mixed finite element discretization is used to solve the multi‐phase equations with a discontinuous finite element representation of the granular temperature and continuity equations. A new solution method is proposed to solve the coupled momentum and continuity equations based on Arnoldi iteration. Two fluidized beds are modelled, one in the bubbling regime and the other in the slugging regime. These simulations are compared with experiments. Copyright © 2001 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.     

Non‐conforming finite elements for axisymmetric charged droplet deformation dynamics and Coulomb explosions

The deformations of both electrically charged and uncharged incompressible axisymmetric Newtonian viscous liquid droplets acting under the effects of surface tension (but not gravity) are studied using a non‐conforming, discontinuous Galerkin finite element procedure with moving meshes. The full Navier–Stokes equations are discretized and solved in an Eulerian manner with a simple predictor–corrector Lagrangian updating of the free boundary location coincident with the droplet surface at each solution iteration. By using linear Crouzeix–Raviart basis functions for the velocity and piecewise constant pressures, results are presented both for the simple (oscillatory) relaxation of elongated electrically charged/uncharged droplets to a sphere and for the deformation to steady state/Coulomb explosions of initially slightly oblate/prolate spheroidal droplets charged beyond/to the Rayleigh limit. Copyright © 2012 John Wiley & Sons, Ltd.     

A sensitivity analysis on the parameter of the GLS method for a second‐gradient theory of incompressible flow

Using a non‐conforming C⁰‐interior penalty method and the Galerkin least‐square approach, we develop a continuous–discontinuous Galerkin finite element method for discretizing fourth‐order incompressible flow problems. The formulation is weakly coercive for spaces that fail to satisfy the inf‐sup condition and consider discontinuous basis functions for the pressure field. We consider the results of a stability analysis through a lemma which indicates that there exists an optimal or quasi‐optimal least‐square stability parameter that depends on the polynomial degree used to interpolate the velocity and pressure fields, and on the geometry of the finite element in the mesh. We provide several numerical experiments illustrating such dependence, as well as the robustness of the method to deal with arbitrary basis functions for velocity and pressure, and the ability to stabilize large pressure gradients. We believe the results provided in this paper contribute for establishing a paradigm for future studies of the parameter of the Galerkin least square method for second‐gradient theory of incompressible flow problems. Copyright © 2015 John Wiley & Sons, Ltd.     

Time‐dependent finite element simulations of a shear‐thinning viscoelastic fluid with application to blood flow

A new finite element method is developed to simulate time‐dependent viscoelastic shear‐thinning flows characterized by the generalized Oldroyd‐B model. The focus of the algorithm is improved stability through a free‐energy dissipative scheme by using low‐order piecewise‐constant finite element approximations for stress. The algorithm is further modified by incorporating a pressure‐projection method, a DG‐upwinding scheme, a symmetric interior penalty DG method to solve the elliptic pressure‐update equation and a geometric multigrid preconditioner. The improved stability and cost to accuracy is compared when using higher order discontinuous bilinear approximation, where in addition, we consider the influence of a slope limiter for these elements. The algorithm is applied to the 2D start‐up‐driven cavity problem, and the stability of the free energy is illustrated and compared between element choices. An application of the model to modelling blood in small arterioles and channels is considered by simulating pulsatile blood flow through a stenotic arteriole. The individual influences of viscoelasticity and shear‐thinning within the generalized Oldroyd‐B model are investigated by comparing results to the Newtonian, generalized Newtonian and Oldroyd‐B models. Copyright © 2014 John Wiley & Sons, Ltd.     

A discontinuous Galerkin‐based sharp‐interface method to simulate three‐dimensional compressible two‐phase flow

A numerical method for the simulation of compressible two‐phase flows is presented in this paper. The sharp‐interface approach consists of several components: a discontinuous Galerkin solver for compressible fluid flow, a level‐set tracking algorithm to follow the movement of the interface and a coupling of both by a ghost‐fluid approach with use of a local Riemann solver at the interface. There are several novel techniques used: the discontinuous Galerkin scheme allows locally a subcell resolution to enhance the interface resolution and an interior finite volume Total Variation Diminishing (TVD) approximation at the interface. The level‐set equation is solved by the same discontinuous Galerkin scheme. To obtain a very good approximation of the interface curvature, the accuracy of the level‐set field is improved and smoothed by an additional P_NP_M‐reconstruction. The capabilities of the method for the simulation of compressible two‐phase flow are demonstrated for a droplet at equilibrium, an oscillating ellipsoidal droplet, and a shock‐droplet interaction problem at Mach 3. Copyright © 2015 John Wiley & Sons, Ltd.     

A spectral‐element discontinuous Galerkin thermal lattice Boltzmann method for conjugate heat transfer applications

We present a spectral‐element discontinuous Galerkin thermal lattice Boltzmann method for fluid–solid conjugate heat transfer applications. Using the discrete Boltzmann equation, we propose a numerical scheme for conjugate heat transfer applications on unstructured, non‐uniform grids. We employ a double‐distribution thermal lattice Boltzmann model to resolve flows with variable Prandtl (Pr) number. Based upon its finite element heritage, the spectral‐element discontinuous Galerkin discretization provides an effective means to model and investigate thermal transport in applications with complex geometries. Our solutions are represented by the tensor product basis of the one‐dimensional Legendre–Lagrange interpolation polynomials. A high‐order discretization is employed on body‐conforming hexahedral elements with Gauss–Lobatto–Legendre quadrature nodes. Thermal and hydrodynamic bounce‐back boundary conditions are imposed via the numerical flux formulation that arises because of the discontinuous Galerkin approach. As a result, our scheme does not require tedious extrapolation at the boundaries, which may cause loss of mass conservation. We compare solutions of the proposed scheme with an analytical solution for a solid–solid conjugate heat transfer problem in a 2D annulus and illustrate the capture of temperature continuities across interfaces for conductivity ratio γ > 1. We also investigate the effect of Reynolds (Re) and Grashof (Gr) number on the conjugate heat transfer between a heat‐generating solid and a surrounding fluid. Steady‐state results are presented for Re = 5?40 and Gr = 10⁵?10⁶. In each case, we discuss the effect of Re and Gr on the heat flux (i.e. Nusselt number Nu) at the fluid–solid interface. Our results are validated against previous studies that employ finite‐difference and continuous spectral‐element methods to solve the Navier–Stokes equations. Copyright © 2016 John Wiley & Sons, Ltd.     

Structural–acoustic sensitivity analysis of radiated sound power using a finite element/ discontinuous fast multipole boundary element scheme

The complete interaction between the structural domain and the acoustic domain needs to be considered in many engineering problems, especially for the acoustic analysis concerning thin structures immersed in water. This study employs the finite element method to model the structural parts and the fast multipole boundary element method to model the exterior acoustic domain. Discontinuous higher‐order boundary elements are developed for the acoustic domain to achieve higher accuracy in the coupling analysis. Structural–acoustic design sensitivity analysis can provide insights into the effects of design variables on radiated acoustic performance and thus is important to the structural–acoustic design and optimization processes. This study is the first to formulate equations for sound power sensitivity on structural surfaces based on an adjoint operator approach and equations for sound power sensitivity on arbitrary closed surfaces around the radiator based on the direct differentiation approach. The design variables include fluid density, structural density, Poisson's ratio, Young's modulus, and structural shape/size. A numerical example is presented to demonstrate the accuracy and validity of the proposed algorithm. Different types of coupled continuous and discontinuous boundary elements with finite elements are used for the numerical solution, and the performances of the different types of finite element/continuous and discontinuous boundary element coupling are presented and compared in detail. Copyright © 2016 John Wiley & Sons, Ltd.     

Adaptive non‐conformal mesh refinement and extended finite element method for viscous flow inside complex moving geometries

Two different techniques to analyze non‐Newtonian viscous flow in complex geometries with internal moving parts and narrow gaps are compared. The first technique is a non‐conforming mesh refinement approach based on the fictitious domain method (FDM), and the second one is the extended finite element method (XFEM). The refinement technique uses one fixed reference mesh, and to impose continuity across non‐conforming regions, constraints using Lagrangian multipliers are used. The size of elements locally in the high shear rate regions is reduced to increase accuracy. FDM is shown to have limitations; therefore, XFEM is applied to decouple the fluid from the internal moving rigid bodies. In XFEM, the discontinuous field variables are captured by using virtual degrees of freedom that serve as enrichment and by applying special integration over the intersected elements. The accuracy of the two methods is demonstrated by direct comparison with results of a boundary‐fitted mesh applied to a two‐dimensional cross section of a twin‐screw extruder. Compared with non‐conforming FDM, XFEM shows a considerable improvement in accuracy around the rigid body, especially in the narrow gap regions. Copyright © 2011 John Wiley & Sons, Ltd.     

Time‐accurate stabilized finite‐element model for weakly nonlinear and weakly dispersive water waves

Introduction of a time‐accurate stabilized finite‐element approximation for the numerical investigation of weakly nonlinear and weakly dispersive water waves is presented in this paper. To make the time approximation match the order of accuracy of the spatial representation of the linear triangular elements by the Galerkin finite‐element method, the fourth‐order time integration of implicit multistage Padé method is used for the development of the numerical scheme. The streamline‐upwind Petrov–Galerkin (SUPG) method with crosswind diffusion is employed to stabilize the scheme and suppress the spurious oscillations, usually common in the numerical computation of convection‐dominated flow problems. The performance of numerical stabilization and accuracy is addressed. Treatments of various boundary conditions, including the open boundary conditions, the perfect reflecting boundary conditions along boundaries with irregular geometry, are also described. Numerical results showing the comparisons with analytical solutions, experimental measurements, and other published numerical results are presented and discussed. Copyright © 2007 John Wiley & Sons, Ltd.     

A Petrov–Galerkin finite element model for one‐dimensional fully non‐linear and weakly dispersive wave propagation

A new finite element method is presented to solve one‐dimensional depth‐integrated equations for fully non‐linear and weakly dispersive waves. For spatial integration, the Petrov–Galerkin weighted residual method is used. The weak forms of the governing equations are arranged in such a way that the shape functions can be piecewise linear, while the weighting functions are piecewise cubic with C²‐continuity. For the time integration an implicit predictor–corrector iterative scheme is employed. Within the framework of linear theory, the accuracy of the scheme is discussed by considering the truncation error at a node. The leading truncation error is fourth‐order in terms of element size. Numerical stability of the scheme is also investigated. If the Courant number is less than 0.5, the scheme is unconditionally stable. By increasing the number of iterations and/or decreasing the element size, the stability characteristics are improved significantly. Both Dirichlet boundary condition (for incident waves) and Neumann boundary condition (for a reflecting wall) are implemented. Several examples are presented to demonstrate the range of applicabilities and the accuracy of the model. Copyright © 2001 John Wiley & Sons, Ltd.     

The improved space–time conservation element and solution element scheme for two‐dimensional dam‐break flow simulation

A new numerical scheme, namely space–time conservation element and solution element (CE/SE) method, has been used for the solution of the two‐dimensional (2D) dam‐break problem. Distinguishing from the well‐established traditional numerical methods (such as characteristics, finite difference, finite element, and finite‐volume methods), the CE/SE scheme has many non‐traditional features in both concept and methodology: space and time are treated in a unified way, which is the most important characteristic for the CE/SE method; the CEs and SEs are introduced, both local and global flux conservations in space and time rather than space only are enforced; an explicit scheme with a stagger grid is adopted. Furthermore, this scheme is robust and easy to implement. In this paper, an improved CE/SE scheme is extended to solve the 2D shallow water equations with the source terms, which usually plays a critical role in dam‐break flows. To demonstrate the accuracy, robustness and efficiency of the improved CE/SE method, both 1D and 2D dam‐break problems are simulated numerically, and the results are consistent with either the analytical solutions or experimental results. Copyright © 2011 John Wiley & Sons, Ltd.     

Finite analytic boundary condition for a higher‐order finite analytic scheme

A higher‐order finite analytic scheme based on one‐dimensional finite analytic solutions is used to discretize three‐dimensional equations governing turbulent incompressible free surface flow. In order to preserve the accuracy of the numerical scheme, a new, finite analytic boundary condition is proposed for an accurate numerical solution of the partial differential equation. This condition has higher‐order accuracy. Thus, the same order of accuracy is used for the boundary. Boundary conditions were formulated and derived for fluid inflow, outflow, impermeable surfaces and symmetry planes. The derived boundary conditions are treated implicitly and updated with the solution of the problem. The basic idea for the derivation of boundary conditions was to use the discretized form of the governing equations for the fluid flow simplified on the boundaries and flow information. To illustrate the influence of the higher‐order effects at the boundaries, another, lower‐order finite analytic boundary condition, is suggested. The simulations are performed to demonstrate the validity of the present scheme and boundary conditions for a Wigley hull advancing in calm water. Copyright © 2005 John Wiley & Sons, Ltd.     

A linearity‐preserving cell‐centered scheme for the heterogeneous and anisotropic diffusion equations on general meshes

In this paper a finite volume scheme for the heterogeneous and anisotropic diffusion equations is proposed on general, possibly nonconforming meshes. This scheme has both cell‐centered unknowns and vertex unknowns. The vertex unknowns are treated as intermediate ones and are expressed as a linear weighted combination of the surrounding cell‐centered unknowns, which reduces the scheme to a completely cell‐centered one. We propose two types of new explicit weights which allow arbitrary diffusion tensors, and are neither discontinuity dependent nor mesh topology dependent. Both the derivation of the scheme and that of new weights satisfy the linearity‐preserving criterion which requires that a discretization scheme should be exact on linear solutions. The resulting new scheme is called as the linearity‐preserving cell‐centered scheme and the numerical results show that it maintain optimal convergence rates for the solution and flux on general polygonal distorted meshes in case that the diffusion tensor is taken to be anisotropic, at times heterogeneous, and/or discontinuous. Copyright © 2010 John Wiley & Sons, Ltd.     

A new numerical model for simulations of wave transformation,breaking and long‐shore currents in complex coastal regions

In this paper, we propose a model based on a new contravariant integral form of the fully nonlinear Boussinesq equations in order to simulate wave transformation phenomena, wave breaking, and nearshore currents in computational domains representing the complex morphology of real coastal regions. The aforementioned contravariant integral form, in which Christoffel symbols are absent, is characterized by the fact that the continuity equation does not include any dispersive term. A procedure developed in order to correct errors related to the difficulties of numerically satisfying the metric identities in the numerical integration of fully nonlinear Boussinesq equation on generalized boundary‐conforming grids is presented. The Boussinesq equation system is numerically solved by a hybrid finite volume–finite difference scheme. The proposed high‐order upwind weighted essentially non‐oscillatory finite volume scheme involves an exact Riemann solver and is based on a genuinely two‐dimensional reconstruction procedure, which uses a convex combination of biquadratic polynomials. The wave breaking is represented by discontinuities of the weak solution of the integral form of the nonlinear shallow water equations. The capacity of the proposed model to correctly represent wave propagation, wave breaking, and wave‐induced currents is verified against test cases present in the literature. The results obtained are compared with experimental measures, analytical solutions, or alternative numerical solutions. Copyright © 2015 John Wiley & Sons, Ltd.     

The discontinuous profile method for simulating two‐phase flow in pipes using the single component approximation

Godunov‐type algorithms are very attractive for the numerical solution of discontinuous flows. The reconstruction of the profile inside the cells is crucial to scheme performance. The non‐linear generalization of the discontinuous profile method (DPM) presented here for the modelling of two‐phase flow in pipes uses a discontinuous reconstruction in order to capture shocks more efficiently than schemes using continuous functions. The reconstructed profile is used to define the Riemann problem at cell interfaces by averaging of the components of the variable in the base of eigenvectors over their domain of dependence. Intercell fluxes are computed by solving the Riemann problem with an approximate‐state solver. The adapted treatment of boundary conditions is essential to ensure the quality of the computational results and a specific procedure using virtual cells at both extremities of the computational domain is required. Internal boundary conditions can be treated in the same way as external ones. Application of the DPM to test cases is shown to improve the quality of computational results significantly. Copyright © 2001 John Wiley & Sons, Ltd.