A coupled continuous and discontinuous finite element method for the incompressible flows

In this paper, we develop a coupled continuous Galerkin and discontinuous Galerkin finite element method based on a split scheme to solve the incompressible Navier–Stokes equations. In order to use the equal order interpolation functions for velocity and pressure, we decouple the original Navier–Stokes equations and obtain three distinct equations through the split method, which are nonlinear hyperbolic, elliptic, and Helmholtz equations, respectively. The hybrid method combines the merits of discontinuous Galerkin (DG) and finite element method (FEM). Therefore, DG is concerned to accomplish the spatial discretization of the nonlinear hyperbolic equation to avoid using the stabilization approaches that appeared in FEM. Moreover, FEM is utilized to deal with the Poisson and Helmholtz equations to reduce the computational cost compared with DG. As for the temporal discretization, a second‐order stiffly stable approach is employed. Several typical benchmarks, namely, the Poiseuille flow, the backward‐facing step flow, and the flow around the cylinder with a wide range of Reynolds numbers, are considered to demonstrate and validate the feasibility, accuracy, and efficiency of this coupled method. Copyright © 2016 John Wiley & Sons, Ltd.     

Parallel finite element computation of unsteady incompressible flows

A parallel semi-explicit iterative finite element computational procedure for modelling unsteady incompressible fluid flows is presented. During the procedure, element flux vectors are calculated in parallel and then assembled into global flux vectors. Equilibrium iterations which introduce some ‘local implicitness’ are performed at each time step. The number of equilibrium iterations is governed by an implicitness parameter. The present technique retains the advantages of purely explicit schemes, namely (i) the parallel speed-up is equal to the number of parallel processors if the small communication overhead associated with purely explicit schemes is ignored and (ii) the computation time as well as the core memory required is linearly proportional to the number of elements. The incompressibility condition is imposed by using the artificial compressibility technique. A pressure-averaging technique which allows the use of equal-order interpolations for both velocity and pressure, this simplifying the formulation, is employed. Using a standard Galerkin approximation, three benchmark steady and unsteady problems are solved to demonstrate the accuracy of the procedure. In all calculations the Reynolds number is less than 500. At these Reynolds numbers it was found that the physical dissipation is sufficient to stabilize the convective term with no need for additional upwind-type dissipation. © 1998 John Wiley & Sons, Ltd.     

A second‐order time accurate finite element method for quasi‐incompressible viscous flows

A finite element method for quasi‐incompressible viscous flows is presented. An equation for pressure is derived from a second‐order time accurate Taylor–Galerkin procedure that combines the mass and the momentum conservation laws. At each time step, once the pressure has been determined, the velocity field is computed solving discretized equations obtained from another second‐order time accurate scheme and a least‐squares minimization of spatial momentum residuals. The terms that stabilize the finite element method (controlling wiggles and circumventing the Babuska–Brezzi condition) arise naturally from the process, rather than being introduced a priori in the variational formulation. A comparison between the present second‐order accurate method and our previous first‐order accurate formulation is shown. The method is also demonstrated in the computation of the leaky‐lid driven cavity flow and in the simulation of a crossflow past a circular cylinder. In both cases, good agreement with previously published experimental and computational results has been obtained. Copyright © 2010 John Wiley & Sons, Ltd.     

Hybrid finite element/volume method for 3D incompressible flows with heat transfer

An implicit hybrid finite element (FE)/volume solver has been extended to incompressible flows coupled with the energy equation. The solver is based on the segregated pressure correction or projection method on staggered unstructured hybrid meshes. An intermediate velocity field is first obtained by solving the momentum equations with the matrix-free implicit cell-centred finite volume (FV) method. The pressure Poisson equation is solved by the node-based Galerkin FE method for an auxiliary variable. The auxiliary variable is used to update the velocity field and the pressure field. The pressure field is carefully updated by taking into account the velocity divergence field. Our current staggered-mesh scheme is distinct from other conventional ones in that we store the velocity components at cell centres and the auxiliary variable at vertices. The Generalized Minimal Residual (GMRES) matrix-free strategy is adapted to solve the governing equations in both FE and FV methods. The presented 2D and 3D numerical examples show the robustness and accuracy of the numerical method.     

A high-order characteristics/finite element method for the incompressible Navier-Stokes equations

In this paper we consider a discretization of the incompressible Navier-Stokes equations involving a second-order time scheme based on the characteristics method and a spatial discretization of finite element type. Theoretical and numerical analyses are detailed and we obtain stability results abnd optimal eror estimates on the velocity and pressure under a time step restriction less stringent than the standard Courant-Freidrichs-Levy condition. Finally, some numerical results obtained wiht the code N3S are shown which justify the interest of this scheme and its advantages with respect to an analogous first-order time scheme. © 1997 John Wiley & Sons, Ltd.     

New stabilized finite element method for time‐dependent incompressible flow problems

A new stabilized finite element method is considered for the time‐dependent Stokes problem, based on the lowest‐order P₁?P₀ and Q₁?P₀ elements that do not satisfy the discrete inf–sup condition. The new stabilized method is characterized by the features that it does not require approximation of the pressure derivatives, specification of mesh‐dependent parameters and edge‐based data structures, always leads to symmetric linear systems and hence can be applied to existing codes with a little additional effort. The stability of the method is derived under some regularity assumptions. Error estimates for the approximate velocity and pressure are obtained by applying the technique of the Galerkin finite element method. Some numerical results are also given, which show that the new stabilized method is highly efficient for the time‐dependent Stokes problem. Copyright © 2009 John Wiley & Sons, Ltd.     

A hybrid vertex-centered finite volume/element method for viscous incompressible flows on non-staggered unstructured meshes

This paper proposes a hybrid vertex-centered finite volume/finite element method for solution of the two dimensional (2D) incompressible Navier-Stokes equations on unstructured grids.An incremental pressure fractional step method is adopted to handle the velocity-pressure coupling.The velocity and the pressure are collocated at the node of the vertex-centered control volume which is formed by joining the centroid of cells sharing the common vertex.For the temporal integration of the momentum equations,an implicit second-order scheme is utilized to enhance the computational stability and eliminate the time step limit due to the diffusion term.The momentum equations are discretized by the vertex-centered finite volume method (FVM) and the pressure Poisson equation is solved by the Galerkin finite element method (FEM).The momentum interpolation is used to damp out the spurious pressure wiggles.The test case with analytical solutions demonstrates second-order accuracy of the current hybrid scheme in time and space for both velocity and pressure.The classic test cases,the lid-driven cavity flow,the skew cavity flow and the backward-facing step flow,show that numerical results are in good agreement with the published benchmark solutions.     

A Hermite finite element method for incompressible fluid flow

We describe some Hermite stream function and velocity finite elements and a divergence‐free finite element method for the computation of incompressible flow. Divergence‐free velocity bases defined on (but not limited to) rectangles are presented, which produce pointwise divergence‐free flow fields (∇· u _h≡0). The discrete velocity satisfies a flow equation that does not involve pressure. The pressure can be recovered as a function of the velocity if needed. The method is formulated in primitive variables and applied to the stationary lid‐driven cavity and backward‐facing step test problems. Copyright © 2009 John Wiley & Sons, Ltd.     

A mixed‐interpolation finite element method for incompressible thermal flows of electrically conducting fluids

A new mixed‐interpolation finite element method is presented for the two‐dimensional numerical simulation of incompressible magnetohydrodynamic (MHD) flows which involve convective heat transfer. The proposed method applies the nodal shape functions, which are locally defined in nine‐node elements, for the discretization of the Navier–Stokes and energy equations, and the vector shape functions, which are locally defined in four‐node elements, for the discretization of the electromagnetic field equations. The use of the vector shape functions allows the solenoidal condition on the magnetic field to be automatically satisfied in each four‐node element. In addition, efficient approximation procedures for the calculation of the integrals in the discretized equations are adopted to achieve high‐speed computation. With the use of the proposed numerical scheme, MHD channel flow and MHD natural convection under a constant applied magnetic field are simulated at different Hartmann numbers. The accuracy and robustness of the method are verified through these numerical tests in which both undistorted and distorted meshes are employed for comparison of numerical solutions. Furthermore, it is shown that the calculation speed for the proposed scheme is much higher compared with that for a conventional numerical integration scheme under the condition of almost the same memory consumption. Copyright © 2016 John Wiley & Sons, Ltd.     

Development of a hybrid finite volume/element solver for incompressible flows

In this paper, we report our development of an implicit hybrid flow solver for the incompressible Navier–Stokes equations. The methodology is based on the pressure correction or projection method. A fractional step approach is used to obtain an intermediate velocity field by solving the original momentum equations with the matrix‐free implicit cell‐centred finite volume method. The Poisson equation derived from the fractional step approach is solved by the node‐based Galerkin finite element method for an auxiliary variable. The auxiliary variable is closely related to the real pressure and is used to update the velocity field and the pressure field. We store the velocity components at cell centres and the auxiliary variable at cell vertices, making the current solver a staggered‐mesh scheme. Numerical examples demonstrate the performance of the resulting hybrid scheme, such as the correct temporal convergence rates for both velocity and pressure, absence of unphysical pressure boundary layer, good convergence in steady‐state simulations and capability in predicting accurate drag, lift and Strouhal number in the flow around a circular cylinder. Copyright © 2007 John Wiley & Sons, Ltd.     

Enhancement of the accuracy of the finite volume particle method for the simulation of incompressible flows

A finite volume particle (FVP) method for simulation of incompressible flows that provides enhanced accuracy is proposed. In this enhanced FVP method, a dummy neighbor particle is introduced for each particle in the calculation and used for the discretization of the gradient model and Laplacian model. The error‐compensating term produced by introducing the dummy neighbor particle enables higher order terms to be calculated. The proposed gradient model and Laplacian model are applied in both pressure and pressure gradient calculations. This enhanced FVP scheme provides more accurate simulations of incompressible flows. Several 2‐dimensional numerical simulations are given to confirm its enhanced performance.     

The analysis of unsteady incompressible flows by a three-step finite element method

This paper describes a three-step finite element method and its applications to unsteady incompressible fluid flows. Stability analysis of the one-dimensional pure convection equation shows that this method has third-order accuracy and an extended numerical stability domain in comparison with the Lax--Wendroff finite element method. The method is cost-effective for incompressible flows because it permits less frequent updates of the pressure field with good accuracy. In contrast with the Taylor-Galerkin method, the present method does not contain any new higher-order derivatives, which makes it suitable for solving non-linear multidimensional problems and flows with complicated boundary conditions. The three-step finite element method has been used to simulate unsteady incompressible flows. The numerical results obtained are in good agreement with those in the literature.     

A least-squares finite element method for time-dependent incompressible flows with thermal convection

The time-dependent Navier–Stokes equations and the energy balance equation for an incompressible, constant property fluid in the Boussinesq approximation are solved by a least-squares finite element method based on a velocity–pressure–vorticity–temperature–heat-flux ( u –P–ω–T– q ) formulation discretized by backward finite differencing in time. The discretization scheme leads to the minimization of the residual in the l²-norm for each time step. Isoparametric bilinear quadrilateral elements and reduced integration are employed. Three examples, thermally driven cavity flow at Rayleigh numbers up to 10⁶, lid-driven cavity flow at Reynolds numbers up to 10⁴ and flow over a square obstacle at Reynolds number 200, are presented to validate the method.     

Spectral/hp penalty least‐squares finite element formulation for unsteady incompressible flows

In this paper, we present spectral/hp penalty least‐squares finite element formulation for the numerical solution of unsteady incompressible Navier–Stokes equations. Pressure is eliminated from Navier–Stokes equations using penalty method, and finite element model is developed in terms of velocity, vorticity and dilatation. High‐order element expansions are used to construct discrete form. Unlike other penalty finite element formulations, equal‐order Gauss integration is used for both viscous and penalty terms of the coefficient matrix. For time integration, space–time decoupled schemes are implemented. Second‐order accuracy of the time integration scheme is established using the method of manufactured solution. Numerical results are presented for impulsively started lid‐driven cavity flow at Reynolds number of 5000 and transient flow over a backward‐facing step. The effect of penalty parameter on the accuracy is investigated thoroughly in this paper and results are presented for a range of penalty parameter. Present formulation produces very accurate results for even very low penalty parameters (10–50). Copyright © 2008 John Wiley & Sons, Ltd.     

A new finite element formulation for both compressible and nearly incompressible fluid dynamics

A new finite element formulation designed for both compressible and nearly incompressible viscous flows is presented. The formulation combines conservative and non‐conservative dependent variables, namely, the mass–velocity (density * velocity), internal energy and pressure. The central feature of the method is the derivation of a discretized equation for pressure, where pressure contributions arising from the mass, momentum and energy balances are taken implicitly in the time discretization. The method is applied to the analysis of laminar flows governed by the Navier–Stokes equations in both compressible and nearly incompressible regimes. Numerical examples, covering a wide range of Mach number, demonstrate the robustness and versatility of the new method. Copyright © 2000 John Wiley & Sons, Ltd.     

Stabilized finite element formulation for incompressible flow on distorted meshes

Flow computations frequently require unfavourably meshes, as for example highly stretched elements in regions of boundary layers or distorted elements in deforming arbitrary Lagrangian Eulerian meshes. Thus, the performance of a flow solver on such meshes is of great interest. The behaviour of finite elements with residual‐based stabilization for incompressible Newtonian flow on distorted meshes is considered here. We investigate the influence of the stabilization terms on the results obtained on distorted meshes by a number of numerical studies. The effect of different element length definitions within the elemental stabilization parameter is considered. Further, different variants of residual‐based stabilization are compared indicating that dropping the second derivatives from the stabilization operator, i.e. using a streamline upwind Petrov–Galerkin type of formulation yields better results in a variety of cases. A comparison of the performance of linear and quadratic elements reveals further that the inconsistency of linear elements equipped with residual‐based stabilization introduces significant errors on distorted meshes, while quadratic elements are almost unaffected by moderate mesh distortion. Copyright © 2008 John Wiley & Sons, Ltd.     

Co-located equal-order control-volume finite element method for two-dimensional axisymmetric incompressible fluid flow

The formulation of a control-volume-based finite element method (CVFEM) for axisymmetric, two-dimensional, incompressible fluid flow and heat transfer in irregular-shaped domains is presented. The calculation domain is discretized into torus-shaped elements and control volumes. In a longitudinal cross-sectional plane, these elements are three-node triangles, and the control volumes are polygons obtained by joining the centroids of the three-node triangles to the mid-points of the sides. Two different interpolation schemes are proposed for the scalar-dependent variables in the advection terms: a flow-oriented upwind function, and a mass-weighted upwind function that guarantees that the discretized advection terms contribute positively to the coefficients in the discretized equations. In the discretization of diffusion transport terms, the dependent variables are interpolated linearly. An iterative sequential variable adjustment algorithm is used to solve the discretized equations for the velocity components, pressure and other scalar-dependent variables of interest. The capabilities of the proposed CVFEM are demonstrated by its application to four different example problems. The numerical solutions are compared with the results of independent numerical and experimental investigations. These comparisons are quite encouraging.     

A finite element method for compressible viscous fluid flows

This work presents a mixed three‐dimensional finite element formulation for analyzing compressible viscous flows. The formulation is based on the primitive variables velocity, density, temperature and pressure. The goal of this work is to present a ‘stable’ numerical formulation, and, thus, the interpolation functions for the field variables are chosen so as to satisfy the inf–sup conditions. An exact tangent stiffness matrix is derived for the formulation, which ensures a quadratic rate of convergence. The good performance of the proposed strategy is shown in a number of steady‐state and transient problems where compressibility effects are important such as high Mach number flows, natural convection, Riemann problems, etc., and also on problems where the fluid can be treated as almost incompressible. Copyright © 2010 John Wiley & Sons, Ltd.     

Local pressure preconditioning method for steady incompressible flows

The convergence and accuracy characteristics of the preconditioned incompressible Euler and Navier–Stokes equations are studied. An object-oriented C++ numerical code has been developed for solving the inviscid and viscous, steady, incompressible flows problems. The code is based on the cell-centred finite volume method. In this scheme, two-dimensional incompressible Euler and Navier–Stokes equations are modified by a robust artificial compressibility (AC) and a local preconditioning matrix of pressure-sensor type. The preconditioned equations are solved with the Jameson's numerical approach, i.e. artificial dissipation and artificial viscosity terms under the form of a fourth- and second-order derivative, respectively. An explicit four-stage Runge–Kutta integration algorithm is applied to obtain the steady-state condition. The computed results include the steady-state solution of flow past the NACA-hydrofoils and a circular cylinder in free stream, for which the numerical results are compared with numerical works of other researchers. Good agreement is observed. The effects of AC parameter, artificial viscosity and dissipation factor, and local preconditioning coefficient on convergence rate and solution accuracy are tested by computing flow over the NACA0012 hydrofoil. In addition, some important design criteria of a preconditioner, such as stiffness reduction, hyperbolicity, symmetrisability, accuracy preservation for M → 0, and M-property have been examined analytically.