A three‐dimensional Cartesian cut cell method for incompressible viscous flow with irregular domains

A three‐dimensional Cartesion cut cell method is presented for the simulations of incompressible viscous flows with irregular domains. A new model (referred to as ‘6+N’ model) is proposed to describe arbitrarily shaped cut cells and treat all the cells as polyhedrons with 6+N faces. The finite volume discretization of the Navier–Stokes equation is then implemented by using the ‘6+N’ model to separate the surface flux integrals into two parts, that is, the fluxes through the basic face of the hexahedron and those through the cutting surfaces. The previously proposed Kitta Cube algorithm and volume computer‐aided design platform (J. Comput. Aided. Des. 2005; 37(4): 1509–1520. Doi:10.1016/j.cad.2005.03.006) are adopted to generate cut cells and provide shape data and physical attributes for the numerical analysis. A modified SIMPLE‐based smoothing pressure correction scheme is applied to suppress checkerboard pressure oscillations caused by the collocated arrangement of velocities and pressure. The calculation accuracy of the numerical method expressed by L₁ and L _∞ norm errors is first demonstrated by the simulation of a pipe flow. Then its feasibility, efficiency, and potential in engineering applications are verified by applying it to solve natural convections between concentric spheres and between eccentric spheres. The heat transfer patterns in eccentric spheres are also obtained by using the numerical method. Copyright © 2011 John Wiley & Sons, Ltd.     

Effects of element order and interface reconstruction in FEM/volume‐of‐fluid incompressible flow simulation

In previous studies, the moment‐of‐fluid interface reconstruction method showed dramatic accuracy improvements in static and pure advection tests over existing methods, but this did not translate into an equivalent improvement in volume‐tracked multimaterial incompressible flow simulation using low‐order finite elements. In this work, the combined effects of the spatial discretization and interface reconstruction in flow simulation are examined. The mixed finite element pairs, Q₁Q₀ (with pressure stabilization) and Q₂P _{? 1} are compared. Material order‐dependent and material order‐independent first and second‐order accurate interface reconstruction methods are used. The Q₂P _{? 1} elements show significant improvements in computed flow solution accuracy for single material flows but show reduced convergence using element‐average piecewise constant density and viscosity in volume‐tracked simulations. In general, a refined Q₁Q₀ grid, with better material interface resolution, provided an accuracy similar to the Q₂P _{? 1} element grid with a comparable number of degrees of freedom. Moment‐of‐fluid shows more benefit from the higher‐order accurate flow simulation than the LVIRA, Youngs', and power diagram interface reconstruction methods, especially on unstructured grids, but does not recover the dramatic accuracy improvements it has shown in advection tests. Published 2012. This article is a US Government work and is in the public domain in the USA.     

Using staggered grids with characteristic boundary conditions when solving compressible reactive Navier–Stokes equations

A specific (hybrid) arrangement of variables is discussed to solve reactive compressible Navier–Stokes equations on staggered‐like grids with high‐order finite difference schemes. The objective is to improve the numerical flow solution at boundaries. Hybrid arrangement behaviour is compared with ‘pure’ colocated and staggered strategies. Classical Fourier analysis shows accuracy to be significantly improved in the hybrid case. One‐dimensional laminar flame test demonstrates increased robustness (in terms of mesh resolution), whereas computation of 1D exiting pressure wave propagation gives evidence that the method also improves accuracy in the prediction of non‐reflecting outflows, compared e.g. with the fully staggered scheme of (J. Comput. Phy. 2003). Multidimensional extension is illustrated through turbulent 2D planar and 3D expanding flames simulations. Copyright © 2011 John Wiley & Sons, Ltd.     

Numerical simulation of transpiration cooling through porous material

Transpiration cooling using ceramic matrix composite materials is an innovative concept for cooling rocket thrust chambers. The coolant (air) is driven through the porous material by a pressure difference between the coolant reservoir and the turbulent hot gas flow. The effectiveness of such cooling strategies relies on a proper choice of the involved process parameters such as injection pressure, blowing ratios, and material structure parameters, to name only a few. In view of the limited experimental access to the subtle processes occurring at the interface between hot gas flow and porous medium, reliable and accurate simulations become an increasingly important design tool. In order to facilitate such numerical simulations for a carbon/carbon material mounted in the side wall of a hot gas channel that are able to capture a spatially varying interplay between the hot gas flow and the coolant at the interface, we formulate a model for the porous medium flow of Darcy–Forchheimer type. A finite‐element solver for the corresponding porous medium flow is presented and coupled with a finite‐volume solver for the compressible Reynolds‐averaged Navier–Stokes equations. The two‐dimensional and three‐dimensional results at Mach number Ma = 0.5 and hot gas temperature T_HG=540 K for different blowing ratios are compared with experimental data. Copyright © 2014 John Wiley & Sons, Ltd.     

A finite element variational multiscale method for incompressible flows based on the construction of the projection basis functions

In this article, we present a finite element variational multiscale (VMS) method for incompressible flows based on the construction of projection basis functions and compare it with common VMS method, which is defined by a low‐order finite element space L_h on the same grid as X_h for the velocity deformation tensor and a stabilization parameter α. The best algorithmic feature of our method is to construct the projection basis functions at the element level with minimal additional cost to replace the global projection operator. Finally, we give some numerical simulations of the nonlinear flow problems to show good stability and accuracy properties of the method. Copyright © 2011 John Wiley & Sons, Ltd.     

Stabilized finite‐element computation of compressible flow with linear and quadratic interpolation functions

The unsteady compressible flow equations are solved using a stabilized finite‐element formulation with C⁰ elements. In 2D, the performance of three‐noded linear and six‐noded quadratic triangular elements is compared. In 3D, the relative performance is evaluated for 6‐noded linear and 18‐noded quadratic wedge elements. Results are compared for the solutions to Euler, laminar, and turbulent flows at different Mach numbers for several flow problems. The finite‐element meshes considered for comparison have same location of nodes for the linear and quadratic interpolations. For the turbulent flow, the Spalart–Allmaras model is used for closure. It is found that the quadratic elements yield better performance than the linear elements. This is attributed to accurate representation of the stabilization terms that involve second‐order derivatives in the formulation. When these terms are dropped from the formulation with quadratic interpolation, the numerical results are similar to those obtained with linear interpolation. The absence of these terms result in added numerical diffusion that seems to give the effect of a relatively reduced Reynolds number. For the same location of nodes, the computations with the linear triangular and wedge elements are approximately 20% and 100% faster than those with quadratic triangular and wedge elements, respectively. However, if the same quadrature rule for numerical integration is used for both interpolations, the computations with quadratic elements are approximately 20% and 45% faster in 2D and 3D, respectively. Copyright © 2014 John Wiley & Sons, Ltd.     

Acoustic simulation using a novel approach for reducing dispersion error

It is well‐known that the traditional finite element method (FEM) fails to provide accurate results to the Helmholtz equation with the increase of wave number because of the ‘pollution error’ caused by numerical dispersion. In order to overcome this deficiency, a gradient‐weighted finite element method (GW‐FEM) that combines Shepard interpolation and linear shape functions is proposed in this work. Three‐node triangular and four‐node tetrahedral elements that can be generated automatically are first used to discretize the problem domain in 2D and 3D spaces, respectively. For each independent element, a compacted support domain is then formed based on the element itself and its adjacent elements sharing common edges (or faces). With the aid of Shepard interpolation, a weighted acoustic gradient field is then formulated, which will be further used to construct the discretized system equations through the generalized Galerkin weak form. Numerical examples demonstrate that the present algorithm can significantly reduces the dispersion error in computational acoustics. Copyright © 2016 John Wiley & Sons, Ltd.     

An efficient and stable finite element solver of higher order in space and time for nonstationary incompressible flow

In this paper, we present fully implicit continuous Galerkin–Petrov (cGP) and discontinuous Galerkin (dG) time‐stepping schemes for incompressible flow problems which are, in contrast to standard approaches like for instance the Crank–Nicolson scheme, of higher order in time. In particular, we analyze numerically the higher order dG(1) and cGP(2) methods, which are super convergent of third, resp., fourth order in time, whereas for the space discretization, the well‐known LBB‐stable finite element pair of third‐order accuracy is used. The discretized systems of nonlinear equations are treated by using the Newton method, and the associated linear subproblems are solved by means of a monolithic (geometrical) multigrid method with a blockwise Vanka‐like smoother treating all components simultaneously. We perform nonstationary simulations (in 2D) for two benchmarking configurations to analyze the temporal accuracy and efficiency of the presented time discretization schemes w.r.t. CPU and numerical costs. As a first test problem, we consider a classical ‘flow around cylinder’ benchmark. Here, we concentrate on the nonstationary behavior of the flow patterns with periodic oscillations and examine the ability of the different time discretization schemes to capture the dynamics of the flow. As a second test case, we consider the nonstationary ‘flow through a Venturi pipe’. The objective of this simulation is to control the instantaneous and mean flux through this device. Copyright © 2013 John Wiley & Sons, Ltd.     

Managing false diffusion during second‐order upwind simulations of liquid micromixing

Liquid mixing is an important component of many microfluidic concepts and devices, and computational fluid dynamics (CFD) is playing a key role in their development and optimization. Because liquid mass diffusivities can be quite small, CFD simulation of liquid micromixing can over predict the degree of mixing unless numerical (or false) diffusion is properly controlled. Unfortunately, the false diffusion behavior of higher‐order finite volume schemes, which are often used for such simulations, is not well understood, especially on unstructured meshes. To examine and quantify the amount of false diffusion associated with the often recommended and versatile second‐order upwind method, a series of numerical simulations was conducted using a standardized two‐dimensional test problem on both structured and unstructured meshes. This enabled quantification of an ‘effective’ false diffusion coefficient (D_false) for the method as a function of mesh spacing. Based on the results of these simulations, expressions were developed for estimating the spacing required to reduce D_false to some desired (low) level. These expressions, together with additional insights from the standardized test problem and findings from other researchers, were then incorporated into a procedure for managing false diffusion when simulating steady, liquid micromixing. To demonstrate its utility, the procedure was applied to simulate flow and mixing within a representative micromixer geometry using both unstructured (triangular) and structured meshes. Copyright © 2016 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.     

On the difficulties and remedies in enforcing the div = 0 condition in the finite element analysis of thermal plumes with strongly temperature-dependent viscosity

A finite element (FE) analysis of experimentally observed creeping thermal plumes in a medium whose viscosity is strongly temperature-dependent is performed. Such plumes are considered to play an important role in numerous geological processes and numerical modelling is often the only option to study their physics. Initial simulations by means of the general-purpose Galerkin finite element package NACHOS-II demonstrated serious deficiencies of the method in modelling plumes with large viscosity contrasts, in spite of several options for the solution (mixed or penalty formulation) and the elements (continuous or discontinous pressure). In agreement with observations from FE simulations of isothermal Stokes flow in other studies, we have isolated the violation of the div = 0 or incompressibility constraint as the major culprit in the failure of the FE method. It is demonstrated that the a posteriori computed discrete divergence (DDIV) can be used as a diagnostic tool to evaluate the reliability of the FE solution and to rank the solution and element options provided in the NACHOS code. On the basis of these considerations, the combination of the mixed method with a Q₂-P₁ (discontinuous pressure) element turns out to be the most suitable for the present plume problem but is still unable to sufficiently enforce the div = 0 condition. With a goal to remedy this detrimental behaviour, several FE modifications and new approaches have been taken. These include: (i) use of a new scaling option for the governing equations which has the effect of equilbrating the stiffness matrices and thus improving their condition; (ii) implementation of several iterative solution techniques such as the iterated penalty and the Uzawa algorithm for the augmented Langrangian to better accommodate the dual role of the pressure; (iii) use of a multistep Newton method to better handle the high non-linearity of the coupled flow/transport problem. Although each of these options (or a combination of them) is able to improve on the quality of FE solution, the most startling amelioration has been gained with option (iii). Use of the latter resulted in very satisfactory modelling of the experimentally observed plumes.     

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 Dorodnitsyn finite element formulation for laminar boundary layer flow

The Dorodnitsyn boundary later formulation is given a finite element interpretation and found to generate very accurate and economical solutions when combined with an implicit, non-iterative marching scheme in the downstream direction. The algorithm is of order (Δ²u, Δx) whether linear or quadratic elements are used across the boundary layer. Solutions are compared with a Dorodnitsyn spectral formulation and a conventional finite difference formulation for three Falkner-Skan pressure gradient cases and the flow over a circular cylinder. With quadratic elements the Dorodnitsyn finite element formulation is approximately five times more efficient than the conventional finite difference formulation.     

A spectral/hp least‐squares finite element analysis of the Carreau–Yasuda fluids

A least‐squares finite element model with spectral/hp approximations was developed for steady, two‐dimensional flows of non‐Newtonian fluids obeying the Carreau–Yasuda constitutive model. The finite element model consists of velocity, pressure, and stress fields as independent variables (hence, called a mixed model). Least‐squares models offer an alternative variational setting to the conventional weak‐form Galerkin models for the Navier–Stokes equations, and no compatibility conditions on the approximation spaces used for the velocity, pressure, and stress fields are necessary when the polynomial order (p) used is sufficiently high (say, p > 3, as determined numerically). Also, the use of the spectral/hp elements in conjunction with the least‐squares formulation with high p alleviates various forms of locking, which often appear in low‐order least‐squares finite element models for incompressible viscous fluids, and accurate results can be obtained with exponential convergence. To verify and validate, benchmark problems of Kovasznay flow, backward‐facing step flow, and lid‐driven square cavity flow are used. Then the effect of different parameters of the Carreau–Yasuda constitutive model on the flow characteristics is studied parametrically. Copyright © 2016 John Wiley & Sons, Ltd.     

Orthogonality of modal bases in hp finite element models

In this paper, we exploit orthogonality of modal bases (SIAM J. Sci. Comput. 1999; 20 :1671–1695) used in hp finite element models. We calculate entries of coefficient matrix analytically without using any numerical integration, which can be computationally very expensive. We use properties of Jacobi polynomials and recast the entries of the coefficient matrix so that they can be evaluated analytically. We implement this in the context of the least‐squares finite element model although this procedure can be used in other finite element formulations. In this paper, we only develop analytical expressions for rectangular elements. Spectral convergence of the L₂ least‐squares functional is verified using exact solution of Kovasznay flow. Numerical results for transient flow over a backward‐facing step are also presented. We also solve steady flow past a circular cylinder and show the reduction in computational cost using expressions developed herein. Copyright © 2007 John Wiley & Sons, Ltd.     

Numerical analysis of the rotating viscous flow approaching a solid sphere

A numerical simulation is performed to investigate the flow induced by a sphere moving along the axis of a rotating cylindrical container filled with the viscous fluid. Three‐dimensional incompressible Navier–Stokes equations are solved using a finite element method. The objective of this study is to examine the feature of waves generated by the Coriolis force at moderate Rossby numbers and that to what extent the Taylor–Proudman theorem is valid for the viscous rotating flow at small Rossby number and large Reynolds number. Calculations have been undertaken at the Rossby numbers (R_o) of 1 and 0.02 and the Reynolds numbers (R_e) of 200 and 500. When R_o=O(1), inertia waves are exhibited in the rotating flow past a sphere. The effects of the Reynolds number and the ratio of the radius of the sphere and that of the rotating cylinder on the flow structure are examined. When R_o ? 1, as predicted by the Taylor–Proudman theorem for inviscid flow, the so‐called ‘Taylor column’ is also generated in the viscous fluid flow after an evolutionary course of vortical flow structures. The initial evolution and final formation of the ‘Taylor column’ are exhibited. According to the present calculation, it has been verified that major theoretical statement about the rotating flow of the inviscid fluid may still approximately predict the rotating flow structure of the viscous fluid in a certain regime of the Reynolds number. Copyright © 2004 John Wiley & Sons, Ltd.     

A pressure‐based unstructured grid method for all‐speed flows

An all‐speed algorithm based on the SIMPLE pressure‐correction scheme and the ‘retarded‐density’ approach has been formulated and implemented within an unstructured grid, finite volume (FV) scheme for both incompressible and compressible flows, the latter involving interaction of shock waves. The collocated storage arrangement for all variables is adopted, and the checkerboard oscillations are eliminated by using a pressure‐weighted interpolation method, similar to that of Rhie and Chow [Numerical study of the turbulent flow past an airfoil with trailing edge separation. AIAA Journal 1983; 21 : 1525]. The solution accuracy is greatly enhanced when a higher‐order convection scheme combined with adaptive mesh refinement (AMR) are used. Copyright © 2000 John Wiley & Sons, Ltd.     

Stability of some low‐order approximations for the Stokes problem

Two‐level low‐order finite element approximations are considered for the inhomogeneous Stokes equations. The elements introduced are attractive because of their simplicity and computational efficiency. In this paper, the stability of a Q₁(h)–Q₁(2h) approximation is analysed for general geometries. Using the macroelement technique, we prove the stability condition for both two‐ and three‐dimensional problems. As a result, optimal rates of convergence are found for the velocity and pressure approximations. Numerical results for three test problems are presented. We observe that for the computed examples, the accuracy of the two‐level bilinear approximation is compared favourably with some standard finite elements. Copyright © 2007 John Wiley & Sons, Ltd.     

An efficient high order direct ALE ADER finite volume scheme with a posteriori limiting for hydrodynamics and magnetohydrodynamics

In this paper, we present a new family of direct arbitrary–Lagrangian–Eulerian (ALE) finite volume schemes for the solution of hyperbolic balance laws on unstructured meshes in multiple space dimensions. The scheme is designed to be high‐order accurate both in space and time, and the mesh motion, which provides the new mesh configuration at the next time step, is taken into account in the final finite volume scheme that is based directly on a space‐time conservation formulation of the governing PDE system. To improve the computational efficiency of the algorithm, high order of accuracy in space is achieved using the a posteriori MOOD limiting strategy that allows the reconstruction procedure to be carried out with only one reconstruction stencil for any order of accuracy. We rely on an element‐local space‐time Galerkin finite element predictor on moving curved meshes to obtain a high‐order accurate one‐step time discretization, while the mesh velocity is computed by means of a suitable nodal solver algorithm that might also be supplemented with a local rezoning procedure to improve the mesh quality. Next, the old mesh configuration at time level tⁿ is connected to the new one at t^n + 1 by straight edges, hence providing unstructured space‐time control volumes, on the boundary of which the numerical flux has to be integrated. Here, we adopt a quadrature‐free integration, in which the space‐time boundaries of the control volumes are split into simplex sub‐elements that yield constant space‐time normal vectors and Jacobian matrices. In this way, the integrals over the simplex sub‐elements can be evaluated once and for all analytically during a preprocessing step. We apply the new high‐order direct ALE algorithm to the Euler equations of compressible gas dynamics (also referred to as hydrodynamics equations) as well as to the magnetohydrodynamics equations and we solve a set of classical test problems in two and three space dimensions. Numerical convergence rates are provided up to fifth order of accuracy in 2D and 3D for both hyperbolic systems considered in this paper. Finally, the efficiency of the new method is measured and carefully compared against the original formulation of the algorithm that makes use of a WENO reconstruction technique and Gaussian quadrature formulae for the flux integration: depending on the test problem, the new class of very efficient direct ALE schemes proposed in this paper can run up to ≈12 times faster in the 3D case. Copyright © 2016 John Wiley & Sons, Ltd.