A staggered control volume scheme for unstructured triangular grids

The purpose of this work is to introduce and validate a new staggered control volume method for the simulation of 2D/axisymmetric incompressible flows. The present study introduces a numerical procedure for solving the Navier–Stokes equations using the primitive variable formulation. The proposed method is an extension of the staggered grid methodology to unstructured triangular meshes for a control volume approach which features ease of handling of irregularly shaped domains. Two alternative elements are studied: transported scalars are stored either at the sides of an element or at its vertices, while the pressure is always stored at the centre of an element. Two interpolation functions were investigated for the integration of the momentum equations: a skewed mass-weighted upwind function and a flow-oriented exponential shape function. The momentum equations are solved over the covolume of a side or of a vertex and the pressure–velocity coupling makes use of a localized linear reconstruction of the discontinuous pressure field surrounding an element in order to obtain the pressure gradient terms. The pressure equation is obtained through a discretization of the continuity equation which uses the triangular element itself as the control volume. The method is applied to the simulation of the following test cases: backward-facing step flow, flow over a two-dimensional obstacle and flow in a pipe with sudden contraction of cross-sectional area. All numerical investigations are compared with experimental data from the literature. A grid convergence and error analysis study is also carried out for flow in a driven cavity. Results compared favourably with experimental data and so the new control volume scheme is deemed well suited for the prediction of incompressible flows in complex geometries. © 1997 John Wiley & Sons, Ltd.     

Numerical analysis of conservative unstructured discretisations for low Mach flows

Unstructured meshes allow easily representing complex geometries and to refine in regions of interest without adding control volumes in unnecessary regions. However, numerical schemes used on unstructured grids have to be properly defined in order to minimise numerical errors. An assessment of a low Mach algorithm for laminar and turbulent flows on unstructured meshes using collocated and staggered formulations is presented. For staggered formulations using cell‐centred velocity reconstructions, the standard first‐order method is shown to be inaccurate in low Mach flows on unstructured grids. A recently proposed least squares procedure for incompressible flows is extended to the low Mach regime and shown to significantly improve the behaviour of the algorithm. Regarding collocated discretisations, the odd–even pressure decoupling is handled through a kinetic energy conserving flux interpolation scheme. This approach is shown to efficiently handle variable‐density flows. Besides, different face interpolations schemes for unstructured meshes are analysed. A kinetic energy‐preserving scheme is applied to the momentum equations, namely, the symmetry‐preserving scheme. Furthermore, a new approach to define the far‐neighbouring nodes of the quadratic upstream interpolation for convective kinematics scheme is presented and analysed. The method is suitable for both structured and unstructured grids, either uniform or not. The proposed algorithm and the spatial schemes are assessed against a function reconstruction, a differentially heated cavity and a turbulent self‐igniting diffusion flame. It is shown that the proposed algorithm accurately represents unsteady variable‐density flows. Furthermore, the quadratic upstream interpolation for convective kinematics scheme shows close to second‐order behaviour on unstructured meshes, and the symmetry‐preserving is reliably used in all computations. Copyright © 2016 John Wiley & Sons, Ltd.     

A high‐precision unstructured adaptive mesh technique for gas–liquid two‐phase flows

Adaptive mesh techniques are used widely in the numerical simulations of fluid flows, and the simulation results with high accuracies are obtained by appropriate mesh adaptations. However, gas–liquid two‐phase flows are still difficult to be simulated on adaptive meshes, especially on unstructured adaptive meshes, because the physical phenomena near gas–liquid interfaces are highly complicated and in general, not modeled appropriately on adaptive meshes. In this paper, a high‐precision unstructured adaptive mesh technique for gas–liquid two‐phase flows is developed and verified/validated. In the unstructured adaptive mesh technique, the PLIC algorithm is employed to simulate interfacial dynamic behaviors and, therefore, the reconstruction method for the interfaces in refined cells is developed, which satisfies the gas and liquid volume conservations and geometrical conservations of interfaces. In addition, the physics‐based consideration is performed on the momentum calculations near interfaces, and the calculation method with gas and liquid momentum conservations is developed. For verification, the slotted‐disk revolution problem is solved. As a result, the unstructured adaptive mesh technique succeeds in reproducing the slotted‐disk shape accurately and well maintaining the shape after one full‐revolution. The dam‐break problem is also simulated and the momentum conservative calculation method succeeds in providing physically appropriate results, which show good agreements with experimental data. Therefore, it is confirmed that the developed unstructured adaptive mesh technique is very efficient to simulate gas–liquid two‐phase flows accurately. Copyright © 2010 John Wiley & Sons, Ltd.     

A semi‐implicit scheme for 3D free surface flows with high‐order velocity reconstruction on unstructured Voronoi meshes

In this paper, we present a computationally efficient semi‐implicit scheme for the simulation of three‐dimensional hydrostatic free surface flow problems on staggered unstructured Voronoi meshes. For each polygonal control volume, the pressure is defined in the cell center, whereas the discrete velocity field is given by the normal velocity component at the cell faces. A piecewise high‐order polynomial vector velocity field is then reconstructed from the scalar normal velocities at the cell faces by using a new high‐order constrained least‐squares reconstruction operator. The reconstructed high‐order piecewise polynomial velocity field is used for trajectory integration in a semi‐Lagrangian approach to discretize the nonlinear convective terms in the governing PDE. For that purpose, a high‐order Taylor method is used as ODE integrator. The resulting semi‐implicit algorithm is extensively validated on a large set of different academic test problems with exact analytical solution and is finally applied to a real‐world engineering problem consisting of a curved channel upstream of two micro‐turbines of a hydroelectric power plant. For this realistic case, some experimental reference data are available from field measurements. Copyright © 2012 John Wiley & Sons, Ltd.     

A gas‐kinetic scheme for the simulation of turbulent flows on unstructured meshes

This paper presents a numerical method for simulating turbulent flows via coupling the Boltzmann BGK equation with Spalart–Allmaras one equation turbulence model. Both the Boltzmann BGK equation and the turbulence model equation are carried out using the finite volume method on unstructured meshes, which is different from previous works on structured grid. The application of the gas‐kinetic scheme is extended to the simulation of turbulent flows with arbitrary geometries. The adaptive mesh refinement technique is also adopted to reduce the computational cost and improve the efficiency of meshes. To organize the unstructured mesh data structure efficiently, a non‐manifold hybrid mesh data structure is extended for polygonal cells. Numerical experiments are performed on incompressible flow over a smooth flat plate and compressible turbulent flows around a NACA 0012 airfoil using unstructured hybrid meshes. These numerical results are found to be in good agreement with experimental data and/or other numerical solutions, demonstrating the applicability of the proposed method to simulate both subsonic and transonic turbulent flows. Copyright © 2016 John Wiley & Sons, Ltd.     

A staggered conservative scheme for every Froude number in rapidly varied shallow water flows

This paper proposes a numerical technique that in essence is based upon the classical staggered grids and implicit numerical integration schemes, but that can be applied to problems that include rapidly varied flows as well. Rapidly varied flows occur, for instance, in hydraulic jumps and bores. Inundation of dry land implies sudden flow transitions due to obstacles such as road banks. Near such transitions the grid resolution is often low compared to the gradients of the bathymetry. In combination with the local invalidity of the hydrostatic pressure assumption, conservation properties become crucial. The scheme described here, combines the efficiency of staggered grids with conservation properties so as to ensure accurate results for rapidly varied flows, as well as in expansions as in contractions. In flow expansions, a numerical approximation is applied that is consistent with the momentum principle. In flow contractions, a numerical approximation is applied that is consistent with the Bernoulli equation. Both approximations are consistent with the shallow water equations, so under sufficiently smooth conditions they converge to the same solution. The resulting method is very efficient for the simulation of large‐scale inundations. Copyright © 2003 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 Godunov‐type fractional semi‐implicit method based on staggered grid for dam‐break modeling

A semi‐implicit finite volume model based upon staggered grid is presented for solving shallow water equation. The model employs a time‐splitting scheme that uses a predictor–corrector method for the advection term. The fluxes are calculated based on a Riemann solver in the prediction step and a downwind scheme in the correction step. A simple TVD scheme is employed for shock capturing purposes in which the Minmond limiter is used for flux functions. As a consequence of using staggered grid, an ADI method is adopted for solving the discretized equations for 2‐D problems. Several 1‐D and 2‐D flows have been modeled with satisfactory results when compared with analytical and experimental test cases. The model is also capable of simulating supercritical as well as subcritical flow. Copyright © 2010 John Wiley & Sons, Ltd.     

A hybrid scheme based on finite element/volume methods for two immiscible fluid flows

We have successfully extended our implicit hybrid finite element/volume (FE/FV) solver to flows involving two immiscible fluids. 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‐centered 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. This updating strategy can be rigorously proven to be able to eliminate the unphysical pressure boundary layer and is crucial for the correct temporal convergence rate. Our current staggered‐mesh scheme is distinct from other conventional ones in that we store the velocity components at cell centers and the auxiliary variable at vertices. The fluid interface is captured by solving an advection equation for the volume fraction of one of the fluids. The same matrix‐free FV method, as the one used for momentum equations, is used to solve the advection equation. We will focus on the interface sharpening strategy to minimize the smearing of the interface over time. We have developed and implemented a global mass conservation algorithm that enforces the conservation of the mass for each fluid. Copyright © 2009 John Wiley & Sons, Ltd.     

A second-order upwind finite-volume method for the Euler solution on unstructured triangular meshes

A scheme for the numerical solution of the two-dimensional (2D) Euler equations on unstructured triangular meshes has been developed. The basic first-order scheme is a cell-centred upwind finite-volume scheme utilizing Roe's approximate Riemann solver. To obtain second-order accuracy, a new gradient based on the weighted average of Barth and Jespersen's three-point support gradient model is used to reconstruct the cell interface values. Characteristic variables in the direction of local pressure gradient are used in the limiter to minimize the numerical oscillation around solution discontinuities. An Approximate LU (ALU) factorization scheme originally developed for structured grid methods is adopted for implicit time integration and shows good convergence characterisitics in the test. To eliminate the data dependency which prohibits vectorization in the inversion process, a black-gray-white colouring and numbering technique on unstructured triangular meshes is developed for the ALU factorization scheme. This results in a high degree of vectorization of the final code. Numerical experiments on transonic Ringleb flow, transonic channel flow with circular bump, supersonic shock reflection flow and subsonic flow over multielement aerofoils are calculated to validate the methodology.     

A stable and accurate projection method on a locally refined staggered mesh

In this paper, a robust projection method on a locally refined mesh is proposed for two‐ and three‐dimensional viscous incompressible flows. The proposed method is robust not only when the interface between two meshes is located in a smooth flow region but also when the interface is located in a flow region with large gradients and/or strong unsteadiness. In numerical simulations, a locally refined mesh saves many grid points in regions of relatively small gradients compared with a uniform mesh. For efficiency and ease of implementation, we consider a two‐level blocked structure, for which both of the coarse and fine meshes are uniform Cartesian ones individually. Unfortunately, the introduction of the two‐level blocked mesh results in an important but difficult issue: coupling of the coarse and fine meshes. In this paper, by properly addressing the issue of the coupling, we propose a stable and accurate projection method on a locally refined staggered mesh for both two‐ and three‐dimensional viscous incompressible flows. The proposed projection method is based on two principles: the linear interpolation technique and the consistent discretization of both sides of the pressure Poisson equation. The proposed algorithm is straightforward owing to the linear interpolation technique, is stable and accurate, is easy to extend from two‐ to three‐dimensional flows, and is valid even when flows with large gradients cross the interface between the two meshes. The resulting pressure Poisson equation is non‐symmetric on a locally refined mesh. The numerical results for a series of exact solutions for 2D and 3D viscous incompressible flows verify the stability and accuracy of the proposed projection method. The method is also applied to some challenging problems, including turbulent flows around particles, flows induced by impulsively started/stopped particles, and flows induced by particles near solid walls, to test the stability and accuracy. Copyright © 2010 John Wiley & Sons, Ltd.     

Parallel implementation of a non‐hydrostatic model for free surface flows with semi‐Lagrangian advection treatment

The parallel implementation of an unstructured‐grid, three‐dimensional, semi‐implicit finite difference and finite volume model for the free surface Navier–Stokes equations (UnTRIM ) is presented and discussed. The new developments are aimed to make the code available for high‐performance computing in order to address larger, complex problems in environmental free surface flows. The parallelization is based on the mesh partitioning method and message passing and has been achieved without negatively affecting any of the advantageous properties of the serial code, such as its robustness, accuracy and efficiency. The key issue is a new, autonomous parallel streamline backtracking algorithm, which allows using semi‐Lagrangian methods in decomposed meshes without compromising the scalability of the code. The implementation has been carefully verified not only with simple, abstract test cases illustrating the application domain of the code but also with advanced, high‐resolution models presently applied for research and engineering projects. The scheme performance and accuracy aspects are researched and discussed. Copyright © 2008 John Wiley & Sons, Ltd.     

A Hybrid Domain Decomposition and Multigrid Method for the Acceleration of Compressible Viscous Flow Calculations on Unstructured Triangular Meshes

This paper is concerned with the formulation and the evaluation of a hybrid solution method that makes use of domain decomposition and multigrid principles for the calculation of two-dimensional compressible viscous flows on unstructured triangular meshes. More precisely, a non-overlapping additive domain decomposition method is used to coordinate concurrent subdomain solutions with a multigrid method. This hybrid method is developed in the context of a flow solver for the Navier-Stokes equations which is based on a combined finite element/finite volume formulation on unstructured triangular meshes. Time integration of the resulting semi-discrete equations is performed using a linearized backward Euler implicit scheme. As a result, each pseudo time step requires the solution of a sparse linear system. In this study, a non-overlapping domain decomposition algorithm is used for advancing the solution at each implicit time step. Algebraically, the Schwarz algorithm is equivalent to a Jacobi iteration on a linear system whose matrix has a block structure. A substructuring technique can be applied to this matrix in order to obtain a fully implicit scheme in terms of interface unknowns. In the present approach, the interface unknowns are numerical fluxes. The interface system is solved by means of a full GMRES method. Here, the local system solves that are induced by matrix-vector products with the interface operator, are performed using a multigrid by volume agglomeration method. The resulting hybrid domain decomposition and multigrid solver is applied to the computation of several steady flows around a geometry of NACA0012 airfoil.     

Implicit schemes for unsteady Euler equations on triangular meshes

An implicit finite element method is presented for the solution of steady and unsteady inviscid compressible flows on triangular meshes under transonic conditions. The method involves a first-order time-stepping scheme with a finite element discretization that reduces to central differencing on a rectangular mesh. On a solid wall the slip condition is prescribed and the pressure is obtained from an approximation of the normal momentum equation. With this solver no artificial viscosity is added to ensure the success of the calculation. Numerical examples are given for steady and unsteady cases.     

A finite volume unstructured multigrid method for efficient computation of unsteady incompressible viscous flows

An unstructured non‐nested multigrid method is presented for efficient simulation of unsteady incompressible Navier–Stokes flows. The Navier–Stokes solver is based on the artificial compressibility approach and a higher‐order characteristics‐based finite‐volume scheme on unstructured grids. Unsteady flow is calculated with an implicit dual time stepping scheme. For efficient computation of unsteady viscous flows over complex geometries, an unstructured multigrid method is developed to speed up the convergence rate of the dual time stepping calculation. The multigrid method is used to simulate the steady and unsteady incompressible viscous flows over a circular cylinder for validation and performance evaluation purposes. It is found that the multigrid method with three levels of grids results in a 75% reduction in CPU time for the steady flow calculation and 55% reduction for the unsteady flow calculation, compared with its single grid counterparts. The results obtained are compared with numerical solutions obtained by other researchers as well as experimental measurements wherever available and good agreements are obtained. Copyright © 2004 John Wiley & Sons, Ltd.     

Comparing analytical and numerical solution of nonlinear two and three‐dimensional hydrostatic flows

New test cases for frictionless, three‐dimensional hydrostatic flows have been derived from some known analytical solutions of the two‐dimensional shallow water equations. The flow domain is a paraboloid of revolution and the flow is determined by the initial conditions, the nonlinear advective terms, the Coriolis acceleration and by the hydrostatic pressure. Wetting and drying is also included. Some specific properties of the exact solutions are discussed under different hypothesis and relative importance of the forcing terms. These solutions are proposed for testing the stability, the accuracy and the efficiency of numerical models to be used for simulating environmental hydrostatic flows. The computed solutions obtained with a semi‐implicit finite difference—finite volume algorithm on unstructured grid are compared with the corresponding analytical solutions in both two and three space dimension. Excellent agreement are obtained for the velocity and for the resulting water surface elevation. Comparison of the computed inundation area also shows a good agreement with the analytical solution with degrading accuracy observed when the inundation area becomes relatively large and for long simulation time. Copyright © 2006 John Wiley & Sons, Ltd.     

High-order ALE method for the Navier–Stokes equations on a moving hybrid unstructured mesh using flux reconstruction method

The flux reconstruction (FR) formulation can unify several popular discontinuous basis high-order methods for fluid dynamics, including the discontinuous Galerkin method, in a simple, efficient form. An arbitrary Lagrangian–Eulerian (ALE) extension to the high-order FR scheme is developed here for moving mesh fluid flow problems. The ALE Navier–Stokes equations are derived by introducing a grid velocity. The conservation law are spatially discretised on hybrid unstructured meshes using Huynh’s scheme (Huynh 2007) on anisotropic elements (quadrilaterals) and using Correction Procedure via Reconstruction scheme on isotropic elements (triangles). The temporal discretisation uses both explicit and implicit treatments. The mesh movement is described by node positions given as a time series, instead of an analytical formula. The geometric conservation law is tested using free stream preservation problem. An isentropic vortex propagation test case is performed to show the high-order accuracy of the developed method on both moving and fixed hybrid meshes. Flow around an oscillating cylinder shows the capability of the method to solve moving boundary viscous flow problems, with the numeric method further verified by comparison of the result on a smoothly deforming mesh and a rigid moving mesh.     

Analytical interface reconstruction algorithms in the PLIC-VOF method for 3D polyhedral unstructured meshes

Piecewise linear interface calculation (PLIC) schemes have been extensively employed in the volume-of-fluid (VOF) method for interface capturing in numerical simulations of multiphase flows. Polyhedral unstructured meshes are often adopted due to their geometric flexibility and superiority in gradient calculation. Four analytical interface reconstruction algorithms in the PLIC-VOF method for arbitrary convex polyhedral cells have been proposed in this study. The plane interface at a given orientation within a polyhedral cell is located by four different analytical techniques. They have been tested successfully for six different geometric shapes that are common in polyhedral meshes. The computational efficiencies of the algorithms have been compared with two other published schemes in the literature. The proposed algorithms have been shown to yield smaller truncation errors with reduction in computational complexity. A numerical simulation of a 3D dam-breaking problem has been successfully performed using the proposed interface reconstruction scheme on a polyhedral mesh. The percentage of the overall computational time consumed has been assessed to justify its optimization in a real multiphase flow simulation.     

A parallel implicit domain decomposition algorithm for the large eddy simulation of incompressible turbulent flows on 3D unstructured meshes

We present a parallel fully implicit algorithm for the large eddy simulation (LES) of incompressible turbulent flows on unstructured meshes in three dimensions. The LES governing equations are discretized by a stabilized Galerkin finite element method in space and an implicit second-order backward differentiation scheme in time. To efficiently solve the resulting large nonlinear systems, we present a highly parallel Newton-Krylov-Schwarz algorithm based on domain decomposition techniques. Analytic Jacobian is applied in order to obtain the best achievable performance. Two benchmark problems of lid-driven cavity and flow passing a square cylinder are employed to validate the proposed algorithm. We then apply the algorithm to the LES of turbulent flows passing a full-size high-speed train with realistic geometry and operating conditions. The numerical results show that the algorithm is both accurate and efficient and exhibits a good scalability and parallel efficiency with tens of millions of degrees of freedom on a computer with up to 4096 processors. To understand the numerical behavior of the proposed fully implicit scheme, we study several important issues, including the choices of linear solvers, the overlapping size of the subdomains, and, especially, the accuracy of the Jacobian matrix. The results show that an exact Jacobian is necessary for the efficiency and the robustness of the proposed LES solver.     

A numerical stabilization framework for viscoelastic fluid flow using the finite volume method on general unstructured meshes

A robust finite volume method for viscoelastic flow analysis on general unstructured meshes is developed. It is built upon a general‐purpose stabilization framework for high Weissenberg number flows. The numerical framework provides full combinatorial flexibility between different kinds of rheological models on the one hand, and effective stabilization methods on the other hand. A special emphasis is put on the velocity‐stress‐coupling on colocated computational grids. Using special face interpolation techniques, a semi‐implicit stress interpolation correction is proposed to correct the cell‐face interpolation of the stress in the divergence operator of the momentum balance. Investigating the entry‐flow problem of the 4:1 contraction benchmark, we demonstrate that the numerical methods are robust over a wide range of Weissenberg numbers and significantly alleviate the high Weissenberg number problem. The accuracy of the results is evaluated in a detailed mesh convergence study.