Mixed-element Unstructured Meshes Research Articles

Hybrid discontinuous Galerkin-finite volume techniques for compressible flows on unstructured meshes

In this paper we develop a family of arbitrarily high-order non-oscillatory hybrid Discontinuous Galerkin(DG)-Finite Volume(FV) schemes for mixed-element unstructured meshes. Their key ingredient is a switch between a DG method and a FV method based on the CWENOZ scheme when invalid solutions are detected by a troubled cell indicator checking the unlimited DG solution. Therefore, the high order of accuracy offered by DG is preserved in smooth regions of the computational domain, while the robustness of FV is utilized in regions with strong gradients. The high-order CWENOZ variant used has the same spatial order of accuracy as the DG variant, while representing one of the most compact applications on unstructured meshes, therefore simplifying the implementation, reducing the computational overhead associated with large stencils of the original WENO reconstruction without sacrificing the desirable non-oscillatory properties of the schemes. We carefully investigate several parameters associated with the switching between DG and FV methods including the troubled cell indicators in a priori fashion. For the first time in the literature, we investigate the definition of the bounds for an admissible solution, the frequency by which we use the troubled cell indicators, and the evolution of the percentage of troubled cells for unsteady test problems. The 2D and 3D Euler equations are solved for well established test problems and compared with computational or experimental reference solutions. All the methods have been implemented and deployed within the UCNS3D open-source high-order unstructured Computational Fluid Dynamics (CFD) solver. The present coupling has the potential to improve the shortcomings of both FV-DG in a computational efficient manner. The improved accuracy and robustness provided is a characteristic of paramount importance for industrial-scale CFD applications, and favours the extension to other systems of governing equations.

Numerical Study of Fuselage Impact on Acoustic Characteristics of a Helicopter Rotor

The paper presents the results of the simulation of unsteady turbulent flow generated by helicopter main rotor in the presence of the fuselage with an emphasis on the analysis of the influence of the fuselage on the rotor-induced flow and the rotor-generated acoustic field. The Reynolds-averaged Navier–Stokes equations with the Spalart–Allmaras turbulence model are used to simulate the Caradonna–Tung rotor and ROBIN fuselage interaction in hovering flight. The governing equations are discretized using the vertex-centered control volume method on mixedelement unstructured meshes with the sliding mesh technology to treat the rotor. The acoustic field generated by the rotor+fuselage interaction is comparatively analyzed against the case of an isolated rotor. It is found that the presence of fuselage significantly changes the rotor-generated acoustics. In particular, the presence of the fuselage noticeably distorts the directivity of acoustic radiation and increases the overall sound pressure level under the fuselage up to 20 dB, emphasizing the importance of the influence of fuselage on the helicopter acoustics.

Method of averaged element splittings for diffusion terms discretization in vertex-centered framework

Method of averaged element splittings is proposed for the discretization of the diffusion terms of the Navier – Stokes equations on mixed-element unstructured meshes. It is applicable when mesh functions are defined in nodes. This method is a linear method, which has much in common with the classical P1-Galerkin method. In the case of simplicial meshes, these methods coincide, but the new method yields a 7-point approximation of the 3D Laplace operator on a Cartesian mesh. For a model elliptic problem we provide both theoretical and numerical accuracy estimates. The numerical evidence shows that accuracy of the new method is the same as of the P1-Galerkin method in 2D and possibly slightly worse in 3D. However, we only prove the order 3/2 in 2D and the first order in 3D, both in L2. The new method has an important advantage in the case of implicit time integration based on the Newton method, which implies solving linear algebraic systems with flux Jacobian. It allows to truncate flux Jacobian to a 7-point stencil for a much wider range of Reynolds and Courant numbers without loss of iterations convergence, compared to the P1-Galerkin method.

Arbitrary high order central non-oscillatory schemes on mixed-element unstructured meshes

In this paper we develop a family of very high-order central (up to 6th-order) non-oscillatory schemes for mixed-element unstructured meshes. The schemes are inherently compact in the sense that the central stencils employed are as compact as possible, and that the directional stencils are reduced in size therefore simplifying their implementation. Their key ingredient is the non-linear combination in a CWENO style similar to Dumbser et al [1] of a high-order polynomial arising from a central stencil with lower-order polynomials from directional stencils. Therefore, in smooth regions of the computational domain the optimum order of accuracy is recovered, while in regions of sharp-gradients the larger influence of the reconstructions from the directional stencils suppress the oscillations. It is the compactness of the directional stencils that increases the chances of at least one of them lying in a region with smooth data, that greatly enhances their robustness compared to classical WENO schemes. The two variants developed are CWENO and CWENOZ schemes, and it is the first time that such very-high-order schemes are designed for mixed-element unstructured meshes. We explore the influence of the linear weights in each of the schemes, and assess their performance in terms of accuracy, robustness and computational cost through a series of stringent 2D and 3D test problems. The results obtained demonstrate the improved robustness that the schemes offer, a parameter of paramount importance for and their potential use for industrial-scale engineering applications.

Robust Conservative Level Set Method for 3D Mixed-Element Meshes — Application to LES of Primary Liquid-Sheet Breakup

AbstractIn this article we detail the methodology developed to construct an efficient interface description technique — the robust conservative level set (RCLS) — to simulate multiphase flows on mixed-element unstructured meshes while conserving mass to machine accuracy. The approach is tailored specifically for industry as the three-dimensional unstructured approach allows for the treatment of very complex geometries. In addition, special care has been taken to optimise the trade-off between accuracy and computational cost while maintaining the robustness of the numerical method. This was achieved by solving the transport equations for the liquid volume fraction using a WENO scheme for polyhedral meshes and by adding a flux-limiter algorithm. The performance of the resulting method has been compared against established multiphase numerical methods and its ability to capture the physics of multiphase flows is demonstrated on a range of relevant test cases. Finally, the RCLS method has been applied to the simulation of the primary breakup of a flat liquid sheet of kerosene in co-flowing high-pressure gas. This quasi-DNS/LES computation was performed at relevant aero-engine conditions on a three-dimensional mixed-element unstructured mesh. The numerical results have been validated qualitatively against theoretical predictions and experimental data. In particular, the expected breakup regime was observed in the simulation results. Finally, the computation reproduced faithfully the breakup length predicted by a correlation based on experimental data. This constitutes a first step towards a quantitative validation.

High Order Schemes on Three-Dimensional General Polyhedral Meshes — Application to the Level Set Method

AbstractIn this article, we detail the methodology developed to construct arbitrarily high order schemes — linear and WENO — on 3D mixed-element unstructured meshes made up of general convex polyhedral elements. The approach is tailored specifically for the solution of scalar level set equations for application to incompressible two-phase flow problems. The construction of WENO schemes on 3D unstructured meshes is notoriously difficult, as it involves a much higher level of complexity than 2D approaches. This due to the multiplicity of geometrical considerations introduced by the extra dimension, especially on mixed-element meshes. Therefore, we have specifically developed a number of algorithms to handle mixed-element meshes composed of convex polyhedra with convex polygonal faces. The contribution of this work concerns several areas of interest: the formulation of an improved methodology in 3D, the minimisation of computational runtime in the implementation through the maximum use of pre-processing operations, the generation of novel methods to handle complex 3D mixed-element meshes and finally the application of the method to the transport of a scalar level set.

WENO schemes on arbitrary mixed-element unstructured meshes in three space dimensions

The paper extends weighted essentially non-oscillatory (WENO) methods to three dimensional mixed-element unstructured meshes, comprising tetrahedral, hexahedral, prismatic and pyramidal elements. Numerical results illustrate the convergence rates and non-oscillatory properties of the schemes for various smooth and discontinuous solutions test cases and the compressible Euler equations on various types of grids. Schemes of up to fifth order of spatial accuracy are considered.

Uniformly high-order schemes on arbitrary unstructured meshes for advection–diffusion equations

The paper presents a linear high-order method for advection–diffusion conservation laws on three-dimensional mixed-element unstructured meshes. The key ingredient of the method is a reconstruction procedure in local computational coordinates. Numerical results illustrate the convergence rates for the linear equation and a non-linear hyperbolic system with diffusion terms for various types of meshes.

WENO Schemes for Mixed-Element Unstructured Meshes

WENO Schemes for Mixed-Element Unstructured Meshes

A Parallel Adaptation Package For Three-Dimensional Mixed-Element Unstructured Meshes

Parallel mesh adaptation methods are developed to treat decomposed, mixed-element unstructured meshes comprised of any combination of basic element types. Emphasis is placed on developing conforming mesh modification methods that are solver-independent. Specificdevelopmentsincludetheimplementationofatreatmentforviscous,highaspectratio near wall tetrahedra, and cell subdivision methods for pyramids, prisms, and hexahedra. Rebalancingoftheadaptedgrid,andparticularlyissuesassociatedwithprocessorassignment of parent/child cell sets, is addressed. The new methods are demonstrated on decomposed, mixed-element meshes employing up to 64 processors. The resulting parallel adaptation package is a powerful, versatile tool for obtaining grid-converged, steady state results, and may readily be applied to other unstructured flow solvers.

Adaptive meshing techniques for viscous flow calculations on mixed element unstructured meshes

Adaptive meshing techniques for viscous flow calculations on mixed element unstructured meshes

Adaptive meshing techniques for viscous flow calculations on mixed element unstructured meshes

An adaptive refinement strategy based on hierarchical element subdivision is formulated and implemented for meshes containing arbitrary mixtures of tetrahedra, hexahedra, prisms and pyramids. Special attention is given to keeping memory overheads as low as possible. This procedure is coupled with an algebraic multigrid flow solver which operates on mixed-element meshes. Inviscid flows as well as viscous flows are computed on adaptively refined tetrahedral, hexahedral, and hybrid meshes. The efficiency of the method is demonstrated by generating an adapted hexahedral mesh containing 3 million vertices on a relatively inexpensive workstation.