Scheme For Unstructured Grids Research Articles

Investigating a High-Order Viscous Flux Scheme for Unstructured Grids

A viscous flux formulation at interior faces and at Dirichlet boundary conditions in the framework of unstructured vertex-centered k-exact finite-volume schemes is presented. It is based on a geometric decomposition of the viscous flux into orthogonal and nonorthogonal parts. Special emphasis is put on the truncation errors and the spectral properties of the scheme, and a connection is established to the well-known α-damping scheme. The latter is also used to introduce discretization coefficients into the method that allow a fourth and a sixth order of accuracy in space on Cartesian grids. The effect of the scheme is presented in terms of canonical diffusion problems, as well as test cases for laminar wall-bounded flows. It is shown that the method preserves a second-order accuracy for the viscous operator, even on highly distorted unstructured grids and in the presence of boundaries and that it strongly enhances the solution accuracy with respect to a conventional scheme for viscous fluxes.

Economically high‐order unstructured‐grid methods: Clarification and efficient FSR schemes

AbstractIn this article, we clarify reconstruction‐based discretization schemes for unstructured grids and discuss their economically high‐order versions, which can achieve high‐order accuracy under certain conditions at little extra cost. The clarification leads to one of the most economical approaches: the flux‐and‐solution‐reconstruction (FSR) approach, where highly economical schemes can be constructed based on an extended ‐scheme combined with economical flux reconstruction formulas, achieving up to fifth‐order accuracy (sixth‐order with zero dissipation) when a grid is regular. Various economical FSR schemes are presented and their formal orders of accuracy are verified by numerical experiments.

Further Development and Application of High-Order Spectral Volume Methods for Compressible Flows

The present work investigates the high-order spectral finite volume (SFV) method with emphasis on applicability aspects for compressible flows. The intent is to improve the understanding and implementation of numerical techniques related to high-order unstructured grid schemes. In that regard, a hierarchical moment limiter and high-order mesh capability are developed for a two-dimensional Euler SFV solver. The limiter formulation and geometry interpreter for high-order mesh creation are new contributions for the SFV method. Literature test cases are evaluated to assess the interaction of curved mesh, limiter and spatial reconstruction features of the SFV scheme. An order of accuracy study is presented along with steady and unsteady problems with strong shock waves and other discontinuities typical of compressible flows. Moreover, second, third and fourth-order spatial resolution analyses are explored and the SFV results are compared with results from different numerical methods.

PyFR: An open source framework for solving advection–diffusion type problems on streaming architectures using the flux reconstruction approach

High-order numerical methods for unstructured grids combine the superior accuracy of high-order spectral or finite difference methods with the geometric flexibility of low-order finite volume or finite element schemes. The Flux Reconstruction (FR) approach unifies various high-order schemes for unstructured grids within a single framework. Additionally, the FR approach exhibits a significant degree of element locality, and is thus able to run efficiently on modern streaming architectures, such as Graphical Processing Units (GPUs). The aforementioned properties of FR mean it offers a promising route to performing affordable, and hence industrially relevant, scale-resolving simulations of hitherto intractable unsteady flows within the vicinity of real-world engineering geometries. In this paper we present PyFR, an open-source Python based framework for solving advection–diffusion type problems on streaming architectures using the FR approach. The framework is designed to solve a range of governing systems on mixed unstructured grids containing various element types. It is also designed to target a range of hardware platforms via use of an in-built domain specific language based on the Mako templating engine. The current release of PyFR is able to solve the compressible Euler and Navier–Stokes equations on grids of quadrilateral and triangular elements in two dimensions, and hexahedral elements in three dimensions, targeting clusters of CPUs, and NVIDIA GPUs. Results are presented for various benchmark flow problems, single-node performance is discussed, and scalability of the code is demonstrated on up to 104 NVIDIA M2090 GPUs. The software is freely available under a 3-Clause New Style BSD license (see www.pyfr.org). Program summaryProgram title: PyFR v0.1.0Catalogue identifier: AETY_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AETY_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: New style BSD licenseNo. of lines in distributed program, including test data, etc.: 12733No. of bytes in distributed program, including test data, etc.: 214183Distribution format: tar.gzProgramming language: Python, CUDA and C.Computer: Variable, up to and including GPU clusters.Operating system: Recent version of Linux/UNIX.RAM: Variable, from hundreds of megabytes to gigabytes.Classification: 6.5, 12.External routines: Python 2.7, numpy, PyCUDA, mpi4py, SymPy, MakoNature of problem:Compressible Euler and Navier–Stokes equations of fluid dynamics; potential for any advection–diffusion type problem.Solution method:High-order flux reconstruction approach suitable for curved, mixed, unstructured grids.Unusual features:Code makes extensive use of symbolic manipulation and runtime code generation through a domain specific language.Running time:Many small problems can be solved on a recent workstation in minutes to hours.

Energy stable flux reconstruction schemes for advection–diffusion problems on triangles

The Flux Reconstruction (FR) approach unifies several well-known high-order schemes for unstructured grids, including a collocation-based nodal discontinuous Galerkin (DG) method and all types of Spectral Difference (SD) methods, at least for linear problems. The FR approach also allows for the formulation of new families of schemes. Of particular interest are the energy stable FR schemes, also referred to as the Vincent–Castonguay–Jameson–Huynh (VCJH) schemes, which are an infinite family of high-order schemes parameterized by a single scalar. VCJH schemes are of practical importance because they provide a stable formulation on triangular elements which are often required for numerical simulations over complex geometries. In particular, VCJH schemes are provably stable for linear advection problems on triangles, and include the collocation-based nodal DG scheme on triangles as a special case. Furthermore, certain VCJH schemes have Courant–Friedrichs–Lewy (CFL) limits which are approximately twice those of the collocation-based nodal DG scheme. Thus far, these schemes have been analyzed primarily in the context of pure advection problems on triangles. For the first time, this paper constructs VCJH schemes for advection–diffusion problems on triangles, and proves the stability of these schemes for linear advection–diffusion problems for all orders of accuracy. In addition, this paper uses numerical experiments on triangular grids to verify the stability and accuracy of VCJH schemes for linear advection–diffusion problems and the nonlinear Navier–Stokes equations.

Simulation for the propulsion of a micro-hydro robot with an unstructured grid

The flow mechanism of contractive and dilative motion was numerically investigated to obtain a propulsive force in a highly viscous fluid. The computing program for the analysis of complicated motions was numerically developed with a cell-centered, unstructured grid scheme. The developed program was validated by the well-known equation of an oscillating plane below viscous fluid for an unsteady problem, which is known as Stokes’ second problem. Validation has continued through comparison with the experimental results. In this case, sinusoidal motion was applied to the validation, instead of trochoidal motion, because it was very difficult to actually simulate trochoidal motion in this experiment. Finally, the validation and comparison with the nodal-point scheme was accomplished by Stokes’ problem, which is the famous problem at a low Reynolds number. The validated code was applied to contractive and dilative motion in a narrow tube, whose motion was embodied by trochoidal movement. In a highly viscous fluid, such as a very sticky honey or a swamp, the computed results show that a viscous force can be used for propulsion instead of a dynamic force. From the present results, it was found that a propulsive force can be obtained by contractive and dilative motion at a low Reynolds number, which can be applied to the propulsion of micro-robots in a highly viscous fluid such as a blood vessel or a swamp. This research could also be considered fundamental research for the propulsion of micro-hydro robots, which are expected to be actively studied in the future in accord with further development of nanotechnology.

High‐order ENO and WENO schemes for unstructured grids

AbstractThis work describes the implementation and analysis of high‐order accurate schemes applied to high‐speed flows on unstructured grids. The class of essentially non‐oscillatory schemes (ENO), that includes weighted ENO schemes (WENO), is discussed in the paper with regard to the implementation of third‐ and fourth‐order accurate methods. The entire reconstruction process of ENO and WENO schemes is described with emphasis on the stencil selection algorithms. The stencils can be composed by control volumes with any number of edges, e.g. triangles, quadrilaterals and hybrid meshes. In the paper, ENO and WENO schemes are implemented for the solution of the dimensionless, 2‐D Euler equations in a cell centred finite volume context. High‐order flux integration is achieved using Gaussian quadratures. An approximate Riemann solver is used to evaluate the fluxes on the interfaces of the control volumes and a TVD Runge–Kutta scheme provides the time integration of the equations. Such a coupling of all these numerical tools, together with the high‐order interpolation of primitive variables provided by ENO and WENO schemes, leads to the desired order of accuracy expected in the solutions. An adaptive mesh refinement technique provides better resolution in regions with strong flowfield gradients. Results for high‐speed flow simulations are presented with the objective of assessing the implemented capability. Copyright © 2007 John Wiley & Sons, Ltd.

A simple strategy for constructing bounded convection schemes for unstructured grids

AbstractDue to the great geometrical flexibility, popularity for unstructured grid methods in fluid dynamics has been increasing in recent years. In parallel with this interest there is a need for bounded second or higher order convection schemes which can be implemented easily in the unstructured setting. In the present work a simple strategy for achieving convective boundedness in the context of a vertex‐centered unstructured finite volume algorithm is demonstrated. Testing is carried out on an inviscid oblique step problem using both structured and unstructured grid arrangements. Further testing for numerical diffusion is done using a distorted grid in a two dimensional channel. The proposed scheme is straightforward to implement and is found to perform well for the cases considered. The overall algorithm converges well and the limiter appears to introduce little extra numerical diffusion beyond that inherently present in the base scheme. Copyright © 2004 John Wiley & Sons, Ltd.

TVD schemes for unstructured grids

A number of approaches have evolved over the last decade for the implementation of total variational diminishing (TVD) schemes within an unstructured grid finite volume method framework. Unfortunately none of these approaches has been comprehensive enough to permit the general implementation of TVD-based schemes in unstructured grids, and/or accurate enough to recover the exact TVD formulation in structured grids. In this paper we propose a simple method that allows the implementation of the full spectrum of TVD schemes in unstructured grids, while recovering their exact formulation on structured grids. Four schemes implemented using this approach, TVD-MINMOD, TVD-MUSCL (monotonic upstream-centered scheme for conservation laws, MUSCL), TVD-SUPERBEE, TVD-OSHER, are tested and compared to Bruner’s TVD formulation [Parallelization of the Euler equations on unstructured grids, AIAA paper 97-1894, 1995], and to the Barth and Jesperson linear reconstruction scheme [The design and application of upwind schemes on unstructured meshes, AIAA paper 89-0366, 1989] by solving four pure advection problems. Results indicate that the Bruner formulation yields, for the same original TVD scheme, overly diffusive results when compared to the current method. The BJ-MUSCL and TVD-MUSCL are shown to be comparable and more accurate than the OSHER scheme. The SUPERBEE performs best though showing tendency for stepping the modeled profile. In all tests the current method is found to retain the behavior of the structured grid TVD formulation.

Peristaltic 운동을 이용한 추진에 대한 연구

고점성 유체내에서의 추진력을 얻기 위하여 Peristatic 운동에 의한 추진을 실험적, 수치적 방법으로 연구하였다. 운동을 수치적으로 해석하기 위하여 비 정규격자를 사용한 셀 중심 법을 이용하여 해석하였다. 실험을 위하여 작은 수조를 만들었으며 모형을 끌 수 있는 전차와 고점성 액체인 그리세린을 사용하여 Peristaltic 운동을 구현하였다. 여러 조건 하에서 실험을 수행하였으며, 여러 실험 결과 중에서 가장 비교가 용이한 정지 상태에서 실험한 결과에 대하여 개발된 프로그램을 이용한 해석 결과와 비교하였다. 해석 결과는 실험 결과와 좋은 일치를 보였다. Peristaltic 운동은 압력 차를 이용하여 추진력이 얻어짐을 계산을 통하여 보였으며 더 많은 계산을 통하여 최적 운동조건이나 Peristatic 운동이 효과적인 영역(레이놀즈 수) 등을 찾아낼 수 있으리라 생각된다. Flow mechanism of peristaltic motion is numerically and experimentally investigated to obtain a propulsive force in highly viscous fluid. Computing code for the analysis of the motions is developed with cell-centered unstructured grid scheme. The computed results by the developed code is compared with the experimental results which have been carried out to find out the propulsion mechanism in highly viscous fluid. The computed results shows good correlation with the experimental results and further the propulsive force can be obtained by sinusoidal motion which makes a pressure difference on waving surface. The more computation with variation of Reynolds number and parameters of motion is expected for finding a proper working range.

Global kinetic energy conservation with unstructured meshes

AbstractConserving kinetic energy is essential in Large Eddy Simulation (LES) in order that subgrid scale modelling (SGS) is not hidden by numerical errors. High‐order schemes, which are not available for unstructured grids, are sufficient but not a necessary condition for energy conservation. The problem addressed here is directly ensuring energy conservation through a careful discretization of the momentum and mass equations, and it is shown that this can be achieved with a low‐order scheme for unstructured grids. Copyright © 2002 John Wiley & Sons, Ltd.

비정규 격자를 이용한 극소 로봇의 추진 해석

Flow mechanism of contractive and dilative motion is numerically investigated to obtain a propulsive force in highly viscous fluid, which is the simulation of the propulsion in micro-organisms. The computing code for the analysis of complicated motions is developed with cell-centered unstructured grid scheme. The developed code is validated by the well-known problems of cavity flow and oscillating wall. The validated code is applied to the contractive and dilative motion in narrow tube. The computed results are compared with nodal points scheme. By the present results, it is found that propulsive force can be obtained by the contractive and dilative motion through simulation with the developed code.

Numerical simulation of fluid flows using an unstructured finite volume method with adaptive tri‐tree grids

AbstractA tri‐tree grid generation procedure is developed together with a finite volume method on the unstructured grid for solving the Navier–Stokes equations. A hierarchic numbering system for the data structure is used. The grid is adapted by adding and removing cell elements dependent on the vorticity magnitude. A special treatment is developed to ensure good quality triangular elements around the cylinder boundary. The adopted finite volume method is based on the cell‐centred scheme. The pressure–velocity coupling is treated using the SIMPLE algorithm. A modified QUICK scheme for unstructured grids is derived. The developed method is used to simulate the flow past a single and multiple cylinders at low Reynolds number. The obtained results are in good agreement with the published data. Copyright © 2002 John Wiley & Sons, Ltd.

A Numerical Simulation of the Diffusion of Near Infrared Light Through Brain Tissue

A numerical simulation of the transport of light energy in the near infrared region of the spectrum through human brain tissue is presented. This simulation models the use of near infrared spectroscopic techniques to quantify the levels of oxygen present in brain tissue. Successful application of the technique requires knowledge of the optical pathlength in the tissue, and it is the goal of this simulation to quantify the relationship of the optical pathlength and the oxygenation state of the tissue. Both implicit and explicit finite element schemes for unstructured grids are implemented and discussed. Several application simulations using three tissue grids of varying degrees of physiological accuracy are then conducted, and figures for the optical pathlength of light through the tissue at varying levels of oxygenation are computed. These results are then used to develop a quantitative relationship between the pathlength and the absorption parameter for the three tissue models which we explore.

Hybrid grid approach to study dynamic stall

A hybrid structured and unstructured grid scheme is developed to study dynamic stall over an airfoil in a wind tunnel. The unsteady Navier-Stokes equations are solved implicity using this hybrid grid scheme. Results are compared with experimental data. Additionally, they reveal informative details of the flowfield physics associated with dynamic stall