Spectral/hp element methods: recent developments, applications, and perspectives

The spectral/hp element method combines the geometric flexibility of the classical h-type finite element technique with the desirable numerical properties of spectral methods, employing high-degree piecewise polynomial basis functions on coarse finite element-type meshes. The spatial approximation is based upon orthogonal polynomials, such as Legendre or Chebychev polynomials, modified to accommodate C0-continuous expansions. Computationally and theoretically, by increasing the polynomial order p, high-precision solutions and fast convergence can be obtained and, in particular, under certain regularity assumptions an exponential reduction in approximation error between numerical and exact solutions can be achieved. This method has now been applied in many simulation studies of both fundamental and practical engineering flows. This paper briefly describes the formulation of the spectral/hp element method and provides an overview of its application to computational fluid dynamics. In particular, it focuses on the use the spectral/hp element method in transitional flows and ocean engineering. Finally, some of the major challenges to be overcome in order to use the spectral/hp element method in more complex science and engineering applications are discussed

High-order DG solvers for under-resolved turbulent incompressible flows: A comparison of L2L^2 and HH(div) methods

The accurate numerical simulation of turbulent incompressible flows is a challenging topic in computational fluid dynamics. For discretisation methods to be robust in the under-resolved regime, mass conservation as well as energy stability are key ingredients to obtain robust and accurate discretisations. Recently, two approaches have been proposed in the context of high-order discontinuous Galerkin (DG) discretisations that address these aspects differently. On the one hand, standard $L^2$-based DG discretisations enforce mass conservation and energy stability weakly by the use of additional stabilisation terms. On the other hand, pointwise divergence-free $H(\operatorname{div})$-conforming approaches ensure exact mass conservation and energy stability by the use of tailored finite element function spaces. The present work raises the question whether and to which extent these two approaches are equivalent when applied to under-resolved turbulent flows. This comparative study highlights similarities and differences of these two approaches. The numerical results emphasise that both discretisation strategies are promising for under-resolved simulations of turbulent flows due to their inherent dissipation mechanisms.Comment: 24 pages, 13 figure

CFD code comparison for 2D airfoil flows

The current paper presents the effort, in the EU AVATAR project, to establish the necessary requirements to obtain consistent lift over drag ratios among seven CFD codes. The flow around a 2D airfoil case is studied, for both transitional and fully turbulent conditions at Reynolds numbers of 3 × 106 and 15 × 106. The necessary grid resolution, domain size, and iterative convergence criteria to have consistent results are discussed, and suggestions are given for best practice. For the fully turbulent results four out of seven codes provide consistent results. For the laminar-turbulent transitional results only three out of seven provided results, and the agreement is generally lower than for the fully turbulent case

Stratified shallow flow modelling

Environmental hydraulics covers a very wide range of applications including free surface flows in rivers. estuaries and lakes. To find engineering solutions to environmental hydraulics problems. 3D numerical modelling is nowadays widely used. However. the computation of sharp spatial gradients (such as found in stratified estuaries and lakes. around plumes near outfalls along rivers and coasts or in exchange areas of high shear). and the modelling of these processes along steep bathymetric slopes (such as found at the edge of dredged channels or of the continental shelf) remains a challenge. In addition. crude assumptions (such as the hydrostatic assumption) are often made to the primary differential equations in order to simplify the problem and enable long term prediction of environmental hydraulic changes. In this thesis. a robust adaptive mesh displacement (AMD) method is implemented and validated against the lock exchange case in particular. The AMD method aims at vertically focusing nodes within each water column to capture sharp gradients. while reducing the number of nodes or requiring prior knowledge of the flow structure. Second. a direct computation of dynamic pressure is introduced based on the equation of vertical momentum and validated against the analytical potential flow theory solution of a source-sink pair. Dynamic pressure is necessary to model destratification recirculation devices. or flow over dredge channel. or solitary waves. for instance. This direct computation method makes the hydrostatic assumption redundant. Third. a new advection scheme is implemented. whose main advantage is simplicity averaging over Riemann problems without solving them. while excessive numerical viscosity is compensated for by using high-resolution MUSCL type reconstruction. Recommendations are made in this thesis to extend the advection scheme developed herein for tracer advection to the non-linear shallow water equations. to the diffusion terms and to turbulence closure laws within the same finite element framework

Numerical Simulation of Selected Two-Dimensional and Three-Dimensional Fluid-Structure Interaction Problems Using OpenFOAM Technology

Fluid-structure interaction (FSI) problems are increasing in various engineering fields. In this thesis, different cases of FSI in two- and three-dimensions (2D and 3D) are simulated using OpenFOAM and foam-extend. These packages have been used to create a coupling between fluid and solid. The vortex-induced vibration (VIV) phenomenon of flow past a circular cylinder is studied using PIMPLE algorithm for pressure-velocity coupling. This VIV study is restricted to incompressible flow simulation at a Reynolds number (Re) of 100. The changes of drag and lift coefficient values depend on the study case and the spring-mass-damper system for the flow past a free oscillatory cylinder. The free vibrating cylinder examined in one-degree-of-freedom (1DOF) and two-degrees-of-freedom (2DOF) systems with linear damping and spring properties. Both will affect the behaviour of the cylinder within the flow with some noticeable differences. The response time of the cylinder and the drag coefficient are the most affected by the spring and damper. Besides the vortex-induced vibration test cases, the two-dimensional and three-dimensional fluid-structure interaction benchmarking is also studied. A partitioned solution method for strongly coupled solver with independent fluid and solid meshes for transient simulation has been applied. The fluid domain dynamics is governed by the incompressible Navier-Stokes equations; however, the structural field is described by the nonlinear elastodynamic equations. Fluid and solid domains are discretised by finite volume method (FVM) in space and time. A strong coupling scheme for partitioned analysis of the thin-walled shell structure exposed to wind-induced vibration (WIV) is presented. The achievement of the 3D membrane roof coupling scheme is studied by applying the 2D model. Additionally, numerical models for the slender shell structures coupling and the 3D flows indicate possible applications of the presented work. The computational fluid dynamics (CFD) simulation results revealed that even the flow is considered as a laminar, turbulence modelling or more refined meshes should be used to capture the generation and release of vortices. A partitioned solution procedure for FSI problems in the building aeroelasticity area is also studied. An illustrative real-world model on the coupled behaviour of membrane structure under wind flow influence is given. A four-point tent subjected to wind motion is a typical application of this work applying with various physical factors that are a necessity for the thin membrane structure. The fluid domain is described by the incompressible Navier-Stokes equations at a Reynolds number of Re = 3,750. However, the motion of the solid field is modeled by total Lagrangian strategy for nonlinear elastic deformation. The FSI simulation, particularly 3D problems require in very long calculation time. Some limitations of the FSI solver in foam-extend package called fsiFoam is discussed. All solvers that used in this thesis are considered to be applied to a wide use of the implementation of FSI models, despite some problems in parallelisation, particularly in the latest FSI solver version. The analysis results are presented to demonstrate accuracy, convergence, and stability

Unstructured mesh based models for incompressible turbulent flows

A development of high resolution NFT model for simulation of incompressible flows is presented. The model uses finite volume spatial discretisation with edge based data structure and operates on unstructured meshes with arbitrary shaped cells. The key features of the model include non-oscillatory advection scheme Multidimensional Positive Definite Advection Transport Algorithm (MPDATA) and non-symmetric Krylov-subspace elliptic solver. The NFT MPDATA model integrates the Reynolds Average Navier Stokes (RANS) equations. The implementation of the Spalart-Allmaras one equations turbulence model extends the development further to turbulent flows. An efficient non-staggered mesh arrangement for pressure and velocity is employed and provides smooth solutions without a need of artificial dissipation. In contrast to commonly used schemes, a collocated arrangement for flow variables is possible as the stabilisation of the NFT MPDATA scheme arises naturally from the design of MPDATA. Other benefits of MPDATA include: second order accuracy, strict sign-preserving and full multidimensionality. The flexibility and robustness of the new approach is studied and validated for laminar and turbulent flows. Theoretical developments are supported by numerical testing. Successful quantitative and qualitative comparisons with the numerical and experimental results available from literature confirm the validity and accuracy of the NFT MPDATA scheme and open the avenue for its exploitation for engineering problems with complex geometries requiring flexible representation using unstructured meshes

On the use of spectral element methods for under-resolved simulations of transitional and turbulent flows

The present thesis comprises a sequence of studies that investigate the suitability of spectral element methods for model-free under-resolved computations of transitional and turbulent flows. More specifically, the continuous and the discontinuous Galerkin (i.e. CG and DG) methods have their performance assessed for under-resolved direct numerical simulations (uDNS) / implicit large eddy simulations (iLES). In these approaches, the governing equations of fluid motion are solved in unfiltered form, as in a typical direct numerical simulation, but the degrees of freedom employed are insufficient to capture all the turbulent scales. Numerical dissipation introduced by appropriate stabilisation techniques complements molecular viscosity in providing small-scale regularisation at very large Reynolds numbers. Added spectral vanishing viscosity (SVV) is considered for CG, while upwind dissipation is relied upon for DG-based computations. In both cases, the use of polynomial dealiasing strategies is assumed. Focus is given to the so-called eigensolution analysis framework, where numerical dispersion and diffusion errors are appraised in wavenumber/frequency space for simplified model problems, such as the one-dimensional linear advection equation. In the assessment of CG and DG, both temporal and spatial eigenanalyses are considered. While the former assumes periodic boundary conditions and is better suited for temporally evolving problems, the latter considers inflow / outflow type boundaries and should be favoured for spatially developing flows. Despite the simplicity of linear eigensolution analyses, surprisingly useful insights can be obtained from them and verified in actual turbulence problems. In fact, one of the most important contributions of this thesis is to highlight how linear eigenanalysis can be helpful in explaining why and how to use spectral element methods (particularly CG and DG) in uDNS/iLES approaches. Various aspects of solution quality and numerical stability are discussed by connecting observations from eigensolution analyses and under-resolved turbulence computations. First, DG’s temporal eigenanalysis is revisited and a simple criterion named "the 1% rule" is devised to estimate DG’s effective resolution power in spectral space. This criterion is shown to pinpoint the wavenumber beyond which a numerically induced dissipation range appears in the energy spectra of Burgers turbulence simulations in one dimension. Next, the temporal eigenanalysis of CG is discussed with and without SVV. A modified SVV operator based on DG’s upwind dissipation is proposed to enhance CG’s accuracy and robustness for uDNS / iLES. In the sequence, an extensive set of DG computations of the inviscid Taylor-Green vortex model problem is considered. These are used for the validation of the 1% rule in actual three-dimensional transitional / turbulent flows. The performance of various Riemann solvers is also discussed in this infinite Reynolds number scenario, with high quality solutions being achieved. Subsequently, the capabilities of CG for uDNS/iLES are tested through a complex turbulent boundary layer (periodic) test problem. While LES results of this test case are known to require sophisticated modelling and relatively fine grids, high-order CG approaches are shown to deliver surprisingly good quality with significantly less degrees of freedom, even without SVV. Finally, spatial eigenanalyses are conducted for DG and CG. Differences caused by upwinding levels and Riemann solvers are explored in the DG case, while robust SVV design is considered for CG, again by reference to DG’s upwind dissipation. These aspects are then tested in a two-dimensional test problem that mimics spatially developing grid turbulence. In summary, a point is made that uDNS/iLES approaches based on high-order spectral element methods, when properly stabilised, are very powerful tools for the computation of practically all types of transitional and turbulent flows. This capability is argued to stem essentially from their superior resolution power per degree of freedom and the absence of (often restrictive) modelling assumptions. Conscientious usage is however necessary as solution quality and numerical robustness may depend strongly on discretisation variables such as polynomial order, appropriate mesh spacing, Riemann solver, SVV parameters, dealiasing strategy and alternative stabilisation techniques.Open Acces

High-order incompressible computational fluid dynamics on modern hardware architectures

In this thesis, a high-order incompressible Navier-Stokes solver is developed in the Python-based PyFR framework. The solver is based on the artificial compressibility formulation with a Flux Reconstruction (FR) discretisation in space and explicit dual time stepping in time. In order to reduce time to solution, explicit convergence acceleration techniques are developed and implemented. These techniques include polynomial multigrid, a novel locally adaptive pseudo-time stepping approach and novel stability-optimised Runge-Kutta schemes. Choices regarding the numerical methods and implementation are motivated as follows. Firstly, high-order FR is selected as the spatial discretisation due to its low dissipation and ability to work with unstructured meshes of complex geometries. Be- ing discontinuous, it also allows the majority of computation to be performed locally. Secondly, convergence acceleration techniques are restricted to explicit methods in order to retain the spatial locality provided by FR, which allows efficient harnessing of the massively parallel compute capability of modern hardware. Thirdly, the solver is implemented in the PyFR framework with cross-platform support such that it can run on modern heterogeneous systems via an MPI + X model, with X being CUDA, OpenCL or OpenMP. As such, it is well-placed to remain relevant in an era of rapidly evolving hardware architectures. The new software constitutes the first high-order accurate cross-platform imple- mentation of an incompressible Navier-Stokes solver via artificial compressibility. The solver and the convergence acceleration techniques are validated for a range of turbu- lent test cases. Furthermore, performance of the convergence acceleration techniques is assessed with a 2D cylinder test case, showing speed-up factors of over 20 relative to global RK4 pseudo-time stepping when all of the technologies are combined. Fi- nally, a simulation of the DARPA SUBOFF submarine model is undertaken using the solver and all convergence acceleration techniques. Excellent agreement with previ- ous studies is obtained, demonstrating that the technology can be used to conduct high-fidelity implicit Large Eddy Simulation of industrially relevant problems at scale using hundreds of GPUs.Open Acces

Computational Aerodynamics on unstructed meshes

New 2D and 3D unstructured-grid based flow solvers have been developed for simulating steady compressible flows for aerodynamic applications. The codes employ the full compressible Euler/Navier-Stokes equations. The Spalart-Al Imaras one equation turbulence model is used to model turbulence effects of flows. The spatial discretisation has been obtained using a cell-centred finite volume scheme on unstructured-grids, consisting of triangles in 2D and of tetrahedral and prismatic elements in 3D. The temporal discretisation has been obtained with an explicit multistage Runge-Kutta scheme. An "inflation" mesh generation technique is introduced to effectively reduce the difficulty in generating highly stretched 2D/3D viscous grids in regions near solid surfaces. The explicit flow method is accelerated by the use of a multigrid method with consideration of the high grid aspect ratio in viscous flow simulations. A solution mesh adaptation technique is incorporated to improve the overall accuracy of the 2D inviscid and viscous flow solutions. The 3D flow solvers are parallelised in a MIMD fashion aimed at a PC cluster system to reduce the computing time for aerodynamic applications. The numerical methods are first applied to several 2D inviscid flow cases, including subsonic flow in a bump channel, transonic flow around a NACA0012 airfoil and transonic flow around the RAE 2822 airfoil to validate the numerical algorithms. The rest of the 2D case studies concentrate on viscous flow simulations including laminar/turbulent flow over a flat plate, transonic turbulent flow over the RAE 2822 airfoil, and low speed turbulent flows in a turbine cascade with massive separations. The results are compared to experimental data to assess the accuracy of the method. The over resolved problem with mesh adaptation on viscous flow simulations is addressed with a two phase mesh reconstruction procedure. The solution convergence rate with the aspect ratio adaptive multigrid method and the direct connectivity based multigrid is assessed in several viscous turbulent flow simulations. Several 3D test cases are presented to validate the numerical algorithms for solving Euler/Navier-Stokes equations. Inviscid flow around the M6 wing airfoil is simulated on the tetrahedron based 3D flow solver with an upwind scheme and spatial second order finite volume method. The efficiency of the multigrid for inviscid flow simulations is examined. The efficiency of the parallelised 3D flow solver and the PC cluster system is assessed with simulations of the same case with different partitioning schemes. The present parallelised 3D flow solvers on the PC cluster system show satisfactory parallel computing performance. Turbulent flows over a flat plate are simulated with the tetrahedron based and prismatic based flow solver to validate the viscous term treatment. Next, simulation of turbulent flow over the M6 wing is carried out with the parallelised 3D flow solvers to demonstrate the overall accuracy of the algorithms and the efficiency of the multigrid method. The results show very good agreement with experimental data. A highly stretched and well-formed computational grid near the solid wall and wake regions is generated with the "inflation" method. The aspect ratio adaptive multigrid displayed a good acceleration rate. Finally, low speed flow around the NREL Phase 11 Wind turbine is simulated and the results are compared to the experimental data