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

Xu, Hanlin; Cantwell, Chris D.; Monteserin, Carlos; Eskilsson, Claes; Engsig-Karup, Allan Peter; Sherwin, Spencer J.

doi:10.1007/s42241-018-0001-1

Cited by 87 publications

(35 citation statements)

References 140 publications

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“…We want to conclude our contribution with some speculations on the relation between high-order and low-order flow solvers. In recent years high-order space discretisation was proposed as an efficient means for the simulation of challenging flow problems, like incompressible Navier-Stokes flows at high Reynolds numbers or real-world applications in computational fluid dynamics [63,43]. The potential benefits of high-order discretisations are suggested to be twofold [63]:…”

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confidence: 99%

On high-order pressure-robust space discretisations, their advantages for incompressible high Reynolds number generalised Beltrami flows and beyond

Gauger¹,

Linke²,

Schroeder³

2019

The SMAI Journal of Computational Mathematics

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An improved understanding of the divergence-free constraint for the incompressible Navier-Stokes equations leads to the observation that a semi-norm and corresponding equivalence classes of forces are fundamental for their nonlinear dynamics. The recent concept of pressure-robustness allows to distinguish between space discretisations that discretise these equivalence classes appropriately or not. This contribution compares the accuracy of pressure-robust and non-pressure-robust space discretisations for transient high Reynolds number flows, starting from the observation that in generalised Beltrami flows the nonlinear convection term is balanced by a strong pressure gradient. Then, pressurerobust methods are shown to outperform comparable non-pressure-robust space discretisations. Indeed, pressure-robust methods of formal order k are comparably accurate than non-pressure-robust methods of formal order 2k on coarse meshes. Investigating the material derivative of incompressible Euler flows, it is conjectured that strong pressure gradients are typical for non-trivial high Reynolds number flows. Connections to vortex-dominated flows are established. Thus, pressure-robustness appears to be a prerequisite for accurate incompressible flow solvers at high Reynolds numbers. The arguments are supported by numerical analysis and numerical experiments.Indeed, the lack of pressure-robustness has been a rather hot research topic in the beginning of the history of finite element methods for CFD [55,31,62,38,25,35] -sometimes called poor mass conservation -and continued to be investigated for many years [33,57,34,60], often in connection with the so-called grad-div stabilisation [32,54,17,40,3,22]. Also, in the geophysical fluid dynamics community and in numerical astrophysics well-balanced schemes have been proposed to overcome similar issues for related Euler and shallow-water equations, especially in connection to nearly-hydrostatic and nearly-geostrophic flows; cf., for example, [20,21,11,44].However, only recently it was understood better that exactly the relaxation of the divergence constraint for incompressible flows, which was invented in classical mixed methods in order to construct discretely inf-sup stable discretisation schemes, introduces the lack of pressure-robustness, since it leads to a poor discretisation of the Helmholtz-Hodge projector [49]. The reason is that the relaxation of the divergence constraint implies a relaxation of the L 2 -orthogonality between discretely divergence-free velocity test functions and arbitrary gradient fields. PRESSURE-ROBUSTNESS, HIGH REYNOLDS NUMBERS, BELTRAMI FLOWSWe only briefly remark that the question of an appropriate discretisation of the nonlinear convection term is intimately connected to the issue of numerical convection stabilisation techniques like upwinding or SUPG [56,14]. With the help of generalised Beltrami flows, we will demonstrate that in real-world flows the nonlinear convection term can be strong, even if the dynamics of the flow is not convectiondominated ...

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confidence: 99%

On high-order pressure-robust space discretisations, their advantages for incompressible high Reynolds number generalised Beltrami flows and beyond

Gauger¹,

Linke²,

Schroeder³

2019

The SMAI Journal of Computational Mathematics

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“…Finite element methods [24][25][26][27] and the finite volume method [28,29] are very effective tools to solve some partial differential equations (PDEs) on complex geometries, which is applied in a wide range of engineering and biomedical disciplines [30][31][32][33][34]. In this paper, the finite volume method is implemented to calculate the NS equation.…”

Section: Resultsmentioning

confidence: 99%

Effects of Single-arc Blade Profile Length on the Performance of a Forward Multiblade Fan

Wei

Ying²,

Xu³

et al. 2019

Processes

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The effects of single-arc blade profile length on the performance of a forward multiblade fan are investigated in this paper by computational fluid dynamics and experimental measurement. The present work emphasizes that the use of a properly reduced blade inlet angle (β1A) and properly improved blade outlet angle (β2A) is to increase the length blade profile, which suggests a good physical understanding of internal complex flow characteristics and the aerodynamic performance of the fan. Numerical results indicate that the gradient of the absolute velocity among the blades in model-L (reducing the blade inlet angle and improving blade outlet angle) is clearly lower than that of the baseline model and model-S (improving the blade inlet angle and reducing blade outlet angle), where a number of secondary flows arise on the exit surface of baseline model and model-S. However, no secondary flow occurs in model-L, and the flow loss at the exit surface of the volute (scroll-shaped flow patterns) for model-L is obviously lower than that of the baseline model at the design point. The comparison of the test results further shows that to improve the blade profile length is to increase the static pressure and the efficiency of the static pressure, since the improved static pressure of the model-L rises as much as 22.5 Pa and 26.2%, and the improved static pressure efficiency of the model-L rises as much as 5 % at the design flow rates. It is further indicated that increasing the blade working area provides significant physical insight into increasing the static pressure, total pressure, the efficiency of the static pressure and the total pressure efficiency.

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“…A large number of numerical methods are widely used to solve the classical Oberbeck-Boussinesq equations [ 31 , 32 , 33 , 34 , 35 , 36 ]. The finite element methods [ 36 ], finite difference method [ 34 ] and the finite volume method [ 35 ] are traditional macroscopic methods for Computational Fluid Dynamics (CFD) calculation. The lattice Boltzmann method (LBM) is a computational fluid dynamics method based on mesoscopic simulation scale [ 37 , 38 , 39 , 40 , 41 ].…”

Section: Convection Diffusion Equation and Numerical Methodsmentioning

confidence: 99%

Time Evolution Features of Entropy Generation Rate in Turbulent Rayleigh-Bénard Convection with Mixed Insulating and Conducting Boundary Conditions

Wei

Shen

Wang

et al. 2020

Entropy

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Time evolution features of kinetic and thermal entropy generation rates in turbulent Rayleigh-Bénard (RB) convection with mixed insulating and conducting boundary conditions at Ra = 109 are numerically investigated using the lattice Boltzmann method. The state of flow gradually develops from laminar flow to full turbulent thermal convection motion, and further evolves from full turbulent thermal convection to dissipation flow in the process of turbulent energy transfer. It was seen that the viscous, thermal, and total entropy generation rates gradually increase in wide range of t/τ < 32 with temporal evolution. However, the viscous, thermal, and total entropy generation rates evidently decrease at time t/τ = 64 compared to that of early time. The probability density function distributions, spatial-temporal features of the viscous, thermal, and total entropy generation rates in the closed system provide significant physical insight into the process of the energy injection, the kinetic energy, the kinetic energy transfer, the thermal energy transfer, the viscous dissipated flow and thermal dissipation.

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

Cited by 87 publications

References 140 publications

On high-order pressure-robust space discretisations, their advantages for incompressible high Reynolds number generalised Beltrami flows and beyond

On high-order pressure-robust space discretisations, their advantages for incompressible high Reynolds number generalised Beltrami flows and beyond

Effects of Single-arc Blade Profile Length on the Performance of a Forward Multiblade Fan

Time Evolution Features of Entropy Generation Rate in Turbulent Rayleigh-Bénard Convection with Mixed Insulating and Conducting Boundary Conditions

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