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Large eddy simulation of the flow over a circular cylinder at Reynolds number Re D = 2580 has been studied with a high-order unstructured spectral dif-ference method. Grid and polynomial refinement studies were carried out to as-sess numerical errors. The mean and fluctuating velocity fields in the wake of a circular cylinder were compared with PIV...
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Citations
... The in-plane cell count has been set high to 201×10 3 in order to compensate for the low order of the finite-volumes spatial discretisation. The number of in-plane degrees of freedom considered is comparable to that used by Sarwar and Mellibovsky [22], somewhat larger at 548 × 8 2 ≃ 351 × 10 3 but considering a slightly higher flow regime = 2000, and way larger than all other numerical studies reported [4,7,8,[26][27][28]. A lower bound for the Kolmogorov length scale in the near wake has been estimated, from the average dissipation rate of turbulent kinetic energy and assuming isotropic turbulence, at about ≃ 0.02, which is commensurate with the typical cell size deployed in the region. ...
... The SD method was also used for high-fidelity simulations of 3D vortex-induced vibrations on quadrilateral mesh [115]. In addition, several works concerned the validation of the SD method capability to compute unsteady solutions of compressible flows by means of LES [116][117][118][119][120][121]. Extensions of the SD method were proposed to handle shocks [122]. ...
This thesis examines the extension of the Spectral Difference (SD) method on unstructured hybrid grids involving simplex cells (triangles, tetrahedra) and prismatic elements. The Spectral Difference method is part of high-order spectral discontinuous numerical methods. These methods rely on piecewise continuous polynomial approximation to obtain high-order accuracy with a good parallel efficiency. The standard SD scheme is first presented in the one-dimensional case and then for tensor-product elements (quadrangles and hexahedra). The treatment of simplex cells using Raviart-Thomas elements is detailed for triangles (in 2D) and tetrahedra (in 3D), followed by the implementation for prismatic elements. The linear stability of the Spectral Difference method using Raviart-Thomas elements (SDRT) is studied on triangles and tetrahedra. The SDRT scheme stability is strongly dependent on the interior flux points location. On triangles, the SDRT implementation based on interior flux points located at Williams-Shunn-Jameson quadrature points is found stable up to the fourth-order of accuracy but shown as spatially unstable for higher orders. Nevertheless, it is shown that this implementation can be stabilized for fifth- and sixth-order schemes using suitable temporal integration schemes. This approach being submitted to strict conditions, an optimization of the interior flux points location is conducted to determine spatially stable SDRT formulations for orders higher than four. The optimization process leads to spatially stable schemes up to the sixth-order of accuracy. Finally, the stability analysis on tetrahedra proves that the SDRT scheme based on the interior flux points located at Shunn-Ham quadrature points is stable up to the third-order. The SD/SDRT numerical method is validated on several academic cases for first and second-order Partial Differential Equations (linear advection equation, Euler equations, Navier-Stokes equations). Both proposed implementations (based either on Williams-Shunn-Jameson quadrature points or optimization points) are used. Numerical experiments involve grids composed of quadratic triangles, linear tetrahedral elements as well as 2D hybrid meshes.
... The numerical stability of the method was verified by Van den Abeele et al. [30] and Jameson [31], showing that the linear stability did not depend on the placement scheme of the solution points inside the elements but on that of the flux collocation points. Finally, it has successfully been used to perform LES and DNS alike [32][33][34]. Vanharen et al. [35] have assessed the accuracy of the SD method for various polynomial orders and compared it with spectral-like resolution 6 th -order compact FD schemes [36,37]. They showed that the 6 th -order SD method is at least as accurate as large-stencil FD methods while offering the compactness suited to unstructured grids. ...
... Inflow conditions were prescribed using Eqs. (32) and non-reflecting boundary were used at the outflow boundaries. The wall was treated as a 300 K isothermal no-slip TDIBC with B = 1. ...
Characteristic boundary conditions for the Navier-Stokes equations (NSCBC) are implemented for the first time with discontinuous spectral methods, namely the spectral difference and flux reconstruction. The implementation makes use of the resolution by these methods of the strong form of the Navier-Stokes equations by applying these conditions through a flux balance regularization which takes the form of a generalized element-compact correction polynomial. It is shown to be at least as effective as similar implementations in finite volume solvers, and sustains arbitrarily-high orders of accuracy on hexahedral-based unstructured meshes. Further, Navier-Stokes time-domain impedance boundary conditions are derived and implemented as a NSCBC sub-class. They account for the diffusive process at the wall and are shown to properly resolve the broadband response of porous materials under normal and grazing flow conditions. The ability of these NSCBC in preventing the appearance of spurious reflections at the boundaries is demonstrated through a varied series of bench-marking simulations. They effectively shield the inner computational domain from any far-field unphysical contamination. Overall, this work enables the use of strong discontinuous spectral methods to study unsteady problems on complex geometries.
... In the present work, and in view of applications to more realistic geometries, the performances of the SEDM are assessed on selected academic test cases featuring mild to high levels of grid distortion or fully unstructured mesh topologies, as well as some level of additional physical complexity (e.g., curved boundaries, detached flows). LES studies involving the SD method have been considered by a number of authors using either explicit SGS modeling [12][13][14][15][16] or relying on the numerical dissipation to mimic a dissipative SGS model [17][18][19]. ...
In this paper, the Spectral-Element Dynamic Model (SEDM), suited for Large-Eddy Simulation (LES) using Discontinuous Finite Element Methods (DFEM), is assessed using unstructured meshes. Five test cases of increasing complexity are considered, namely, the Taylor-Green vortex at Re = 5000, the turbulent channel flow at Reτ = 587, the circular cylinder in cross-flow at ReD = 3900, the square cylinder in cross-flow at ReD = 22400 and the channel with periodic constrictions at Reh = 10595. Various discretization parameters such as the grid spacing, polynomial degree and numerical flux are assessed and very accurate results are reported in all cases. This consistency in the results demonstrates the versatility of the SEDM approach and its ability to gage the actual resolution and quality of the mesh and, accordingly, to introduce an amount of sub-grid dissipation which is adapted to the spatial discretization considered.
... The study of numerical dissipation is therefore of capital importance when considering LES, also because the modeled energy drain from filtered scales toward sub-grid scales (SGS) is usually expressed as a dissipative operator [6,7], making these two sources of dissipation compete with each other. LES studies involving the SD method have be considered either using explicit SGS modeling [8][9][10][11][12][13] or relying on the numerical dissipation to mimic a dissipative SGS model [14][15][16]. This technique, termed Implicit-LES (ILES) [17] can yield accurate results if the numerical dissipation shows the same structure as the subgrid dissipation. ...
In this paper, the numerical dissipation properties of the Spectral Difference (SD) method are studied in the context of vortex dominated flows and wall-bounded turbulence, using uniform and distorted grids. First, the validity of using the SD numerical dissipation as the only source of subgrid dissipation (the so-called Implicit-LES approach) is assessed on regular grids using various polynomial degrees (namely, p = 3, p = 4, p = 5) for the Taylor-Green vortex flow configuration at Re = 5 000. It is shown that the levels of numerical dissipation greatly depend on the order of accuracy chosen and, in turn, lead to an incorrect estimation of the viscous dissipation levels. The influence of grid distortion on the numerical dissipation is then assessed in the context of finite Reynolds number freely-decaying and wall-bounded turbulence. Tests involving different amplitudes of distortion show that highly skewed grids lead to the presence of small-scale, noisy structures, emphasizing the need of explicit subgrid modeling or regularization procedures when considering coarse, high-order SD computations on unstructured grids. Under-resolved, high-order computations of the turbulent channel flow at Reτ = 1000 using highly-skewed grids are considered as well and present a qualitatively similar agreement to results obtained on a regular grid.
... Moreover, well-documented data of this case are available, mostly for Re D 3900 and Re D 2580 (Re D is the Reynolds number based on the cylinder diameter D and freestream velocity U). The latter case has more consistent results from different researchers [6][7][8][9][10]. However, for the former case, controversial near-wake results were found both experimentally and numerically [11]. ...
... ILES studies at this Reynolds number were widely used to validate high-order numerical methods. Mohammad et al. [9] obtained satisfactory results using ILES based on a high-order spectral difference method. In their simulations, the prediction of recirculation length was dependent on the fineness of the grid. ...
... The computational domain and the coordinate system used in the present paper for the cylinder flow are shown in Fig. 2. Here, x, y, and z denote the streamwise, transverse, and spanwise directions, respectively. The span of the computational domain is πD, which is identical to those in the previous studies [8][9][10]. The transverse a) b) c) Fig. 1 One-dimensional energy spectra for DHIT using different schemes. ...
Implicit large-eddy simulation studies of low-speed circular cylinder flows at subcritical Reynolds numbers of both 2580 and 3900 are conducted. Focus is placed on the controversial profile shape of the streamwise velocity around one diameter downstream from the cylinder. The minimum dispersion and controllable dissipation scheme is employed for the reconstruction and the simple low-dissipation AUSM scheme is used as the Riemann solver. The combined numerical method performs well in the implicit large-eddy simulation of the separated flows. With the numerical method’s feature of controllable dissipation, the nonmonotonic effect of the numerical dissipation on the predicted length of the recirculation bubble is studied and explained. When the numerical scheme is excessively dissipative, the increasing of dissipation tends to turn the flow into laminar vortex shedding, leading to a greatly underestimated recirculation bubble and a V-shape profile. However, when the numerical dissipation is small, its mechanism of suppressing the perturbations magnified by the numerical dispersion will become important. Increasing the dissipation will then increase the predicted recirculation size by delaying the shear-layer transition and tends to show a U-shape profile result. For implicit large-eddy simulation, the dispersion should be minimized and the dissipation should be controlled to an appropriate level.
... The studies based on more advanced LES models, such as the dynamic or the scale-similarity approaches (which require an explicit filtering of the solution) are found to provide excellent results [43][44][45], especially when high-orders of accuracy are considered. The ILES approach is often applied considering low orders of accuracy and/or low Reynolds configurations in order to ensure a global dissipation equivalent or greater than the dissipation originating from sub-grid scales [46][47][48][49][50][51]. Although this approach can yield good results for low Reynolds number flows, it has been demonstrated that at very high Reynolds number the dissipation stemming from the discretization of the convective flux for the DG and SD methods is not sufficient to provide a physical representation of the flows at higher orders of accuracy [52,53]. ...
A spectral dynamic modeling procedure for Large-Eddy Simulation is introduced in the context of discontinuous finite element methods. The proposed sub-grid scale model depends on a turbulence sensor built from the computation of a polynomial energy spectrum in each of the discretization elements. The evaluation of the energy decay gives an estimation of the quality of the resolution in each element and allows for adapting the intensity of the sub-grid dissipation locally. This approach is simple, robust, efficient and it is shown that the sub-grid model adapts to the amount of numerical dissipation in order to provide an accurate representation of the true sub-grid stresses. The present approach is tested for the large-eddy simulation of transitional, fully-developed and wall-bounded turbulence. In particular, results are reported for the Taylor-Green vortex and periodic turbulent channel flows at moderate Reynolds number. For these configurations, the new model shows an accurate description of turbulent phenomena at relatively coarse resolutions.
... Therefore, the performance of the SD method for turbulent flow simulations had been investigated either by an unresolved DNS type of simulation (Implicit Large Eddy Simulation, ILES, without modeling the contribution from the unresolved scales) as presented by Van den Abeele et al. [64,46], or through a full LES approach by coupling of the SD method with the local eddy-viscosity (WALE) model by Parsani et al. [65,66,67]. More ILES turbulent flow simulations can be found in Abrar et al.in [68] by simulating high Reynolds number flow over a circular cylinder, and Liang et al. [69] who had simulated compressible turbulent channel flow at Re=400. ...
In the field of Computational fluid dynamics, compact higher order discretization on
unstructured grids can be a possible avenue for improving the predictive capabilities of
numerical flow simulations of modern flow problems with reduced cost and effort. Therefore,
a two dimensional Navier-Stokes flow solver on unstructured quadrilateral grids has been
developed, associated with an elliptic structured grid generation module. The solver utilizes
the higher order Spectral Difference method as a spatial discretization approach which has
a conservative differential formulation. In this method two sets of points are used, namely
solution points for the conservative variables (unknowns) and flux points for the flux vector
components. Using the two sets of points, different polynomials can be constructed to repre-
sent the solution and flux variables inside each cell, and higher order accuracy is achieved by
raising those polynomial orders. In addition, an efficient isoparamteric mapping is employed
to transform the physical element to a standard computational one and allows for higher order
curved wall representations. Various inviscid flux treatments approaches are considered and
implemented in the solver. These include the Steger-Warming flux vector splitting, Rusanov
and Roe approximate Riemann solvers. Viscous flux discretization is acheived using the
Bassi and Rebay original approach. For the time integration, we employ explicit Runge-Kutta
schemes .
Flow cases with different physics and diverse geometries are considered to demonstrate the
solver’s ability to tackle practical aerodynamic flow problems. These cases include inviscid
subsonic flow around circular cylinder and NACA0012 airfoil. In addition, viscous flow cases
including laminar flow around a circular cylinder, and BATR airfoil-spoiler configuration
are computed. The solver’s accuracy and convergence through both h-refinement and p-
refinement is demonstrated up to fifth order of spatial accuracy, and bicubic curved elements.
Comparison of the computed results with published numerical and experimental data indicates
good agreement. Additionally, for the BATR airfoil-spoiler case an investigation of such
complex flow physics and effect of both angle of attack and spoiler deflections on the
aerodynamic characteristics of the flow are presented. Furthermore, the solver offers the
flexibility of working with both structured and unstructured grids. The solver is developed
using the C++ programming language on windows operating systems, and utilizes OpenMp
parallel directives to achieve better performance and parallelization on multi-core machines.
... It is identical to the staggered grid multi-domain method proposed in Kopriva and Kolias 25 , and Kopriva 26 for 2D quadrilaterals and 3D hexahedral elements. Liang et al. 27 has simulated the laminar flow past two side by side cylinders, and Mohammad et al., 28 investigated turbulent flow past a circular cylinder using the SD method. In addition, the performance of the SD method has been investigated by Liang et al. 29 to solve the compressible turbulent channel flow at Re = 400 without modeling the contribution from the unresolved scales in the flowfield. ...
The flowfield around airfoil with deflected spoiler has been simulated numerically using the higher order spectral difference method by solving the compressible two-dimensional, full Navier-Stokes equations on unstructured quadrilateral grids. Firstly, the spatial order of accuracy of the numerical method is verified. In addition, the unsteady flow over a circular cylinder has been simulated to ensure the accuracy of the time integration scheme implemented in the present work. Numerical results for the airfoil-spoiler problem are at angles of attack 0-8 degrees, spoiler deflection angles 0,15,30,60 degrees, and they are compared with experimental and other numerical results. Investigation of the effect of spoiler deflection angles on the averaged aerodynamic forces, pressure distributions, unsteady shedding behavior, shedding frequencies/Strouhl numbers, has been illustrated. The present work demonstrates the potential capability and flexibility of the higher order Spectral Difference method to solve high Reynolds number problems with complex geometries.
... The first test case aims to examine the capability of the present high-order finite-element methods for capturing three-dimensional unsteady flow structures and viscous boundary effects. This example involves viscous flow over a circular cylinder at a Reynolds number of 2580 based on the diameter of the cylinder [52]. The numerical simulation is conducted at a free-stream Mach number of 0.2 using various orders of the DG and SUPG discretizations in conjunction with the second-order backward difference time discretization. ...
... The geometry definition and the computational mesh containing 68,629 unstructured tetrahedral elements are displayed in Fig. 4. The x-axis is along the flow streamwise direction and the z-axis is along the spanwise direction. The dimensions in the streamwise, spanwise and transverse directions correspond to the work in [52]. Uniform flow is specified at the domain inlet and an outflow boundary condition with fixed pressure is set at the outlet. ...
... A more quantitative study on capturing the wake statistics is performed next. In particular, the mean velocities and Reynolds stresses at various locations in the wake region are compared with the experimental data [52]. Fig. 6(a) displays the mean streamwise velocity profiles normalized by free-stream velocity, using various orders of DG schemes. ...
In this paper, high-order finite-element discretizations consisting of discontinuous Galerkin (DG) and streamline upwind/Petrov Galerkin (SUPG) methods are investigated and developed for solutions of two- and three-dimensional compressible viscous flows. Both approaches treat the discretized system fully implicitly to obtain steady state solutions or to drive unsteady problems at each time step. A modified Spalart and Allmaras (SA) turbulence model is implemented and is discretized to an order of accuracy consistent with the main Reynolds Averaged Navier–Stokes (RANS) equations using the present high-order finite-element methods. To accurately represent the real geometry configurations for viscous flows, high-order curved boundary meshes are generated via a Computational Analysis PRogramming Interface (CAPRI), while the interior meshes are deformed subsequently through a linear elasticity solver. The mesh movement procedure effectively avoids the generation of invalid elements that can occur due to the projection of curved physical boundaries and thus allows high-aspect-ratio curved elements in viscous boundary layers. Several numerical examples, including large-eddy simulations of viscous flow over a three-dimensional circular cylinder and turbulent flows over a NACA 4412 airfoil and a high-lift multi-element airfoil at high angles of attack, are considered to show the capability of the present high-order finite-element schemes in capturing typical viscous effects such as flow separation and to compare the accuracy between the high-order DG and SUPG discretization methods.
