Massively Parallel Spectral Difference Solver for Simulating Vortex-Induced Vibrations of Circular Cylinders

A high-order spectral difference (SD) method for solving the Navier-Stokes equations on moving, deformable unstructured grids has been developed [1]. In this paper, the SD method and the artificial compressibility method (ACM) are integrated with a dual time-stepping scheme to model unsteady incompressible viscous flow past an inline tube bundle of cylinders equally sized (diameter = d) and spaced (spacing = 2.1*d) over an unstructured grid. Flow simulation results are obtained using a fourth-order space accurate SD method. Two forced oscillation cases are considered; (1) 1st cylinder oscillation and (2) 2nd cylinder oscillation. The Reynolds number used for both cases is 100 and the flow is laminar. Forced oscillation is performed in the tranverse direction, and the subsequent altering of the flow physics of the system is studied. The frequency of vortex shedding behind each cylinder is the same. Root mean square results show that the lift coefficient is greatest for the 4th inline cylinder in both cases. Furthermore, a reduction in both lift and drag coefficients is seen from case (1) to case (2).

A collocated-grid spectral difference method for compressible flows

A high-order spectral difference (SD) method has been developed with graphics processing units (GPUs) using compute unified device architecture (CUDA). It solves the three-dimensional Navier-Stokes equations on unstructured hexahedral grids with RungeKutta time integration. The method is efficient since operations are completed in a onedimensional fashion and the equations are solved in differential form, removing explicit surface and volume integral calculations. Additionally, solution and flux reconstructions are completed locally per cell, increasing the parallelization of the implementation. Due to this efficiency, the application of GPU computing is appealing. This paper presents the SD method implementation with GPU CUDA computing and presents accuracy studies with isotropic vortex propagation and Couette flow, verifies the high-order accuracy of the solver with a numerical sensitive aero-acoustic problem, and compares the developed solver and a high-order finite difference solver with a case presented in the 1st International Workshop on High-Order CFD Methods. Finally, the GPU solver is compared to a similar central processing unit (CPU) solver, where speed-ups ranging from 20-40x faster are illustrated.

Numerical simulations of 2-DOF vortex-induced vibration of a circular cylinder in two and three dimensions: A comparison study

Suppression of the vortex-induced vibration of a circular cylinder surrounded by eight rotating wake-control cylinders

A high-order accurate flow solution method for quadrilateral elements is developed and tested. The method employs the flux reconstruction (FR) approach proposed by Huynh. Ideal orders of accuracy are achieved for several problems on structured-based grids while some deterioration is observed on purely unstructured grids. Cares for curved wall boundary are also shown. The localized artificial diffusivity (LAD) method is incorporated to capture discontinuities that is crucial for compressible flow simulations. The FR+LAD works very well for 1D test cases but multi-dimensional shock problems requires further modifications to confine the artificial diffusivity to the shock location. Hybrid scheme of the FR and finite difference method taking advantage of 1D-feature alleviates the instability. I. Introduction Expanding demands for the computational fluid dynamics (CFD) requires more reliable numerical schemes. Specifically, turbulent flow simulations, acoustic simulation, combustion simulations and their combined problems. The key words in such simulations may be high accuracy and high resolution. In structured grid CFD, a compact finite difference scheme satisfies both requirements at higher level than existing other methods. If we put more emphasis on easier handling of complex geometries and flow-data dependent grid generation (grid adaptation), then unstructured grid method should also be explored and it should depart from 2nd-order scheme with limiter. Recent efforts in this area are development of the discontinuous Galerkin (DG) method, 1,2 the spectral finite volume (SV) method 3 and the staggered-grid multi-domain method spectral finite difference (SG) method 4,5 and its evolutional method called a spectral difference method. 6 The common and maybe the only way for those unstructured methods is to add degrees of freedom in each cell to achieve high order since the wide stencil of the original grid is not possible due to unstructured nature. The ENO-type method that uses original degrees of freedom in the grids and extends adaptive stencils to the flow solutions is also explored but the difficulty of extending stencils due to undirectional nature of unstructured grids and the unsuitability for parallel computing prohibits wide applications. A flux reconstruction (FR) scheme 7,8 studied in the paper is the new one among the former group. It uses differential form of conservation laws like the SD method but it is more general and includes equivalent counter part of the schemes in the group. In this paper, the performance of the FR is tested by a newly developed Navier-Stokes solver. Realistic flow applications of the FR in the Navier-Stokes equations are seldom seen since it is relatively new scheme. The Localized artificial diffusivity (LAD) 9–13 is incorporated for the shock capturing in the FR. The last item is motivated by the challenge and success of combining the SD and LAD in the recent study. 11,12 Wang 14 developed the lifting collocation penalty (LCP) method to extend the FR to triangular or tetrahedral grids and applied the LCP-FR combined method for hybrid unstructured grids. In the present study, only quadrilateral cells are treated to directly apply the LAD method that was developed on structured grids. The shock capturing by the FR+LAD high order scheme is tried.

Towards an understanding of marine fouling effects on VIV of circular cylinders: Aggregation effects

On the vortex-induced vibration of a low mass ratio circular cylinder near a planar boundary

In this study the noise generated by the 30P30N highlift airfoil is simulated with a high order computational aeroacoustics code. To handle the complex geometry of the airfoil, the high order spectral difference method based on unstructured mesh is used for spatial discretization. The multi-time-step method based on Adam-Bashforth scheme is utilized for time marching, and a speed-up ratio greater than 10 is obtained comparing with the single time step method. The detached eddy simulation method is conducted for turbulence modeling. Radiation and outflow non-reflective boundary conditions are used in the far field and downstream. In this simulation, the inflow Mach number is 0.17, and the Reynolds number based on the inflow velocity and the chord length of the airfoil is 1.7e6. Three attack angles, which are 4, 5.5, and 8.5 degrees, respectively, are considered in this study. The dynamic pressure on the surface of the slat is sampled and compared with the experimental data by other researchers, and a good agreement is obtained. The far field noise, computed with the permeable Ffowcs Williams-Hawkings (FW-H) integration method, is analyzed and compared with the experimental data. The noise source is analyzed and identified with a correlation analysis of the dynamic pressure on the slat and the near field noise.

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 R e = 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 R e τ = 1000 using highly-skewed grids are considered as well and present a qualitatively similar agreement to results obtained on a regular grid.

The work focuses on the development of a 2D quadrilateral element based Spectral Difference solver for viscous flow calculations, and the application of the p-multigrid method and implicit time-stepping to accelerate convergence. This paper extends the previous work by Liang et al (2009) on the p-multigrid method for 2D inviscid compressible flow, to viscous flows. The high-order spectral difference solver for unstructured quadrilateral meshes is based on the formulation of Sun et al for unstructured hexahedral elements. The p-multigrid method operates on a sequence of solution approximations of different polynomial orders ranging from one upto four. An efficient preconditioned Lower-Upper Symmetric Gauss-Seidel (LU-SGS) implicit scheme is also implemented. The spectral difference method is applied to a variety of inviscid and viscous compressible flow problems. The speed-up using the p-Multigrid and Implicit time-stepping techniques is also demonstrated.

Two-degree-of-freedom vortex-induced vibrations of a circular cylinder in the vicinity of a stationary wall

A study on the numerical dissipation of the Spectral Difference method for freely decaying and wall-bounded turbulence

An efficient implicit lower-upper symmetric Gauss-Seidel (LU-SGS) solution algorithm has been developed for a high order multi-domain spectral difference method on unstructured hexahedral grids. The LU-SGS solver is preconditioned by the block element matrix, and the system of equations is then solved with an exact LU decomposition approach. In addition, the effects of several parameters on the convergence rate in the implicit scheme have been investigated for both external and internal flows. The implicit scheme has shown a speed up factor of more than an order of magnitude relative to the multi-stage explicit Runge-Kutta scheme for several demonstration problems.

This paper presents a study focused on solving hyperbolic conservation laws using arbitrary-order spectral difference (SD) methods. The study is structured around several crucial aspects. Firstly, in order to ensure the maximum principle for scalar equations and positivity preservation for Euler systems, we adopt the concept of flux limiters. This adoption leads to the development of the structure-preserving SD scheme with a flux limiter (SDFL), which is proven to preserve the original high-order accuracy. However, the SDFL scheme with lower order might lack conservational properties, despite its strong performance in short-term simulations. Consequently, we have developed a specific variant of the SDFL scheme with conservational properties, referred to as the CSDFL scheme. Secondly, we introduce a modified WENO-ZQ (MWENO-ZQ) reconstruction to suppress spurious oscillations when simulating problems with strong discontinuities. Finally, we conduct extensive numerical experiments to validate the effectiveness of the proposed high-order SD (SDFL,CSDFL) methods with MWENO-ZQ reconstruction. The results demonstrate the robustness and efficiency of these techniques in solving problems involving strong discontinuities, low pressure, and low density.