High-Order Structure-Preserving Spectral Difference Methods for Hyperbolic Conservation Laws

A collocated-grid spectral difference method for compressible flows

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).

This paper presents high-fidelity simulations of the Vortex-Induced Vibration (VIV) phenomena using a new computational model based on the high-order spectral difference (SD) method on unstructured grids. The SD method has shown promise in the past as a highly accurate, yet sufficiently fast method for solving unsteady viscous compressible flows. A Riemann solver is used to compute the inviscid fluxes at the cell interfaces, and the viscous fluxes are computed using an averaging mechanism based on the fluxes from the two cells that share the interface. A third-order Runge-Kutta scheme is used to advance time. In this viscous, compressible flow solver, the displacement of an elastically mounted bluff body has been coupled to the lift force created by vortex shedding. The solver is validated by correlating the lift and drag coefficients with previous published results of two cases: single rigid cylinder, and single elastically mounted cylinder. The rigid cylinder case validates the accuracy of the fluid solver. The elastically mounted case validates the fluid-structure coupling. We simulate the phenomenon of wake galloping with two cylinders after the solver has been validated using single cylinder VIV phenomenon.

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

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.

This study presents a high-fidelity direct numerical simulation (DNS) framework tailored for investigating turbulent flows through complex porous structures. It employs a compressible Navier–Stokes solver based on the spectral difference (SD) method, with immersed boundary conditions (IBCs) implemented via the Brinkman penalisation technique and integrated using a Strang splitting approach. A pressure gradient scaling (PGS) strategy is incorporated to improve computational efficiency. To provide realistic inflow conditions, synthetic turbulence is injected at the inlet using a random Fourier modes method. The methodology is validated in several stages. First, the IBC approach is tested against results from a body-fitted mesh, showing strong agreement in the mean velocity field. Next, the effectiveness of the PGS technique is demonstrated by comparing scaled and unscaled simulations, both of which yield consistent velocity fields and spectral content. Finally, the full DNS-SD framework is benchmarked against finite volume method results from the literature, successfully reproducing key turbulence characteristics, including two-point correlations. The validated solver is ultimately applied to simulate turbulent flow through a complex porous geometry. The results illustrate the robustness of the approach and highlight its potential for advancing the understanding of turbulence in porous materials.

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.

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.

In this study, the spectral difference (SD) method has been extended to deal with the Reynolds-Averaged Navier- Stokes (RANS) equations coupled with the Spalart-Allmaras (SA) and k ω turbulence models. The description of limiting or stabilization techniques adopted in order to prevent the simulation from blowing up are given in this paper. The reliability, robustness and accuracy of the implementation have been assessed by computing several validation cases: the flow over flat plate(Re “ 5ˆ 10), the flow past a backward-facing step (Re “ 37400) and the flow around NACA0012 airfoil at different angles of attack (α “ 0, 10). Finally the developed SD solver is applied to simulate the tone noise generated by subsonic cavity flow at moderate Reynolds number.

The acoustic attenuation coefficient of soft biological tissue has been observed to have an increasing linear-with-frequency attenuation characteristic with a slope, denoted by β, that varies with the disease condition of the liver. Hence, it would be diagnostically useful to estimate the value of β from reflected ultrasound signals. Two approaches for estimating β are examined: the spectral-shift approach, which estimates β from the downward shift experienced by the propagating pulse spectrum with penetration into the liver, and the spectral-difference approach, which estimates β from the slope of the log spectral differences. While the spectral-shift approach requires the propagating pulse to have a Gaussian-shaped spectrum, the spectral-difference method does not require a specific spectral form. A mathematical model is developed to simulate the random ultrasound signals reflected from the liver. The bias and variance properties of the β estimators are determined by using the simulated signals and compared as a function of the data window size. The results indicate that, while the accuracy of both approaches is equivalent for large data windows, the frequency-shift approach is more accurate than the spectral-difference approach for most practical cases.

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.

The flux reconstruction approach to high-order methods is robust, efficient, simple to implement, and allows various high-order schemes, such as the nodal discontinuous Galerkin method and the spectral difference method, to be cast within a single unifying framework. Utilizing a flux reconstruction formulation, it has been proved (for one-dimensional linear advection) that the spectral difference method is stable for all orders of accuracy in a norm of Sobolev type, provided that the interior flux collocation points are located at zeros of the corresponding Legendre polynomials. In this article the aforementioned result is extended in order to develop a new class of one-dimensional energy stable flux reconstruction schemes. The energy stable schemes are parameterized by a single scalar quantity, which if chosen judiciously leads to the recovery of various well known high-order methods (including a particular nodal discontinuous Galerkin method and a particular spectral difference method). The analysis offers significant insight into why certain flux reconstruction schemes are stable, whereas others are not. Also, from a practical standpoint, the analysis provides a simple prescription for implementing an infinite range of energy stable high-order methods via the intuitive flux reconstruction approach.

High fidelity numerical simulation of airfoil thickness and kinematics effects on flapping airfoil propulsion

Fully-discrete spatial eigenanalysis of discontinuous spectral element methods: Insights into well-resolved and under-resolved vortical flows

Abstract. We developed a synoptic climatology for Lake El'gygytgyn, Chukotka Russia, and explored modern climate trends affecting air temperatures there to aid in paleoclimate reconstructions of a 3.6 million-year-old sediment core taken from the lake. Our self-organized mapping (SOM) approach identified 35 synoptic weather patterns, based on sea level pressure, that span the range of synoptic patterns influencing the study domain over the 1961–2009 NCEP/NCAR analysis period. We found strong seasonality in modern weather patterns, with summer weather primarily characterized by weak low pressure systems over the Arctic Ocean or Siberia and winter weather primarily characterized by strong high pressure over the Arctic Ocean and strong low pressure in the Pacific Ocean. In general, the primary source of variation in air temperatures came from the dominant patterns in each season, which we identify in the text, and nearly all of the dominant weather patterns here have shown increasing temperatures. We found that nearly all of the warming in mean annual temperature over the past 50 yr (about 3 °C) occurred during sub-freezing conditions on either side of summer (that is, spring and fall). Here we found that the most summer-like weather patterns (low pressures to the north) in the shoulder seasons were responsible for much of the change. Finally, we compared the warmest 15 yr of the record (1995–2009) to the coolest (1961–1975) and found that changes in thermodynamics of weather were about 3 to 300 times more important than changes in frequency of weather patterns in controlling temperature variations during spring and fall, respectively. That is, in the modern record, general warming (local or advected) is more important by orders of magnitude than changes in storm tracks in controlling air temperature at Lake El'gygytgyn. We conclude with a discussion of how these results may be relevant to the paleoclimate reconstruction efforts and how this relevancy could be tested further.