Artificial Diffusivity Scheme Research Articles

Impact of the Stabilized Finite Element Method on Acoustic and Vortical Perturbations in Thermoacoustic Systems

Abstract This paper seeks to advance linear stability analyses of thermoacoustic systems conducted with the stabilized finite element method (sFEM). Specifically, this work analyzes and quantifies the impact of the streamline-upwind-Petrov–Galerkin (SUPG) artificial diffusion scheme on (eigen)mode shapes and damping rates of the isentropic linearized Euler equations (LEEs) in frequency space. The LEE (eigen)mode shapes are separated into acoustic and vortical perturbation components via a Helmholtz decomposition and their sensitivity on the employed stabilization scheme is investigated separately. The regions where numerical stabilization mainly acts on the perturbation types are identified and explanations for the observations are provided. A methodology is established, which allows the quantification of the impact of artificial diffusion on the acoustic field in terms of a damping rate. This nonphysical damping rate is used to determine the physically meaningful, acoustic LEE damping rate, which is corrected by the contribution of artificial diffusion. Hence, the presented method eliminates a main shortcoming of LEE eigenfrequency analyses with the sFEM and, as a result, provides more accurate information on the stability of thermoacoustic systems.

On a robust and accurate localized artificial diffusivity scheme for the high-order flux-reconstruction method

The high-order flux reconstruction (FR) method on unstructured hexahedral grids is coupled with the localized artificial diffusivity (LAD) scheme for aiming at accurate simulation of shock-turbulence interaction. In order to overcome known robustness issues particularly with the high-order (r>0) derivative formulation, important properties of the artificial bulk viscosity (ABV) profile affecting the solution are investigated first. For the purpose of comparison, an approximated Gaussian filter and modified restriction-prolongation (RP) filters are developed and tested for the present FR-LAD approach. Then, we propose a multidimensional RP filter on unstructured quadrilateral and hexahedral grids to address the issues on non-Cartesian grids. It is shown that a simple extension of a one-dimensional RP filter for two-dimensional grids may result in insufficient smoothing of ABV with r=2, and that the proposed multidimensional RP filter can provide smooth ABV with both r=0 and r=2. The proposed FR-LAD scheme is tested for typical shock-related problems, including the 1D shock tube, 1D shock-entropy wave interaction, 2D steady shock flows, and 2D shock-vortex interaction. The FR-LAD scheme has favorable properties of subcell shock capturing with the length scale of O(h/p) and superior preservation of high-order accuracy for smooth flows. Finally, LES of an overexpanded supersonic jet is performed to demonstrate its capability for practical applications. The proposed FR-LAD scheme can be an attractive candidate for LES or DNS of compressible flows involving shocks, contact discontinuities, turbulence, and their interactions on unstructured meshes.

Localized artificial diffusivity scheme for deflagrations and detonation waves

The ability of a localized artificial diffusivity (LAD) method to capture reactive flow discontinuities, such as deflagrations and detonations, using high-order compact differencing scheme is evaluated in this paper. Different types of steady and unsteady problems under 1D and 2D conditions are tested. For the laminar premixed flame, the flame speed and numerical flame thickness are used when evaluating the accuracy of the LAD results. As the grid spacing increases, the physical behaviour of the flame is maintained using LAD. It is shown that the ability to obtain accurate flame structure are controlled by the artificial diffusivity terms for both the thermal conductivity and species diffusivity. The artificial species diffusivity term is found to affect the L2 error of the solution when the mesh is progressively refined. However, the error reduces according to the order of the discretization error for the schemes being used when the LAD terms are switched off. The choice of the model constants for the LAD terms, selected previously for non-reacting cases, are found appropriate as the flame thickness and chemical reaction rate are found to adjust such that the flame speed is well predicted. The finding is verified further using a 2D configuration involving vortex-flame interaction. The ability of LAD to retain the flame properties allows for the flame response to vortex interaction to be captured at larger grid sizes. For the CJ detonation wave, the structure of the detonation wave is well captured without spurious oscillations. The amplitude of wiggles near a shock or contact discontinuity and the number of grid points used to capture the discontinuity are largely independent of the mesh size. A 2D CJ detonation configuration is also investigated and shows the ability of the LAD method to capture the transverse variations of the detonation wave when subjected to an initial perturbation. The study demonstrates that the LAD method does not show any major detrimental effect on the discontinuity capturing under the conditions examined for both deflagration and detonation wave.

Modulation to compressible homogenous turbulence by heavy point particles. I. Effect of particles’ density

In this paper, two-way interactions between heavy point particles and forced compressible homogenous turbulence are simulated by using a localized artificial diffusivity scheme and a Eulerian–Lagrangian approach. The initial turbulent Mach number is around 1.0 and the Taylor Reynolds number is around 110. Seven different simulations of 106 particles with different particle densities (or initial Stokes numbers) are considered. The statistics of the compressible turbulence, such as the turbulence Mach number, kinetic energy, dilatation, and the kinetic energy spectra, from different simulations are compared with each other and with the one-way undisturbed case. Our results show that the turbulence is suppressed if the two-way coupling backward interactions are considered, and the suppression is more obvious if the density of particles is higher. The kinetic energy spectrum at larger initial Stokes number (higher density) exhibits a reduction at low wave numbers and an augmentation at high wave numbers, which is similar to those obtained in incompressible cases. The probability density functions of dilatation and normal upstream Mach number of shocklets also show that the modulation to the shocklet statistics is more apparent for particles with higher density.

An artificial diffusivity discontinuous Galerkin scheme for discontinuous flows

In this paper a discontinuous Galerkin (DG) scheme based on artificial diffusivity is developed for discontinuous flows. The artificial diffusivity model takes the formulation in [Kawai S, Lele SK. Localized artificial diffusivity scheme for discontinuity capturing on curvilinear meshes. J Comput Phys 2008; 227: 9498–526], and to compute the high-order derivatives therein with relatively low order DG schemes (less than fifth order), a novel method which is feasible for unstructured grids is proposed, which incorporates a filter into the differentiation process. Convergence tests show that the computed 1st, 2nd and 3rd derivatives using the proposed method are able to achieve second order accuracy for one- and two-dimensional cases. Several typical test cases are simulated to assess the ability of the artificial diffusivity DG scheme in terms of accuracy and stability.

Large-Eddy Simulation of Jet Mixing in Supersonic Crossflows

Large-eddy simulation of an underexpanded sonic jet injection into supersonic crossflows is performed to obtain insights into key physics of the jet mixing. A high-order compact differencing scheme with a recently developed localized artificial diffusivity scheme for discontinuity-capturing is used. Progressive mesh refinement study is conducted to quantify the broadband range of scales of turbulence that are resolved in the simulations. The simulations aim to reproduce the flow conditions reported in the experiments of Santiago and Dutton [Santiago, J. G., and Dutton, J. C., Velocity Measurements of a Jet Injected into a Supersonic Crossflow, Journal of Propulsion and Power, Vol.132,1997, pp. 264―273] and elucidate the physics of the jet mixing. A detailed comparison with these data is shown. Statistics obtained by the large-eddy simulation with turbulent crossflow show good agreement with the experiment, and a series of mesh refinement studies shows reasonable grid convergence in the predicted mean and turbulent flow quantities. The present large-eddy simulation reproduces the large-scale dynamics of the flow and jet fluid entrainment into the boundary-layer separation regions upstream and downstream of the jet injection reported in previous experiments, but the richness of data provided by the large-eddy simulation allows a much deeper exploration of the flow physics. Key physics of the jet mixing in supersonic crossflows are highlighted by exploring the underlying unsteady phenomena. The effect of the approaching turbulent boundary layer on the jet mixing is investigated by comparing the results of jet injection into supersonic crossflows with turbulent and laminar crossflows.

Assessment of localized artificial diffusivity scheme for large-eddy simulation of compressible turbulent flows

The localized artificial diffusivity method is investigated in the context of large-eddy simulation of compressible turbulent flows. The performance of different artificial bulk viscosity models are evaluated through detailed results from the evolution of decaying compressible isotropic turbulence with eddy shocklets and supersonic turbulent boundary layer. Effects of subgrid-scale (SGS) models and implicit time-integration scheme/time-step size are also investigated within the framework of the numerical scheme used. The use of a shock sensor along with artificial bulk viscosity significantly improves the scheme for simulating turbulent flows involving shocks while retaining the shock-capturing capability. The proposed combination of Ducros-type sensor with a negative dilatation sensor removes unnecessary bulk viscosity within expansion and weakly compressible turbulence regions without shocks and allows it to localize near the shocks. It also eliminates the need for a wall-damping function for the bulk viscosity while simulating wall-bounded turbulent flows. For the numerical schemes used, better results are obtained without adding an explicit SGS model than with SGS model at moderate Reynolds number. Inclusion of a SGS model in addition to the low-pass filtering and artificial bulk viscosity results in additional damping of the resolved turbulence. However, investigations at higher Reynolds numbers suggest the need for an explicit SGS model. The flow statistics obtained using the second-order implicit time-integration scheme with three sub-iterations closely agrees with the explicit scheme if the maximum Courant–Friedrichs–Lewy is kept near unity.

Localized artificial diffusivity scheme for discontinuity capturing on curvilinear meshes

A simple and efficient localized artificial diffusivity scheme is developed for the purpose of capturing discontinuities on curvilinear and anisotropic meshes using a high-order compact differencing scheme. The artificial diffusivity is dynamically localized in space to capture different types of discontinuities such as a shock wave, contact surface or material discontinuity. The method is intended for use with large-eddy simulation of compressible transitional and turbulent flows. The method captures the discontinuities on curvilinear and anisotropic meshes with minimum impact on the smooth flow regions. The amplitude of wiggles near a discontinuity and the number of grid points used to capture the discontinuity do not depend on the mesh size. The comparisons between the proposed method and high-order shock-capturing schemes illustrate the advantage of the method for the simulation of flows involving shocks, turbulence and their interactions. The multi-dimensional formulation is tested on a variety of 1D and 2D, steady and unsteady, different types of discontinuity-related problems on curvilinear and anisotropic meshes. A simplification of the method which reduces the computational cost does not show any major detrimental effect on the discontinuity capturing under the conditions examined.

Spectral Element Method with Adaptive Artificial Diffusion for Solving the Radiative Transfer Equation

An adaptive isotropically artificial diffusion (AISO) scheme is developed for the spectral element method to mitigate the ray effects encountered in the solution of radiative transfer problems. By considering the coupled manner of ray effects and false scattering, the artificial diffusion coefficient is determined heuristically from both local angular discretization scale and local spatial discretization scale. The scheme is easily and efficiently implemented under the spectral or finite-element method framework. The performance of various artificial diffusion schemes is studied and compared. The streamwise artificial diffusion schemes are only responsible for stabilization of strong convection-induced instability, while the isotropically artificial diffusion scheme shows very good performance in mitigating the “wiggles” in both low- and high-order spatial approximation. Numerical experiments show that the spectral element method with the AISO scheme is stable, high-order-accurate, and effective for solving radiative transfer in simple and complex geometries, and also robust for mitigating ray effects of various origins.

Finite-Element Analysis of Corona Discharge Onset in Air With Artificial Diffusion Scheme and Under Fowler–Nordheim Electron Emission

A complete finite-element analysis method for corona discharge process, which is governed and coupled by charge transport equation and electric field equation, was presented. The charge transport equation of first order was transformed into second order by utilizing the artificial diffusion scheme. The two second-order equations were analyzed by the finite-element formulation which is well-developed for second-order equations. The Fowler-Nordheim injection boundary condition was adopted for charge transport equation. After verifying the numerical results by comparing to the analytic solutions using parallel plane electrodes with one carrier system, we extended the result to blade-plane electrodes in 2-D xy geometry with three carriers system. Radius of the sharp tip was taken to be 50 mum. When this sharp geometry was solved by utilizing the space discretizing methods, the very sharp tip was found to cause a singularity in electric field and space charge distribution around the tip. To avoid these numerical difficulties in the finite-element method, finer meshes, a higher-order shape function, and artificial diffusion scheme were employed

Effect of spatial discretization schemes on numerical solutions of viscoelastic fluid flows

This study examines the effect of discretization schemes for the convection term in the constitutive equation on numerical solutions of viscoelastic fluid flows. For this purpose, a temporally evolving mixing layer, a two-dimensional vortex pair interacting with a wall, and a fully developed turbulent channel flow are selected as test cases, and eight different discretization schemes are considered. Among them, the first-order upwind difference scheme (UD) and artificial diffusion scheme (AD), which are commonly used in the literature, show most stable and smooth solutions even for highly extensional flows. However, the stress fields are smeared too much by these schemes and the corresponding flow fields are quite different from those obtained by higher-order upwind difference schemes. Among higher-order upwind difference schemes investigated in this study, a third-order compact upwind difference scheme (CUD3) with locally added AD shows stable and most accurate solutions for highly extensional flows even at relatively high Weissenberg numbers.

Artificial diffusion concept for moving conductor eddy current problems with edge elements

We report a way to condition the global stiffness matrix resulting from the Galerkin edge element formulation, in order to make the bi-conjugate gradient (BICG) solver converge for high Peclet number situations. The concept of artificial diffusion is introduced and extended to edge element cases. A general expression of the artificial diffusion scheme for 3D edge elements is derived and a numerical validation in 2D case is presented. It was observed that although nearly identical results were achieved both with or without introducing artificial parameters for a 2D test problem, the former provided a better conditioned global stiffness matrix to ensure the BICG's convergence.