Stable, entropy-consistent, and localized artificial-diffusivity method for capturing discontinuities

In this work, a localized artificial-viscosity/diffusivity method is proposed for accurately capturing discontinuities in compressible flows. There have been numerous efforts to improve the artificial diffusivity formulation in the last two decades, through appropriate localization of the artificial bulk viscosity for capturing shocks. However, for capturing contact discontinuities, either a density or internal energy variable is used as a detector. An issue with this sensor is that it not only detects contact discontinuities, but also falsely detects the regions of shocks and vortical motions. Using this detector to add artificial mass/thermal diffusivity for capturing contact discontinuities is hence unnecessarily dissipative. To overcome this issue, we propose a sensor similar to the Ducros sensor (for shocks) to detect contact discontinuities, and further localize artificial mass/thermal diffusivity for capturing contact discontinuities. The proposed method contains coefficients that are less sensitive to the choice of the flow problem. This is achieved by improved localization of the artificial diffusivity in the present method. A discretely consistent dissipative flux formulation is presented and is coupled with a robust low-dissipative scheme, which eliminates the need for filtering the solution variables. The proposed method also does not require filtering for the discontinuity detector/sensor functions, which is typically done to smear out the artificial fluid properties and obtain stable solutions. Hence, the challenges associated with extending the filtering procedure for unstructured grids is eliminated, thereby, making the proposed method easily applicable for unstructured grids. Finally, a straightforward extension of the proposed method to two-phase flows is also presented.Comment: 24 pages, 11 figures, Under review in the Physical Review Fluids journa

A Discontinuity-Capturing Methodology for Two-Phase Inviscid Compressible Flow

The explicit filtering approach is applied to the quasi-conservative five-equation model of compressible two-phase flows to capture the interface between each fluid and a shock wave. The basic idea of the present filter is to combine a low-order linear filter with a high-order one via a proper discontinuity sensor and optimum linear weights. The capability of the proposed filter in capturing the contact discontinuity and damping the grid-to-grid oscillations is analysed. Various one-dimensional and two-dimensional test cases are performed, namely the interface advection of gas-gas flow, the shockinterface interaction, the gas-liquid Riemann problem, and the inviscid shock-bubble interaction. The numerical results reveal that the present filtering method can accurately capture the propagation of the shock waves and interfaces. Additionally, it produces less spurious oscillations compared with the existing 2nd-order discontinuity-capturing filter

Smooth and compactly supported viscous sub-cell shock capturing for Discontinuous Galerkin methods

In this work, a novel artificial viscosity method is proposed using smooth and compactly supported viscosities. These are derived by revisiting the widely used piecewise constant artificial viscosity method of Persson and Peraire as well as the piecewise linear refinement of Klöckner et al. with respect to the fundamental design criteria of conservation and entropy stability. Further investigating the method of modal filtering in the process, it is demonstrated that this strategy has inherent shortcomings, which are related to problems of Legendre viscosities to handle shocks near element boundaries. This problem is overcome by introducing certain functions from the fields of robust reprojection and mollififers as viscosity distributions. To the best of our knowledge, this is proposed for the first time in this work. The resulting $C_0^\infty$ artificial viscosity method is demonstrated to provide sharper profiles, steeper gradients and a higher resolution of small-scale features while still maintaining stability of the method

An a posteriori, efficient, high-spectral resolution hybrid finite-difference method for compressible flows

Versión aceptada de https://doi.org/10.1016/j.cma.2018.02.013[Abstract:] A high-order hybrid method consisting of a high-accurate explicit finite-difference scheme and a Weighted Essentially Non-Oscillatory (WENO) scheme is proposed in this article. Following this premise, two variants are outlined: a hybrid made up of a Finite Difference scheme and a compact WENO scheme (CRWENO 5), and a hybrid made up of a Finite Difference scheme and a non-compact WENO scheme (WENO 5). The main difference with respect to similar schemes is its a posteriori nature, based on the Multidimensional Optimal Order Detection (MOOD) method. To deal with complex geometries, a multi-block approach using Moving Least Squares (MLS) procedure for communication between meshes is used. The hybrid schemes are validated with several 1D and 2D test cases to illustrate their accuracy and shock-capturing properties.This work has been partially supported by the Ministerio de Economía y Competitividad (grant #DPI2015-68431-R) of the Spanish Government and by the Consellería de Educación e Ordenación Universitaria of the Xunta de Galicia (grant #GRC2014/039), cofinanced with FEDER funds and the Universidade da Coruña.Xunta de Galicia; GRC2014/03

Effectiveness of Shock Capturing Methods in the Discontinuous Galerkin/Flux Reconstruction Scheme for Computational Fluid Dynamics

The development of various numerical methods capable of accurately simulating fluid flow has evolved greatly over time. In the past years, discontinuous Galerkin methods have seen great interest for problems such as Large Eddy Simulations, Aeroacoustics, incompressible and even compressible flows. These methods attractiveness some from their ability to easily increase the order of accuracy thus yielding more precise solutions. These methods use higher order polynomials, which can easily be increased or decreased within the element, while allowing for discontinuities between elements. When shocks and discontinuities are present in a simulation, particular attention must be taken to avoid Gibbs phenomenon within the elements. This phenomenon occurs when steep gradients in the solution are present causing the solution to have erratic oscillations typically associated with the higher order terms of the integrating polynomial. These oscillations in turn lead to non-physical solutions such as negative pressures and therefore need to be controlled. A variety of methods have been developed to mitigate the oscillatory behavior of discontinuous Galerkin methods when steep gradients are present, a very promising method is the addition of artificial viscosity in order to diminish the effects of the non-physical oscillations. Adding a viscous term to the conservation equations being solved can inevitably lead to inaccurate solutions if it is added in excessive amounts. The balance between damping of the non-physical oscillations and minimizing the amount of artificial viscosity added can if the location of the shocks in the flow field is known. This intricate balance is achieved by ensuring that the functions used to find the areas of concern are not overlapping shock regions with smooth regions and when viscosity is added it is important that it is limited to ensure that it will not completely dissipate the real solution

Positivity-preserving discontinuous spectral element methods for compressible multi-species flows

We introduce a novel positivity-preserving, parameter-free numerical stabilisation approach for high-order discontinuous spectral element approximations of compressible multi-species flows. The underlying stabilisation method is the adaptive entropy filtering approach (Dzanic and Witherden, J. Comput. Phys., 468, 2022), which is extended to the conservative formulation of the multi-species flow equations. We show that the straightforward enforcement of entropy constraints in the filter yields poor results around species interfaces and propose an adaptive, parameter-free switch for the entropy bounds based on the convergence properties of the pressure field which drastically improves its performance for multi-species flows. The proposed approach is shown in a variety of numerical experiments applied to the multi-species Euler and Navier--Stokes equations computed on unstructured grids, ranging from shock-fluid interaction problems to three-dimensional viscous flow instabilities. We demonstrate that the approach can retain the high-order accuracy of the underlying numerical scheme even at smooth extrema, ensure the positivity of the species density and pressure in the vicinity of shocks and contact discontinuities, and accurately predict small-scale flow features with minimal numerical dissipation.Comment: Submitted for revie

LES of Temporally Evolving Mixing Layers by an Eighth-Order Filter Scheme

An eighth-order filter method for a wide range of compressible flow speeds (H.C. Yee and B. Sjogreen, Proceedings of ICOSAHOM09, June 22-26, 2009, Trondheim, Norway) are employed for large eddy simulations (LES) of temporally evolving mixing layers (TML) for different convective Mach numbers (Mc) and Reynolds numbers. The high order filter method is designed for accurate and efficient simulations of shock-free compressible turbulence, turbulence with shocklets and turbulence with strong shocks with minimum tuning of scheme parameters. The value of Mc considered is for the TML range from the quasi-incompressible regime to the highly compressible supersonic regime. The three main characteristics of compressible TML (the self similarity property, compressibility effects and the presence of large-scale structure with shocklets for high Mc) are considered for the LES study. The LES results using the same scheme parameters for all studied cases agree well with experimental results of Barone et al. (2006), and published direct numerical simulations (DNS) work of Rogers & Moser (1994) and Pantano & Sarkar (2002)