Accurate, efficient and monotonic numerical methods for multi-dimensional compressible flows

Kim, Kyu Hong; Kim, Chongam

doi:10.1016/j.jcp.2005.02.021

Cited by 131 publications

(127 citation statements)

References 20 publications

(33 reference statements)

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“…The compressible equations are discretised by the characteristics-based method, as detailed in Eberle (1987); Bagabir and Drikakis (2004); Drikakis (2003). High resolution is achieved by the Monotonic Upstream-Centered Scheme for Conservation Laws (MUSCL) scheme in its Total Variation Diminishing form (Van Leer 1977) in conjunction with the fifth-order accurate limiter (Kim and Kim 2005) and low-Mach corrections (Thornber et al 2008). The fifth-order version of the MUSCL scheme has been found to provide accurate results for a broad range of flows (Drikakis et al 2009).…”

Section: Methodsmentioning

confidence: 99%

Computing multi-mode shock-induced compressible turbulent mixing at late times

et al. 2015

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Both experiments and numerical simulations pertinent to the study of self-similarity in shock-induced turbulent mixing often do not cover sufficiently long enough times for the mixing layer to become developed in a fully turbulent manner. When the Mach number of the flow is sufficiently low, numerical simulations based on the compressible flow equations tend to become less accurate due to inherent numerical cancellation errors. This paper concerns a numerical study of the late time behaviour of single-shocked Richtmyer-Meshkov Instability (RMI) and associated compressible turbulent mixing using a new technique that addresses the above limitation. The present approach exploits the fact that RMI is a compressible flow during the early stages of the simulation and incompressible at late times. Therefore, depending on the compressibility of the flow field the most suitable model, compressible or incompressible, can be employed. This motivates the development of a hybrid compressible-incompressible solver that removes the low-Mach number limitations of the compressible solvers, thus allowing numerical simulations of late time mixing. Simulations have been performed for a multi-mode perturbation at the interface between two fluids of densities corresponding to an Atwood number of 0.5, and results are presented for the development of the instability, mixing parameters and turbulent kinetic energy spectra. The results are discussed in comparison with previous compressible simulations, theory and experiments.

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Section: Methodsmentioning

confidence: 99%

Computing multi-mode shock-induced compressible turbulent mixing at late times

et al. 2015

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“…As the outcomes in two-dimensional flows [5], MLP is also proved to bring enhanced convergence and accuracy improvement simultaneously in three-dimensional compressible flows. …”

Section: Resultsmentioning

confidence: 99%

“…In order to find out the criterion of oscillation control for multiple dimensions, Kim and Kim [5] extended the one-dimensional monotonic condition to two dimensions and presented the two-dimensional limiting condition successfully. With the limiting condition, a multi-dimensional limiting process (MLP) is proposed which gives more accurate results for two-dimensional Euler and Navier-Stokes equations.…”

Section: Introductionmentioning

confidence: 99%

Multi-dimensional Limiting Process for Two- and Three-dimensional Flow Physics Analyses

Yoon

Kim

2009

Computational Fluid Dynamics 2006

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In this paper, we derive a limiting condition for three-dimensional compressible flows and present the multi-dimensional limiting process for three-dimensions. The basic idea of the multi-dimensional limiting condition is that the vertex values interpolated at a grid point should be within the maximum and minimum cell-average values of neighboring cells for the monotonic distribution. By applying the MLP (Multi-dimensional Limiting Process), we can achieve monotonic characteristic, which results in the substantial enhancement of solution accuracy and convergence behavior. IntroductionSince the late 1970s, numerous ways to control oscillations have been studied and several limiting concepts have been proposed. Most representatives would be TVD [1, 2], TVB [4] and ENO [3]. The concept of TVD (Total Variation Diminishing) was proven to be extremely successful in solving hyperbolic conservation laws. Most oscillation-free schemes have been based on the mathematical analysis of one-dimensional convection equation and applied to systems of equations with the help of some linearization step. They are also applied to multi-dimensional applications with dimensional splitting. Though they may work successfully in many cases, it is insufficient or almost impossible to control oscillations near shock discontinuity in multiple space dimensions. For that reason, the need of oscillation control method for multi-dimensional applications is obvious.In order to find out the criterion of oscillation control for multiple dimensions, Kim and Kim [5] extended the one-dimensional monotonic condition to two dimensions and presented the two-dimensional limiting condition successfully. With the limiting condition, a multi-dimensional limiting process (MLP) is proposed which gives more accurate results for two-dimensional Euler and Navier-Stokes equations. It is this approach which prompts the work of the present paper. Basically, it extends the idea of MLP to three dimensions. Thus, in this paper, we derive a three-dimensional limiting condition and present the multi-dimensional limiting process for two-and three-dimensional situations.

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“…In regards to the ILES, three grid levels where created having 43 3 , 64 3 and 96 3 cells. The extrapolation methods employed for the simulations are the second-order MUSCL scheme with Van Albada slope limiter (M2-VA), the third-order MUSCL with Kim and Kim limiter [24] (M3-KK), and the third-order MUSCL scheme with the Drikakis and Zoltak limiter [25] (M3-DD). The second-order strong stability-preserving Runge-Kutta scheme is used for temporal discretisation for the ILES approach.…”

Section: Numerical Model and Flow Diagnosticsmentioning

confidence: 99%

Investigation of Numerical Dissipation in Classical and Implicit Large Eddy Simulations

2017

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Abstract:The quantitative measure of dissipative properties of different numerical schemes is crucial to computational methods in the field of aerospace applications. Therefore, the objective of the present study is to examine the resolving power of Monotonic Upwind Scheme for Conservation Laws (MUSCL) scheme with three different slope limiters: one second-order and two third-order used within the framework of Implicit Large Eddy Simulations (ILES). The performance of the dynamic Smagorinsky subgrid-scale model used in the classical Large Eddy Simulation (LES) approach is examined. The assessment of these schemes is of significant importance to understand the numerical dissipation that could affect the accuracy of the numerical solution. A modified equation analysis has been employed to the convective term of the fully-compressible Navier-Stokes equations to formulate an analytical expression of truncation error for the second-order upwind scheme. The contribution of second-order partial derivatives in the expression of truncation error showed that the effect of this numerical error could not be neglected compared to the total kinetic energy dissipation rate. Transitions from laminar to turbulent flow are visualized considering the inviscid Taylor-Green Vortex (TGV) test-case. The evolution in time of volumetrically-averaged kinetic energy and kinetic energy dissipation rate have been monitored for all numerical schemes and all grid levels. The dissipation mechanism has been compared to Direct Numerical Simulation (DNS) data found in the literature at different Reynolds numbers. We found that the resolving power and the symmetry breaking property are enhanced with finer grid resolutions. The production of vorticity has been observed in terms of enstrophy and effective viscosity. The instantaneous kinetic energy spectrum has been computed using a three-dimensional Fast Fourier Transform (FFT). All combinations of numerical methods produce a k −4 spectrum at t * = 4, and near the dissipation peak, all methods were capable of predicting the k −5/3 slope accurately when refining the mesh.

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Accurate, efficient and monotonic numerical methods for multi-dimensional compressible flows

Cited by 131 publications

References 20 publications

Computing multi-mode shock-induced compressible turbulent mixing at late times

Computing multi-mode shock-induced compressible turbulent mixing at late times

Multi-dimensional Limiting Process for Two- and Three-dimensional Flow Physics Analyses

Investigation of Numerical Dissipation in Classical and Implicit Large Eddy Simulations

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