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

Abstract

<Abstract> 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.

Stable and accurate interface capturing advection schemes

“…To fulfil the above requirements, our choice moves towards multidimensional limiting process (MLP) strategies ( [40,29]) that are generalizations of the limiters to the multidimensional case, thus allowing to keep control of the gradient direction. For time integrators we will simply use second-order Runge-Kutta RK2 schemes.…”

“…Multidimensional limiting process or MLP has been introduced by different authors [40,29] in order to provide an improved accuracy for multidimensional problems, especially for highspeed computational fluid dynamics. It is a natural extension of the one-dimensional slopelimiting process that takes into account the local neighbouring information for both gradient reconstruction and limitation.…”

“…For example, if conservation properties are mandatory, levelset methods are not the right candidate family: even if today we find volume conservative levelset methods, mass conservation may not be strictly fulfilled, what can be not accurate enough for some highly compressible flows or flows with high ratios of density. After decades of sustained developments and research in this fields, interface capturing (IC) methods are still an active field of investigation (see the recent references [8,9,10,2,7,29,31,40]). They can show advantages like a natural extension to an arbitrary number of fluids, phases of materials, the possibility to deal with complex topologies and configurations (triple points, ...) in a rather easy way, the simplicity of code development, debugging and optimization.…”

Compressive properties of Min-mod-type limiters in modelling shockwave-containing flows

“…Kim and Kim [29] proposed a MLP (Multi-dimensional Limiting Process) method which demonstrates a fine feature of controlling numerical oscillations in multi-space dimensions with very desirable properties in terms of accuracy, efficiency and robustness. Yoon and Kim [30] modified the aforementioned MLP and refined it for three-dimensional applications without assuming local gradients, which made an excellent improvement on the solution accuracy, convergence, as well as the robustness for the steady/unsteady flows.…”