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Performance analysis of high accuracy multi-dimensional limiting process
SUN Di, YAN Chao , YU Jian, QU Feng, HUA Jun
School of Aeronautic Science and Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100191, China
Abstract:The conventional limiting process is mostly based on one-dimensional structure, which cannot keep monotonic features of quantities under conditions of multi-dimensional discontinuities, leading to non-physical oscillations. In order to overcome the structure defects of the conventional methods, multi-dimensional limiting process (MLP) is a high accuracy limiter whose basic idea is that the vertex values interpolated at a grid point should be within the maximum and minimum cell-average values of neighboring cells through multi-dimensional correction. The major advantage of MLP is to avoid multi-dimensional oscillatory effectively and ensure solving accuracy. According to a set of test cases including one-dimensional shock tube, non-viscous vortex flow and shock boundary-layer interaction, the performance of MLP with high accuracy was analyzed, it is verified that third-order MLP can avoid multi-dimensional oscillatory effectively both in continuous and discontinuous area. Compared with higher-order WENO (weighted essentially non-oscillatory) schemes, the third-order MLP maintains several desirable characteristics, such as simple algorithm, simple implementation, improving the solving accuracy, monotonicity and convergence. For these properties, MLP can be applied to study complicated flow in engineering and scientific research, and is expected to have a bright application future.
Key words: high accuracy multi-dimensional limiting process     multi-dimensional limiting process     monotonicity     shock boundary-layer interaction     WENO schemes

1 计算方法

1.1 控制方程及空间离散方法

1.2 MUSCL(TVD)限制器的一般形式

MUSCL格式的具体形式[14, 15]

1.3 高阶多维限制器的构造

5阶限制器的表达形式为

 图 1 格点值与格心值分布示意图Fig. 1 Distribution schematic of cell-vertex value and cell-center value of grid

2 算例及结果分析

2.1 Sod问题

 图 2 密度分布曲线Fig. 2 Density distribution curves
2.2 二维涡流动问题

 图 3 初始流场密度分布Fig. 3 Density distribution of initial vortex flow field
 图 4 沿AB线的压力分布Fig. 4 Pressure distribution along line AB

 图 5 熵随时间变化的曲线Fig. 5 Variation curves of entropy with change of time

 图 6 不同格式的网格收敛性比较Fig. 6 Comparison of grid convergence with different schemes
2.3 激波边界层干扰

 图 7 压力等值线图Fig. 7 Pressure contours
 图 8 压强分布曲线Fig. 8 Pressure distribution curves

 图 9 壁面压强分布曲线Fig. 9 Pressure distribution curves along wall surface
3 结 论

1) MLP3限制器与常见二阶精度TVD限制器相比具有明显的优势:与minmod限制器相比,MLP3有较高的精度,且通过多维限制函数,它避免了过多的耗散,可以较为精确地捕捉到激波等间断;与superbee限制器相比,MLP3具有良好的保单调性以避免非物理解的产生.

2) 二维涡流动算例表明:较之于高阶WENO格式及传统TVD限制器,MLP3限制器更容易实现,且在花费更小计算量的同时保持强鲁棒性及高精度.

3) 激波边界层干扰算例表明:超声速有黏流动计算时，MLP3限制器的求解精度高于同阶WENO格式,与5阶WENO格式结果相似.因此MLP3具有较高的黏性分辨率和保单调特性.

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#### 文章信息

SUN Di, YAN Chao, YU Jian, QU Feng, HUA Jun

Performance analysis of high accuracy multi-dimensional limiting process

Journal of Beijing University of Aeronautics and Astronsutics, 2015, 41(3): 437-442.
http://dx.doi.org/10.13700/j.bh.1001-5965.2014.0185