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1. 北京航空航天大学 航空科学与工程学院, 北京 100083;
2. 中国空间技术研究院 载人航天总体部, 北京 100094

Performance analysis of a new-type third-order TVD limiter
ZHAO Yatian1, YAN Chao1, SUN Di1, QU Feng2
1. School of Aeronautic Science and Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100083, China;
2. General Department of Manned Spacecraft, China Academy of Space Technology, Beijing 100094, China
Received: 2016-04-06; Accepted: 2016-07-22; Published online: 2016-10-28 15:16
Foundation item: National Natural Science Foundation of China (11402016)
Corresponding author. YAN Chao, E-mail: yanchao@buaa.edu.cn
Abstract: For numerical scheme in computational fluid dynamics (CFD), limiter technology is an important factor affecting computational accuracy and stability. Although the present classical second-order total variation diminishing (TVD) limiters with a wide application can well satisfy the computing requirements, its performance not only largely differs but also cannot be properly weighted between resolution and dissipation. Therefore, a new third-order TVD interpolation limiter (T-3 limiter) has been studied and compared with three classical limiters. First, through one-dimensional Riemann problem, it has been found that T-3 limiter is simultaneously characterized by both high intermittent resolution and excellent stability; then, by numerical simulation of hypersonic flow over a double-cone body and X-33 configuration, it has been found that T-3 limiter boasts the capability of portraying complex flow and good aerothermodynamic calculation performance.
Key words: limiter     computational fluid dynamics (CFD)     shocks     double-cone disturbance flow     aerothermodynamic

1 计算方法 1.1 控制方程

 (1)

1.2 二阶TVD限制器的一般形式

 (2)

 (3)

minmod限制器：

 (4)

superbee限制器：

 (5)

double minmod限制器：

 (6)
1.3 三阶TVD插值构造原理

 (7)

 (8)

2 数值模拟结果与分析

2.1 Sod问题

 (9)

 图 1 密度分布曲线 Fig. 1 Density distribution curves
2.2 双锥绕流

 图 2 流场结构示意图[14] Fig. 2 Schematic diagram of structure of flow field[14]
 图 3 对称面等马赫线分布 Fig. 3 Mach contours in symmetry plane

 图 4 壁面压强沿母线分布曲线 Fig. 4 Distribution curves of wall surface pressure along generating line

 限制器 x/L 分离点 再附点 分离区长度 minmod 0.597 5 1.103 5 0.506 double minmod 0.565 8 1.164 1 0.598 superbee 0.505 0 1.151 1 0.646 T-3 0.449 1 1.095 0 0.646

 限制器 峰值位置 (x/L) 位置误差/% 压强峰值 (p/p∞) 峰值误差/% 实验 1.424 5 104.698 minmod 1.358 5 4.633 0 94.307 9.925 double minmod 1.429 6 0.358 0 94.689 9.560 superbee 1.437 5 0.913 0 107.886 3.045 T-3 1.498 5 1.825 0 102.831 1.783

T-3与superbee限制器对双锥绕流流场结构计算最为准确。但superbee由于耗散小，导致振荡大，流场结构不够清晰。T-3不仅得到的对称面等马赫线流场结构清晰，而且子午线的压强分布与实验值误差最小。

2.3 X-33外形飞行器

 图 5 40°迎角下对称面等马赫线图和壁面压强云图 Fig. 5 Mach contours in symmetry plane and wall surface pressure contours at α=40°

 图 6 40°迎角下热流云图 Fig. 6 Contours of heat transfer at α=40°
 图 7 迎风区子午线热流分布与实验值的对比 Fig. 7 Comparison between windward centerline heat flow distribution and experimental data

3 结论

1) 限制器的选取需要兼顾分辨率和计算稳定性2个方面，T-3限制器与superbee相比，数值色散小，具有良好的稳定性和收敛性以避免非物理解的产生。与double minmod相比，构造原理相同，但计算精度更高。与minmod相比，具有好的间断分辨率且通过限制函数避免了过多的数值耗散。

2) 一维Sod激波管算例表明，T-3、minmod、superbee和double minmod均能较为准确捕捉激波、接触间断和膨胀波等结构。

3) 双锥绕流算例表明，对于复杂流动，T-3限制器性能较为优越。T-3分辨率与superbee相当，且由于耗散比superbee大，得到的流场结构更加稳定清晰。

4) 对X-33外形的热流计算表明，T-3限制器能合理预测驻点位置，较为准确地刻画热流分布的规律，表现出了较好的高超声速流动适用性及气动热预测能力。

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#### 文章信息

ZHAO Yatian, YAN Chao, SUN Di, QU Feng

Performance analysis of a new-type third-order TVD limiter

Journal of Beijing University of Aeronautics and Astronsutics, 2017, 43(4): 800-805
http://dx.doi.org/10.13700/j.bh.1001-5965.2016.0266