不可压流计算的Projection 方法和SIMPLE 方法间桥梁的搭建麦箭
倪明玖
中国科学院研究生院物理科学学院
不可压流计算方法设计的一个核心问题是发展合适的离散方法，满足不可压约束条件。原始变量法中，获得很好应用的有MAC 方法 [1]，Projection 方法 [2-5]和SIMPLE 方法[6-8]。MAC 是显式法，已应用到湍流DNS (Direct Numerical Simulation)和多流体界面流的数值模拟。Projection 方法和SIMPLE 方法是隐式或半隐式算法。隐式算法较之显式法的优越性是时间步长可以大大增加。Projection 方法已经广泛应用于非定常流的模拟，而SIMPLE 类方法则成功地模拟定常流，特别是传热有关的流动计算。SIMPLE 类方法也被用来解非定常流的计算。分析这些原始变量法 (特别是Projection 方法和SIMPLE 方法)间的关系将很有意义。
本文将在Projection 方法和SIMPLE 方法间搭建一个桥梁。首先通过对压力梯度项进行非线形离散(()
n n p p ∇−+∇+n n I θθ1)，基于矩阵分析方法构造了一类具有二阶时间精度的Projection 方法。包括一类三步和一类四步Projection 方法。这里是非线形系数矩阵，它可能是网格尺寸、时间步长、流动速度的函数。并将标准的SIMPLE 方法写成简洁的矢量表达式。一些经典的二阶精度Projection 方法和 SIMPL
E 类方法是本文构建的一类Projection 方法的特例。而SIMPLEC 方法更是完全等价于经典的三步Projection 方法。本文并证明标准的 SIMPLE 方法及SIMPLEC 方法，应用到非定常流的计算中，有二阶时间精度。这样，本文在这两类方法间搭建了一桥梁。
n θ通过模拟两维Taylor 涡，数值验证SIMPLE 方法和本文构造的Projection 方法是二阶时间精度算法(见图一)。用Projection 方法，结合全守恒（质量、动量和动能守恒）四阶空间离散格式，对直管道二维充分发展通道湍流进行了直接数值模拟，并和谱方法的结果[9]进行了比较（见图二）。结合Level Set 方法[10]和 VOF（Volume of Fluid）方法[11]，本文应用Projection 方法模拟了多流体界面流，并和实验结果[12，13]及数值解[14]进行了比较（见图三和图四）。结合电流守恒格式[15]，本文也模拟了强磁场作用下的金属流体MHD 流，并和分析解[16]进行了比较。
参考文献
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By SIMPLE Method using 2nd-order update of temporal term
By classical four-step Projection Method with a Runge-Kutta technique ud Figure 1 The Maximum Error of Velocity u over the Refined Meshes
y
+
024681012
14161820(a). Dimensionless mean velocity and
temperature
y
0.00.5
1.0
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2.0
2.5
3.0
(b). Distribution of rms velocity and temperature
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Gain
y
+
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0.040.080.120.16
0.20
Loss
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+
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300-0.20-0.15-0.10
-0.050.00
0.05
0.10
0.15
0.20
0.25(c). Budget of turbulence kinetic
(d). Budget of temperature variance 22
θϑ′=k
Figure 2 DNS results for fully developed channel thermal field
(a)Zero level t contour from time=0.2 to time=3.0 with time interval 0.2
(b) Location of water front at y=0
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教育故事幼儿园(c) Height of wetted wall at x=0
Figure 3： History of water front location on solid surfaces in the dam break
