Loose and Strong Coupling Methods for Flow/Kinematics Coupled Simulations and Stability Analysis
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摘要: 为了更加精确地模拟流动/运动耦合问题, 建立了耦合动态混合网格生成非定常流场计算和六自由度运动方程求解的一体化计算方法, 并在统一框架内同时实现了松耦合与紧耦合方法.通过圆柱涡致自激振荡(vortex induced vibration, VIV)的模拟, 对不同时间精度的松耦合和紧耦合算法的优劣及适用范围进行了评估和分析; 通过引入附加质量的概念, 对耦合算法的稳定性进行了理论分析.研究表明:在流体的密度与物体的密度接近时, 松耦合方法是不稳定的, 必须采用紧耦合方法.最后利用耦合算法对二维鱼体的自主游动和钝锥三自由度自由飞过程进行了数值模拟, 证实了理论分析的结论.Abstract: In order to simulate the flow/kinematics coupled problem more accurately, an integrated method which couples the dynamic hybrid grid generation, unsteady flow simulation, and computation of flight mechanics equations was presented. The loose coupling method and the strong coupling method (or fully-implicit method) were achieved in a unified framework. The vortex induced vibration (VIV) of a cylinder was simulated to compare different coupling algorithms and investigate their performance. Furthermore, the stability condition was studied for these coupling algorithms. Finally, the S-type starting with self-propelled swimming of a 2D model fish and a blunted cone 3-dimension free-flight process were simulated with the present coupling algorithms. The numerical and theoretical results validate that the loose coupling methods will result in instability when the added-mass is very close to the real mass of the solid. In that case, the strong coupling method will be a better choice.
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图 2 松耦合数据交互示意图
Figure 2. Sketch of data exchanges between CFD and RBD in loose coupling computation
图 3 紧耦合数据交互示意图
Figure 3. Sketch of data exchanges between CFD and RBD in fully implicit coupling computation
图 6 4种CFD/RBD方法lg(2ymax)随Δt的变化
Figure 6. lg(2ymax) vs. time steps (computed by four types of second-order CFD/RBD coupling algorithms)
图 8 不同耦合方式计算的lg(2ymax) vs. lg(2π2St2ζm/ρ∞r2)
Figure 8. Displacements in the y-direction vs. the reduced damping coefficient (ζ) with different coupling algorithms
图 9 μs=4/π时松耦合方法计算得到的位移变化曲线
Figure 9. Displacements in the x and y directions by loose coupling algorithm when μs=4/π
图 10 μs=1时松耦合方法计算的位移变化曲线
Figure 10. Displacements in the x and y directions by loose coupling algorithm when μs=1
图 11 S型启动过程中的动态混合网格
Figure 11. Hybrid dynamic meshes during the S-type starting process
图 12 鱼体质心在水平方向的位移及速度变化情况
Figure 12. Translations and velocities in horizontal direction
图 14 固定质心方式阻力随摆动波速的变化情况
Figure 14. Drag vs. wave speed when the center of gravity is fixed
图 15 鱼体启动过程中若干典型位置流场的涡量云图
Figure 15. Vorticity contours during the S-type starting process
表 1 统一形式飞行力学离散控制方程中各种格式的系数
Table 1. Parameters in different coupling algorithms for the solution of flight dynamics equations
temporal scheme order a b c d e classification 1st-forward-difference formula (FDF1) 1 1 0 0 1 0 loose coupling 1st-backward-difference formula (BDF1) 1 1 0 1 0 0 strong coupling 2nd-forward-difference formula (FDF2) 2 1 0 0 3/2 -1/2 loose coupling 2nd-backward-difference formula (BDF2) 2 4/3 -1/3 2/3 0 0 strong coupling Crank-Nicolson (C-N) 2 1 0 1/2 1/2 0 strong coupling 3rd-Adams-Moulton (A-M3) 3 1 0 5/12 8/12 -1/12 strong coupling 
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表 2 不同Δt下各种耦合方式计算得到的lg(2ymax)
Table 2. lg(2ymax) computed by different coupling algorithms with different time steps
Δt N-S temporal precision 1st-forward-difference formula 1st-backward-difference formula 2nd-forward-difference formula 2nd-backward-difference formula Crank-Nicolson 3rd-Adams-Moulton 0.20 1st order diverged -1.2097 -0.9841 -0.9353 -0.9496 -0.9569 2nd order diverged -0.9293 -0.5616 -0.6369 -0.6070 -0.5994 0.15 1st order diverged -1.0650 -0.8116 -0.7980 -0.8007 -0.8023 2nd order -0.0292 -0.8857 -0.6001 -0.6483 -0.6305 -0.6257 0.10 1st order -0.3423 -0.9116 -0.6970 -0.7020 -0.6996 -0.6993 2nd order -0.3252 -0.8277 -0.6319 -0.6538 -0.6460 -0.6437 0.05 1st order -0.5118 -0.7682 -0.6540 -0.6576 -0.6564 -0.6560 2nd order -0.5222 -0.7522 -0.6496 -0.6549 -0.6531 -0.6525 0.02 1st order -0.6020 -0.6964 -0.6512 -0.6519 -0.6517 -0.6516 2nd order -0.6080 -0.6978 -0.6548 -0.6556 -0.6554 -0.6553 0.01 1st order -0.6296 -0.6755 -0.6531 -0.6532 -0.6532 -0.6532 2nd order -0.6328 -0.6775 -0.6556 -0.6558 -0.6558 -0.6558
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表 3 鱼体运动及质量参数
Table 3. Parameters of 2D fish swimming
fish body length i.e. reference length l wave length λ wave velocity c mass of 2D fish m 0.1 m 0.1 m 0.1 m/s 1.0 kg/m
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表 4 3种方法得到的鱼体巡游速度
Table 4. Cruising speeds calculated by three different methods
methods cruising speed/ wave speeds fixed mass center 0.66 free swimming: 1DOF 0.66 free swimming: 3DOFs 0.57
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表 5 钝锥质量特性
Table 5. Mass characteristics of the blunt cone
mass m inertia moment of x-axis Ixx inertia moment of y-axis Iyy inertia moment of z-axis Izz 246.9 g 335 g·cm2 335 g·cm2 580.89 g·cm2
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表 6 钝锥自由飞过程计算条件
Table 6. Flow parameters of the blunt cone free-flight
Mach number M Reynolds number Re reference length Lref reference temperature T∞ reference pressure p∞ reference density ρ∞ reference velocity U∞ reference time tref 5 8.45×106 1 m 69.17 K 942.86Pa 0.04714 kg/m3 836.645 m/s Lref / U∞ = 1.195×10-3 s
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