DEVELOPMENT OF A NUMERICAL WAVE TANK WITH A CORRECTED SMOOTHED PARTICLE HYDRODYNAMICS SCHEME TO REDUCE NONPHYSICAL ENERGY DISSIPATION
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摘要: 目前, 无网格光滑粒子流体动力学SPH粒子法在波浪与结构物相互作用研究方面得到广泛应用, 但该方法模拟波浪远距离传播时, 常常面临严重的能量耗散问题, 导致波高非物理性降低, 给大范围海域、长时间作用下的波-物耦合作用研究带来一定困难. 对此, 本文采用一种核函数修正算法, 在确保粒子间相互作用对称性的同时, 改进压力梯度离散项的计算精度, 设法解决SPH方法中能量非物理性耗散的难题. 相较于前人减缓能量非物理性衰减的方法, 本文的修正SPH算法避免了自由液面搜索等复杂处理过程, 并能保证动量守恒特性. 数值结果中, 采用振荡液滴、规则波、不规则波等算例, 验证本修正SPH算法的准确性和有效性. 结果表明, 该修正SPH算法能准确模拟振荡液滴形态变化, 且动能保持较好守恒性. 通过数值水池与物理水池两者规则波与不规则波结果的对比分析表明, 基于本文修正SPH算法建立的数值波浪水池具有较好的抗能量衰减效果, 能实现长时间、远距离波浪传播的准确模拟. 此外, 本算法能在低光滑长度系数条件下, 实现精确模拟, 将极大缩减三维SPH模拟的时间, 从而节约计算成本.Abstract: So far, the smoothed particle hydrodynamics (SPH) method has been widely applied in the study of the interactions between water waves and structures. However, nonphysical energy dissipation is a still problem which challenges the simulation of wave-body interactions at large-scale and long duration. For example, in the SPH simulation of wave propagation to a long distance, the wave height could gradually become much smaller than the one generated near the wave maker. To tackle this problem, in this work a kernel correction algorithm is applied to the pressure gradient term in the SPH model, aiming to prevent nonphysical energy dissipation in long time simulations. The kernel correction algorithm is able to ensure the symmetry of the interaction between particle pairs, and therefore, compared with other corrective methods, the present corrected algorithm ensures the conservation of linear momentum and also avoids the complicated treatment at the free surface. Two numerical cases, i.e., the oscillating droplet and wave propagation in a numerical wave tank, are presented to test the accuracy and validity of present corrected SPH algorithm. For the oscillating droplet case, the corrected algorithm is shown to accurately simulate the evolution of the droplet shape, and the kinetic energy is dissipated much slower than traditional SPH models. Through the simulations of regular and irregular wave propagations as well as validations with experimental data, the capability of the corrected SPH algorithm to reduce nonphysical energy attenuation is demonstrated, even for wave propagation at long-term and long-distance conditions. In addition, this algorithm will be shown to be optimal for the SPH simulation at small smooth length, which contributes to save SPH computational cost significantly at three dimensional simulations.
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图 3 不同粒子分辨率下角动量
${M_A}$ 的时历曲线Figure 3. Time evolution of the angular momentum
${M_A}$ with different particle resolutions
图 5 αs = 2.0, 不同粒子分辨率时机械能衰减率
${E_{per}}$ 时历曲线Figure 5. Time evolution of the decay rate of mechanical energy at different particle resolutions with αs = 2.0
图 6 αs = 1.3(上)和αs = 2.0(下)时, 振荡液滴形态: 理论值(虚线)和SPH结果(压力云图)
Figure 6. With smoothing length coefficient αs =1.3 (top) and αs = 2.0 (bottom), the shapes of oscillating droplet: theoretical results (dash line) and SPH results (pressure contour)
图 7 δ-SPH与δ-SPHC动能比较: 总体图(上)与局部放大图(下), 其中黑矩形框为放大区域
Figure 7. The comparison between time evolutions of kinetic energy between δ-SPH (top) and δ-SPHC (bottom), an enlarged zone view of the results in the dark rectangle is shown in the bottom panel
图 9 不同算法和光滑长度系数下, 振荡液滴机械能衰减率时历曲线
Figure 9. Time evolution of the decay rate of mechanical energy for the oscillating droplet with different SPH models and different smoothing length coefficient
图 11 造波推板横向位移随时间变化: (a)周期为T = 1 s的规则波, (b)不规则波
Figure 11. Paddle motions of the regular wave with (a) period T = 1 s and (b) irregular wave
图 12 SPH数值水池示意图, 红竖线表示波高仪, 选取距离波高仪一个粒子间距内自由面粒子的最大高度为波面高度
Figure 12. Schematic diagram of the numerical wave tank, the red vertical line represents wave gage. Maximum height of the free-surface particle within one-particle distance from the red line is measured to represent the wave elevation
图 13
${\alpha _s}$ =2.0时, 不同粒子分辨率下波面高度的δ-SPH模拟值与试验值对比Figure 13. Comparison of the wave elevations between δ-SPH results and experimental data, SPH results with different particle resolutions and
${\alpha _s} = 2.0$ are provided
图 14 在
${\alpha _s}$ = 1.3,$d/\Delta x$ = 60条件下, 传统δ-SPH模拟的波面形态Figure 14. Wave surface simulated by δ-SPH method with
${\alpha _s}$ = 1.3 and$d/\Delta x$ = 60
图 15 在
${\alpha _s} = 1.3$ ,$d/\Delta x$ = 60条件下, 传统δ-SPH模型计算的不同位置波面高度时历曲线Figure 15. Wave elevation probed at different positions simulated by δ-SPH method with
${\alpha _s} = 1.3$ and$d/\Delta x$ = 60
图 16 在
${\alpha _s} = 2.0$ ,$d/\Delta x$ = 60条件下, 传统δ-SPH模拟的波面形态Figure 16. Wave surface simulated by δ-SPH method with
${\alpha _s}$ = 2.0 and$ d/\Delta x $ = 60
图 17 在
${\alpha _s}$ = 2.0,$ d/\Delta x $ = 60条件下, 传统δ-SPH模拟的不同位置波面高度时历曲线Figure 17. Wave elevation probed at different positions simulated by δ-SPH method with
${\alpha _s}$ = 2.0 and$d/\Delta x$ = 60
图 18 在αs = 1.3和
$d/\Delta x$ = 15, 30, 60条件下, δ-SPHC计算的波面高度时历曲线与试验值对比Figure 18. Comparison between δ-SPHC results and experimental data for wave elevation: SPH results with αs = 1.3 and
$d/\Delta x$ = 15, 30, 60
图 19 在αs = 2.0和
$d/\Delta x$ =15, 30, 60条件下, δ-SPHC计算的波面高度时历曲线与试验值对比Figure 19. Comparison between δ-SPHC results and experimental data for wave elevation: SPH results with αs = 2.0 and
$d/\Delta x$ = 15, 30, 60
图 20
${x_{{\rm{probe}}}}$ =2.37 m处, 不同SPH算法下的波面高度数值结果与试验值对比Figure 20. Comparison between different SPH simulations and experimental data for wave elevations at
${x_{{\rm{probe}}}}$ = 2.37 m
图 21 规则波压力云图(δ-SPHC,
${\alpha _s}$ = 2.0,$d/\Delta x$ = 60)Figure 21. Pressure contour of regular wave (δ-SPHC,
${\alpha _s}$ = 2.0,$d/\Delta x$ = 60)
图 22
${\alpha _s}$ =1.3,$d/\Delta x$ = 60时, δ-SPHC算法模拟的波面形态Figure 22. The wave surface simulated by δ-SPHC with
${\alpha _s}$ = 1.3 and$d/\Delta x$ = 60
图 23
${\alpha _s}$ =1.3,$d/\Delta x$ =60时, δ-SPHC模拟的不同位置波面高度时历曲线Figure 23. Wave elevations probed at different positions simulated by δ-SPHC method with
${\alpha _s}$ = 1.3 and$d/\Delta x$ = 60
图 24
${\alpha _s}$ =2.0,$d/\Delta x$ =60时, δ-SPHC算法模拟的波面形态Figure 24. The wave surface simulated by δ-SPHC with
${\alpha _s}$ = 2.0 and$d/\Delta x$ =60
图 25
${\alpha _s}$ = 2.0,$d/\Delta x$ = 60时, δ-SPHC模拟的不同位置波面高度时历曲线Figure 25. Wave elevations probed at different positions simulated by δ-SPHC method with
${\alpha _s}$ = 2.0 and$d/\Delta x$ = 60
图 26 在
${x_{{\rm{probe}}}}$ = 6.37 m处, 规则波波面高度数值模拟结果与试验结果Figure 26. Time evolution of the regular wave elevation at
${x_{{\rm{probe}}}}$ = 6.37 m: SPH results and experimental data
图 27 δ-SPHC在不同黏性系数α时,
${x_{{\rm{probe}}}}$ = 6.37 m处波面高度时历曲线Figure 27. Time history of wave elevation at
${x_{{\rm{probe}}}}$ = 6.37 m simulated by δ-SPHC with different α
图 28
${\alpha _s}$ =1.3和不同粒子分辨率条件下, 不规则波δ-SPHC模拟结果的收敛性分析Figure 28. Convergence analysis of irregular waves simulated by δ-SPHC with αs=1.3 at different particle resolutions
图 29 不规则波压力云图(δ-SPHC,
${\alpha _s}$ =2.0,$d/\Delta x$ =60)Figure 29. Pressure contour of irregular wave (δ-SPHC,
${\alpha _s}$ =2.0,$d/\Delta x$ =60)
图 30
$ x_{\text {probe }} $ = 6.37处, 不规则波的波面高度时历曲线Figure 30. Time evolution of the irregular wave elevation measured at
${x_{{\rm{probe}}}}$ = 6.37 m表 1 δ-SPH算法与δ-SPHC算法计算效率比较
Table 1. Comparison of computational efficiency between δ-SPH and δ-SPHC schemes
Scheme CPU time
of single step/st2/t1 δ-SPH t1 = 0.5963 1.09 δ-SPHC t2 = 0.6576 1.09 
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表 2 不同算法在
${{\boldsymbol{x}}_{{\bf{probe}}}}$ = 6.37 m 处平均波高对比Table 2. Comparison of average wave heights probed at 6.37 m simulated with different methods
Method Exp. δ-SPH, αs = 1.3 δ-SPH, αs = 2.0 δ-SPHC, αs = 1.3 δ-SPHC, αs = 2.0 wave height H/m 0.0428 0.0224 0.0397 0.0419 0.0427 relative error $\varepsilon $/% − 47.66 7.24 2.10 0.23
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