An immersed boundary method based on the lattice Boltzmann approach in three dimensions, with application

We have proposed a 3D immersed boundary (IB) method using the D3Q19 lattice Boltzmann model to represent the flow model. Our numerical study indicates that the hybrid method is first-order accurate, which is comparable to the situation for other versions of the IB method in general. The newly developed method is employed to simulate a viscous flow past a compliant sheet fixed at its midline in a 3D channel. Our numerical results indicate that the drag of the flexible sheet scales approximately as the inflow speed in the ranges of the dimensionless parameters used in the simulations. This is in sharp contrast with the drag of a rigid body in a viscous flow (drag scales as the square of the inflow speed). Our numerical results for the flexible-sheet–fluid interaction are based on simulations with dimensionless parameters in certain ranges: , . We would like to point out that when the Reynolds number or the dimensionless bending rigidity are too large, the numerical method becomes unstable. When is sufficiently large, small deformations of the immersed boundary may generate sufficiently large force to induce overshoot of the immersed boundary leading to instability. When the Reynolds number is sufficiently large, inertia forces dominate over viscous forces. This gives rise to rapid relaxation of the immersed solid boundary towards tangential equilibrium, which engenders an even stricter constraint on the time step size for an explicit IB method such as ours to maintain numerical stability. Moreover, as increases, flow gradients are more confined, i.e. thinner boundary layers formed, vortical flows are tighter spirals, etc. So, the flow variations grow larger. Spatial resolution deteriorates and a solution can grow unboundedly. Therefore finer spatial resolution and a smaller time step size are needed to correctly resolve the flow. The possible synergistic interaction of these two factors may render the stability analysis more difficult. Significant efforts along this direction are certainly needed in the future. Note that except for the convergence check, the results presented here are based on simulations performed on a grid of 120×60×60. The flexible sheet is 20×40 (width by height). Initially the spaces between the sheet edges and the walls are 10 along the direction (top to bottom) and 20 along the direction (front to rear). While the distance between the sheet edges and the walls in the direction remains roughly the same, the distance along the direction increases as the sheet bends with the flow. The final distance when the flow reaches a quasi-steady state varies from simulation to simulation depending on the specific parameters used. One may wonder what the wall effects on the simulation results are. The wall effects for the 2D version of this problem have been investigated quantitatively by Zhu [83] . We believe that the wall effects in 3D should be approximately the same in general. That is, compared to the unbounded case (the ratio of the sheet width to the channel width is zero), a finite size of the flow channel may delay the vortex formation and shedding and slightly increase the drag coefficient of the sheet. In the case of a viscous flow past a rigid object, the vortex shedding is usually seen when is of the order of 100. But here in our case, vortex shedding is not apparent at this range of . Presumably this is because the sheet is deformable. The existence of a second dimensionless parameter, i.e. the flexure modulus , may quantitatively modify the role of . It has been shown in the 2D equivalent [83] of the 3D problem here in our work that the critical for vortex shedding increases as the decreases, i.e., the flexibility of the sheet will delay the vortex shedding compared to that for a rigid sheet. Together with the similar effect of a finite flow channel, this may explain why the vortex shedding is not intensive for our simulations at round 100. Notice that the drag coefficient in our case (a flexible sheet) is in general approximately one order of magnitude greater than that of a plate placed normal to an incompressible viscous flow [91] . To explain this seeming contradiction, we would like to point out first that the flexible sheet in our simulations is in fact not inextensible: it stretches with the local flow and this can be seen in the bottom panel of Fig. 5 . A stretched flexible sheet absorbs more elastic energy from the flowing fluid and therefore experiences more drag than an inextensible flexible body. Secondly, the drag reduction induced by body flexibility discussed in our paper means that the drag of a flexible body is reduced when it bends and streamlines with the local flow because of its flexibility. In our case it appears that the drag coefficient decreases with the degree of body flexibility . This is because a more flexible sheet tends to bend more and represent itself as a smaller obstacle to the mainstream flow, and thus to reduce resistance from the flow. We hasten to add that this does not necessarily mean that the drag of a flexible body must be less than that of a corresponding rigid body of the same geometry. In some cases just the opposite is found. Here are two more examples. Existing laboratory experimental data [92,93] show that the drag coefficient of a rigid smooth plate aligned with a viscous flow is approximately 10 times less than that of a flexible stationary flag of the same geometry. A very recent work by [94] has found something similar: the drag coefficient of a flexible flapping biofilm is greater than that of a stiffer biofilm in a viscous flow. An intuitive explanation may be as follows: when a viscous fluid is moving past a flexible body, the local body surface, especially the edges of the body, may move more easily with the local flow compared to the rigid case. This kind of “extra” local movement of a flexible body which is absent in the case of a rigid body generates more disturbances to the local flow and thus causes more energy dissipation due to fluid viscosity. Consequently the flexible body may experience more resistance from the flow compared to a rigid equivalent. This effect becomes more important as the body becomes more flexible. This being said, however, analytical or quantitative results seems to be out of the question at this point. Certainly this issue is worthy of further study in the future.The immersed boundary (IB) method originated by Peskin has been popular in modeling and simulating problems involving the interaction of a flexible structure and a viscous incompressible fluid. The Navier–Stokes (N–S) equations in the IB method are usually solved using numerical methods such as FFT and projection methods. Here in our work, the N–S equations are solved by an alternative approach, the lattice Boltzmann method (LBM). Compared to many conventional N–S solvers, the LBM can be easier to implement and more convenient for modeling additional physics in a problem. This alternative approach adds extra versatility to the immersed boundary method. In this paper we discuss the use of a 3D lattice Boltzmann model (D3Q19) within the IB method. We use this hybrid approach to simulate a viscous flow past a flexible sheet tethered at its middle line in a 3D channel and determine a drag scaling law for the sheet. Our main conclusions are: (1) the hybrid method is convergent with first-order accuracy which is consistent with the immersed boundary method in general; (2) the drag of the flexible sheet appears to scale with the inflow speed which is in sharp contrast with the square law for a rigid body in a viscous flow.

我们提出了一种使用D3Q19格子玻尔兹曼模型的三维浸入边界（IB）方法来表示流动模型。我们的数值研究表明，这种混合方法具有一阶精度，这与一般情况下其他版本的IB方法的情况相当。新开发的方法被用于模拟在三维通道中流经固定在中线的柔性薄片的粘性流。我们的数值结果表明，在模拟中使用的无量纲参数范围内，柔性薄片的阻力大致与流入速度成正比。这与粘性流中刚体的阻力（阻力与流入速度的平方成正比）形成鲜明对比。我们关于柔性薄片 - 流体相互作用的数值结果是基于在某些范围内的无量纲参数的模拟：[此处可能缺失具体参数值]。我们想指出的是，当雷诺数[此处可能缺失具体符号]或无量纲抗弯刚度[此处可能缺失具体符号]过大时，数值方法会变得不稳定。当[此处可能缺失具体符号]足够大时，浸入边界的小变形可能会产生足够大的力，导致浸入边界超调，从而引起不稳定。当雷诺数足够大时，惯性力超过粘性力。这导致浸入固体边界迅速向切向平衡松弛，这对像我们这样的显式IB方法的时间步长提出了更严格的约束，以保持数值稳定性。此外，随着[此处可能缺失具体符号]的增加，流动梯度更加受限，即形成更薄的边界层，涡旋流是更紧密的螺旋等。所以，流动变化更大。空间分辨率变差，解可能会无限制地增长。因此，需要更精细的空间分辨率和更小的时间步长来正确解析流动。这两个因素可能的协同作用可能会使稳定性分析更加困难。未来肯定需要在这个方向上做出重大努力。需要注意的是，除了收敛性检查外，这里呈现的结果是基于在120×60×60的网格上进行的模拟。柔性薄片是20×40（宽×高）。最初，薄片边缘与壁之间在[此处可能缺失具体坐标轴方向说明]方向（从上到下）的间距为10，在[此处可能缺失具体坐标轴方向说明]方向（从前到后）的间距为20。虽然薄片边缘与壁在[此处可能缺失具体坐标轴方向说明]方向的距离大致保持不变，但在[此处可能缺失具体坐标轴方向说明]方向的距离随着薄片随流动弯曲而增加。当流动达到准稳态时的最终距离因模拟中使用的具体参数而异。人们可能想知道壁面对模拟结果有什么影响。朱[83]已经对这个问题的二维版本的壁面效应进行了定量研究。我们认为三维中的壁面效应一般应该大致相同。也就是说，与无界情况（薄片宽度与通道宽度之比为零）相比，有限尺寸的流动通道可能会延迟涡旋的形成和脱落，并略微增加薄片的阻力系数。在粘性流流经刚体的情况下，当[此处可能缺失具体符号]约为100时通常会看到涡旋脱落。但在我们的情况中，在这个[此处可能缺失具体符号]范围内涡旋脱落不明显。推测这是因为薄片是可变形的。第二个无量纲参数，即弯曲模量[此处可能缺失具体符号]的存在，可能会定量地改变[此处可能缺失具体符号]的作用。在我们工作中这里的三维问题的二维等效问题[83]中已经表明，涡旋脱落的临界[此处可能缺失具体符号]随着[此处可能缺失具体符号]的减小而增加，即薄片的柔性会比刚性薄片延迟涡旋脱落。结合有限流动通道的类似效应，这可能解释了为什么在我们的模拟中当[此处可能缺失具体符号]约为100时涡旋脱落不强烈。注意，在我们的情况（柔性薄片）中，阻力系数一般比垂直于不可压缩粘性流放置的[此处可能缺失具体物体描述]板大约一个数量级[91]。为了解释这种看似矛盾的情况，我们首先想指出，我们模拟中的柔性薄片实际上不是不可伸长的：它随着局部流动而拉伸，这可以在图5的底部面板中看到。拉伸的柔性薄片从流动的流体中吸收更多的弹性能量，因此比不可伸长的柔性体受到更多的阻力。其次，我们论文中讨论的由物体柔性引起的阻力减少意味着，由于其柔性，当柔性体随着局部流动弯曲并与流线一致时，其阻力会减少。在我们的情况中，似乎阻力系数[此处可能缺失具体符号]随着物体柔性程度[此处可能缺失具体符号]的增加而减小。这是因为更柔性的薄片往往更容易弯曲，并将自己呈现为对主流流动更小的障碍，从而减少来自流动的阻力。我们赶紧补充说，这并不一定意味着柔性体的阻力必须小于相同几何形状的相应刚体的阻力。在某些情况下，情况恰恰相反。这里还有两个例子。现有的实验室实验数据[92,93]表明，与粘性流对齐的刚性光滑板的阻力系数大约是相同几何形状的柔性静止旗帜的阻力系数的1/10。[94]最近的一项工作也发现了类似的情况：在粘性流中，柔性拍动生物膜的阻力系数大于更硬的生物膜的阻力系数。一种直观的解释可能如下：当粘性流体流经柔性体时，局部物体表面，特别是物体的边缘，与刚体情况相比，可能更容易随着局部流动移动。这种柔性体特有的“额外”局部运动在刚体情况下不存在，它会对局部流动产生更多干扰，从而由于流体粘性导致更多的能量耗散。因此，与刚体等效物相比，柔性体可能会受到来自流动的更多阻力。随着物体变得更柔性，这种效应变得更加重要。话虽如此，然而，在这一点上分析或定量结果似乎是不可能的。当然，这个问题在未来值得进一步研究。 由佩斯金提出的浸入边界（IB）方法在模拟涉及柔性结构和粘性不可压缩流体相互作用的问题中很受欢迎。IB方法中的纳维 - 斯托克斯（N - S）方程通常使用诸如快速傅里叶变换（FFT）和投影方法等数值方法来求解。在我们的工作中，N - S方程是通过一种替代方法——格子玻尔兹曼方法（LBM）来求解的。与许多传统的N - S求解器相比，LBM更容易实现，并且在对问题中的附加物理进行建模时更方便。这种替代方法为浸入边界方法增加了额外的通用性。在本文中，我们讨论了在IB方法中使用三维格子玻尔兹曼模型（D3Q19）。我们使用这种混合方法来模拟在三维通道中流经固定在中线的柔性薄片的粘性流，并确定薄片的阻力缩放定律。我们的主要结论是：（1）混合方法收敛，具有一阶精度，这与一般的浸入边界方法一致；（2）柔性薄片的阻力似乎与流入速度成正比，这与粘性流中刚体的平方定律形成鲜明对比。