Stabilising pipe flow by a baffle designed using energy stability
Stabilising pipe flow by a baffle designed using energy stability
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DOI：
10.1017/jfm.2020.602
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发表时间：
2020-09
影响因子：
3.7
通讯作者：
Zijing Ding;Elena Marensi;A. Willis;R. Kerswell
中科院分区：
文献类型：
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作者：
Zijing Ding;Elena Marensi;A. Willis;R. Kerswell
摘要先前的实验(Kühnen等人，Flow Turb.燃烧，第100卷，2018年，第919-943页)和数值(Marensi等人，流体力学杂志，第863卷，2019年，第850-875页)的研究已经证明，在雷诺数为O(10^4)$时，流向局部挡板可以完全重新考虑管道流动湍流。优化挡板的设计涉及到解决一个复杂的变分问题，这个问题是围绕着Navier-Stokes方程的时间步进湍流解而建立的，很难求解。取而代之的是，我们研究了一种简单得多的“谱”方法，其基础是最大化折流板修正的层流的能量稳定性。随之而来的最优化问题与推导湍流中能量耗散率上限的变分程序有很大的相似之处(例如，Pending&Kerswell，J.Fluid Mech，Vol.477,2003，pp.363-379)，因此可以使用那里开发的经过精心磨练的技术来解决这里的问题。挡板由线性阻力$-F(黑体符号{x})\粗体符号{u}$(具有$F(\黑体符号{x})\ge 0\\所有\粗体符号{x}$)建模，其中挡板的范围由$L_{\Alpha}$范数约束，在$1\leq\Alpha\leq 2$范围内探索各种选择。渐近分析表明，最优挡板始终是轴对称和流向独立的，仅保留径向相关性。在所有情况下出现的最佳挡板具有与实验中发现的相似的结构：挡板延缓了管道中心的流动，导致靠近壁面的流动变得更快，从而减少了那里的湍流剪切。数值模拟表明，所设计的挡板能在中等雷诺数(3500美元)下有效地对湍流进行再氨化，并确定了一种节能方案。在Re=2400时的直接数值模拟也表明，可以通过将能量稳定性设计的挡板截断到有限长度来实现减阻。
Abstract Previous experimental (Kühnen et al., Flow Turb. Combust., vol. 100, 2018, pp. 919–943) and numerical (Marensi et al., J. Fluid Mech., vol. 863, 2019, pp. 850–875) studies have demonstrated that a streamwise-localised baffle can fully relaminarise pipe flow turbulence at Reynolds numbers of $O(10^4)$. Optimising the design of the baffle involves tackling a complicated variational problem built around time stepping turbulent solutions of the Navier–Stokes equations which is difficult to solve. Here instead, we investigate a much simpler ‘spectral’ approach based upon maximising the energy stability of the baffle-modified laminar flow. The ensuing optimal problem has much in common with the variational procedure to derive an upper bound on the energy dissipation rate in turbulent flows (e.g. Plasting & Kerswell, J. Fluid Mech., vol. 477, 2003, pp. 363–379) so well-honed techniques developed there can be used to solve the problem here. The baffle is modelled by a linear drag force $-F(\boldsymbol {x}) \boldsymbol {u}$ (with $F(\boldsymbol {x}) \ge 0 \ \forall \boldsymbol {x}$) where the extent of the baffle is constrained by an $L_{\alpha }$ norm with various choices explored in the range $1 \leq \alpha \leq 2$. An asymptotic analysis demonstrates that the optimal baffle is always axisymmetric and streamwise independent, retaining just radial dependence. The optimal baffle which emerges in all cases has a similar structure to that found to work in experiments: the baffle retards the flow in the pipe centre causing the flow to become faster near the wall thereby reducing the turbulent shear there. Numerical simulations demonstrate that the designed baffle can relaminarise turbulence efficiently at moderate Reynolds numbers ($Re \le 3500$), and an energy saving regime has been identified. Direct numerical simulation at $Re=2400$ also demonstrates that the drag reduction can be realised by truncating the energy-stability-designed baffle to finite length.