Integral relations for the skin-friction coefficient of canonical flows
Integral relations for the skin-friction coefficient of canonical flows
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DOI：
10.1017/jfm.2022.444
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发表时间：
2022-05
影响因子：
3.7
通讯作者：
P. Ricco;M. Skote
中科院分区：
文献类型：
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作者：
P. Ricco;M. Skote
摘要我们证明了Fukagata等人S(Phys.流体，第14卷，第11期，2002年，第73-76页)自由流边界层的恒等式被简化为von Kármán动量积分方程式，当用来获得恒等式的积分的上界为渐近大时，动量积分方程式联系着表面摩擦系数和动量厚度。如果使用有限的上界，则恒等式的项错误地依赖于上界本身。与槽道和管流不同，在自由流边界层情况下，雷诺应力对壁面剪应力的影响是无法量化的，因为雷诺应力从恒等式中消失了。通过对流动量方程进行附加积分而获得的无限多个备选恒等式也都简化为von Kármán方程。对于槽道流动，发现了相似的恒等式，其中物理项对壁面剪应力的相对影响取决于连续积分的次数，这表明层流和湍流对壁面摩擦系数的贡献仅在Fukagata等人发现的原始恒等式中被区分。(物理。流体，第14卷，第11期，2002年，第73-76页)。在大量积分的限制下，这些恒等式退化为壁面摩擦系数的定义，对于槽道和管流，发现了一种新的双重积分恒等式。此外，在Renard&Deck(流体力学杂志，第790卷，2016，pp.339-367)的研究之后，我们将表面摩擦系数唯一地分解为积分厚度沿流向方向变化的总和。我们利用了能量厚度和惯性厚度，该厚度由与平均流壁法向对流有关的厚度和与平均流向速度的流向不均匀有关的厚度组成。因此，流向动量方程的不同项对摩擦阻力的贡献可以用这些积分厚度来量化。
Abstract We show that the Fukagata et al.'s (Phys. Fluids, vol. 14, no. 11, 2002, pp. 73–76) identity for free-stream boundary layers simplifies to the von Kármán momentum integral equation relating the skin-friction coefficient and the momentum thickness when the upper bound in the integrals used to obtain the identity is taken to be asymptotically large. If a finite upper bound is used, the terms of the identity depend spuriously on the bound itself. Differently from channel and pipe flows, the impact of the Reynolds stresses on the wall-shear stress cannot be quantified in the case of free-stream boundary layers because the Reynolds stresses disappear from the identity. The infinite number of alternative identities obtained by performing additional integrations on the streamwise momentum equation also all simplify to the von Kármán equation. Analogous identities are found for channel flows, where the relative influence of the physical terms on the wall-shear stress depends on the number of successive integrations, demonstrating that the laminar and turbulent contributions to the skin-friction coefficient are only distinguished in the original identity discovered by Fukagata et al. (Phys. Fluids, vol. 14, no. 11, 2002, pp. 73–76). In the limit of large number of integrations, these identities degenerate to the definition of skin-friction coefficient and a novel twofold-integration identity is found for channel and pipe flows. In addition, we decompose the skin-friction coefficient uniquely as the sum of the change of integral thicknesses with the streamwise direction, following the study of Renard & Deck (J. Fluid Mech., vol. 790, 2016, pp. 339–367). We utilize an energy thickness and an inertia thickness, which is composed of a thickness related to the mean-flow wall-normal convection and a thickness linked to the streamwise inhomogeneity of the mean streamwise velocity. The contributions of the different terms of the streamwise momentum equation to the friction drag are thus quantified by these integral thicknesses.