The spatial stability of multiple solutions for fully developed laminar flow through a curved tube and through a straight tube in the presence of buoyant effects was studied using a three‐dimensional numerical representation of the fully elliptic Navier–Stokes and energy equations. Several recent numerical studies have reported the existence of multiple solutions for fully developed laminar flow in curved tubes; see, for example, Yang and Keller [Appl. Numer. Math. 2, 257 (1986)] or Daskopoulos and Lenhoff [J. Fluid Mech. 203, 125 (1989)]. A similar solution multiplicity for the case of fully developed laminar flow in a heated straight tube has been discussed by Nandakumar et al. [J. Fluid Mech. 152, 145 (1985)]. Each of these studies has reported solutions of a two‐ and four‐cell type, which are symmetric about the midplane of the tube and invariant in the axial (streamwise) flow direction. In addition, the flow visualization study of Cheng and Yuen [J. Heat Trans. 109, 55 (1987)] indicates that two‐ and four‐cell solutions can be obtained experimentally for the flow developing in a curved tube with a 180° bend. The present study investigates the stability of two‐ and four‐cell solutions with respect to perturbations that are asymmetric about the midplane of the tube. Perturbed two‐ and four‐cell solutions are imposed as entrance boundary conditions for three‐dimensional simulations of the full tube cross section and the evolution of the flow is observed as it progresses downstream from the entrance plane. For the conditions explored, present results indicate that the two‐cell solutions are stable, and asymmetric perturbations are observed to decay as the flow travels downstream. Four‐cell solutions are unstable when asymmetric perturbations are introduced, and growth of the perturbations eventually results in a complex three‐dimensional transition from four‐cell to two‐cell flow. Details of the transition flow are described and transition mechanisms are identified.

中文翻译：

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关于管流在空间上受力的空间稳定性

的 空间的 稳定性 多个 解决方案 为全面发展 层流 通过弯曲管和通过直管在存在浮力效应的情况下，使用全椭圆形Navier-Stokes和能量的三维数值表示来研究 方程。 最近几个 数值研究 报告存在多个 解决方案 为全面发展 层流在弯管中; 例如，参见Yang和Keller [Appl。Numer。数学。2，257（1986）]或Daskopoulos和Lenhoff [J. 流体机械。203，125（1989）。一个相似的解 充分发展的情况下的多样性 层流Nandakumar等 人已经讨论了在加热的直管中进行加热。[J. 流体机械。152，145（1985）]。这些研究均已报告解决方案 两室和四室类型的，相对于管的中平面对称，轴向（沿流）不变 流方向。除此之外流可视化对郑和元的研究[J．传热。109，55（1987）]表示两格和四格解决方案 可以通过实验获得 流在具有180°弯曲的弯曲管中显影。本研究调查稳定性 两格和四格 解决方案 关于 摄动相对于管的中平面不对称。扰动的两格和四格解决方案 被强加为入口 边界条件 用于全管横截面的三维模拟以及 流当它从入口平面向下游前进时被观察到。对于探索的条件，目前的结果表明两单元解决方案 稳定且不对称 摄动 观察到随着 流向下游行驶。四格解决方案 不对称时不稳定 摄动 被介绍，并且增长 摄动 最终导致从四单元到两单元的复杂三维过渡 流。 过渡细节 流 描述和确定过渡机制。