The dynamics of a laminar flow in a symmetric channel with a sudden expansion
The dynamics of a laminar flow in a symmetric channel with a sudden expansion
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DOI：
10.1017/s0022112001004086
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发表时间：
2001-06-10
影响因子：
3.7
通讯作者：
Rusak, Z
中科院分区：
文献类型：
--
作者：
Hawa, T;Rusak, Z
分叉分析，线性稳定性研究，并直接数值模拟的动态的二维，不可压缩，层流在一个对称的长通道突然扩张与直角和扩张比Did（d是宽度的通道进口部分和D是宽度的出口部分）。定常流方程的分岔分析集中在临界雷诺数Re-c（D/d）附近的流动状态，当Re大于或等于Re-c（D/d）时，除了基本对称状态外，还出现非对称状态。非对称态在Re处的分岔具有叉状性质，非对称扰动的增长类似于根Re-Re-c（D/d）。稳定性分析是基于线性化的运动方程的无穷小的二维扰动的稳定的对称以及非对称状态的演变。对于对称和非对称平衡态，在Re-c（D/d）处都存在中性稳定的非对称扰动模。用渐近方法证明了当Re <Re-c（D/d）时，对称状态存在渐近稳定的扰动模。当Re> Re-c（D/d）时，对称态对这种非对称扰动模是不稳定的。当Re> Re-c（D/d）时，非对称系统存在一个渐近稳定的扰动模。直接数值模拟的理论方法的指导下。为了提高数值模拟的精度，在展开角附近的区域中还加入了与Moffatt（1964）的渐近解的匹配。小振幅和大振幅的扰动在流动中的动态描述和从对称到非对称状态的过渡被证明。模拟澄清的线性稳定性结果和时间渐近行为的流量之间的关系。目前的分析提供了一个理论基础，以前的实验和数值计算结果，并揭示了更多的光从对称到非对称状态的粘性流动在一个扩展的通道。这是从对称状态到稳定的非对称平衡状态的演化过程，当来流的雷诺数大于Re-c（D/d）时，对称状态失去稳定性。稳定性的损失是粘性耗散、底部对称流扰动的下游对流和二维非对称扰动诱导的上游对流之间相互作用的结果。
Bifurcation analysis, linear stability study, and direct numerical simulations of the dynamics of a two-dimensional, incompressible, and laminar flow in a symmetric long channel with a sudden expansion with right angles and with an expansion ratio Did (d is the width of the channel inlet section and D is the width of the outlet section) are presented. The bifurcation analysis of the steady flow equations concentrates on the flow states around a critical Reynolds number Re-c(D/d) where asymmetric states appear in addition to the basic symmetric states when Re greater than or equal to Re-c(D/d). The bifurcation of asymmetric states at Re, has a pitchfork nature and the asymmetric perturbation grows like root Re-Re-c(D/d). The stability analysis is based on the linearized equations of motion for the evolution of infinitesimal two-dimensional disturbances imposed on the steady symmetric as well as asymmetric states. A neutrally stable asymmetric mode of disturbance exists at Re-c(D/d) for both the symmetric and the asymmetric equilibrium states. Using asymptotic methods, it is demonstrated that when Re < Re-c(D/d) the symmetric states have an asymptotically stable mode of disturbance. However, when Re > Re-c(D/d), the symmetric states are unstable to this mode of asymmetric disturbance. It is also shown that when Re > Re-c(D/d) the asymmetric states have an asymptotically stable mode of disturbance. The direct numerical simulations are guided by the theoretical approach. In order to improve the numerical simulations, a matching with the asymptotic solution of Moffatt (1964) in the regions around the expansion corners is also included. The dynamics of both small- and large-amplitude disturbances in the flow is described and the transition from symmetric to asymmetric states is demonstrated. The simulations clarify the relationship between the linear stability results and the time-asymptotic behaviour of the flow. The current analyses provide a theoretical foundation for previous experimental and numerical results and shed more light on the transition from symmetric to asymmetric states of a viscous flow in an expanding channel. It is an evolution from a symmetric state, which loses its stability when the Reynolds number of the incoming flow is above Re-c(D/d), to a stable asymmetric equilibrium state. The loss of stability is a result of the interaction between the effects of viscous dissipation, the downstream convection of perturbations by the base symmetric flow, and the upstream convection induced by two-dimensional asymmetric disturbances.