LINEAR GLOBAL INSTABILITY OF THE PLANE FLOW WITH MEANDERING WALL
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摘要: 为便于数值分析, 蜿蜒河流水动力和演变模型中一般隐性假设二次时均流−二次涡的关系与明渠流时均流-明渠湍流的关系相同, 但由于高雷诺数下的DNS算力限制和实验尺度限制, 这种隐含假设是否成立目前尚无相关湍流研究来支撑. 文章试图通过分析明渠湍流和二次湍流发展初期的研究, 侧面揭示其湍流结构的异同. 通过对曲线正交坐标系下的平面二维NS方程使用双参数摄动的方法, 建立了一种求解蜿蜒边界弱非线性层流的摄动解法, 并推导得出一个适用于蜿蜒边界的EOS方程以及其特征值问题的解法. 蜿蜒边界下弱非线性层流解为一系列蜿蜒谐波分量的叠加, 其中线性部分使得两壁产生流速差, 非线性部分随着雷诺数增大呈指数增长. 水流的扰动增长率特征谱的第一模态与直道流相似, 由3条曲线、4个波段合成, 但其长波段和短波段的扰动流场与直道流不同, 所有短波段的扰动流速近似于KH涡. 蜿蜒边界对内部水流扰动有一定的选择性. 偏角幅值越大扰动增长越快; 蜿蜒波数的影响则为先增后减, 有一个使扰动增长最快的蜿蜒波数. 扰动流场由一个典型的TS波和一对波包形式的二次涡叠加而成, 波包只有纵向流速分量, 包络线由蜿蜒波数控制, 波包内是与直道扰动波参数相同的TS波.Abstract: In the hydrodynamic and processing research of meandering river, it is implicitly assumed that the relationship between secondary flow and secondary turbulence is the same as that between mean flow and turbulence in open channel flows. However, there is no relevant turbulence research to support this implicit assumption, due to the limitation of DNS model and PIV measurement at high Reynolds number. The differences and similarities research of turbulent structures development between meandering channel and straight channel flow are benefit to the secondary turbulent flow in meandering rivers. A planar two-dimensional NS equation in orthogonal coordinate system and the two-parameter perturbation method were established to solve the weak nonlinear laminar flow and flow instability problem in the meandering channel. And a governing equation, named with extended Orr-Sommerfeld (EOS) equation was derived to solve the eigenvalue problem of planer flow with meandering boundary. The weak nonlinear laminar flow is combination of a series of meandering harmonic components, in which the linear component causes the velocity difference between the two walls, and the nonlinear component increases exponentially with the increase of Reynolds number. The first modal of the disturbance growth rate spectrum is similar to that of the straight channel flow, which is composed of three type curves and divided four disturbance wave bands. However, the disturbance flow field at the longwave band and the shortwave band is different from that of the straight flow. Specially, the velocity disturbance at shortwave band is similar to that of the Kelvin-Helmholtz vortex, may due to the velocity difference caused by linear component of laminar. The two meandering parameters have a certain selectivity to the internal disturbance in channel. The larger the angular amplitude is, the faster the disturbance grows. With the increase of the meandering wavenumber, the disturbance growth rate increases at first and then decreases. The disturbed flow field is formed by superposition of a typical TS wave and a pair of wave packets. The wave packet pair has only longitudinal velocity components, with two envelopes controlled by the boundary wavenumber and interior TS wave with the same parameters as TS wave in the wave packet.
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图 1 正弦派生曲线边界平面示意图
Figure 1. Sketch of the orthogonal curvilinear coordinate system with meandering wall
图 2 弱非线性层流解中分解量
${{\bar{{\boldsymbol{q}}}}_{00}},{{\bar{{\boldsymbol{q}}}}_{11}},{{\bar{{\boldsymbol{q}}}}_{20}}$ Figure 2. The weakly nonlinear laminar solution
${{\bar{{\boldsymbol{q}}}}_{00}},{{\bar{{\boldsymbol{q}}}}_{11}},{{\bar{{\boldsymbol{q}}}}_{20}} $ 
图 3 参数(R, αc)对线性层流解
${\bar {\boldsymbol{q}}_{{11}}}$ 的影响Figure 3. Counter of the linear laminar solution
${\bar {\boldsymbol{q}}_{{11}}}$ effected by parameters (R, αc)
图 4 参数(R, αc)对非线性层流解
${\bar {\boldsymbol{q}}_{{20}}}$ 的影响Figure 4. Counter of the nonlinear laminar solution
${\bar {\boldsymbol{q}}_{{20}}}$ effected by parameters (R, αc)
图 5 直道流(OS方程)的扰动增长率特征谱和横向扰动流速分布
Figure 5. Spectrum of growth rate and profile of transverse velocity from OS equation
图 6 蜿蜒边界下(EOS方程)的扰动增长率特征谱和横向扰动流速分布
Figure 6. Spectrum of growth rate and profile of transverse velocity from EOS equation
图 7 蜿蜒边界下扰动增长率和相速度的云图
Figure 7. Contour of growth rate and phase speed of disturbance between meandering walls
图 8 蜿蜒波数αc对扰动特征值的影响
Figure 8. Effect of meander wavenumber αc on growth rate and phase speed
图 9 偏角幅值θc对扰动特征值的影响
Figure 9. Effect of angular amplitude θc on growth rate and phase speed
表 1 无量纲参数及其物理意义汇总
Table 1. Dimensionless parameters and physical significance
Dimensionless parameters Formula Physical significance αc 2πB*/M* meandering wavenumber θc θc meandering amplitude R $U_m^* $B*/ν* Reynolds number F $U_m^* $/( g*B*)1/2 Froude number cm αcθc curvature amplitude ε u'/ u0 disturbance to mean flow 
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