The generation of internal waves by vibrating elliptic cylinders. Part 2. Approximate viscous solution
The generation of internal waves by vibrating elliptic cylinders. Part 2. Approximate viscous solution
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通过振动椭圆柱产生内波。
DOI：
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发表时间：
1997
影响因子：
3.7
通讯作者：
G. Keady
中科院分区：
文献类型：
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作者：
D. Hurley;G. Keady
给出了通过椭圆柱的直线振动在粘性 Boussinesq 流体中产生内部重力波的近似理论。引入了参数 λ，该参数与围绕圆柱体的振荡边界层的厚度与其横截面的典型尺寸之比的平方成正比。当 λ[Lt ]1 时（或等效地当雷诺数 R[Gt ]1 时），圆柱体表面的粘性边界条件可以用无粘性边界条件代替 λ 中的一阶。针对 λ[Lt ]1 的情况提出了粘性解决方案，其中第 1 部分（Hurley 1997）中发现的流函数的傅里叶表示通过在被积函数中包含一个考虑粘性耗散的因子进行修改。在极限 λ→0 下，所提出的解决方案在流场中的每一点都变成无粘性解决方案。为了便于表述，首先考虑半径为a的圆柱体的情况，并且在上面的λ的定义中我们将a作为其横截面的典型尺寸。对所提出的近似解的精度进行了分析和数值研究，得出的结论是，如果 λ 足够小，则在整个流场中它都是准确的，但在特性接触圆柱体附近的小区域除外，其中粘性效应占主导地位。计算表明，在沿波束距圆柱体中心距离 s 处，典型波束中心线上的速度与（恒定）无粘性值一致（在 1% 以内），前提是 λs/a 小于约 10−3。该结果被解释为表明，当 λs/a 约为 10−3 时，源自接触圆柱体（无粘速度奇异的地方）的特性的粘性效应到达光束的中心线。对于较大的 s 值，整个光束的粘性效应非常显着，并且光束的速度分布会发生变化，直到达到 Thomas 和 Stevenson (1972) 获得的相似解给出的值（当 λs/a 约为 2 时，误差在 1% 左右）。对于较大的 λs/a 值，应用它们的相似性解决方案。马卡洛夫等人在一篇重要论文中。 （1990）给出了与我们非常相似的圆柱体的近似解。然而，当粘度为零时，它不会减少到无粘性。最后表明，经过小的修改后，我们对圆柱体的结果适用于所有椭圆柱体。
An approximate theory is given for the generation of internal gravity waves in a viscous Boussinesq fluid by the rectilinear vibrations of an elliptic cylinder. A parameter λ which is proportional to the square of the ratio of the thickness of the oscillatory boundary layer that surrounds the cylinder to a typical dimension of its cross-section is introduced. When λ[Lt ]1 (or equivalently when the Reynolds number R[Gt ]1), the viscous boundary condition at the surface of the cylinder may to first order in λ be replaced by the inviscid one. A viscous solution is proposed for the case λ[Lt ]1 in which the Fourier representation of the stream function found in Part 1 (Hurley 1997) is modified by including in the integrands a factor to account for viscous dissipation. In the limit λ→0 the proposed solution becomes the inviscid one at each point in the flow field. For ease of presentation the case of a circular cylinder of radius a is considered first and we take a to be the typical dimension of its cross-section in the definition of λ above. The accuracy of the proposed approximate solution is investigated both analytically and numerically and it is concluded that it is accurate throughout the flow field if λ is sufficiently small, except in a small region near where the characteristics touch the cylinder where viscous effects dominate. Computations indicate that the velocity on the centreline on a typical beam of waves, at a distance s along the beam from the centre of the cylinder, agrees, within about 1%, with the (constant) inviscid values provided λs/a is less than about 10−3. This result is interpreted as indicating that those viscous effects which originate from the characteristics that touch the cylinder (places where the inviscid velocity is singular) reach the centreline of the beam when λs/a is about 10−3. For larger values of s, viscous effects are significant throughout the beam and the velocity profile of the beam changes until it attains, within about 1% when λs/a is about 2, the value given by the similarity solution obtained by Thomas & Stevenson (1972). For larger values of λs/a, their similarity solution applies. In an important paper Makarov et al. (1990) give an approximate solution for the circular cylinder that is very similar to ours. However, it does not reduce to the inviscid one when the viscosity is taken to be zero. Finally it is shown that our results for a circular cylinder apply, after small modifications, to all elliptical cylinders.