首页> 外文学位 >The viscous N vortex problem: A generalized Helmholtz/Kirchhoff approach.
【24h】

The viscous N vortex problem: A generalized Helmholtz/Kirchhoff approach.

机译:粘性N涡旋问题:广义Helmholtz / Kirchhoff方法。

获取原文
获取原文并翻译 | 示例

摘要

In this work a new model is proposed to describe the motion of N localized vortex structures in two dimensional incompressible viscous flow. We will use the velocity-vorticity representation, commonly known as the vorticity equation, of incompressible flow which has the particular advantage in two dimensions of reducing to a scalar equation.;An early model of two-dimensional inviscid flow in terms of moving point vortices was developed by Kirchhoff and Helmholtz. While the Helmholtz-Kirchhoff point vortex model captures many of the basic physical phenomena observed in 2D rotational flows, experiments with even simple vortex configurations exhibit complications far beyond the point vortex predictions. This new model replaces the Dirac point vortex with a Hermite expansion to represent each vortex. This is a natural choice of representation as the leading order term in the expansion is a Gaussian which is both a common regularization of a delta function and, even more importantly, is an exact solution of the 2D vorticity equation. Furthermore these Hermite functions have recently been shown by Gallay and Wayne to be related to invariant manifolds in the phase space of the vorticity equation which govern its long time asymptotics. With mild restrictions on the initial data we prove that the expansions are convergent for all time. The model reduces the 2D vorticity equation to a system of quadratic ordinary differential equations (ODEs) that track the centers of each vortex along with the evolution of each Hermite moment. As a first application of this approach a single Hermite expansion is used to model a co-rotating pair of vortices to greatly improve the accuracy of the far field acoustic pressure.;Additionally, leading-order corrections to the frequency of rotation due to viscosity and finite core size are derived in the case of two well separated like-signed vortices. Finally, combinatorial formulas for the coefficients of the resulting ODEs are derived to improve the numerical implementation of the model. With these combinatorial formulae classic numerical experiments such as vortex merger are computed with our new model and results are discussed.
机译:在这项工作中,提出了一个新的模型来描述二维不可压缩粘性流中N个局部涡结构的运动。我们将使用不可压缩流的速度-涡度表示形式(通常称为涡度方程),它在将二维简化为标量方程方面具有特殊优势。;就移动点涡而言,二维无粘性流的早期模型由Kirchhoff和Helmholtz开发。虽然Helmholtz-Kirchhoff点涡模型捕获了在2D旋转流中观察到的许多基本物理现象,但即使是简单涡结构的实验也显示出远远超出点涡预测的复杂性。这个新模型用Hermite展开替换了Dirac点涡,以表示每个涡。这是表示的自然选择,因为展开中的前导项是高斯函数,既是增量函数的常见正则化,更重要的是,它是2D涡度方程的精确解。此外,最近,Gallay和Wayne已证明这些Hermite函数与控制其长时间渐近性的涡度方程相空间中的不变流形有关。在对初始数据的适度限制下,我们证明了扩展在所有时间内都是收敛的。该模型将2D涡度方程简化为二次常微分方程(ODE)系统,该系统跟踪每个涡旋的中心以及每个Hermite矩的演变。作为此方法的第一个应用,单个Hermite展开用于模拟同向旋转的一对涡流,从而大大提高了远场声压的精度。此外,由于粘度和在两个良好分离的类似符号涡旋的情况下,得出了有限的磁芯尺寸。最后,导出了所得ODE系数的组合公式,以改进模型的数值实现。利用这些组合公式,利用我们的新模型计算了经典的数值实验,例如涡旋合并,并讨论了结果。

著录项

  • 作者

    Uminsky, David T.;

  • 作者单位

    Boston University.;

  • 授予单位 Boston University.;
  • 学科 Mathematics.
  • 学位 Ph.D.
  • 年度 2009
  • 页码 96 p.
  • 总页数 96
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类 数学;
  • 关键词

  • 入库时间 2022-08-17 11:38:26

相似文献

  • 外文文献
  • 中文文献
  • 专利
获取原文

客服邮箱:kefu@zhangqiaokeyan.com

京公网安备:11010802029741号 ICP备案号:京ICP备15016152号-6 六维联合信息科技 (北京) 有限公司©版权所有
  • 客服微信

  • 服务号