Numerical approach for bifurcation of nonlinear problem

非线性问题分岔的数值方法

• 批准号: 09640295
• 负责人: SHOJI Mayumi
• 金额: $ 1.41万
• 依托单位: Nihon University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1999
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640295/
• 关键词: bifurcation; gravity waves; vorticity; Biburcation; progriesive wave; corticity; progressive wave

We have the following results on incompressible fluid.The first one is on the two-dimensional stagnation-point solution of the Navier-Stokes equations. On it Childress et al. ('89) investigated an unsteady example, namely they show some numerical examples of finite time blow-up for large Reynolds number, They also gave the critical Reynolds number numerically. However we have obtained different results using modified formulation. We had no blow-up and examined it by two methods, finite difference scheme and spectral mathod. ([1])The second one is on bifurcation problem of gravity waves with constant vorticity. Our object is to see the global bifurcation structure, combining the results with our results for capillary-gravity waves we have obtained before. The results on this work are as below :1. It is found that bifurcation structures are unchanged qualitatively as vorticity varies.2. As for symmetric waves, we conjectured numerically how many kinds of mode n bifurcation solutions exist. We checked it for n=1〜6 by simulations.3. We gave the information about the flow beneath the free surface by plotting stream-lines in the fluid region. It can be seen that eddy appears for positive vorticity and it expands as the vorticity becomes larger.4. Zufiria ('87) gave non-symmetric solutions for gravity waves of infinite depth numerically. We attempted to follow them by the nonsymmetric version of our algorithm, but we couldn't find any. We believe there might be no non-symmetric solutions for the case of infinite depth, but for the case of finite depth. We will further study it.

关于不可压缩流体，我们得到了以下结果。第一个结果是关于Navier-Stokes方程的二维驻点解。在它上面，奇尔德里斯等人。(‘89)研究了一个非定常的例子，即他们给出了大雷诺数的有限时间爆破的数值例子，并给出了临界雷诺数。然而，我们使用改进的配方得到了不同的结果。我们没有爆破性，并用两种方法进行了检验，即有限差分格式和谱方法。([1])第二个是关于常涡度重力波的分叉问题。我们的目标是看到全球分叉结构，结合我们的结果与我们之前获得的毛细重力波的结果。本文的研究结果如下：1.研究发现，分叉结构在性质上不随涡度变化而变化。对于对称波，我们在数值上推测了存在多少种模式n分岔解。通过模拟验证了n=1~6。我们通过绘制流体区的流线，给出了关于自由表面下流动的信息。可以看出，正涡度时会出现涡旋，并且随着涡度的增大，涡旋会逐渐扩展。Zufiria(‘87)用数值方法给出了无限深重力波的非对称解。我们试图用我们的算法的非对称版本来跟踪它们，但我们找不到任何一个。我们认为，对于无限深度的情况，可能没有非对称的解，但是对于有限深度的情况，可能没有非对称的解。我们会进一步研究。

项目成果

M.Shoji: "Bifurcation of rotational water waves"Proc. Of FBP'99, GAKUTO Int. Ser. Math. Sci. and Appl.,. (to appear).

M.Shoji：“旋转水波的分叉”Proc。

H.Okamoto and M.Shoji: "The spectial method for unseteady two-dimensional Navier-Stokes equations" Proc.of Third China-Japan Seminar or Numerical Mathematics. 253-260 (1998)

H.Okamoto和M.Shoji：“不稳定二维纳维-斯托克斯方程的特殊方法”第三届中日数值数学研讨会论文集。

H. Okamoto and M. Shoji: "The spectral method for unsteady two-dimensional Navier-Stokes equations"Proceedings of Third China-Japan Seminar on Numerical Mathematics, eds. Z.-c. Shi and M. Mori, Science Press. 253-260 (1998)

H.冈本、M. Shoji：“非定常二维纳维-斯托克斯方程的谱法”，第三届中日数值数学研讨会论文集，主编。

M. Shoji: "Bifurcation of rotational water waves"Proc. of FBP'99, GAKUTO Int. Ser. Math. Sci. and Appl.,. (to appear).

M. Shoji：“旋转水波的分叉”Proc。

H.Okamoto and M.Shoji: "The spectral method for unsteady two-dim. Navier-Stokes equations"Proc. Of Third China-Japan Seminar on Num. Math.. 253-260 (1998)

H.Okamoto 和 M.Shoji：“非定常二维纳维-斯托克斯方程的谱法”Proc。