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Study on the finite element scheme for fluid flows with free interface

自由界面流体流动有限元格式研究

基本信息

批准号：
10640111
负责人：
OHMORI Katsushi
金额：
$ 2.11万
依托单位：
Toyama University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1998
资助国家：
日本
起止时间：
1998 至 1999
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640111/
关键词：
Finite element method Free interface Incompressible fluid Two-fluid flows Surface tension Convergence of the interface

项目摘要

This study has been carried out during 1998-1999 in order to develop and analyze the finite element scheme for fluid flows with free interface, which are often found in nature and manay industrial processes.In 1998, we have considered a finite element scheme for two-fluid flows with low ratio of densities. We assume that two fluids are both viscous, incompressible and immiscible. As the mathematical model for this problem we use the one-fluid model assuming thr Boussinesq approximation to the Navier-Stokes equations. The interface is considered as the 0-level Set of the pseudo-density function which is the Solution of the transport equation. We have proposed a mixed finite element scheme with P1 iso P2/P1 element for these equations. Especially, we have also proposed the re-initialization technique in the finite element scheme for the transport equation by using a double well potential.In 1999 we have considered the finite element method for two-fluid flows with free interface including surface tension effect. Here we use the true two-fluid flow system in order to deal the problems with the high ratios of density and viscosity. In general, the surface tension effect is represented by the line integral on the interface. In our study the surface tension effect is interpreted as a body force spread across the interfacial region with a finite thickness.On the other hand, as the mathematical analysis of the finite element scheme for two-fluid flows we have considered the convergence of the approximate interface. In fact, estimating the LィイD1pィエD1(Ω)-norm of difference between the measure of positive value of the pseudo-density function and its approximation under the convergence of the finite element solution, we have prove it. Here we have used the Heaviside operator. The mathematical analysis of the total finite element scheme, however, is our theme in the near future.

本研究是在1998-1999年期间进行的，目的是发展和分析自然界和许多工业过程中经常遇到的具有自由界面的流体流动的有限元格式，1998年，我们考虑了低密度比双流体流动的有限元格式。我们假设两种流体都是粘性的、不可压缩的、不互溶的。作为这个问题的数学模型，我们使用的单流体模型假设的Boussinesq近似的Navier-Stokes方程。界面被认为是赝密度函数的0-水平集，赝密度函数是输运方程的解。本文对这类方程提出了一个P_1 ~ isoP_2/P_1元的混合有限元格式。特别地，我们还提出了双势阱输运方程有限元格式中的重新初始化技术。1999年我们考虑了考虑表面张力效应的自由界面双流体流动的有限元方法。在这里，我们使用真正的双流体流动系统，以处理与密度和粘度的高比例的问题。通常，表面张力效应由界面上的线积分表示。在我们的研究中，表面张力效应被解释为一个体积力分布在有限厚度的界面区域上，另一方面，作为二流体流动有限元格式的数学分析，我们考虑了近似界面的收敛性。实际上，在有限元解收敛的条件下，我们已经证明了伪密度函数的正测度与其近似值之差的L D1 p D1（Ω）-范数的估计，这里我们使用了Heaviside算子。然而，总的有限元方案的数学分析，是我们在不久的将来的主题。