scholarly journals The Immersed Interface Method for the Navier–Stokes Equations with Singular Forces

2001 ◽  
Vol 171 (2) ◽  
pp. 822-842 ◽  
Author(s):  
Zhilin Li ◽  
Ming-Chih Lai
Keyword(s):  
Stokes Equations ◽  
Navier Stokes ◽  
Immersed Interface Method ◽  
Immersed Interface ◽  
Navier Stokes Equations

scholarly journals New Finite Difference Methods Based on IIM for Inextensible Interfaces in Incompressible Flows

2011 ◽  
Vol 1 (2) ◽  
pp. 155-171 ◽  
Author(s):  
Zhilin Li ◽  
Ming-Chih Lai
Keyword(s):  
Surface Tension ◽  
Finite Difference ◽  
Incompressible Flows ◽  
Stokes Equations ◽  
Finite Difference Methods ◽  
Navier Stokes ◽  
Immersed Interface Method ◽  
Immersed Interface ◽  
Navier Stokes Equations ◽  
Difference Methods

AbstractIn this paper, new finite difference methods based on the augmented immersed interface method (IIM) are proposed for simulating an inextensible moving interface in an incompressible two-dimensional flow. The mathematical models arise from studying the deformation of red blood cells in mathematical biology. The governing equations are incompressible Stokes or Navier-Stokes equations with an unknown surface tension, which should be determined in such a way that the surface divergence of the velocity is zero along the interface. Thus, the area enclosed by the interface and the total length of the interface should be conserved during the evolution process. Because of the nonlinear and coupling nature of the problem, direct discretization by applying the immersed boundary or immersed interface method yields complex nonlinear systems to be solved. In our new methods, we treat the unknown surface tension as an augmented variable so that the augmented IIM can be applied. Since finding the unknown surface tension is essentially an inverse problem that is sensitive to perturbations, our regularization strategy is to introduce a controlled tangential force along the interface, which leads to a least squares problem. For Stokes equations, the forward solver at one time level involves solving three Poisson equations with an interface. For Navier-Stokes equations, we propose a modified projection method that can enforce the pressure jump condition corresponding directly to the unknown surface tension. Several numerical experiments show good agreement with other results in the literature and reveal some interesting phenomena.

scholarly journals An immersed interface method for the Navier-Stokes equations on irregular domains

PAMM ◽  
2007 ◽  
Vol 7 (1) ◽  
pp. 1025401-1025402
Author(s):  
Zhilin Li ◽  
Ming-Chih Lai ◽  
Kazufumi Ito
Keyword(s):  
Stokes Equations ◽  
Navier Stokes ◽  
Irregular Domains ◽  
Immersed Interface Method ◽  
Immersed Interface ◽  
Navier Stokes Equations

scholarly journals The Sensitivity Analysis for the Flow Past Obstacles Problem with Respect to the Reynolds Number

2012 ◽  
Vol 4 (1) ◽  
pp. 21-35 ◽  
Author(s):  
Kazufumi Ito ◽  
Zhilin Li ◽  
Zhonghua Qiao
Keyword(s):  
Sensitivity Analysis ◽  
Reynolds Number ◽  
Critical Reynolds Number ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Irregular Domains ◽  
Time Step ◽  
Immersed Interface Method ◽  
Navier Stokes Equations ◽  
Flow Past

AbstractIn this paper, numerical sensitivity analysis with respect to the Reynolds number for the flow past obstacle problem is presented. To carry out such analysis, at each time step, we need to solve the incompressible Navier-Stokes equations on irregular domains twice, one for the primary variables; the other is for the sensitivity variables with homogeneous boundary conditions. The Navier-Stokes solver is the augmented immersed interface method for Navier-Stokes equations on irregular domains. One of the most important contribution of this paper is that our analysis can predict the critical Reynolds number at which the vortex shading begins to develop in the wake of the obstacle. Some interesting experiments are shown to illustrate how the critical Reynolds number varies with different geometric settings.

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