Boundary and Interface Conditions for High-Order Finite-Difference Methods Applied to the Euler and Navier–Stokes Equations

1999 ◽  
Vol 148 (2) ◽  
pp. 621-645 ◽  
Author(s):  
Jan Nordström ◽  
Mark H Carpenter
Keyword(s):  
Finite Difference ◽  
Stokes Equations ◽  
Finite Difference Methods ◽  
High Order ◽  
Navier Stokes ◽  
Interface Conditions ◽  
Navier Stokes Equations ◽  
Difference Methods ◽  
High Order Finite Difference

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.

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