Systematic Derivation of Jump Conditions for the Immersed Interface Method in Three-Dimensional Flow Simulation

• Published in: SIAM journal on scientific computing Vol. 27; no. 6; pp. 1948 - 1980
• Main Authors: Xu, Sheng, Wang, Z. Jane
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2006
• Subjects: Accuracy; Boundary conditions; Computational methods in fluid dynamics; Exact sciences and technology; Fluid dynamics; Fundamental areas of phenomenology (including applications); Mathematics; Navier-Stokes equations; Numerical analysis; Numerical analysis. Scientific computation; Partial differential equations, initial value problems and time-dependant initial-boundary value problems; Physics; Sciences and techniques of general use; Simulation; Velocity; Discontinuity; 65M06; 76D05. 76M20; Immersed boundary method; three-dimensional Navier-Stokes equations; Incompressible flow; Singular force; Flow simulation; Cartesian grid methods; Numerical analysis; Three dimensional flow; Navier Stokes equation; Cartesian grid method; jump conditions; immersed interface method; Viscous flow
• ISSN: 1064-8275; 1095-7197
• DOI: 10.1137/040604960

In this paper, we systematically derive jump conditions for the immersed interface method [SIAM J. Numer. Anal., 31 (1994), pp. 1019-1044; SIAM J. Sci. Comput., 18 (1997), pp. 709-735] to simulate three-dimensional incompressible viscous flows subject to moving surfaces. The surfaces are represented as singular forces in the Navier--Stokes equations, which give rise to discontinuities of flow quantities. The principal jump conditions across a closed surface of the velocity, the pressure, and their normal derivatives have been derived by Lai and Li [Appl. Math. Lett., 14 (2001), pp. 149-154]. In this paper, we first extend their derivation to generalized surface parametrization. Starting from the principal jump conditions, we then derive the jump conditions of all first-, second-, and third-order spatial derivatives of the velocity and the pressure. We also derive the jump conditions of first- and second-order temporal derivatives of the velocity. Using these jump conditions, the immersed interface method is applicable to the simulation of three-dimensional incompressible viscous flows subject to moving surfaces, where near the surfaces the first- and second-order spatial derivatives of the velocity and the pressure can be discretized with, respectively, third- and second-order accuracy, and the first-order temporal derivatives of the velocity can be discretized with second-order accuracy.