A smoothing technique for discrete delta functions with application to immersed boundary method in moving boundary simulations

Calculation of fluid–structure interaction: methods, refinements, applications

“…Furthermore, the IBM allows computation of a body's movement in the fixed Eulerian frame of reference, negating the need to introduce a secondary frame that moves with the body (the 'body-fixed' frame), thereby circumventing the need to perform conversions between more than one frame of references. Several different variants of the IMB method have evolved since the 1970s, which can be grossly divided into direct or feedback forcing methods (Fadlun et al, 2000;Kempe and Fröhlich, 2012;Kim et al, 2001;Lai and Peskin, 2000;Shin et al, 2008;Tseng and Ferziger, 2003;Uhlmann, 2005a;Yang and Balaras, 2006;Yang et al, 2009;Yu and Shao, 2007), sharp-interface cut-cell methods (Seo and Mittal, 2011;Udaykumar et al, 2001) and hybrid versions of the two (Gilmanov and Sotiropoulos, 2005). Regarding circular cylinder flows, the studies of Borazjani and Sotiropoulos (2009), Gazzola et al (2011), Kim and Choi (2006), Kolomenskiy and Scheider (2009) and Yang et al (2008) constitute a selection of studies in which flow past various arrangements of single or tandem cylinders that were either stationary, freely vibrating or a combination of the two, have been investigated using the IBM.…”

“…Seo and Mittal (2011) for sharp interfaces, and Peskin (1977) and Uhlmann (2005a) for distributed forcing methods). Yang et al (2009) used smoothed discrete delta functions to reduce non-physical oscillations that occur in freshly cleared cells, replicating the case of a cylinder undergoing prescribed transverse oscillations in Re = 185 flow that had previously been investigated by Uhlmann (2005a) and Guilmineau and Queutey (2002), as well as the free VIV case of Leontini et al (2006); the resulting time traces of lift and drag forces were characterised by noticeably fewer oscillations than when non-smoothed functions were used. Shiels et al (2001) conducted simulations of a cylinder that was free to vibrate in the transverse direction in a flow of Re = 100 using a viscous-vortex method, with a cylinder of zero mass, stiffness and damping.…”

“…F is the force at the selected number of Lagrangian locations in space, N L , and V L is an assigned volume around each location, which is of the order of the cube of the Eulerian grid spacing, h. Here, the smooth regularised delta function due to Yang et al (2009) is used and the Lagrangian force is computed according to Uhlmann (2005a) 6:…”

“…While Uhlmann's (2005a) original method and successive similar methods; for example, Kempe and Fröhlich (2012), and Yang et al (2009) used Lagrangian forcing points only on the immersed boundary itself, the method described above also places Lagrangian points inside the immersed boundary (Uhlmann, 2005b). In the following, several test case flows are simulated using both types of Lagrangian-point placement allowing a direct assessment of validity and accuracy of the two forcing point strategies.…”

“…The cylinder is forced to displace in a sinusoidal manner in transverse direction to the flow with an amplitude of A = 0·2D and a frequency of f = 0·8f s in which f s is the natural frequency of the stationary cylinder at Re = 185. The structural motion of the cylinder with f = 0·156U/D is prescribed as y(t) = −A sin(2/πft), identical to what was utilised in Uhlmann (2005a) and Yang et al (2009). The same domain size and boundary conditions as in the stationary cylinder simulations are used.…”