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Immersed Boundary Methods

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Computational Fluid Dynamics

Abstract

Analysis of flow over bodies with complex surface geometry can pose a challenge in terms of spatial discretization. Creating a high quality boundary fitted mesh can not only be difficult but also time consuming especially when there are complex flow structures over intricate boundary geometry that need to be resolved in the simulations. This is especially true for bodies encountered in engineering applications such as fluid flow around an automotive engine and undercarriage as well as aircraft landing gears. Furthermore, if we have problems involving fluid-structure interaction, the location of the moving or deforming interface needs to be determined numerically. In such cases, the need to re-mesh the flow field around the body at every time step can introduce an added computational burden. Similar situation arises when one attempts to simulate particle-laden flows in which the interface between different phases needs to be tracked and resolved accurately.

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Notes

  1. 1.

    For a collocated spatial discretization, a stronger requirement of \(\sum _{i=\text {odd}} d(x/\Delta s - i) = \sum _{i=\text {even}} d(x/\Delta s - i)= 1/2\) is imposed to avoid checkerboard oscillation from naively transmitting the boundary force onto the Eulerian grid.

  2. 2.

    The exponential form of the discrete delta function is infinitely continuous which is useful for spectral methods to avoid the introduction of spatial oscillations.

  3. 3.

    The translational invariance would need a stronger statement of \(\sum _{i} d(x_1/\Delta s - i) d(x_2/\Delta s - i) = \text {fnc}(x_1/\Delta s-x_2/\Delta s)\) for all x.

  4. 4.

    The discrete Navier–Stokes equations can be formulated as a Karush–Kuhn–Tucker (KKT) system [25, 35] (see: Eqs. (3.26) and (5.45) with \(D = -G^{T}\)).

  5. 5.

    In Fadlun et al. [9], the third-order Runge–Kutta method is used.

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Authors and Affiliations

  1. Osaka University, Suita, Osaka, Japan

    Takeo Kajishima

  2. Florida State University, Tallahassee, FL, USA

    Kunihiko Taira

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  1. Takeo Kajishima
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  2. Kunihiko Taira
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Correspondence to Takeo Kajishima or Kunihiko Taira .

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Kajishima, T., Taira, K. (2017). Immersed Boundary Methods. In: Computational Fluid Dynamics. Springer, Cham. https://doi.org/10.1007/978-3-319-45304-0_5

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  • DOI: https://doi.org/10.1007/978-3-319-45304-0_5

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