An immersed interface solver for the 2-D unbounded Poisson equation and its application to potential flow

Marichal, Yves; Chatelain, Philippe; Winckelmans, Grégoire

doi:10.1016/j.compfluid.2014.03.012

Cited by 12 publications

(8 citation statements)

References 37 publications

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“…Immersed interface methods are well-suited for simulations involving moving and deforming geometry, including fluid-structure interaction problems that are often discretized with lower-order immersed boundary or penalization methods. There are also promising developments in extending the IIM to non-smooth geometries with thin features, cusps, and acute interior corners [25,24], which would further broaden the range of flows that can be simulated with the current method. The accuracy of surface pressure and shear distributions can be greatly increased through the use of multi-resolution adaptive grids [52], which allow computational elements to be concentrated around immersed surfaces.…”

Section: Discussionmentioning

confidence: 99%

“…To tackle such problems, a common approach is to prescribe the solution outside of the problem domain (typically to zero value), and then treat the irregular domain boundary as a jump discontinuity. In this case the jump discontinuity is no longer physically constrained, and the value of the jump in each derivative f (k) (x α ) must be calculated directly from the function f (x) by evaluating a one-sided finite difference stencil [22,25,26]. To illustrate this procedure, consider the same function f discussed above, now with f (x) = 0 for x > x α to model a domain boundary (Figure 1b).…”

Section: Specializing the Explicit Jump Iimmentioning

confidence: 99%

“…To extend the above methodology to the full velocity reconstruction problem, the unknown constants ψk must be added as additional unknowns, and the circulation constraints must be discretized to determine their values. Instead of imposing these constraints directly on immersed solid bodies, we choose to follow (25) and specify the circulation around the boundary of a rectangular region R k that encloses each solid body B k . This avoids integration over any immersed surfaces, and allows us to take advantage of a conservation property inherent in the standard 5point Laplacian.…”

Section: It Is Also Helpful To Define the Inclusion Operatormentioning

confidence: 99%

“…Recently, the IIM has also been integrated within vortex particle-mesh methods, which rely on a combined Lagrangian-Eulerian approach to integrate the incompressible Navier-Stokes equations. In [25] a traditional Lighthill splitting approach was introduced to handle the vorticity boundary condition, leading to first-order temporal and second-order spatial accuracy. Subsequently, the same group employed a Thom-like finite difference boundary condition to achieve high-order accuracy in time [26], and an extension to 3D [27].…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

An immersed interface method for the 2D vorticity-velocity Navier-Stokes equations with multiple bodies

Gabbard,

Gillis,

Chatelain

et al. 2021

Preprint

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We present an immersed interface method for the vorticity-velocity form of the 2D Navier Stokes equations that directly addresses challenges posed by multiply connected domains, nonconvex obstacles, and the calculation of force distributions on immersed surfaces. The immersed interface method is re-interpreted as a polynomial extrapolation of flow quantities and boundary conditions into the obstacle, reducing its computational and implementation complexity. In the flow, the vorticity transport equation is discretized using a conservative finite difference scheme and explicit Runge-Kutta time integration. The velocity reconstruction problem is transformed to a scalar Poisson equation that is discretized with conservative finite differences, and solved using an FFT-accelerated iterative algorithm. The use of conservative differencing throughout leads to exact enforcement of a discrete Kelvin's theorem, which provides the key to simulations with multiply connected domains and outflow boundaries. The method achieves second order spatial accuracy and third order temporal accuracy, and is validated on a variety of 2D flows in internal and free-space domains.

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Section: Discussionmentioning

confidence: 99%

Section: Specializing the Explicit Jump Iimmentioning

confidence: 99%

Section: It Is Also Helpful To Define the Inclusion Operatormentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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An immersed interface method for the 2D vorticity-velocity Navier-Stokes equations with multiple bodies

Gabbard,

Gillis,

Chatelain

et al. 2021

Preprint

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“…Note that the IIM introduces a similar modification to the Poisson equation (Marichal et al. 2014) and the Sherman–Morrison–Woodbury decomposition can produce an expression for the discrete vortex sheet strength (Gillis et al. 2019) that is equivalent to the formula we obtain by using the Schur complement method.…”

Section: Introductionmentioning

confidence: 99%

Planar potential flow on Cartesian grids

Beckers

Eldredge

2022

J. Fluid Mech.

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Potential flow has many applications, including the modelling of unsteady flows in aerodynamics. For these models to work efficiently, it is best to avoid Biot–Savart interactions. This work presents a grid-based treatment of potential flows in two dimensions and its use in a vortex model for simulating unsteady aerodynamic flows. For flows consisting of vortex elements, the treatment follows the vortex-in-cell approach and solves the streamfunction–vorticity Poisson equation on a Cartesian grid after transferring the circulation from the vortices onto the grid. For sources and sinks, an analogous approach can be followed using the scalar potential. The combined velocity field due to vortices, sinks and sources can then be obtained using the Helmholtz decomposition. In this work, we use several key tools that ensure the approach works on arbitrary geometries, with and without sharp edges. First, the immersed boundary projection method is used to account for bodies in the flow and the resulting body-forcing Lagrange multiplier is identified as the bound vortex sheet strength. Second, sharp edges are treated by decomposing the vortex sheet strength into a singular and non-singular part. To enforce the Kutta condition, the non-singular part can then be constrained to remove the singularity introduced by the sharp edge. These constraints and the Poisson equation are formulated as a saddle-point system and solved using the Schur complement method. The lattice Green's function is used to efficiently solve the discrete Poisson equation with unbounded boundary conditions. The method and its accuracy are demonstrated for several problems.

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An IB-LBM implementation for fluid-solid interactions with an MLS approximation for implicit coupling

Wang

Yan

Liu

2018

Applied Mathematical Modelling

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An immersed interface solver for the 2-D unbounded Poisson equation and its application to potential flow

Cited by 12 publications

References 37 publications

An immersed interface method for the 2D vorticity-velocity Navier-Stokes equations with multiple bodies

An immersed interface method for the 2D vorticity-velocity Navier-Stokes equations with multiple bodies

Planar potential flow on Cartesian grids

An IB-LBM implementation for fluid-solid interactions with an MLS approximation for implicit coupling

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