A penalization method to take into account obstacles in incompressible viscous flows

Angot, Philippe; Bruneau, Charles‐Henri; Fabrie, Pierre

doi:10.1007/s002110050401

Cited by 717 publications

(670 citation statements)

References 11 publications

(14 reference statements)

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“…where the √ η behavior is consistent with previous studies [1,7]. In order to discretize (1.3), it is possible to use a Fourier pseudo-spectral method, where collocation on a regular grid of N points is used to evaluate the product χv.…”

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confidence: 80%

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Approximation of the Laplace and Stokes operators with Dirichlet boundary conditions through volume penalization: a spectral viewpoint

2014

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Abstract. We report the results of a detailed study of the spectral properties of Laplace and Stokes operators, modified with a volume penalization term designed to approximate Dirichlet conditions in the limit when a penalization parameter, η, tends to zero. The eigenvalues and eigenfunctions are determined either analytically or numerically as functions of η, both in the continuous case and after applying Fourier or finite difference discretization schemes. For fixed η, we find that only the part of the spectrum corresponding to eigenvalues λ η −1 approaches Dirichlet boundary conditions, while the remainder of the spectrum is made of uncontrolled, spurious wall modes. The penalization error for the controlled eigenfunctions is estimated as a function of η and λ. Surprisingly, in the Stokes case, we show that the eigenfunctions approximately satisfy, with a precision O(η), Navier slip boundary conditions with slip length equal to √ η. Moreover, for a given discretization, we show that there exists a value of η, corresponding to a balance between penalization and discretization errors, below which no further gain in precision is achieved. These results shed light on the behavior of volume penalization schemes when solving the Navier-Stokes equations, outline the limitations of the method, and give indications on how to choose the penalization parameter in practical cases.

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confidence: 80%

“…Due to this analogy, the method is sometimes called Brinkman penalization. It has been mathematically justified in academic cases [1,7], and its use has rapidly spread to various domains of scientific computing.…”

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confidence: 99%

Approximation of the Laplace and Stokes operators with Dirichlet boundary conditions through volume penalization: a spectral viewpoint

2014

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“…It can be shown that the solution of problem (42) converges slowly to the solution of the full Navier-Stokes problem in the channel domain as η → 0. 6 By plotting the tangential velocity as a function of strain rate at the channel wall location, Nguyen van yen, Farge, and Schneider 1 show that the penalisation method can be used to approximate noslip boundary conditions. More specifically, they demonstrate that their results are close to those for a Navier boundary condition with slip length s L approximately equal to 4/Re.…”

Section: B Comparison Of the Penalisation Methods With The Influence mentioning

confidence: 99%

The effect of slip length on vortex rebound from a rigid boundary

2013

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The problem of a dipole incident normally on a rigid boundary, for moderate to large Reynolds numbers, has recently been treated numerically using a volume penalisation method by Nguyen van yen, Farge, and Schneider [Phys. Rev. Lett. 106, 184502 (2011)]. Their results indicate that energy dissipating structures persist in the inviscid limit. They found that the use of penalisation methods intrinsically introduces some slip at the boundary wall, where the slip approaches zero as the Reynolds number goes to infinity, so reducing to the no-slip case in this limit. We study the same problem, for both no-slip and partial slip cases, using compact differences on a Chebyshev grid in the direction normal to the wall and Fourier methods in the direction along the wall. We find that for the no-slip case there is no indication of the persistence of energy dissipating structures in the limit as viscosity approaches zero and that this also holds for any fixed slip length. However, when the slip length is taken to vary inversely with Reynolds number then the results of Nguyen van yen et al. are regained. It therefore appears that the prediction that energy dissipating structures persist in the inviscid limit follows from the two limits of wall slip length going to zero, and viscosity going to zero, not being treated independently in their use of the volume penalisation method.

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“…A description of the numerical method can be found in Kolomenskiy et al (Kolomenskiy et al, 2011). Briefly, the no-slip boundary condition at the solid boundary is modeled using the volume penalization method (Angot et al, 1999). The penalized Navier-Stokes equations are solved using a classical Fourier pseudo-spectral method, the computational domain being therefore a rectangular box with periodic boundary conditions imposed at its six faces.…”

Section: Tests Of Force Balancementioning

confidence: 99%

Force balance in the take-off of a pierid butterfly: relative importance and timing of leg impulsion and aerodynamic forces

Bimbard¹,

Kolomenskiy²,

Bouteleux³

et al. 2013

Journal of Experimental Biology

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SUMMARYUp to now, the take-off stage has remained an elusive phase of insect flight that was relatively poorly explored compared with other maneuvers. An overall assessment of the different mechanisms involved in force production during take-off has never been explored. Focusing on the first downstroke, we have addressed this problem from a force balance perspective in butterflies taking off from the ground. In order to determine whether the sole aerodynamic wing force could explain the observed motion of the insect, we have firstly compared a simple analytical model of the wing force with the acceleration of the insect's center of mass estimated from video tracking of the wing and body motions. Secondly, wing kinematics were also used for numerical simulations of the aerodynamic flow field. Similar wing aerodynamic forces were obtained by the two methods. However, neither are sufficient, nor is the inclusion of the ground effect, to predict faithfully the body acceleration. We have to resort to the leg forces to obtain a model that best fits the data. We show that the median and hind legs display an active extension responsible for the initiation of the upward motion of the insect's body, occurring before the onset of the wing downstroke. We estimate that legs generate, at various times, an upward force that can be much larger than all other forces applied to the insect's body. The relative timing of leg and wing forces explains the large variability of trajectories observed during the maneuvers. Supplementary material available online at

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A penalization method to take into account obstacles in incompressible viscous flows

Cited by 717 publications

References 11 publications

Approximation of the Laplace and Stokes operators with Dirichlet boundary conditions through volume penalization: a spectral viewpoint

Approximation of the Laplace and Stokes operators with Dirichlet boundary conditions through volume penalization: a spectral viewpoint

The effect of slip length on vortex rebound from a rigid boundary

Force balance in the take-off of a pierid butterfly: relative importance and timing of leg impulsion and aerodynamic forces

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