A second order penalized direct forcing for hybrid Cartesian/immersed boundary flow simulations

Abstract

In this paper, we propose a second order penalized direct forcing method to deal with fluidstructure interaction problems involving complex static or time-varying geometries. As this work constitutes a first step toward more complicated problems, our developments are restricted to Dirichlet boundary condition in purely hydraulic context. The proposed method belongs to the class of immersed boundary techniques and consists in immersing the physical domain in a Cartesian fictitious one of simpler geometry on fixed grids. A penalized forcing term is added to the momentum equation to take the boundary conditions around/inside the obstacles into account. This approach avoids the tedious task of re-meshing and allows us to use fast and accurate numerical schemes. In contrary, as the immersed boundary is described by a set of Lagrangian points that does not generally coincide with those of the Eulerian grid, numerical procedures are required to reconstruct the velocity field near the immersed boundary. Here, we develop a second order linear interpolation scheme and we compare it to a simpler model of order one. As far as the governing equations are concerned, we use a particular fractional-step method in which the penalized forcing term is distributed both in prediction and correction equations. The accuracy of the proposed method is assessed through 2-D numerical experiments involving static and rotating solids. We show in particular that the numerical rate of convergence of our method is quasi-quadratic.

Numerical modeling of an in-vessel flow limiter using an immersed boundary approach

“…Some perspectives can be outlined about the improvement of this numerical model in order to proceed to the geometry optimization of the design of the flow-limiter fins. In particular, we can mentioned the space-interpolation scheme across the boundary interface to reach the second order [18] and the definition of immersed-wall laws for RANS/large-eddy simulations.…”

“…Let us notice that for Dirichlet boundary conditions and elliptic problems, the theoretical rate of convergence in space of the Q 1 -finite element method with non-boundary-fitted meshes is O(h 1 ) in L 2 (Ω) norm, with h the space step [17]. As a whole, contributions to the validation of this IB approach can be found in [8] and [18]. On one side, in the context of dilatable two-phase flow elliptic problems, the work mentioned in [8] validates the ISI method with respect to body-fitted finite-element computations and to the JEBC method (an IB method using a finite-volume discretization).…”

“…A first-order rate of convergence in space is numerically reached. On the other side, in the context of incompressible one-phase flow Navier-Stokes equation, the work mentioned in [18] gives elements of validation for a finite-volume first-order penalty method very similar to the ISI method. Again, a first-order rate of convergence in space is numerically reached on the test case of a laminar flow around a static cylinder of diameter D (Reynolds number = 20).…”

A sharp immersed boundary method based on penalization and its application to moving boundaries and turbulent rotating flows

“…However, most of published methods do not reach second order accuracy in space. To the author's knowledge, the Sub-Mesh Penalty Method of Sarthou et al [16], the second order Penalty model of Introïni et al [17], and the work of Chantalat et al [18] are the only models based on penalisation which offer improvements to tackle the rasterisation effect on Cartesian grids. In addition, the active zone for the forcing term is generally also diffused by the use of mask functions combined with Heaviside functions, as it is the case in [19] and in [14].…”

FluSI: A Novel Parallel Simulation Tool for Flapping Insect Flight Using a Fourier Method with Volume Penalization

“…As animals forage following odor traces, this technique can potentially be attractive for insect flight simulations as well. An interesting recent development has been proposed in [13] in the context of finite-difference discretizations. Their idea is to modify the fractional step projection scheme such that the Neumann boundary condition, as introduced in [15], appears in the pressure Poisson equation.…”