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A high-order in space spectral-element methodology for the solution of a strongly coupled fluid-structure interaction (FSI) problem is developed. A methodology is based on a partitioned solution of incompressible fluid equations on body-fitted grids, and nonlinearly-elastic solid deformation equations coupled via a fixed-point iteration approach with Aitken relaxation. A comprehensive verification strategy of the developed methodology is presented, including h-, p-and temporal refinement studies. An expected order of convergence is demonstrated first separately for the corresponding fluid and solid solvers, followed by a self-convergence study on a coupled FSI problem (self-convergence refers to a convergence to a reference solution obtained with the same solver at higher resolution). To this end, a new three-dimensional fluid-structure interaction benchmark is proposed for a verification of the FSI codes, which consists of a fluid flow in a channel with one rigid and one flexible wall. It is shown that, due to a consistent problem formulation, including initial and boundary conditions, a high-order spatial convergence on a fully coupled FSI problem can be demonstrated. Finally, a developed framework is applied successfully to a Direct Numerical Simulation of a turbulent flow in a channel interacting with a compliant wall, where the fluid-structure interface is fully resolved.
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A partitioned numerical scheme for fluid–structure interaction with slip
We present a loosely coupled, partitioned scheme for solving fluid–structure interaction (FSI) problems with the Navier slip boundary condition. The fluid flow is modeled by the Navier–Stokes equations for an incompressible, viscous fluid, interacting with a thin elastic structure modeled by the membrane or Koiter shell type equations. The fluid and structure are coupled via two sets of coupling conditions: a dynamic coupling condition describing balance of forces, and a kinematic coupling condition describing fluid slipping tangentially to the moving fluid–structure interface, with no penetration in the normal direction. Problems of this type arise in, e.g. , FSI with hydrophobic structures or surfaces treated with a no-stick coating, and in biologic FSI involving rough surfaces of elastic tissues or tissue scaffolds. We propose a novel, efficient partitioned scheme where the fluid sub-problem is solved separately from the structure sub-problem, and there is no need for sub-iterations at every time step to achieve stability, convergence, and its first-order accuracy. We derive energy estimates, which prove that the proposed scheme is unconditionally stable for the corresponding linear problem. Moreover, we present convergence analysis and show that under a time-step condition, the method is first-order accurate in time and optimally convergent in space for a Finite Element Method-based spatial discretization. The theoretical rates of convergence in time are confirmed numerically on an example with an explicit solution using the method of manufactured solutions, and on a benchmark problem describing propagation of a pressure pulse in a two-dimensional channel. The effects of the slip rate and fluid viscosity on the FSI solution are numerically investigated in two additional examples: a 2D cylindrical FSI example for which an exact Navier slip Poiseuille-type solution is found and used for comparison, and a squeezed ketchup bottle example with gravity enhanced flow. We show that the Navier-slip boundary condition increases the outflow mass flow rate by 21% for a bottle angled at 45 degrees pointing downward, in the direction of gravity.
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- Award ID(s):
- 1853340
- PAR ID:
- 10332163
- Editor(s):
- Grandmont, C.; Hillairet, M.; Matin, S.; Muha, B.; Vergarra, Ch.
- Date Published:
- Journal Name:
- Mathematical Modelling of Natural Phenomena
- Volume:
- 16
- ISSN:
- 0973-5348
- Page Range / eLocation ID:
- 8
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript
- Journal Article:
- https://doi.org/10.1051/mmnp/2020051
