Procedure for 2D fluid–structure interaction simulation

Abstract

A numerical technique for the solution of the structural dynamics equations of motion is presented. The structural dynamics mass and momentum conservation equations are solved using a control volume technique which is secondorder accurate in space along with a dual time-step scheme that is second order accurate in time. The momentum conservation equation is written in terms of the Piola-Kirchoff stresses and the displacement velocity components. The stress tensor is related to the Lagrangian strain and displacement tensors using the St. Venant-Kirchoff constitutive relationship. Source terms are included to account for surface pressure and body forces. Verification of the structural dynamics solution procedure is presented for a two-dimensional vibrating cantilever beam. In addition, the structural dynamics solution procedure has been implemented into a general purpose two dimensional conjugate heat transfer solution procedure that uses a similar dual time-step control volume technique to solve the fluid mass, energy, and Navier-Stokes equations as well as the structural energy heat conduction equation. The resulting overall solution procedure allows for solutions to fluid/structure, fluid/thermal, or fluid/thermal/structure interaction problems. Verification of the multidisciplinary procedure is performed using a cylinder with a flexible solid protruding downstream that mimics a cylinder-flag configuration. The approach is a proof of concept for compressible flow with continuum based solids. The methods are currently being extended to 3D flow fields and solids.

Computational structural dynamics general solution procedure using finite volumes

“…7 The proposed solver demonstrated excellent agreement with theoretical vibratory response of a cantilever beam and good agreement with literature. 7,9 The second test case was a highly flexible square cantilever plate subject to a gravitational load. The first two modal frequencies predicted by the presented structural solver show good agreement with the commercial code ANSYS-Discovery AIM.…”

“…The maximum X and Y deformation of point A is 0.0381 m and 0.142 m, respectively. Zorn and Davis 7 reported the maximum displacement of 0.022 m and 0.120 m, while Turek and Hron 9 reported 0.029 m and 0.130 m. The discrepancy between maximum reported the displacements by Zorn and Davis, 7 Turek and Hron, 9 and the current study is caused by the three-dimensional effects of the strain equations.…”

“…The 2nd Piola-Kirchhoff stress, S ij , is related to the strain through the St. Venant-Kirchhoff constitutive relationship, as shown in equation (6). The 1st and the 2nd Piola-Kirchhoff stresses are related using the configuration array as shown in equation (7). The 1st Piola-Kirchhoff stress, P ij , allows a direct mapping of forces to the referential configuration.…”

“…Note that the current study calculates h ij based on nodal displacement, rather than nodal velocity as proposed by Gottfried and Fleeter, 8 per the findings of Zorn and Davis. 7 Gottfried and Fleeter 8 recommended Q hg to be between 0.05 and 0.15. In the current study, all simulations use an hourglass coefficient of 0.05.…”

“…Figure5. Three-dimensional (3D) cantilever beam case displacement time histories compared with two-dimensional (2D) studies 7,9. …”