A Variational Projection Scheme for Nonmatching Surface-to-Line Coupling between 3D Flexible Multibody System and Incompressible Turbulent Flow

This paper is concerned with the partitioned iterative formulation to simulate the fluid-structure interaction of a nonlinear multibody system in an incompressible turbulent flow. The proposed formulation relies on a three-dimensional (3D) incompressible turbulent flow solver, a nonlinear monolithic elastic structural solver for constrained flexible multibody system and the nonlinear iterative force correction scheme for coupling of the turbulent fluid-flexible multibody system with nonmatching interface meshes. While the fluid equations are discretized using a stabilized Petrov-Galerkin formulation in space and the generalized-$\alpha$ updates in time, the multibody system utilizes a discontinuous space-time Galerkin finite element method. We address two key challenges in the present formulation. Firstly, the coupling of the incompressible turbulent flow with a system of nonlinear elastic bodies described in a co-rotated frame. Secondly, the projection of the tractions and displacements across the nonmatching 3D fluid surface elements and the one-dimensional line elements for the flexible multibody system in a conservative manner. Through the nonlinear iterative correction and the conservative projection, the developed fluid-flexible multibody interaction solver is stable for problems involving strong inertial effects between the fluid-flexible multibody system and the coupled interactions among each multibody component. The accuracy of the proposed coupled finite element framework is validated against the available experimental data for a long flexible cylinder undergoing vortex-induced vibration in a uniform current flow condition. Finally, a practical application of the proposed framework is demonstrated by simulating the flow-induced vibration of a realistic offshore floating platform connected to a long riser and an elastic mooring system

An adaptive variational procedure for the conservative and positivity preserving Allen-Cahn phase-field model

We present an adaptive variational procedure for unstructured meshes to capture fluid-fluid interfaces in two-phase flows. The two phases are modeled by the phase-field finite element formulation, which involves the conservative Allen-Cahn equation coupled with the incompressible Navier-Stokes equations. The positivity preserving variational formulation is designed to maintain the bounded and stable solution of the Allen-Cahn equation. For the adaptivity procedure, we consider the residual-based error estimates for the underlying differential equations of the two-phase system. In particular, the adaptive refinement/coarsening is carried out by the newest vertex bisection algorithm by evaluating the residual error indicators based on the error estimates of the Allen-Cahn equation. The coarsening algorithm avoids the storage of the tree data structures for the hierarchical mesh, thus providing the ease of numerical implementation. The proposed nonlinear adaptive partitioned procedure aims at reducing the amount of coarsening while maintaining the convergence properties of the underlying nonlinear coupled differential equations. We investigate the adaptive phase-field finite element scheme through the spinodal decomposition in a complex curved geometry and the volume-conserved interface motion driven by the mean curvature flow for two circles in a square domain. We then assess the accuracy and efficiency of the proposed procedure by modeling the free-surface motion in a sloshing tank. The mesh adaptivity remarkably reduces the degrees of freedom and the computational cost by nearly half for similar accuracy and improves the mass conservation. Finally, we apply the adaptive numerical framework to solve the application of a dam-breaking problem with topological changes.Comment: 33 pages, 20 figures, 1 tabl

A positivity preserving and conservative variational scheme for phase-field modeling of two-phase flows

We present a positivity preserving variational scheme for the phase-field modeling of incompressible two-phase flows with high density ratio and using meshes of arbitrary topology. The variational finite element technique relies on the Allen-Cahn phase-field equation for capturing the phase interface on a fixed mesh with a mass conservative and energy-stable discretization. Mass is conserved by enforcing a Lagrange multiplier which has both temporal and spatial dependence on the solution of the phase-field equation. The spatial part of the Lagrange multiplier is written as a mid-point approximation to make the scheme energy-stable. This enables us to form a conservative, energy-stable and positivity preserving scheme. The proposed variational technique reduces spurious and unphysical oscillations in the solution while maintaining second-order spatial accuracy. To model a generic two-phase free-surface flow, we couple the Allen-Cahn phase-field equation with the Navier-Stokes equations. Comparison of results between standard linear stabilized finite element method and the present variational formulation shows a remarkable reduction of oscillations in the solution while retaining the boundedness of the phase-indicator field. We perform a standalone test to verify the accuracy and stability of the Allen-Cahn two-phase solver. Standard two-phase flow benchmarks such as Laplace-Young law and sloshing tank problem are carried out to assess the convergence and accuracy of the coupled Navier-Stokes and Allen-Cahn solver. Two- and three-dimensional dam break problem are then solved to assess the scheme for the problem with topological changes of the air-water interface on unstructured meshes. Finally, we demonstrate the phase-field solver for a practical problem of wave-structure interaction in offshore engineering using general three-dimensional unstructured meshes.Comment: 35 pages, 22 figures, 5 table

Transverse flow-induced vibrations of a sphere in the proximity of a free surface: A numerical study

We present a numerical study on the transverse flow-induced vibration (FIV) of an elastically mounted sphere in the vicinity of a free surface at subcritical Reynolds numbers. To begin, We verify and analyze the mode transitions and the motion trajectories of a fully submerged sphere vibrating freely in all directions for the Reynolds number up to $30\,000$. Next, the response dynamics of a transversely vibrating sphere is studied for three values of normalized immersion ratio ($h^*=h/D$, where $h$ is the distance from the top of the sphere to undisturbed free-surface level and $D$ is the sphere diameter), at $h^*=1$ (fully submerged sphere with no free-surface effect), $h^*=0$ (where the top of the sphere touches the free surface) and $h^*=-0.25$ (where the sphere pierces the free surface). At the lock-in range, we observe that the amplitude response at $h^*=0$ is decreased significantly compared to the case at $h^*=1$. It is found that the vorticity flux is diffused due to the free-surface boundary and the free surface acts as a sink of energy that leads to a reduction in the transverse force and amplitude response. When the sphere pierces the free surface at $h^*=-0.25$, the amplitude response at the lock-in state is found to be greater than all the submerged cases studied with the maximum peak-to-peak amplitude of $\sim2D$. We find that the interaction of the piercing sphere with the air-water interface causes a relatively large surface deformation and has a significant impact on the synchronization of the vortex shedding and the vibration frequency. Increased streamwise vorticity gives rise to a relatively larger transverse force to the piercing sphere at $h^*=-0.25$, resulting in greater positive energy transfer per cycle to sustain the large-amplitude vibration. Lasty, we study the sensitivity of FIV response on the mass ratio, $m^*$, and Froude number, $Fr$, at the lock-in state.Comment: 45 pages, 32 figure