B07, Science College, Department of Applied Mathematics, Feng Chia University

Aim and Scope:

Fluid-structure interaction problem is an important problem in computational fluid dynamics and poses a challenge for today’s numerical methods in this field. It has many applications in engineering such as energy harvesting, civil and environmental engineering, hydraulics engineering, just name a few. In old days, the most common method to simulate the flow with a complicated solid boundary is to use a body-fitted technique with grids fitting and clustering along the complex boundary. Most of time, the solid object may not be at rest and it requires further technique to deal with a moving object. The Arbitrary Lagrangian Eu- lerian (ALE) numerical method is a popular approach to accommodate the complicated fluid-structure interface varying with time. In the Eulerian coordinate frame, fluids flow through the static computational mesh. While in the Lagrangian coordinate, the mesh moves with the solid. Nevertheless, mesh updating or re-meshing is computationally expensive for the ALE algorithm. In addition to the ALE algorithm, the immersed boundary method is becoming popular since it was first introduced by Peskin due to its capability to handle simulations for a moving complex boundary with lower computational cost and memory requirements than the conventional body-fitted method. In this method, a fixed Cartesian grid and a Lagrangian grid are employed for fluids and immersed solid object, respectively. The interaction between fluids and the immersed solid boundary is linked through the spreading of the singular force from the Lagrangian grid to the Cartesian grid and the interpolation of the velocity from the Cartesian grid to the Lagrangian grid using a discrete Dirac delta function. Instead of using a delta function to distribute force from the Lagrangian grid to the Cartesian grid, Mohd-Yusof introduced a novel immersed boundary method, namely the direct-forcing immersed boundary method (DFIB). This method uses a virtual forcing term determined by the difference between the interpolated velocities at the boundary points and the desired boundary velocities. The idea of DFIB has been applied successfully in many applications, and many variations of DFIB have developed since then. This one-day mini-workshop aims to study several state-of-art DFIB methods featuring (1) a joint method of level set and DFIB to investigate fluid-structure interaction problem with a free air-water interface, (2) a two-stage DFIB method with prediction and correction steps to largely shorten numerical time step compared with most other DFIB methods.

Invited Speakers:

1. Chin-Biao Liao (廖清標)

**Title:** A joint method of level set and direct-forcing immersed boundary for fluid-structure interaction with free air-water surface (I) & (II)

**Abstract:** Fluid-structure interaction problems have been a challenging subject for computational fluid dynamics, not to mention if the problem involves a free surface between two fluids such as water and air. While level set methods have been celebrating huge success than ever in simulating two-phase flows, various novel and efficient immersed boundary methods have been recently applied successfully to solve difficult fluid-structure interaction problems. However, few methods were reported to solve the fluid-structure interaction problems involving free surface, which can be seen as a kind of three-phase flow problems. Those methods are difficult to implement, which motivated us to develop an efficient joint method of level set and immersed boundary for it. Here incompressible Navier-Stokes equations together with the level set equation are discretized by finite difference method in simple staggered Cartesian grids, with the accuracy of order in space is 2nd order, featuring central difference for the viscous term and 2nd-order upwind scheme for convection term. Projection method is employed here with pressure Poisson equation to replace divergence-free condition. 2nd-order Adams-Bashforth method is used for the time-integration of convective and viscous terms, and forward Euler method for pressure and force terms. A 5th-order WENO scheme is used to solve the level set equation to track the interface of two fluids with surface tension that requests very high accuracy. As to the solid part, an immersed boundary method with direct forcing is applied to simulate a solid object moving in two fluids. Volume of solid interpolation is employed to obtain further accuracy in space for flow near the solid. The whole method is validated with a 2D uniform flow past a circular cylinder with vortex shedding. Coefficients of drag and lift, Strouhal number were computed and are all very accurate compared with literatures. Several cases of fluid-structure interaction computational examples are shown to demonstrate the excellence of the current method. In conclusion, our current method is efficient, straight-forward, robust and successful in solving fluid-structure interaction problems with free surface involved.

2. Suh-Yuh Yang (楊肅煜)

**Title: **A simple direct-forcing immersed boundary projection method with prediction-correction for fluid-solid interaction problems (I) & (II)

**Abstract:** In this work, we propose a simple and novel direct-forcing immersed boundary (IB) projection method in conjunction with a prediction-correction (PC) process for simulating the dynamics of fluid-solid interaction problems, in which each immersed solid object can be stationary or moving in the fluid with a prescribed velocity. The method is mainly based on the introduction of a virtual force which is distributed only on the immersed solid bodies and appended to the fluid momentum equations to accommodate the internal boundary conditions at the immersed solid boundaries. More specifically, we first predict the virtual force on the immersed solid domain by using the difference between the prescribed solid velocity and the computed velocity, which is obtained by applying the Choi-Moin projection scheme to the incompressible Navier-Stokes equations on the entire domain including the portion occupied by the solid bodies. The predicted virtual force is then added to the fluid momentum equations as an additional forcing term and we employ the same projection scheme again to correct the velocity field, pressure and virtual force. Although this method is a two-stage approach, the computational cost of the correction stage is rather cheap, since the associated discrete linear systems need to be solved in the correction stage are same with that in the prediction stage, except the right-hand side data terms. Such a PC procedure can be iterated to form a more general method, if necessary. The current two-stage direct-forcing IB projection method has the advantage over traditional one-stage direct-forcing IB projection methods, consisting of the prediction step only, by allowing much larger time step, since traditional methods generally request quite small time step for flow field relaxed and adjusted to the solid body movement even using implicit scheme. Numerical experiments of several benchmark problems are performed to illustrate the simplicity and efficient performance of the newly proposed method. Convergence tests show that the accuracy of the velocity field is super-linear in space in all the 1-norm, 2-norm, and maximum norm. We also find that our numerical results are in very good agreement with the previous works in the literature and one correction at each time step appears to be good enough for the proposed PC procedure.