Simulation of a highly elastic structure interacting with a two-phase flow
© Svenning et al.; licensee Springer. 2014
Received: 30 April 2012
Accepted: 5 March 2014
Published: 3 June 2014
The aim of this paper is to present and validate a modeling framework that can be used for simulation of industrial applications involving fluid structure interaction with large deformations.
Fluid structure interaction phenomena involving elastic structures frequently occur in industrial applications such as rubber bushings filled with oil, the filling of liquid in a paperboard package or a fiber suspension flowing through a paper machine. Simulations of such phenomena are challenging due to the strong coupling between the fluid and the elastic structure. In the literature, this coupling is often achieved with an arbitrary Lagrangian Eulerian framework or with smooth particle hydrodynamics methods. In the present work, an immersed boundary method is used to couple a finite volume based Navier-Stokes solver with a finite element based structural mechanics solver for large deformations.
The benchmark of an elastic rubber beam in a rolling tank partially filled with oil is simulated. The simulations are compared to experimental data as well as numerical simulations published in the literature. 2D simulations performed in the present work agree well with previously published data. Our 3D simulations capture effects neglected in the 2D case, showing excellent agreement with previously published experiments.
The good agreement with experimental data shows that the developed framework is suitable for simulation of industrial applications involving fluid structure interaction. If the structure is made of a highly elastic material, e.g. rubber, the simulation framework must be able to handle the large deformations that may occur. Immersed boundary methods are well suited for such applications, since they can efficiently handle moving objects without the need of a body-fitted mesh. Combining them with a structural mechanics solver for large deformations allows complex fluid structure interaction problems to be studied.
Keywordsfluid structure interaction immersed boundary methods sloshing tank FSI benchmark
Numerical simulations of highly elastic structures deforming in a free surface flow are challenging since the fluid-structure coupling is strong. The geometrically nonlinear response of the structure and the need to accurately resolve the free surface further increases the complexity of the simulations. The coupling between the fluid and the structure can be handled in different ways. A popular approach is the Arbitrary Lagrangian Eulerian (ALE) method , where the grid is deformed when the structure moves. Simulations with Smooth Particle Hydrodynamics (SPH) [2, 3] and Particle Finite Element Methods (PFEM)  are also reported in the literature. Immersed Boundary Methods (IBM) allow the flow around deforming objects in the flow to be resolved without the need of a body-fitted mesh. IBMs are therefore well suited for Fluid Structure Interaction (FSI) applications with large structural displacements. The original IBM developed by Peskin  was explicitly formulated and only first-order accurate in space. Majumdar et al.  developed a more stable method, which is implicitly formulated and second-order accurate in space. However, this method suffers from problems with mass conservation and pressure oscillations. To resolve these issues, Mark et al. [7, 8] developed a second-order accurate hybrid IBM. The IBM developed by Mark et al. has been validated for simulation of fiber suspension flows with elastic fibers in .
FSI simulations can be performed in a monolithic or a partitioned way. Using a monolithic approach implies that all equations are solved simultaneously in the same matrix. In the partitioned approach, the different equations are solved separately and coupling algorithms are employed. Using the partitioned approach without coupling iterations between the fluid and the structure solutions is attractive in terms of computational efficiency. However, this approach often results in instabilities due to the added mass effect if the simulation time is long enough . Gauss-Seidel iterations as well as quasi-Newton  techniques have been proposed to deal with these problems.
The aim of this paper is to present and validate a modeling framework that can be used for simulation of FSI in industrial applications. To achieve this, the partitioned approach with Gauss-Seidel iterations is used. The fluid-structure coupling is handled with the IBM developed by Mark et al.  and the structure is modeled as a St. Venant-Kirchhoff material, thus taking large deformations into account.
In the present work, a finite volume discretization on a Cartesian octree grid is used to solve the Navier-Stokes equations. A finite element discretization in total Lagrangian formulation is used to predict the motion of the structure. The fluid and structure models together with the FSI coupling are described in the following.
where is the oil density and is the air density. In this way the mass is conserved but the interface may be diffusive. Therefore, it is important to use a shock capturing convective scheme. Hence, in this work the shock capturing scheme CICSAM developed by Ubbink  is employed. To further reduce the diffusion of the interface and improve the computational speed adaptive grid refinements along the interface are employed. The Backward Euler scheme is used for the temporal discretization.
where F is the deformation gradient and . Large deformations can then be taken into account by using hyperelastic material models, that provide the relation between the second Piola-Kirchhoff stress and its work conjugate strain measure, the Green strain E.
where denotes the Kroenecker delta.
It is interesting to note that equation (10) is similar to the corresponding equation for linear elasticity, but in equation (10) the Green strain appears instead of the small strain tensor and the second Piola-Kirchhoff stress appears instead of the Cauchy stress. It should be mentioned that there are other material models for rubber, e.g. the Mooney-Rivlin model . Such models offer higher accuracy at large strains, but require more input data for calibration. In the cases considered in the present work, the strains remain relatively small and the St. Venant-Kirchhoff model is therefore sufficient.
The Finite Element Method (FEM) is used to discretize the equations governing the motion of the solid. Isoparametric basis functions are used and the integrals are evaluated with Gaussian quadrature. Full integration is used in the simulations in the present work. The nonlinear system of equations is solved with Newton’s method, so that asymptotic second order convergence in the iterations is achieved. Using Newton’s method requires computation of the consistent tangent stiffness matrix. This topic is well described in many books on FEM for structural mechanics, see e.g. [16, 17]. A pure displacement formulation is employed (in contrast to mixed formulations sometimes used, see e.g. ). Therefore, incompressible solid materials are modeled as nearly incompressible by setting Poisson’s ratio to a value close to, but not equal to, 0.5. Hexahedral elements with 20-nodes are used in the simulations in this paper. This element has nodes on the edge midpoints, but not on the face centers or in the element center. The basis functions for the 20-node hex element, as well as an interesting discussion on reduced integration for that element, are given in .
where is the velocity, is the acceleration and Δt is the time step length. γ and β are constants that control the accuracy and numerical dissipation of the scheme. Setting and gives the trapezoidal rule. The scheme is unconditionally stable for linear problems if .
FSI simulations can be performed in a monolithic or a partitioned way. Using a monolithic approach implies that all equations are solved simultaneously in the same matrix. In the partitioned approach, the different equations are solved separately and coupling algorithms are employed. In the present work, the partitioned approach is employed and the simulations are performed without coupling iterations when possible. Gauss-Seidel iterations are used when necessary for stability reasons.
In this work the mirroring IBM  is used to model the presence of moving solid objects, without the need of a body-fitted mesh. In the method the fluid velocity is set to the local velocity of the object with an immersed boundary condition. To set this boundary condition a cell type is assigned to each cell in the fluid domain. The cells are marked as fluid cells, internal cells or mirroring cells depending on the position relative to the IB . The velocity in the internal cells is set to the velocity of the immersed object with a Dirichlet boundary condition. The mirroring cells are used to construct implicit boundary conditions that are added to the operator for the momentum equations. This results in a fictitious fluid velocity field inside the immersed object. Mass conservation is ensured by excluding the fictitious velocity field in the discretized continuity equation. The result is a robust method that is second order accurate in space. A complete description of the method can be found in . The force exerted on the solid by the fluid is computed by numerically integrating the fluid traction vector over the fluid-solid interface.
Results and discussion
The tank has two holes in the upper wall, so that zero pressure can be prescribed there. When 2D simulations are performed, symmetry boundary conditions are used on the faces with normal in the y-direction and no slip conditions are enforced on the remaining walls. When 3D simulations are performed, no slip is enforced on all walls. The beam is clamped at the point A. When 2D simulations are performed, all nodes of the solid mesh are locked in the y-direction, leading to a plane strain assumption.
In the simulations, the gravitation vector was rotated instead of rotating the whole domain. The centrifugal forces, arising from the fact that the simulation is performed in an accelerating coordinate system, have been neglected. This is justified because the angular velocity of the motion is small. As will be seen, good results are obtained with this approximation.
The 2D simulations presented in Figure 6 were performed without coupling iterations. However, Gauss-Seidel iterations were used in the 3D simulation to get a stable solution.
A framework for simulation of highly deforming elastic structures in a two-phase flow is proposed and validated. The Navier-Stokes solver utilizes an immersed boundary method to efficiently handle moving geometries without the need of a body-conforming mesh and a volume of fluids method. The discretization is performed on an adaptive octree grid allowing grid refinements around the structure and the oil-air interface. By coupling the Navier-Stokes solver with a structural dynamics solver for large deformations a robust framework for three-dimensional fluid-structure interaction applications is realized. The good agreement with previously published data demonstrates the accuracy.
This work was supported in part by the Sustainable Production Initiative and the Production Area of Advance at Chalmers. The support is gratefully acknowledged.
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