Issue 29

A. De Rosis et alii, Frattura ed Integrità Strutturale, 29 (2014) 343-350; DOI: 10.3221/IGF-ESIS.29.30 344 mesoscopic point of view, where the flow behavior emerges by the space-time evolution of a function describing the density of a set of fictitious particles. These particles move upon an Eulerian grid which is kept fixed during the overall analysis. Among the possible advantages of the LB method over the macroscopic-based one, the most immediate is the greater computational efficiency, especially in interaction problems involving moving structures. It has been demonstrated that the solution given by the LB method is equivalent to the one of the Navier-Stokes equations for an incompressible flow with a second order of accuracy [3]. In order to account for the presence of a solid body immersed in the lattice fluid background, the Immersed Boundary (IB) method is adopted [4,5]. With respect to the well-consolidated bounce-back rule [6,7], such method is characterized by superior properties in terms of stability. According to a recent work carried out by the authors, it preserves a high level of accuracy and it leads to a quite general implementation of the numerical algorithm [8]. A Volume-Of-Fluid-based approach (VOF) [9,10] is used to take trace of the liquid-gas interface movements due to the mass advection. In particular, the model uses the VOF to update the nodes fill levels (in terms of mass) and a pressure boundary to restore the missing distribution functions at the liquid-gas interface. When moving structures are considered, the non-linear structure dynamics is computed by adopting the Time Discontinuous Galerkin (TDG) scheme. This choice over standard Newmark algorithms is motivated by its superior properties in terms of stability, accuracy and convergence [11-13]. In this paper, the LB, VOF, IB and TDG methods are combined within a staggered explicit coupling algorithm called FELBA (Finite Element Lattice Boltzmann Analysis), which has been validated by the authors for different applications, including mechanics [14,15], industry [16], flapping flight [17,18], and even shallow waters [19], among the others. In particular, the above ingredients are combined together in order to predict the flow physics arising in two different cases: a dam break and two solid structures obstructing the development of a flow in a channel. Structures are idealized by Euler-Bernoulli finite elements capable of undergoing large displacements, according to the corotational formulation [20,21]. The accuracy of the coupling algorithm is evaluated in terms of the energy that is artificially introduced in the system at the fluid-structure interface. Here, scenarios characterized by different values of the Reynolds number are investigated. Findings in terms of the three components of the tip displacement are given for the above-mentioned configurations, together with considerations about the artificially introduced interface energy. As regards the simulation of the dam break phenomenon, findings from a numerical analysis are compared to literature results, showing the effectiveness of the proposed approach. P ROBLEM STATEMENT he flow physics is computed by solving the LB equation, that is f i ( x + c i  t, t+  t) = f i ( x ,t) +  [f i ( x ,t)-f i eq ( x ,t)] with i =0,…,8, (1) where f i are the particle distribution functions, x is the position, t is the time,  is the relaxation frequency,  t is the time step and c i are the lattice vectors in the adopted D2Q9 model [2]. Notice that the relaxation frequency is strictly related to the fluid viscosity, i.e.  =(1/  -0.5)/3. The equilibrium distribution functions f i eq are computed as a second-order expansion in the local Mach number [2]. The presence of an immersed body is accounted for by adding at the right-hand side of Eq. (1) the quantity 3 c i w i g i ( x ,t) , where g i ( x ,t) is a term computed via an implicit velocity-correction based IB method [22]. Once the corrected Eq. (1) is solved, the macroscopic density  and fluid velocity v are computed as  =  i f i , v =  i f i c i /  (2) On the other hand, deformable solids are idealized as corotational beam finite elements. The resultant non-linear equation of the solid motion is integrated in time by adopting the TDG scheme according to the implementation proposed in [13]. As discussed in [16], the TDG scheme has shown superior properties with respect to the adoption of standard Newmark algorithms for solving fluid-structure interaction problems. As mentioned above, in order to handle fluid-structure interface condition, the IB method is adopted. In this way, the solid body is represented by a set of Lagrangian points, independent from the Eulerian fluid mesh, thus allowing a quite general implementation of the algorithm of computation. If a free surface flow is considered, the difference in terms of density and viscosity between the two phases composing the system is usually very high. Thus, it is possible to affirm that the behavior of the overall system is governed by the phase with the higher density [9]. The phase characterized by the lower density is considered only as a surface boundary condition, thus it is neglected in the rest of the domain. The adopted free surface model represents a combination of the LB and VOF methods, the latter being responsible of the interface tracking. This method requires a particular treatment T