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Recent advances in EdgeCFD on wave-structure interaction and turbulence modeling

Renato N. Elias 1, Alvaro L. G. A. Coutinho 1, Milton A. Gonçalves Jr. 1, Adriano M. A. Cortês 1, José L. Drummond Alves 2, Nestor O. Guevara Jr. 2, Carlos E. Silva 2, Bruno Correa 2, Fernando A. Rochinha 3, Gabriel M. Guerra Bernadá 3, Erb F. Lins 3 and Daniel F. de Carvalho e Silva 4

1 High Performance Computing Center, COPPE/Federal University of Rio de Janeiro, Rio de Janeiro, RJ, Brazil

2 Laboratory for Computational Methods in Engineering, COPPE/Federal University of Rio de Janeiro, Rio de Janeiro, RJ, Brazil

3 Mechanical Engineering Program, COPPE/Federal University of Rio de Janeiro, Rio de Janeiro, RJ, Brazil

4 PETROBRAS Research Center Rio de Janeiro, RJ, Brazil

AbstractThe computation of free surface flows is challenging since waves are highly nonlinear and commonly present merging, frag-mentation and cusps, leading to the use of interface capturing ALE approaches. Turbulence is also important on several offshore applications. In this work we report recent advances of EdgeCFD to simulate these problems. EdgeCFD is a fully implicit 3D incompressible stabilized parallel edge-based finite element flow solver associated to the Volume-of-Fluid (VoF) method. The VoF marking function is also solved by a fully implicit parallel edge-based finite element formulation. Turbulence in EdgeCFD is treated within the Large Eddy Simulation (LES) framework. The performance and accuracy of EdgeCFD is tested in the simu-lation waves and in the interaction between waves and a semisubmersible structure. We also access the performance of several LES models on the simulation of the flow around a cylinder at Reynolds 3,900.

Keywords Turbulence, free surface flows, wave simulation, stabilized finite element method, large eddy simulation.

1 Introduction

Fuel or water sloshing in tanks, waves breaking in ships, offshore platforms motions, wave action on harbors and coastal areas are some of the problems in hydrodynamics in which complex flows involving waves and free surface occur. The main computational challenge when solving such highly nonlinear problem is determining the evolution of the water/air interface location. There are a large number of numerical methods devoted to the computation of free-surface problems. These methods are frequently classified as interface tracking and interface capturing methods.

Vol. 9 No. 1 pp. 49-58 June 2014 Marine Systems & Ocean Technology 49

Manuscript submitted to MS&OT on August 05, 2013. Accepted on February 03, 2014. Editor: Marcelo A. S. Neves. Article posted online on August 05, 2013 on August 26, 2014: URL: www.sobena.org.br/msot/volume.htm .

Interface tracking methods are based on a Lagrangian framework where the moving interface or boundary is explicitly tracked by the computational grid or by the particles of meshless methods which must be deformed or moved in order to follow the fluid flow. The deforming-spatial-domain/stabilized space-time finite element formulation (DSD/SST) proposed by Tezduyar et al. (1992, 2000) and Tezduyar (2001) is a mesh-based example of interface tracking method. Particle methods, such as those of Violeau and Issa (2007) and Koshizuka et al. (1995) are examples of smoothed particle hydrodynamics (SPH) methods to the simulation of free-surface problems. However, these methods still present a high computational cost since they need to compute the interaction between the particles using search algorithms. As a compromise between the advantages offered by mesh based and meshless methods, Del Pin et al (2007) introduced the particle finite element method (PFEM) for free-surface flows. In this method the critical parts of the continuum are discretized with particles while the remaining parts are treated by a Lagrangian finite element formulation. Another technique mixing Lagrangian and Eulerian flavors was proposed in Takizawa et al., 2006). In this work the authors enhanced the Constrained Interpolation Profile method (CIP) for solving hyperbolic equations with a meshless Soroban grid. The resulting formulation was used to treat fluid-object and fluid-structure interaction in the presence of free-surfaces.

As a cost effective alternative to interface tracking methods, interface-capturing methods have emerged. Interface capturing methods are Eulerian in their concept, thus they rely on a unique and fixed computational grid to capture the interface evolution. In this class of methods the interface is represented by a scalar function which marks the regions filled with the fluids involved. In other words, the interface position is implicitly captured in a scalar marking function value and the interface evolution is determined by the additional cost of solving an advection equation for the marker. As opposed to interface tracking methods, interface-capturing methods require little effort to represent all complicated features of moving interfaces. Additionally, the parallel implementation and post-processing of interface capturing methods are straightforward. The main drawback of interface capturing methods is the need to average the fluid properties at the interface cells (elements) due to the discontinuity of the Eulerian representation of the interface. Moreover, the accuracy and computational cost of interface capturing methods are typically associated to grid resolution, properties of the marking function chosen to represent the interface and numerical methods for solving the fluid flow and marking function advection. The well-known volume-of-fluid scheme (VoF), firstly proposed by Hirt and Nichols (1981) for Cartesian grids, is an interface capturing technique that employs a step function ranging from 0 to 1 to represent the fraction of fluid within the grid cells. In this sense, the partially filled cells represent the interface. The main issues associated to VoF methods include the difficulty in advecting a discontinuous step function and the accurate modeling of surface tension effects. Level set methods (Sethian, 1999) implement free-surface flows in a different manner than VoF by changing the marking function employed to represent the interface. Therefore, the fluids are associated to the range of the distance function signs while the interface is implicitly represented by the zero level set. However, the level set method suffers when the distance

function loses its properties and must be rebuilt. In fact, the success of level set method lies in its ability of building and keeping a signed distance function without losing its properties. The enhanced-discretization interface-capturing method, firstly proposed by Tezduyar et al. (1997) and the work of Lohner et al. (2006), is an example of unstructured grid formulations based on the finite element method to solve free-surface flows using a VoF marking function.

Another important issue in flow simulation is the accurate prediction of turbulent structures. The Navier-Stokes equations represent the mathematical model for both the laminar and turbulent flows and, despite the increasing computational power available, the complete resolution of this equation, in all time and spatial scales of turbulent flow, is a challenging task. For high Reynolds number, the Large Eddy Simulation (LES) is a more affordable choice. In this approach, the flow scales are decomposed into large and fine scales. The large scales represent the flow structures that are well approximated by the time and spatial discretization used. The structures or fluctuations that cannot be adequately represented by the discretization should be modeled and their effect accounted in the large-scale solution. This approach is well suited for turbulent flow since large scales has a great dependence of global geometry and the small scales are, usually, homogeneous, localized and independent of large flow structures.

In the LES context, the Residual Based Variational Multiscale method (RB-VMS) has emerged as a choice to the modeling of the subscale information (Gravemeier et al., 2007, Hughes, 2004). In this model, the equation for the small scales is obtained from a variational formulation of Navier-Stokes equations. The fine scales model will result as directly related to large-scale residua. The model obtained this way is consistent: once the large scales adequately represent all scales of flow, the residua will vanish and the fine scales model has no effect on the flow. Another great advantage of this model is its clear analogies with standard stabilized methods for advection-dominated flows. This analogy allows a straightforward extension of computer codes already implemented within the stabilized framework (Lins et al., 2009, Lins et al., 2010).

In this work we use our VoF edge-based solver (Elias and Coutinho, 2007), to deal with complex wave phenomena. The main characteristics of our solver are: Streamline-Upwind/Petrov-Galerkin (SUPG), Pressure-Stabilizing/Petrov-Galerkin (PSPG) and Least Squares Incompressibility Constraint (LSIC) stabilized finite element formulation; implicit time marching scheme with adaptive time stepping control; advanced Inexact Newton solvers; edge-based data structures to save memory and improve performance; support to message passing and shared memory parallel programming models; and large eddy simulation extensions using either the classical Smagorinsky model or RB-VMS Also, we extend our code to include a Residual Based Eddy Viscosity Model in addition to the already implemented RB-VMS LES solver.

The remainder of this paper is organized as follows. The next section presents the incompressible flow, LES modeling, the interface-capturing governing equations and the implementation of a relaxation zone, respectively; in the following, we show

50 Marine Systems & Ocean Technology Vol. 9 No. 1 pp. 49-58 June 2014

Recent advances in EdgeCFD on wave-structure interaction and turbulence modelingRenato N. Elias, Alvaro L. G. A. Coutinho, Milton A. Gonçalves Jr., Adriano M. A. Cortês, José L. Drummond Alves, Nestor O. Guevara Jr., Carlos E. Silva,

Bruno Correa, Fernando A. Rochinha, Gabriel M. Guerra Bernadá,Erb F. Lins and Daniel F. de Carvalho e Silva

results obtained for the simulation of three specific progressive waves and the interaction of the progressive waves with a fixed semisubmersible structure. Our results have been compared with theoretical and experimental data. In this work, the movement of a piston-like wave generator generates the waves. Finally, the LES results are compared with experimental data available. The final remarks and conclusions are summarized in the last section.

2 Governing equations and finite element formulation

2.1 Incompressible fluid flow

Let be the spatial domain, where nsd is the number of space dimensions. Let denote the boundary of W. We consider the following velocity-pressure formulation of the Navier-Stokes equations governing the incompressible flow of two immiscible fluids within an Arbitrary Lagrangian-Eulerian frame (Donea and Huerta, 2003):

(1)

(2)

where r and u are the density and velocity, umesh is the mesh velocity, f is the body force vector carrying the gravity acceleration effect and s is the stress tensor given as

(3)

where p is the pressure, I is the identity tensor, T is the deviatoric stress tensor

(4)

and e(u) is the strain rate tensor defined as

(5)

The essential and natural boundary conditions associated with Eqs.1, 2 can be imposed at different portions of the boundary and represented by,

(6)

(7)

where g and h are complementary subsets of . The above equations are discretized in space by either a stabilized finite element method (Tezduyar and Sathe, 2007), or a Residual-Based Variational Multiscale Method (Lins et al. 2009, 2010). In both cases edge-based data structures are utilized.

2.2 Large eddy simulation

In the present work a large eddy simulation (LES) approach to turbulence is considered by the use of three models. The first one is the classic Smagorinsky (1963) turbulence model. In this model, the viscosity m is augmented by a subgrid-scale viscosity mSGS proportional to the norm of the local strain rate tensor and to a filter width h defined here as the cubic root of the element volume,

(8)

where CS is the Smagorinsky constant, taken as 0.1.

The two other models use the Variational Multiscale Formulation (Gravemeier et al., 2007). In this framework, it is assumed that the velocity and pressure fields have the decomposition in large and small scales as shown in Eqs. 7~8,

(9)

(10)

This can be seen as a direct decomposition in the approximation spaces used in the variational formulation. In this way, the coarse scale solution uh can be associated with the finite element (FEM) approximation, obtained by the discretization of the spatial domain in elements with characteristic size h. In Bazilevs et al. (2007) the model for subgrid velocity u’ is derived from the variational FEM solution and has algebraic form given as

(11)

where ResM (uh, ph ) is the residua of Navier-Stokes momentum equation and ResC (uh, ph ) is the residua of continuity equation. This form has a very deep connection with stabilized methods (Gravemeier et al., 2007, Hughes, 2004). In fact, it was shown in Lins et al. (2009, 2010) that a stabilized FEM could, with minor modifications of computer code structure, be adapted to incorporate this LES model. However, it was observed in Wang and Oberai (2010) that the RB-VMS formulation, despite accurately representing the cross-stress terms (uhu’), under predicts the contribution of Reynolds stress terms (u’u’). With this issue in mind, an alternative LES model was addressed in Oberai et al. (2012) and Liu (2012) and tested using spectral codes in incompressible and compressible turbulent flows. In this model, the concept of eddy viscosity is added to the RB-VMS formulation by the use of

(12)

In the above equation we clearly see that the turbulent viscosity depends directly of the subgrid velocity u’. The constant Cs is not arbitrary; we can calculate it by computing the model dissipation and equating it with the total dissipation, considering homogeneous isotropic turbulence (in this paper we set this value as Cs = 0.15

(Oberai et al., 2012)). Note that this model is inherently dynamic and consistent, as it vanishes when the coarse scales adequately represents the velocity and pressure fields (u’ = 0). When used with RB-VMS it gives better results than the former only (Liu, 2012). Through this paper, we refer to this model as the Residual Based Eddy Viscosity Model (RB-EVM).


Recent advances in EdgeCFD on wave-structure interaction and turbulence modelingRenato N. Elias, Alvaro L. G. A. Coutinho, Milton A. Gonçalves Jr., Adriano M. A. Cortês, José L. Drummond Alves, Nestor O. Guevara Jr., Carlos E. Silva, Bruno Correa, Fernando A. Rochinha, Gabriel M. Guerra Bernadá,Erb F. Lins and Daniel F. de Carvalho e Silva

2.3 Interface capturing

In volume-of-fluid (VoF) method (Hirt and Nichols, 1981), a scalar marking function can be employed to capture the position of the interface between the fluids by simply using the fluids fraction relationship.

The volume-of-fluid can be stated as: assuming the value 1 in regions filled with fluid A, e.g., water, and the value 0 in regions filled with fluid B, e.g., air, the position of the fluid interface will be defined by the isovalue contour f (x,t) = fc , where fc

[0,1] . The value fc = 0.5 is usually assumed. Finally, the function f (x) driven by a velocity field u, satisfies the following transport equation, given in conservative form, as

(13)

In the volume-of-fluid (VoF) formulation the fluid density and viscosity, employed in the fluid flow solution, are interpolated across the interface as follows:

(14)

(15)

where subscripts A and B denote the values corresponding to each fluid. Eq. 16 is discretized by Streamline-Upwind/Petrov-Galerkin formulation with discontinuity capturing as in Brooks and Hughes (1982) and Galeão and do Carmo (1988). To avoid non-physical results, values lying outside the range [0,1] are truncated with the following function:

(16)

A complete description of the stabilized finite element method and the solution procedure used in this work are shown in Elias et al. (2008).

2.4 The piston-like wave generator

The working scheme of the wave generator can be viewed in Fig.1

Fig. 1 Working scheme of a piston-like wave generator. Lateral view.

The wave generator is simulated by its defining boundary (GC ) movement. In this plane the description of the problem is purely Lagrangian. The piston-like wave generator moves in a single direction (normal to the face) and the generated wave propagates itself in this same direction. The generator’s stroke is given by 2A, where A is the amplitude of the piston’s displacement. The velocity of the fluid particles over the wave generator is the temporal derivative of the prescribed

space function of the nodes that belong to the piston, which is a boundary condition of the problem.The mesh in the ALE region, WALE , is movable. In this region, all the nodes move in the same direction of the wave generator, gradually, in the sense of the nodes over the interface GI do not move. This is done to avoid mesh entangling. Thus, nodes closer to the wave generator suffer a larger displacement than the ones closer to the interface GI . In the region WE the mesh is fixed and the description of the problem is purely Eulerian. The displacement of the nodes belonging to the interface GC and to the region WALE is given by an oscillatory temporal function. The damping effect is produced by an exponential factor. A periodic function at the piston plane is prescribed, simulating a piston-type wave generator, imposing the following movement to the nodes:

(17)

(18)

(19)

and

(20)

(21)

(22)

where x, y and z are the position coordinates of the nodes; w and T are the piston angular frequency and its period, respectively, while t refers to the time and j is the phase angle. Also, r0 = r (t - dt), where dt is the time step.

2.5 Relaxation zone

It is useful to create computational tanks to simulate and evaluate wave problems. Every time a generated wave hit one side of the tank, defined as a solid wall, the interaction between the wave and the wall reflects the incoming wave. The reflected wave can interfere with the next incoming wave or even with the wave generator. One way of avoiding this is to introduce a relaxation zone, implemented to diminish and retard the reflection effect that occurs when the incoming waves reach the opposite side of tank.

The numerical implementation of the relaxation zone is based on Jacobsen et al (2012). The method requires the definition of a region that damps the incoming waves, playing the role of a numerical beach. This can be accomplished by applying the following damping functions to the velocity field,

(23)

(24)

where fi = u,v,w are the velocity components. For all the simulations we take a = 1.5. The function aR is the damping suffered by the variable fi . The velocity at the end of the relaxation zone, fitarget

, is null. The function cR is defined simply as:




(25)

The function cR is linear and serves to restrict the domain of action of Eq. 29. It is defined in the wave propagation direction. The coordinates xi and xf are respectively the relaxation zone beginning and end. Fig. 2 illustrates the behavior of function aR with respect to cR.

Fig. 2 Behavior of aR with respect to cR.

3 Test problems

3.1 Wave calibration

This case represents the simulation of production and propagation of three progressive regular waves, from a piston-type wave generator. The computational domain is presented in Fig. 3. It has 400 meters length, 120 meters height and 150 meters width. Fig. 3 also presents the initial condition.

Fig. 3 Initial condition. L = 400m, h = d = 60m and w = 150m.

The interface GI , which limits the ALE region in the right side, is 100m from the wave generator placed on face GC . All the nodes of in this sub-domain (including faces GC and GI ) move according to Eqs. 16-18 and, consequently, the generated wave propagation is in the longitudinal direction x. The fluid properties are: rwater = 1000kg / m3, mwater = 0.01kg / (ms), rair = 1kg / m3 and mair = 0.001kg / (ms). Viscosities are considered higher than they really are to keep the relation between non-dimensional groups and to avoid unstable regions in the air phase.

The three waves used in the tests have the periods and heights listed in Table 1, from Iwanowski et al. (2009).

Table 1 Properties of the three waves used in the tests.

InTable 1, T is the wave period and H is the wave height. The linear wave theory (Journée and Massie, 2001), acceptable for those waves, gives the relation between the piston movement and the generated wave by:

(26)

where

(27)

and s is the stroke, za is the wave elevation and Note that H = 2za. The wave number k is defined as:

(28)

with l is the wave length. The amplitude of the wave generator and properties of the expected generated waves are shown in Table 2.

Table 2 Wave generator amplitude and properties of the expected waves.

The determination of the wavelengths is based on the linear wave theory. The periods of the wave generator and of the generated wave are assumed to be the same. Amplitude and period of the wave generator are input data to the solver, as well as the phase angle j and the size of the ALE region.

The no-slip condition is prescribed on the seabed. In the sides of the domain we have slip conditions. In the top of the domain the pressure is null, simulation an opening to the atmosphere. In the wave generator (inlet), are prescribed the piston velocity components. In the opposite side (outlet) the velocity components are null, like a closed box. In addition, we implemented the relaxation zone close to the outlet. In the presented cases, the size of the relaxation zone is 100m.

The time step is fixed to 10-2 seconds and the simulation time is 60 seconds. Linear GMRES tolerance is set to 10-5 for both the Navier-Stokes equations and scalar marking function equation. Non-linear convergence tolerance is set to 10-3. The mesh has 208,354 nodes and 1,222,489 tetrahedral elements. It was observed that with this set-up the Navier-Stokes non-liner flow solver converges usually in 4 iterations while the scalar marking function equation converges in 6 iterations. Fig. 4 presents some snapshots of the generated medium wave.



Fig. 4 Generation and propagation of the short wave. Free surface elevation at (a) t = 10s, (b) t = 20s, (c) t = 30s and (d) t = 40s.

Table 3 summarizes the wave properties obtained from the numerical simulations, where we can see that good agreement with the linear theory is obtained for all three waves.

Table 3 Numerical properties of the three waves.

Fig. 5 epicts for all three cases the numerical results obtained for the spectral density of the generated waves.

Fig. 5 Numerical results. Spectral Density x Frequency.

Note that the maxima of the functions in Fig. 5 occur at the matching frequency of each one of the waves (f = 1/T ).

3.2 Impact over a fixed semi-submersible structure

This case refers to the interaction of a wave with a fixed semi-submersible structure placed at the center of the domain already used in the prior simulations. The mesh here has 222,205 nodes and 1,295,949 tetrahedral elements.

One of the main objectives of the development of this work is to deal with green water problems. Thus, viscous effects in the presented example are important in the interaction between the incoming water mass and the solid structure. Although we have good approximate solutions obtained based on potential theory, the Navier-Stokes equations are more suitable to address these issues. All the fluid properties were kept. Fig. 6 shows the semi-submersible structure, taken from Bellezi and Cheng (2012).

Fig. 6 Semi-submersible structure.

The simulation time for all cases is 50 seconds. Time step and the same computational set-up are the same as in the previous examples. Fig. 7 shows snapshots for the interaction between the long wave and the semi-submersible structure.




Fig. 7 Interaction between the long wave and the semi-submersible structure. Free surface at (a) t = 15s, (b) t = 20s, (c) t = 25s and (d) t = 30s.

Three pressure sensors, again as in Bellezi and Cheng (2012), are placed on the first column, as shown in Fig. 8

Fig. 8 Pressure sensors placed on 1st column of the semi-submersible structure.

The results provided by the pressure sensor on the base of the first column are presented in Fig. 9

Fig. 9 Impact over the semi-submersible structure. Medium wave. Pressure on the base of first column.

The wave run-up at the first column is in Fig. 10

Fig. 10 Interaction between the medium wave and the semi-submersible structure. First column run-up.

It is important to remark that the profile in Fig. 10 and the minimum and maximum values in the plots agree reasonably well with the results in Iwanowski (2009), Bellezi and Cheng (2012).

3.3 Large Eddy Simulation of flow over cylinder at Re=3,900

In this test, the RB-EVM is applied to the solution of the turbulent flow around a circular cylinder at . The formulations compared are: SUPG/PSGP/LSIC stabilization with LES using Eq. 12 called RB-EVM, SUPG/PSGP/LSIC stabilization with LES using Eq. 8, dubbed LES-Smagorinsky and the Residual Based Variational Multiscale Method using equation (Lins et al., 2010), RB-VMS.

None of the methods above uses any form of wall damping or functions to compute flow quantities near solid boundaries. Our objective is comparing all of the methodologies in the most straightforward way.

The dimensions of the flow domain are: height 8D, width 50D and length 50D, where D is the cylinder diameter. The cylinder was displaced 25D from the inflow region along the centerline of the domain. The mesh used is shown in Fig. 11. The mesh has 4,000,896 linear tetrahedral and 692,208 nodes.



Fig. 11 Cylinder in cross flow problem. Top left: Flow domain. Top Right: Full Mesh. Bottom: detail of mesh refinement around the cylinder.

The boundary conditions are set as follows: no-slip condition at the cylinder surface, u = 1 at domain inlet, zero-normal velocity and zero-shear stress at the lateral boundaries, traction-free conditions at the outflow boundary. The numerical solution parameters are: constant time step of 0.01 with a total number of steps equal to 20,000; the nonlinear loops are stopped after a decrease of three orders of magnitude of the relative and absolute residua. The maximum tolerance for the Inexact Newton solver is set to 0.1. The number of Krylov vectors for the nodal block-diagonal preconditioned GMRES solver is 35.

Results are expressed for the drag and lift coefficients in Fig. 12. Note that the two formulations based on residua exhibit characteristics typical of vortex shedding in turbulent flow and are alike, both in frequency. The random variation typical of turbulent flows also appears to be captured by the two methodologies. Furthermore, the fluctuations of this coefficient are broader than in the LES-Smagorinsky and RB-VMS method. Since this method adds little momentum diffusion, the gradients of the velocity field in the boundary layer on the cylinder surface are higher, which results in a higher drag. The small dissipation introduced by this method also reduces the fluctuations in the wake region of the cylinder.

Fig. 12 Drag (above) and Lift coefficient for RB-EVM, RB-VMS and Smagorinsky LES models.

In Table 4 it is shown the computed mean integral for different methods, specifically the values of drag coefficients and the Strouhal number. As can be seeing the values for the RB-VMS method show good agreement with the experimental results. The RB-EVM also can predict fairly well the Strouhal number.

Table 4 Computed mean integral quantities for different methods

In Figs. 13 and 14, velocity, pressure and eddy viscosity fields for all three formulations in a time step are shown. For other time, instants a similar behavior is observed. As can be seen, the fluctuations are very similar. Note the damping of fluctuations in the wake region due to increased diffusion in LES-Smagorinsky. The use of RB-EVM adds much less eddy viscosity to flow as can be noted in Fig.14.

Fig. 13 Velocity magnitude (above) and Pressure field at time t = 150




Fig. 14 Eddy Viscosity field at time.

3 Conclusions

In this work we have given a demonstration of recent developments incorporated into our incompressible flow solver for problems involving wave interactions with moving physical boundaries. We have simulated waves generated from a piston-like wave maker and their interactions with a semi-submersible structure. We also have implemented a relaxation zone close to the outflow to diminish and retard the reflection effect of the incoming waves. The main conclusions are that some important parameters in wave generator movement and in the wave behavior, as wavelength and height, are in good agreement with linear wave theory. In the problem of the wave interaction with a semi-submersible structure, we have computed the pressure over the 1st column of the structure. We noticed the numerical run-up on the 1st column presents a similar profile as found in other works. We have shown the implementation of a LES Model based on the Residual Based Variational Multiscale method and the Eddy Viscosity approach. For the turbulent flow around a cylinder at Reynolds 3,900, this model has a performance equivalent to the formulation already implemented (RB-VMS). We also have shown that it can be easily implemented as a consistent LES model into existing FEM codes. Subsequent tests, considering different degrees of mesh refinement, comparison with others experimental data and flow geometries must be performed in order to more accurately evaluate the performance and accuracy of this formulation on unstructured grid codes.

Acknowledgement

The authors would like to thank the financial support of the PETROBRAS Research Center and the MCT/CNPq Brazilian Council for Scientific Research. The High-Performance

Computing Center (NACAD) at the Federal University of Rio de Janeiro provided the computational resources for this research.

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Recent advances in EdgeCFD on wave-structure interaction ... paper4.pdf · Recent advances in EdgeCFD on wave-structure interaction and turbulence ... by the zero level set. However

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