A. L. Braun and A. M. Awruch Numerical Simulation of the Wind A. L. Braun Action on a Long-Span Bridge Deck PPGEC/UFRGS A numerical model to study the aerodynamic and aeroelastic bridge deck behavior is Av. Osvaldo Aranha, 99 – 3o andar presented in this paper. The flow around a rigid fixed bridge cross-section, as well as the 90035-190 Porto Alegre, RS. Brazil flow around the same cross-section with torsional motion, are investigated to obtain the www.cpgec.ufrgs.br aerodynamic coefficients, the Strouhal number and to determine the critical wind speed [email protected] originating dynamic instability due to flutter. The two-dimensional flow is analyzed employing the pseudo-compressibility approach, with an Arbitrary Lagrangean-Eulerian (ALE) formulation and an explicit two-step Taylor-Galerkin method. The finite element method (FEM) is used for spatial discretization. The structure is considered as a rigid A. M. Awruch body with elastic restrains for the cross-section rotation and displacement components. PPGEC/UFRGS & PROMEC/UFRGS The fluid-structure interaction is accomplished applying the compatibility and equilibrium Av. Sarmento Leite, 425 conditions at the fluid-solid interface. The structural dynamic analysis is performed using 90050-170 Porto Alegre, RS. Brazil the classical Newmark’s method. www.mecanica.ufrgs.br/promec Keywords: Fluid-structure interaction, Finite Element Method (FEM), Large Eddy [email protected] Simulation (LES), aeroelasticity, aerodynamics explicit two-step Taylor-Galerkin method with an Arbitrary Introduction Lagrangean-Eulerian (ALE) description. A similar Taylor-Galerkin formulation was used by Tabarrok & Su (1994) and by Rossa & Wind tunnel tests for assessment of aerodynamic and aeroelastic Awruch (2001), but with a semi-implicit scheme. The (ALE) scheme informations in the study of bridge girders performance are was first presented by Hirt et al. (1974) in a numerical work. Since numerically simulated in this work. The usual way to obtain these this first paper many other authors used this description with the informations is using representative models in a wind tunnel. same concepts. The classical Smagorinsky’s model, similar to that However, with the improvement in computers technology and presented by Kuroda (1997), was employed for the sub-grid scales computational fluid dynamics (CFD) algorithms, many of these simulation. The finite element method was used for spatial problems can also be analyzed by numerical simulation.1 discretization. The structure was considered as a rigid body with Long-span bridges, such as suspension bridges for example, elastic restrains for the cross-section rotation and displacement must be designed to support, from a static point of view, the mean components. The coupling between fluid and structure was wind forces (using the drag, lift and pitching moment coefficients). performed applying the compatibility and equilibrium equations at Besides, considering that such structures show low damping and the interface. The structural dynamic analysis was accomplished low stiffness, they are subjected to aeroelastic phenomena, such as using the classical Newmark’s method (Bathe, 1996). Examples are flutter, galloping and vortex shedding induced vibrations. Only the presented to illustrate the capability of the computational method. first case will be studied in the present work. The term aeroelasticity is used when the aerodynamic forces Governing Equations for the Flow Simulation produce some kind of structural instability as a consequence of the interaction between these forces and the structural motion. The The governing equations, considering the pseudo- flutter phenomenon is a type of aeroelastic instability that begins compressibility approach in an isothermic process, Large Eddy when the effective damping (structural + aerodynamic) becomes Simulation (LES) with Smagorinsky’s model for turbulent flows and negative. an Arbitrary Lagrangean-Eulerian (ALE) description, are: Kawahara & Hirano (1983) were one of the first authors to a) Momentum equations: analyze numerically the wind action on a bridge cross-section. They used the Finite Element Method (FEM) to obtain the aerodynamic ∂v ∂vi ∂vi 1 ∂p ∂ j ∂vi ∂vk coefficients as functions of the angle of attack of the wind and the + ()v j − wj + δij − ()ν +ν t + + λ δij = 0 ∂t ∂x ρ ∂x ∂x ∂x ∂x ∂x Strouhal number. Kuroda (1997) employed two different numerical j j j i j k procedures to study the approaching span of the Great Belt East (i, j, k = 1, 2) in Ù (1) Bridge: the Finite Element Method and the Finite Difference Method (FDM), and Large Eddy Simulation (LES) with the ∂v 2 1 1 ∂vi j Smagorinsky’s model for the turbulent flow. He also presented the being νt = ()CS ∆ ()2Sij Sij 2 with Sij = + and 2 ∂x ∂x results referring to the aerodynamic coefficients for various angles j i 1/2 of attack and the Strouhal number for both numerical methods Ä = (Äx Äy ) , where Äx and Äy are the element dimensions in the employed in his analysis. Larsen & Walther (1997) analyzed several global axis direction x and y, respectively. bridge decks observing their aeroelastic behaviour using a numerical b) Mass conservation equation: code based on Discrete Vortex Simulation (DVS), presenting the respective critical flutter velocity (using the flutter derivatives). ∂v Recently, (2002) applied a direct method for the flutter ∂p ∂p 2 j Selvam et al. + ()v j − wj + ρc = 0 (j = 1, 2) in Ù (2) analysis. They used (FEM) and (LES). ∂t ∂x j ∂x j In this work, the analysis of the flow of a slightly compressible fluid in a two-dimensional flow domain was carried out using an p 2 which is obtained considering ∂ = c . ∂ρ Paper accepted August, 2003. Technical Editor: Aristeu da Silveira Neto.. The boundary conditions of Eqs. (1) and (2) are the following: 352 / Vol. XXV, No. 4, October-December 2003 ABCM Numerical Simulation of the Wind Action on a Long-Span Bridge Deck v = w (i = 1, 2) on the solid boundary à (3) n+ 1 i i vS 2 4) Calculate vi with: v = vˆ on the boundary Ãv or 1 a 1 1 2 n+ 2 n+ 2 ~ n+ 2 1 Ät ∂Äp p = ˆp on the boundary Ãp (4) vi = vi − (9) ñ 8 ∂xi ∂v ó n n 1 n − p ∂vi j ∂vk ij j 5) Calculate v + v Äv with: äij + ()í + í t + + ë n j = = Si i = i + i ñ ∂x ∂x ∂x ñ j i k (i,j,k=1,2) in à (5) ∂v ó ∂vi 1 ∂p ∂ ∂vi j Ävi = Ät− rj − äij + í + + ∂x ñ ∂x ∂x ∂x ∂x j j j j i In these equations, vi and p (the velocity components and the (10) 1 n+ 2 pressure, respectively) are the unknowns. The viscosities í ì = ñ ∂vk ë äij ∂xk and ë ÷ , the specific mass ñ and the sound velocity c, are the = ñ ì n+1 n fluid properties. The eddy viscosity í t depends of 6) Calculate p = p + Äp with: t = ñ derivatives of the filtered velocity components, of the element n+ 1 dimensions and of the Smagorinsky’s constant C . For a purely ∂v 2 S ∂p 2 j Äp = Ät− rj − ñc (11) Eulerian description, the mesh motion velocity w at each nodal ∂x ∂x j j point, with components wi , is equal to zero. Now, for a purely Lagrangean description, the mesh motion velocity at each nodal These expressions must be employed after applying the classical point is equal to the fluid velocity, i.e. vi = wi (i = 1, 2). Finally, in Galerkin technique into the finite element method (MEF) context. an Arbitrary Lagrangean-Eulerian formulation, w ≠ 0 and w ≠ v . As the scheme is explicit, the resulting system is conditionally On the boundaries à and à , prescribed values for velocity stable, with a stability condition given by: va p and pressure, vˆ and ˆp , respectively, must be specified, while on Äx Ät < á i (i = 1,..., NTE) (12) ˆ i c v Ãó the boundary force t must be in equilibrium with the stress + i tensor components ó . In Eq. (5), n is the direction cosine ij j where á (which is a real number less than one) is a safety between a vector perpendicular to à and the axis x . ó j coefficient, Äxi and vi are the i-th element characteristic Initial conditions for the pressure and the velocity components at dimension and the velocity, respectively, and NTE is the total t = 0 must be given. number of elements. Although variable time step could be adopted (Teixeira & a The Algorithm for the Flow Simulation Awruch , 2001), in this work an unique value of Ät will be used for the whole process, adopting the smallest one from those obtained by Expanding the governing equations in a Taylor’s series up to Eq. (12). second order terms, the algorithm for the flow simulation contains the following steps (Braun, 2002): 1 The Fluid-Structure Coupling ~ n+ 2 1) Calculate vi with: In the present work, the structure is idealized as a two- dimensional rigid body. Displacement and rotations take place on 1 the plane formed by the axis x1 and x2; the body is restricted by ~n+ 2 n Ät ∂vi 1 ∂p vi = vi + − rj − äij + dampers and springs, as indicated in Fig. 1. 2 x ñ x ∂ j ∂ j n (6) ∂v 2 n ∂ ∂vi j ∂vk Ät ∂ vi í + + ë äij + rjrk ∂x ∂x ∂x ∂x 4 ∂x x j j i k j k whererj = (v j − w j ) and v = (ν +ν t ) . n+ 1 2) Calculate p 2 with: n 1 ∂v 2 n n+ 2 n Ät ∂p 2 j Ät ∂ p (7) p = p + − rj − ñc + rirj 2 ∂xj ∂xj 4 ∂xj∂xi Figure 1.