PSI - Issue 8

N. Di Domenico et al. / Structural Integrity Procedia 00 (2017) 000–000

7

N. Di Domenico et al. / Procedia Structural Integrity 8 (2018) 422–432

428

k i = 0

X ( t ) CFD = X CFD 0 +

ξ ( t ) i ∆ X i

(15)

Where X CFD 0 is the vector of undeformed mesh nodal positions, k is the number of modes taken into account, ξ ( t ) i the modal coordinate calculated at time t for mode i and ∆ X i the vector of displacements introduced by the i-th modal shape for each node. Mesh deformations are then achieved by linearly superimposing the action of each modal shape amplified by its modal coordinate. The CFD calculation is at this point started and the FSI simulation can begin. For transient analyses, using this weak approach, the investigation is performed considering the loads frozen in the time step. Modal forces, employed in equation (7), are computed on prescribed surfaces by projecting the nodal forces (fluid pressure and shear) onto the modal shape. Modal coordinates are calculated by using equation (7). At each time step the mesh is updated by employing equation (15) and the analysis re-executed until simulation end. By using the modal superimposition method the FEM solver can be used just once at initialization, avoiding the complex and resource consuming task of data exchange. For very large models (millions cells), moreover, pressure mapping and mesh update can be very time consuming, making the modal superposition method 10-12 times faster than two-way in transient analysis. Reducing the amount of data exchanged between solvers is not only a speed but also a reliability matter: transient simulations can last hours, days or more and so a robust method that reduces all the possible liabilities is strongly advised. The error introduced by the structural modes embedding and associated with truncation errors has however to be considered (especially for steady cases). Can be demonstrated, as shown in Biancolini et al. (2016), that few modes are needed in order to obtain accurate results. It must be taken into account also the validity range of this method, restricted only to linear structures being them the limitation of modal theory. By employing the proposed method a broad set of problems can be tackled. Steady FSI can be carried to account for structure elasticity (aircraft wings, propeller blades, racing), transient simulations with prescribed motions can be simulated (flapping devices or modes acceleration for Reduced Order Models in flutter analysis) and transient simulation with vibrations excited by the flow can be investigated (forced response or the computation of damped frequencies). Transient FSI using the proposed modal approach was successfully employed by Dupain (2015) and Groth (2016) to study the flow behavior of tube arrays in water crossflow as empirically investigated in Weaver and Abd-Rabbo (1985).

4. Application

Numerical accuracy of the proposed FSI method was demonstrated studying the vortex shedding induced vibrations of a literature testcase for which experimental data are available (Ausoni (2009)). The investigated body is a NACA 0009 hydrofoil operating at zero angle of attack in a uniform high speed flow with Re h = 16 . 1 · 10 3 − 96 . 6 · 10 3 with the trailing edge thickness as reference length h .

y

Perfect embedding

C ref

x

z

y 1

x 1

h=3.22

B=150

Pivot embedding

L=100 mm

Fig. 2: Blunt trailing edge NACA 0009 hydrofoil (Ausoni (2009))