Application of finite elements for computational aeroelasticity

Ralf Unger

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In this thesis, a coupled multiphysical system is considered, whereas the focus is upon aeroelastic problems. For a consistent fo...

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In this thesis, a coupled multiphysical system is considered, whereas the focus is upon aeroelastic problems. For a consistent formulation of such coupled systems, an energy based variational formulation is chosen to describe initially the structural and ﬂuid subsystem by Hamilton’s principle. Both basic ﬂuid model equations - inviscid and viscous ﬂuid models - are employed by this weak variational energy principle. This procedure allows to describe the coupled problem by the classical direct two-ﬁeld approach as well as by a novel indirect three-ﬁeld approach. To discretize the entire system consistently with ﬁnite elements, the CBS scheme is employed for the ﬂuid domain described by the Navier-Stokes equation in ALE frame of reference. This allows the ﬂuid domain to be temporally deformable, which is essential for aeroelastic computations. The CBS scheme is veriﬁed for a wide range of typical ﬂuid problems ranging from inviscid, viscous, incompressible and turbulent ﬂows. A good agreement with data published in literature and with the further solver TAU are found, which underlines the applicability of the CBS scheme for diﬀerent ﬂuid ﬂow models. The DG-CBS scheme as a novel and attractive approach has been derived from the continuous version. One important advantage of the DG version is the design of the element edge ﬂux to be locally conservative. For the example of the laminar ﬂow over the NACA0012 airfoil as well as for the panel ﬂutter problem, a comparison of the CBS and DG-CBS scheme is made on structured ﬂuid grids including grid convergence studies. With biquadratic, more accurate results in terms of the ﬂutter frequency are obtained with DG-CBS scheme. Moreover, no global system of linear equations needs to be solved at the computational expense of addidtional element edge ﬂux calculations with the DG version. This might be attractive for ﬂuid grids with a high number of degrees of freedom. Consequently, the whole coupled system is further discretized with ﬁnite elements including the structural subdomain, the deformation of the ﬂuid grid and the transfer scheme. For the ﬂuid grid deformation, it is found, that all of the presented stiﬀness evaluation methods perform similarly. The stiﬀness strategy based on the wall distance and the characteristic length is recommended to be used for the simple testcases with the unstructured grid. For a structured grid around an airfoil, the best grids are obtained with the stiﬀness methods based on the wall distance. Thus, for general ﬂuid grid deformations, the method, which use a combination of the wall distance and the characteristic length, can be recommended and is hence applied for the panel ﬂutter problem. Based on the uniﬁed weak variational coupling schemes, several data transfer schemes are introduced, which share the property of load and energy conservation. With a h-reﬁnement of the integration grid, a signiﬁcant reduction of the transfer error is observed for low-curved interface meshes. The decrease of the transfer error is limited by the facetting error, which is identiﬁed for highly curved interface meshes and for a realistic wing conﬁguration. For the panel ﬂutter problem at Ma∞ = 1.0 and rp = 170, the Galerkin and the dual-Lagrange based transfer as well as the conservative interpolation gives similar results in terms of the frequency and amplitude of the LCO. With its local accuracy together with a global load conservation property and due to the eﬃciency of a matrix-free transfer scheme, the dual-Lagrange based transfer is an attractive approach for the data transmission of the coupled system. A smooth transfer scheme is proposed, which uses the novel three-ﬁeld coupling approach with a higher spatial order discretization of the connectivity frame. Regarding the time integration and equilibrium iteration, the three-ﬁeld approach is assessed for a strongly coupled problem. With the use of the Newton-GMRES iteration scheme, the number of DN cycles is reduced for the three-ﬁeld approach. Moreover, the same coupling matrices are identiﬁed for the three-ﬁeld approach, which alreay appeared within the iteration process of two-ﬁeld approach. This allows the application of a simple staggered time integration scheme for the panel ﬂutter problem. The comparison of the two- and three-ﬁeld approach shows that both, the frequency and the amplitude of the LCO, are only marginally aﬀected. However, the smooth data transfer leads to a clean ﬂuid solution without artiﬁcial shocks, which has been observed with the two-ﬁeld approach and a small number structural elements at the interface. Furthermore, a consistent time integration approach for the structure is proposed, so that both subsystems use the same temporal discretization. Here, similar results in terms of the LCO’s frequency and amplitude are obtained, when the Newmark or the consistent three-point backward diﬀerence scheme for the structural time integration of the panel is applied. Thus, the panel ﬂutter problem using a simple staggered time integration scheme with the consistent time integration for the ﬂuid and structural subsystem and with the proposed three-ﬁeld approach could be analyzed in detail running numerous simulations. At subsonic ﬂow conditions, the panel shows a static deﬂection behavior in up- or downward direction depending on Ma∞ and rp, but indepentent of rm. On the other hand, the panel exhibits a LCO and the critical values of the dynamic pressure strongly depend on the mass ratio. For low values of rm, a supersonic dip in the stability boundary is observed. It is shown, that the frequency of the LCO increases with increasing Mach number, dynamic pressure and mass ratio. Moreover, a linear dependency between the frequency and the amplitude of the LCO for high mass ratios and low Mach numbers is found. Turbulence modeling with the aid of the CBS scheme in the context of an aeroelastic problem is employed in this thesis. The Spalart-Allmaras turbulence model in conjunction with the CBS scheme is primarily veriﬁed with data found in literature and with the ﬂowsolver TAU for pure compressible ﬂuid ﬂow over an airfoil. For the panel ﬂutter problem, the turbulent boundary layer leads to an additional damping behavior. The frequency of the LCO is unaﬀected by the Reynolds number, but a dependency regarding Re is noticed for the amplitude and the mean deformation. Finally, a strong shift of the critical dynamic pressure to higher values could be observed for the stability chart, which is caused by the damping inﬂuence. Subsequent work regarding this thesis certainly involves the investigation of the panel ﬂutter phenomenon in three dimensions. This is motivated by the good performance of the CBS scheme in 3D found in literature. Another topic, which should be considered further, is the application of the three-ﬁeld coupling approach for more than two subdomains, e.g. ﬂuid-ﬂuid-structure or ﬂuid-structure-structure interaction. In this context, the performance of the presented coupling scheme in conjunction with an incompressible ﬂuid could be a subject for research. Herein, the avoidance of the added mass eﬀect due to an artiﬁcial compressibility within the CBS scheme is an interesting aspect. Moreover, attempts to improve the standard ﬁnite element methodology by a NURBS based isogeometric analysis can be observed in literature, see [CHB09] and the references therein. A NURBS b…