Flutter Prediction by a Cartesian Mesh Euler Method with Small Perturbation Gridless Boundary Conditions

Abstract

A method for the prediction of transonic flutter by the Euler equations on a Cartesian mesh is presented. Surface boundary conditions are applied using a perturbation of a gridless treatment in such a manner that solutions are obtained on a stationary mesh. For steady problems, the gridless method applies surface boundary conditions using a weighted average of the flow properties within a cloud of nodes in the vicinity of the surface. Weight functions are derived based on a least squares fitting of the surrounding cloud of nodes.For unsteady calculations, a perturbation of the weight functions is incorporated to account for a fluctuating surface normal direction. Additionally, a varying surface normal velocity is introduced into the boundary treatment. The nature of the method provides for efficient and accurate solution of transient problems in which surface deflections are small (i.e. flutter calculations) without the need for a deforming mesh. Problem setup also requires minimal effort due to the use of patched embedded Cartesian grids. Although deviations in the airfoil angle of attack are small, the mean angle of attack can be large, being prescribed by the far field boundary condition. Results show good agreement with available experimental data and wing flutter computations using traditional moving mesh algorithms considering a two-degree of freedom structural wing model.

Aeroelastic simulations using gridless boundary condition and small perturbation techniques

“…Small perturbation boundary conditions are not used to mimic the geometry but are used only to track the movements of the airfoil or wings. Kirshman and Liu [15] also developed a similar approach in their studies for two-dimensional airfoils. As applied to flutter simulations where the unsteady motion or deformation of the flow boundary is small with respect to the mean position, the small perturbation approximation proves to be general and accurate.…”

“…Euler fluxes for the neighbors of cut cells near the solid surfaces are directly computed using the reflected points which are set to satisfy either the moving or non-moving wall boundary conditions. As in Kirshman and Liu [15] and Koh et al [8], throughout the rest of the flow field, a standard finite volume formulation for the Euler equations is used. Overall the meshless approach uses unrelated points for the approximation thus solving the problems of grid generation near boundaries.…”

“…The above mentioned computations by Kirshman and Liu [15] for two-dimensional flutter simulation make use of a set of shape functions over a cloud of gridless nodal points. Unsteady surface motions are introduced by a small perturbation of the shape functions.…”

A meshless finite point method for three‐dimensional analysis of compressible flow problems involving moving boundaries and adaptivity

“…[11] for a comparative analysis of popular discretization approaches). Moreover, applications to unsteady problems accounting for body motion have also been explored with success in a meshless context through small perturbation boundary conditions or domain deformation techniques, see for instance [12][13][14]. All in all, meshless features have been increasingly exploited and numerous advantages over conventional discretization techniques have been brought to light.…”

Development and applications of the Finite Point Method to compressible aerodynamics problems

“…Meshless procedures for dealing with unsteady problems accounting for body motion have been successfully applied in the literature; see for instance some approaches using small perturbation boundary conditions and domain deformation techniques in (Anandhanarayanan, 2010;Kirshman & Liu, 2004;Wang, Chen & Periaux, 2009). In this chapter, a general solution approach is proposed with basis on the h-adaptive methodology previously presented in Chapter 6.…”