Physical and spurious modes in mixed finite element formulation for the Galbrun equation

Sound propagation in moving media can be described by the Galbrun equation for the oscillating component of the fluid displacement. A displacement based finite element formulation using standard Lagrangian elements produces spurious modes, which renders it unfeasible for any numerical purpose. Herein, the quadratic eigenvalue problem for the mixed formulation in 2D using Mini elements and Taylor-Hood elements is set up and solved. Solution confirms that both element types are suitable for low Mach numbers and under certain conditions. Although the formulation is not free from spurious results, it is shown that physical and spurious modes are well separated for low Mach numbers in non-dissipative systems. Vorticity modes, such as those arising from the linearized Euler equations, could not be identified. If absorbing walls are considered, separation of physical and spurious modes becomes less clear. Then, eigenvalues of both types of modes are located closer to each other in the complex plane. Examples include the one-dimensional duct problem, for which the spurious modes are discussed for the energy conserving problem, and an annular duct under two conditions: first subjected to a shear flow and with a rigid boundary, and secondly with an absorbing boundary, which allows investigating the dissipative case.