Implementation of a Quasi-Three-Dimensional Nonreflecting Blade Row Interface for Steady and Unsteady Analysis of Axial Turbomachines

Abstract

Higher order nonreflecting blade row interfaces are today widely used for performing both steady and unsteady simulations of the flow withing axial turbomachines. In this paper, a quasithree-dimensional nonreflecting interface based on the exact, two-dimensional nonreflecting boundary condition for a single frequency and azimuthal wave number developed by Giles is presented. The formulation has been implemented to work for both steady simulations as well as unsteady simulations employing the nonlinear Harmonic Balance method. The theory behind the construction of the nonreflecting interface is presented and details on the numerical implementation is also provided. The implementation is verified for two dimensional wave propagation along a straight cascade. It is shown that the interface correctly absorbs incoming waves, but also found that the chosen implementation strategy may be ill-posed. A simple solution to stabilize the implementation is therefore implemented, but future work should seek a more generic solution to this problem.

A Nonreflecting Formulation for Turbomachinery Boundaries and Blade Row Interfaces

“…It is also shown how the chosen nonreflecting formulation can be implemented consistently to both turbomachinery boundaries and to blade row interfaces. The latter application was previously considered by the present authors with the same nonreflecting formulation that is considered in this work [10]. The implementation is finally verified for a set of two-dimensional wave-propagation problems.…”

“…( 8) will be used to decompose a flow perturbation with a known frequency and azimuthal wave number into incoming and outgoing waves. In order to do this, one assumes that each of the waves vary harmonically in the axial direction and thereby can be written in the following form [1,7] q (x, r, θ, t) = q(r)e i(ωt−k x x−k z z) (10) Here, k x is the axial wavenumber of the wave and k z z = mθ, where k z = m/r is the azimuthal wavenumber of the wave and z = rθ. The variable substitution z = rθ may also be applied in Eq.…”

“…It should also be pointed out that the modification of ω only applies when the eigenvalues/eigenvectors are being computed, not for calculating the temporal evolution of the wave in Eq. (10). The group velocity is also still used for computing the direction of propagation of the convected waves.…”

“…The implementation could then be verified by comparing the numerically computed wave with the analytical solution defined by Eq. (10).…”