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.
NomenclatureA bleed = area of bleed in physical model A region = area of applied boundary condition D = bleed hole diameter δ = 99% boundary-layer thickness p = imposed static back pressure p plenum = plenum static pressure p ∞ = freestream static pressure y = nondimensional wall distance Φ = bleed region porosity
The Harmonic Balance method is well suited for analyzing unsteadiness in turbomachinery flows comprised of a few dominant frequencies. A harmonic condition is imposed on the temporal derivatives through a Fourier transform operation. The solution is then reinterpreted as a time-domain problem, where several instances of time (lying within the largest period) are solved for simultaneously with the enforcement of the time-harmonic condition providing coupling between time instances. A discontinuous Galerkin discretization is used together with overset grids to provide higher-order spatial accuracy and flexibility in representing complex geometry. In this work, the discontinuous Galerkin infrastructure is extended for unsteady problems with a Harmonic Balance method and a Diagonally Implicit Runge-Kutta time-integrator. Verification results are presented for both time integration approaches in addition to results for a turbine blade with unsteadiness driven by a prescribed unsteady inlet boundary condition. Comparisons of results from the Harmonic Balance and Diagonally Implicit Runge-Kutta approaches are very close, with some small discrepancies that require further investigation. Significantly, rapid convergence from the Newton solver is obtained for the Harmonic Balance approach applied to the Euler equations for the turbine blade problem. Solutions converged by 8–10 orders of magnitude are obtained in between 5 and 16 Newton steps.
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