Towards a High Order Throughflow: Part II—Investigation of the Nonlinear Harmonic Method Coupled With an Immersed Boundary Method for the Modeling of the Circumferential Stresses

Abstract

Capturing a level of modeling of the flow inside a multi-stage turbomachine, such as unsteadiness for example, can be done at different degrees of details, either by capturing all deterministic features of the flow with a pure unsteady method or by settling for an approximated solution at a lower computational cost. The harmonic methods stand in this second category. Amongst them the “Nonlinear Harmonic Method” from He revealed the most efficient. This method consists of solving the fully nonlinear 3D steady problem and a linearized perturbation system in the frequency domain. As it has been shown by the authors that the circumferential variations exhibit a harmonic behavior, it is proposed here to adapt this method to the through-flow model, where the main nonlinear system would be the common throughflow equations and the auxiliary system would give access to the circumferential stresses. As the numerical local explicit impermeability conditions are unsupported by Fourier series, the adaptation of this technique to the throughflow model passes through a reformulation of the blade effect by a smooth force field as in the “Immersed Boundary Method” from Peskin. A simple example of an inviscid flow around a cylinder will illustrate the preceding developments, bringing back the mean effect of the circumferential non uniformities into the meridional flow.