A penalty formulation for the throughflow modeling of turbomachinery

Abstract Despite the widespread use of fully three-dimensional Computational Fluid Dynamics (CFD) techniques, axisymmetric flow models still represent a key tool in turbomachinery design. In this paper, a novel method for the numerical solution of axisymmetric flow models for turbomachinery is presented and demonstrated for two cases, both relevant for the industrial perspective. The key feature of the proposed method is that the flow tangency condition to the blade mean line is enforced with a penalty term, which can also be interpreted as an immersed boundary technique. Differently from the techniques commonly applied for the solution of this kind of flows, the proposed method does not require additional constraint equations to be solved in bladed regions and is therefore very simple to be implemented into existing axisymmetric CFD codes. The techniques employed to deal with flows of high incidence angle (misalignments between the actual flow and the direction imposed by the blade) and to account for the aerodynamic losses and the blade blockage are also thoroughly discussed. The method has been implemented as an extension of the zFlow code for the numerical solution of the Euler equations in cylindrical coordinates, which is based on an hybrid finite element/finite volume space approximation, an implicit time integration scheme, and can deal with fluids of arbitrarily complex equations of state. The details of the numerical method are fully described in the paper. The performance of the developed code is demonstrated by the results obtained in the simulation of a single-stage axial fan, compared against a fully three-dimensional CFD simulation. The potentialities for more complex transonic flow conditions are finally demonstrated by the calculation of a double-stage low pressure steam turbine.