Adaption of Giles Non-Local Non-Reflecting Boundary Conditions for a Cell-Centered Solver for Turbomachinery Applications

Abstract

In contrast to external flow aerodynamics, where one-dimensional Riemann boundary conditions can be applied far up- and downstream, the handling of non-reflecting boundary conditions for turbomachinery applications poses a greater challenge due to small axial gaps normally encountered. For boundaries exposed to non-uniform flow in the vicinity of blade rows, the quality of the simulation is greatly influenced by the underlying non-reflecting boundary condition and its implementation. This paper deals with the adaptation of Giles’ well-known exact non-local boundary conditions for two-dimensional steady flows to a cell-centered solver specifically developed for turbomachinery applications. It is shown that directly applying the theory originally formulated for a cell-vertex scheme to a cell-centered solver may yield an ill-posed problem due to the necessity of having to reconstruct boundary face values before actually applying the exact non-reflecting theory. In order to ensure well-posedness, Giles’ original approach is adapted for cell-centered schemes with a physically motivated reconstruction of the boundary face values, while still maintaining the non-reflecting boundary conditions. The extension is formulated within the original framework of determining the circumferential distribution of one-dimensional characteristics on the boundary. It is shown that, due to approximations in the one-dimensional characteristic reconstruction of boundary face values, the new approach can only be exact in the limiting case of cells with a vanishing width in the direction normal to the boundary if a one-dimensional characteristic reconstruction of boundary face values is used. To overcome the dependency on the width of the last cell, the new boundary condition is expressed explicitly in terms of a two-dimensional modal decomposition of the flow field. In this formulation, vanishing modal amplitudes for all incoming two-dimensional modes can easily be accomplished for a converged solution. Hence we are able to ensure perfectly non-reflecting boundary conditions under the same conditions as the original approach. The improvements of the new method are demonstrated for both a subsonic turbine and a transonic compressor test case.

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

“…In addition to this, we have adopted the regularization proposed by Frey et al [16] which ensures that waves corresponding to acoustic resonance are not allowed to exist. Our final implementation is to some extent similar to the work reported by Robens et al [17] in which phase shifting of the waves from the interior cells is employed to obtain a definition of the exterior solution. The eigenvectors, which scale the perturbations of the primitive flow variables induced by the corresponding wave, have in this work been chosen to be the same as those reported by Kersken et al [14].…”

“…(29) 6: for l = 1 to 2 do [5][l] = α * ext [5]e −i(k x,5 ∆x)n(l−1) 27: end for 28: for l = 1 to 2 do 29:q abs (x l ) = T(r)α * abs [1 : 5][l] 30: end for such, only convected waves which are are not resolved properly by the computational mesh will be excluded from the nonreflecting analysis, and step 3 of Algorithm 1 is therefore not expected to yield any significant errors. Note that the phase-shift employed here only is necessary for cell-centered solvers, since for a node centered scheme the information is already available at the boundary [17].…”

A Nonreflecting Formulation for Turbomachinery Boundaries and Blade Row Interfaces

“…Phase extrapolation has also been used by e.g. Robens et al [17]. The procedure used for an outlet is almost equivalent to the one used for the inlet, with the difference that there will be four outgoing waves and one incoming wave.…”

Consistent Non-Reflecting Boundary Conditions for Both Steady and Unsteady Flow Simulations in Turbomachinery Applications

“…The derivation of the residual Jacobian ∂R ∂q can be found in [10]. For the adaption of Giles' original boundary condition to a cell-centred solver, where the boundary condition is applied between two pseudo-time updates, we also need to modify the update of the mean charactersistics at the faces due the pseudo-time change of the outgoing characteristics in the interior [16]. We define another residual…”