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TRACE – Non-Reflecting Boundary Conditions
For the numerical simulation of turbomachinery flow, non-reflecting boundary conditions at the inlet and outlet boundaries are indispensable for the realistic representation of fluid physics. In the case of stationary simulations, non-reflecting boundary conditions are also required at the coupling planes of the grid rows, also known as mixing planes.
A family of such non-reflecting boundary conditions for turbomachinery simulation has been proposed by Giles [1988, 1990, 1991]. The conditions are based on a characteristic analysis of the flow state at the boundary based on the linearized two-dimensional Euler equations. Due to the wave nature of the conservation equations, incoming and outgoing disturbance waves are distinguished in a modal approach, so that flow conditions can be prescribed at the boundaries in such a way that artificial reflections are prevented.
Stationary Non-Reflecting Boundary Conditions
The original formulation of the Giles boundary conditions was done for a vertex-centered solution scheme in which the boundary of the computational domain is formed by vertices of the computational mesh. The flow state is therefore known on the boundary. For the application of the Giles boundary conditions in a cell-centered solution scheme, as it is used in the CFD method TRACE of the DLR Institute of Propulsion Technology, the theory has to be adapted. In a research project, a physically correct reconstruction of the flow variables on the boundary, taking into account the characteristic wave propagation, was derived (Robens). In the flow solver TRACE two reconstruction methods were implemented, which differ in how the propagation of the flow between the cell centre of the boundary cell and the boundary itself is modelled: On the one hand, a simplified modelling based on the linearized, one-dimensional Euler equations (referred to as Characteristic Boundary Conditions) and on the other hand a more precise modelling based on the linearized, two-dimensional Euler equations (referred to as Modal Boundary Conditions). The validation on relevant test cases showed the higher accuracy of the two-dimensional, modal reconstruction.
Higher Order Transient Non-Reflecting Boundary Conditions
The transient, non-reflecting boundary conditions described by Giles on the basis of the linearized, two-dimensional Euler equations are non-local due to the involved integral transformations of the flow state: On the boundary, a spatial Fourier transformation in circumferential direction and a temporal Laplace transformation are performed. Especially the temporal non-locality is a major hurdle for the implementation of these boundary conditions in a time domain solver, because the history of the flow state would have to be stored on the boundary. By approximating the exact theory, Giles succeeded in localizing the boundary conditions in time. However, the resulting loss of accuracy of the boundary conditions can lead to non-negligible reflections in case of aeroelastic and/or aeroacoustic simulations. In a research project, higher-order non-reflecting boundary conditions were implemented in the time domain solution module of TRACE (Henninger ). For this purpose, the exactly non-reflecting boundary conditions were localized in time by introducing additional functions on the boundary according to Hagstrom et al . The accuracy order of the boundary conditions and the associated runtime costs can be controlled by the user via the number of additional functions considered. The validation by means of turbomachine-specific test cases showed the superiority of the higher-valued Hagstrom boundary conditions compared to the approximate Giles boundary conditions for aeroacoustics and aeroelastic problems.
Henninger, S. (2019). "Zeitbereichsimplementierung höherwertiger nichtreflektierender Randbedingungen für die Simulation instationärer Turbomaschinenströmungen", Dissertation, nstitute of Jet Propulsion and Turbomachinery, RWTH Aachen University, 2019
Robens, S. (2015). "Stationäre nicht-lokale Randbedingungen für zell-zentrierte Schemata und integrale Bilanzierung von Casing-Treatments in Turboverdichtern", Dissertation, Institute of Jet Propulsion and Turbomachinery, RWTH Aachen University, 2015
Giles, M.B. (1988). "Non-reflecting boundary conditions for the Euler equations", Techn. Ber. MIT, CFDL-TR-88-1.
Giles, M.B. (1990). "Nonreflecting boundary conditions for Euler equation calculations", AIAA Journal 28, pp. 2050–2058.