Keywords: CFD, flow modeling, pulse charging, quasi-three-dimensional (Q-3D) computation, turbines, turbocharger, turbomachinery, unsteady compressible flow.

The basic idea of the Q-3D model is the replacement of a 3D model with two special 2D models. The first one is the so called S2 model, which solves the basic equations on a streamsheet between the blades, reaching from hub to casing. The second one solves the equations on one or more S1 streamsheets between hub and casing reaching from blade to blade. The coupling between the two models is done by geometry and flow quantities, which are calculated in one model and used as input for the other model. The S1 model delivers the circulation of the blades, the S2-model the geometry of the streamsheet and the corresponding streamsheet thickness. The iteration between S2 and S1 is stopped, when the changes in blade circulation and streamsheet geometry are small.

The numerical method used [1] is a time marching method, which solves the Euler or Navier-Stokes equations with different turbulence models in structured multiblock grids. The spatial discretization is performed with a cell centered finite volume method. The fluxes are computed with a central scheme and Jameson's artificial viscosity [2] or with the AUSM method [3]. The time discretization is performed with a multi stage Runge Kutta scheme. The time step size is determined for steady problems by a local time stepping method and by a dual time stepping method [4] for unsteady problems.

The special preprocessing of the Q-3D method is demonstrated by means of the first example, the turbine stage of a turbocharger with outlet casing. A local S1 coordinate system is used, which is angle preserving and also valid for pure radial streamsheets. The grid generator [5] solves two elliptic differential equations for the coordinates and allows the prescription of point distance and grid angle for the first grid cells. All boundary conditions are prescribed in dummy cells, that surround each mesh block. To get the dummy cell values for the non moving blocks of the stator and the moving blocks of the rotor a so called CHIMERA Interpolation [6] is used.

The two main results for the first example are: the influence of the gas exhaust casing on the turbine stage performance is well predicted with the Q-3D method despite the unsufficient flow simulation in the casing itself. The measured and computed unsteady torque of the turbine is in excellent agreement.

The coupling between areodynamics and mechanics is successfully validated with the second example [7], a turbine stage. Measured and computed displacement of the rotor blades agree very well. The CHIMERA Interpolation is also validated for this example. The mass loss due to the interpolation is in the same range like the overall value.

For the last example [8] - a stage compressor cascade - grid dependency is checked by comparing the pressure distribution on the initial guide vane blades. The solution is grid independent with meshes of about 15'000 cells for each channel. But the unsteady phenomena, especially the vortex shedding at the rotor wake are still grid dependent which is shown by a Fourier decomposition of the mass flow in the wake region of the rotor.

The ultimate goal of the method is to close the gap between 2D and 3D methods continuously. This is possible by introducing further S1- and S2 streamsheets and an unsteady coupling between the S1-S2 equation sets. The simple S2-models used in the present examples will be replaced by an unsteady Navier-Stokes solver.

1

O. Schäfer, B. Phillipsen, and B. Breuhaus, "Last Advances in Numerical Simulation of Aerodynamic Forces on Turbine Blades of Turbochargers for Pulse Charged Engines", CIMAC Congress, pages 1705-1719, Copenhagen, 1998.

2

A. Jameson, W. Schmidt, and E. Turkel, "Numerical Solutions of the Euler Equations by Finite Volume Methods using Runge-Kutta-Time-Stepping Schemes", AIAA paper 81-1259, 1981.

3

M. Liou, "Progesss towards an Improved CFD Method: AUSM+", AIAA-95-1701-CP, 1995.

4

N.D. Melson, M.D. Sanetrik, and H.L. Atkins, "Time-accurate Navier-Stokes Calculations with Multigrid Acceleration", 6th Copper Mountain Conference on Multigrid Methods, pages 423-439, 1993.

5

R.L. Sorenson, "A Computer Program to Generate Two-Dimensional Grids about Airfoils and other Shapes by the Use of Poisson Equations", Technical Memorandum 81198, NASA, 1980.

6

T. Steger and J. Benek, "On the Use of Composite Schemes in Computational Aerodynamics", Computational Methods in Applied Mechanics and Engineering, 64, 1-3, 1987. doi:10.1016/0045-7825(87)90045-4

7

ADTurB, "Final Technical Report", Brite/EuRam-Framework IV,BRPR-CT95-0124, Report Number: ADTB-RR-0010, 2000.

8

H. Gallus, "Einfluss von instationärer Strömung und Turbulenz auf die Grenzschichten und auf die Druckverteilungen von Beschaufelungen moderner mehrstufiger Axialverdichter", FVV-Vorhaben Nr.601: Stufeninteraktion, 1997.

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