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Research Papers

Toward a High Order Throughflow––Investigation of the Nonlinear Harmonic Method Coupled With an Immersed Boundary Method for the Modeling of the Circumferential Stresses

[+] Author and Article Information
J. P. Thomas

Turbomachinery Group, University of Liège, Chemin des Chevreuils 1, B-4000 Liège, Belgiumjp.thomas@ulg.ac.be

O. Léonard

Turbomachinery Group, University of Liège, Chemin des Chevreuils 1, B-4000 Liège, Belgiumo.leonard@ulg.ac.be

This assumption of small perturbations will be discussed later in the contribution.

The exact solution of an isolated cylinder is obtained analytically by the singularities method. When this cylinder is part of a row, a periodicity condition must be satisfied. To this aim, the method of images is employed. It consists of correcting the solution of the isolated cylinder by the effect of neighbors located as on a blade cascade. When doing this, the impermeability of the main cylinder is only approximative. Nevertheless, this method gives the advantage of an analytical solution and is chosen for comparison purpose.

J. Turbomach 134(1), 011017 (May 31, 2011) (9 pages) doi:10.1115/1.4003256 History: Received October 06, 2010; Revised October 20, 2010; Published May 31, 2011; Online May 31, 2011

Capturing a level of modeling of the flow inside a multistage turbomachine, such as unsteadiness for example, can be done at different levels of detail, 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. Among them, the “nonlinear harmonic method” (NLHM) from He and Ning [1998, “Efficient Approach for Analysis of Unsteady Viscous Flows in Turbomachines,” AIAA J., 36, pp. 2005–2012] 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 the NLHM to the throughflow 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 relies on a reformulation of the blade effect by a smooth force field as in the “immersed boundary method” from Peskin [2002, “The Immersed Boundary Method,” Acta Numerica, 11, pp. 1–39]. A simple example of an inviscid flow around a cylinder will illustrate the preceding developments, bringing back the mean effect of the circumferential nonuniformities into the meridional flow.

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Copyright © 2012 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

Deterministic boundary conditions

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Figure 2

Computation domains with standard formulation of the blades: nonperiodic continuous domain (top) and periodic discontinuous domain (bottom)

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Figure 3

Conformal (left) and nonconformal (right) grids

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Figure 4

Transmission of information from the Lagrangian grid to the Cartesian grid (Mittal and Iaccarino (19))

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Figure 5

Distribution of the blade force from the Lagrangian grid to the Cartesian grid (right) and distribution function (left)

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Figure 6

Axial and circumferential components of the distributed force

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Figure 7

Average of the amplitudes of the Fourier modes

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Figure 8

Kinetic energy of the perturbations for different numbers of modes

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Figure 9

Mach number of the mean flow

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Figure 10

Circumferential evolution of the axial velocity: full pitch view (top) and zoom over the fluid region (bottom)

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Figure 11

Absolute velocity field: 20 modes computed (top) and exact

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Figure 12

Static pressure field: 20 modes computed (top) and exact

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