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IMA Journal of Applied Mathematics Advance Access originally published online on December 15, 2008
IMA Journal of Applied Mathematics 2009 74(2):230-249; doi:10.1093/imamat/hxn036
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© The Author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

On the temporal stability of steady-state quasi-1D bubbly cavitating nozzle flow solutions

Senay Pasinlioglu{dagger}

Faculty of Science and Letters, Department of Mathematics, Istanbul Technical University, 34469 Maslak, Istanbul, Turkey

Can F. Delale{ddagger}

Faculty of Aeronautics and Astronautics, Istanbul Technical University, 34469 Maslak, Istanbul, Turkey and TÜBITAK Feza Gürsey Institute, PO Box 6, 81220 Çengelköy, Istanbul, Turkey

Günter H. Schnerr§

Lehrstuhl für Fluidmechanik—Fachgebiet Gasdynamik, Technische Universität München Boltzmannstrasse 15, D-85747 Garching, Germany

{dagger} Email: pasinliogl{at}itu.edu.tr

{ddagger} Corresponding author. Email: delale{at}itu.edu.tr

§ Email: schnerr{at}flm.mw.tu-muenchen.de

Received on October 15, 2007; Revision received June 26, 2008. Accepted on October 23, 2008

Quasi-1D unsteady bubbly cavitating nozzle flows are considered by employing a homogeneous bubbly liquid flow model, where the non-linear dynamics of cavitating bubbles is described by a modified Rayleigh–Plesset equation. The various damping mechanisms are considered by a single damping coefficient lumping them together in the form of viscous dissipation and by assuming a polytropic law for the expansion and compression of the gas. The complete system of equations, by appropriate uncoupling, are then reduced to two evolution equations, one for the flow speed and the other for the bubble radius when all damping mechanisms are considered by a single damping coefficient. The evolution equations for the bubble radius and flow speed are then perturbed with respect to flow unsteadiness resulting in a coupled system of linear partial differential equations (PDEs) for the radius and flow speed perturbations. This system of coupled linear PDEs is then cast into an eigenvalue problem and the exact solution of the eigenvalue problem is found by normal mode analysis in the inlet region of the nozzle. Results show that the steady-state cavitating nozzle flow solutions are stable only for perturbations with very small wave numbers. The stable regions of the stability diagram for the inlet region of the nozzle are seen to be broadened by the effect of turbulent wall shear stress.

Keywords: bubbly cavitating flows; steady-state solutions; temporal stability.


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