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

S. Pasinlioglu, C.F. Delale, Günter Schnerr

Research output: Contribution to journalArticleAcademicpeer-review

6 Citations (Scopus)

Abstract

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.
Original languageEnglish
Pages (from-to)230-249
JournalIMA journal of applied mathematics
Volume74
Issue number2
DOIs
Publication statusPublished - 2008

Fingerprint

Nozzle
Nozzles
Damping
Bubble
Partial differential equations
Linear partial differential equation
Radius
Eigenvalue Problem
Evolution Equation
Bubbly Flow
Perturbation
Shear stress
Viscous Dissipation
Wall Shear Stress
Compaction
Liquid Flow
Normal Modes
Modified Equations
Coefficient
Coupled System

Keywords

  • IR-86553
  • temporal stability
  • steady-state solutions
  • Bubbly cavitating flows
  • METIS-245248

Cite this

Pasinlioglu, S. ; Delale, C.F. ; Schnerr, Günter. / On the temporal stability of steady-state quasi-1D bubbly cavitating nozzle flow solutions. In: IMA journal of applied mathematics. 2008 ; Vol. 74, No. 2. pp. 230-249.
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abstract = "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.",
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On the temporal stability of steady-state quasi-1D bubbly cavitating nozzle flow solutions. / Pasinlioglu, S.; Delale, C.F.; Schnerr, Günter.

In: IMA journal of applied mathematics, Vol. 74, No. 2, 2008, p. 230-249.

Research output: Contribution to journalArticleAcademicpeer-review

TY - JOUR

T1 - On the temporal stability of steady-state quasi-1D bubbly cavitating nozzle flow solutions

AU - Pasinlioglu, S.

AU - Delale, C.F.

AU - Schnerr, Günter

PY - 2008

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N2 - 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.

AB - 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.

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