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Research Papers: Multiphase Flows

An Approximate Computational Method for the Fluid Stiction Problem of Two Separating Parallel Plates With Cavitation

[+] Author and Article Information
Rudolf Scheidl

Institute of Machine Design and
Hydraulic Drives,
Johannes Kepler University Linz,
Altenberger Straße 69,
Linz 4020, Austria
e-mail: rudolf.scheidl@jku.at

Christoph Gradl

Institute of Machine Design and
Hydraulic Drives,
Johannes Kepler University Linz,
Altenberger Straße 69,
Linz 4020, Austria
e-mail: christoph.gradl@jku.at

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received June 17, 2015; final manuscript received October 29, 2015; published online February 17, 2016. Assoc. Editor: Kwang-Yong Kim.

J. Fluids Eng 138(6), 061301 (Feb 17, 2016) (12 pages) Paper No: FE-15-1407; doi: 10.1115/1.4032299 History: Received June 17, 2015; Revised October 29, 2015

Stiction forces exerted by a fluid in a thin, quickly widening gap to its boundaries can become a strongly limiting factor of the performance of technical devices, like compressor valves or hydraulic on–off valves. In design optimization, such forces need to be properly and efficiently modeled. Cavitation during parts of a stiction process plays a strong role and needs to be taken into account to achieve a meaningful model. The paper presents an approximate calculation method which uses qualitative solution properties of the non cavitating stiction problem, in particular of its level curves and gradient lines. In this method, the formation of the cavitation boundaries is approximated by an elliptic domain. The pressure distribution along its principle axis is described by a directly integrable differential equation, the evolutions of its boundaries is guided just by pressure boundary conditions when the cavitation zone expands and by a nonlinear differential equation when it shrinks. The results of this approximate model agree quite well with the solutions of a finite volume (FV) model for the fluid stiction problem with cavitation.

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Figures

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Fig. 1

The fluid stiction problem for a circular parallel gap: a fluid filled gap of two separating plates causes a stiction force FS; a cavitation zone (radial extension RC) may arise causing substantial changes for FS; at the right-hand side measured (FS) and computed values values (FADV) of the stiction force and of the computed cavitation zone extension (RC) are shown (values taken from Ref. [4], radius R: 16.5 mm, fluid viscosity ηV: 0.043 Pa · s)

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Fig. 2

Evolution of the nondimensional gap height h/h0 of a circular stiction gap loaded with a constant force

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Fig. 3

The situation with an expanding cavitation zone ΩC; the diagram at the right depends on β; the cavitation boundary coordinate αC(t) for a certain β follows from Eq. (4); its corresponding gap height (hC(αC)) enters the condition for the evolution of the boundary in the shrinking phase according to Eq. (5)

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Fig. 4

The solution of the Stefan problem can be described by level sets and gradient lines

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Fig. 5

gαα and gββ in the neighborhood of the minimum pressure point as function of the local principal coordinates (ξ,η); only one quadrant is shown; parameter values k = 1, a = 3, b = 1

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Fig. 6

log(E(α,β)) in the neighborhood of the minimum pressure point as function of the local principal coordinates (ξ,η); only one quadrant is shown; parameter values k = 1, a = 3, b = 1

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Fig. 7

Qualitative picture for the case of a rectangular domain (level and gradient lines of the pressure in the first quadrant are shown) at shrinking cavitation zone development; the level curves with small α are close to ellipses; shrinking starts at maximum cavitation zone (αCmax); the shrinking speed at ξ-axis (α˙Cξ) is higher than at η-axis α˙Cη)

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Fig. 8

The solution of the cavitation problem is given by the variable u, from which pressure p and the evolution of the local cavitation height hN can be computed by the functions sg(u) and sg(-u), respectively

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Fig. 9

The mesh for the FV model and the resulting Stefan problem pressure distribution of a circular domain; for symmetry reasons, only a quarter domain is shown

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Fig. 10

Results of the nonlinear FV and the approximate semi-analytical model of the circular domain fluid stiction problem. Force as well as cavitation boundary development of both models agree.

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Fig. 11

The mesh for the FV model and the resulting Stefan problem pressure distribution of an elliptic domain; for symmetry reasons only a quarter of the domain is shown

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Fig. 12

Results of the nonlinear FV and the approximate semi-analytical model of the elliptic domain

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Fig. 13

The mesh for the FV model and the resulting Stefan problem pressure distribution of two rectangular domains with different aspect ratios (1/2 and 1/3); for symmetry reasons, only a quarter of the domain is shown

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Fig. 14

Results of the nonlinear FV and the approximate semi-analytical model of the rectangular domain, aspect ratio ½

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Fig. 15

Results of the nonlinear FV and the approximate semi-analytical model of the rectangular domain, aspect ratio 1/3

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