Research Papers: Gas Turbines: Structures and Dynamics

The Depth-Averaged Numerical Simulation of Laminar Thin-Film Flows With Capillary Waves

[+] Author and Article Information
Bruce Kakimpa

Gas Turbine and Transmissions Research
Centre (G2TRC),
University of Nottingham,
Nottingham, NG7 2RD, UK
e-mail: bruce.kakimpa@nottingham.ac.uk

Herve Morvan

Gas Turbine and Transmissions Research
Centre (G2TRC),
University of Nottingham,
Nottingham, NG7 2RD, UK
e-mail: herve.morvan@nottingham.ac.uk

Stephen Hibberd

Associate Professor
Gas Turbine and Transmissions Research
Centre (G2TRC),
University of Nottingham,
Nottingham, NG7 2RD, UK
e-mail: stephen.hibberd@nottingham.ac.uk

1Corresponding author.

Contributed by the Structures and Dynamics Committee of ASME for publication in the JOURNAL OF ENGINEERING FOR GAS TURBINES AND POWER. Manuscript received March 15, 2016; final manuscript received April 6, 2016; published online May 24, 2016. Editor: David Wisler.

J. Eng. Gas Turbines Power 138(11), 112501 (May 24, 2016) (10 pages) Paper No: GTP-16-1105; doi: 10.1115/1.4033471 History: Received March 15, 2016; Revised April 06, 2016

Thin-film flows encountered in engineering systems such as aero-engine bearing chambers often exhibit capillary waves and occur within a moderate to high Weber number range. Although the depth-averaged simulation of these thin-film flows is computationally efficient relative to traditional volume-of-fluid (VOF) methods, numerical challenges remain particularly for solutions involving capillary waves and in the higher Weber number, low surface tension range. A depth-averaged approximation of the Navier–Stokes equations has been used to explore the effect of surface tension, grid resolution, and inertia on thin-film rimming solution accuracy and numerical stability. In shock and pooling solutions where capillary ripples are present, solution stability, and accuracy are shown to be highly sensitive to surface tension. The common practice in analytical studies of enforcing unphysical low Weber number stability constraints is shown to stabilize the solution by artificially damping capillary oscillations. This approach, however, although providing stable solutions is shown to adversely affect solution accuracy. An alternative grid resolution-based stability criterion is demonstrated and used to obtain numerically stable shock and pooling solutions without recourse to unphysical surface tension values. This allows for the accurate simulation of thin-film flows with capillary waves within the constrained parameter space corresponding to physical material and flow properties. Results obtained using the proposed formulation and solution strategy show good agreement with available experimental data from literature for low Re coating flows and moderate to high Re falling wavy film flows.

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Fig. 1

Thin-film rimming flow geometry and coordinate system used

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Fig. 2

Thin-film rimming flow solution classification into; (a) shear dominated smooth flow, (b) a transitional shock flow regime where shear and gravity are in balance, and (c) gravity dominated pool flow

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Fig. 3

Thin-film flow dimensionless parameters

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Fig. 4

(a) Film height profiles and (b) momentum source terms for smooth solutions in cases A1–A3

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Fig. 5

(a) Film height profiles in cases B1–B3 and (b) momentum source terms for the stable shock solution in case B2

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Fig. 6

(a) Film height profiles in cases C1 and C2 and (b) momentum source terms for the stable pool solutions in case C2

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Fig. 7

Polar plot of film solution illustrating the artificial smoothing of pool solutions due to the use of very high surface tension coefficients

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Fig. 8

Results of the grid sensitivity studies for (a) case B2 (Δx=1.15 mm), case B2f1 (Δx=0.58 mm), cases B2f2 (Δx=0.29 mm), case B2c (Δx=2.30 mm), and (b) case B1 (Δx=1.15 mm), case B1f3 (Δx=0.14 mm)

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Fig. 9

Sensitivity of small-wavelength disturbances in stable cases B1 (σ=0.0245N/m) and B2 (σ=0.5N/m) to surface tension

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Fig. 10

Effect of inertia representation on (a) smooth solution A1, (b) shock solution B2, and (c) pool solution C2

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Fig. 11

Experimental measurements from Ref. [15] together with equivalent film thickness predictions for (a) case 1 and (b) case 2 as defined in Table 3

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Fig. 12

ETFM predictions for solitary waves traveling down a 6.4 deg slope with Re = 29 and inlet forcing frequency of 1.5 Hz. ETFM model results for both simplified and full inertia representations are shown together with equivalent experimental measurements from Ref. [16].



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