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The initial development of a jet caused by fluid, body and free surface interaction with a uniformly accelerated advancing or retreating plate. Part 1. The principal flow

  • M. T. Gallagher (a1), D. J. Needham (a1) and J. Billingham (a2)

Abstract

The free surface and flow field structure generated by the uniform acceleration (with dimensionless acceleration  $\unicode[STIX]{x1D70E}$ ) of a rigid plate, inclined at an angle $\unicode[STIX]{x1D6FC}\in (0,\unicode[STIX]{x03C0}/2)$ to the exterior horizontal, as it advances ( $\unicode[STIX]{x1D70E}>0$ ) or retreats ( $\unicode[STIX]{x1D70E}<0$ ) from an initially stationary and horizontal strip of inviscid incompressible fluid under gravity, are studied in the small-time limit via the method of matched asymptotic expansions. This work generalises the case of a uniformly accelerating plate advancing into a fluid as studied by Needham et al. (Q. J. Mech. Appl. Maths, vol. 61 (4), 2008, pp. 581–614). Particular attention is paid to the innermost asymptotic regions encompassing the initial interaction between the plate and the free surface. We find that the structure of the solution to the governing initial boundary value problem is characterised in terms of the parameters $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D707}$ (where $\unicode[STIX]{x1D707}=1+\unicode[STIX]{x1D70E}\tan \unicode[STIX]{x1D6FC}$ ), with a bifurcation in structure as $\unicode[STIX]{x1D707}$ changes sign. This bifurcation in structure leads us to question the well-posedness and stability of the governing initial boundary value problem with respect to small perturbations in initial data in the innermost asymptotic regions, the discussion of which will be presented in the companion paper Gallagher et al. (J. Fluid Mech. vol. 841, 2018, pp. 146–166). In particular, when $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D707})\in (0,\unicode[STIX]{x03C0}/2)\times \mathbb{R}^{+}$ , the free surface close to the initial contact point remains monotone, and encompasses a swelling jet when $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D707})\in (0,\unicode[STIX]{x03C0}/2)\times [1,\infty )$ or a collapsing jet when $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D707})\in (0,\unicode[STIX]{x03C0}/2)\times (0,1)$ . However, when $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D707})\in (0,\unicode[STIX]{x03C0}/2)\times \mathbb{R}^{-}$ , the collapsing jet develops a more complex structure, with the free surface close to the initial contact point now developing a finite number of local oscillations, with near resonance type behaviour occurring close to a countable set of critical plate angles $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D6FC}_{n}^{\ast }\in (0,\unicode[STIX]{x03C0}/2)$ ( $n=1,2,\ldots$ ).

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Copyright

This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.

Corresponding author

Email address for correspondence: d.j.needham@bham.ac.uk

References

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Gallagher, M. T.2015 The initial development of a jet caused by fluid, body and free surface interaction. PhD thesis, University of Birmingham.
Gallagher, M. T., Needham, D. J. & Billingham, J. 2018 The initial development of a jet caused by fluid, body and free surface interaction with a uniformly accelerated advancing or retreating plate. Part 2. Well-posedness and stability of the principal flow. J. Fluid Mech. 841, 146166.
Greenhow, M. 1987 Wedge entry into initially calm water. Appl. Ocean Res. 9 (4), 214223.
Greenhow, M. & Lin, W.1983 Nonlinear free surface effects: experiments and theory. Tech. Rep. DTIC Document.
Howison, S. D., Ockendon, J. R. & Wilson, S. K. 1991 Incompressible water-entry problems at small deadrise angles. J. Fluid Mech. 222, 215230.
Iafrati, A. & Korobkin, A. A. 2005 Starting flow generated by the impulsive start of a floating wedge. J. Engng Maths 51 (2), 99126.
King, A. C. & Needham, D. J. 1994 The initial development of a jet caused by fluid, body and free-surface interaction. Part 1. A uniformly accelerating plate. J. Fluid Mech. 268, 89101.
Needham, D. J. 2012 The initial development of a jet caused by fluid, body and free surface interaction. Part 4. The large-time structure. IMA J. Appl. Maths 77 (4), 451472.
Needham, D. J., Billingham, J. & King, A. C. 2007 The initial development of a jet caused by fluid, body and free-surface interaction. Part 2. An impulsively moved plate. J. Fluid Mech. 578, 6784.
Needham, D. J., Chamberlain, P. G. & Billingham, J. 2008 The initial development of a jet caused by fluid, body and free surface interaction. Part 3. An inclined accelerating plate. Q. J. Mech. Appl. Maths 61 (4), 581614.
Norkin, M. & Korobkin, A. A. 2011 The motion of the free-surface separation point during the initial stage of horizontal impulsive displacement of a floating circular cylinder. J. Engng Maths 70 (1), 239254.
Sedov, L. I., Chu, C. K., C, H., Seckler, B. & Gillis, J. 1965 Two-Dimensional Problems in Hydrodynamics and Aerodynamics. Interscience.
Tassin, A., Korobkin, A. A. & Cooker, M. J. 2014 On analytical models of vertical water entry of a symmetric body with separation and cavity initiation. Appl. Ocean Res. 48, 3341.
Van Dyke, M. 1964 Perturbation Methods in Fluid Mechanics. Applied Mathematics and Mechanics. vol. 8. Academic Press.
Wu, G. X. 2001 Initial pressure distribution due to jet impact on a rigid body. J. Fluid Struct. 15 (2), 365370.
Yang, S. A. & Chwang, A. T.1989 Nonlinear viscous waves produced by an impulsively moving plate. Tech. Rep. IIHR Report No. 332. Iose Institute of Hydraulic Research, The University of Iowa.
Yang, S. A. & Chwang, A. T. 1992 An experimental study of nonlinear waves produced by an accelerating plate. Phys. Fluids A 4 (11), 24562465.
Yilmaz, O., Korobkin, A. A. & Iafrati, A. 2013 The initial stage of dam-break flow of two immiscible fluids. Linear analysis of global flow. Appl. Ocean Res. 42, 6069.
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Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
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