Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-12-02T21:53:43.307Z Has data issue: false hasContentIssue false
  • English
  • Français

Three-dimensional free-surface flow over arbitrary bottom topography

Published online by Cambridge University Press:  03 May 2018

Nicholas R. Buttle ,
Ravindra Pethiyagoda ,
Timothy J. Moroney  and
Scott W. McCue Open the ORCID record for Scott W. McCue [Opens in a new window]
Nicholas R. Buttle
Affiliation:
School of Mathematical Sciences, Queensland University of Technology, Brisbane QLD 4001, Australia
Ravindra Pethiyagoda
Affiliation:
School of Mathematical Sciences, Queensland University of Technology, Brisbane QLD 4001, Australia
Timothy J. Moroney
Affiliation:
School of Mathematical Sciences, Queensland University of Technology, Brisbane QLD 4001, Australia
Scott W. McCue*
Affiliation:
School of Mathematical Sciences, Queensland University of Technology, Brisbane QLD 4001, Australia
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider steady nonlinear free surface flow past an arbitrary bottom topography in three dimensions, concentrating on the shape of the wave pattern that forms on the surface of the fluid. Assuming ideal fluid flow, the problem is formulated using a boundary integral method and discretised to produce a nonlinear system of algebraic equations. The Jacobian of this system is dense due to integrals being evaluated over the entire free surface. To overcome the computational difficulty and large memory requirements, a Jacobian-free Newton–Krylov (JFNK) method is utilised. Using a block-banded approximation of the Jacobian from the linearised system as a preconditioner for the JFNK scheme, we find significant reductions in computational time and memory required for generating numerical solutions. These improvements also allow for a larger number of mesh points over the free surface and the bottom topography. We present a range of numerical solutions for both subcritical and supercritical regimes, and for a variety of bottom configurations. We discuss nonlinear features of the wave patterns as well as their relationship to ship wakes.

JFM classification

Waves/Free-surface Flows: Surface gravity waves Wakes/Jets: Wakes Waves/Free-surface Flows: Waves/Free-surface Flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 846 , 10 July 2018 , pp. 166 - 189
Check for updates
Copyright
© 2018 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Binder, B. J., Blyth, M. G. & Balasuriya, S. 2014 Non-uniqueness of steady free-surface flow at critical Froude number. Europhys. Lett. 105, 44003.Google Scholar
Binder, B. J., Blyth, M. G. & McCue, S. W. 2013 Free-surface flow past arbitrary topography and an inverse approach for wave-free solutions. IMA J. Appl. Maths 78, 685–696.CrossRefGoogle Scholar
Binder, B. J., Dias, F. & Vanden-Broeck, J.-M. 2006 Steady free-surface flow past an uneven channel bottom. Theor. Comput. Fluid Dyn. 20, 125–144.CrossRefGoogle Scholar
Broutman, D., Rottman, J. W. & Eckermann, S. D. 2010 A simplified Fourier method for nonhydrostatic mountain waves. J. Atmos. Sci. 60, 2686–2696.Google Scholar
Brown, P. N. & Saad, Y. 1990 Hybrid Krylov methods for nonlinear systems of equations. SIAM J. Sci. Stat. Comput. 11, 450–481.CrossRefGoogle Scholar
Chapman, S. J. & Vanden-Broeck, J.-M. 2006 Exponential asymptotics and gravity waves. J. Fluid Mech. 567, 299–326.Google Scholar
Chardard, F., Dias, F., Nguyen, H. Y. & Vanden-Broeck, J.-M. 2011 Stability of some stationary solutions to the forced KdV equation with one or two bumps. J. Engng Maths 70, 175–189.Google Scholar
Chuang, J. M. 2000 Numerical studies on non-linear free surface flow using generalized Schwarz–Christoffel transformation. Intl J. Numer. Meth. Fluids 32, 745–772.Google Scholar
Darmon, A., Benzaquen, M. & Raphaël, E. 2014 Kelvin wake pattern at large Froude numbers. J. Fluid Mech. 738, R3.Google Scholar
Dias, F. 2014 Ship waves and Kelvin. J. Fluid Mech. 746, 1–4.CrossRefGoogle Scholar
Dias, F. & Vanden-Broeck, J.-M. 1989 Open channel flows with submerged obstructions. J. Fluid Mech. 206, 155–170.Google Scholar
Dias, F. & Vanden-Broeck, J.-M. 2002 Generalised critical free-surface flows. J. Engng Maths 42, 291–301.CrossRefGoogle Scholar
Eckermann, S. D., Lindeman, J., Broutman, D., Ma, J. & Boybeyi, Z. 2010 Momentum fluxes of gravity waves generated by variable Froude number flow over three-dimensional obstacles. J. Atmos. Sci. 67, 2260–2278.CrossRefGoogle Scholar
Ellingsen, S. Å. 2014 Ship waves in the presence of uniform vorticity. J. Fluid Mech. 742, R2.CrossRefGoogle Scholar
Forbes, L. K. 1985 On the effects of non-linearity in free-surface flow about a submerged point vortex. J. Engng Maths 19, 139–155.Google Scholar
Forbes, L. K. 1989 An algorithm for 3-dimensional free-surface problems in hydrodynamics. J. Comput. Phys. 82, 330–347.CrossRefGoogle Scholar
Forbes, L. K. & Hocking, G. C. 2005 Flow due to a sink near a vertical wall, in infinitely deep fluid. Comput. Fluids 34, 684–704.CrossRefGoogle Scholar
Forbes, L. K. & Schwartz, L. W. 1982 Free-surface flow over a semicircular obstruction. J. Fluid Mech. 114, 299–314.CrossRefGoogle Scholar
Gazdar, A. S. 1973 Generation of waves of small amplitude by an obstacle placed on the bottom of a running stream. J. Phys. Soc. Japan 34, 530–538.Google Scholar
He, J., Zhang, C., Zhu, Y., Wu, H., Yang, C.-J., Noblesse, F., Gu, X. & Li, W. 2014 Comparison of three simple models of Kelvin’s ship wake. Eur. J. Mech. (B/Fluids) 49, 12–19.Google Scholar
Higgins, P. J., Read, W. W. & Belward, S. R. 2006 A series-solution method for free-boundary problems arising from flow over topography. J. Engng Maths 54, 345–358.Google Scholar
Hindmarsh, A. C., Brown, P. N., Grant, K. E., Lee, S. L., Serban, R., Shumaker, D. E. & Woodward, C. S. 2005 SUNDIALS: suite of nonlinear and differential/algebraic equation solvers. ACM Trans. Math. Softw. 31, 363–396.CrossRefGoogle Scholar
Hocking, G. C., Holmes, R. J. & Forbes, L. K. 2013 A note on waveless subcritical flow past a submerged semi-ellipse. J. Engng Maths 81, 1–8.Google Scholar
King, A. C. & Bloor, M. I. G. 1990 Free-surface flow of a stream obstructed by an arbitrary bed topography. Q. J. Mech. Appl. Maths 43, 87–106.Google Scholar
Knoll, D. A. & Keyes, D. E. 2004 Jacobian-free Newton–Krylov methods: a survey of approaches and applications. J. Comput. Phys. 193, 357–397.CrossRefGoogle Scholar
Lamb, H. 1932 Hydrodynamics. Cambridge University Press.Google Scholar
Li, Y. & Ellingsen, S. Å. 2016 Ship waves on uniform shear current at finite depth: wave resistance and critical velocity. J. Fluid Mech. 791, 539–567.Google Scholar
Lustri, C. J., McCue, S. W. & Binder, B. J. 2012 Free surface flow past topography: a beyond-all-orders approach. Eur. J. Appl. Maths 23, 441–467.CrossRefGoogle Scholar
McCue, S. W. & Forbes, L. K. 2002 Free-surface flows emerging from beneath a semi-infinite plate with constant vorticity. J. Fluid Mech. 461, 387–407.Google Scholar
Miao, S. & Liu, Y. 2015 Wave pattern in the wake of an arbitrary moving surface pressure disturbance. Phys. Fluids 27, 122102.CrossRefGoogle Scholar
Noblesse, F., He, J., Zhu, Y., Hong, L., Zhang, C., Zhu, R. & Yang, C. 2014 Why can ship wakes appear narrower than Kelvin’s angle? Eur. J. Mech. (B/Fluids) 46, 164–171.CrossRefGoogle Scholar
Părău, E. I. & Vanden-Broeck, J.-M. 2002 Nonlinear two-and three-dimensional free surface flows due to moving disturbances. Eur. J. Mech. (B/Fluids) 21, 643–656.Google Scholar
Părău, E. I. & Vanden-Broeck, J.-M. 2011 Three-dimensional waves beneath an ice sheet due to a steadily moving pressure. Phil. Trans. R. Soc. Lond. A 369, 2973–2988.Google ScholarPubMed
Părău, E. I., Vanden-Broeck, J.-M. & Cooker, M. J. 2005a Nonlinear three-dimensional gravity–capillary solitary waves. J. Fluid Mech. 536, 99–105.Google Scholar
Părău, E. I., Vanden-Broeck, J.-M. & Cooker, M. J. 2005b Three-dimensional gravity-capillary solitary waves in water of finite depth and related problems. Phys. Fluids 17, 122101.CrossRefGoogle Scholar
Părău, E. I., Vanden-Broeck, J.-M. & Cooker, M. J. 2007a Nonlinear three-dimensional interfacial flows with a free surface. J. Fluid Mech. 591, 481–494.CrossRefGoogle Scholar
Părău, E. I., Vanden-Broeck, J.-M. & Cooker, M. J. 2007b Three-dimensional capillary-gravity waves generated by a moving disturbance. Phys. Fluids 19, 082102.Google Scholar
Pethiyagoda, R., McCue, S. W., Moroney, T. J. & Back, J. M. 2014a Jacobian-free Newton–Krylov methods with GPU acceleration for computing nonlinear ship wave patterns. J. Comput. Phys. 269, 297–313.Google Scholar
Pethiyagoda, R., McCue, S. W. & Moroney, T. J. 2014b What is the apparent angle of a Kelvin ship wave pattern? J. Fluid Mech. 758, 468–485.CrossRefGoogle Scholar
Pethiyagoda, R., McCue, S. W. & Moroney, T. J. 2015 Wake angle for surface gravity waves on a finite depth fluid. Phys. Fluids 27, 061701.Google Scholar
Pethiyagoda, R., McCue, S. W. & Moroney, T. J. 2017 Spectrograms of ship wakes: identifying linear and nonlinear wave signals. J. Fluid Mech. 811, 189–209.Google Scholar
Pethiyagoda, R., Moroney, T. J., MacFarlane, G. J., Binns, J. R. & McCue, S. W. 2018a Time-frequency analysis of ship wave patterns in shallow water: modelling and experiments. Ocean Engng 158, 123–131.Google Scholar
Pethiyagoda, R., Moroney, T. J. & McCue, S. W. 2018b Efficient computation of two-dimensional steady free-surface flows. Intl J. Numer. Meth. Fluids 86, 607–624.Google Scholar
Rabaud, M. & Moisy, F. 2013 Ship wakes: Kelvin or Mach angle? Phys. Rev. Lett. 110, 214503.Google Scholar
Saad, Y. & Schultz, M. H. 1986 GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7, 856–869.CrossRefGoogle Scholar
Scullen, D. C.1998 Accurate computation of steady nonlinear free-surface flows. PhD thesis, Department of Applied Mathematics, University of Adelaide.Google Scholar
Smeltzer, B. K. & Ellingsen, S. Å. 2017 Surface waves on currents with arbitrary vertical shear. Phys. Fluids 29, 047102.CrossRefGoogle Scholar
Soomere, T. 2007 Nonlinear components of ship wake waves. Appl. Mech. Rev. 60, 120–138.Google Scholar
Teixeira, M. A. C. 2014 The physics of orographic gravity wave drag. Frontiers Phys. 2, 43.Google Scholar
Torsvik, T., Soomere, T., Didenkulova, I. & Sheremet, A. 2015 Identification of ship wake structures by a time-frequency method. J. Fluid Mech. 765, 229–251.Google Scholar
Wade, S. L., Binder, B. J., Mattner, T. W. & Denier, J. P. 2017 Steep waves in free-surface flow past narrow topography. Phys. Fluids 29, 062107.CrossRefGoogle Scholar
Wehausen, J. V. & Laitone, E. V. 1960 Surface Waves. Springer.Google Scholar
Zhang, C., He, J., Zhu, Y., Yang, C.-J., Li, W., Zhu, Y., Lin, M. & Noblesse, F. 2015 Interference effects on the Kelvin wake of a monohull ship represented via a continuous distribution of sources. Eur. J. Mech. (B/Fluids) 51, 27–36.CrossRefGoogle Scholar
Zhang, Y. & Zhu, S. 1996a A comparison study of nonlinear waves generated behind a semicircular trench. Proc. R. Soc. Lond. A 452, 1563–1584.Google Scholar
Zhang, Y. & Zhu, S. 1996b Open channel flow past a bottom obstruction. J. Engng Maths 30, 487–499.Google Scholar
Zhu, Y., He, J., Zhang, C., Wu, H., Wan, D., Zhu, R. & Noblesse, F. 2015 Farfield waves created by a monohull ship in shallow water. Eur. J. Mech. (B/Fluids) 49, 226–234.Google Scholar
Supplementary material: File

Buttle et al. supplementary material

Buttle et al. supplementary material 1

Download Buttle et al. supplementary material(File)
File 230.5 KB