Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-26T05:42:15.593Z Has data issue: false hasContentIssue false
  • English
  • Français

Potential flow over a submerged rectangular obstacle: Consequences for initiation of boulder motion

Part of: Geometric function theory Coupling of solid mechanics with other effects Incompressible inviscid fluids Incompressible viscous fluids

Published online by Cambridge University Press:  05 September 2019

J. G. HERTERICH Open the ORCID record for J. G. HERTERICH [Opens in a new window]  and
F. DIAS Open the ORCID record for F. DIAS [Opens in a new window]
J. G. HERTERICH
Affiliation:
School of Mathematics and Statistics, University College Dublin, Belfield, Dublin 4, Ireland emails: james.herterich@ucd.ie; frederic.dias@ucd.ie Earth Institute, University College Dublin, Belfield, Dublin 4, Ireland
F. DIAS
Affiliation:
School of Mathematics and Statistics, University College Dublin, Belfield, Dublin 4, Ireland emails: james.herterich@ucd.ie; frederic.dias@ucd.ie CMLA, ENS Paris–Saclay, CNRS, Université Paris–Saclay, 94235 Cachan, France Earth Institute, University College Dublin, Belfield, Dublin 4, Ireland
Get access Rights & Permissions [Opens in a new window]

Abstract

Steady two-dimensional fluid flow over an obstacle is solved using complex variable methods. We consider the cases of rectangular obstacles, such as large boulders, submerged in a potential flow. These may arise in geophysics, marine and civil engineering. Our models are applicable to initiation of motion that may result in subsequent transport. The local flow depends on the obstacle shape, slowing down in confining corners and speeding up in expanding corners. The flow generates hydrodynamic forces, drag and lift, and their associated moments, which differ around each face. Our model replaces the need for ill-defined drag and lift coefficients with geometry-dependent functions. We predict smaller flow velocities to initiate motion. We show how a joint-bound boulder can be transported against gravity, and analyse the influence of a wake region behind an isolated boulder.

Keywords

Complex variable methodsconformal mappingincompressible inviscid fluidsgeophysicsfluid–solid interaction

MSC classification

Primary: 86-08: Computational methods 74F10: Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
Secondary: 30C20: Conformal mappings of special domains 76B99: None of the above, but in this section 76D10: Boundary-layer theory, separation and reattachment, higher-order effects
Type
Papers
Information
European Journal of Applied Mathematics , Volume 31 , Issue 4 , August 2020 , pp. 646 - 681
Check for updates
Copyright
© Cambridge University Press 2019

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

Abd-el Malek, M. B., Hanna, S. N. & Kamel, M. T. (1991) Approximate solution of gravity flow from a uniform channel over triangular bottom for large froude number. Appl. Math. Model. 15(1), 25–32.CrossRefGoogle Scholar
Abd-el Malek, M. B. & Masoud, S. Z. (1988) Linearized solution of a flow over a ramp. Appl. Math. Model. 12(4), 406–410.CrossRefGoogle Scholar
Abramowitz, M., Stegun, I. A., et al. (1972) Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, vol. 9. Dover, New York.Google Scholar
Acheson, D. J. (1990) Elementary Fluid Dynamics. Clarendon Press, Wotton-under-Edge.Google Scholar
Akrish, G., Rabinovitch, O. & Agnon, Y. (2016) Extreme run-up events on a vertical wall due to nonlinear evolution of incident wave groups. J. Fluid Mech. 797, 644–664.CrossRefGoogle Scholar
Akrish, G., Schwartz, R., Rabinovitch, O. & Agnon, Y. (2016) Impact of extreme waves on a vertical wall. Nat. Hazards 84(2), 637–653.CrossRefGoogle Scholar
Alexander, J. & Cooker, M. J. (2016) Moving boulders in flash floods and estimating flow conditions using boulders in ancient deposits. Sedimentology 63(6), 1582–1595.CrossRefGoogle Scholar
Balta, S. & Smith, F. T. (2018) Fluid flow lifting a body from a solid surface. Proc. Roy. Soc. A, 474(2219), 20180286.CrossRefGoogle ScholarPubMed
Batchelor, G. K. (1967) An Introduction to Fluid Dynamics. Cambridge University Press, Cambridge.Google Scholar
Belward, S. R. & Forbes, L. K. (1993) Fully non-linear two-layer flow over arbitrary topography. J. Eng. Math. 27(4), 419–432.CrossRefGoogle Scholar
Binder, B. J., Vanden-Broeck, J.-M. & Dias, F. (2005) Forced solitary waves and fronts past submerged obstacles. Chaos: An Interdisciplinary J. Nonlinear Sci., 15(3), 037106.CrossRefGoogle ScholarPubMed
Biolchi, S., Furlani, S., Antonioli, F., Baldassini, N., Deguara, J. C., Devoto, S., Di Stefano, A., Evans, J., Gambin, T., Gauci, R., et al. (2016) Boulder accumulations related to extreme wave events on the eastern coast of Malta. Nat. Hazards Earth Syst. Sci. 16, 737–756.CrossRefGoogle Scholar
Bot, P., Rabaud, M., Thomas, G., Lombardi, A. & Lebret, C. (2016) Sharp transition in the lift force of a fluid flowing past nonsymmetrical obstacles: evidence for a lift crisis in the drag crisis regime. Phys. Rev. Lett. 117(23), 234501.CrossRefGoogle ScholarPubMed
Boutros, Y. Z., Abd-el Malek, M. B. & Masoud, S. Z. (1986) Linearized solution of a flow over a nonuniform bottom. J. Computat. Appl. Math. 16(1), 105–116.CrossRefGoogle Scholar
Brilleslyper, M. A., Dorff, M. J., McDougall, J. M., Rolf, J. S., Schaubroek, L. E., Stankewitz, R. L. & Stephenson, K. (2012) Explorations in Complex Analysis. MAA, Washington.CrossRefGoogle Scholar
Carrier, G. F., Krook, M. & Pearson, C. E. (1966) Functions of a Complex Variable: Theory and Technique. McGraw-Hill, New York.Google Scholar
Costa, J. E. (1983) Paleohydraulic reconstruction of flash-flood peaks from boulder deposits in the Colorado Front Range. GSA Bull. 94(8), 986–1004.2.0.CO;2>CrossRefGoogle Scholar
Cox, R., Ardhuin, F., Dias, F., Autret, R., Beisiegel, N., Earlier, C. S., Herterich, J. G., Kennedy, A. B., Paris, R. L., Raby, A. & Weiss, R. Boulders deposited by storm waves can be misinterpreted as tsunami-related because commonly used hydrodynamic equations are flawed (Under Review).Google Scholar
Cox, R., Watkins, O. G.Jahn, K. L. & Cox, P. (2018) Extraordinary boulder transport by storm waves (west of Ireland, winter 2013–2014) and criteria for analysing coastal boulder deposits. Earth Sci. Rev. 177, 623–636.CrossRefGoogle Scholar
Cox, R., Zentner, D. B., Kirchner, B. J. & Cook, M. S. (2012) Boulder ridges on the Aran Islands (Ireland): recent movements caused by storm waves, not tsunamis. J. Geol. 120(3), 249–272.CrossRefGoogle Scholar
Cox, S. J. & Cooker, M. J. (1999) The motion of a rigid body impelled by sea-wave impact. Appl. Ocean Res. 21(3), 113–125.CrossRefGoogle Scholar
Cullen, N. D. & Bourke, M. C. (2018) Clast abrasion of a rock shore platform on the Atlantic coast of Ireland. Earth Surf. Process. Landf. 43(12), 2627–2641.CrossRefGoogle Scholar
Cummins, C. P. & Dias, F. (2017) A new model of viscous dissipation for an oscillating wave surge converter. J. Eng. Math. 103(1), 195–216.CrossRefGoogle Scholar
Dias, F., Elcrat, A. R. & Trefethen, L. N. (1987) Ideal jet flow in two dimensions. J. Fluid Mech. 185, 275–288.CrossRefGoogle Scholar
Dias, F. & Vanden-Broeck, J.-M. (1989) Open channel flows with submerged obstructions. J. Fluid Mech. 206, 155–170.CrossRefGoogle Scholar
Dias, F. & Vanden-Broeck, J. M. (2002) Generalised critical free-surface flows. J. Eng. Math. 42(3), 291–301.CrossRefGoogle Scholar
Driscoll, T. A. & Trefethen, L. N. (2002). Schwarz–Christoffel Mapping, vol. 8. Cambridge University Press, Cambridge.CrossRefGoogle Scholar
Einstein, H. A. & El-Samni, E.-S. A. (1949). Hydrodynamic forces on a rough wall. Rev. Mod. Phys. 21(3), 520.CrossRefGoogle Scholar
Elcrat, A. R. & Trefethen, L. N. (1986) Classical free-streamline flow over a polygonal obstacle. J. Computat. Appl. Math. 14(1–2), 251–265.CrossRefGoogle Scholar
Feng, J., Huang, P. Y. & Joseph, D. D. (1995). Dynamic simulation of the motion of capsules in pipelines. J. Fluid Mech. 286, 201–227.CrossRefGoogle Scholar
Fichaut, B. & Suanez, S. (2011) Quarrying, transport and deposition of cliff-top storm deposits during extreme events: Banneg Island, Brittany. Marine Geol. 283(1), 36–55.CrossRefGoogle Scholar
Forbes, L. K. (1988) Critical free-surface flow over a semi-circular obstruction. J. Eng. Math. 22(1), 3–13.CrossRefGoogle Scholar
Forbes, L. K. & Schwartz, L. W. (1982). Free-surface flow over a semicircular obstruction. J. Fluid Mech. 114, 299–314.CrossRefGoogle Scholar
Funada, T. & Joseph, D. D. (2001). Viscous potential flow analysis of Kelvin–Helmholtz instability in a channel. J. Fluid Mech. 445, 263–283.CrossRefGoogle Scholar
Goto, K., Miyagi, K., Kawamata, H. & Imamura, F. (2010). Discrimination of boulders deposited by tsunamis and storm waves at Ishigaki Island, Japan. Marine Geol. 269(1), 34–45.CrossRefGoogle Scholar
Hansom, J. D., Barltrop, N. D. P. & Hall, A. M. (2008) Modelling the processes of cliff-top erosion and deposition under extreme storm waves. Marine Geol. 253(1), 36–50.CrossRefGoogle Scholar
Helley, E. J. (1969) Field measurement of the initiation of large bed particle motion in Blue Creek near Klamath, California. US Government Printing Office, Washington.CrossRefGoogle Scholar
Herterich, J. G., Cox, R. & Dias, F. (2018). How does wave impact generate large boulders? Modelling hydraulic fracture of cliffs and shore platforms. Marine Geol. 399, 34–46.CrossRefGoogle Scholar
Herterich, J. G. & Dias, F. Extreme long waves over a varying bathymetry. J. Fluid Mech. (In Press). doi: https://doi.org/10.1017/jfm.2019.618.CrossRefGoogle Scholar
Hoerner, S. F. (1965). Fluid-Dynamic Drag: Theoretical, Experimental and Statistical Information. Hoerner Fluid Dynamics.Google Scholar
Hoffman, J. (2005) Computation of mean drag for bluff body problems using adaptive DNS/LES. SIAM J. Sci. Comput. 27(1), 184–207.CrossRefGoogle Scholar
Hoffman, J. (2006) Simulation of turbulent flow past bluff bodies on coarse meshes using General Galerkin methods: drag crisis and turbulent Euler solutions. Computat. Mech. 38(4–5), 390–402.CrossRefGoogle Scholar
Hoffman, J. & Johnson, C. (2010) Resolution of d’Alembert’s paradox. J. Math. Fluid Mech. 12(3), 321–334.CrossRefGoogle Scholar
Hutter, K. & Wang, Y. (2016) Fluid and Thermodynamics: Volume 1: Basic Fluid Mechanics. Springer, Switzerland.Google Scholar
Imamura, F., Goto, K. & Ohkubo, S. (2008) A numerical model for the transport of a boulder by tsunami. J. Geophys. Res. Oceans 113(C1).CrossRefGoogle Scholar
Joseph, D. D., Liao, T. Y. & Hu, H. H. (1993) Drag and moment in viscous potential flow. Eur. J. Mech. Ser. B Fluids 12, 97–106.Google Scholar
Joulin, G., Denet, B. & El-Rabii, H. (2010) Potential-flow models for channelled two-dimensional premixed flames around near-circular obstacles. Phys. Rev. E 81(1), 016314.CrossRefGoogle ScholarPubMed
Kennedy, A. B., Mori, N., Yasuda, T., Shimozono, T., Tomiczek, T., Donahue, A., Shimura, T. & Imai, Y. (2017) Extreme block and boulder transport along a cliffed coastline (Calicoan Island, Philippines) during Super Typhoon Haiyan. Marine Geol. 383, 65–77.CrossRefGoogle Scholar
Kennedy, A. B., Mori, N., Zhang, Y., Yasuda, T., Chen, S.-E., Tajima, Y., Pecor, W. & Toride, K. (2016) Observations and modeling of coastal boulder transport and loading during Super Typhoon Haiyan. Coastal Eng. J. 58(01), 1640004.CrossRefGoogle Scholar
Kober, H. (1957) Dictionary of Conformal Representations. Dover, New York.Google Scholar
Nandasena, N. A. K., Paris, R. & Tanaka, N. (2011) Reassessment of hydrodynamic equations: minimum flow velocity to initiate boulder transport by high energy events (storms, tsunamis). Marine Geol. 281(1), 70–84.CrossRefGoogle Scholar
Nandasena, N. A. K. & Tanaka, N. (2013) Boulder transport by high energy: Numerical model-fitting experimental observations. Ocean Eng. 57, 163–179.CrossRefGoogle Scholar
Noormets, R., Crook, K. A. W. & Felton, E. A. (2004) Sedimentology of rocky shorelines: 3.: Hydrodynamics of megaclast emplacement and transport on a shore platform, Oahu, Hawaii. Sediment. Geol. 172(1), 41–65.CrossRefGoogle Scholar
Nott, J. (1997) Extremely high-energy wave deposits inside the Great Barrier Reef, Australia: Determining the cause-tsunami or tropical cyclone. Marine Geol. 141(1-4), 193–207 (1997).CrossRefGoogle Scholar
Nott, J. (2003) Waves, coastal boulder deposits and the importance of the pre-transport setting. Earth Planet. Sci. Lett. 210(1), 269–276.CrossRefGoogle Scholar
Ockendon, J. R., Howison, S., Lacey, A. & Movchan, A. (2003) Applied Partial Differential Equations. Oxford University Press, Oxford.Google Scholar
Rovere, A., Casella, E., Harris, D. L., Lorscheid, T., Nandasena, N. A. K., Dyer, B., Sandstrom, M. R., Stocchi, P., D’Andrea, W. J. & Raymo, M. E. (2017) Giant boulders and Last Interglacial storm intensity in the North Atlantic. Proc. Nat. Acad. Sci. 114(46), 12144–12149.CrossRefGoogle ScholarPubMed
Ryu, Y. & Chang, K.-A. (2008). Green water void fraction due to breaking wave impinging and overtopping. Exp. Fluids 45(5), 883–898.CrossRefGoogle Scholar
Ryu, Y., Chang, K.-A. & Mercier, R. (2007). Runup and green water velocities due to breaking wave impinging and overtopping. Experiments in Fluids 43(4), 555–567 (2007).CrossRefGoogle Scholar
Scheffers, A., Scheffers, S., Kelletat, D. & Browne, T. (2009) Wave-emplaced coarse debris and megaclasts in Ireland and Scotland: boulder transport in a high-energy littoral environment. J. Geol. 117(5), 553–573.CrossRefGoogle Scholar
Sheremet, A., Staples, T., Ardhuin, F., Suanez, S. & Fichaut, B. (2014) Observations of large infragravity wave runup at Banneg Island, France. Geophys. Res. Lett. 41(3), 976–982.CrossRefGoogle Scholar
Sherry, M. J., Jacono, D. L., Sheridan, J., Mathis, R. & Marusic, I. (2009) Flow separation characterisation of a forward facing step immersed in a turbulent boundary layer. In: TSFP Digital Library Online. Begel House Inc.Google Scholar
Shusser, M. & Weihs, D. (1996) Stability of source-vortex and doublet flows. Phys. Fluids 8(11), 3197–3199.CrossRefGoogle Scholar
Simpson, R. L. (1989) Turbulent boundary-layer separation. Ann. Rev. Fluid Mech. 21(1), 205–232.CrossRefGoogle Scholar
Smith, F. T. (1979). Laminar flow of an incompressible fluid past a bluff body: the separation, reattachment, eddy properties and drag. J. Fluid Mech. 92(1), 171–205.CrossRefGoogle Scholar
Smith, F. T., Brighton, P. W. M., Jackson, P. S. & Hunt, J. C. R. (1981) On boundary-layer flow past two-dimensional obstacles. J. Fluid Mech. 113, 123–152.CrossRefGoogle Scholar
Smith, F. T. & Wilson, P. L. (2013). Body-rock or lift-off in flow. J. Fluid Mech. 735, 91–119.CrossRefGoogle Scholar
Stewartson, K. (1981) D’Alembert’s paradox. SIAM Rev. 23(3), 308–343.CrossRefGoogle Scholar
Trefethen, L. N. (1980). Numerical computation of the Schwarz–Christoffel transformation. SIAM J. Sci. Stat. Comput. 1(1), 82–102.CrossRefGoogle Scholar
Tuck, E. O. (1965). The effect of non-linearity at the free surface on flow past a submerged cylinder. J. Fluid Mech. 22(2), 401–414.CrossRefGoogle Scholar
Vanden-Broeck, J.-M. & Dias, F. (1991). Nonlinear free-surface flows past a submerged inclined flat plate. Phys. Fluids A Fluid Dyn. 3(12), 2995–3000.CrossRefGoogle Scholar
Veldman, A. E. P. (2001) Matched asymptotic expansions and the numerical treatment of viscous-inviscid interaction. J. Eng. Math. 39(1), 189–206.CrossRefGoogle Scholar
Verhoff, A. (2010) Two-dimensional potential flow solutions with separation. J. Fluid Mech. 657, 238–264.CrossRefGoogle Scholar
Verhoff, A. (2012) Generalized Poisson Integral Formula applied to potential flow solutions for free and confined jets with secondary flow. Comput. & Fluids 54, 18–38.CrossRefGoogle Scholar
Viotti, C., Carbone, F. & Dias, F. (2014) Conditions for extreme wave runup on a vertical barrier by nonlinear dispersion. J. Fluid Mech. 748, 768–788.CrossRefGoogle Scholar
Vollmer, S. & Kleinhans, M. G. (2007) Predicting incipient motion, including the effect of turbulent pressure fluctuations in the bed. Water Resour. Res. 43(5).CrossRefGoogle Scholar
Weiss, R. (2012) The mystery of boulders moved by tsunamis and storms. Marine Geol. 295, 28–33.CrossRefGoogle Scholar
Weiss, R. & Diplas, P. (2015) Untangling boulder dislodgement in storms and tsunamis: Is it possible with simple theories? Geochem. Geophys. Geosyst. 16(3), 890–898.CrossRefGoogle Scholar
Williams, D. M. & Hall, A. M. (2004) Cliff-top megaclast deposits of Ireland, a record of extreme waves in the North Atlantic - storms or tsunamis? Marine Geol. 206(1), 101–117.Google Scholar