Nonlinear Higher Order Spectral Solution for a Two-Dimensional Moving Load on Ice

• Published in: Journal of Fluid Mechanics
• Main Authors: Bonnefoy, Félicien, Meylan, Michael, Ferrant, Pierre
• Other Authors: Laboratoire de mécanique des fluides (LMF), École Centrale de Nantes (ECN)-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics Auckland, University of Auckland Auckland, Laboratoire de recherche en Hydrodynamique, Énergétique et Environnement Atmosphérique (LHEEA)
• Format: Article in Journal/Newspaper
• Language: English
• Published: HAL CCSD 2009
• Subjects: [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph]; [SPI.MECA.MEFL]Engineering Sciences [physics]/Mechanics [physics.med-ph]/Fluids mechanics [physics.class-ph]; Ice Sheet
• Online Access: https://hal.science/hal-00798968; https://hal.science/hal-00798968/document; https://hal.science/hal-00798968/file/bonnefoy_etal09.pdf; https://doi.org/10.1017/S0022112008004849

International audience We calculate the nonlinear response of an infinite ice sheet to a moving load in the time domain in two dimensions, using a higher-order spectral method. The nonlinearity is due to the moving boundary, as well as the nonlinear term in Bernoulli's equation and the elastic plate equation. We compare the nonlinear solution with the linear solution and with the nonlinear solution found by Parau & Dias (J. Fluid Mech., vol. 460, 2002, pp. 281–305). We find good agreement with both solutions (with the correction of an error in the Parau & Dias 2002 results) in the appropriate regimes. We also derive a solitary wavelike expression for the linear solution – close to but below the critical speed at which the phase speed has a minimum. Our model is carefully validated and used to investigate nonlinear effects. We focus in detail on the solution at a critical speed at which the linear response is infinite, and we show that the nonlinear solution remains bounded. We also establish that the inclusion of nonlinearities leads to significant new behaviour, which is not observed in the linear solution.