1 / 13

Weakly nonlinear analysis of dunes by the use of a sediment transport formula

This study presents a sediment transport formula that incorporates the pressure gradient for the weakly nonlinear analysis of dunes. It explores the transition of dune formation, the effect of bed inclination, and the influence of pressure gradient on bedload transport. The results show the occurrence of the subcritical bifurcation in the transition, even with a reasonably large effect of bed inclination. Additionally, it is found that the pressure gradient reduces the gravity effect in the bedload formula.

vway
Download Presentation

Weakly nonlinear analysis of dunes by the use of a sediment transport formula

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Weakly nonlinear analysis of dunes by the use of a sediment transport formula incorporating the pressure gradient Satomi Yamaguchi (Port and airport Institute, Japan) Norihiro Izumi (Tohoku University, Japan)

  2. Transition of dune formation Water depth variation during flood (Simons and Richardson, 1961) Hysteresis in the transition Weakly nonlinear stability analysis (Yamaguchi & Izumi, 2003) -The subcritical bifurcationappears in the transition

  3. Transition of dune formation Weakly nonlinear analysis (Yamaguchi & Izumi, 2003) - the subcritical bifurcationin the transition - effect of bed inclinationin sediment transport formulas [ the gravity effect ] - Fredsoe’s formula The subcritical bifurcation appears when the effect of bed inclination is small. -Kovacs-Paker’s formula The subcritical bifurcation does not appear. | The gravity effect on inclined bed is estimated to be large.

  4. Effect of pressure gradient Pressure distribution on dune form (Raudkivi, 1963) Effect of bed inclination [the gravity] [resistance due to the pressure gradient] In this study, a bedload transport formula incorporating the pressure gradient on the basis of Kovacs-Paker’s formula for the weakly nonlinear analysis of dunes.

  5. Effect of pressure gradient on bedload transport The coordinate system - on the basis of Kovacs-paker’s formula - 2D (bed inclination in the streamwise direction)

  6. Effect of pressure gradient on bedload transport Force balance on a sediment particle drag force gravity force resistance due to pressure gradient dynamic Coulomb friction force the force balance in the tangential direction to the bed Sediment particle velocity vp

  7. Effect of pressure gradient on bedload transport Force balance on the bedload layer bed shear stress gravity force the sum of the pressurep1- p4 grain stress fluid shear stress at the bottom () the force balance in the tangential direction to the bed Volume of particle in the bedload layerx

  8. Effect of pressure gradient on bedload transport Bedload sediment transport rate volume of particle in the bedload layer the coordinate system critical Shields shear stress particle velocity - Effect of pressure gradient on bedload sediment transport : increase : decrease - Balance between the gravity and pressure gradient tangential to the bed

  9. Application to analysis of dunes - Formulations flow (2D Reynolds equations ) the coordinate system time variation of bed elevation - Linear stability analysis - Weakly nonlinear stability analysis

  10. Application to analysis of dunes - Linear stability analysis W : growth rate, k: wave number - Weakly nonlinear stability analysis the growth rate expansion method Landau equation Landau constant a a1 < 0: supercritical bifurcation a1 > 0: subcritical bifurcation | hysteresis in the transition b Fc Subcritical bifurcation

  11. Results of the linear stability analysis Instability diagram S = 0.002 (average bed slope) mc = 0.84 (dynamic Coulomb friction coefficient) t*co= 0.047(the critical Shields shear stress for a horizontal bed) stable unstable neutral curve Kovacs-Parker’s formula the present formula incorporating pressure gradient

  12. Results of the weakly nonlinear stability analysis The present formula incorporating pressure gradient Kovacs-Parker’s formula Landau constant a1 < 0a1 > 0 supercritical bifurcation subcritical bifurcation | hysteresis in the transition Balance between the gravity and pressure gradient phase difference(from the phase of bed form) out of phase | The effect of the gravity is reduced by the pressure gradient gravity : 0.500p : -0.546p pressure gradient the phase of linear solutions

  13. Conclusions - We propose a bedload formula incorporating pressure gradient for the analysis of dunes. - The weakly nonlinear analysis with the use of the proposed formula shows that the subcritical bifurcation occurs in the transition between dune-covered and flat beds even if the effect of bed inclination is reasonably large. - It is found that the pressure gradient reduces the gravity effect in the bedload formula.

More Related