August  2014, 34(8): 3219-3239. doi: 10.3934/dcds.2014.34.3219

Pressure beneath a traveling wave with constant vorticity

1. 

Department of Mathematics, Pennsylvania State University, University Park, PA 16802, United States

2. 

Mathematics Department, Seattle University, Seattle, WA 98122, United States

Received  July 2013 Revised  September 2013 Published  January 2014

The main focus of this paper is to derive a direct relationship between the surface of an inviscid traveling gravity wave in two dimensions, and the pressure at the bottom of the fluid without approximation, including the effects of constant vorticity. Using this relationship, we reconstruct both the pressure and streamlines throughout the fluid domain. We compare our numerical results with various analytical results (such as the bounds presented in [7-10])as well as known numerical results (see [16]).
Citation: Vishal Vasan, Katie Oliveras. Pressure beneath a traveling wave with constant vorticity. Discrete & Continuous Dynamical Systems - A, 2014, 34 (8) : 3219-3239. doi: 10.3934/dcds.2014.34.3219
References:
[1]

M. J. Ablowitz and T. S. Haut, Spectral formulation of the two fluid Euler equations with a free interface and long wave reductions,, Analysis and Applications, 6 (2008), 323. doi: 10.1142/S0219530508001213.

[2]

A. Ali and H. Kalisch, Reconstruction of the pressure in long-wave models with constant vorticity,, European Journal of Mechanics B / Fluids, 37 (2013), 187. doi: 10.1016/j.euromechflu.2012.09.009.

[3]

A. C. L. Ashton and A. S. Fokas, A non-local formulation of rotational water waves,, Journal of Fluid Mechanics, 689 (2011), 129. doi: 10.1017/jfm.2011.404.

[4]

D. Clamond and A. Constantin, Recovery of steady periodic wave profiles from pressure measurements at the bed,, Journal of Fluid Mechanics, 714 (2013), 463. doi: 10.1017/jfm.2012.490.

[5]

A. Constantin, M. Ehrnström and E. Wahlén, Symmetry of steady periodic gravity water waves with vorticity,, Duke Math. J., 140 (2007), 591. doi: 10.1215/S0012-7094-07-14034-1.

[6]

A. Constantin and J. Escher, Symmetry of steady periodic surface water waves with vorticity,, J. Fluid Mech., 498 (2004), 171. doi: 10.1017/S0022112003006773.

[7]

A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity,, Communications on Pure and Applied Mathematics, 57 (2004), 481. doi: 10.1002/cpa.3046.

[8]

A. Constantin and W. Strauss, Rotational steady water waves near stagnation,, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 365 (2007), 2227. doi: 10.1098/rsta.2007.2004.

[9]

A. Constantin and W. Strauss, Pressure beneath a Stokes wave,, Communications on Pure and Applied Mathematics, 63 (2010), 533. doi: 10.1002/cpa.20299.

[10]

A. Constantin and E. Varvaruca, Steady periodic water waves with constant vorticity: Regularity and local bifurcation,, Arch. of Rat. Mech. and Anal., 199 (2011), 33. doi: 10.1007/s00205-010-0314-x.

[11]

B. Deconinck and K. Oliveras, The instability of periodic surface gravity waves,, J. Fluid Mech., 675 (2011), 141. doi: 10.1017/S0022112011000073.

[12]

D. Henry, On the pressure transfer function for solitary water waves with vorticity,, Mathematische Annalen, 357 (2013), 23. doi: 10.1007/s00208-013-0899-0.

[13]

J. Ko and W. Strauss, Effect of vorticity on steady water waves,, Journal of Fluid Mechanics, 608 (2008), 197. doi: 10.1017/S0022112008002371.

[14]

J. Ko and W. Strauss, Large-amplitude steady rotational water waves,, European Journal of Mechanics-B/Fluids, 27 (2008), 96. doi: 10.1016/j.euromechflu.2007.04.004.

[15]

K. Oliveras, V. Vasan, B. Deconinck and D. Henderson, Recovering the water-wave profile from pressure measurements,, SIAM J. of Applied Mathematics, 72 (2012), 897. doi: 10.1137/110853285.

[16]

A. F. Teles da Silva and D. H. Peregrine, Steep, steady surface waves on water of finite depth with constant vorticity,, Journal of Fluid Mechanics, 195 (1988), 281. doi: 10.1017/S0022112088002423.

[17]

V. Vasan and B. Deconinck, The inverse water wave problem of bathymetry detection,, Journal of Fluid Mechanics, 714 (2013), 562. doi: 10.1017/jfm.2012.497.

[18]

E. Wahlen, Steady periodic capillary waves with vorticity,, Ark. Mat., 44 (2006), 367. doi: 10.1007/s11512-006-0024-7.

[19]

E. Wahlen, Steady water waves with a critical layer,, Journal of Differential Equations, 246 (2009), 2468. doi: 10.1016/j.jde.2008.10.005.

[20]

P. I. Plotnikov and J. F. Toland, The Fourier Coefficients of Stokes Waves,, Nonlinear Problems in Mathematical Physics and Related Topics, 1 (2002), 303. doi: 10.1007/978-1-4615-0777-2_18.

show all references

References:
[1]

M. J. Ablowitz and T. S. Haut, Spectral formulation of the two fluid Euler equations with a free interface and long wave reductions,, Analysis and Applications, 6 (2008), 323. doi: 10.1142/S0219530508001213.

[2]

A. Ali and H. Kalisch, Reconstruction of the pressure in long-wave models with constant vorticity,, European Journal of Mechanics B / Fluids, 37 (2013), 187. doi: 10.1016/j.euromechflu.2012.09.009.

[3]

A. C. L. Ashton and A. S. Fokas, A non-local formulation of rotational water waves,, Journal of Fluid Mechanics, 689 (2011), 129. doi: 10.1017/jfm.2011.404.

[4]

D. Clamond and A. Constantin, Recovery of steady periodic wave profiles from pressure measurements at the bed,, Journal of Fluid Mechanics, 714 (2013), 463. doi: 10.1017/jfm.2012.490.

[5]

A. Constantin, M. Ehrnström and E. Wahlén, Symmetry of steady periodic gravity water waves with vorticity,, Duke Math. J., 140 (2007), 591. doi: 10.1215/S0012-7094-07-14034-1.

[6]

A. Constantin and J. Escher, Symmetry of steady periodic surface water waves with vorticity,, J. Fluid Mech., 498 (2004), 171. doi: 10.1017/S0022112003006773.

[7]

A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity,, Communications on Pure and Applied Mathematics, 57 (2004), 481. doi: 10.1002/cpa.3046.

[8]

A. Constantin and W. Strauss, Rotational steady water waves near stagnation,, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 365 (2007), 2227. doi: 10.1098/rsta.2007.2004.

[9]

A. Constantin and W. Strauss, Pressure beneath a Stokes wave,, Communications on Pure and Applied Mathematics, 63 (2010), 533. doi: 10.1002/cpa.20299.

[10]

A. Constantin and E. Varvaruca, Steady periodic water waves with constant vorticity: Regularity and local bifurcation,, Arch. of Rat. Mech. and Anal., 199 (2011), 33. doi: 10.1007/s00205-010-0314-x.

[11]

B. Deconinck and K. Oliveras, The instability of periodic surface gravity waves,, J. Fluid Mech., 675 (2011), 141. doi: 10.1017/S0022112011000073.

[12]

D. Henry, On the pressure transfer function for solitary water waves with vorticity,, Mathematische Annalen, 357 (2013), 23. doi: 10.1007/s00208-013-0899-0.

[13]

J. Ko and W. Strauss, Effect of vorticity on steady water waves,, Journal of Fluid Mechanics, 608 (2008), 197. doi: 10.1017/S0022112008002371.

[14]

J. Ko and W. Strauss, Large-amplitude steady rotational water waves,, European Journal of Mechanics-B/Fluids, 27 (2008), 96. doi: 10.1016/j.euromechflu.2007.04.004.

[15]

K. Oliveras, V. Vasan, B. Deconinck and D. Henderson, Recovering the water-wave profile from pressure measurements,, SIAM J. of Applied Mathematics, 72 (2012), 897. doi: 10.1137/110853285.

[16]

A. F. Teles da Silva and D. H. Peregrine, Steep, steady surface waves on water of finite depth with constant vorticity,, Journal of Fluid Mechanics, 195 (1988), 281. doi: 10.1017/S0022112088002423.

[17]

V. Vasan and B. Deconinck, The inverse water wave problem of bathymetry detection,, Journal of Fluid Mechanics, 714 (2013), 562. doi: 10.1017/jfm.2012.497.

[18]

E. Wahlen, Steady periodic capillary waves with vorticity,, Ark. Mat., 44 (2006), 367. doi: 10.1007/s11512-006-0024-7.

[19]

E. Wahlen, Steady water waves with a critical layer,, Journal of Differential Equations, 246 (2009), 2468. doi: 10.1016/j.jde.2008.10.005.

[20]

P. I. Plotnikov and J. F. Toland, The Fourier Coefficients of Stokes Waves,, Nonlinear Problems in Mathematical Physics and Related Topics, 1 (2002), 303. doi: 10.1007/978-1-4615-0777-2_18.

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