9. DALRYMPLE, R. A. , "A Numerical Model for Periodic Finite Amplitude 

 Waves on a Rotational Fluid," Journal of Computational Physics^ 

 Bruges, Belgium, Vol. 24, No. 1, May 1977, pp. 29-42. 



Keywords. Currents, Vertical Shear; Numerical Model; Theory; Waves, 

 Finite-Amplitude; Waves, Nonlinear. 



Discussion. An iterative finite -difference model is developed to 

 describe two-dimensional periodic gravity waves on the surface of a 

 fluid containing vorticity in the form of a vertical shear current 

 (i.e., a steady horizontal current whose local velocity varies with 

 elevation). A coordinate transformation due to Dubreil-Jacotin has been 

 used to map the fluid domain into a rectangle. The full nonlinear 

 constant pressure free surface boundary condition is used iteratively 

 until convergence is achieved. A comparison is made to an analytical 

 model for a linear shear current, and results are also shown for a mean 

 flow with a seventh power law velocity distribution. (author's 

 abstract) 



The key to the finite-difference model is the mapping of the wave 

 domain into a rectangle by posing the problem with y as a function of x 

 and ^ . Although the resulting governing equation is more difficult, 

 the domain becomes a rectangle with a base equal to the length of the 

 wave and a height equal to the value of the surface streamline. From 

 symmetry, only half the wavelength need be studied. 



The usual boundary conditions pertain: no flow through the bottom, 

 a periodic solution in the direction of wave travel, and a constant 

 pressure streamline at the surface. This last free-surface condition is 

 attacked by iterating the Bernoulli equation along the free-surface 

 streamline. To fix the free surface, three constraints are applied: a 

 mean sea level constraint to ensure that the mean free surface is fixed, 

 a dynamic free-surface boundary condition constrained by the Bernoulli 

 constant, and a wave height constraint to ensure convergence. 



The author tests his finite-difference scheme against two cases: a 

 linear velocity distribution and an approximation to the seventh power 

 law. Both tests are for the same wave conditions (height = 0.61 meter; 

 depth = 3.0 meters; period = 10 seconds), and both are for an opposing 

 current with an approximately equal discharge, having peak surface 

 velocities of -0.91 meter per second (linear) and -0.61 meter per second 

 (seventh power). The combined wave-current interaction gives a velocity 

 in the seventh power case that is about 30 percent greater in magnitude 

 than the linear case. The seventh power law case has a relative maximum 

 in the velocity distribution located close to the bottom. 



Coastal Engineering Significance. From the example offered by the 

 author, it is clear that the realistic currents approximated by the 

 seventh power law cannot be simply superimposed on the wave particle 

 velocity to get the resulting motion. This is in contrast to currents 



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