The Study of Water Waves Breaking

December 7th 2006

Contents

•  Introduction •  Shallow Water Waves •  Deep Water Waves •  Beach Waves

•  Breaking Waves •  Solutions

•  Analytical •  Experimental •  Computational

•  Conclusion

Introduction: Water Waves

•  Stokes wave train generated by an oscillating plunger

•  28 wavelengths away from the generation point, the waves disintegrate

The Waves Become Unstable

‘In’sight

•  In the following slides the depth is 22% of the wavelength.

•  White particles are photographed over one period demonstrating the trajectories of the water particles

•  A standing wave is created by propagating waves at the left and then having the waves reflected at the far right

•  The radius of the circles traced by the particles’ trajectories decreases exponentially with depth.

Shallow Water Waves •  In shallow water waves, the particle

trajectories are elliptical versus the circular paths found in deep water waves

Deep Water Waves •  Deep water waves are those whose

wavelength is less than the height of the body of water

Beach Waves •  Biesel noted that, as ocean waves approach

the beach the water gets shallower. This causes: •  The amplitude of the waves to increase •  The wave speed decreases •  The wavelength decreases •  The water particles tend to align there elliptical

trajectories with the bottom’s slope

Circles -> Ellipses

Floor

Breaking Waves

•  The changes in the properties of the wave lead to structural instability in a non-linear manner. As the wave approaches the beach the bottom is slowed down while the top part continues forward.

•  Thusly, the wave breaks.

Solutions

•  Analytical •  Experimental •  Computational

Analytical

•  A parametric solution for a breaking wave has been developed by Longuet and Higgins.

•  This solution only describes the flow up to the moment of impact. Another solution involving turbulence is required to describe the aftermath.

Parametric representation; branch points

•  Consider the flow to be incompressible, irrotational and in two dimensions

•  x = + i and z = x +iy are the particle position coordinates

•  Assume x and z are analytic functions of complex and or t

•  x can be expresssed as a function of z if is eliminated

•  The following suffixed terms represent the partial differentiation with respect to or t

•  W* is the particle velocity, where W=X/Z

Say z = 0 at =0, where z=z0 then,

near here, z-z0~1/2(-0)2z

-0 (z- z0)1/2, then, x-x0 ~ (-0)(z-z0)1/2

Boundary Condition From Bernoulli; -2p = (xt-Wzt) + c.c. + WW* - g(z+z*) – 2,

-2 Dp/Dt = (xtt-Wztt) + 2K(xt-Wzt) + K2(x-Wz) + c.c. – g(W+W*)– 2,

where

K = D/Dt = W*-zt . z

at the boundary condition of the free moving surface, p = 0, Dp/Dt = 0 so,

zt = W*, K= 0

then,

-2 Dp/Dt = (xtt-Wztt) + c.c. – g(W+W*)– 2,

We’ve assumed that the flow is Lagrangian and is real at the free surface. Now,

ztt – g = irz

where r is some function of and t which is real on the boundary

it was found that the particle acceleration is,

a = D/Dt zt(*) = ztt(*) + K*zt*(*),

K= [zt(*)-zt()]/z ,

Frames of Reference ztt – g = irz

is a non-homogeneous linear diferential equation for z(,t) with solutions z0(,t) and z0 + z1(,t). then,

ztt = irz z1 = z0 -1/2 gt2,

The Stokes Corner Flow

Velocity potential X = - 1/12 g2(-t)3 = 2/3 ig1/2z3/2

Longuet and Higgins proved that at the tip of the plunging wave there’s an interior flow which

is the focus of a rotating hyperbolic flow.

Upwelling Flow

i = ½ t (ztt-g) = z, z = - ½ gt z = - ½ gt, W = zt*(-) = ½ g, x = ¼ g2t, x = - 1/8 g2t2 = - z2/2t The free surface is the y-axis

Velocity potential

= - x2 – y2 2t

the streamlines are = - xy = constant,

t For t > 0 the flow represents a decelerated upwelling, in which the vertical and horizontal components of flow are given by

x = - x/t, y = y/t at x = 0 the pressure is constant -py = vt +(uvx+vvy), where (u,v) = (x,y) à py = 0

-p = t + ½ (x2 +y

2) – gx

Now, we want the solution to the homogeneous boundary condition. This will describe the flows complementary to the upwelling flow.

½ tztt = z when +* = 0 Make z a polynomial, z = bn

n + bn-1n-1 + … + b0; bn is a function of time

bnn + is of the form At + B and A and B are constant

P0 = t, P1 = t + t2, P2 = t2 + 2t2 + 2/3 t3, P3 = t2 + 3t22 + 2t3 1/3 t4.

Q0 = 1, Q1 = + 2tln|t|, Q2 = 2 + 4tln|t| + 4t2(ln|t| - 3/2), Q3 = 3 + 6t2ln|t| + 12t2(ln|t| - 3/2) + 4t3(ln|t| - 7/3)

z = n (AnPn + BnQn)

Physical Meaning

To understand these flows; consider a linear, cubic, and quadratic flow.

Comparison with Observation

Only the front face of the wave is being described

Experimental

•  The following is a numerical approach involving coefficients that were experimentally determined.

The Setup

The wave’s velocity fields were measured laser Doppler velocimeter (LDV) and particle image velocimetry (PIV)

Computational

•  The computation of a breaking wave acts as a good test to see if the numerical model accurately depicts nature.

A Popular Model In 1804 Gerstner Developed a wave model whose particle

(x,z) coordinates are mapped as so:

x = x0 –Rsin(Kx0-t) z = z0 +Rcos(Kx0-t)

Where,

R= R0eKz0 R0 = particle trajectory radius = 1/K K= number of waves = angular speed z0 = A2/4 A= 2R = 2/K

z

x

Biesel improved on this model to account for the particle trajectory’s tendency toward an elliptical shape.

Improving still on Biesel’s model was Founier-Reeves

x = x0 + Rcos()Sxsin() + Rsin()Szcos() z = z0 – Rcos()Szcos() + Rsin()Sxsin() Sx = (1-e-kxh)-1,Sz(1 – e-Kzh) sin() = sin()e-K0h = - t + 0

x0K(x)x K(x) = K/(tanh(Kh))1/2

Here: K0 – relates depth to to the angle of the particles elliptical trajectory Kx – is the enlargement factor on the major axis of the ellipse Kz – is the reduction factor of the minor axis These variables range between 0 and 1. They are used to tune the model in order to avoid unreal results that arise from a negatively sloping beach.

Waves

Beach Floor

(-)

Conclusion

An accurate model of a wave crashing on the beach could yield beneficial information for coastal structures such as boats, or break-walls.

Also, Perhaps a more accurate wave model

could assist in the design of surf boards

References •  Cramer, M.s. "Water Waves Introduction." Fluidmech.Net. 2004.

Cambridge University. 6 Dec. 2006 <http://www.fluidmech.net/tutorials/ocean/w_waves.htm>.

•  Crowe, Clayton T., Donald F. Elger, and John A. Roberson. Engineering Fluid Mechanics. 7th ed. United States: John Wiley & Sons, Inc., 2001. 350.

•  Gonzato, Jean-Christophe, and Bertrand Le Saec. "A Phenomenological Model of Coastal Scenes Based on Physical Considerations." Laboratoire Bordelais De Recherche En Informatique.

•  Lin, Pengzhi, and Philip L. Liu. "A Numerical Study of Breaking Waves in the Surf Zone." Journal of Fluid Mechanics 359 (1998): 239-264.

•  Longuet, and Higgins. "Parametric Solutions for Breaking Waves." Journal of Fluid Mechanics 121 (1982): 403-424.

•  Richeson, David. "Water Waves." June 2001. Dickinson College. 6 Dec. 2006 <http://users.dickinson.edu/~richesod/waves/index.html>.

•  Van Dyke, Milton. An Album of Fluid Mechanics. 10th ed. Stanford, California: The Parabolic P, 1982.

Hydraulic Jump