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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(x!t-Wz!t) + 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.
i! = !
½ t (ztt-g) = z!,


                                    Upwelling Flow
z = - ½ gt!
z! = - ½ gt,
W = zt*(-!) = ½ g!,
x! = ¼ g2t!,
x = - 1/8 g2t!2 = - 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 +!y2) – 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-1!n-1 + … + b0; bn is a function of time
bn!n + is of the form At + B and A and B are constant




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

    Q0 = 1,
    Q1 = ! + 2tln|t|,
    Q2 = !2 + 4t!ln|t| + 4t2(ln|t| - 3/2),
    Q3 = !3 + 6t!2ln|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:
                                          z
x = x0 –Rsin(Kx0-!t)
z = z0 +Rcos(Kx0-!t)                              x


Where,

R= R0eKz0 R0 = particle trajectory radius = 1/K
K= number of waves           != angular speed
z0 = !A2/4!         A= 2R           != 2!/K
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 + !0x0K(x)!x
K(x) = K!/(tanh(K!h))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

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Wave mechanics

  • 1. The Study of Water Waves Breaking December 7th 2006
  • 2. Contents •  Introduction •  Shallow Water Waves •  Deep Water Waves •  Beach Waves •  Breaking Waves •  Solutions •  Analytical •  Experimental •  Computational •  Conclusion
  • 3. Introduction: Water Waves •  Stokes wave train generated by an oscillating plunger •  28 wavelengths away from the generation point, the waves disintegrate
  • 4. The Waves Become Unstable
  • 5. 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.
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  • 9. 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
  • 10. 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
  • 11. 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.
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  • 16. 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.
  • 17. 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!
  • 18. 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
  • 19. Boundary Condition From Bernoulli; -2p = (xt-Wzt) + c.c. + WW* - g(z+z*) – 2!, -2 Dp/Dt = (xtt-Wztt) + 2K(x!t-Wz!t) + 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!,
  • 20. 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,
  • 21. 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.
  • 22. i! = ! ½ t (ztt-g) = z!, Upwelling Flow z = - ½ gt! z! = - ½ gt, W = zt*(-!) = ½ g!, x! = ¼ g2t!, x = - 1/8 g2t!2 = - 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 +!y2) – gx
  • 23. 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-1!n-1 + … + b0; bn is a function of time bn!n + is of the form At + B and A and B are constant P0 = t, P1 = t! + t2, P2 = t!2 + 2t2! + 2/3 t3, P3 = t!2 + 3t2!2 + 2t3! 1/3 t4. Q0 = 1, Q1 = ! + 2tln|t|, Q2 = !2 + 4t!ln|t| + 4t2(ln|t| - 3/2), Q3 = !3 + 6t!2ln|t| + 12t2!(ln|t| - 3/2) + 4t3(ln|t| - 7/3) z = !n (AnPn + BnQn)
  • 24. Physical Meaning To understand these flows; consider a linear, cubic, and quadratic flow.
  • 25. Comparison with Observation Only the front face of the wave is being described
  • 26. Experimental •  The following is a numerical approach involving coefficients that were experimentally determined.
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  • 28. The Setup The wave s velocity fields were measured laser Doppler velocimeter (LDV) and particle image velocimetry (PIV)
  • 29. Computational •  The computation of a breaking wave acts as a good test to see if the numerical model accurately depicts nature.
  • 30. A Popular Model In 1804 Gerstner Developed a wave model whose particle (x,z) coordinates are mapped as so: z x = x0 –Rsin(Kx0-!t) z = z0 +Rcos(Kx0-!t) x Where, R= R0eKz0 R0 = particle trajectory radius = 1/K K= number of waves != angular speed z0 = !A2/4! A= 2R != 2!/K
  • 31. 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 + !0x0K(x)!x K(x) = K!/(tanh(K!h))1/2
  • 32. 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
  • 33. 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
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  • 37. 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.
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