- 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.
- 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.
- 15. Solutions • Analytical • Experimental • Computational
- 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.
- 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
- 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.
- 44. Hydraulic Jump