A Numerical Simulation of a Plunging Breaking Wave

A Numerical Simulation of a Plunging Breaking Wave Paul Adams1, Kevin George1, Mike Stephens1, Kyle A. Brucker2, Thomas O’Shea2 and Douglas Dommermuth2 1 Unclassified Data Analysis and Assessment Center, U.S. Army Engineering Research and Development Center, MS 39180 2 Science Applications International Corporation 10260 Campus Point Drive, San Diego, CA 92121 November, 22 2009 Abstract This article describes the fluid dynamics video, “A Numerical Sim- ulation of a Plunging Breaking Wave”, which was submitted to the gallery of fluid motion at the 2009 APS/DFD conference. The simula- tion was of a deep-water plunging breaking wave. It was a two-phase calculation which used a Volume of Fluid (VOF) method to simulate the interface between the two immiscible fluids. Surface tension and viscous effects were not considered. The initial wave was generated by applying a spatio-temporal pressure forcing on the free surface. The video shows the 50% isocontour of the volume fraction from several different perspectives. Significant air entrainment is observed as well as the presence of stream-wise vortex structures. 1 Formulation The computational domain, shown schematically in Fig. 1, moves with the linear crest speed of the wave, U, and has dimensions [2π, π/2, 2π], with [1024, 256, 512] grid points. Periodic boundary conditions are used in the horizontal directions and free-slip boundary conditions in the vertical. The Froude number, Fr = U(k/g)1/2, is unity, where g is the acceleration due to gravity and k is the wave number. The density ratio between the two fluids is 1 arXiv:0910.2580v1 [physics.flu-dyn] 14 Oct 2009 Figure 1: Schematic of computational domain. ⃗g = [0, 0, −g]. 1000:1. The total distance traveled by the wave, X in the laboratory frame, is equivalent to Ut in the simulation. Gravity acts in the −z direction. 2 Numerical Method Let u∗ i denote the three-dimensional velocity field, ρ the density, and p the pressure, all functions of space (x∗ i ) and time (t∗), where an asterisk denotes a dimensional quantity. Time, space, velocity, density and pressure are normalized as follows: t = t∗Uk , xi = x∗ i k , ui = u∗ i U , ρ = ρ∗ ρw , p = p∗ ρwU2 . (1) where ρw is the density of water, k is the wave number, and U = (g/k)1/2. In the Volume of Fluid method (Rider et al. (1994)) the fraction of fluid that is inside a cell is denoted by φ. By definition, φ = 0 for a cell that is totally filled with air, and φ = 1 for a cell that is totally filled with water. The density expressed in terms of φ is ρ = (1 −λ)φ + φ (2) where λ is the density ratio between air and water. 2 The non-dimensional governing equations are: Momentum: ∂ui ∂t + ∂(ukui) ∂xk = −1 ρ ∂p ∂xi − 1 Fr2 δi3 + pa ρ δ (φ) , (3) VOF: ∂φ ∂t + uk ∂φ ∂xk = 0. (4) The relevant non-dimensional parameter is the surface Froude number, Fr = U(k/g)1/2. The temporal integration is handled with an explicit RK2 scheme, and the advective terms with the flux based limited QUICK scheme of Leonard (1997). The VOF algorithm uses the operator-split method of Puckett et al. (1997). As discussed in Dommermuth et al. (1998) the di- vergence of the momentum equation, Eq. (3), combined with the solenoidal constraint ∂ui/∂xi = 0, provide a Poisson equation for the dynamic pres- sure. The last term on the r.h.s. of Eq. (3) is the atmospheric pressure forcing term. In the simulation discussed here the following forcing function was used for t < 4π pa = 1 2 [A0cos (x) + Ar]  1 −cos  t 2  (5) where 0.1A0 =  1 V Z V dV Ar (xi) 2 (6) Here, A0 = 0.02 and Ar is a uniform random disturbance that has been passed through a low-pass filter. 3 Flow description Fig. 2 is a plot the the total energy, E(t) = Z V dV ρ(t) (Ui(t)Ui(t) + gz) , (7) over time. Four distinct stages are evident in Fig. 2, they are: A. Atmo- spheric forcing; B. Potential flow before breaking; C. Breaking which con- sists of plunging, spilling and splash-up events; and D. Potential flow after breaking. 3 Figure 2: Total Energy, R V dV ρ (UiUi + gz) 4 Video The full size video, mpeg2 encoded, is approximately 75Mb (download). The web size video, mpeg1 encoded is approximately 10Mb (download). The videos are also available at www.saic.com/nfa. Scenes 1 and 2 in the video start near the end of stage A (see Fig. 2) and show the evolution through stages B and C and end early in stage D. Scene 3 starts after the initial plunging event, early in stage C and ends late in stage C. Scene 4 starts near the end of stage B, and ends in the middle of stage C. Scene 5 (Vortex Tubes) starts near the end of stage B and pauses/ends early in stage C. In scenes 1, 2 and 3 the 50% isocontour of the volume is shown. In scenes 4 and 5 the magnitude of velocity projected onto the free surface is shown in selected regions. SCENES: 1. Tank View– A perspective view with walls added to provide context to the breaking events. After the initial plunging event, a jet is formed (splash-up). After the secondary jet impinges on the free surface the remaining

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