Meandering flows are wave driven much like streaming flows. The interaction of the wavy portion of the flow with the vortical portion of the flow gives rise to meandering flows. Meandering flows, like streaming flows, are not turbulent. The nonlinear wave interactions give sum and difference frequencies in meandering flows. The difference frequencies have long been associated with streaming flows in the ocean (Longuet-Higgins, 1953). Meandering flows occur in the atmosphere as well as the ocean. Streaming flows are a subset of meandering flows. Meandering flows are expected to vary over long distances due to nonlinear wave interactions, variations in currents and winds, etc. In the upper ocean, (Dommermuth, 2020d) shows that slow variations of the meandering wind-drift interacting with the earth’s magnetic field induces a magnetic disturbance in the upper ocean and lower atmosphere. Dommermuth (2018b,c,d,e) show how the wind drift affects wave breaking. Here, the effects of variations in the meandering wind drift and meandering wind on the surface charge density are considered. Blanchard (1963); Gathman (1986) show that the surface charge density is due to the electrification of the atmosphere by (1) bubbles bursting on the ocean surface due to the effects of wave breaking and (2) sea spray torn off the crests of waves by wind shear. The variations in surface charge density are expected due to variations in wave breaking and wind shear occurring over long spatial scales due to meandering flows. This paper along with Dommermuth (2020d) provide experimentalists with bases for measuring effects of meandering flows on the magnetic and electric fields in the ocean and atmosphere. Experiments can be performed to confirm the existence of meandering flows and to quantify the mixing of the upper ocean and lower atmosphere. During fair weather the electric field that is generated by the effect of meandering flows on the surface charge density is about 10% of the potential gradient at altitudes 3km above the ocean surface. The effects of the electric disturbance attenuate slowly with altitude at a rate that is very similar to the potential gradient. The frequencies of oscillations are about 0.2-1Hz due to nonlinear sum-frequency wave interactions. During a storm the electric field that is generated by the meandering flows would become increasingly violent with significant energy being radiated within specific frequency bands in an organized manner.
The Ocean’s Heartbeat (OH) is an inverse energy cascade that occurs through interactions between surface gravity waves and organized vortical structures in the atmosphere and the ocean. The vortical wake of spilling breaking waves can generate the inverse energy cascade even in the absence of wind and current shear. Standing waves are generated as the vortical portions of the flow modulate the generation and evolution of surface gravity waves and vice versa. Resonances occur between the standing waves and coherent structures in the ocean. Monopolar, dipolar, tripolar, and quadrupolar vortical structures (OH structures) are shed out the back of spilling breaking waves. Intense OH structures generate knots in the free-surface elevation in the wake of spilling breaking waves. OH structures surf the crests of spilling breaking waves slightly behind the whitecaps. The results of numerical simulations give credence to two conjectures: 1) OH standing waves can generate microseisms even when opposing wave groups are not present and 2) The OH inverse energy cascade is present in satellite altimetry of sea surface height measurements.
The interaction of Stokes waves with log profiles of the wind in the atmosphere and the wind drift in the ocean is studied. Data assimilation is used to nudge the wavy portion of the base flow toward a Stokes wave and the vortical portion of the base flow toward log profiles. The data assimilation framework allows for free-surface vorticity and the turbulent diffusion of free-surface vorticity into the atmosphere and into the ocean for the vortical portion of the flow.
The results of numerical simulations show that coherent structures form on the free surface and diffuse upward into the atmosphere and downward into the ocean even in the absence of stratification. As the crests of the Stokes waves pass over the coherent structures, the amplitudes of the coherent structures abruptly increase and the phases of the coherent structures abruptly change. As the friction velocity in the water increases, the coherent structures surf the crests of steep Stokes waves. The resulting turbulent wakes that form behind the wave crests induce meandering flows with vertical structures that are helical in the atmosphere and the ocean. Cross sections of the meandering flows transverse to the wind form layers with a diagonal pattern that reflects the helical shedding into the wake. The successive passing of coherent structures beneath the wave crests gives rise to a multitude of spatial and temporal variations. In view of the important contributions that the coherent structures and meandering flows make to the mixing of the atmosphere and the ocean, the physics associated with these spatial and temporal variations are herein named the Ocean’s Heartbeat. As discussed in this paper, the Ocean’s Heartbeat can be heard using electromagnetic principles.
Windrows form at the interstices of the coherent structures even in the absence of breaking waves. As the steepnesses of the Stokes waves increase, the rate of lateral spreading of the windrows in- creases, and the windrows become more rectilinear, less sinuous, and less diffuse. The numerical simulations show good agreement for the following experimental observations: 1) Convergence zones form beneath windrows, 2) Divergence zones form between windrows, 3) Downwelling oc- curs beneath windrows; 4) Upwelling occurs between windrows; 5) Streamwise velocities are enhanced beneath windrows; 6) Streamwise velocities are diminished between windrows; and 7) Y-Junctions that point upwind form as windrows merge. Although Y-Junctions that point down- wind have been observed, their importance has not been previously recognized. The numerical results show that Y-Junctions that point downwind form as windrows split.
The kinetic energy of the coherent structures and meandering flows increases linearly with respect to time in correspondence with forced two-dimensional turbulence and the formation of an inverse energy cascade. Also, in correspondence with forced two-dimensional turbulence, the enstrophy of the vertical component of vorticity is constant on average in a surface-following coordinate system in planes that are parallel to the free surface in both the atmosphere and the ocean. The simulation with the longest duration shows evidence of energy condensation as the length scales of the coherent structures approach the size of the computational domain. The flux of energy into the vortical portion of the flow increases as the wave steepness increases. The flux of energy into the vortical portion of the flow also increases as the friction velocities in the atmosphere and ocean increase relative to the phase speed of the Stokes wave.
The linear growth rate of energy in the vortical portion of the flow is comparable to the initial exponential growth rate of wind-driven ocean waves for steep Stokes waves with intermediate age. The physical scales in this study correspond to a region where there is significant scatter in Plant (1982)’s wave-growth measurements for inverse wave ages less than u∗/c_o < 0.4, where u∗ is the friction velocity in the air and c_o is the phase speed of the Stokes wave. The inverse energy cascade can be so strong that modulation of the waves through a feedback mechanism occurs. As the waves are modulated by the vortical portion of the flow, the inverse energy cascade momentarily breaks down and then reestablishes itself. It is conjectured that growing seas jump back and forth between states of two and three-dimensional turbulence as is evident in the growth of energy and the oscillations in entrophy. During this phase, wave breaking occurs in such a manner that windrows do not break up, which supports Dommermuth (2020)’s conjecture that spilling breaking occurs in lanes. The spilling breaking waves and coherent structures work in concert to form windrows! Numerical simulations with larger domains are required to clarify the physics.
Preliminary results indicate that the growth rate of the kinetic energy in the vortical portion of the flow for fixed friction velocity in the water scales according to the turbulent diffusion of the initial free-surface vorticity, which is expressed in terms of a non-dimensional Ocean’s Heartbeat number, β_v ∼ R_H. R_H = ω^s/(k^2ν^s) ≈ 500, where k is the wavenumber of the Stokes wave, ν^s is the two-dimensional eddy viscosity evaluated on the free surface, and ω^s is the initial vorticity on the free surface at the crest of the Stokes wave in an irrotational flow. For a steady flow, the initial free-surface vorticity is expressed in terms of the surface curvature (κ) and the total tangential velocity (u_t) evaluated on the free surface: ω_s = −2κ u_t. ω_s quantifies both the initial enstrophy for the initial boundary value problems starting from rest and the ongoing production of turbulence through interactions with shear. For variations of the wave friction velocity with fixed wave steepness, it is conjectured that conservation of wave action should be considered.
The Ocean’s Heartbeat number R_H reflects scaling in accordance with forced two-dimensional turbulence. The forcing here is provided by continuously nudging the wavy portion of the flow to an irrotational Stokes wave while the vortical portion of the flow is nudged toward log profiles in the atmosphere and the ocean. The effects of the friction velocities in the atmosphere and the ocean are included indirectly in the eddy viscosity.
The formation of Langmuir circulations is associated with an inverse energy cascade. The forma- tion of windrows with large lateral spreading is the manifestation of this inverse energy cascade. Meandering flows form large coherent structures in the atmosphere and ocean due to this inverse energy cascade. Meandering flows are fundamental mechanisms for mixing in the ocean and atmosphere over large spatial and temporal scales. The Ocean’s Heartbeat is an extraordinary mechanism by which the wavy portion of the flow strongly forces the vortical portions of the flow in the atmosphere and the ocean.
Dommermuth, D. G., Rhymes, L. E., and Rottman, J. W., “Direct Simulations of Breaking Ocean Waves with Data Assimilation,” OCEANS, 2013, San Diego, California, USA, 2013. https://www.researchgate.net/publication/269573554
Dommermuth, D. G., Lewis, C. D., Tran, V. H., and Valenciano, M. A., “Direct Simulations of Wind- Driven Breaking Ocean Waves with Data Assimilation,” Proceedings of the 30th Symposium on Naval Hydrodynamics, Hobart, Tasmania, Australia, 2014. https://www.researchgate.net/publication/266396605
Dommermuth, D.G., “The Ocean’s Heartbeat: An Inverse Energy Cascade that Mixes the Lower Atmosphere and Upper Ocean,” PowerPoint presentation, Air-Sea Interactions and Implications for Offshore Wind Energy, virtual event, February 10-11, 2022. https://www.researchgate.net/publication/358510653
Dommermuth, D.G., “The Ocean’s Heartbeat: An Inverse Energy Cascade that Mixes the Lower Atmosphere and Upper Ocean,” 34th Symposium on Naval Hydrodynamics, Washington, D.C., USA, June 26 – July 1, 2022. https://www.researchgate.net/publication/361556689
Dommermuth, D.G., “Frameworks for Studying the Ocean’s Heartbeat,” PowerPoint presentation, 35th Symposium on Naval Hydrodynamics, Nantes, France, July 8-12, 2024. https://www.researchgate.net/publication/382268957
Dommermuth, D. G., “The Generation of Electric Fields by Meandering Flows,” ResearchGate preprint, Oct 2020. https://www.researchgate.net/publication/344787449(Please see more recent technical reports on the electric field that is induced by the transport of space charge density by the meandering wind.)
Dommermuth, D. G., “The Electric and Magnetic Fields due to the Transport of Space Charge Density by the Meandering Wind over the Ocean Surface,” ResearchGate preprint, Sep 2021. https://www.researchgate.net/publication/354665883(Please see more recent technical reports on the electric field that is induced by the transport of space charge density by the meandering wind.)
Dommermuth, D.G., “The Electric and Magnetic Fields due to the Transport of Space Charge Density by the Meandering Wind over the Ocean Surface: New Evidence of an Inverse Energy Cascade in the Lower Atmosphere,” ResearchGate preprint, Sep 2021. https://www.researchgate.net/publication/354935485
Dommermuth, D.G., “The Electric and Magnetic Fields due to Magnetic Induction by Meandering Flows in the Oceanic and Atmospheric Boundary Layers: New Evidence of an Inverse Energy Cascade in the Upper Ocean,” ResearchGate preprint, Oct 2021. https://www.researchgate.net/publication/355215804
Dommermuth, D.G., “A Parametric Study of the Electric Field in the Atmosphere due to the Transport of Space Charge Density by the Meandering Wind over the Ocean Surface,” ResearchGate preprint, Nov 2021. https://www.researchgate.net/publication/356002487
Maxima Scripts for Meandering Flows
Dommermuth, D.G., “Maxima Coding for Solving the Electric and Magnetic Fields due to the Transport of Space Charge Density over the Ocean Surface: New Evidence of an Inverse Energy Cascade in the Lower Atmosphere,” ResearchGate code, Sep 2021. https://www.researchgate.net/publication/354935522
Dommermuth, D.G., “A Maxima Script for Solving the Electric and Magnetic Fields due to Magnetic Induction by Meandering Flows in the Oceanic and Atmospheric Boundary Layers: New Evidence of an Inverse Energy Cascade in the Upper Ocean,” ResearchGate code, Oct 2021. https://www.researchgate.net/publication/355209396
Fortran Codes for Meandering Flows
Dommermuth, D.G., “A Fortran Code for Calculating Electric and Magnetic Fields due to the Transport of Space Charge Density by the Meandering Wind over the Ocean Surface: New Evidence of an Inverse Energy Cascade in the Lower Atmosphere,” ResearchGate code, Sep 2021. https://www.researchgate.net/publication/354935467
Dommermuth, D.G., “A Fortran Code for Calculating the Electric and Magnetic Fields due to Magnetic Induction by Meandering Flows in the Oceanic and Atmospheric Boundary Layers: New Evidence of an Inverse Energy Cascade in the Upper Ocean,” ResearchGate code, Oct 2021. https://www.researchgate.net/publication/355209298
Dommermuth, D.G., “F90 Coding for Calculating the Magnetic Fields due to Magnetic Induction by Meandering Drift Currents,” ResearchGate code, Oct 2021. https://www.researchgate.net/publication/355651566
Dommermuth, D.G., “F90 Coding for a Parametric Study of the Electric Field in the Atmosphere due to the Transport of Space Charge Density by the Meandering Wind over the Ocean Surface,” ResearchGate code, Nov 2021. https://www.researchgate.net/publication/356002311
The Effect of the Wind Drift on Wave Growth, Wave Breaking, and the Production of Turbulence
Dommermuth, D. G., “The Effect of Wind-Drift Currents on the Production of Turbulent Kinetic Energy During Wave Breaking,” ResearchGate preprint, 2018. https://www.researchgate.net/publication/325139479
Helmholtz Decompositions into Wavy and Vortical Portions
Dommermuth, D. G., “The laminar interactions of a pair of vortex tubes with a free surface,” J. Fluid Mech., Vol. 246, 1993, pp. 91–115. https://doi.org/10.1017/S0022112093000059
Mui, R. C. and Dommermuth, D. G., “The vortical structure of a near-breaking gravity-capillary wave,” Journal of Fluids Engineering, Vol. 117, 1994,355–361. https://doi.org/10.1115/1.2817269
Dommermuth, D. G., Novikov, E.A., and Mui, C.Y., “The Interaction of Surface Waves with Turbulence,” The Proceedings of the Symposium on Free-Surface Turbulence, ASME Fluids Engineering Division Summer Meeting, Lake Tahoe, California, USA, 1994. https://www.researchgate.net/publication/271527603
Entrainment and Mixing due to Plunging Breaking Waves
Brucker, K. A., O’Shea, T. T., Dommermuth, D. G., and Adams, P., “Three-dimensional simulations of deep-water breaking waves,” Proceedings of the 28th Symposium on Naval Hydrodynamics, Pasadena, California, USA, 2010. https://www.researchgate.net/publication/266619197
Dommermuth, D. G., “The Entrainment and Mixing of Air due to a Rectilinear Vortex Moving Parallel to a Free Surface,” ResearchGate preprint, Jun 2020. https://www.researchgate.net/publication/342247893
Dommermuth et al. (2014); Dommermuth (2020b) show that windrows form due to the action of breaking waves. Foam, biological material, flotsam, and jetsam surf the fronts of breaking waves. The surfing action of the breaking waves scrubs the free surface clean like a wedge plow on a train clears snow. Dommermuth et al. (2014); Dommermuth (2020b) hypothesize that each successive pass of a breaking wave increases the length of the windrows. There is a sweet spot for forming long rectilinear windrows whereby there is surfing followed by spilling of foam, flotsam, jetsam, etc. off to the sides of the whitecaps. Dommermuth (2020c) shows that windrows are less likely to persist if the whitecaps are long crested. When whitecaps are too long transverse to the wind, successive passes of breaking waves tend to break up the windrows. Conversely, if whitecaps are too narrow transverse to the wind, there is no surfing to line up foam, flotsam, jetsam, etc. Here, numerical simulations confirm that successive surfing and scrubbing events lead to the formation of windrows. For Type 1 Windrows, successive spilling breaking waves progress down the middle of the breaking wave that proceeded them, whereas for Type 2 Windrows, successive spilling breaking waves progress down the middle of the windrow band of the breaking wave that proceeded them. The windrows that are formed by Type 1 interactions are thick and rectilinear. The windrows that are formed by Type 2 interactions are thinner, clumpy, and serpentine. Type 2 Windrows have twice as many bands as Type 1. Random fluctuations to the lateral positions of the spilling breakers show that Type 1 and Type 2 Windrows are persistent over a broad range. However, windrows will not form if the lateral positions of spilling breaking waves are uniformly distributed relative to each other.
Dommermuth et al. (2014) shows that windrows form due to the action of breaking waves. Foam, biological material, flotsam, and jetsam surf the fronts of breaking waves. The surfing action of the breaking waves scrubs the free surface clean like a wedge plow on a train clears snow. Each successive pass of a breaking wave increases the length of the windrows. This surfing mechanism differs from Langmuir (1938)’s original hypothesis that the “seaweed accumulated in streaks because of transverse surface currents converging toward the streaks.” There is a sweet spot for forming long rectilinear windrows whereby there is surfing followed by spilling of foam, flotsam, jetsam, etc. off to the sides of the whitecaps. Windrows are less likely to persist if the whitecaps are long crested. When whitecaps are too long transverse to the wind, successive passes of breaking waves tend to break up the windrows. Conversely, if whitecaps are too narrow transverse to the wind, there is no surfing to line up foam, flotsam, jetsam, etc.
A rectilinear vortex with a core filled with air models a plunging breaking wave whose tip has pinched off a pocket of air. For a rectilinear vortex moving parallel to the free surface, air is entrained as the free surface wraps around the axial vorticity. The amount of air that is entrained depends on a Froude number that is based on the circulation of vorticity. Cross-axis vorticity is formed as a result of instabilities in the axial vorticity. Roll-wave instabilities form on the outer face of the sheet of water that wraps around the axial vorticity. For sufficiently high Froude numbers, both the axial and cross-axis vortex tubes entrain air. The air that is entrained is driven by pressure gradients toward the centers of the axial and cross-axis vortex tubes where the pressures are minimal.
Smoothing is required in two-phase Volume-of-Fluid (VOF) simulations at high Reynolds numbers to prevent tearing of the free surface due to a discontinuity in the tangential velocity between the air and the water across the free surface. The free-surface boundary layer is not resolved in VOF simulations at high Reynolds numbers with large density jumps such as air and water. Under these circumstances, the tangential velocity is discontinuous across the free-surface interface and the normal component is continuous. As a result of the discontinuity, unphysical tearing of the free surface tends to occur.
As discussed by Fu, Ratcliffe, O’Shea, Brucker, Graham, and Wyatt (2010) and Dommermuth, Fu, O’Shea, Brucker, and Wyatt (2010), Favre-like filtering can be used to alleviate this problem by forcing the air velocity at the interface to be driven by the water velocity in a physical manner. These papers are available at http://tinyurl.com/ybh4rw4w and http://tinyurl.com/yaedzwqe. The filtering is similar to using smoothing in High-Order Spectral (HOS) simulations (see Dommermuth and Yue, 1987). This type of filtering is called density-weighted velocity smoothing (DWVS) by Dommermuth.
Sussman (2003) and Sussman, Smith, Hussiani, Ohta, and Zhi-Wei (2007) use extrapolation to push the water velocity into the air. With extrapolation, there is the possibility of overshooting the velocity. Also, the tests that Sussman (2003) and Sussman, et al. (2007) perform do not consider the formation of thin jets, which is a more demanding test. However, the method is promising and deserves further investigation.
Fu, et al. (2010) discuss the use of jets formed by certain types of standing waves to test DWVS. The test is based on the studies of Longuet-Higgins and Dommermuth (2001). As Fu, et al. (2001) show, the test is easily set up in two-phase simulations of standing waves.
The test is illustrated using four different cases. An animation of the results is available at the top of this blog entry. The upper left quadrant is a 128^3 simulation with DWVS, the lower left quadrant is a 256^3 with DWVS, the upper right quadrant is a 128^3 simulation without smoothing, and the lower right quadrant is a 256^3 simulation without smoothing. The simulations without smoothing breakup over time, especially the high-resolution simulation, whereas the simulations with DWVS converge.
The figure below is from Fu, et al. (2001). The figure shows three different resolutions with DWVS to a boundary-integral method. For the highest-resolution DWVS simulation there is very good agreement with the boundary-integral method with a physical breakup at the tips of the free-surface jets where vortices are shed into the air. Details of the comparison are provided in Fu, et al. (2001).
The viscosity in DNS simulations at low Reynolds number inhibits the tearing of the free surface. Examples of DNS simulations at low Reynolds numbers include Deike, Popinet, and Melville (2015), Deike, Melville, and Popinet (2016), and Deike, Pizzo, and Melville (2017), but these approaches cannot be extended to wavelengths greater than 10 to 20cm because of limitations associated with resolving the free-surface boundary layer and severe Courant restrictions. Often DNS simulations like Deike, Melville, and Popinet (2016) use an artificially low Reynolds number to stabilize their simulations. Even with an artificially high viscosity, the tearing that is observed in some DNS simulations is probably due to poor resolution of the free-surface boundary layer and the subsequent jump in the tangential velocities.
As Dommermuth, D.G., Lewis, C.D., Tran, V.H., and Valenciano, M.A. (2014) show, simulations at high Reynolds numbers are desirable to study (1) the formation of windrows, Langmuir circulations, and wind streaks; (2) wave growth due to the effects of the wind boundary layer in the air and the drift layer in the water; (3) meandering of the cross wind and cross drift; and (4) spilling breaking waves. Dommermuth, et al. (2014) is available at http://tinyurl.com/y79sm9rp. Videos associated with Dommermuth, et al. (2014) are available at http://tinyurl.com/y8madvbw.
References:
Brucker, K. A., O’Shea, T. T., Dommermuth, D. G., and Adams, P. (2010) “Three-dimensional simulations of deep-water breaking waves,” Proceedings of the 28th Symposium on Naval Hydrodynamics, Pasadena, California, USA.
Dommermuth, D. G. and Yue, D. K. (1987) A high-order spectral method for the study of nonlinear gravity waves, J. Fluid Mech., vol. 187, 267–288.
Dommermuth, D. G., Fu, T. C., O’Shea, T. T., Brucker, K. A., and Wyatt, D. C., (2010) Numerical prediction of a seaway, Proceedings of the 28th Symposium on Naval Hydrodynamics, Pasadena, California, USA.
Dommermuth, D.G., Lewis, C.D., Tran, V.H., and Valenciano, M.A. (2014) “Simulations of wind-driven breaking ocean waves with data assimilation,” Proceedings of the 30th Symposium on Naval Hydrodynamics, Hobart, Tasmania, Australia.
Deike, L., S. Popinet, and W. Melville (2015), Capillary effects on wave breaking, J. Fluid Mech., 769, 541–569.
Deike, L., W. Melville, and S. Popinet (2016), Air entrainment and bubble statistics in breaking waves, J. Fluid Mech., 801, 91–129.
Deike, L., Pizzo, N. E., and Melville, W. K. (2017) Lagrangian transport by breaking surface waves. J. Fluid Mech. (submitted).
Fu, T.C., Ratcliffe, T., O’Shea, T.T., Brucker, K.A., Graham, R.S., Wyatt, D.C., and Dommermuth, D.G. (2010) A comparison of experimental measurements and computational predictions of a deep-v planing hull, Proceedings of the 28th Symposium on Naval Hydrodynamics, Pasadena, California, USA, 2010.
Longuet-Higgins, M. and Dommermuth, D. (2001) Vertical jets from standing waves, Proc. R. Soc. Lond., Vol. A 457, 2137–2149.
Sussman, M. (2003) A second-order coupled levelset and volume of fluid method for computing growth and collapse of vapor bubbles, J. of Comput. Phys., 187, 110–136.
Sussman, M., Smith, K.M., Hussaini, M.Y., Ohta, M., and Zhi-Wei, R. (2007) A sharp interface method for incompressible two-phase flows. J. Comput. Phys., 221, 469-505.
Numerical simulations of nonlinear progressive waves are prone to developing spurious high-frequency standing waves unless the flow field is given sufficient time to adjust (Dommermuth, 2000). An adjustment scheme allows the natural development of nonlinear self-wave (locked modes) and inter-wave (free modes) interactions.
For High-Order Spectral (HOS) simulations, an adjustment procedure allows nonlinear free-surface simulations to be initialized with linear solutions. For two-phase volume-of-fluid simulations like the Numerical Flow Analysis (NFA) code, nonlinear waves are generated from rest by gradually applying a surface stress. NFA requires a different approach than HOS because the necessary two-phase solutions for progressive waves are not readily available. Dommermuth, Fu, O’Shea, Brucker, and Wyatt (2010) show that the adjustment procedure can be used to initialize fully-nonlinear simulations of regular waves and irregular seas. The paper is available at http://tinyurl.com/yaedzwqe.
The figure below shows an adjusted NFA solution in comparison to an exact Stokes wave up to the sixth harmonic. Note that there are no oscillations in the Fourier harmonics.
Deike, Popinet, and Melville (2015), Deike, Melville, and Popinet (2016), and Deike, Pizzo, and Melville (2017) use third-order Stokes solutions to initialize their two-phase VOF simulations. As discussed by Dommermuth (2000) and Dommermuth, et al. (2010), third-order Stokes solutions lead to the formation of spurious high-frequency standing waves for simulations of both regular waves and irregular seas and are inappropriate for fully-nonlinear simulations of waves. The gradual application of a surface stress (see Dommermuth, et al., 2010) eliminates problems with initializing two-phase VOF simulations with third-order solutions.
References:
Dommermuth, D. G. (2000) The initialization of nonlinear waves using an adjustment scheme, Wave Motion, 32, 307–317.
Dommermuth, D. G., Fu, T. C., O’Shea, T. T., Brucker, K. A., and Wyatt, D. C., (2010) Numerical prediction of a seaway, Proceedings of the 28th Symposium on Naval Hydrodynamics, Pasadena, California, USA.
Deike, L., S. Popinet, and W. Melville (2015), Capillary effects on wave breaking, J. Fluid Mech., 769, 541–569.
Deike, L., W. Melville, and S. Popinet (2016), Air entrainment and bubble statistics in breaking waves, J. Fluid Mech., 801, 91–129.
Deike, L., Pizzo, N. E., and Melville, W. K. (2017) Lagrangian transport by breaking surface waves. J. Fluid Mech. (submitted).
The figure below shows fissioning of a wave packet using the High-Order Spectral (HOS) method and the Numerical Flow Analysis (NFA) code with comparisons to experiments. The experiments are reported in Su (1982). Details of the HOS simulation are available in Dommermuth and Yue (1987). Details of the NFA simulation are available in Dommermuth, Fu, Brucker, O’Shea, and Wyatt (2010). The paper is available at http://tinyurl.com/yaedzwqe.
Su (1982) studied experimentally the evolution of wave groups that had initially square envelopes. For wave steepnesses ranging from 0.09 to 0.28, he measured the free-surface elevation at eight stations down the tank. For wave steepnesses greater than 0.14, he observed intense two-dimensional breaking at distances between ten and twenty carrier wavelengths from the wavemaker. Fifteen to twenty-five wavelengths from the wavemaker, crescent-shaped breaking waves often developed, and from twenty to forty-five wavelengths away, two-dimensional spilling breaking was common. The initial packet has 5 waves.
NFA is a two-phase formulation with volume-of-fluid code interface tracking. NFA solves the full three- dimensional Navier-Stokes equations using an implicit sub-grid scale (SGS) model. NFA allows wave overturning and wave breaking. The results here show that NFA can accurately model subtle wave nonlinearities.
For the NFA simulation, a wave packet is generated using a surface stress. A fixed frame of reference is used. Unlike the HOS simulations, the NFA simulations allow breaking. For these NFA simulations, the breaking manifests itself as coaming at the crests of the waves at the front of the wave group. At about forty wavelengths from the wavemaker, we confirm the experimental observation that the wave group fissions into two packets. The NFA simulations are able to predict a weak wave nonlinearity that occurs over one hundred wave periods. The agreement between results of the numerical simulations and experimental measurements suggests that VOF simulations could be used to study the long-time evolution of a seaway complete with wave breaking.
References:
Dommermuth, D. G. and Yue, D. K. (1987) A high-order spectral method for the study of nonlinear gravity waves, J. Fluid Mech., vol. 187, 267–288.
Dommermuth, D. G., Fu, T. C., O’Shea, T. T., Brucker, K. A., and Wyatt, D. C. (2010) Numerical prediction of a seaway, Proceedings of the 28th Symposium on Naval Hydrodynamics, Pasadena, California, USA
Su, M. (1982) Evolution of groups of gravity waves with moderate to high steepness, Phys. Fluids, vol. 25, 2167–2174.
