5 results
Sum-frequency triad interactions among surface waves propagating through an ice sheet
- Max W. Pierce, Yuming Liu, Dick K.P. Yue
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- Journal:
- Journal of Fluid Mechanics / Volume 980 / 10 February 2024
- Published online by Cambridge University Press:
- 06 February 2024, A45
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We study nonlinear resonant wave–wave interactions which occur when ocean waves propagate into a thin floating ice sheet. Using multiple-scale perturbation analysis, we obtain theoretical predictions of the wave amplitude evolution as a function of distance travelled past the ice edge for a semi-infinite ice sheet. The theoretical predictions are supported by a high-order spectral (HOS) method capable of simulating nonlinear interactions in both open water and the ice sheet. Using the HOS method, the amplitude evolution predictions are extended to multiple (coupled) triad interactions and a single ice sheet of finite length. We relate the amplitude evolution to mechanisms with strong frequency dependence – ice bending strain, related to ice breakup, as well as wave reflection and transmission. We show that, due to sum-frequency interactions, the maximum strain in the ice sheet can be more than twice that predicted by linearised theory. For an ice sheet of finite length, we show that nonlinear wave reflection and transmission coefficients depend on a parameter in terms of wave steepness and ice length, and can have values significantly different than those from linear theory. In particular, we show that nonlinear sum-frequency interactions can appreciably decrease the total wave energy transmitted past the ice sheet. This work has implications for understanding the occurrence of ice breakup, wave attenuation due to scattering in the marginal ice zone and the resulting ice floe size distribution.
Fundamental time scales of bubble fragmentation in homogeneous isotropic turbulence
- Declan B. Gaylo, Kelli Hendrickson, Dick K.P. Yue
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- Journal:
- Journal of Fluid Mechanics / Volume 962 / 10 May 2023
- Published online by Cambridge University Press:
- 03 May 2023, A25
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We investigate the fundamental time scales that characterise the statistics of fragmentation under homogeneous isotropic turbulence for air–water bubbly flows at moderate to large bubble Weber numbers, $We$. We elucidate three time scales: $\tau _r$, the characteristic age of bubbles when their subsequent statistics become stationary; $\tau _\ell$, the expected lifetime of a bubble before further fragmentation; and $\tau _c$, the expected time for the air within a bubble to reach the Hinze scale, radius $a_H$, through the fragmentation cascade. The time scale $\tau _\ell$ is important to the population balance equation (PBE), $\tau _r$ is critical to evaluating the applicability of the PBE no-hysteresis assumption, and $\tau _c$ provides the characteristic time for fragmentation cascades to equilibrate. By identifying a non-dimensionalised average speed $\bar {s}$ at which air moves through the cascade, we derive $\tau _c=C_\tau \varepsilon ^{-1/3} a^{2/3} (1-(a_{max}/a_H)^{-2/3})$, where $C_\tau =1/\bar {s}$ and $a_{max}$ is the largest bubble radius in the cascade. While $\bar {s}$ is a function of PBE fragmentation statistics, which depend on the measurement interval $T$, $\bar {s}$ itself is independent of $T$ for $\tau _r \ll T \ll \tau _c$. We verify the $T$-independence of $\bar {s}$ and its direct relationship to $\tau _c$ using Monte Carlo simulations. We perform direct numerical simulations (DNS) at moderate to large bubble Weber numbers, $We$, to measure fragmentation statistics over a range of $T$. We establish that non-stationary effects decay exponentially with $T$, independent of $We$, and provide $\tau _r=C_{r} \varepsilon ^{-1/3} a^{2/3}$ with $C_{r}\approx 0.11$. This gives $\tau _r\ll \tau _\ell$, validating the PBE no-hysteresis assumption. From DNS, we measure $\bar {s}$ and find that for large Weber numbers ($We>30$), $C_{\tau }\approx 9$. In addition to providing $\tau _c$, this obtains a new constraint on fragmentation models for PBE.
Modelling entrainment volume due to surface-parallel vortex interactions with an air–water interface
- Kelli Hendrickson, Xiangming Yu, Dick K.P. Yue
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- Journal:
- Journal of Fluid Mechanics / Volume 938 / 10 May 2022
- Published online by Cambridge University Press:
- 14 March 2022, A12
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We consider the entrainment volume that results from the quasi-two-dimensional interactions of rising surface-parallel vorticity with an air–water interface. Based on systematic (three-dimensional) direct numerical simulations (DNS) of the canonical problem of a rectilinear vortex pair impinging on and entraining air at the free surface, we develop a phenomenological model to predict the resulting entrainment volume in terms of four key parameters. We identify a new parameter, a circulation flux Froude number $Fr^2_\Xi =|\varGamma |W/a^2\,g$, that predicts the dimensionless volume $\forall$ of entrained air initiated by a coherent vortical structure of circulation $\varGamma$, effective radius $a$, vertical rise velocity $W$ with gravity $g$. For $Fr^2_\Xi$ below some critical value $Fr^2_{\Xi cr}$, no air is entrained. For $Fr^2_\Xi >Fr^2_{\Xi cr}$, the average initial entrainment $\overline {\forall }_o$ scales linearly with ($Fr^2_\Xi -Fr^2_{\Xi cr}$). We also find that $\overline {\forall }_o$ is linearly dependent on circulation Weber number $We_{\varGamma }$ for a range of vortex Bond number $5 \lesssim Bo_{\varGamma } \lesssim 50$, and parabolically dependent on circulation Reynolds $Re_{\varGamma }$ for $Re_{\varGamma }\lesssim 2580$. Outside of these ranges, surface tension and viscosity have little effect on the initial entrainment volume. For the canonical rectilinear vortex problem, the simple model predicts $\overline {\forall }_o$ extremely well for individual coherent structures over broad ranges of $Fr^2_\Xi$, $We_{\varGamma }$, $Bo_{\varGamma }$ and $Re_{\varGamma }$. We evaluate the performance of this parameterisation and phenomenological entrainment model for air entrainment due to the complex periodic vortex shedding and quasi-steady wave breaking behind a fully submerged horizontal circular cylinder. For the range of parameters we consider, the phenomenological model predicts the event-by-event dimensionless entrainment volume measured in the DNS satisfactorily for this complex application.
Hydrodynamics of large wave energy converter arrays with random configuration variations
- Grgur Tokić, Dick K.P. Yue
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- Journal:
- Journal of Fluid Mechanics / Volume 923 / 25 September 2021
- Published online by Cambridge University Press:
- 21 July 2021, R1
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We study the effect of random perturbations of body positions in large uniformly spaced arrays of axisymmetric wave energy converters (WECs). We perform systematic computational simulations of ensembles of randomized array configurations that are obtained by introducing zero-mean position perturbations (characterized by randomness parameter $\varepsilon$) to line arrays of uniform spacing $d$. Of special interest are the conditions under which these randomized arrays can extract more energy than the underlying uniform arrays. We find that random WEC arrays achieve substantial energy extraction gains over the same number of isolated bodies in monochromatic and irregular incident seas. Introducing $\varepsilon >0$ acts to smooth out the uneven performance of uniformly spaced arrays over varying incident wavenumber. In the low-scattering regime, the standard deviation of array gain grows with the square of wavenumber, for which we provide a theoretical explanation. We show that the uniform line array with spacing optimized for a given incident spectrum generally outperforms randomized arrays of any mean $d$ and $\varepsilon$ in that spectrum, and we offer a heuristic explanation. This holds for a wide range of incident spectra.
Effects of power-law entrainment on bubble fragmentation cascades
- Declan B. Gaylo, Kelli Hendrickson, Dick K.P. Yue
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- Journal:
- Journal of Fluid Mechanics / Volume 917 / 25 June 2021
- Published online by Cambridge University Press:
- 28 April 2021, R1
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We consider the evolution of the bulk bubble-size distribution $N(a,t)$ of large bubbles (Weber number ${\textit {We}}\gg 1$) under free-surface entrainment described generally by an entrainment size distribution $I(a)$ with power-law slope $\gamma$ and large-radius cutoff $a_{max}$. Our main focus is the interaction between turbulence-driven fragmentation and free-surface entrainment, and, for simplicity, we ignore other mechanisms such as degassing, coalescence and dissolution. Of special interest are the equilibrium bulk distribution $N_{eq}(a)$, with local power-law slope $\tilde {\beta }_{eq}(a)$, and the time scale $\tau _c$ to reach this equilibrium after initiation of entrainment. For bubble radii $a\ll a_{max}$, we find two regimes for the dependence of $N_{eq}(a)$ on the entrainment distribution. There is a weak injection regime for $\gamma \ge -4$, where $\tilde {\beta }_{eq}(a)=-10/3$ independent of the entrainment distribution; and a strong injection regime for $\gamma <-4$, where the power-law slope depends on $\gamma$ and is given by $\tilde {\beta }_{eq}(a)=\gamma +2/3$. The weak regime provides a general explanation for the commonly observed $-10/3$ power law originally proposed by Garrett et al. (J. Phys. Oceanogr., vol. 30 (9), 2000, pp. 2163–2171), and suggests that different weak entrainment mechanisms can all lead to this result. For $a\sim a_{max}$, we find that $N_{eq}(a)$ exhibits a steepening deviation from a power law due to fragmentation and entrainment, similar to what has been observed, but here absent other mechanisms such as degassing. The evolution of $N(a,t)$ to $N_{eq}(a)$ is characterised by the critical time $\tau _c \propto C_f \varepsilon ^{-1/3} {a_{max}}^{2/3}$, where $\varepsilon$ is the turbulence dissipation rate and $C_f$ is a new constant that quantifies the dependence on the daughter size distribution in a fragmentation event. For typical breaking waves, $\tau _c$ can be quite small, limiting the time $t\lesssim \tau _c$ when direct measurement of $N(a,t)$ might provide information about the underlying entrainment size distribution.