Frequency dispersion of small-amplitude capillary waves in viscous fluids

This work presents a detailed study of the dispersion of capillary waves with small amplitude in viscous fluids using an analytically derived solution to the initial value problem of a small-amplitude capillary wave as well as direct numerical simulation. A rational parametrization for the dispersion of capillary waves in the underdamped regime is proposed, including predictions for the wave number of critical damping based on a harmonic-oscillator model. The scaling resulting from this parametrization leads to a self-similar solution of the frequency dispersion of capillary waves that covers the entire underdamped regime, which allows an accurate evaluation of the frequency at a given wave number, irrespective of the fluid properties. This similarity also reveals characteristic features of capillary waves, for instance that critical damping occurs when the characteristic time scales of dispersive and dissipative mechanisms are balanced. In addition, the presented results suggest that the widely adopted hydrodynamic theory for damped capillary waves does not accurately predict the dispersion when viscous damping is significant, and an alternative definition of the damping rate, which provides consistent accuracy in the underdamped regime, is presented.

Figure 1

DNS results of the amplitude of a capillary wave of case A with wave number $k={10}^{-4}{k}_{\mathrm{c}}$ as a function of dimensionless time $\tau =t{\omega }_0$ for different mesh spacings $\mathrm{\Delta }x$. The analytical solution according to Eq. (12) is shown as solid circles. The inset shows the amplitude for a particular time interval.

Figure 2

Comparison of the dimensionless frequency $\omega /{\omega }_{\mathrm{m}}$ as a function of the dimensionless wave number $k/k_{\mathrm{c}}$ obtained with AIVS and DNS for case A. The inset shows the frequency at small wave numbers, and the solid line represents a spline fit of the AIVS result.

Figure 3

Dimensionless undamped frequency ${\widehat{\omega }}_0={\omega }_0/{\omega }_{0,\mathrm{c}}$ as a function of dimensionless wave number $k/k_{\mathrm{c}}^{*}$, based on the proposed critical wave number given in Eq. (23), and $k/k_{\mathrm{c}}^{\mathrm{B}}$, based on the critical wave number given in Eq. (16).

Figure 4

Damping ratio $\zeta$ as a function of dimensionless wave number $\widehat{k}=k/k_{\mathrm{c}}^{*}$ for the considered cases obtained with AIVS, where point $(1,1)$ represents critical damping. Note that $\zeta \ge 1$ for $\widehat{k}<1$ and $\zeta \le 1$ for $\widehat{k}>1$ are not part of the physical parameter space.

Figure 5

Dimensionless frequency $\widehat{\omega }=\omega t_{\mathrm{vc}}$ as a function of dimensionless wave number $\widehat{k}=k/k_{\mathrm{c}}^{*}$ obtained with AIVS and DNS. The inset shows the frequency of capillary waves with small wave numbers.

Figure 6

Dimensionless damping rate $\widehat{\mathrm{\Gamma }}=\mathrm{\Gamma }\tilde{\rho }/\left(\tilde{\mu }{k}^2\right)$ as a function of dimensionless wave number $\xi =k/k_{\mathrm{c}}^{*,\mathrm{h}.\mathrm{o}.}$ obtained with AIVS for cases A ($\beta =0.0625$) and D ($\beta =0$) and with DNS for case E ($\beta =7\times {10}^{-5}$). The inset shows the dimensionless damping rate at small wave numbers.

Figure 7

Relative error $\varepsilon$ [see Eq. (29)] of the frequency predictions given in Eqs. (28), (30), and (31) with respect to the frequencies obtained from AIVS and DNS as a function of dimensionless wave number $\widehat{k}$.