Dispersion and viscous attenuation of capillary waves with finite amplitude
Fabian Denner Gounséti Paré Stéphane Zaleski





Department of Mechanical Engineering, Imperial College London,
Exhibiton Road, London, SW7 2AZ, United Kingdom
Sorbonne Universités, UPMC Univ Paris 06, CNRS, UMR 7190, Institut
 Jean Le Rond d'Alembert, F-75005 Paris, France

We present a comprehensive study of the dispersion of capillary waves
with finite amplitude, based on direct numerical simulations.
The presented results show an increase of viscous attenuation and, consequently,
a smaller frequency of capillary waves with increasing initial wave amplitude.
Interestingly, however, the critical wavenumber as well as the wavenumber at
which the maximum frequency is observed 
remain the same for a given two-phase system, irrespective of the wave
amplitude.
By devising an empirical correlation that describes the effect of the wave
amplitude on the viscous attenuation, the dispersion of capillary waves with finite
initial amplitude is shown to be, in very good approximation, self-similar
throughout the entire underdamped regime and independent of the fluid
properties. The results also shown that analytical solutions for capillary waves
with infinitesimal amplitude are applicable with reasonable accuracy
for capillary waves with moderate amplitude.


Dispersion and viscous attenuation of capillary waves with
finite amplitude
    Fabian Denner1f.denner09@imperial.ac.uk
Gounséti Paré2 Stéphane Zaleski2

    
=============================================================================



§ INTRODUCTION

Waves on fluid interfaces for which surface tension is the main restoring and
dispersive mechanism, so-called capillary waves, play a key role in many
physical phenomena, natural processes and engineering applications.
Prominent examples are the heat and mass transfer between the atmosphere and the
ocean <cit.>, capillary wave turbulence
<cit.> and the stability of liquid and
capillary bridges <cit.>.

The dispersion relation for a capillary wave with small amplitude on a fluid
interface between two inviscid fluids is <cit.>

    ω_0^2 = σ k^3/ρ̃ ,

where ω_0 is the undamped angular frequency, σ is the surface
tension coefficient, k is the wavenumber and ρ̃ =ρ_a +
ρ_b is the relevant fluid density, where subscripts a
and b denote properties of the two interacting bulk phases. The
dispersion relation given in Eq. (<ref>) is only valid for waves with
infinitesimal amplitude <cit.>.

In reality, however, capillary waves typically have a finite amplitude.
<cit.> was the first to provide an exact solution for progressive
capillary waves of finite amplitude in fluids of infinite depth. The
frequency of capillary waves with finite amplitude a (measured from the
equilibrium position to the wave crest or trough) and wavelength
λ is given as
<cit.>

    ω = ω_0  (1 + π^2   a^2/λ^2)^-1/4 .

The solution of <cit.> was extended to capillary waves on liquid
films of finite depth by <cit.> and to general gravity and
capillary waves by <cit.>.
However, these studies neglected viscous stresses, in order to make an
analytical solution feasible. 

Since capillary waves typically have a short wavelength (otherwise the influence
of gravity also has to be considered) and because viscous stresses act
preferably at small lengthscales <cit.>,
understanding how viscous stresses affect the dispersion of capillary waves is
crucial for a complete understanding of the associated processes and for
optimising the related applications. Viscous stresses are known to attenuate the
wave motion, with the frequency of capillary waves in viscous fluids being
ω = ω_0 + i Γ.
This complex frequency leads to three damping regimes: the underdamped regime
for k < k_c, critical damping for k = k_c and the
overdamped regime for k > k_c. A wave with critical wavenumber
k_c requires the shortest time to return to its equilibrium state
without oscillating, with the real part of its complex angular frequency
vanishing, Re(ω) = 0. Critical damping, thus, represents the
transition from the underdamped (oscillatory) regime, with k < k_c
and Re(ω) > 0, to the overdamped (non-oscillatory) regime, with
k>k_c and Re(ω) = 0.
Based on the linearised Navier-Stokes equations, in this context usually
referred to as the weak damping assumption, the dispersion relation of capillary
waves in viscous fluids is given as <cit.>

    ω_0^2 + (i ω + 2 ν k^2)^2
     - 4 ν^2  k^4 √(1+ i ω/ν k^2) = 0 ,

where ν = μ/ρ is the kinematic viscosity and μ is the dynamic
viscosity. 

The damping rate based on Eq. (<ref>) is
Γ = 2 ν k^2, applicable for k ≪√(ω_0/ν)
<cit.>. 
Note that Eq. (<ref>) has been derived for a
single fluid with a free surface <cit.>.

Previous analytical and numerical studies showed that the
damping coefficient Γ is not a constant, but is dependent on the
wavenumber and changes significantly throughout the underdamped regime
<cit.>.
<cit.> recently proposed a consistent scaling for
small-amplitude capillary waves in viscous fluids, which leads to a self-similar
characterisation of the frequency dispersion of capillary waves in the entire
underdamped regime. The results reported by <cit.> also suggest that the weak damping
assumption is not appropriate when viscous stresses dominate the dispersion of
capillary waves, close to critical damping.
With regards to finite-amplitude capillary waves in viscous fluids, the
interplay between wave amplitude and viscosity as well as the effect of the
amplitude on the frequency and critical wavelength have yet to be studied and
quantified.

In this article, direct numerical simulation (DNS) is applied to study the
dispersion and viscous attenuation of freely-decaying capillary waves with
finite amplitude in viscous fluids.
The presented results show a nonlinear increase in viscous attenuation and,
hence, a lower frequency for an increasing initial amplitude of capillary waves.
Nevertheless, the critical wavenumber for a given two-phase system is found to
be independent of the initial wave amplitude and is accurately predicted by the
harmonic oscillator model proposed by <cit.>.
An empirical correction to the characteristic viscocapillary timescale is
proposed that leads to a self-similar solution for the dispersion of finite-amplitude
capillary waves in viscous fluids.

In Sect. <ref> the characterisation of capillary waves is
discussed and Sect. <ref> describes the computational
methods used in this study. In Sect. <ref>, the dispersion of
capillary waves with finite amplitude is studied and
Sect. <ref> analyses the validity of linear wave theory
based on an infinitesimal wave amplitude. The article is summarised and
conclusions are drawn in Sect. <ref>.



§ CHARACTERISATION OF CAPILLARY WAVES


Assuming that no gravity is acting, the fluids are free of surfactants and
inertia is negligible, only two physical mechanisms govern the dispersion of
capillary waves; surface tension (dispersion) and viscous stresses
(dissipation). 
The main characteristic of a capillary wave in viscous fluids is its frequency

    ω = ω_0 + iΓ = ω_0  √(1-ζ^2) ,

with ζ = Γ/ω_0 being the damping ratio. 
In the underdamped regime (for k<k_c) the damping ratio is ζ <
1, ζ =1 for critical damping (k=k_c) and ζ >1 in the overdamped regime 
(for k>k_c).
As recently shown by <cit.>, the dispersion of capillary
waves can be consistently parameterised by the
critical wavenumber k_c together with an appropriate timescale.

The wavenumber at which capillary waves are critically damped, the so-called
critical wavenumber, is given as
<cit.>

    k_c = 2^2/3/l_vc (1.0625 - β)  ,

where the viscocapillary lengthscale is

    l_vc = μ̃^2/σ ρ̃ ,

with  μ̃ = μ_a+μ_b, and

    β = ρ_aρ_b/ρ̃^2ν_aν_b/ν̃^2

is a property ratio, with ν̃ = ν_a+ν_b. Note
that l_vc follows from a balance of capillary and viscous timescales
<cit.>. Based on the governing mechanisms, the characteristic
timescale of the dispersion of capillary waves is the viscocapillary timescale
<cit.>

    t_vc = μ̃^3/σ^2  ρ̃ .

Defining the dimensionless wavenumber as k̂=k/k_c and the
dimensionless frequency as ω̂ = ω t_vc results in a
self-similar characterisation of the dispersion of capillary waves with small
(infinitesimal) amplitude <cit.>, i.e. there exists a
single dimensionless frequency ω̂ for every dimensionless wavenumber k̂.



§ COMPUTATIONAL METHODS


The incompressible flow of isothermal, Newtonian fluids is governed by the
momentum equations

    ∂  u_i/∂ t + u_j ∂
     u_i/∂ x_j  = - 1/ρ∂ p/∂ x_i
     + ∂/∂ x_j[ ν(∂
     u_i/∂ x_j + ∂ u_j/∂ x_i)] 
     + f_σ,i/ρ

and the continuity equation

    ∂ u_i/∂ x_i = 0  ,

where x≡ (x,y,z) denotes a Cartesian coordinate system, t
represents time, u is the velocity, p is the pressure and f_σ
is the volumetric force due to surface tension acting at the fluid
interface. 
The hydrodynamic balance of forces acting at the fluid
interface is given as <cit.>

    (p_a - p_b. + . σ κ)
    m̂_i = [ μ_a(. ∂ u_i/∂
    x_j|_a + . ∂ u_j/∂ x_i|_a)
    . - . μ_b( . ∂ u_i/∂
    x_j|_b + . ∂ u_j/∂ x_i|_b) ] m̂_j - ∂σ/∂ x_i ,

where κ is the curvature and m̂ is the unit normal
vector (pointing into fluid b) of the fluid interface.
In the current study the surface tension coefficient σ is taken
to be constant and, hence, ∇σ = 0. <cit.> performed
extensive molecular dynamics simulations, showing that hydrodynamic theory is applicable to capillary waves
in the underdamped regime as well as at critical damping.



 §.§ DNS methodology


The governing equations are solved numerically in a single linear system of
equations using a coupled finite-volume framework with collocated variable
arrangement <cit.>, resolving all relevant scales in space and time.
The momentum equations, Eq. (<ref>), are discretised using a Second-Order Backward Euler
scheme for the transient term and convection is discretised using central
differencing <cit.>. The continuity equation,
Eq. (<ref>), is discretised using the 
momentum-weighted interpolation method for two-phase flows proposed by
<cit.>, providing an accurate and robust pressure-velocity coupling.

The Volume-of-Fluid (VOF) method <cit.> is adopted to capture the
interface between the immiscible bulk phases.
The local volume fraction of both phases is represented by the colour function
γ, with the interface located in mesh cells with a colour function value
of 0 < γ < 1. The local density ρ and viscosity μ are
interpolated using an arithmetic average based on the colour function γ
<cit.>, e.g. ρ(x) =
ρ_a [1-γ(x)] + ρ_bγ(x) for density.
The colour function γ is advected by the linear advection equation

    ∂γ/∂ t + u_i ∂γ/∂ x_i
    = 0  ,

which is discretised using a compressive VOF method <cit.>.

Surface tension is modelled as a surface force per unit volume, described by
the CSF model <cit.> as
f_s =  σ κ ∇γ.
The interface curvature is computed as
κ = h_xx/(1+h_x^2)^3/2,
where h_x and h_xx represent the first and second derivatives of the
height function h of the colour function with respect to the x-axis of
the Cartesian coordinate system, calculated by means of central differences.
No convolution is applied to smooth the colour function field or the surface
force <cit.>.

A standing capillary wave with wavelength λ and initial amplitude a_0
in four different two-phase systems is simulated. The fluid properties of the
considered cases, which have previously also been considered in the study on the
dispersion of small-amplitude capillary waves in viscous fluids by
<cit.>, are given in Table <ref>.
The computational domain, sketched in Fig. <ref>, has the dimensions
λ× 3λ and is represented by an equidistant Cartesian mesh
with mesh spacing Δ x = λ/100, which has previously been shown to
provide an adequate spatial resolution <cit.>. The applied
computational time-step is Δ t = (200  ω_0)^-1, which satisfies
the capillary time-step constraint <cit.> and results in a Courant
number of 𝐶𝑜 = Δ t   |u|/Δ x < 10^-2.
The domain boundaries oriented parallel to the interface are treated as
free-slip walls, whereas periodic boundary conditions are applied at the other
domain boundaries. 



The flow field is initially stationary and no gravity is acting.





 §.§ Analytical initial-value solution


The analytical initial-value solution (AIVS) for small-amplitude capillary waves
in viscous fluids, as proposed by <cit.> based
on the linearised Navier-Stokes equations, for the special cases of a single
fluid with a free-surface (i.e. ρ_b = μ_b = 0)
<cit.> and for two-phase systems with equal bulk phases of
equal kinematic viscosity (i.e. ν_a=ν_b)
<cit.> is considered as reference solution. Since the AIVS is
based on the linearised Navier-Stokes equations, it is only valid in the limit
of infinitesimal wave amplitude a_0 → 0. In the present
study, the AIVS is computed at time intervals Δ t = (200  ω_0)^-1, i.e. with 200 solutions per undamped period, which provides a sufficient
temporal resolution of the evolution of the capillary wave.



 §.§ Validation

The dimensionless frequency ω̂=ω t_vc as a function
of dimensionless wavenumber k̂=k/k_c for Case A with initial wave
amplitude a_0 = 0.01 λ is shown in Fig. <ref>,
where the results obtained with the DNS methodology described in
Sect. <ref> are compared against AIVS, see Sect. <ref>, as
well as results obtained with the open-source DNS code Gerris <cit.>.
The applied DNS methodology is in very good agreement with the results
obtained with Gerris and is excellent agreement with
the analytical solution up to k̂≈ 0.9.
For k̂>0.9 the very small amplitude at the first extrema (|a_1| ∼
10^-6) is indistinguishable from the error caused by the underpinning
modelling assumption and numerical discretisation errors. Hence, the applied
numerical method can provide accurate and reliable results for k̂≤
0.9, as previously reported in Ref. <cit.>.
Figure <ref> shows DNS result of the dimensionless
frequency as a function of dimensionless wavenumber for Case A with an initial amplitude of a_0 = 0.1
λ, compared against results obtained with Gerris, exhibiting a very
good agreement.

Note that Gerris has previously been successfully applied to a variety of
related problems, such as capillary wave turbulence <cit.>
and capillary-driven jet breakup <cit.>.





§ DISPERSION AND DAMPING OF FINITE-AMPLITUDE CAPILLARY WAVES


The damping ratio ζ as a function of dimensionless wavenumber k̂
for Cases A and D with different initial wave amplitudes a_0 is shown in
Fig. <ref>. The damping ratio increases with
increasing amplitude for any given wavenumber k̂<1. This trend is
particularly pronounced for smaller wavenumbers, i.e. longer wavelength.
Irrespective of the initial amplitude a_0 of the capillary wave,
however, critical damping (ζ=1) is observed at k̂=k/k_c=1,
with k_c defined by Eq. (<ref>). Hence, the
critical wavenumber and, consequently, the characteristic lengthscale
l_vc remain unchanged for different initial
amplitudes of the capillary wave.


As a result of the increased damping for capillary waves with larger initial
amplitude, the frequency of capillary waves with large initial amplitude is
lower than the frequency of capillary waves with the same wavenumber but
smaller initial amplitude, as observed in
Fig. <ref>, which shows the dimensionless frequency
ω̂ = ω t_vc as a function of dimensionless wavenumber
k̂ = k/k_c for Cases A and D with different initial wave
amplitudes.
Interestingly, the wavenumber at which the maximum frequency is observed,
k_m≈ 0.751 k_c, is unchanged by the initial
amplitude of the capillary wave. This concurs with the earlier observation that
the critical wavenumber is not dependent on the initial wave amplitude.
Furthermore, comparing Figs. <ref> and
<ref> suggests that there exists a single
dimensionless frequency ω̂ = ω t_vc for any given
dimensionless wavenumber k̂ and initial wave amplitude a_0.


Based on the DNS results for the considered cases and different initial
amplitudes, an amplitude-correction to the viscocapillary
timescale t_vc can be devised. Figure <ref>
shows the correction factor C=t_vc^∗/t_vc, where
t_vc^∗ is the viscocapillary timescale obtained from DNS results
with various initial amplitudes, as a function of the dimensionless initial
amplitude â_0=a_0/λ. This correction factor is well approximated
at k̂ = 0.75 by the correlation 

    C ≈ -4.5  â_0^3 + 5.3  â_0^2 + 0.18  â_0 + 1  ,

as seen in Fig. <ref>. The amplitude-corrected
viscocapillary timescale then readily follows as

    t_vc^∗≈  C   t_vc .

Thus, the change in frequency as a result of a finite initial wave amplitude is
independent of the fluid properties. Note that this correction is particularly
accurate for moderate wave amplitudes of a_0 ≲ 0.1 λ.


By redefining the dimensionless frequency ω̂^∗ = ω
t_vc^∗ with the amplitude-corrected viscocapillary timescale
t_vc^∗ as defined in Eq. (<ref>), a
(approximately) self-similar solution of the dispersion of capillary waves with
initial wave amplitude a_0 can be obtained, as seen in
Fig. <ref>. Thus, for every dimensionless wavenumber
k̂ there exists, in good approximation, only one dimensionless frequency
ω̂^∗. 
The maximum frequency is ω̂_m^∗≈ 0.488 at
k̂_m≈ 0.751, as previously reported for small-amplitude
capillary waves <cit.>.




§ VALIDITY OF LINEAR WAVE THEORY


As observed and discussed in the previous section, an increasing initial wave
amplitude results in a lower frequency of the capillary wave. 
The influence of the amplitude is small if
μ̃=0. According to the analytical solution derived by
<cit.> for inviscid fluids, see
Eq. (<ref>), the frequency error

    ε(k̂, â_0) =
    |ω_AIVS(k̂)
    -ω(k̂, â_0)|/ω_AIVS(k̂) ,

where ω_AIVS is the frequency according to the AIVS solution, is
ε = 0.06% for a_0 = 0.05 λ and ε = 2.32% for a_0 = 0.10 λ.

The influence of the initial wave amplitude on the frequency of capillary waves
increases significantly in viscous fluids. As seen in
Fig. <ref> for Case A with k̂ = 0.01, the time to the
first extrema increases noticeably for increasing initial wave amplitude a_0. However, this frequency shift
diminishes for subsequent extrema as the wave decays
rapidly and the wave amplitude reduces.
The frequency error ε associated with a finite initial wave
amplitude, shown in Fig. <ref>, is approximately constant for the considered range
of dimensionless wavenumbers and is, hence, predominantly a function of the wave
amplitude. For an initial amplitude of a_0 = 0.05
λ the frequency error is ε≈ 2.5%, as observed in
Fig. <ref>, and rises to ε≈ 7.5% for a_0 =
0.10 λ. 




§ CONCLUSIONS


The dispersion and viscous attenuation of capillary waves is considerably
affected by the wave amplitude. Using direct numerical simulation in conjunction
with an analytical solution for small-amplitude capillary waves, we have studied
the frequency and the damping ratio of capillary waves with finite amplitude
in the underdamped regime, including critical damping. 

The presented numerical results show that capillary waves with a given
wavenumber experience a larger viscous attenuation and exhibit a lower frequency
if their initial amplitude is increased. 



Interestingly, however, the critical wavenumber of a capillary wave
in a given two-phase system is independent of the wave amplitude. Similarly, the
wavenumber at which the maximum frequency of the capillary wave is observed
remains unchanged by the wave amplitude, although the maximum frequency depends
on the wave amplitude, with a smaller maximum frequency for increasing wave
amplitude.
Consequently, the viscocapillary lengthscale
l_vc is independent of the wave amplitude and k∼
l_vc^-1 irrespective of the wave amplitude. The reported reduction
in frequency for increasing wave amplitude has been consistently observed for all
considered two-phase systems, meaning that a larger amplitude leads to an
increase of the viscocapillary timescale t_vc. The viscocapillary
timescale has been corrected for the wave amplitude with an empirical
correlation, which leads to an approximately self-similar solution of the
dispersion of capillary waves with finite amplitude in
arbitrary viscous fluids.

Comparing the frequency of finite-amplitude capillary waves with the analytical
solution for infinitesimal amplitude, we found that the analytical
solution based on the infinitesimal-amplitude assumption is applicable with
reasonable accuracy (ε≲ 2.5 %) for capillary waves with an
amplitude of a_0 ≲ 0.05 λ.


The financial support from the Engineering and
Physical Sciences Research Council  through Grant
No. EP/M021556/1 is gratefully acknowledged. Data supporting this publication
can be obtained from https://doi.org/10.5281/zenodo.259434 under a Creative
Commons Attribution license.


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