Direct numerical simulations of capillary Luc Deike, Daniel Fuster, Michaël Berhanu, Eric Falcon

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Luc Deike, Daniel Fuster, Michaël Berhanu, Eric Falcon. Direct numerical simulations of capil- lary wave turbulence. Physical Review Letters, American Physical Society, 2014, 112, pp.234501. ￿10.1103/PhysRevLett.112.234501￿. ￿hal-00999653￿

HAL Id: hal-00999653 https://hal.archives-ouvertes.fr/hal-00999653 Submitted on 3 Jun 2014

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Luc Deike,1, 2 Daniel Fuster,3 Michael Berhanu,1 and Eric Falcon1 1Univ Paris Diderot, Sorbonne Paris Cit´e, MSC, UMR 7057 CNRS, F-75 013 Paris, France, EU 2Scripps Institution of , University of California San Diego, La Jolla, California 3Institut Jean le Rond d’Alembert, Universit´ePierre et Marie Curie, UMR 7190, Paris, France, EU (Dated: June 3, 2014) This work presents Direct Numerical Simulations of capillary wave turbulence solving the full 3D Navier Stokes equations of a two-phase flow. When the interface is locally forced at large scales, a statistical stationary state appears after few forcing periods. Smaller wave scales are generated by nonlinear interactions, and the spectrum is found to obey a power law in both wave number and in good agreement with weak turbulence theory. By estimating the mean flux from the dissipated power, the Kolmogorov-Zakharov constant is evaluated and found to be compatible with the exact theoretical value. The time scale separation between linear, nonlinear interaction and dissipative times is also observed. These numerical results confirm the validity of weak turbulence approach to quantify out-of equilibrium wave statistics.

PACS numbers: 47.35.-i, 05.45.-a, 47.52.+j, 47.27.-i

Wave turbulence aims to provide a general descrip- number, the Kolmogorov-Zakharov spectrum reads [17] tion of a set of weakly nonlinear interacting . The KZ 1/2 −3/4 −15/4 theoretical framework of weak-wave turbulence has been Sη(k) = Ck ǫ (γ/ρ) k , (1) widely applied to very different physical situations such as gravity and capillary waves at the surface of a liq- with ǫ the mean energy flux, γ the surface tension KZ uid, internal waves in the and the atmosphere, and ρ the liquid density. Ck is the non-dimensional flexural waves on a plate, optical waves and magneto- Kolmogorov-Zakharov (KZ) constant that can be explic- hydrodynamical waves [1–3]. Experimental and numeri- itly calculated [17, 18]. This direct has cal results in various systems show that the description been widely explored experimentally finding good agree- provided by weak turbulence has limitations. For in- ment with theory for the frequency (or wave-number) stance, the existence of at all scales has an scaling of the wave spectrum [19–25]. Recent works ad- influence on the energy spectrum measured in capillary dress the influence of dissipation on wave turbulence [4] wave turbulence [4] as well as in flexural wave turbulence and the coexistence of anisotropic structures and wave on plates [5, 6]. In the case of ocean gravity waves, wave turbulence [25]. Numerical simulations is a powerful tool breaking is known to be the main dissipation source ap- to answer to these questions, notably it allows to access pearing at various scales [7–9]. The existence and influ- quantities difficult to measure experimentally. In com- ence of coherent structures among a set of random waves parison to the wide literature available for experimental is also an open question in wave turbulence [10–12]. Di- measurements, there are only a few numerical studies on rect Numerical Simulations (DNS) are an appealing tool capillary wave turbulence. These studies can be typi- to quantify the influence of different processes on wave cally classified in two different groups: kinetic equation turbulence and test the various theoretical hypotheses simulations [1, 26, 27] and weakly nonlinear hamiltonian separately. dynamics simulations [18, 28]. Both approaches remain limited to weakly nonlinear situations and cannot inves- Capillary wave is one of the simplest systems to study tigate the possible influence of air and water flow on the wave turbulence. However, this regime often interacts waves. The reason why more complete models have not with turbulence in experiments and these been tested is that solving the full Navier-Stokes equa- mutual interactions remain an open question [4, 13]. The tions in multiphase flow is a numerical challenge. Only numerical investigation of purely capillary wave turbu- thanks to the recent development of numerical methods lence finds application probing the validity range of weak [29] it is now possible to perform long wave turbulence turbulence theory in experiments, and to improve our simulations, in order to obtain representative statistics, understanding of the influence of capillary waves at the with relative high resolution in terms of wave number. ocean surface regarding dissipation, air-water exchanges In this letter, we present the first observation of the [9, 14, 15], or microwave remote sensing techniques of the direct energy cascade in capillary wave turbulence from ocean surface [15, 16]. the numerical solution of the full 3D Navier-Stokes equa- tions. The numerical method and physical configuration The main result of capillary wave turbulence is the are first introduced and the stationary state is charac- existence of a direct energy cascade. In terms of the terized. We show the ability of the simulations to cap- spatial spectrum Sη(k), where k is the wave ture the propagation of capillary waves. The obtained

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(a) (b) k0 = kp, k1 = 1.4kp, k2 = 1.2kp, k3 = 0.8kp, k4 = 0.6kp, with kp = 2π/(0.4L) the central forcing wave number and ω = (γ/ρ)k3 tanh(kh) the corresponding frequency i p i given by the linear relation of capillary waves. The forcing is located in space through the gaussian 2 2 2 function α(x, y) = exp (−(x − xc) − (y − yc) )/2r , with r = 0.15L, xc = yc = 0.25L. Note that the forcing area size has no influence on the generated . We expect capillary waves to propagate according to 0.08 1 the linear dispersion relation ω2 = (γ/ρ)k3 [34]. The (c) (d) 0.07 maximal resolution in the simulations presented here 0.06 −1 10

0.05 ) corresponds to 256×256 grid on the interface. Adap- η σ /

0.04 η (t) (J) (t) t tive mesh refinement is used in order to decrease the E −2 0.03 Pdf( 10 resolution in the bulk and to reduce the computational 0.02 0.01 time. However, despite the use of adaptive mesh −3 0 10 0 5 10 15 20 −4 −3 −2 −1 0 1 2 3 4 refinement techniques, the high computational cost of η/ σ tfp η the method has impeded to refine the grid to a level where the numerical viscosity naturally introduced by FIG. 1. (a,b) Snapshot of the wave interface η(x, y) at time the numerical schemes becomes negligible compared tfp = 0.3 and tfp = 9.1. (c) Total wave energy Et as a function time tfp. A stationary state is reached for tfp > 5 to the physical viscosity [35]. Note that the effect of and the mean value is indicated by dashed line. (d) PDF of artificial numerical dissipation is also present in previous the wave height η/ση . Dashed line is the normalized gaussian. numerical computations presented in the literature with pseudo-spectral methods [18, 28]. Figure 1 (a,b) shows two snapshots of the interface space-time wave spectrum is compared with the classi- η(x, y) during the first period of forcing (Fig. 1a) and cal dispersion relation. The wave spectrum both in wave after few forcing periods (Fig. 1b). The forcing area number or frequency is found to obey a power law in good is clearly visible on Fig. 1a (top left corner), where we agreement with the weak turbulence theory from which see waves propagate from the circular region where the the KZ constant can be estimated. The nonlinear inter- source is applied and from others corners due to the pe- action time scale τnl is estimated and is found to scale riodic boundary conditions. After a few forcing wave −3/4 as τnl ∼ k , in agreement with wave turbulence the- periods, (Fig. 1b) the wave field displays random fea- ory. Finally, we show that the wave turbulence inertial tures with a wide range of spatial scales. The wave range is defined by τl ≪ τnl ≪ τd, where τl is the linear field nonlinearity is estimated by the typical steepness propagation time and τd the dissipation time. r = σηkp ≈ 0.3, with ση the rms value of the wave ampli- We solve the 3D two-phase Navier Stokes equations tude. Figure 1c shows the total wave energy Et = Es +Ec accounting for surface tension and viscous effects using as a function of dimensionless time tfp. The kinetic and the open source solver Gerris [29, 30]. This solver potential energy on the domain volume Ω are computed has been successfully used in multiphase problems like from the liquid velocity v, and the surface area, A(x, y), as E = 1 ρv2dΩ and E = γA(x, y). After a short tran- atomization, the growth of instabilities at the interface c R 2 s [31], wave breaking [32] or splashing [33]. The physical sient state, the wave energy reaches a stationary state properties of the two phases are those of air and water. (tfp > 5) where the wave energy fluctuates around a Gravity is not present. The simulation domain is a cube constant mean value (displayed by a dashed line in Fig. of length L = 1 m with periodic boundary conditions 1b). Wave statistics are obtained in the time interval in the x and y horizontal directions. In the vertical t = [10 : 20]fp, with fp = ωp/(2π). We now focus on the direction z slipped wall (symmetry) conditions are statistical and dynamical properties of the waves during imposed. The interface between the liquid and gas phase this stationary regime. Figure 1d shows the probability is initially placed at the half plane z = 0 (the water density function (PDF) of the wave height η/ση during depth is h = 0.5L). It is perturbed locally introducing this stationary stage. Gaussian statistics are observed, a source in the momentum equation. This source is ob- meaning that the nonlinear effects are weak enough to not tained from the linear wave solution [34] corresponding induce a significant asymmetry on the capillary waves. to a forcing on the interface elevation η: η(x, y, t) = The space-time wave height spectrum Sη(ω/ωp, k/kp) 4 is shown in Fig. 2 (a). Energy is found to be localized α(x, y) η0 cos (k~i · ~x − ωit); and on the liquid veloc- Pi=0 in the Fourier space around the linear dispersion rela- ity ~v (~v = ∇~ φ, φ the velocity potential): φ(x, y, z, t) = tion of capillary waves. The local maxima of the spec- 4 ωi cosh ki(z+h) ~ α(x, y) i=0 −η0 i i cos (ki · ~x − ωit), trum for each frequency (crosses) fall relatively well on P  k cosh(k h)  where η0 is the wave amplitude. The forcing modes are the theoretical dispersion relation curve. A slight mis- 3

25 1

−4 (a) 10 0 20 S (k) k15/4 η

−1 )

3 −6 50 -3/4 1/2 10 (γ/ρ) ε 15 −2 (k) (m (k) η p S k/k −3 −8 10 10 10 0.5 1 −4 k/ k 10 p −10 5 10 −5 0.2 1 10 20 k/ k −2−2 p 10 0 −6 0 10 20 30 40 50 ω/ω p −15 −5 −15 −5 (c) (b) −6 −6 −4 −10 10 17/6 −10 −7 −7 S (ω) ω η −8 −8 −5 −5 s) 50 1/6 1/2 −9 2 p

p −9

/k (γ/ρ) ε /k y

y 0 −10 0 −10 k k −11 −11 (m ) −6 5 ω 10 5 −12 −12 ( 10 −13 η 10 −13 10 S −14 −14

15 −15 15−15 −10 −5 0 5 10 15 −15 −15 −10 −5 0 5 10 15 k /k k /k 1 x p x p −8 0.2 1 10 20 10 ω / ω p

FIG. 2. (a): Sη(ω/ωp, k/kp). (−−): Linear dispersion re- 2 γ 3 0.5 1 10 50 lation ω = ρ k . (white +): spectrum maxima. (b,c): ω / ω ∗ ∗ p Sη(kx/kp, ky/kp) at a fixed ω = ω/ωp = 6 (b) and ω = ∗ ∗ ω/ωp = 18 (c). Circles in solid line indicate k (ω ) given by the dispersion relation. The wave field is found isotropic. FIG. 3. (a): Sη(k/kp). (b): Sη(ω/ωp). Theoreti- −− ∼ −15/4 Colors are Sη log-scaled. cal KZ spectrum ( ) is respectively Sη k −17/6 and Sη ∼ ω . Inset: compensated spec- 15/4 1/2 −3/4 trum, respectively Sη(k)(k) /(ǫ (γ/ρ) ) and 17/6 1/2 1/6 Sη(ω)ω /(ǫ (γ/ρ) ). Vertical dot-dashed line in- match between theoretical and numerical values occurs dicates the forcing scale range. at high which is attributed to the numerical dispersion (see the supplementary materials [35] for de- tails) that is linked to the lack of resolution for the highest tained by integrating over all frequencies the space-time frequencies. While forcing is only applied at low k, the spectrum depicted in Fig. 2. It exhibits a power-law energy spectrum spreads over a large range of wave num- regime within an inertial range 1 . k/k . 6 from the bers showing that nonlinear transfers among the different p forcing scale to a dissipative cut-off length, beyond which scales have taken place. Moreover, the dispersion relation the spectrum departs from the power law. The frequency broadens in frequency as the wave number increases, first spectrum (obtained by integrating over k) also exhibits due to non-linear interaction between the waves and then a power-law in the range 1 . ω/ω . 8 (see Fig. 3b). at high frequencies (ω/ω > 20) due to dissipative effects, p p The inertial range spreads over roughly one decade in as discussed later on. Note also that for simulations with frequency (the range is smaller in terms of wave num- lower values of σ the space-time spectrum non-linear η bers). The observed power laws within the inertial range broadening is reduced (not shown). Figures 2 (b) and (c) are found to be in good agreement with wave turbulence depict the spatial spectrum S (k /k , k /k ) at two dif- − − η x p y p theory scalings: S (k) ∼ k 15/4, S (ω) ∼ ω 17/6 [17]. ferent frequencies ω∗. An isotropic wave field is observed η η The compensated spectrum is shown in the inset in both where the energy is localized around a circle of radius cases (Fig. 3a and b). The flat spectrum observed within ∗ ∗2 ∗2 k = qkx + ky well predicted by the linear dispersion the inertial range confirms the good agreement between relation ω∗(k∗). Again a significant broadening on the DNS and wave turbulence theory. This limited inertial spectrum is observed for high wave numbers. These ob- range is a consequence of the finite resolution of the in- servations still holds regardless of the frequency and the terface, therefore it may be enlarged at expenses of larger forcing amplitude. Thus, numerical computations cap- computational resources (see [35] for details). ture well capillary waves that propagate at various scales It is now possible to estimate numerically the KZ con- in an isotropic wave field. stant from Eq. (1) and the compensated spectrum: KZ 15/4 1/2 −3/4 Figure 3a depicts the spatial spectrum Sη(k/kp), ob- Ck = Sη(k)k /[ǫ (γ/ρ) ] (inset Fig. 3a), or 4

KZ 17/6 1/2 1/6 Cω = Sη(ω)ω /[ǫ (γ/ρ) ] (inset Fig. 3b). To 1 this end, we evaluate the mean energy flux ǫ using the 10 measure of the dissipated power D = 2ρν Ω Sij Sij dΩ, 1 R with Sij the deformation tensor Sij = 2 ij (∂vi)/(∂xj ), i = {x, y, z} [36]. The mean energy fluxP is then defined by ǫ ≡ hDi/(Aρ), where A is the surface area and h.i designed an average over time. A comparable value of (s)

l 0 the flux is obtained using the dissipation spectrum as τ 10 , ,

γ 2 emp emp nl τ 0 in [4], ǫd = dωdk k Sη(k, ω)/τ where τ = τd p ρ Sη (k*,ω/ω )/Sη (k*) R emp num 1 for k/kp < 6 and τ = τ for k/kp > 6, so that d −1 τ emp includes τ = 1/(2νk2) [34] the viscous linear dis- 10 d −2 num 10 sipation, and τd the total (numerical and physical) −3 10 ω/ω dissipation time valid only at small scales (k/kp > 6) −4 p 100 5 10 15 20 25 30 35 −1 [35]. We obtain from the DNS the following estimation 10 0 1 of the KZ constant, CKZ = 16 and CKZ = 34. Thus 10 10 ω k k/k KZ KZ p Ck /Cω ≈ 2, while the theoretical ratio given by the relation Sη(k)dk = Sη(ω)dω is 3/2. The KZ constant FIG. 4. Time scale evolutions as a function of k/kp.(◦): Cnk can be also defined from the wave action spectrum 2 KZ KZ τnl(k) = 1/∆ω;(−): τl = 1/ω;(−): τd = 1/(2νk ) viscous equation, and Ck = 2πCnk [18]. Using the Ck value num dissipation and (− · −): τd numerical dissipation function from our simulations and the latter equation leads to num A k A ≈ (1/τd = kmax /τd, with 100 an empirical non di- Cnk = 5 ± 1, the uncertainty coming from using either ∼ −3/4 KZ KZ mension parameter [35]). Theoretical prediction τnl k the value of C or C . Our estimation of Cnk is ∗ 0 ∗ k ω (−−). Inset: Sη(k , ω) normalized by its maximum Sη (k ) ∗ of the same order of magnitude as 9.85 the theoretical as a function of ω/ωp, for k /kp = 7.6. (−−): gaussian fit, 2 2 value found in [18]. The difference between these two exp [−(ω − ωc) /∆ω ] with ∆ω = 2π 0.15 and ωc = 2π 0.66 − values can be due to the short inertial range induced rad 1. by the numerical dissipation. Note that in the previous Hamiltonian simulations, the KZ constant was found two times smaller than ours [18]. This difference is probably 2 related to the limited length of the inertial range and τd = 1/(2νk ) [34], while the dot-dashed line displays the the numerical dissipation at small scales, our numerical extrapolated empirical (physical and numerical) dissipa- num methods allowing for a better resolution of these scales. tion τd , determined by measuring the decay rate of a 2D freely decaying capillary wave for various spatial reso- One of weak turbulence key hypothesis regards the lution [35]. Thus when the nonlinear time scale becomes time scale separation. The linear wave propagation time of the same order as the total dissipation time, the cas- is indeed supposed to be much smaller than the time scale cade progressively ends and dissipation is responsible for of the nonlinear energy exchanges. This latter must also the broadening of the spectrum, as already observed in be small compared to the dissipative time. We will now Fig. 2. We note that the numerical dissipation does not evaluate the various time scales involved in the problem. affect the capillary wave turbulence cascade within the As already discussed, Sη(k, ω) broadens around the linear inertial range, while the physical dissipation would be- dispersion relation curve. The interaction nonlinear time come dominant for very high resolution far beyond cur- scale τnl(k) is linked to this broadening [2, 37, 38]. This rent computational resources. As expected by the weak time is defined by τnl(k) = 1/∆ω(k), where ∆ω(k) is the turbulence theory, the power law spectrum observed is inverse of the spectrum width at a given wave number shown to fall within the range defined by the double in- num k. As shown in the inset of Fig. 4, ∆ω(k) is extracted equality τl ≪ τnl ≪ τd . from Sη(k, ω) using the rms value of a Gaussian fit with In conclusion, this work presents DNS of capillary wave respect to ω at a given k∗. Then, iterating this protocol turbulence where the two-phase 3D Navier Stokes equa- ∗ to all k allow us to determine τnl(k). Figure 4 shows tions have been solved. The wave height spectrum is that τnl(k) is found to be close to the theoretical scaling found to exhibit frequency and wave number power-laws −3/4 of capillary wave turbulence τnl ∼ k [17] within the in good agreement with weak turbulence theory. We also inertial range 1 < k/kp < 6, and then strongly decreases observe a clear time scale separation between linear and for larger k. The linear propagation time τl = 1/ω is also nonlinear times. These numerical results confirm the va- shown on Fig. 4 where we see a clear time scale sepa- lidity of weak turbulence approach to quantify out-of- ration τl ≪ τnl within the inertial range. Close to the equilibrium wave statistics. It also opens new perspec- forcing scales, both times are of the same order of mag- tives in order to better understand wave turbulence sys- nitude. 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