A viscous effect of wave attractor in geometry with underwater peak

Abstract

The influence of the viscosity on a wave attractor flow has been previously studied, particularly in relation to the widening of the hydrodynamical structures. In this work, we simulate an attractor flow with a peculiar bottom shape that includes an underwater hill. During the simulation, we discovered a side structure appearing beyond the wave attractor. We determined that the appearance of this structure is connected to viscosity. In this article, we consider the behavior of this newly found structure. Additionally, we discuss the challenges of energy accumulation and the estimation of the Reynolds number, which is a non-trivial problem in the context of wave attractor flows.

1 Introduction

A wave attractor is a phenomenon where running wave coherent structures form instead of standing modes in a closed geometry. Wave focusing leads to a significant increase in motion amplitude, which can produce turbulence (Scolan et al. 2013; Brouzet et al. 2017a). The presence of wave attractors in the ocean can influence impurity transport, sedimentation processes, surface waves, and energy accumulation (da Silva et al. 2012; Scolan et al. 2013; Elistratov et al. 2020). Soon after the discovery of this phenomenon, the principal conditions for internal wave attractor formation were formulated (Maas and Lam 1995): a slope, periodic external forcing, and vertical stratification. Since then, wave attractors have been widely studied experimentally (Maas et al. 1997; Maas 2001, 2009; Manders and Maas 2003; Hazewinkel et al. 2008; Swart et al. 2010; Scolan et al. 2013; Sibgatullin et al. 2015; Dossmann et al. 2016) as well as numerically (Grisouard et al. 2008; Brouzet et al. 2014, 2016a, b; Jouve and Ogilvie 2014; Sibgatullin et al. 2015; Elistratov et al. 2020; Ryazanov et al. 2021). Comparisons (Grisouard et al. 2008; Hazewinkel et al. 2011) between numerical approaches and experimental data show good agreement.

Wave attractors are studied preliminary in simple-shaped model vessels: parabolic or trapezium (Maas 2001; Hazewinkel et al. 2008; Brouzet et al. 2014, 2016a, b; Sibgatullin et al. 2015; Elistratov et al. 2020; Ryazanov et al. 2021). Attractor in natural basins can hardly be found in such geometries hence an attractor from shelf to shelf would be enormous; it is more likely that attractor beams reflect from shelf and underwater mountains, like in Luzon Strait (Wang et al. 2015). The feature that distinguishes such a peak from a shelf slope is a finite height. The attractor in inviscid liquids will form the same structure both in the case of slope and bottom mountain if the attractor fits the domain form, which is obvious (Fig. 1, dashed blue line); however, attractors in viscous media have a tendency to expand and their beams to widen. The question is what will happen if the peak has such height that the skeleton is reflected from it, but the widened attractor does but not completely (light blue area on Fig. 1 shown for a slope case; peak case behavior is a question)?

Fig. 1

Attractor skeleton (blue dashed line) is the same both for the bottom peak and the whole slope (black dashed line). The light blue area represents a viciously widened attractor in the case of the slope. The attractor shape for the bottom peak case is a subject of investigation

The structure of the article is as follows: First, we describe the numerical setup used in the simulation. Next, we present the results obtained, focusing on the attractor structure formed in a geometry with an underwater peak and its velocity characteristics relation to the viscosity. Ultimately, we discuss the energy accumulation and Reynolds number estimation in wave attractor flows.

2 Numerical set-up

To investigate the behavior of wave attractors reflecting from a peak near its top, we simulate a flow in the geometry shown in Fig. 2. The 2D formulation, as verified in previous studies (Grisouard et al. 2008; Hazewinkel et al. 2011; Brouzet et al. 2014, 2016a; Sibgatullin et al. 2015) captures the main attractor features while conserving computational resources. The right wall of the domain is slightly inclined, ensuring there is only one attractor between the left wall and the peak (Fig. 3). The mesh depicted is a 48 × 48 spectral element mesh, originally box-shaped and smoothly deformed.

Fig. 2

Problem domain

Fig. 3

Ray-tracing, ${\omega }_{0}=0.718$ (1/s)

We solved the Navier-Stokes equations using the Boussinesq approximation

$$\frac{\partial\vec{v}}{\partial t}+\left(\vec{v}\cdot\nabla\right)\vec{v}=-\frac1{\rho_m}\nabla\widetilde p+\frac{\rho_s}{\rho_m}\vec{g}+\nu\Delta\vec{v},$$

(1)

salt transport equation:

$$\frac{\partial {\rho }_{s}}{\partial t}+\left(\vec{v}\cdot \nabla \right){\rho }_{s}={\lambda }_{s}\Delta {\rho }_{s},$$

(2)

$$\rho ={\rho }_{m}+{\rho }_{s},$$

(3)

and continuity equation for incompressible flow:

$$\text{div} \vec{v}=0,$$

(4)

where $\widetilde{p}$ denotes the pressure without the hydrostatical part $g{\rho }_{m}$, ${g}$ is the gravitational acceleration, ${\rho }_{m}$ is fresh water density, ${\rho }_{s}$ is dissolved salt density, and ${\lambda }_{s}$ is the salt diffusivity coefficient.

To provide the external forcing for wave attractor formation (Maas and Lam 1995) we simulate a tool known as a wave maker (Gostiaux et al. 2007). This tool generates periodic boundary perturbations in a given shape using a system of shafts and eccentrics (Mercier et al. 2008, 2010). It is also used for numerical simulations (Brouzet et al. 2017a, b; Ryazanov et al. 2021) and can be easily placed at one of the plane boundaries. In our case, the wave maker is situated on the top side, changing the upper wall profile according to the following equation:

$$s\left( {x,t} \right) = - a\sin \left( {2{\uppi }x/L} \right)\cos \left( {\omega_{0} t} \right),$$

(5)

where L is the domain full length along the x-axis (60 cm on Fig. 2).

For amplitudes a < < H, the changes in domain shape are so small that the perturbation can be transferred to the velocity boundary condition:

$$\left. {\vec{v}} \right|_{y = H} = \dot{s}\left( {x,t} \right)\vec{e}_{y} = a\omega_{0} \sin \left( {2{\uppi }x/L} \right)\sin \left( {\omega_{0} t} \right)\vec{e}_{y} ,$$

(6)

on a constant domain. This small trick allows for a significant reduction in computational costs by eliminating the need for the moving meshes. The only issue is ensuring that the mass flux should be equal to zero. This problem is fixed by selecting a spatial profile such that the full length L consists of an integer number of spatial sine waves (2πx/L in Eq. (6)). During the simulation, we selected an extremely small a = 0.002 cm to avoid turbulent effects.

On the other boundaries, a slip condition is applied: $\left. {\vec{v}} \right|_{b} = \vec{0}.$

Salinity is bounded by the impermeability condition: $\frac{\partial \rho}{{\partial n}} = 0.$

Initially, the liquid is assumed to be steady, hence:

$$\left. {\vec{v}} \right|_{t=0} = \vec{0},$$

and initial salinity distribution was linear:

$$\left. {\rho_{m} } \right|_{t = 0} = \Delta \rho \left( {1 - y/H} \right),$$

so that $\Delta \rho = 0.04\rho_{m}$ (for N ≡ 1), and smoothed in the near-border regions in order to satisfy the impermeability condition.

Considering the Navier-Stokes Eq. (1) and omitting convective and viscous parts one can derive the dispersion relation for a stratified fluid (Maas and Lam 1995):

$$\omega /N = \cos \theta ,$$

(7)

where $N = \sqrt { - \frac{g\partial \rho }{{\rho \partial y}}}$ is the buoyancy frequency, ω is the wave frequency, θ is the angle between the group velocity vector and the vertical direction. This equation, first, yields a peculiar reflection law that conserves the angle between the wave and gravity. Second, it forms the basis for the ray-tracing procedure (Maas et al. 1997), which is used to determine the attractor shape (’skeleton’) based on the external frequency ω ₀ from Eq. (6). Figure 3 shows the attractor found using this procedure for $\omega_{0} = 0.718$ (1/s). Different line colors represent different starting points, illustrating that a ray, irrespective of its starting point, tends to the single attractor in this geometry.

For further work, we introduce non-dimensional time $t/T_{0}$ ($T_{0} = 1/f_{0}$, $f_{0} = \omega_{0} /(2{\uppi })$).

3 Numerical simulation results

As we can see on Fig. 3, all the rays tend to the rhomboidal attractor; however, the ray-tracing method is derived for non-viscous liquid. In a viscous one, the attractor beams widen as $\propto \nu^{1/3}$ (Brouzet et al. 2017b). If the beams reflect from the whole plane, the patterns of reflected light just widen, but in the case of the reflection from a peak there can be a situation when the attractor is so wide that some parts of it are out of the peak.

To comprehend what happens in such cases, we ran a numerical simulation in the discussed geometry with real-water viscosity ($\nu = \nu_{0} = 0.01$ cm²/s). A vertical velocity component $v_{y}$ snapshot is represented in Fig. 4. Beyond the predicted attractor, another unexpected structure is clearly visible. Although this structure forms a closed poly-line and has the same angle with vertical as the main attractor, we failed to identify it using the ray-tracing method. The likely reason is that ray-tracing visualizes a closed figure that attracts the rays (which, in a real liquid, are internal wave beams). The observed structure does not attract these beams; instead, they are drawn to the main (rhomboid) attractor, making the other structure a ’distractor’ rather than an attractor. Nevertheless, to avoid confusion with the main attractor predictable by ray-tracing, we will refer to this unexpected structure as a ’side’ attractor.

Fig. 4

$v_{y}$ snapshot (unit: cm/s). Blue and red dots mark the midpoints of the rays of the main and ’side’ attractors, respectively

The simulation of attractors with different viscosities led us to the conclusion that the viscosity results in a less intense side attractor (Fig. 5). This is not surprising because when viscosity $\nu\to 0$, the attractor increasingly resembles the inviscid line-wise skeleton produced by ray tracing. As we increase viscosity, the main and viscous attractors merge owing to the viscous widening of their beams (Fig. 5c).

Fig. 5

$v_{y}$ momentary snapshots (unit: cm/s) for different viscosities: a $\nu /\nu_{0} = 0.01$; b $\nu /\nu_{0} = 0.2$; c $\nu /\nu_{0} = 5$. Blue and red dots mark are the same as those shown in Fig. 4

Upon revealing this fact, we investigated the amplitude at the midpoint of the first array of the attractor (marked by blue and red dot marks in Figs. 4 and 5). The amplitude is considered in the steady-state regime (Fig. 6). The main attractor amplitude settling (or saturation) behaves logarithmically (Fig. 7). The last two points were excluded because the saturation time ceased to change, possibly owing to the merge with a side attractor.

Fig. 6

$v_{y}$ evolution in the marked points for different viscosities: a $\nu /\nu_{0} = 0.01$; b $\nu /\nu_{0} = 0.2$; c $\nu /\nu_{0} = 5$

Fig. 7

Main attractor amplitude settling time

Figure 8 represents the behavior of the velocity amplitudes for both structures as a function of viscosity. The amplitude of the main attractor decreases, which is expected because the attractor becomes more ’blurry’ while the energy input remains constant. Similarly, the amplitude of the side attractor also decreases. Except at low viscosities, the velocity for both structures decreases according to a power law with the same exponent.

Fig. 8

Vertical velocity amplitude on the main and side attractors

Their ratio is depicted in Fig. 9. At high viscosities, this ratio stabilizes at a constant value, indicating the merging (Fig. 5c) of the two structures (and this corresponds to the same decay power observed in Fig. 8). At low viscosities, the amplitude ratio seems to follow a power law again, with the side attractor velocity amplitude decreasing more slowly than that of the main attractor. As the amplitude decrease is connected to the attractor widening (we remind that we simulated non-turbulent (’laminar’) attractors, so the amplitude decrease with viscosity cannot be caused by turbulence formation–but rather related to widening), we would expect that the side attractor to also widen more slowly.

Fig. 9

Velocity amplitude relation

To examine the widening process, we will monitor the attractor width. We estimate the attractor width by measuring the ray width at half of its maximum value. First, we applied a Hilbert transform to the vertical velocity. Next, we took a slice orthogonal to the attractor ray (Fig. 10, white line) and interpolated the data along this slice line. The graph of the Hilbert-transform modulus of the velocity versus the slice coordinate (Fig. 11) contains peaks–one for the main attractor and one for the side attractor. The modulus of Hilbert transform at each spatial point serves as the temporal envelope of the oscillations (Fig. 12). Each peak width at half of its height serves as an estimation of the corresponding attractor width, which will be referred to as half-width.

Fig. 10

$v_{y}$ Hilbert transform (unit: cm/s) and slice segment

Fig. 11

$v_{y}$ Hilbert transform, half-width estimation

Fig. 12

Hilbert transform acts like an envelope. The black line is the Hilbert transform of $v_{y}$, and the color lines are $v_{y}$ momentary slices

Since the Hilbert-transform slice is stable over time (Fig. 13), our method for measuring the half-width is correct. The results for the width across different viscosities are represented in Fig. 14. The middle part of this dependency can be described by a power law, although the power is slightly different from the theoretical 1/3 for an attractor in a trapezium (Brouzet et al. 2017b). The unruly behavior at high viscosities can be explained by the merging of the hydrodynamical structures (see Figs. 15 and 16).

Fig. 13

Hilbert-transform modulus is stable over time for both structures

Fig. 14

Attractor half-width vs. viscosity

Fig. 15

$\nu /\nu_{0} = 5$: start of structures merge

Fig. 16

$\nu /\nu_{0} = 10$: complete merge

4 Reynolds number and energy accumulation

Another aspect we would like to consider is the Reynolds number. When simulating a visible wave attractor (i.e., one that is not obscured by turbulence, allowing for a discernible structure in velocity snapshots), it is important to select the set-up parameters appropriately, as turbulence is known to arise at sufficiently high Reynolds numbers. The attractor behaves quite complex depending on viscosity, and the main challenge is that the Reynolds number cannot be determined from set-up values, like in a jet problem. This is because the spatial sizes of the hydrodynamical structures and the velocity values depend on the problem parameters but cannot be calculated directly from them.

The Reynolds number does not behave as $\propto 1/\nu$ with changes in viscosity because characteristic length and speed also depend on viscosity. By estimating the characteristic length as the main attractor’s half-width and the flow speed as the Hilbert-transform maximum velocity ($v_{x}$ and $v_{y}$ behave the same way depending on viscosity, otherwise the attractor structure would not have the same shape for different viscosities), we can derive the following relation for the Reynolds number dependence on the viscosity. For this estimation, we consider only $v_{y}$, where $v_{y}^{a}$ is a steady $v_{y}$ amplitude (Fig. 6):

$${\text{Re}} = \frac{Lu}{\nu } = \frac{{w_{1/2} v_{y}^{a} }}{\nu } \propto \frac{{\nu^{0.251} \nu^{ - 0.406} }}{\nu } \propto \frac{1}{{\nu^{1.155} }}.$$

This means that the Reynolds number increases faster than $\propto 1/\nu$, which may lead to more intensive turbulence as viscosity is reduced. This factor must be taken into account when simulating model attractors in vessels of ordinary sizes and low viscosity which is frequently done to reproduce the full-scale attractor behavior or to thin its ray width. Additionally, external forcing must be reduced by more than $\nu$ if one wants to maintain the same Reynolds number.

An important feature of a flow with a wave attractor is the energy accumulation (Elistratov et al. 2020) owing to the increase in motion amplitude. For attractors it is typical (Drijfhout and Maas 2007; Scolan et al. 2013; Brouzet et al. 2015) to consider kinetic energy in its relative form:

$$\overline{E }=E/{E}_{w}, \; E=\iiint \left({v}_{x}^{2}+{v}_{y}^{2}\right)\text{d}V, \;{E}_{w}={a}^{2}{\omega }_{0}^{2}/2.$$

(8)

Here, the normalization factor $E_{w}$ is the mean energy density of a vessel oscillating like a rigid body (or the energy amplitude if the liquid oscillates with the amplitude and frequency of the wave maker at every point).

Figure 17 depicts the evolution of relative kinetic energy for $\nu = 0.1\nu_{0}$ as an example. As a typical value, we consider the mean energy over the last 50T ₀ (see the black line, local time-averaged energy obtained with empirical mode decomposition (Huang et al. 1998; Elistratov et al. 2020) as $\overline{E} - \sum\nolimits_{i = 1}^{3} {IMF_{i} } \left[ {\overline{E}} \right]$, IMF is intrinsic mode), where time-mean is stable.

Fig. 17

Relative kinetic energy $\overline{E}$ (Eq. (8)), $\nu /\nu_{0} = 0.1$

The dependence of such typical energies on viscosity is plotted in Fig. 18. Most values fit an exact power-law dependence. We would expect a definite power value because both the attractor width and its amplitude also exhibit power-law behavior. An estimation can be made (÷ indicates the range since the side and main attractors behave slightly differently):

$${\overline E}_\text{est}{\propto}\,w_{1/2}\left(v_y\right)^2{\propto}\,\nu^{0.235\div272}\left(\nu^{-0.406}\right)^2{\propto}\,\nu^{-\left(0.577\div0.540\right)}.$$

Fig. 18

Mean relative kinetic energy vs. viscosity

However, the real approximation −0.393 (Fig. 18) does not fit the interval [−0.577, −0.540]. This means that something hinders the dissipation, and something accumulates energy besides the both attractor structures. It is worth noting that for low viscosities, the power is even higher (it is negative and smaller in absolute values), which excludes the influence of the background.

A similar issue was observed in the ’basic’ attractor (Maas and Lam 1995; Maas et al. 1997; Ermanyuk et al. 2015; Brouzet et al. 2017b) in the trapezium geometry (Fig. 19). The approximations of half-width and velocity amplitude vs. viscosity follow an ideal power law (Figs. 20a and b); however, the energy approximation decreases faster than expected (expected $\overline{E}_{{{\text{est}}}} \propto \nu^{0.392} \nu^{{2 \times \left( { - 0.376} \right)}} = \nu^{ - 0.36}$, in fact $\overline{E} \propto \nu^{ - 0.404}$). The first point that stands out is that the half-width follows a power of 0.392, which is slightly different from the expected 1/3; the second point is the increased power for energy decay, which is less than expected (greater in absolute value). This contrasts with the complex double-attractor system from the current study, where the energy viscosity-power relation is higher (smaller in absolute value).

Fig. 19

’Basic’ attractor, $\nu /\nu_{0} = 0.5$: $v_{y}$ slice (unit: cm/s)

Fig. 20

Characteristics of the basic attractor in a trapezium basin: a half-width; b velocity amplitude; c relative kinetic energy $\overline{E}\;$ (Eq. (8))

The Reynolds number in trapezium geometry behaves as follows: $\text{Re}={w}_{1/2}{v}_{y}^{a}\frac{1}\nu\propto\frac{1}{\nu^{0.984}}$, which is close to more common $1/\nu$.

The secret of the additional energy accumulation in the double-attractor system appears in the form of the side attractor. At the intersection points of its rays, marked in Fig. 21, the velocity has a greater amplitude than at other points of the side attractor. This amplitude becomes more significant with higher viscosity (in comparison with the main attractor). This ray-crossing may be responsible for the additional energy accumulation, preventing its dissipation and softening the energy decrease curve shown in Fig. 18.

Fig. 21

$v_{y}$ Hilbert transform (unit: cm/s), geometry with bottom peak: a $\nu /\nu_{0} = 0.1$; b $\nu /\nu_{0} = 2$. The red arrow marks side attractor rays intersection

The fact that attractor ray-crossing can play a role in energy accumulation makes such complex-shaped attractors important to investigate.

5 Discussion and conclusions

In this work, we have simulated a wave attractor flow in a domain with a peculiar shape (featuring a bottom peak), which can occur in real basins. In addition to the expected attractor, we discovered another hydrodynamical structure that tends to disappear when the viscosity approaches zero. This qualitatively new effect arises from the reflection of attractor rays from the bottom peak near its top and their widening due to viscosity causing their ’scattering’ beyond the peak. This new structure, which cannot be predicted by typical ray-tracing algorithms, can contribute to energy accumulation. It was found that the relative kinetic energy decrease in such a system is slower than estimated, unlike the basic attractor in a trapezium basin. This phenomenon is explained by energy accumulation near the intersection points of the side attractor rays. This suggests that the complex shapes of the attractor play an important role in energy accumulation and need to be thoroughly investigated. Our study revealed that changes in viscosity alter the Reynolds number as $\text{Re}\propto 1/{\nu}^{1.155}$, i.e., faster than $1/\nu$. By contrast, an ordinary attractor in a trapezium domain it behaves like $\propto 1/\nu$.

Furthermore, our investigation shows that the basin form may lead to special effects in the wave attractor flow, and sharp bottom hills should be taken into account. The latter may lead to the formation of additional hydrodynamical structures, making it inappropriate to replace them with full-height walls. The complexity in such geometries arises because the side attractor that appears in the flow and significantly influences the energy cannot be predicted by ray-tracing methods and can only be identified by CFD simulations, which can be costly in the case of 3D simulation simulations.