Numerical study of experimentally inspired stratied turbulence forced by waves Jason Reneuve1Cl ement Savaro1G eraldine Davis1

2025-05-02 0 0 9.28MB 33 页 10玖币
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Numerical study of experimentally inspired stratified turbulence
forced by waves
Jason Reneuve,1Cl´ement Savaro,1G´eraldine Davis,1
Costanza Rodda,1Nicolas Mordant,1and Pierre Augier1, ∗
1Laboratoire des Ecoulements G´eophysiques et Industriels,
Universit´e Grenoble Alpes, CNRS, Grenoble-INP, F-38000 Grenoble, France
Stratified flows forced by internal waves similar to those obtained in the Coriolis
platform (LEGI, Grenoble, France) [1] are studied by pseudospectral triply-periodic
simulations. The experimental forcing mechanism consisting in large oscillating ver-
tical panels is mimicked by a penalization method. The analysis of temporal and
spatiotemporal spectra reveals that the flow for the strongest forcing in the experi-
ments is composed of two superposed large and quasi-steady horizontal vortices, of
internal waves in box modes and of much weaker waves outside the modes. Spa-
tial spectra and spectral energy budget confirm that the flow is in an intermediate
regime for very small horizontal Froude number Fhand buoyancy Reynolds num-
ber Rclose to unity. Since the forcing frequency ωfis just slightly smaller than
the Brunt-V¨ais¨al¨a frequency N, there are energy transfers towards slower waves and
large vortices, which correspond to an upscale energy flux over the horizontal.
Two other experimentally feasible sets of parameters are investigated. A larger
amplitude forcing shows that it would indeed be possible to produce in huge appara-
tus like the Coriolis platform stratified turbulence forced by waves for small Fhand
buoyancy Reynolds number Rof order 10. Forcing slower waves for ωf= 0.40N
leaves space between ωfand Nfor “down-time-scale” transfers through weakly non-
linear interactions with temporal spectra consistent with ω−2slope. However, for
this set of parameters, the large scales of the flow are strongly dissipative and there
is no downscale energy cascade.
I. INTRODUCTION
Statistical results obtained from oceanic measurements are usually interpreted as the
signature of an internal wave field. Gravito-inertial waves can propagate into the oceans
because of the stable density stratification and the Earth rotation. These two effects are
characterized by two frequencies, the Brunt-V¨ais¨al¨a frequency Nand the Coriolis frequency
f, respectively. In the oceans, Nis usually much larger than f. The velocity and temperature
spectra measured at different times and locations were observed to present some similarities.
Temporal spectra scale as ω−2between the Coriolis frequency fand the Brunt-V¨ais¨al¨a
frequency N. Vertical spectra tend to scale as kz
−2at large scales and as N2kz
−3(“saturated
spectra”) at intermediate scales. A turbulent k−5/3slope is observed only at small scales
along the vertical (dropped spectra) and at much larger scales along the horizontal (towed
∗pierre.augier@univ-grenoble-alpes.fr
arXiv:2210.02855v1 [physics.flu-dyn] 6 Oct 2022
2
spectra). The kz
−5/3spectrum at small scales corresponds to weakly stratified isotropic
turbulence. In contrast, the kh
−5/3spectrum at much larger scales together with the steeper
vertical spectra cannot be due to isotropic turbulence and another interpretation should be
found.
Garrett & Munk (GM) [2] showed that a simple model of a continuous superposition
of internal waves is consistent with different types of measurements. Even though the
GM model is fully empirical and not based on a physical understanding of the underlying
dynamics, it is a remarkable result that the oceanic spectra can be modeled with only internal
waves, i.e. that the different spectra are consistent with the dispersion and polarization
relations. These spectra are often called “wave spectra” but they are actually the footprint
of the full dynamics, which can also involve non-wavy flows.
Indeed, flows influenced by stable stratification and system rotation can also contain a
non-wavy “balanced” part [3]. For example, in the case without rotation, the equation for
the vertical vorticity ωzdoes not contain any linear terms, which implies that the so-called
”horizontal vortices” (associated with ωzand horizontal velocity) are not linearly coupled
with the buoyancy. In the large majority of recent numerical simulations of stratified turbu-
lence, the associated ”vortical” (more precisely toroidal) energy is not small and horizontal
vortices play a key role. Moreover, wave-vortex interactions appear to be more efficient than
wave-wave interactions to drive a forward cascade [4].
There had been a lot of debate about the dynamical regime producing the GM spectra
and different physical explanations have been proposed (for reviews, see [4,5]). The success
of the GM model seems to indicate that the oceanic spectra are due to a kind of internal wave
turbulence. However, such spectra have not been reproduced with internal waves, neither
with laboratory experiments nor with numerical simulations. The large-scale k−5/3
hspectrum
tends to indicate that there could be an anisotropic turbulent cascade with an energy flux
through the horizontal scales. Recently, the theory of Weak Wave Turbulence (WWT),
which assumes that the flow consists only of weakly interacting waves, has been used to
derive solutions corresponding to the standard GM spectra and to observed variabilities
around this historical model [5,6].
An alternative dynamical explanation of the oceanic spectra has been proposed by Lind-
borg, Brethouwer and Riley [7,8]. They show that many oceanic measurements could
actually be compatible with “strongly stratified turbulence” [9,10]. This regime is strongly
nonlinear and involves, similarly to isotropic turbulence, both toroidal (associated with ωz)
and poloidal, horizontally divergent, modes. In the inertial range, the energy is approxi-
mately equipartitioned between these two modes. We will follow [11] and call this particular
regime “LAST” (for Layered Anisotropic Stratified Turbulence) to avoid confusion with
other somehow turbulent regimes in stratified fluids. The LAST regime is associated with
a downscale energy cascade and is obtained only when the large scales are simultaneously
strongly stratified (small horizontal Froude number Fh) and weakly influenced by viscosity
(large buoyancy Reynolds number R=ReFh2) [12].
Flows in the LAST regime have been obtained in high resolution idealized simulations.
In contrast, we are not aware of studies reporting forced dissipative flows composed of
a continuum of internal waves (without vortical modes) associated with spectra similar to
oceanic ones. Let us note that reproducing this regime might require very large experimental
apparatus or very large simulations that were until recently unfeasible. Moreover, most
3
numerical simulations of stratified turbulence were not designed to produce a pure internal
wave field without vortices, as described in the GM models. We can mention few studies
focussing on this subject.
•Waite & Bartello [4] carried out simulations of stratified turbulence forced in waves
with kz∼kh(fast waves for which ω'0.7N). They concluded that they have been
unsuccessful at reproducing the observed spectra and that the effects of adding vortices
can be dramatic.
•Lindborg & Brethouwer [9] completed this study by forcing in waves with kzkh
(slow waves for which ωN) such that the vertical Froude number Fv=UN/lvis
of order unity. For some parameters, they observed a clear downscale energy cascade.
Their spatial and spatiotemporal spectra show that the dynamics of the inertial range
corresponds to the LAST regime and that inertial waves dominate only for the very
large horizontal scales.
•Le Reun et al. [13] simulate some forced dissipative flows made of internal gravity
waves but the buoyancy Reynolds number is quite small. Therefore, these waves
should be dissipated at large horizontal scales and there is no need for a downscale
energy cascade bringing energy at small horizontal scales.
•Calpe Linares et al. [14] studied two-dimensional stratified turbulence forced by fast
internal waves. They verified that the conditions on Fhand Ralso determined the
regime for 2d stratified turbulence. For values corresponding to the oceanic dynamics
(Fh1 and R  1), the dynamics corresponds to a strongly nonlinear regime which
cannot be interpreted in the framework of the WWT theory.
Recent experiments [1,15] have managed to generate strongly stratified turbulent flows by
forcing large scale internal waves into a large scale tank (the Coriolis platform in Grenoble)
filled with stratified salty water. A continuum of waves was observed, and key elements of
WWT phenomenology were identified. However, some discrepancies with the phenomenol-
ogy were noticed, such as a flat frequency spectra or finite size effects in the form of res-
onant modes. Frequency spectra qualitatively consistent with GM spectra were observed
at frequencies greater than the forcing and extending at frequencies greater than N. This
observation suggests the occurence of strongly non linear wave turbulence [15].
In this article, we present the results of numerical simulations that were designed in direct
inspiration of the experiments of [1]. In particular, we model with a immersed boundary
method the experimental forcing mechanism which uses oscillating panels on the boundaries
of the fluid domain to generate waves. This forcing scheme is uncommon for numerical
simulations, as it is local in space and forces only waves, as opposed to most numerical
forcing schemes used in stratified turbulence. We focus is this study on sets of physical
parameters that were or could be obtained in real experiments in huge apparatus like the
Coriolis platform. The paper is not aimed at simulating the details of the experiment but
only its major characteristics.
This article is organized as follow. The numerical methods are presented in section 2.
Section 3 is dedicated to the analysis of simulations corresponding to a set of parameters
considered in [1]. We then extend in section 4 the experiments of [1] by considering two
other sets of parameters that could be obtained in the Coriolis platform.
4
II. NUMERICAL SETUP
The numerical simulations presented in this article are performed using the pseudospec-
tral solver ns3d.strat from the FluidSim Python package [16–18]. The simulations and the
analysis should be reproducible with Fluidsim version 0.6.1 [19]. Using this solver, we in-
tegrate the three-dimensional Navier-Stokes equations under the Boussinesq approximation
with an added fourth-order hyperviscosity term:
∂tv+ (v·∇)v=bez−1
ρ0
∇p+ν2∇2v+ν4∇4v+fh,(1)
∂tb+ (v·∇)b=−N2vz+κ2∇2b+κ4∇4b, (2)
where vis the velocity, pthe pressure, b=−gδρ/ρ0, with ρ0the mean density and δρ
the departure from the stable linear density stratification. For all simulations, the second-
order viscosity is set to the value for water in usual temperature and pressure conditions,
ν2= 10−6m2/s. In the following, all physical quantities from simulation data are expressed
in SI units.
The fourth-order viscosity ν4is left as a free parameter and adapted to the resolution
of simulations in order to ensure that the energy brought at the smallest simulated scales
by the non linear fluxes are dissipated without accumulation. Similarly, the equation of
motion (2) for the buoyancy field bpresents both second and fourth order diffusive terms,
with corresponding diffusion coefficients κ2and κ4. We can then build two different Prandtl
numbers Pri=νi/κifor i= 2,4. In all simulations, both those Prandtl numbers are set
to unity, such that κ2=ν2and κ4=ν4. The use of both normal and hyperviscosity is an
important tool for the comparison with experiments. It allows us to carry out simulations
with a well defined physical Reynolds number at a coarse resolution and to be able to
quantify the difference with a proper DNS. More specifically, we use the measure of the
turbulent kinetic dissipations εK2and εK4based on both viscosities, and the ratio εK2/εK
where εK=εK2+εK4is the total kinetic energy dissipation, as an indicator of how close the
simulations we perform are to proper DNS of the true Navier-Stokes equations with only
water viscosity. For a set of physical parameters, the needed hyperviscosity decreases when
the resolution is increased and the ratio εK2/εKgrows towards unity.
The geometry and forcing scheme used for the simulations are inspired by the configura-
tion of experiments that were performed in the Coriolis facility in LEGI, Grenoble (France)
[1]. In the experiments, a parallelepipedic domain of size 6 ×6×1 m3is isolated inside
the tank of the facility, using two adjacent fixed walls and two oscillating walls acting as
wavemakers. The wavemakers are set to oscillate around their mid-height horizontal axis,
with frequency ωfand maximum horizontal excursion abeing free parameters of this forcing
mechanism. In the simulations we present here, the wavemakers are modeled using a L2
volume penalization method [20,21], which is schematized in figure 1.
This penalization methods works by introducing a forcing term of the following shape in
the momentum equation (1):
fh=1
τΘ(x)(v∗
h(x, t)−vh),(3)
where v∗
h(x, t) is the target velocity profile imposed by the wavemakers, Θ(x) is an activation
function ensuring that the forcing acts only locally in place of the wavemakers, and τis a
5
FIG. 1. Schematic representation of the L2volume penalization method, shown in the whole
simulation domain (left) and in a vertical cut (right). Wavemakers are modeled by parallepipedic
penalization volumes (in dark blue) close to the vertical boundaries of the simulation domain. In
each of these volumes, we implement a virtual body force designed to impose a prescribed velocity
profile (blue arrows) on the normal component of the fluid velocity field. The vertical periodicity of
the simulation domain is ensured by choosing a sinusoidal velocity profile with one wavelength over
the height Hof the box. Horizontal periodicity is ensured by having the penalization volumes sit
over the vertical boundaries of the domain, so that facing boundaries are imposed the same velocity
profile at all times. The mathematical details of the virtual body force are given in equation (3).
typical timescale at which the forcing acts. The forcing term described by equation (3)
ensures that in the region where Θ(x) = 1, the horizontal velocity field vhfollows the target
velocity v∗
hover timescales that are large compared to τ. In order to give a meaningful
description of the dynamics imposed by wavemakers, we must then choose τto be small
compared to the typical timescales of the flow that we simulate. Typically, we take τ
to be small compared to the Brunt-V¨ais¨al¨a period, i.e. τ=TN/20, with TN= 2π/N.
The activation function Θ(x) is zero outside the wavemakers, and unity inside a region
representing the oscillating panels. For the sake of simplicity, we choose this activation
region of a single panel to be a paralleleliped of the same height as the simulation domain, of
length slightly smaller than the horizontal size of the box in order to avoid interpenetration
of the panels in the corners, and with a thickness equal to the forcing amplitude 2a. In
order to ensure the horizontal periodicity of the forcing term fthat is required by the
pseudospectral solver, the activation region of each panel is centered on the corresponding
vertical boundary of the box, meaning that contrary to the experimental setup, the panels
are not facing motionless walls but rather a virtual copy of themselves. Additionally to
the periodicity constraint, the use of a pseudospectral solver requires that the activation
function Θ is a smooth function of space in order to avoid Gibbs oscillations. To ensure
this, we take Θ to grow from zero to unity as a hyperbolic tangent when crossing the
activation region boundary, over a small scale set arbitrarily to four grid points. Inside the
activation region of each wavemaker, we apply the forcing term only to the fluid velocity
component that is orthogonal to the corresponding vertical boundary, meaning that we
model the panels with a free-slip boundary condition, and that we neglect their vertical
motion. This last approximation is justified by the fact that the experimental values for
aare of the order of a few centimeters, meaning that the angular excursion of a panel is
very small. We thus only prescribe the orthogonal component of the target velocity field to
be v∗
⊥=asin(2πz/Lz)s(t), with Lzthe height of the simulation box, and s(t) a prescribed
摘要：

Numericalstudyofexperimentallyinspiredstrati edturbulenceforcedbywavesJasonReneuve,1ClementSavaro,1GeraldineDavis,1CostanzaRodda,1NicolasMordant,1andPierreAugier1,1LaboratoiredesEcoulementsGeophysiquesetIndustriels,UniversiteGrenobleAlpes,CNRS,Grenoble-INP,F-38000Grenoble,FranceStrati edowsforc...

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