1. Introduction
The energy cascade has been a cornerstone concept in turbulence theory, where energy is supplied to a system at large scales and dissipated at small scales. The classical Kolmogorov picture (K41) says that between those two ends of the spectrum, the energy is transported locally from scale to scale. As the energy travels from the large scales, where it is created, to the small scales, where it is dissipated, the information contained about the large-scale flow eventually gets lost, and the statistics become isotropic (Kolmogorov’s local isotropy hypothesis). In spectral space, the mode-to-mode energy transfer between two modes
$\boldsymbol{\kappa }$
and
$\boldsymbol{\kappa}^{\prime}$
implies a third mode,
$\boldsymbol{\kappa }$
–
$\boldsymbol{\kappa}^{\prime}$
, such that they form a triangle called a triad interaction. As the idea of scale-locality of energy transfer is that only scales similar to a given scale contribute to energy transfer across the latter, it is expected that energy transfer occurs due to local triads, where
$|{\boldsymbol{\kappa }}| \approx |\boldsymbol{\kappa}^{\prime}| \approx |\boldsymbol{\kappa }{-} \boldsymbol{\kappa}^{\prime}|$
.
An early investigation of spectral locality came from Kraichnan (Reference Kraichnan1971). The introduction of a parameter
$v$
, measuring the ratio of the smallest to the middle wavenumber in a triad, in the direct interaction approximation (Kraichnan Reference Kraichnan1959) closure led to the discovery that a significant part of the energy transfer is due to interactions in which
$v \lt 0.5$
. While this may seem contradictory, the locality of a triad, satisfied for
$|{\boldsymbol{\kappa }}| \approx |\boldsymbol{\kappa}^{\prime}| \approx |\boldsymbol{\kappa }{-} \boldsymbol{\kappa}^{\prime}|$
, is not to be confused with the locality of energy transfer, which states that energy transfer should be high between neighbouring scales
$|{\boldsymbol{\kappa }}| \approx |\boldsymbol{\kappa}^{\prime}|$
. Indeed, these early results suggested that non-local triads consisting of two neighbouring scales and a large scale (small wavenumber) are crucial for energy transfer.
Domaradzki and co-workers (Domaradzki & Rogallo Reference Domaradzki and Rogallo1988, Reference Domaradzki and Rogallo1990; Domaradzki Reference Domaradzki1992; Ohkitani & Kida Reference Ohkitani and Kida1992) explicitly calculated band-to-band energy transfer functions using filtered velocity fields obtained from direct numerical simulations (DNS). They showed that the most intense energy transfers occurred between neighbouring wavenumbers, mediated by a small wavenumber, thus establishing the concept of ‘local energy transfer in non-local triad interaction’. While it is in agreement with the classical picture of the energy cascade with local transfers, this point of view clashes with another assumption of the Kolmogorov framework, namely the universality of small scales. Indeed, the statistical independence of small scales from large-scale flow is questioned by the fact that the most intense triad interactions include the large scales. In particular, an extensive body of work about anisotropically forced homogeneous turbulence is presented in Yeung & Brasseur (Reference Yeung and Brasseur1991), Brasseur & Wei (Reference Brasseur and Wei1994), Yeung, Brasseur & Wang (Reference Yeung, Brasseur and Wang1995) and Zhou, Yeung & Brasseur (Reference Zhou, Yeung and Brasseur1996). They found that non-local interactions were mostly responsible for anisotropy distribution in high-wavenumbers shells.
Subsequent studies (Waleffe Reference Waleffe1992; Zhou Reference Zhou1993a , Reference Zhoub ), while confirming the predominance of non-local triads in the energy flux, argued that the mode-to-mode energy transfers are not the appropriate quantities to determine the locality of energy transfer. Instead, integrated quantities such as the shell-to-shell energy transfer functions should be investigated. In this aggregate function, the influence of the non-local interactions producing intense local energy transfers diminishes. Indeed, there is a fixed number of non-local interactions compared to the total number of interactions, and further cancellations due to sign-definite regions are expected. Eyink (Reference Eyink1994, Reference Eyink1995) introduced another definition of local flux to show that local interactions do dominate in the limit of high Reynolds number, in order to justify universality of small-scale statistics. Later, the previous work was extended for non-sharp filters, obtaining bounds on the contribution of non-local interactions to the local energy flux (Eyink Reference Eyink2005; Aluie & Eyink Reference Aluie and Eyink2009; Eyink & Aluie Reference Eyink and Aluie2009).
Following works (Alexakis, Mininni & Pouquet Reference Alexakis, Mininni and Pouquet2005; Mininni, Alexakis & Pouquet Reference Mininni, Alexakis and Pouquet2006; Domaradzki & Carati Reference Domaradzki and Carati2007a , Reference Domaradzki and Caratib ; Domaradzki, Teaca & Carati Reference Domaradzki, Teaca and Carati2009) continue to support the ‘local energy transfer in non-local triad interaction’ perspective via higher Reynolds number simulations, while showing a weakening, but non-vanishing, of these interactions as the Reynolds number is growing (Mininni, Alexakis & Pouquet Reference Mininni, Alexakis and Pouquet2008). With the use again of statistics based on spectral band-truncated velocity fields, they showed the coexistence of non-local interactions and local energy cascade. Furthermore, they highlighted that the extent of locality, i.e. the number of bands involved in intense local energy transfers, is dependent on the numerical forcing conditions.
Furthermore, while not directly looking at the locality question, recent studies (Kang, Protas & Bustamante Reference Kang, Protas and Bustamante2021; Protas, Kang & Bustamante Reference Protas, Kang and Bustamante2024) investigate the ‘path’ of energy across spectral modes, and show that phase correlation in phase space is also local, suggesting that energy is transferred locally between neighbouring modes.
While the locality of energy transfer is established in all works mentioned above, there is controversy about which triad interactions are responsible for it, the main question being ‘are the large scales involved in energy transfer at any other scale?’. We would like to put this question in the general context of the statistical independence of small scales, as assumed by Kolmogorov theory for large enough Reynolds numbers. In this regard, there are still ongoing discussions about deviations from the expected behaviour of universal small-scale statistics. For example, a slower return to isotropy or even a persistence of anisotropy at small scales from large-scale forcing was observed numerically (Biferale & Toschi Reference Biferale and Toschi2001; Biferale & Vergassola Reference Biferale and Vergassola2001) and experimentally (Carter & Coletti Reference Carter and Coletti2017, Reference Carter and Coletti2018). These are usually attributed to moderate Reynolds number effects; see Iyer, Sreenivasan & Yeung (Reference Iyer, Sreenivasan and Yeung2017) for a discussion on the topic.
In the present work, we compute individual mode-to-mode energy transfers directly from DNS, which allows us to distinguish between the energy exchanged by each pair of modes within a triad. We introduce a potential function based on the energy content of the modes involved, and show that it predicts the distribution of intense energy transfers. By performing simulations with forcing applied at intermediate wavenumbers, we demonstrate that the most intense transfers are determined not only by the local or non-local character of the interaction, but mainly by the spectral location of the energy-containing range (ECR). The methodology and definitions are presented in § 2, followed by results for classical large-scale forcing in § 3, and for shifted forcing configurations in § 4.
2. Methodology
2.1. Incompressible flow in spectral space
The Navier–Stokes equations for an incompressible flow are given by
\begin{align} \partial _{t} u_{j} + u_{l} \frac {\partial u_{j}}{\partial x_{l}} & = - \frac {\partial p}{\partial x_{j}} + \nu \frac {\partial ^{2} u_{j}}{\partial x_{l}\, \partial x_{l}}, \end{align}
\begin{align} \frac {\partial u_{j}}{\partial x_{j}} & = 0. \end{align}
Using the Fourier series representation of the velocity field
$u_{j}(\boldsymbol{x})$
as
\begin{equation} u_{j}(\boldsymbol{x}) = \sum _{\boldsymbol{\kappa }} \hat{u}_{j}(\boldsymbol{\kappa })\, \textrm{e}^{i \boldsymbol{\kappa } \boldsymbol{\cdot }\boldsymbol{x}}, \end{equation}
the Navier–Stokes equations are transformed to spectral space, reading
\begin{align} \big(\partial _{t} + \nu \kappa ^{2}\big)\, \hat{u}_{j}(\boldsymbol{\kappa }) & = - P_{\textit{jm}}(\boldsymbol{\kappa })\, \hat{C}_{m}(\boldsymbol{\kappa }), \end{align}
\begin{align} \kappa _{j}\, \hat{u}_{j}(\boldsymbol{\kappa }) & = 0, \end{align}
where the pressure projection tensor
$P_{\textit{jm}}(\boldsymbol{\kappa })$
is given by
\begin{equation} P_{\textit{jm}}(\boldsymbol{\kappa }) = \delta _{\textit{jm}} - \frac {\kappa _{j} \kappa _{m}}{\kappa ^{2}}, \end{equation}
and the convection term is expressed as a convolution product
$\hat{C}_{m}(\boldsymbol{\kappa }) = \sum _{\boldsymbol{\kappa}^{\prime}} \hat{c}_{m}(\boldsymbol{\kappa }, \boldsymbol{\kappa}^{\prime})$
, with the single contributions reading
\begin{equation} \hat{c}_{m}(\boldsymbol{\kappa }, \boldsymbol{\kappa}^{\prime}) = i \kappa _{l}\, \hat{u}_{m}(\boldsymbol{\kappa}^{\prime})\, \hat{u}_{l}(\boldsymbol{\kappa }{-} \boldsymbol{\kappa}^{\prime}). \end{equation}
The kinetic energy of mode
$\boldsymbol{\kappa }$
is defined as
$\hat{E}(\boldsymbol{\kappa }) = 1/2 \, \hat{u}_{j}(\boldsymbol{\kappa })\, \hat{u}_{j}^{*}(\boldsymbol{\kappa })$
, where
$^{*}$
denotes the complex conjugate, and its evolution follows directly from the Navier–Stokes equations as
\begin{equation} \big(\partial _{t} + \nu \kappa ^{2}\big)\, \hat{E}(\boldsymbol{\kappa }) = - \sum _{\boldsymbol{\kappa}^{\prime}} \hat{T}(\boldsymbol{\kappa }, \boldsymbol{\kappa}^{\prime}), \end{equation}
with the mode-to-mode energy transfer rate
\begin{equation} \hat{T}(\boldsymbol{\kappa }, \boldsymbol{\kappa}^{\prime}) = \text{Re} \big \{ P_{\textit{jm}}(\boldsymbol{\kappa })\, \hat{c}_{m}(\boldsymbol{\kappa }, \boldsymbol{\kappa}^{\prime})\, \hat{u}_{j}^{*}(\boldsymbol{\kappa }) \big \} = \text{Re} \big \{ i \kappa _{l}\, \hat{u}_{l}(\boldsymbol{\kappa }{-} \boldsymbol{\kappa}^{\prime})\, \hat{u}_{j}(\boldsymbol{\kappa}^{\prime})\, \hat{u}_{j}^{*}(\boldsymbol{\kappa }) \big \}. \end{equation}
This quantity represents the energy exchange rate between modes
$\boldsymbol{\kappa }$
and
$\boldsymbol{\kappa}^{\prime}$
. A triad interaction consists of three mode-to-mode energy transfers, one between each pair of modes involved. By convention, a negative
$\hat{T}(\boldsymbol{\kappa }, \boldsymbol{\kappa}^{\prime})$
value means energy transfer from
$\boldsymbol{\kappa}^{\prime}$
to
$\boldsymbol{\kappa }$
(gain), and a positive
$\hat{T}(\boldsymbol{\kappa }, \boldsymbol{\kappa}^{\prime})$
value means energy transfer from
$\boldsymbol{\kappa }$
to
$\boldsymbol{\kappa}^{\prime}$
(loss).
The quantity of reference to discuss the locality of energy transfer is the shell-to-shell transfer function. A shell
$K$
is defined as a set of modes defined by a radius
$k$
and an increment
$\Delta k$
as
$K = \left \{ \boldsymbol{\kappa}^{\prime} \mid k \le |\boldsymbol{\kappa}^{\prime}| \lt k + \Delta k \right \}$
. The shell-to-shell transfer function, i.e. the total amount of energy transfer between shells
$K$
and
$K^{\prime}$
, is given by integration of
$\hat{T}$
as
\begin{equation} \hat{S}(K, K^{\prime}) = \sum _{\boldsymbol{\kappa } \in K} \sum _{\boldsymbol{\kappa}^{\prime} \in K^{\prime}} \hat{T}(\boldsymbol{\kappa }, \boldsymbol{\kappa}^{\prime}). \end{equation}
This requires knowledge of the whole
$(\boldsymbol{\kappa }, \boldsymbol{\kappa}^{\prime})$
space, which is six-dimensional and therefore not directly computable. Instead, we will present computations of
$\hat{T}$
for given sampling wavenumbers
$\boldsymbol{\kappa }^{(s)}$
, for all
$\boldsymbol{\kappa}^{\prime}$
. We argue here that thanks to isotropy, the mode-to-mode energy transfer
$\hat{T}(\boldsymbol{\kappa }^{(s)}, \boldsymbol{\kappa}^{\prime})$
is similar up to a rotation for two different sampling points if they have the same norm, as they imply summation over a shell. Therefore, we will compute the mode-to-shell energy transfer as
\begin{equation} \hat{S}(\boldsymbol{\kappa }, K^{\prime}) = \sum _{\boldsymbol{\kappa}^{\prime} \in K^{\prime}} \hat{T}(\boldsymbol{\kappa }, \boldsymbol{\kappa}^{\prime}), \end{equation}
which on average is representative of
$\hat{S}(K, K^{\prime})$
. The expected value is recovered upon multiplication with the number of modes in shell
$K$
. We will present mode-to-shell transfer functions normalised by their maximum absolute value.
2.2. Locality of energy transfers
Historically, two notions of locality have emerged. The first, which we will refer to as ‘Kraichnan locality’ (Kraichnan Reference Kraichnan1967, Reference Kraichnan1971), derives from the assumption of universality of small scales and energy cascade. To satisfy this hypothesis, the scales involved in the energy transfer across a given scale have to be similar (
$|{\boldsymbol{\kappa }}| \approx |\boldsymbol{\kappa}^{\prime}| \approx |\boldsymbol{\kappa }{-} \boldsymbol{\kappa}^{\prime}|$
), as shown in figure 1(
a).
Scale-local energy transfer (ET) within (a) a local interaction and (b) a non-local interaction.
The second notion comes from the first statistics gathered from numerical simulations dedicated to the locality topic. Without access to the triad interaction statistics, Domaradzki & Rogallo (Reference Domaradzki and Rogallo1990) gathered information using transfer functions based on the quantity
$T(k|p,q)$
, describing the effect on scales of wavenumber
$k$
of all triad interactions involving wavenumbers of sizes
$p$
and
$q$
. They showed that most of the energy transfer was due to triads with a leg in the large scales (see figure 1
b), hence introducing the concept of ‘local energy transfer in non-local interactions’. These results, while acknowledging the locality of energy transfer, clash with the classical theory as the most active interactions involve a large scale.
While the two proposals agree on the locality of the energy cascade, there is controversy about which interactions are responsible for it. Theoretical studies predict an upper bound on the contributions of non-local interactions (Eyink Reference Eyink1995, Reference Eyink2005; Aluie & Eyink Reference Aluie and Eyink2009), while numerical results point to non-local interactions with a leg in the large scales being determinant for energy transfer. Mininni et al. (Reference Mininni, Alexakis and Pouquet2008) numerically showed a weakening but non-vanishing influence of non-local interactions against increasing
$Re$
.
In the present work, we present results obtained from DNS with moderate Reynolds number that confirm the dominance of local energy transfer in non-local interactions at low Reynolds number observed previously numerically. We propose that these kinds of interactions will always dominate in intensity (but not in number), and introduce for that purpose the potential of the mode-to-mode energy transfer, i.e. the absolute maximum energy transfer between modes
$\boldsymbol{\kappa }$
and
$\boldsymbol{\kappa}^{\prime}$
. We derive this expression based on dimensional arguments from the expression for the mode-to-mode energy transfer
$\hat{T}(\boldsymbol{\kappa }, \boldsymbol{\kappa}^{\prime})$
(2.9), relating the velocity modes involved to their energy content:
\begin{equation} \hat{Q}(\boldsymbol{\kappa }, \boldsymbol{\kappa}^{\prime}) = |\boldsymbol{\kappa }|\, \hat{E}(\boldsymbol{\kappa })^{1/2}\, \hat{E}(\boldsymbol{\kappa}^{\prime})^{1/2}\, \hat{E}(\boldsymbol{\kappa }{-} \boldsymbol{\kappa}^{\prime})^{1/2}. \end{equation}
Note that
$\hat{T}(\boldsymbol{\kappa }, \boldsymbol{\kappa}^{\prime})$
(and corresponding
$\hat{Q}$
) represents the energy transfer rate between
$\boldsymbol{\kappa }$
and
$\boldsymbol{\kappa}^{\prime}$
, and that the mode
$\boldsymbol{\kappa }{-} \boldsymbol{\kappa}^{\prime}$
does not directly exchange energy with
$\boldsymbol{\kappa }$
within this individual mode-to-mode transfer. Further noticing the detailed conservation property
$\hat{T}(\boldsymbol{\kappa }, \boldsymbol{\kappa}^{\prime}) = - \hat{T}(\boldsymbol{\kappa}^{\prime}, \boldsymbol{\kappa })$
(Onsager Reference Onsager1949; Eyink & Sreenivasan Reference Eyink and Sreenivasan2006) (see Appendix D), each mode has a clearly defined physical interpretation. The modes
$\boldsymbol{\kappa }$
and
$\boldsymbol{\kappa}^{\prime}$
exchange energy at a rate given by
$\hat{T}$
with the influence of
$\boldsymbol{\kappa }{-} \boldsymbol{\kappa}^{\prime}$
. We call
$\boldsymbol{\kappa }$
the primary (or sampling) mode,
$\boldsymbol{\kappa}^{\prime}$
the reactant mode, and
$\boldsymbol{\kappa }{-} \boldsymbol{\kappa}^{\prime}$
the catalyst mode.
Of course, for a given
$\boldsymbol{\kappa}^{\prime}_{0}$
, we mean not that the energy transfer
$\hat{T}(\boldsymbol{\kappa }, \boldsymbol{\kappa }- \boldsymbol{\kappa}^{\prime}_{0})$
is identically zero, but that it should be considered as a separate energy transfer for which
$\boldsymbol{\kappa }- \boldsymbol{\kappa}^{\prime}_{0}$
is the reactant mode (exchanging energy) and
$\boldsymbol{\kappa}^{\prime}_{0}$
is the catalyst mode (not exchanging energy). While both mode-to-mode transfers involve the same triangle of modes and are usually considered as a single triad interaction in the literature, we make the distinction based on which modes can gain/lose energy and which cannot. This allows us to study separately the energy exchange rates associated with the two legs involving the primary mode in a given interaction. This approach is similar to that of Alexakis et al. (Reference Alexakis, Mininni and Pouquet2005) and Mininni et al. (Reference Mininni, Alexakis and Pouquet2006, Reference Mininni, Alexakis and Pouquet2008) described in the next subsection.
To investigate
$\hat{Q}$
, let us assume a classical scenario, with the energy-containing range (ECR) confined at the largest scales. For that purpose, we use the model spectrum from Pope (Reference Pope2000), determined by the energy-containing scale
$L = L_{0} = 2\pi$
, the viscosity
$\nu = 10^{-7}$
and
$Re_{\lambda } = 10^{5}$
, as drawn in figure 2(
a).
(a) Model kinetic spectrum with energy at largest scales, (b) resulting mode-to-mode potential
$\hat{Q}$
, and (c) in log scale to show the decay of potential. Note that (c) is centred around
$\kappa _{x}/2$
to highlight the symmetry of
$\hat{Q}$
.
With a fixed primary mode
$\boldsymbol{\kappa }$
, the potential depends on the product of the square root of the energy of both the reactant and catalyst modes. This function is symmetric and exhibits two peaks, as highlighted in figure 2(
b). They correspond to regions of high potential
$\hat{Q}$
, where either the reactant (green) or the catalyst (blue) modes have high energy content (i.e. are in the ECR). Note that the potential is determined by the spectrum and its scaling with
$|\boldsymbol{\kappa }|$
, and that a scaling exponent other than
$-5/3$
may result in a different distribution of regions with high potential.
The first region of interest (blue) concerns energy transfers for which the catalyst mode has a high energy content and will enhance the energy transfer between
$\boldsymbol{\kappa }$
and
$\boldsymbol{\kappa}^{\prime}$
. As the energy is at the largest scales, the dominant interactions will be for
$\boldsymbol{\kappa }{-} \boldsymbol{\kappa}^{\prime} \approx 0$
, so for
$\boldsymbol{\kappa } \approx \boldsymbol{\kappa}^{\prime}$
. Therefore this region corresponds to the non-local interactions showcasing intense local energy transfer, as reported in the literature (Domaradzki & Rogallo Reference Domaradzki and Rogallo1990). Assuming for now that
$\hat{Q}$
is a good estimator for
$\hat{T}$
in this region, we can reformulate the locality statement: the dominant local energy transfer observed is not only due to the non-local nature of the interactions, but is rather because the non-local interactions involve a catalyst mode in the ECR. This shifts the understanding of the locality of the intense energy transfers from the locality/non-locality of a triad to the energy content of its modes, as hinted in Alexakis et al. (Reference Alexakis, Mininni and Pouquet2005) and Mininni et al. (Reference Mininni, Alexakis and Pouquet2006) in their discussion about the dependence of the energy transfer function to the forcing range. This proposal will also let us discuss the cases where the ECR is not located at the largest scales in § 4. This spectral region is labelled the catalytic region.
The other region of high energy transfer potential (green) is the case where the reactant mode has a high energy content. Due to the symmetry of the definition of
$\hat{Q}$
, it corresponds to non-local energy transfer in non-local interactions. In other words, a mode
$\boldsymbol{\kappa }$
has the potential to directly exchange energy with modes in the ECR, wherever it is. While it has been reported that energy transfer functions show a slight positive exchange with the large scales (Domaradzki & Rogallo Reference Domaradzki and Rogallo1990), these kinds of energy transfer have never been observed, and would go against the established picture of the scale-localness of the energy cascade. We will discuss dampening of these mode-to-mode transfers in § 3.2, and residual non-local energy transfers in § 4.3. This spectral region is labelled the reacting region.
Assuming that both modes are in the inertial range, such that
$\hat{E}(\boldsymbol{\kappa }) \propto |\boldsymbol{\kappa }|^{-5/3}$
, the scaling of the potential follows
\begin{equation} \hat{Q}(\boldsymbol{\kappa }, \boldsymbol{\kappa}^{\prime}) \propto |\boldsymbol{\kappa}^{\prime}|^{-5/6}\, |\boldsymbol{\kappa }{-} \boldsymbol{\kappa}^{\prime}|^{-5/6}. \end{equation}
Furthermore, if
$|\boldsymbol{\kappa}^{\prime}| \gg |\boldsymbol{\kappa }|$
, then
$|\boldsymbol{\kappa }{-} \boldsymbol{\kappa}^{\prime}| \approx |\boldsymbol{\kappa}^{\prime}|$
, and the scaling reduces to
\begin{equation} \hat{Q}(\boldsymbol{\kappa }, \boldsymbol{\kappa}^{\prime}) \propto |\boldsymbol{\kappa}^{\prime}|^{-5/3}. \end{equation}
This scaling behaviour is highlighted in figure 2( c), between the regions of high potential and the regions where either the reactant or catalyst mode is in the dissipation range.
2.3. Transfer functions from the literature
As mentioned in the Introduction, the numerical research on the topic of locality (Domaradzki & Rogallo Reference Domaradzki and Rogallo1990; Alexakis et al. Reference Alexakis, Mininni and Pouquet2005) used a second-order quantity to study locality. To obtain the latter, researchers calculated the third-order correlator
$T_{3}$
, defined as
\begin{equation} T_{3}(K, P, Q) = \int \boldsymbol{u}_{K} \boldsymbol{\cdot }(\boldsymbol{u}_{P} \boldsymbol{\cdot }\boldsymbol{\nabla }) \boldsymbol{u}_{Q}\, {\rm d}\boldsymbol{x}^{3}, \end{equation}
where
$\boldsymbol{u}_{K}$
stands for a velocity field filtered to retain only the modes in shell
$K$
. It represents ‘the rate of energy transfer from energy in a shell
$Q$
to energy in shell
$K$
due to interaction with the velocity field in shell
$P$
’ (Mininni et al. Reference Mininni, Alexakis and Pouquet2006). The major difference between
$T_{3}$
and the mode-to-mode energy transfer
$\hat{T}$
in (2.9) is that using a physical quantity based on filtered velocity fields will capture many interactions. Any interaction such that one mode is in shell
$K$
, another in shell
$Q$
, and the third in shell
$P$
, will be accounted for, such that the results are valid only for groups of triad interactions. Domaradzki & Carati (Reference Domaradzki and Carati2007a
,
Reference Domaradzki and Caratib
) discussed the influence of different spectral filters to obtain the velocity fields required by
$T_{3}$
, and showed only minor dependence on the results.
Note that this approach is similar in spirit to our distinction between reactant and catalyst modes. Indeed, ‘this term [
$T_{3}$
] does not give information about the energy the shell
$P$
receives or gives to shells
$K$
and
$Q$
’ (Mininni et al. Reference Mininni, Alexakis and Pouquet2006). In the notation of Mininni et al. (Reference Mininni, Alexakis and Pouquet2006), the shell
$P$
contains the catalyst modes and therefore contains no information about the transfer between the latter and the primary mode; see discussion in § 2.2.
The shell-to-shell transfer function is obtained by integration over all
$P$
shells, denoted in the literature as
$T_{2}(K, Q) = \sum _{P} T_{3}(K, P, Q)$
(similar to the second-order correlator
$P(k|p) = \sum _{q} T(k|p,q)$
from Domaradzki & Rogallo Reference Domaradzki and Rogallo1990). The summation is similar to the mode-to-shell estimate
$\hat{S}(\boldsymbol{\kappa }, K^{\prime})$
in (2.11).
2.4. Comparison with theoretical predictions
A powerful theoretical tool to quantify the energy transfers is provided by the analytical theories of turbulence introduced in the 1970s, such as the direct interaction approximation (Kraichnan Reference Kraichnan1959) or eddy-damped quasi-normal Markovian (EDQNM) (Orszag Reference Orszag1970) approaches. We chose an estimator
$\hat{H}(\boldsymbol{\kappa }, \boldsymbol{\kappa}^{\prime})$
of the mode-to-mode energy transfer derived from the EDQNM theory as presented in Lesieur (Reference Lesieur2008), which we summarise as
\begin{equation} \hat{H}(\boldsymbol{\kappa }, \boldsymbol{\kappa}^{\prime}) = \theta (\boldsymbol{\kappa }, \boldsymbol{\kappa}^{\prime})\, b(\boldsymbol{\kappa }, \boldsymbol{\kappa}^{\prime}) \frac {|\boldsymbol{\kappa }|}{|\boldsymbol{\kappa}^{\prime}|\,|\boldsymbol{\kappa }{-} \boldsymbol{\kappa}^{\prime}|} \hat{f}(\boldsymbol{\kappa }, \boldsymbol{\kappa}^{\prime}), \end{equation}
with the function
$\hat{f}(\boldsymbol{\kappa }, \boldsymbol{\kappa}^{\prime})$
as
\begin{equation} \hat{f}(\boldsymbol{\kappa }, \boldsymbol{\kappa}^{\prime}) = \hat{E}(\boldsymbol{\kappa }{-} \boldsymbol{\kappa}^{\prime}) \left ( |\boldsymbol{\kappa }|^2\, \hat{E}(\boldsymbol{\kappa}^{\prime}) - |\boldsymbol{\kappa}^{\prime}|^2\, \hat{E}(\boldsymbol{\kappa }) \right ). \end{equation}
The factor
$b(\boldsymbol{\kappa }, \boldsymbol{\kappa}^{\prime})$
depends on the shape of the triad and is given by
\begin{equation} b(\boldsymbol{\kappa }, \boldsymbol{\kappa}^{\prime}) = \frac {|\boldsymbol{\kappa}^{\prime}|}{|\boldsymbol{\kappa }|} \big ( xy + z^{3} \big ), \end{equation}
where
$x$
,
$y$
and
$z$
are the cosines of the angles opposite to
$\boldsymbol{\kappa }$
,
$\boldsymbol{\kappa}^{\prime}$
and
$\boldsymbol{\kappa }{-} \boldsymbol{\kappa}^{\prime}$
, respectively. The triad interaction time
$\theta (\boldsymbol{\kappa }, \boldsymbol{\kappa}^{\prime})$
is simplified for numerical procedure following Domaradzki & Rogallo (Reference Domaradzki and Rogallo1990), resulting in the function
\begin{equation} \theta (\boldsymbol{\kappa }, \boldsymbol{\kappa}^{\prime}) = \big [ \eta (\boldsymbol{\kappa }) + \eta (\boldsymbol{\kappa}^{\prime}) + \eta (\boldsymbol{\kappa }{-} \boldsymbol{\kappa}^{\prime}) \big ]^{-1}, \end{equation}
with the function
\begin{equation} \eta (\boldsymbol{\kappa }) = \nu\, |\boldsymbol{\kappa }|^2 + 0.31 \big [ |\boldsymbol{\kappa }|^3\, \hat{E}(\boldsymbol{\kappa }) \big ]^{1/2} \end{equation}
as proposed by Orszag (Reference Orszag1970). Figure 3 presents slices of the potential
$\hat{Q}$
and the absolute value of the EDQNM estimator
$\hat{H}$
computed from the model spectrum introduced in the previous section; see figure 2(
a). These functions both possess a cylindrical symmetry around the line in the direction of the primary mode
$\boldsymbol{\kappa }$
, here the
$\kappa^{\prime}_{x}$
axis.
(a) Potential
$\hat{Q}(\boldsymbol{\kappa } = [200,0], \boldsymbol{\kappa}^{\prime})$
and (b) EDQNM estimator
$\hat{H}(\boldsymbol{\kappa } = [200,0], \boldsymbol{\kappa}^{\prime})$
computed from the model spectrum for
$\boldsymbol{\kappa } = [200,0]$
.
Note that while the EDQNM estimator
$\hat{H}$
is not symmetric in its arguments, it exhibits a pattern similar to that of the potential. That is, there are mainly two regions where
$|\hat{H}|$
is large, locally around the primary mode as can be expected from the local energy transfer from the nonlocal interactions, but also in the ECR, close to the origin.
Another striking difference is that
$|\hat{H}|$
is much smaller everywhere in the domain than the potential. This is to be expected, as the potential assumes that every quantity in the mode-to-mode energy transfer
$\hat{T}$
is aligned, and their phases are such that the energy transfer is optimal, which cannot be the case. Furthermore,
$|\hat{H}|$
decays much faster than the potential as
$\boldsymbol{\kappa}^{\prime}$
moves away from the primary mode.
Finally, the EDQNM estimator shows specular, i.e. angle-dependent, behaviour around the origin and the primary mode. This is due to the zeros of the function
$\hat{f}$
in (2.17) and the factor
$b$
in (2.18), as shown in figure 4. The physical relevance of these functions will be discussed in §§ 3.1 and 4.2 by comparing
$\hat{H}$
to the values of
$\hat{T}$
obtained from DNS.
(a) Function
$\hat{f}(\boldsymbol{\kappa } = [200,0], \boldsymbol{\kappa}^{\prime})$
and (b) factor
$b(\boldsymbol{\kappa } = [200,0], \boldsymbol{\kappa}^{\prime})$
from the EDQNM estimator computed from the model spectrum for
$\boldsymbol{\kappa } = [200,0]$
.
2.5. Numerical procedure
In order to validate our hypotheses, we performed three sets of DNS with resolution
$512^{3}$
, using a pseudo-spectral code. The three cases differ by the spectral forcing range, where the first corresponds to a low wavenumber forcing (LWF) configuration, with the forcing applied at the largest scales in the domain, while the two others consider forcing ranges shifted to intermediate wavenumbers (intermediate wave forcing, IWF). As a result of the shifted forcing range, the ratio of the integral scale to the Kolmogorov length scale is significantly reduced, resulting in a lower
$Re_{\lambda }$
. The simulation parameters are summarised in table 1.
Simulation parameters.
For each case, several independent simulations were time and ensemble averaged once statistical stationarity has been reached. This results in the averaging of up to 12 000 time frames gathered over a total of 800 eddy turnover times for the IWF20 case, while the LWF and IWF10 cases converged with less time frames.
The mode-to-mode energy transfer
$\hat{T}$
, the potential
$\hat{Q}$
and the EDQNM estimator
$\hat{H}$
were computed directly from (2.9), (2.12) and (2.16), respectively, for a set of sample points chosen along the
$x$
axis, from
$\kappa ^{(s)}_{x} = 10$
to
$\kappa ^{(s)}_{x} = 80$
. The algorithm used to efficiently perform this computation is presented in Appendix C. The obtained energy transfer fields were validated against the energy transfers computed from the pseudo-spectral method, as the sum of all contributions has to be equal to the total energy transfer, i.e.
$\sum _{\boldsymbol{\kappa}^{\prime}} \hat{T}({\boldsymbol{\kappa }}, \boldsymbol{\kappa}^{\prime}) = \hat{C}_{m}({\boldsymbol{\kappa }})\, \hat{u}_{m}^{\ast }({\boldsymbol{\kappa }})$
.
3. Spectral energy transfers in homogeneous isotropic turbulence
In this section, we present the results obtained from the DNS in the classical case of LWF (first row of table 1). Figure 5 presents an example of the distribution for the sampling point
$\boldsymbol{\kappa } = [10,0]$
. Each point represents the value of the mode-to-mode energy transfer
$\hat{T}(\boldsymbol{\kappa }, \boldsymbol{\kappa}^{\prime})$
between the sampling mode and the mode given by the coordinate of the point (reactant mode). For a chosen mode
$\boldsymbol{\kappa}^{\prime}_{0}$
, the mode-to-mode energy transfer to the other mode of the triad
$\boldsymbol{\kappa }-\boldsymbol{\kappa}^{\prime}_{0}$
is read directly from the same figure, as indicated. The remainder of the paper is dedicated to presenting and discussing the distribution of
$\hat{T}$
and the dominance of energy transfers in the vicinity of the sampling mode. All subsequent figures are zoomed in such a way that all cases presented in table 1 can be compared directly.
Distribution of energy transfers
$\hat{T}(\boldsymbol{\kappa }, \boldsymbol{\kappa}^{\prime})$
for the LWF case (
$\kappa _{f} \sim 1$
), for
$\boldsymbol{\kappa } = [10,0]$
. The green ticks indicate the position of the sampling point. For a given
$\boldsymbol{\kappa}^{\prime}_{0}$
, the mode-to-mode transfers corresponding to
$\hat{T}(\boldsymbol{\kappa }, \boldsymbol{\kappa}^{\prime}_{0})$
and
$\hat{T}(\boldsymbol{\kappa }, \boldsymbol{\kappa }{-} \boldsymbol{\kappa}^{\prime}_{0})$
are read as indicated.
3.1. Distribution of spectral energy transfers
In order to discuss the distribution of mode-to-mode energy transfers, let us define the high energy transfer (HET) kernel as the set of modes with large energy transfer intensities.
Figure 6 shows slices in the
$(\kappa^{\prime}_{x}, \kappa^{\prime}_{y})$
plane of the distribution of the mode-to-mode energy transfer
$\hat{T}(\boldsymbol{\kappa }, \boldsymbol{\kappa}^{\prime})$
. We find that for every sampling point, the transfers are dominant in intensity in the vicinity of the latter, i.e. the HET kernel is composed of the neighbours of the sampling point, corresponding to a sphere in three dimensions. These transfers correspond to the ‘scale-local energy transfer in non-local interactions’, as illustrated in figure 1(
b), where the sampling and reactant modes are neighbours, and the catalyst mode is near the origin.
An interesting feature is that the HET kernel is independent of the sampling point wavenumber. Indeed, the latter is composed of approximately the same number of modes forming a spherical region (in three dimensions) around
$\boldsymbol{\kappa }$
. These figures also demonstrate that the energy transfer is scale-local and forward on average. The modes with a wavenumber smaller than
$\boldsymbol{\kappa }$
exhibit a negative
$\hat{T}$
(energy gain), while the modes with larger wavenumber show a positive
$\hat{T}$
(energy loss). Notably, the energy transfer due to convection is similar whether the sampling point is in the inertial or the dissipation range (
$\kappa _{\eta } \approx 42$
).
These features are made clearer when we compare the aggregated mode-to-shell transfer function
$\hat{S}$
for different sampling points, shown in figure 7. The number of shells relevant for the energy transfer is not dependent on
$|\boldsymbol{\kappa }|$
. These results agree with those reported in the literature, where Mininni et al. (Reference Mininni, Alexakis and Pouquet2006) showed a collapse of the transfer functions around the
$|\boldsymbol{\kappa }| - K^{\prime}$
shell.
The distribution of mode-to-mode energy transfers is consistent with previously obtained results and clarifies that the scale-local energy transfer in non-local interactions is the most intense on an individual basis. We now proceed to link these results to the potential
$\hat{Q}$
.
Distribution of energy transfers
$\hat{T}(\boldsymbol{\kappa }, \boldsymbol{\kappa}^{\prime})$
for the LWF case (
$\kappa _{f} \sim 1$
). The green ticks indicate the position of the sampling point. Here, (a)
$\boldsymbol{\kappa } = [20,0]$
, (b)
$\boldsymbol{\kappa } = [40,0]$
, (c)
$\boldsymbol{\kappa } = [60,0]$
, (d)
$\boldsymbol{\kappa } = [80,0]$
.
Shell-to-shell transfer functions
$\hat{S}(\kappa,K^{\prime})$
for the LWF case (
$\kappa _{f} \sim 1$
). The blue shaded area corresponds to the ECR, and the vertical green line indicates the sampling point. Here, (a)
$\boldsymbol{\kappa } = [20,0]$
, (b)
$\boldsymbol{\kappa } = [40,0]$
, (c)
$\boldsymbol{\kappa } = [60,0]$
.
3.2. Potential and effectiveness of energy transfers
Upon the introduction of the potential
$\hat{Q}$
, we mentioned that it is symmetric in its arguments. Indeed, for a given triad with fixed
$\boldsymbol{\kappa}^{\prime}$
and
$\boldsymbol{\kappa }{-} \boldsymbol{\kappa}^{\prime}$
, we have
$\hat{Q}({\boldsymbol{\kappa }}, \boldsymbol{\kappa}^{\prime}) = \hat{Q}({\boldsymbol{\kappa }}, \boldsymbol{\kappa }{-} \boldsymbol{\kappa}^{\prime})$
since it is only based on the energy content of the modes involved. When considering in particular a non-local triad, it means that the potentials of scale-local (between
$\boldsymbol{\kappa }$
and
$\boldsymbol{\kappa}^{\prime}$
) and scale-non-local (between
$\boldsymbol{\kappa }$
and
$\boldsymbol{\kappa }{-} \boldsymbol{\kappa}^{\prime}$
) energy transfers are the same. This is illustrated in figures 8(a,b), which show the regions of high potential, one centred around the origin, and the other centred around the sampling point.
However, the direct comparison of the potential
$\hat{Q}$
to the mode-to-mode energy transfer
$\hat{T}$
reveals that the HET kernel is only composed of the energy transfers for which the catalytic mode is in the ECR, as described in § 2.3, as opposed to the energy transfers for which the reactant mode is in the ECR, which are much smaller. We recover here that the most intense energy transfers occur in a non-local interaction, but that within this interaction, only the local energy transfer is large, and not the non-local one. Since our proposal is that the most intense energy transfers are the ones with the largest potential, and that both local and non-local energy transfers share the same potential, we provide arguments to explain the dampening of the latter.
Comparison of (a,b) potential
$\hat{Q}$
and (c,d) mode-to-mode energy transfer
$\hat{T}$
, for the LWF case (
$\kappa _{f} \sim 1$
). The green ticks indicate the position of the sampling point. Here, (a,c)
$\boldsymbol{\kappa } = [30,0]$
, (b,d)
$\boldsymbol{\kappa } = [60,0]$
.
For this purpose, we introduce the effectiveness of the mode-to-mode energy transfer as the ratio of actual (averaged) transfer
$\hat{T}$
to the potential
$\hat{Q}$
as
\begin{equation} \hat{\eta }({\boldsymbol{\kappa }}, \boldsymbol{\kappa}^{\prime}) = \frac {|\hat{T}({\boldsymbol{\kappa }}, \boldsymbol{\kappa}^{\prime})|}{\hat{Q}({\boldsymbol{\kappa }}, \boldsymbol{\kappa}^{\prime})}. \end{equation}
Figure 9 shows the effectiveness of energy transfers of the sampling points presented in figure 8. The first observation is that the patterns are similar for both sampling points. In fact, these patterns are similar for all sampling points (not shown here), and while we discuss the dependence of the distribution of
$\hat{T}$
with different forcing configurations in the following sections,
$\hat{\eta }$
remains the same in these cases; see Appendix B. This leads us to believe that the effectiveness patterns show some universality, independent of the shape of the energy spectrum.
Energy transfer effectiveness for the LWF case (
$\kappa _{f} \sim 1$
) centred around the sampling point. The green ticks indicate the position of the sampling point. Here, (a)
$\boldsymbol{\kappa } = [30,0]$
, (b)
$\boldsymbol{\kappa } = [60,0]$
.
We identify ‘cones of low effectiveness’ regions along the
$\kappa^{\prime}_{x}$
axis on each side of the sampling point (conical in three dimensions due to axial symmetry), and associate them with effects of the divergence-free condition (2.5). The expression for the mode-to-mode energy transfer
$\hat{T}$
(2.9) contains the factor
$\kappa _{l} \hat{u}_{l}(\boldsymbol{\kappa }{-} \boldsymbol{\kappa}^{\prime})$
, which is zero if
$\boldsymbol{\kappa }$
is normal to
$\hat{u}(\boldsymbol{\kappa }{-} \boldsymbol{\kappa}^{\prime})$
. The divergence-free condition states that the velocity mode is normal to its wavenumber, such that
$\hat{u}(\boldsymbol{\kappa }{-} \boldsymbol{\kappa}^{\prime})$
is normal to
$\boldsymbol{\kappa }{-} \boldsymbol{\kappa}^{\prime}$
. Therefore, if
$\boldsymbol{\kappa }$
and
$\boldsymbol{\kappa}^{\prime}$
are such that
$\boldsymbol{\kappa }$
is close to aligned with
$\boldsymbol{\kappa }{-} \boldsymbol{\kappa}^{\prime}$
(i.e.
$\boldsymbol{\kappa}^{\prime}$
is near the
$\kappa^{\prime}_{x}$
axis), the result is that
$\boldsymbol{\kappa }$
is also close to normal to
$\hat{u}(\boldsymbol{\kappa }{-} \boldsymbol{\kappa}^{\prime})$
, thus effectively dampening the energy transfer, as sketched in figure 10. This behaviour is encompassed in the EDQNM estimator through the geometric factor
$b$
. Figure 4(
b) shows low values of
$b$
along the
$\kappa^{\prime}_{x}$
axis, and values close to 1 on the sides (
$\boldsymbol{\kappa }$
and
$\boldsymbol{\kappa}^{\prime}$
share the same
$\kappa^{\prime}_{x}$
component).
(a) Dampened and (b) favoured energy transfers as a result of the divergence-free condition.
Furthermore, there is a decay of
$\hat{\eta }$
when
$|\boldsymbol{\kappa }{-} \boldsymbol{\kappa}^{\prime}|$
is growing. In the EDQNM estimator, it is the
$\theta (\boldsymbol{\kappa }, \boldsymbol{\kappa}^{\prime})$
function that plays this role. The eddy-damping (or eddy relaxation) parameter is modelled as such since ‘the memory should cut off because phase correlations between Fourier modes (or rather eddies) do not persist in the random convection fields of other eddies. Coherence is destroyed by non-linear scrambling’ (Orszag Reference Orszag1970). In short, neighbouring modes are expected to be phase correlated such that
$|\hat{T}|$
is maximised, and this correlation weakens as the modes are further away. Recent results (Kang et al. Reference Kang, Protas and Bustamante2021; Protas et al. Reference Protas, Kang and Bustamante2024) show that ‘triads involving energy-containing Fourier modes align their phases so as to maximise the energy flux toward small scales’ in the one-dimensional Burgers equation, i.e. that the interactions with neighbouring modes display high levels of phase correlation. It can be expected that this result also holds for the three-dimensional Navier–Stokes equations, which will be the subject of subsequent studies.
These two effects combine such that, while having a large potential, the energy transfers with the reactant mode in the ECR are small (although not always negligible; see § 4.3), as expected by the K41 theory. Indeed, the ECR is usually around the origin (large scales), and therefore for any direction of
$\boldsymbol{\kappa }$
, the corresponding
$\boldsymbol{\kappa }{-} \boldsymbol{\kappa}^{\prime}$
mode will always be in the ‘cone of low effectiveness’, where
$\boldsymbol{\kappa }$
is normal to
$\hat{u}(\boldsymbol{\kappa }{-} \boldsymbol{\kappa}^{\prime})$
.
Having now established that the mode-to-mode energy transfer
$\hat{T}$
is dependent on the potential
$\hat{Q}$
, modulated by dampening factors, we note that the potential
$\hat{Q}$
is itself determined by the energy spectrum. This naturally leads to the question of the dependency of
$\hat{T}$
on the energy spectrum.
4. Influence of large scales on the extent of energy transfer locality
In this section, we analyse the distribution of the mode-to-mode energy transfer
$\hat{T}$
in cases with shifted ECR, produced by forcing in a specific range; see table 1.
(a) Model kinetic spectrum with energy at intermediate scales, and (b) resulting mode-to-mode potential
$\hat{Q}$
.
4.1. Potential with shifted energy-containing range
When the forcing is not applied directly at the largest available scales in the domain, the energy spectrum presents a thermal equilibrium range, with a scaling proportional to
$\kappa ^{2}$
. Figure 11 presents such a spectrum computed again with the model presented in Pope (Reference Pope2000), determined by the energy-containing scale
$L = L_{0}/20$
, the viscosity
$\nu = 10^{-7}$
and
$Re_{\lambda } = 10^{5}$
. Figure 11(b) shows the resulting potential along
$\kappa^{\prime}_{x}$
. Since the energy-containing range is now a shell in spectral space, the regions with high potential (both reacting and catalytic) are now also shells, centred around the origin and the sampling point, respectively.
This means that the mode-to-mode energy transfer, if indeed strongly determined by the potential as proposed in the previous section, will be the most intense not with direct neighbours, but instead with modes slightly further away. The HET kernel will be similar to the imprint of the ECR shifted around the sampling point.
Comparisons of (a,b) potential
$\hat{Q}$
, (c,d) EDQNM estimator
$\hat{H}$
, and (e,f) mode-to-mode energy transfer
$\hat{T}$
, for the IWF10 case (
$\kappa _{f} \sim 10$
). The green ticks indicate the position of the sampling point. Here, (a,c,e)
$\boldsymbol{\kappa } = [30,0]$
,
$\kappa _{f} \sim 10$
, (b,d,f)
$\boldsymbol{\kappa } = [60,0]$
,
$\kappa _{f} \sim 10$
.
4.2. Distribution of spectral energy transfers
Figures 12 and 13 present the distributions of the potential
$\hat{Q}$
, the EDQNM estimator
$\hat{H}$
and the mode-to-mode energy transfer
$\hat{T}$
for the cases IWF10 (
$\kappa _{f} \sim 10$
) and IWF20 (
$\kappa _{f} \sim 20$
), respectively. The sampling points are chosen such that the reacting and catalytic regions (as seen in the potential
$\hat{Q}$
figures) are distinct. The overlapping cases are discussed in § 4.3.
The main observation is that the HET kernel does indeed matches the catalytic region. The most intense transfers correspond to the high potential region with the catalyst mode in the ECR, as indicated by the rings of radii 10 and 20, respectively, for the IWF10 and IWF20 cases, observed in the mode-to-mode energy transfer panels.
This confirms the proposition that
$\hat{T}$
is strongly dependent on the spectral location of the ECR, and allows us to revise the classical argument that the most intense interactions correspond to ‘local energy transfer in non-local interactions’. The intensity of the energy transfer is related not only to whether the interaction is local or non-local, but rather to the energy content of the modes involved. In most simulations, the ECR is at the largest available scales, therefore the intense energy transfers also correspond to non-local interactions as described in the literature (Domaradzki & Rogallo Reference Domaradzki and Rogallo1990; Waleffe Reference Waleffe1992).
Comparisons of (a,b) potential
$\hat{Q}$
, (c,d) EDQNM estimator
$\hat{H}$
, and (e,f) mode-to-mode energy transfer
$\hat{T}$
, for the IWF20 case (
$\kappa _{f} \sim 20$
). The green ticks indicate the position of the sampling point. Here, (a,c,e)
$\boldsymbol{\kappa } = [60,0]$
,
$\kappa _{f} \sim 20$
, (b,d,f)
$\boldsymbol{\kappa } = [80,0]$
,
$\kappa _{f} \sim 20$
.
The comparison between the mode-to-mode energy transfer
$\hat{T}$
and the EDQNM estimator
$\hat{H}$
shows reasonable agreement, in both distribution and intensity. However, and specifically in the IWF20 case, it seems that
$\hat{H}$
is slightly overestimating the intensity of energy transfer to neighbouring modes, which may even surpass the transfer in the catalytic region; see figures 13(b,d,f). As explained in § 3.2, the energy transfer of neighbours is enhanced through the
$\theta$
function in (2.19). The choice of model for the eddy-damping parameter is thus probably the cause of this discrepancy.
It appears clearly that the energy transfer displays the same specular behaviour predicted by the EDQNM estimator and discussed for the LWF case in § 3.2. Indeed, the energy transfers are suppressed along the
$\kappa^{\prime}_{x}$
axis, and the analysis of the effectiveness patterns in Appendix B reveals that the ‘cones of low effectiveness’ are also observed in both IWF10 and IWF20 cases. The external cone (for
$\kappa^{\prime}_{x} \gt \kappa _{x}$
) appears to be wider than the inner one (for
$\kappa^{\prime}_{x} \lt \kappa _{x}$
). This is attributed to the observation that the potential
$\hat{Q}$
is larger towards the larger scales than the smaller scales, and less to the suppression patterns of the divergence-free condition being different on either side of
$\kappa _{x}$
. This behaviour can also be seen in figures 2(
b) and 11(b), where the regions between the peaks of
$\hat{Q}$
show smaller decay than the outer ones.
Finally, the forward nature of the convection term is established for all sampling points, where the energy transfers are negative for
$|\boldsymbol{\kappa}^{\prime}| \lt |{\boldsymbol{\kappa }}|$
, and positive for
$|\boldsymbol{\kappa}^{\prime}| \gt |{\boldsymbol{\kappa }}|$
. It is visible on the EDQNM estimator
$\hat{H}$
panels that the energy transfers are close to zero for
$|\boldsymbol{\kappa}^{\prime}| \approx |{\boldsymbol{\kappa }}|$
. In these regions, the energy transfers are also intense but not sign-definite, and therefore average to zero contribution. This result is expected to hold for the mode-to-mode energy transfer
$\hat{T}$
upon sufficient convergence of the statistics.
Figures 14 and 15 show the shell-to-shell transfer functions for the IWF10 and IWF20 cases, respectively. As can be expected from the distribution of energy transfers presented above, the transfer functions are spread over several shells, with most of the transfers happening with shells within the range
$[\kappa _{x} - \kappa _{f}, \kappa _{x} + \kappa _{f}]$
(indicated in light green).
The locality of energy transfer holds as the exchanges are primarily with neighbouring shells. The extent of locality, which can be defined as the distance between the sampling mode and the peaks of the transfer functions, scales with
$\kappa _{f}$
. Note that this extent of locality is not the same on either side of the sampling mode, which is again linked to the bias of the potential
$\hat{Q}$
towards the large scales. Furthermore, the peaks of
$\hat{S}(\kappa , K^{\prime})$
are not exactly at
$\kappa _{x} \pm \kappa _{f}$
. This was already observed in Mininni et al. (Reference Mininni, Alexakis and Pouquet2006) and attributed to moderate Reynolds number effects. Here, we argue that it is due to the fact that the shell-to-shell is obtained from progressive shell integration through the catalytic region. We have established the specular behaviour of the latter, and it is therefore to be expected that the extent of locality scales with
$\kappa _{f}$
, but is not directly proportional to it.
Shell-to-shell transfer functions
$\hat{S}(\kappa , K^{\prime})$
for the IWF10 case. The blue shaded area correspond to the ECR, and the green shaded area to the range
$[\kappa _{x} - \kappa _{f}, \kappa _{x} + \kappa _{f}]$
; the vertical green line indicates the sampling point.
Shell-to-shell transfer functions
$\hat{S}(\kappa , K^{\prime})$
for the IWF20 case. The blue shaded area correspond to the ECR, and the green shaded area to the range
$[\kappa _{x} - \kappa _{f}, \kappa _{x} + \kappa _{f}]$
; the vertical green line indicates the sampling point.
4.3. Residual non-local energy transfers
We have seen in the previous sections that the energy transfers in the reacting region (direct exchange with the large scales), while having a large potential, are dampened out by geometric and phase correlation factors. However, the suppressing factors may not always be sufficient, and residual non-local energy transfers are observed. This behaviour is barely visible on the
$\hat{H}$
and
$\hat{T}$
panels in figures 12 and 13, with non-zero energy transfers with the ECR, but clearly appears in the shell-to-shell transfer function aggregates
$\hat{S}$
in figures 14 and 15. Indeed, there is non-zero energy transfer with the ECR shell (here corresponding to the forcing range, indicated in blue). Note that this transfer is negative, such that on average, energy is transferred from the large scales directly to the sampling point.
If the sampling mode is close enough to the ECR, then it reveals direct interplay between the reacting and catalytic spectral regions, as shown in figure 16. In figures 16(a,c,e), the two regions of high potential intersect, resulting in dominant direct energy transfer with the large scales. These residual transfers are described as non-local because they represent energy transfer to fixed modes (in the ECR) independently of the sampling mode location, whereas the catalytic region is always shifted around the sampling point. One may still consider them as local in the sense that the modes of the ECR are within the range
$[\kappa _{x} - \kappa _{f}, \kappa _{x} + \kappa _{f}]$
. Progressively, as
$|{\boldsymbol{\kappa }}|$
increases, the amount of non-local direct energy exchange with the ECR decreases to zero.
Such direct energy transfers with the ECR can also be seen in the literature, e.g. in figure 14 of Mininni et al. (Reference Mininni, Alexakis and Pouquet2006). As is the case here, their intensity decreases against growing
$Q$
(here
$K^{\prime}$
).
Comparisons of (a,b) EDQNM estimator
$\hat{H}$
, (c,d) mode-to-mode energy transfer
$\hat{T}$
, and (e,f) resulting shell-to-shell transfer function
$\hat{S}$
,) for the IWF20 case (
$\kappa _{f} \sim 20$
). The green ticks indicate the position of the sampling point. Here, (a,c,e)
$\boldsymbol{\kappa } = [30,0]$
,
$\kappa _{f} \sim 20$
, (b,d,f)
$\boldsymbol{\kappa } = [40,0]$
,
$\kappa _{f} \sim 20$
.
4.4. Summary of the distribution of energy transfers
Figure 17 presents a sketch of the distribution of the mode-to-mode energy transfer
$\hat{T}({\boldsymbol{\kappa }}, \boldsymbol{\kappa}^{\prime})$
, assuming that the residual non-local energy transfers presented in the previous subsection are negligible. Note that in three dimensions, the resulting distribution is axisymmetric around the axis given by the direction of the sampling point. The most intense energy transfers are located in the catalytic interactions region, which is an imprint of the ECR shifted around the sampling point.
The specular behaviour is explained as follows: for
$|\boldsymbol{\kappa}^{\prime}| \lt |{\boldsymbol{\kappa }}|$
, the transfers are on average negative; for
$|\boldsymbol{\kappa}^{\prime}| \approx |{\boldsymbol{\kappa }}|$
, the transfers are not sign-definite and average to zero; and for
$|\boldsymbol{\kappa}^{\prime}| \gt |{\boldsymbol{\kappa }}|$
, the transfers are on average positive. Furthermore, the transfers are suppressed along the axis given by the direction of
$\boldsymbol{\kappa }$
due to geometric factors and a bias of the potential
$\hat{Q}$
towards the larger scales.
Sketch of the spectral distribution of the mode-to-mode energy transfer
$\hat{T}$
with the ECR at some intermediate scales.
4.5. Open questions
What is established in this work is that the most intense mode-to-mode energy transfers are strongly dependent on the spectral location of the most energetic scales. In most cases, the latter are equivalent to the largest scales considered, such that the most intense transfers are associated with interactions described as non-local in the literature (Domaradzki & Rogallo Reference Domaradzki and Rogallo1990; Waleffe Reference Waleffe1992). However, it does not mean that these interactions will dominate the total energy transfer and thus the dynamics of the turbulent cascade. Indeed, as objected in Zhou (Reference Zhou1993a
,Reference Zhou
b
) and Eyink (Reference Eyink1994, Reference Eyink1995), the number of these intense energy transfers is fixed and not dependent on the sampling point wavenumber
$\kappa$
. If we define a local interaction in the spirit of Kraichnan (Reference Kraichnan1971) – i.e. such that the minimum wavenumber in the triad is more than half the sampling point wavenumber
$\kappa$
, and the maximum wavenumber in the triad is less than twice
$\kappa$
– then the set of local interactions
$\mathcal{L}_{\textit{local}}$
is given by the intersection of two spectral domains as
\begin{equation} \mathcal{L}_{\textit{local}}({\boldsymbol{\kappa }}) = \left \{ \boldsymbol{\kappa}^{\prime} \mid 0.5 \kappa \lt |\boldsymbol{\kappa}^{\prime}| \lt 2 \kappa \right \} \cap \left \{ \boldsymbol{\kappa}^{\prime} \mid 0.5 \kappa \lt |\boldsymbol{\kappa }{-} \boldsymbol{\kappa}^{\prime}| \lt 2 \kappa \right \}.\end{equation}
As both domains are spherical shells and their intersection is guaranteed by construction, the total number of local interactions grows like
$\sim \kappa ^{3}$
. If the intensity of the mode-to-mode energy transfers does not decay too fast against growing distance
$|\boldsymbol{\kappa }{-} \boldsymbol{\kappa}^{\prime}|$
, then the local interactions will eventually contribute more to the total energy transfer than the intense non-local interactions. Determining a scaling of
$\hat{T}({\boldsymbol{\kappa }}, \boldsymbol{\kappa}^{\prime})$
on
$|\boldsymbol{\kappa }{-} \boldsymbol{\kappa}^{\prime}|$
from the data is difficult due to the specular behaviour discussed in the previous sections. The next step is then to compare the contribution of the HET kernel and the local interaction region to the total energy transfer against
$\boldsymbol{\kappa }$
for different Reynolds numbers, for which a new set of simulations with wider inertial range is necessary. Yeung et al. (Reference Yeung, Brasseur and Wang1995) indicate that at least a decade of wavenumbers in the inertial range is necessary to see a relevant contribution from local triad interactions.
The decay of residual non-local interactions can be studied in a similar manner. While the present work shows that these residuals become small already at moderate Reynolds numbers, they may explain why earlier results showed how small scales are tied to the large-scale dynamics (Yeung & Brasseur Reference Yeung and Brasseur1991; Biferale & Toschi Reference Biferale and Toschi2001; Biferale & Vergassola Reference Biferale and Vergassola2001).
The computation of individual mode-to-mode energy transfer also opens the possibility of studying in depth the phase correlation between modes. Recent results have shown that neighbouring modes are phase correlated such that the energy transfer is maximised towards the smaller scales with viscous Burgers equation. It remains to be established if this ‘phase correlation locality’ also holds in three-dimensional Navier–Stokes turbulence, and whether it is associated with the catalytic region presented in this paper. If the latter is verified, then it is also interesting to check if the neighbouring modes that are phase correlated to each other are also correlated to the catalyst mode.
The study of phase correlation is still highly relevant. Indeed, as discussed in Orszag (Reference Orszag1970), Leith (Reference Leith1971) and Lesieur (Reference Lesieur2008), the only modelling part in the EDQNM approach (apart from the quasi-normal approximation, which is justified therein) is the eddy-damping parameter in the function
$\theta ({\boldsymbol{\kappa }}, \boldsymbol{\kappa}^{\prime})$
in (2.19), which is artificially added to mimic the destruction of coherence (phase correlation) by the nonlinear convection. Better understanding of the distribution and scaling of the phase correlation can thus lead to improved models for spectral viscosity, which are still relevant today for large eddy simulation subgrid scale modelling.
5. Conclusions
We present direct computation of individual mode-to-mode energy transfers
$\hat{T}(\boldsymbol{\kappa }, \boldsymbol{\kappa}^{\prime})$
from DNS of homogeneous isotropic turbulence. This approach differs from previous investigations, which relied on shell-filtered transfer functions that could not distinguish between the energy exchanged by individual modes within a triad. Our methodology allows for a clear identification of the reactant mode (which exchanges energy with the primary mode) and the catalyst mode (which mediates the interaction without direct energy exchange). The comparison between
$\hat{T}$
and an EDQNM function
$\hat{H}$
shows good agreement in all cases, recovering the forward, scale-local nature of energy transfer consistent with the cascade picture.
We introduce the potential
$\hat{Q}(\boldsymbol{\kappa }, \boldsymbol{\kappa}^{\prime})$
, a dimensional estimate of the maximum energy transfer based on the energy content of the modes involved. The fact that
$\hat{Q}$
successfully predicts the distribution of
$\hat{T}$
in the region around the sampling point allows us to link the intensity of energy transfers to the spectral location of the energy-containing range. This provides a new perspective on the classical finding of ‘local energy transfer in non-local interactions’: such energy transfers dominate not only because they are part of non-local interaction, but because they involve a catalyst mode in the energy-containing range.
The potential also predicts direct, non-local, exchanges with modes in the energy-containing range. Our analysis shows that these transfers are suppressed by geometric factors arising from the divergence-free condition, with phase correlation likely playing an additional damping role, as is the case in the EDQNM theory. Despite this suppression, residual non-local energy transfers persist when the sampling mode is close to the energy-containing range.
To further test our proposal, we perform DNS with forcing at intermediate wavenumbers. In these cases, the region of intense energy transfers shifts accordingly, forming a ring (sphere in three dimensions) around the sampling point whose radius scales with the forcing wavenumber
$\kappa _f$
. The scale-locality of the energy transfer is also recovered upon shell integration of the individual contributions. The analysis of the similar patterns displayed between the mode-to-mode energy transfer
$\hat{T}$
, the potential
$\hat{Q}$
and the EDQNM function
$\hat{H}$
in all presented cases validates the proposal that the energy content of the modes involved in the triad is crucial to the intensity of the energy transfer rates.
While the results confirm that what are known as non-local triad interactions do dominate in intensity, the natural next step is to use a similar approach to recover the scaling of the contribution of such interactions to total energy transfer quantities, which is expected to decrease as the scales get smaller, according to the central assumption of universality of small scales in turbulent flows.
Acknowledgements
The authors would like to thank G. Eyink for discussions pointing us in the right direction concerning expressions for the mode-to-mode energy transfer rate and its symmetries. Numerical simulations have been performed on the Euler cluster of ETH Zürich.
Funding
The authors gratefully acknowledge financial support from the Swiss Federal Office of Energy (SFOE) in the project HPH2HD, contract no. S1/50219601.
Declaration of interests
The authors report no conflict of interest.
Appendix A. Kinetic energy spectra
Figure 18 presents the energy spectra of the cases studied.
Energy spectra for the (a) LWF,
$\kappa _{f} \sim 1$
, (b) IWF10,
$\kappa _{f} \sim 10$
, and (c) IWF20,
$\kappa _{f} \sim 20$
cases.
Appendix B. Effectiveness of mode-to-mode energy transfers
Figure 19 presents the energy transfer effectiveness
$\hat{\eta }$
in (3.1) for the IWF10 and the IWF20 cases.
Energy transfer effectiveness for (a,b) the IWF10 case (
$\kappa _{f} \sim 10$
) and (c,d) the IWF20 case (
$\kappa _{f} \sim 20$
), centred on the sampling point. The green ticks indicate the position of the sampling point. Here, (a,c)
$\boldsymbol{\kappa } = [30,0]$
, (b,d)
$\boldsymbol{\kappa } = [60,0]$
.
Appendix C. Efficient computation of spectral energy transfer
$\hat{T}$
Given a sample point
$\boldsymbol{\kappa }^{(s)}$
, the mode-to-mode energy transfer
$\hat{T}(\boldsymbol{\kappa }^{(s)}, \boldsymbol{\kappa}^{\prime})$
is computed directly as in (2.9). The key step involves obtaining the shifted velocity field
$\hat{\boldsymbol{u}}(\boldsymbol{\kappa }^{(s)} - \boldsymbol{\kappa}^{\prime})$
from the solution variable
$\hat{\boldsymbol{u}}(\boldsymbol{\kappa}^{\prime})$
. In the naive implementation, one can simply loop over all
$\boldsymbol{\kappa}^{\prime}$
in the domain, and compute the velocity field shifted at
$\boldsymbol{\kappa }^{(s)} - \boldsymbol{\kappa}^{\prime}$
. This is highly inefficient and renders the computation unfeasible for large spectral domains. Instead, using the fact that the velocity field is also periodic in spectral space (i.e. aliased due to the use of fast Fourier transform operations), the whole shifted velocity field can be obtained at once using roll and flip operations, as described in the following algorithm (making use of Python negative indexing).
-
(i) Compute
$\hat{\boldsymbol{u}}(-\boldsymbol{\kappa}^{\prime})$
by flipping the original velocity field
$\hat{\boldsymbol{u}}(\boldsymbol{\kappa}^{\prime})$
. Note that there is also a roll operation of one index in every direction, which is required to keep the index
$[0,0,0]$
at the origin of the spectral domain in case of even resolution
$N$
:
$\hat{\boldsymbol{u}}(-\boldsymbol{\kappa}^{\prime}) \leftarrow \texttt {flip} \bigl [\ \texttt {roll}[\hat{\boldsymbol{u}}(\boldsymbol{\kappa}^{\prime}), \ \texttt {shifts} = (-1, -1, -1)]\ \bigr ]$
-
(ii) Compute
$\hat{\boldsymbol{u}}(\boldsymbol{\kappa }^{(s)} - \boldsymbol{\kappa}^{\prime})$
by shifting the velocity field
$\hat{\boldsymbol{u}}(-\boldsymbol{\kappa}^{\prime})$
obtained in the previous step according to the chosen sampling point
$\hat{\boldsymbol{u}}(\boldsymbol{\kappa }^{(s)} - \boldsymbol{\kappa}^{\prime}) \leftarrow \texttt {roll}[\hat{\boldsymbol{u}}(- \boldsymbol{\kappa}^{\prime}), \ \texttt {shifts} = \boldsymbol{\kappa }^{(s)}]$
-
(iii) Dealias
$\hat{\boldsymbol{u}}(\boldsymbol{\kappa }^{(s)} - \boldsymbol{\kappa}^{\prime})$
as the previous shift operation introduces non-physical periodicity in the spectral velocity field. This is done by setting to zero all modes with wavenumbers
$\boldsymbol{\kappa }^{(s)} - \boldsymbol{\kappa}^{\prime}$
outside the spectral domain.
The final mode-to-mode energy transfer
$\hat{T}(\boldsymbol{\kappa }^{(s)}, \boldsymbol{\kappa}^{\prime})$
is then computed as a matrix multiplication of the shifted velocity field
$\hat{\boldsymbol{u}}(\boldsymbol{\kappa }^{(s)} - \boldsymbol{\kappa}^{\prime})$
and the original velocity field
$\hat{\boldsymbol{u}}(\boldsymbol{\kappa}^{\prime})$
(plus summation for the dot product), significantly reducing the computational cost compared to the naive implementation. Note that the same approach is used to obtain the shifted energy field
$\hat{E}(\boldsymbol{\kappa }^{(s)} - \boldsymbol{\kappa}^{\prime})$
for the energy transfer potential (2.12) and the EDQNM function
$\hat{H}$
(2.16).
Appendix D. Detailed conservation property
We derive here the detailed conservation property
$\hat{T}(\boldsymbol{\kappa}^{\prime}, \boldsymbol{\kappa }) = - \hat{T}(\boldsymbol{\kappa }, \boldsymbol{\kappa}^{\prime})$
:
\begin{equation} \begin{aligned} \hat{T}(\boldsymbol{\kappa}^{\prime}, \boldsymbol{\kappa }) &= \text{Re} \left \{ i \kappa^{\prime}_{l}\, \hat{u}_{j}(\boldsymbol{\kappa })\, \hat{u}_{l}(\boldsymbol{\kappa}^{\prime} - \boldsymbol{\kappa })\, \hat{u}_{j}^{*}(\boldsymbol{\kappa}^{\prime}) \right \} \\ &= \text{Re} \left \{ i \kappa^{\prime}_{l}\, \hat{u}_{j}(\boldsymbol{\kappa })\, \hat{u}_{l}^{*}(\boldsymbol{\kappa }{-} \boldsymbol{\kappa}^{\prime})\, \hat{u}_{j}^{*}(\boldsymbol{\kappa}^{\prime}) \right \} \\ &= \text{Re} \left \{ i \kappa _{l}\, \hat{u}_{j}(\boldsymbol{\kappa })\, \hat{u}_{l}^{*}(\boldsymbol{\kappa }{-} \boldsymbol{\kappa}^{\prime})\, \hat{u}_{j}^{*}(\boldsymbol{\kappa}^{\prime}) \right \} \\ &= - \text{Re} \left \{ i \kappa _{l}\, \hat{u}_{j}^{*}(\boldsymbol{\kappa })\, \hat{u}_{l}(\boldsymbol{\kappa }{-} \boldsymbol{\kappa}^{\prime})\, \hat{u}_{j}(\boldsymbol{\kappa}^{\prime}) \right \} \\ &= - \hat{T}(\boldsymbol{\kappa }, \boldsymbol{\kappa}^{\prime}). \end{aligned} \end{equation}
We used the Hermitian symmetry property
$\hat{u}_{j}(-\boldsymbol{\kappa }) = \hat{u}_{j}^{*}(\boldsymbol{\kappa })$
and an extra relation coming from the divergence-free condition (2.5):
\begin{equation} \kappa _{l}\, \hat{u}_{l}(\boldsymbol{\kappa }{-} \boldsymbol{\kappa}^{\prime}) = \kappa^{\prime}_{l}\, \hat{u}_{l}(\boldsymbol{\kappa }{-} \boldsymbol{\kappa}^{\prime}). \end{equation}
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