Comparing local energy cascade rates in isotropic turbulence using structure function and filtering formulations

The classic description of the energy cascade in turbulence postulates that kinetic energy originates from forcing large-scale eddies, is subsequently transferred to smaller-scale eddies (forward cascade), and is eventually dissipated due to viscous effects (Richardson, 1922; Kolmogorov, 1941). In a statistical sense, the sign and magnitude of third-order moments of velocity increments confirm this general direction of the energy cascade, as described by the $4/5$ law governing the global average of the third-order longitudinal velocity increment (Kolmogorov, 1941; Frisch, 1995), $\langle\delta u_{L}(\ell)^{3}\rangle\equiv\langle([{\bf u}({\bf x}+{\bm{\ell}})-{\bf u}({\bf x})]\cdot{{\bm{\ell}}/\ell})^{3}\rangle=-\frac{4}{5}\,\ell\,\langle\epsilon\rangle$, where $\langle...\rangle$ denotes global averaging, $\delta u_{L}(\ell)$ is the longitudinal velocity increment and $\epsilon$ is the viscous dissipation rate, while the displacement $\ell=|{\bm{\ell}}|$ is assumed to be well inside the inertial range of turbulence. In this sense, the quantity $-\frac{5}{4}\langle\delta u_{L}(\ell)^{3}\rangle/\ell$ is often interpreted as a measure of the energy flux going from scales larger than $\ell$ to all smaller scales. Because turbulence is known to be highly intermittent in space and time (Kolmogorov, 1962; Frisch, 1995; Meneveau & Sreenivasan, 1991) there has also been much interest in characterizing the local properties of the energy cascade, i.e. the fluctuations of the energy flux before averaging. However, without statistical averaging, the $4/5$-law is less meaningful, e.g., the quantity $-\frac{5}{4}\delta u_{L}^{3}/\ell$ cannot simply be interpreted as an energy flux locally in space and time. To enable such interpretation, it is necessary to consider explicit angular averaging over all possible directions of the vector $\bm{\ell}$. Such formulations have been developed in prior works by Duchon & Robert (2000), Eyink (2002) and Hill (2001, 2002a). Duchon & Robert (2000) and Eyink (2002) use such equations to study the energy cascade and energy dissipation in the limit of zero viscosity. A review about extensions to the classic Kolmogorov equation is presented by Dubrulle (2019), specifically focusing on the Duchon & Robert (2000) local formulation.

Hill (2001, 2002a) developed a local version of the Kolmogorov equation in which the reference position ${\bf x}$ is symmetrically located halfway between the two points ${\bf x}+{\bf r}/2$ and ${\bf x}-{\bf r}/2$ separated by ${\bf r}$ over which the velocity increment is computed. This equation, which we shall denote as the generalized Kolmogorov-Hill equation (GKHE), describes the evolution of the second-order (squared) velocity difference, a measure of energy content of all scales smaller than $|\bm{r}|$ at a specific physical position $\bf x$. As will be reviewed in §2, scale-space integration over ${\bf r}$ of the GKHE up to some scale $\ell$ in the inertial range and without additional statistical averaging provides a localized description of the energy cascade process. The GKHE also includes effects of viscous dissipation, viscous diffusion, advection, and pressure. A number of prior works have studied various versions of the GKH equation. For isotropic turbulence, Yasuda & Vassilicos (2018) quantified the variability of the energy flux that arises in this equation, while Carbone & Bragg (2020) considered a definition of mean energy flux approximated based on a solenoidal filtered velocity increments and examined its connections to average vortex and strain stretching rates. Besides applications to isotropic homogeneous flow, numerous studies have investigated the application of the statistically averaged GKHE to spatially non-homogeneous flows. For instance, in wall boundedf lows, researchers have explored the energy cascade using a Reynolds decomposition to isolate effects of mean shear and non-homogeneity (Antonia et al., 2000; Danaila et al., 2001, 2004, 2012; Marati et al., 2004; Cimarelli et al., 2013). Investigations have also studied the energy cascade rates in boundary layer bypass transition (Yao et al., 2022), and flow separation (Mollicone et al., 2018). Furthermore, specific attention has been given to the study of inverse cascade in wake flows (Gomes-Fernandes et al., 2015; Portela et al., 2017) and at turbulent/non-turbulent interfaces (Zhou & Vassilicos, 2020; Cimarelli et al., 2021; Yao & Papadakis, 2023).

The notion of transfer, or flux, of kinetic energy across length-scales is of particular practical interest also in the context of large eddy simulation (LES). There the rate of energy cascade is commonly referred to as the subgrid or subfilter-scale (SGS, SFS) rate of dissipation. It is defined as the contraction between the subgrid stress tensor and the filtered strain-rate tensor and arises as a source term in the transport equation for subgrid/subfilter-scale kinetic energy (Piomelli et al., 1991; Meneveau & Katz, 2000). This quantity characterizes the energy transfers between the resolved scale and the residual scale within the inertial range, which is also a local property (Eyink & Aluie, 2009). The SGS dissipation is highly intermittent (Cerutti & Meneveau, 1998), and can be both positive and negative locally, but on average, energy is known to be transferred from large scales to the residual scales (forward cascade). There is considerable literature on the subject starting from the seminal papers by Lilly (1967), Leonard (1975) and Piomelli et al. (1991). Some reviews include Meneveau & Katz (2000); Meneveau (2010); Moser et al. (2021).

Without averaging, it has been a common observation that the SGS/SFS dissipation can be negative which has often been interpreted as indicative of local inverse cascading of kinetic energy, i.e., energy transfering from small to large scales of motion (“backscatter” (Piomelli et al., 1991)). Borue & Orszag (1998) noted that the forward cascade occurs predominantly in regions characterized by strong straining, where the magnitude of negative skewness of the strain tensor and vortex stretching are large. Conversely, backscatter was observed in regions with strong rotation. The relationship between SGS dissipation and stress topology and stress strain alignment geometry is discussed and measured based on 3D PIV measurements by Tao et al. (2002). In a more recent study, Ballouz & Ouellette (2018) investigated the SGS tensor by considering the relative alignment of the filtered shear stress and strain tensors. They found that the energy cascade efficiency is quite low, a trend they attributed to energy being transfered largely between positions in physical space. Quantitatively, in expressing the subgrid-stress tensor as a superposition of all smaller scale Gaussian-filtered velocity gradients, Johnson (2020, 2021) was able to isolate the relative contributions of small-scale strain self-stretching and vortex stretching finding both to be important.

It has been questioned whether it is the local quantity $-\tau_{ij}\tilde{S}_{ij}$ (where $\tau_{ij}$ and $\tilde{S}_{ij}$ are the subgrid-scale stress and resolved strain-rate tensors, respectively), or the work done by the SGS/SFS force, $\tilde{u}_{i}\partial_{j}\tau_{ij}$ (where $\tilde{u}_{i}$ is the resolved velocity), that should be the genuine definition of local energy cascade rate. For instance Kerr et al. (1996) use the latter in their study of correlations of cascade rate and vorticity, and more recently Vela-Martín & Jiménez (2021) uses both quantities in their analysis. Moreover the SGS force plays a central role for optimal LES modeling (Langford & Moser, 1999). The SGS force is invariant to divergence-free tensor fields which therefore do not affect the large-scale dynamics but addition of such a tensor field to $\tau_{ij}$ can certainly affect the usual definition of subgrid-scale dissipation $-\tau_{ij}\tilde{S}_{ij}$. By re-expressing the SGS stress and dissipation terms using an optimization procedure, Vela-Martín (2022) provided arguments that the often observed backscatter does not actually contribute to the energy cascade between scales but rather to the energy flux in the physical space, also suggesting that backscatter does not need to be explicitly modeled in LES.

As can be seen from this partial summary of the literature on backscatter and inverse cascade in the LES filtering approach, no consensus has been reached regarding the possible importance and physical interpretation of local backscatter using the definition based on the inner product of the subgrid stress and filtered strain-rate tensors. Also, the question of inverse cascade has not received much attention from the point of view of the local versions of the Kolmogorov equation in the structure function approach. Therefore, in this paper we first revisit the generalized local structure function formulation (§2.1). We argue that in this formulation the term responsible for the energy cascade can be unambiguously interpreted as a flux of kinetic energy between scales since it appears inside a divergence in scale space. In this sense it differs from the filtering formulation used in LES (reviewed in §2.2) in which typically a fixed filter scale is used and no change in scales is considered, thus making the concept of a “flux in scale space” less clearly defined and open to various interpretations.

With the definition of local cascade rate or energy flux clarified for the structure function approach, we perform a comparative study of both the structure function and the filtering approaches’ energy flux terms in a relatively high Reynolds number DNS database of forced isotropic turbulence at a Taylor-scale Reynolds number of 1,250. The data analysis is greatly facilitated by the availability of these data in a new version of the Johns Hopkins Turbulence Database (JHTDB) System, in which python notebooks access the data directly (see appendix A). The comparisons involve various statistical properties of the energy flux. First, in §3 we provide comparisons of both quantities by means of simple statistical measures such as their mean values, joint probability density distributions and correlation coefficients, comparing both the two definitions of kinetic energy and kinetic energy cascade rate or flux. We then comparatively examine conditional averages based on the local molecular dissipation rate averaged over a ball of size $\ell$, specifically re-examining the Kolmogorov refined similarity hypothesis (KRSH) in §4. Then, in §5, we present comparative conditional averages of kinetic energy flux based on properties of the large-scale velocity gradient field such as the strain and rotation magnitudes, and the $Q$ and $R$ invariants. Particular attention is placed on events of local negative energy flux and whether or not such events can be considered to be of statistical significance. Overall conclusions are presented in §6.

In this section, both the structure function based (GKHE) and filtering (LES) energy equations are reviewed. We focus on the term representing energy cascade (energy flux) in each equation, and describe some of the prior efforts in the literature relating the structure function and filtering approaches.

In this section, we provide comparisons of local kinetic energies in the structure function formalism, $k_{{\rm sf},\ell}$, with that in the filtering formalism, $k_{{\rm sgs},\ell}$. We also compare the local energy cascade rates $\Phi_{\ell}$ and $\Pi_{\ell}$. We consider data from a direct numerical simulation (DNS) of forced isotropic turbulence at $R_{\lambda}=1{,}250$ (the Taylor-scale Reynolds number) that used 8,192³ grid points (Yeung et al., 2012). The analysis is performed at three length-scales in the inertial range, $\ell=30\eta,\,45\eta,\,60\eta$ where $\eta=(\nu^{3}/\langle\epsilon\rangle)^{1/4}$ is the (average) Komogorov scale. Quantities are evaluated directly from their definitions using spherical integration. To compute volume spherically filtered quantities such as $\epsilon_{\ell}$, $k_{{\rm sgs},\ell}$, $k_{{\rm sf},\ell}$ or $\tau_{ij}$ from data, we fix the middle point coordinate ${\bf x}$ in the physical domain. Subsequently, we download data in a cubic domain using the JHTDB’s cutout service in a cube of size equal to $\ell$. The data are then multiplied by a spherical mask (filter) to evaluate local filtered quantities. The velocity components are obtained by utilizing pre-computed Getfunctions from the Johns Hopkins Turbulence database (JHTDB) including spatial interpolation, as explained in more detail in Appendix A. For surface averages, we discretize the outer surface of diameter $\ell$ into 500 points pairs ($+$ and $-$ points) that are approximately uniformly distributed on the sphere. The accuracy of this method of integration has been tested by increasing the number of points used in the discretization. Data on the specified points are obtained from the database using 8th-order Lagrange spatial interpolation.

Overall mean values are obtained at the three scales and are plotted in Fig. 2 (a). The results for kinetic energy for the structure function approach are consistent with the analytical evaluation (see Appendix B). For the SGS kinetic energy, the numerical results fall below the theoretical inertial range prediction, due to effects of the viscous range that reduces the amount of SGS kinetic energy even at scales much larger than the Kolmogorov scale. The average values of the spectral fluxes agree very well with the averaged rate of viscous dissipation.

Figure 3(a) shows the joint PDF of $k_{{\rm sf},\ell}$ and $k_{{\rm sgs},\ell}$ at scale $\ell=45\eta$. The correlation coefficient between both quantities is $\rho_{kk}=0.97$ (Fig. 2 (b)). Similarly, Figure 3(b) shows the joint PDF of $\Pi_{\ell}$ and $\Phi_{\ell}$, also at scale $\ell=45\eta$ for the same dataset. The correlation coefficient between both quantities is measured to be $\rho_{\Phi\Pi}=0.58$ (Fig. 2 (b)), significantly lower than for the energies but still appreciable. It can be seen that negative values occur for both $\Pi_{\ell}$ and $\Phi_{\ell}$, although it appears that $\Phi_{\ell}$ has more variability and larger negative excursions than $\Pi_{\ell}$. As summarized in §1, the relevance of locally negative values of $\Pi_{\ell}$ to the flow physics remains unclear, especially given the fact that upon averaging, the quantity becomes positive. Conversely, the quantity $\Phi_{\ell}$ has a clearer local interpretation, in the sense that locally negative values can clearly be interpreted as kinetic energy (local $\delta u_{i}^{2}/2)$) showing a net flux into a sphere of diameter $\ell$ in scale space, i.e. becoming associated with energy at smaller scales. Its overall average also is positive. An interesting question is whether negative values of $\Pi_{\ell}$ or $\Phi_{\ell}$ survive under some type of statistical averaging. In the next sections we use conditional averaging to quantify the importance of negative values (inverse local cascade, or backscatter).

In this section, we compare conditionally averaged cascade rates/fluxes for both the structure function and filtering formulations, conditioned on $\epsilon_{\ell}$, i.e, $\langle\Phi_{\ell}|\epsilon_{\ell}\rangle$ and $\langle\Pi_{\ell}|\epsilon_{\ell}\rangle$. According to KRSH (Kolmogorov, 1962), the statistical properties of velocity increments depend on the local average dissipation within a sphere of scale $\ell$, rather than being determined by the globally averaged dissipation. Written in terms of the quantities of present interest, the KRSH would read $\langle\Phi_{\ell}|\epsilon_{\ell}\rangle=\epsilon_{\ell}$ since $\Phi_{\ell}$ is fully determined by the velocity increments envisioned in the KRSH. Loosely extending the KRSH arguments to the filtering formalism would suggest $\langle\Pi_{\ell}|\epsilon_{\ell}\rangle=\epsilon_{\ell}$.

In order to assess this hypothesis, we evaluate the conditional averages based on the same dataset described before. Results for $\langle\Phi_{\ell}|\epsilon_{\ell}\rangle$ and $\langle\Pi_{\ell}|\epsilon_{\ell}\rangle$ are shown in Figure 4. Results for the three scales considered are included. As can be seen, the plot shows a close agreement between both $\langle\Phi_{\ell}|\epsilon_{\ell}\rangle$ and $\langle\Pi_{\ell}|\epsilon_{\ell}\rangle$, with $\epsilon_{\ell}$. It is important to note that $\Phi_{\ell}$ and $\Pi_{\ell}$ are conditioned on the exact same values of $\epsilon_{\ell}$. The similarities and differences observed in Figure 4 indicate that $\Phi_{\ell}$ and $\Pi_{\ell}$ share many properties (same conditional averages) but they are not identical. For instance, it is clear from Fig. 3 (b) that the variance of $\Phi_{\ell}$ exceeds that of $\Pi_{\ell}$, even though their mean values are the same.

In general, the behaviors of both $\langle\Phi_{\ell}|\epsilon_{\ell}\rangle$ and $\langle\Pi_{\ell}|\epsilon_{\ell}\rangle$ confirm the validity of the KRSH in the present context. More detailed analysis of the KRSH for $\Phi_{\ell}$ and connections to Eq. 2 are reported in Yao et al. (2023)

In this section, we explore correlations between properties of the velocity gradient tensor filtered at scale $\ell$ and the two definitions of energy cascade rate/flux. The velocity gradient tensor encapsulates information about fluid deformation and rotation and connections to the energy cascade have been studied extensively. Already Bardina et al. (1985) examined the impact of rotation on homogeneous isotropic turbulence (HIT) and observed that rotation decreases the dissipation (cascade) rate while increasing the length scales, suggestive of inverse energy cascade effects. Goto (2008) investigated physical mechanisms underlying forward energy cascade and argued that forward cascade can be triggered in regions characterized by strong strain between two large-scale tubular vortices. The role of the filtered gradient tensor for energy cascade was first explored numerically in Borue & Orszag (1998) and experimentally in Van der Bos et al. (2002) building on the “Clark model” that approximates features of the subgrid-scale tensor using Taylor-series expansion. Recent studies by Johnson (2020, 2021) and Carbone & Bragg (2020) have significantly expanded on such analyses and examined the roles of self-strain amplification and vortex stretching driving the forward energy cascade process. For inverse cascade, a vortex thinning mechanism may be at play (Johnson, 2021).

A first level of characterization of the properties of the velocity gradient tensor are its invariants. To characterize deformation and rotation, we evaluate the strain and rotation invariants from data, defined according to

where $S_{ij}$ and $\Omega_{ij}$ are the symmetric and antisymmetric parts of the velocity gradient tensor $A_{ij}=\partial u_{i}/\partial x_{j}$ and the tilde denotes, as before, spherical tophat filtering over a ball of diameter $\ell$. For consistency with prior literature, these values will be normalized by the overall average $\langle Q_{w}\rangle=\frac{1}{2}\langle\Omega_{\ell}^{2}\rangle$ (equal to $\frac{1}{2}\langle S_{\ell}^{2}\rangle$ in isotropic turbulence).

A more detailed characterization of the statistics of velocity gradients involves the invariants $Q$ and $R$ (Vieillefosse, 1982). It is well-known that the joint probability density function (JPDF) of $Q$-$R$ exhibits a characteristic tear-drop shape (Chong et al., 1990; Meneveau, 2011), from which flow topology information such as vortex stretching and compression can be inferred (Chong et al., 1990; Borue & Orszag, 1998; Lüthi et al., 2009; Danish & Meneveau, 2018). These two invariants (at scale $\ell$) are defined as usual according to

Under the assumption of restricted Euler dynamics (Meneveau, 2011), the transport equation for the velocity gradient tensor leads to ${dQ_{\ell}}/{dt}=-3R_{\ell}$ and ${dR_{\ell}}/{dt}=-\frac{2}{3}Q_{\ell}^{2}$ (Cantwell, 1992). The quantity $R_{\ell}$ can thus be considered as the (negative) rate of change of $Q_{\ell}$ and contains both vortex stretching and strain self-stretching mechanisms (Johnson, 2021). In our comparative investigation of energy cascade rates, conditional averaging based on the four invariant quantities $S^{2}_{\ell}$, $\Omega^{2}_{\ell}$, $Q_{\ell}$ and $R_{\ell}$ will be undertaken.

We begin with qualitative visualizations of the fields in small subsets of the domains analzyed. Panel (a) and (b) of Figure 5 depict a sample instantaneous field of $\Phi_{\ell}$ and $\Pi_{\ell}$ respectively, highlighting regions of strong local forward cascade (indicated by solid red circles) and strong inverse cascade (indicated by dashed circles). The correlation between these two variables is evident, the computed correlation coefficient between the snapshots is 0.64. On both panels (a) and (b) the fluxes are normalized by the global averaged dissipation $\langle\epsilon\rangle$. As already noted based on the joint PDFs, there are differences between $\Phi_{\ell}$ and $\Pi_{\ell}$. The maximum magnitude of the positive cascade rate in $\Phi_{\ell}$ is about twice that of $\Pi_{\ell}$, while the magnitude of the negative cascade rate in $\Phi_{\ell}$ is about 3 to 4 times larger. Since $\langle\Phi_{\ell}\rangle\sim\langle\Pi_{\ell}\rangle\sim\langle\epsilon_{\ell}\rangle\sim\langle\epsilon\rangle$, the significant different maximum values indicates $\Phi_{\ell}$ is more variable and intermittent than $\Pi_{\ell}$. Also, $\Phi_{\ell}$ exhibits somewhat finer-scale spatial features.

In this paper we explore, based on a DNS dataset of isotropic forced turbulence at a relatively high Reynolds number ($R_{\lambda}=1250$), local features of the energy cascade. We compare two common definitions of the spatially local rate of kinetic energy cascade at some scale $\ell$. The first is based on the cubic velocity difference term appearing in the Generalized Kolmogorov-Hill equation (GKHE), in the structure function approach. The second is the subfilter-scale energy flux term in the transport equation for subgrid-scale kinetic energy, i.e., as used in filtering approach often invoked in LES. Particular attention is placed on interpretation and statistical robustness of observations of local negative structure-function energy flux and subfilter-scale energy flux. The notion and relevance of local inverse cascade or “backscatter” has been open to debates in the literature. We argue that the interpretation of $\Phi_{\ell}({\bf x},t)$ as a spatially local energy flux appears unambiguous because it arises naturally from a divergence term in scale space. And, the symmetric formulation of Hill (2001, 2002b) leads to the spherically averaged third-order structure function-based definition of a local cascade rate involving velocities at two points that are treated equally via angular averaging over the sphere.

The data confirm the presence of local instances where $\Phi_{\ell}({\bf x},t)$ is negative, i.e. indicative of local inverse cascade events in 3D turbulence. Flow visualizations show that spatially the inverse cascade events are often located near the core of large-scale vortex structures. Comparable observations for the LES-based energy flux $\Pi_{\ell}({\bf x},t)$ (which also displays negative values at many locations in the flow as is well-known in the LES literature on “backscatter”) show that $\Pi_{\ell}({\bf x},t)$ displays smoother and more blob-like features. Regarding the statistical significance of such observations, local observations from single realizations are extended using conditional averaging. Attention is placed first on relationships between the local cascade rate and the local filtered viscous dissipation rate $\epsilon_{\ell}({\bf x},t)$ that plays a central role in the classic KRSH Kolmogorov (1962). Results show that conditional averaging of both $\Phi_{\ell}({\bf x},t)$ and $\Pi_{\ell}({\bf x},t)$ eliminates negative values and that the conditional averages in fact equate $\epsilon_{\ell}({\bf x},t)$ to very good approximation, entirely consistent with KRSH predictions.

The analysis then focuses on conditional averages of $\Phi_{\ell}$ and $\Pi_{\ell}$ conditioned on properties of the filtered velocity gradient tensor properties, in particular four of its most invariants (strain and rotation rate square magnitude and the two $Q-R$ invariants). We find statistically robust evidence of inverse cascade as measured with $\Phi_{\ell}$ when both the large-scale rotation rate is strong and the large-scale strain rate is weak. When defined using $\Pi_{\ell}$, the conditional averaging based on large-scale strain and rotation rates does not lead to any significant average backscatter. When conditioning based on the $R$ and $Q$ invariants, significant net inverse cascading is observed for $\Phi_{\ell}$ in the “vortex compression” $R>0$, $Q>0$ quadrant. Qualitatively similar, but quantitatively much weaker trends are observed for the conditionally averaged subfilter scale energy flux $\Pi_{\ell}$. We recall that a multiscale decomposition of $\Pi_{\ell}$ in terms of velocity gradients at multiple scales Johnson (2020, 2021) shows that $\Pi_{\ell}<0$ appears associated with a vortex-thinning mechanism occuring at smaller scales interacting with large-scale strains.

In summary, present results show that locally negative values of kinetic energy fluxes at scale $\ell$ are observed for both the structure function and filtering approaches, and at least for the structure function approach, the interpretation as a flux in scale space appears unambiguous. Regarding statistical robustness and potential net impact of such local observations, conditional averaging shows that such inverse cascade becomes the statistically dominant mechanism in regions in which the turbulent motions at scales larger than $\ell$ are of the “vortex compression” ($R>0$ and $Q>0$) type.

Future work should extend conditional averaging to more accurately reflect entire flow structures and their possible connections to local inverse cascade mechanisms. Other pointwise quantities such as helicity can also be explored. It would also be instructive to connect present results with the multiscale decomposition of Johnson (2020, 2021) and thus be able to identify the small-scale mechanisms associated to local backscatter/inverse cascade events. And, further theoretically obtained exact relations between structure function and filtering approaches may yet be found.

We thank Prof. Gregory Eyink for fruitful discussions and comments on this manuscript, Dr. Miguel Encinar for valuable suggestions, and the JHTDB/IDIES staff for their assistance with the database and its maintenance. This work is supported by the National Science Foundation (Grant # CSSI-2103874).

The authors report no conflict of interest.