2.1. Generalised scale-to-scale transfers

In this section we outline the general method for obtaining the scale-space energy, helicity and enstrophy budget equations. These originate from studies focused on the energy and can be found in the classic work of Von Kármán & Howarth (Reference Von Kármán and Howarth1938) and later made general in order to account for anisotropy (Monin & Yaglom Reference Monin and Yaglom1975) and inhomogeneity (Hill Reference Hill2002). As such the evolution equations are commonly referred to as the generalised Kármán–Howarth–Monin (KHM) equations or the Kármán–Howarth–Monin–Hill (KHMH) equations. For the present work, these equations will be used to describe the scale-space dynamics of three specific structure functions: the energy structure function $\delta q^2= \delta u_i \delta u_i$, the helicity structure function $\delta h = \delta u_i \delta \omega _i$ and the enstrophy structure function $\delta \omega ^2 = \delta \omega _i \delta \omega _i$. Here, $\delta u_i$ and $\delta \omega _i$ represent, respectively, velocity and vorticity increments, taken as the difference of that quantity at two points $x_i$ and $x_i'$, i.e. $\delta u_i = u_i-u_i'$ and $\delta \omega _i = \omega _i-\omega _i'$. Henceforth, primed variables denote belonging to a set of spatial points $x_i'$ independent of $x_i$. A detailed derivation and interpretation of the KHMH equations in the scale-space coordinates system is given in Hill (Reference Hill2002) and Marati, Casciola & Piva (Reference Marati, Casciola and Piva2004). In the present study we abstain from using the Reynolds decomposition, instead following a similar approach to Yasuda & Vassilicos (Reference Yasuda and Vassilicos2018) where the instantaneous dynamics of each term at various scales can be analysed.

The derivation of the transfer equations begins with the equations for velocity and vorticity, (2.1) and (2.2). There are three common steps in deriving the budgets for $\delta q^2$, $\delta h$ and $\delta \omega ^2$:

1. (i) Take the difference of (2.1) and (2.2) evaluated on two independent coordinates $x_k$ and $x_k'$. These constitute the two-point equations for the velocity and vorticity differences ($\delta u_i = u_i-u_i'$ and $\delta \omega _i = \omega _i-\omega _i'$, respectively)

   1. (a) For $\delta q^2$: Multiply the equation for $\delta u_i$ by $2\delta u_i$.

   2. (b) For $\delta h$: Add the product between $\delta \omega _i$ and the equation for $\delta u_i$ to the product between $\delta u_i$ and the equation for $\delta \omega _i$.

   3. (c) For $\delta \omega ^2$: Multiply the equation for $\delta \omega _i$ by 2$\delta \omega _i$.

2. (ii) Express all quantities as two-point differences ($\delta$) or sums ($\varSigma$). The strain rate tensor, that appears in the equation for $\delta \omega ^2$, is expressed as $S_{ik} = ({\partial u_i}/{\partial x_k} + {\partial u_k}/{\partial x_i} )/2$ with $\delta S_{ik}=S_{ik}-S_{ik}'$ and $\varSigma S_{ik}=S_{ik}+S_{ik}'$.

3. (iii) Change the coordinate system from $(x_k,x_k')$ to $(X_k,r_k)$, where $X_k \equiv (x_k+x_k')/2$ represents physical space (in the sense that it represents a centroid position) and $r_k \equiv x_k'-x_k$ represents the space of scales (as it relates to a distance vector).

The resulting equations describe the various budgets of the scale-space energy, helicity and enstrophy structure functions. For the energy structure function $\delta q^2$ one obtains:

(2.3)\begin{align} \frac{\partial}{\partial t}\delta q^2 + \frac{\partial \delta u_k\delta q^2 }{\partial r_k} &=-\frac{\partial}{\partial X_k}\left(\frac{\varSigma u_k\delta q^2}{2}\right)- \frac{2}{\rho}\frac{\partial \delta u_k \delta p}{\partial X_k}\nonumber\\ &\quad +\nu\left[2\frac{\partial^2}{\partial r_k^2}+\frac{1}{2}\frac{\partial^2}{\partial X_k^2}\right]\delta q^2 -2\nu\left[\left(\frac{\partial u_i}{\partial x_k}\right)^2+\left(\frac{\partial u_i'}{\partial x_k'}\right)^2 \right] . \end{align}

In shorthand notation:

(2.4)\begin{equation} \mathcal{A}_t^{q^2} + \varPi^{q^2} ={-}\mathcal{T}^{q^2}-\mathcal{T}_p^{q^2}+\mathcal{D}_{\nu}^{q^2}-\mathcal{E}^{q^2}, \end{equation}

where the superscript denotes the energy $q^2$, and the terms of (2.4) reflect the terms presented in (2.3) sequentially. For the helicity structure function $\delta h$ one obtains:

(2.5)\begin{align} &\frac{\partial}{\partial t}\delta h + \frac{\partial}{\partial r_k}\left[\delta u_k\delta h -\frac{1}{2}\delta \omega_k \delta q^2 \right] ={-}\frac{\partial}{\partial X_k}\left[\frac{\varSigma u_k\delta h}{2}-\frac{\varSigma \omega_k\delta q^2}{4}\right]\nonumber\\ &\quad - \frac{1}{\rho}\frac{\partial \delta \omega_k \delta p}{\partial X_k} +\nu\left[2\frac{\partial^2}{\partial r_k^2}+\frac{1}{2}\frac{\partial^2}{\partial X_k^2}\right]\delta h -2\nu\left[\left(\frac{\partial \omega_i}{\partial x_k}\frac{\partial u_i}{\partial x_k}\right)+\left(\frac{\partial \omega_i'}{\partial x_k'}\frac{\partial u_i'}{\partial x_k'}\right) \right] . \end{align}

In shorthand notation:

(2.6)\begin{equation} \mathcal{A}_t^{h} + \varPi^{h} ={-}\mathcal{T}^{h}-\mathcal{T}_p^{h}+\mathcal{D}_{\nu}^{h}-\mathcal{E}^{h}, \end{equation}

where the superscript denotes the helicity $h$ for the respective terms of (2.5). Finally, for the enstrophy $\delta \omega ^2$ one obtains:

(2.7)\begin{align} &\frac{\partial}{\partial t}\delta \omega^2 + \frac{\partial \delta u_k\delta \omega^2}{\partial r_k} =-\frac{\partial}{\partial X_k}\left(\frac{\varSigma u_k\delta \omega^2}{2}\right)+ (\delta \omega_k \delta \omega_i \varSigma S_{ik} + \varSigma \omega_k \delta \omega_i \delta S_{ik})\nonumber\\ &\quad + \nu\left[2\frac{\partial^2}{\partial r_k^2}+\frac{1}{2}\frac{\partial^2}{\partial X_k^2}\right]\delta \omega^2 -2\nu\left[\left(\frac{\partial \omega_i}{\partial x_k}\right)^2+\left(\frac{\partial \omega_i'}{\partial x_k'}\right)^2 \right]. \end{align}

In shorthand notation:

(2.8)\begin{equation} \mathcal{A}_t^{\omega^2} + \varPi^{\omega^2} ={-}\mathcal{T}^{\omega^2}+\mathcal{G}_{S}^{\omega^2}+\mathcal{D}_{\nu}^{\omega^2}-\mathcal{E}^{\omega^2}, \end{equation}

where the superscript denotes the enstrophy $\omega ^2$.

Notice that these three budgets are similarly structured, where:

1. (i) $\mathcal {A}_t$ is an unsteady term and represents the temporal increase or decrease of scale energy/helicity/enstrophy at each instant.

2. (ii) $\varPi$ represents the nonlinear exchanges in scale space of energy/helicity/enstrophy.

3. (iii) $\mathcal {T}$ is the nonlinear turbulent transport in physical space.

4. (iv) $\mathcal {T}_p$ results from the interaction of the pressure and velocity/vorticity fields to produce a pressure transport that acts to transport energy/helicity at a particular scale.

5. (v) $\mathcal {G}_S$ is a generation term in scale space resulting from the coupling between the rate-of-strain and the enstrophy.

6. (vi) $\mathcal {D}_{\nu }$ represents the viscous diffusion of energy/helicity/enstrophy both in scale and physical space.

7. (vii) $\mathcal {E}$ represents the two-point average dissipation rate. This can be seen, for example for the scale-to-scale energy transfers, dividing equation (2.3) by 4 on both sides such that $\mathcal {E}=(\varepsilon + \varepsilon ')/2$, where $\varepsilon = \nu (\partial u_i/\partial x_k)^2$.

We refer to Marati et al. (Reference Marati, Casciola and Piva2004), Danaila et al. (Reference Danaila, Krawczynski, Thiesset and Renou2012), Valente & Vassilicos (Reference Valente and Vassilicos2015), Alves Portela et al. (Reference Alves Portela, Papadakis and Vassilicos2017), Mollicone et al. (Reference Mollicone, Battista, Gualtieri and Casciola2018) and Gatti et al. (Reference Gatti, Chiarini, Cimarelli and Quadrio2020), who give a more detailed interpretation of the various terms in the scale-space framework.

There are several useful observations that follow immediately from deriving the evolution equations for the scale energy, helicity and enstrophy. These equations resemble their one-point counterparts except that they account for nonlinear exchanges across scales through $\mathcal {\varPi }$. Closer inspection reveals several differences amongst the three budgets. It is clear that transfers of helicity (2.5) not only arise from the interactions of the scale to scale helicity $\delta h$ with the velocity increment $\delta u_k$, but also from the interaction of the scale-to-scale energy $\delta q^2$ with the vorticity increment $\delta \omega _k$. This is consistent with the two channels of inter-scale transfer identified in the study of Yan et al. (Reference Yan, Li, Yu, Wang and Chen2020), who highlight distinct helicity transfers through the combined action of vortex twisting and vortex stretching arising naturally from the vorticity equation. The pressure term only has an explicit role in the scale energy and helicity budgets. For the scale enstrophy budget, the pressure transport is effectively ‘curled out’; however a non-conservative generation $\mathcal {G}_S$ term emerges whose role is dynamically distinct compared to the pressure transport.

3.1. Von-Kármán mixing tank

The experimental data set considered in this work was gathered in the Göttingen Turbulence Facility #3. The experimental rig consisted of a steel cylinder of diameter and height equal to $48\ {\rm cm}$ and $58\ {\rm cm}$ respectively. It featured $8$ axial baffles, attached to its inside wall, and two counter-rotating impellers of diameter $25\ {\rm cm}$ driving the flow inside the cylinder. The rotation frequency was set to $0.2\ {\rm Hz}$. Water was used as the working fluid (kinematic viscosity of $\nu = 0.98\ {\rm mm}^2\ {\rm s}^{-1}$). The flow was seeded with PMMA microspheres with mean diameter $6.0\ {\rm m}$ and specific gravity $\rho _{seeding} /\rho = 1.22$ (the resultant particle Stokes number was estimated as $6\times 10^{-5} \ll 1$). A high-speed Nd:YAG pulse laser was used as the source of illumination. Two high-speed Phantom v640 cameras were used to record the flow. A more in-depth description can be found in Knutsen et al. (Reference Knutsen, Baj, Lawson, Bodenschatz, Dawson and Worth2020).

The flow inside a cubic domain of $8.5\times 8.5\times 8.5\ {\rm mm}^3$ located at the centre of the tank (where the mean flow vanishes) was measured using scanning particle image velocimetry (scanning PIV). This measurement technique is extensively discussed in Lawson & Dawson (Reference Lawson and Dawson2014), and here we only include a brief summary of its principles. In the first step, two cameras acquire multiple stereo PIV images, with a very small time separation, as the laser sheet traverses the measurement volume. A volumetric snapshot of the scattered light intensity is subsequently reconstructed from the recorded series of images. Ultimately, cross-correlation of the reconstructed snapshots is performed, yielding a single, volumetric velocity field.

Short, time-resolved sequences of velocity snapshots were captured during the experiment (a correction proposed by Wang et al. Reference Wang, Gao, Wei, Li and Wang2017, was used to reduce the residual divergence of the data). Each sequence consisted of six snapshots with the time separation below one-tenth of the Kolmogorov time scale. A vast collection of $2\times 10^5$ such sequences was gathered throughout the experiment. They were considered independent samples of the velocity field as the time separation between sequences was of the order of the impeller revolution period. A Lagrangian filtering, similar to that proposed by Novara & Scarano (Reference Novara and Scarano2013), was applied to the data in the post-processing phase. Each sequence was used to advect artificial tracers, whose initial positions coincided with the measurement grid points, forward and backward in time. The resultant traces were utilised to evaluate the filtered velocity and Lagrangian acceleration vectors. The approach described by Lawson & Dawson (Reference Lawson and Dawson2015) was subsequently employed to reconstruct pressure fields. The discretised momentum equation was rearranged to form an over-determined set of linear equations used to solve for pressure in the least-square sense.

The experimental data was stored as cubes containing fields for all components of the velocity, material derivatives and pressure field (each cube corresponded to one time-resolved measurement sequence). A fourth-order central difference scheme was used to evaluate spatial derivatives.

3.2. Direct numerical simulation

A direct numerical simulation (DNS) data set of forced homogeneous and isotropic turbulence (HIT) provided by the Johns Hopkins University (JHU) turbulence database was used to provide numerical comparison (Li et al. Reference Li, Perlman, Wan, Yang, Meneveau, Burns, Chen, Szalay and Eyink2008). The DNS is triply periodic on a cubic domain of size length $2{\rm \pi}$ with Reynolds number $Re_{\lambda }=433$ on a regular grid of $1024^3$ points. In time intervals of one second, 256 spatially independent sub-cubes of side length $5\lambda$ ($0.4L$) were extracted from the database at full resolution for a total of 2560 sub-cubes over 10 time units (or $5T_L$, where $T_L=u_{rms}/L$ is the large eddy turnover time scale), for which third-order statistics (such as the structure functions reported in § 3.4.2) were found to satisfactorily converge.

The DNS data were stored similarly to the experimental data, where each cube contained fields for all components of the velocity, the material derivatives (accounting for the numerical forcing) and the pressure field. Fourth-order differentiation in physical space was applied directly to the fields as necessary to calculate gradient quantities used in the analysis. As the JHU database enforces the divergence-free condition via spectral methods, a small but non-negligible divergence residual was found for the gradients in physical space. This error was found to propagate to higher-order derivatives, impacting the energy, helicity and enstrophy budget residuals at approximately 2 %, 5 %, and 15 %, respectively (Appendix A). Though this error is non-negligible, it is not believed to overshadow the correlation-based conclusions drawn in this study that are primarily rooted in phase information. For the Von-Kármán (VK) data, the error propagation of gradients in physical space is more significant as a result of the limited sub-pixel accuracy. This is discussed in more detail in Appendix A but is particularly problematic for enstrophy dissipation (fourth order gradients). As a result, conclusions based on such high-order gradients in the VK data are avoided.

3.3. Characterisation of the data sets

In this section we characterise the data sets to elucidate the statistical nature of the scale-space energy, helicity and enstrophy quantities before investigating their transfers in § 4. Basic statistical quantities for both data sets are reported in table 1.

Table 1. Parameters from the Von-Kármán mixing tank scanning PIV experiment (Knutsen et al. Reference Knutsen, Baj, Lawson, Bodenschatz, Dawson and Worth2020) and the Johns Hopkins University DNS of homogeneous and isotropic turbulence (Li et al. Reference Li, Perlman, Wan, Yang, Meneveau, Burns, Chen, Szalay and Eyink2008). Note the symbols $\Delta x$ and $\Delta t$ denote the grid spacing and temporal spacing of the data sets.

The probability density functions (p.d.f.s) of the dissipation rates of the energy, helicity and enstrophy are shown in figure 1. These distributions are of central importance towards comprehending the scale-by-scale cascades, as both energy and helicity can only be destroyed by viscous dissipation. The wide tails of these p.d.f.s extend to extreme values resulting in large flatness (Van Atta & Antonia Reference Van Atta and Antonia1980) and are indicative of small-scale intermittency, characteristic of turbulent flows (Sreenivasan & Antonia Reference Sreenivasan and Antonia1997; Laval, Dubrulle & Nazarenko Reference Laval, Dubrulle and Nazarenko2001). The dissipation p.d.f.s agree well between data sets for the energy and helicity but deviate at extreme values for the dissipation of enstrophy. This is due to the unavoidable PIV error propagation (see Appendix A).

Figure 1. Premulitplied p.d.f.s of the dissipation of energy (a), helicity (b) and enstrophy (c). The inset of (b) highlights the range of helicity dissipation between ${\pm }5$ standard deviations.

3.4. Ensemble-averaged structure functions

In this section, in addition to the classical energy structure function, the structure functions associated with the helicity and enstrophy are probed; these are central to core theories of turbulence (Zhou Reference Zhou2021). This analysis provides further context on the extension of the scale-space framework beyond that of the scale-space energy budget.

3.4.1. Energy, helicity and enstrophy structure functions

The energy, helicity and enstrophy structure functions are presented in figure 2. Here, and for all scale-space quantities, the overbar denotes both ensemble averaging and averages over all scale-space orientations at a particular scale $r=\|{\boldsymbol r}\|$. We note that here (and elsewhere in this study) the error bars correspond to statistical errors obtained using the convergence of the moments of the p.d.f.s (using a 95 % confidence interval). Though bias errors arising from experimental artefacts and discretisation are also of concern, they do not dominate the uncertainty. This is supported by the small residual in the energy, helicity and enstrophy budgets (Appendix A).

Figure 2. Normalised mean energy structure function $\overline {\delta q^2}$ (a), helicity structure function $\bar {\delta h}$ (b) and enstrophy structure function $\overline {\delta \omega ^2}$ (c).

The energy structure functions in figure 2(a) are compensated by the classical inertial range scaling $(r\overline {\epsilon ^{q^2}})^{2/3}$ (Kolmogorov Reference Kolmogorov1941a). The universal constant for the analogous longitudinal second-order structure function is $C_2\approx 2.0\pm 0.2$ (Sreenivasan Reference Sreenivasan1995; Pope Reference Pope2001); however, the energy structure functions feature a sum over two transverse components (that scale as $\tfrac {4}{3}C_2$) and one longitudinal component. Therefore, in the inertial range, the energy structure function is expected to plateau at a value of approximately $7.3\pm 0.7$. For the VK data, this value is achieved only at the largest available separation. This is likely due to the limited range of separations but may also be attributed to the Reynolds number ($Re_{\lambda }=199$) for which the inertial range is expected to be limited in breadth (Ishihara, Gotoh & Kaneda Reference Ishihara, Gotoh and Kaneda2009). For the JHU data the plateau is more evident; however, it is also considerably limited due to the sizes of the extracted sub-cubes of side length $5\lambda$.

The helicity structure functions are shown in figure 2(b). As the helicity is not a sign-definite quantity, the structure functions were found to converge very slowly due to their characteristically high variation (Kurien, Taylor & Matsumoto Reference Kurien, Taylor and Matsumoto2004). The helicity structure functions are normalised using the inertial range scaling in physical space as $\bar {\delta h} /\overline {\epsilon ^{h}}(r^2/\overline {\epsilon ^{q^2}})^{1/3}$, outlined by Brissaud et al. (Reference Brissaud, Frisch, Léorat, Lesieur and Mazure1973). A developing plateau is observed for the JHU data for $r>\lambda$, in agreement with the energy structure function. Also consistent with its energy structure function and limited range of separations, the VK data does not show a clear plateau for $\bar {\delta h}$. The overlap within uncertainty between the two data sets indicates satisfactory collapse, though curiously this occurs mostly for $r$ in the near-dissipative range. The lack of large separations $r$ in the VK data prevents from evaluating the collapse of the data sets for $r>\lambda$ (in the inertial range).

Despite the lack of explicit helical forcing in the DNS, the scale-space helicity is non-zero for both data sets. This is consistent with the helicity spectrum computed over a wide range of simulations (Chen et al. Reference Chen, Chen and Eyink2003; Mininni, Alexakis & Pouquet Reference Mininni, Alexakis and Pouquet2006) with and without helical forcing (Alexakis Reference Alexakis2017). This can be seen directly through an expansion of $\bar {\delta h}$, i.e. for homogeneous turbulence, $\overline {(u_i-u_i')(\omega _i-\omega _i')}=2\bar {h}-2\overline {u_i\omega _i'}$. This reveals that even when small-scale mirror symmetry holds ($\bar {h}=0$), the non-zero scale space helicity (and spectrum of helicity) arises from the coherence in the velocity–vorticity correlation (Levich & Shtilman Reference Levich and Shtilman1988). This was shown explicitly for isotropic turbulence by Deusebio et al. (Reference Deusebio, Boffetta, Lindborg and Musacchio2014). The present results confirm this non-zero correlation as well as the tendency of the turbulence cascade to restore small-scale mirror symmetry with decreasing scale (Kraichnan Reference Kraichnan1973; Chen et al. Reference Chen, Chen and Eyink2003).

The enstrophy structure functions are presented in figure 2(c), normalised using the energy dissipation rate and the Kolmogorov length scale $\eta =(\nu ^3/\overline {\epsilon ^{q^2}})^{1/4}$ as $(\overline {\epsilon ^{q^2}}/\eta ^2)^{2/3}$. A normalisation using $\overline {\epsilon ^{\omega ^2}}$ was tested but found to give unsatisfactory agreement (likely due to error propagation, see Appendix A). The close relationship between the enstrophy and the dissipative small scales (Jiménez et al. Reference Jiménez, Wray, Saffman and Rogallo1993) implies a constant scaling in the inertial range. Consistent with the energy and helicity structure functions, the enstrophy structure functions reach a maximum only for the largest separations and more conclusively for the JHU data. This reaffirms enstrophy as a quantity that is confined to small scales (Jiménez et al. Reference Jiménez, Wray, Saffman and Rogallo1993; Davidson, Morishita & Kaneda Reference Davidson, Morishita and Kaneda2008; Ishihara, Kaneda & Hunt Reference Ishihara, Kaneda and Hunt2013; Elsinga et al. Reference Elsinga, Ishihara, Goudar, Da Silva and Hunt2017).

3.4.2. Nonlinear flux energy, helicity and enstrophy structure functions

In this section the nonlinear flux (NLF) structure functions, of which the divergence is taken in the nonlinear inter-scale transfers $\varPi$, are explored. A small departure to justify this terminology will be taken here. These structure functions are commonly referred to as ‘third-order structure functions’, but as the origin of ‘third order’ is in the statistical moment of the increment of a single quantity (i.e. $\overline {\delta u^3}$), a different terminology is adopted for improved generality. As noted by Hill (Reference Hill2002), making use of Gauss’ theorem when integrating the nonlinear energy transfer term of (2.3) inside a ball $\mathcal {V}_R$ (defined in the space of scales as $\{\boldsymbol {r}\in \mathbb {R}^{3} : \lvert \boldsymbol {r} \rvert \le R \}$) gives $\unicode{x222D} \varPi ^{q^2} \,\mathrm {d}\mathcal {V}_R = \unicode{x222F} _{\partial \mathcal {V}_R}\overline {\delta u_k\delta q^2} n_k \,\mathrm {d}\mathcal {S}$ with $n_k$ the outward normal vector and $d\mathcal {S}$ the surface of the spherical shell (the boundary of $\mathcal {V}_R$). The orientation average over the spherical shell is identically zero, leaving only a flux in the radial direction. This motivates referring to these quantities as NLF structure functions in the present context. In addition to the central role of NLF structure functions in classical turbulence theory, i.e. the 4/5ths (K41) and 2/15ths laws (Chkhetiani Reference Chkhetiani1996; L'vov, Podivilov & Procaccia Reference L'vov, Podivilov and Procaccia1997), their physical significance is well documented in the context of a spherical scale-space coordinate system (Gomes-Fernandes et al. Reference Gomes-Fernandes, Ganapathisubramani and Vassilicos2015; Valente & Vassilicos Reference Valente and Vassilicos2015; Alves Portela et al. Reference Alves Portela, Papadakis and Vassilicos2017). The NLF structure functions for the experimental and numerical data sets are presented in figure 3.

Figure 3. Normalised orientation-averaged NLF structure functions corresponding to the nonlinear scale-to-scale flux of energy $\overline {\delta q^2}$ (a), helicity $\bar {\delta h}$ via advection (b) and vortex stretching (c), and enstrophy $\overline {\delta \omega ^2}$ (d).

The orientation-averaged NLF energy structure function is shown in figure 3(a), normalised using the classical 4/3rds law (Hill Reference Hill2002). The subscript $r$ is used to denote the radial flux as described above. For $r>\lambda$ the plateau seen in the JHU data suggests classical behavior of $\overline {\delta u_r \delta q^2}$. Consistent with figure 2, the VK data does not show a clear plateau developing for $\overline {\delta u_r \delta q^2}/\tfrac {4}{3}\overline {\varepsilon ^{q^2}}r$ but does reach value close to one at $r\approx \lambda$ (the largest available separation). This may imply the nonlinear inter-scale transfer is dominant at $r=\lambda$ (Yasuda & Vassilicos Reference Yasuda and Vassilicos2018), but it must be noted that such an apparent balance is possible when some inhomogeneous effects are at play (Alves Portela et al. Reference Alves Portela, Papadakis and Vassilicos2017).

The orientation-averaged NLF helicity structure functions are shown in figures 3(b,c), normalised using the product of $r$ and dissipation rate of helicity $\overline {\epsilon ^h}$. This normalisation is chosen to compare the two distinct mechanisms associated with $\mathcal {\varPi }^h$ in (2.5). The first, $\overline {\delta u_k \delta h}$, originates from the advective term and the second, $\overline {\delta \omega _k \delta q^2}$, from the vortex stretching term of (2.2) (Yan et al. Reference Yan, Li, Yu, Wang and Chen2020). It is again seen that, due to its non-positive definiteness and inherently high variation, the nonlinear flux of helicity is very slow to converge, leading to large uncertainty bars. Despite this, the fluxes are found to be non-zero, and the JHU data appears to exhibit a plateau (albeit within large uncertainty bars).

Finally, the normalised NLF enstrophy structure functions can be seen in figure 3(d). Similarly to the enstrophy structure function, a collapse of the data sets is observed in the near-dissipation range between $r/\eta =10$ and $r/\lambda =1$. The flux is seen to decrease as $r^{-3}$ ($r^{-2}$ in the figure due to compensating $-\overline {\delta u_r\delta \omega ^2}$ by $r$) into the inertial range. This power law decrease is consistent with the results of Davidson et al. (Reference Davidson, Morishita and Kaneda2008) (in their case, a scale-by-scale enstrophy flux is defined that is not motivated by the present spherical scale-space coordinate system. This necessitates compensating $-\overline {\delta u_r\delta \omega ^2}$ by $r$ for a one-to-one comparison).

4.1. Correlations within budgets

Normalised correlations ($-$1: perfectly anti-correlated, 0: uncorrelated, and 1: perfectly correlated) were tabulated as outlined in Appendix A. We shall focus on the separations $r=2.5\eta$, $0.5\lambda$ and $\lambda$ that correspond to the dissipative and near-dissipative ranges and the onset of inertial range, respectively. The latter is of particular interest, as it is the scale for which Yasuda & Vassilicos (Reference Yasuda and Vassilicos2018) identified the largest variations in energy transfers (standard deviation) in periodic box turbulence with $Re_{\lambda }=178$, indicative of strong dynamics at that scale.

The correlations of the terms in the scale energy, helicity and enstrophy budgets are presented in figures 4, 5 and 7, respectively. Starting with the energy in figure 4, the correlations of terms are in good agreement across the three considered scales between both the JHU and VK data. At the smallest $r$, the high correlation between the viscous diffusion and dissipation seen in figure 4(a,d) reflects the average balance between $\mathcal {D}_{\nu }^{q^2}$ and $\mathcal {E}^{q^2}$, as noted by (Valente & Vassilicos Reference Valente and Vassilicos2015).

Figure 4. Correlation coefficients between terms of (2.4) for $\delta q^2$ at various separations; results based on VK (a–c) and JHU (d–f) datasets.

Figure 5. Correlation coefficients between terms of (2.6) for $\delta h$ at various separations; results based on VK (a–c) and JHU (d–f) datasets.

The most robust correlation of figure 4 lies between the unsteady transport $\mathcal {A}_t^{q^2}$ and the turbulent transport $\mathcal {T}^{q^2}$. This is taken to be the signature of the random sweeping effect identified by Yasuda & Vassilicos (Reference Yasuda and Vassilicos2018), who found that random sweeping indeed extends from single- to two-point quantities. (Strictly speaking, the random sweeping effect is reflected in the correlation between $\mathcal {A}_t^{q^2}$ and $(\mathcal {T}^{q^2}-\varPi ^{q^2})$. However, in the present analysis, we restrict to correlations between individual terms but remark that the signature of random sweeping is consistent with the results of Yasuda & Vassilicos (Reference Yasuda and Vassilicos2018).) As $\mathcal {T}$ is associated with transport in physical space, its high correlation with $\mathcal {A}_t^{q^2}$ is likely a result of the known preferential anti-alignment between the unsteady acceleration $({\partial }/{\partial t}) u_i$ and the convective acceleration $u_k ({\partial }/{\partial x_k}) u_i$ (Tsinober Reference Tsinober2001).

Turning attention to the nonlinear transfer $\varPi ^{q^2}$, an anti-correlation with the pressure transport term $\mathcal {T}_p^{q^2}$ is seen to persist across all scales investigated here. As the pressure (and resulting pressure transport) results from a volumetric integration of the velocity field over the entire flow domain, the non-local influence of the baffles of the VK tank and the numerical forcing of the JHU data could non-trivially impact the role of the pressure transport. Despite this, a significant anti-correlation between $\mathcal {T}_p^{q^2}$ and $\varPi ^{q^2}$ persists between both data sets, indicating a dynamical link between those quantities.

As seen in figure 5, the correlations between the terms involved in the scale to scale helicity budget show some similarities with those observed for the energy budget. At the smallest scales, the same positive correlation between $\mathcal {D}_{\nu }^h$ and $\mathcal {E}^h$ is seen just as in the energy, reflecting the destruction of helicity by viscous forces. The signature of the random sweeping effect is again identified from the correlation between $\mathcal {A}_t^{h}$ and $\mathcal {T}^{h}$ at all $r$. In contrast to the energy transfers, however, the pressure transport $\mathcal {T}_p^{h}$ appears to have an increasing correlation with the nonlinear transfer $\varPi ^h$ as $r$ decreases.

Before this result is discussed for the helicity cascade in more depth, it is necessary to adopt the framework presented by Yasuda & Vassilicos (Reference Yasuda and Vassilicos2018) for the energy budget. As in their work, the correlation of $\mathcal {T}_p^{q^2}$ and $|\delta \boldsymbol {u}|\, |\delta \boldsymbol {\nabla } p |$ is confirmed to be negligible in the present study. On the other hand, defining the cosine of the angle between $\delta \boldsymbol u$ and $-\delta {\boldsymbol {\nabla }} p$ as

(4.1)\begin{equation} \cos{\phi_u} = (\widehat{\delta\boldsymbol{u}}\boldsymbol{\cdot}\hat{\boldsymbol{r}})(-\widehat{\delta\boldsymbol{\nabla} p}\boldsymbol{\cdot}\hat{\boldsymbol{r}}) + (\hat{\delta\boldsymbol{u}}\times\hat{\boldsymbol{r}})\boldsymbol{\cdot}(-\widehat{\delta\boldsymbol{\nabla} p}\times\hat{\boldsymbol{r}}), \end{equation}

where $\hat {\boldsymbol {\cdot }}$ denotes unit norm, we find that the correlation of $\mathcal {T}_p^{q^2}$ with $\cos {\phi _u}$ is substantial. This is shown in figure 6(a) for the JHU data, but similar results were found for the VK data (these are omitted for brevity).

Figure 6. Mean value of the alignment between unit vectors of the velocity (a) and vorticity (b) increments via ((4.1)) and ((4.2)). The correlations of the nonlinear helicity transfer mechanisms $\varPi ^{h,u}$ and $\varPi ^{h,\omega }$ (see text for definitions) with the pressure transport of helicity $\mathcal {T}_p^h$ are shown in (c). All results are from the JHU data.

From both the dot product and cross product contributions of (4.1), Yasuda & Vassilicos (Reference Yasuda and Vassilicos2018) found that the mean was small but positive. This implies that in the averaged picture, the convergence events (i.e. ${\delta \boldsymbol {u}}\boldsymbol {\cdot }\hat {\boldsymbol {r}}<0$) occur simultaneously with compressing forces (i.e. $-\widehat {\delta \boldsymbol {\nabla } p}\boldsymbol {\cdot }\hat {\boldsymbol {r}}<0$), and vice-versa: divergence events are coupled to expanding pressure forces. Importantly, one may conclude based on the identity between the (scale-space) volume integral $\unicode{x222D} _{\mathcal {V}_R} \varPi ^{q^2} \,\mathrm {d}\mathcal {V}_R$ and surface integral $\unicode{x222F} _{\partial \mathcal {V}_R} \delta \boldsymbol {u}\boldsymbol {\cdot }\hat {\boldsymbol {r}}\delta q^2\,\mathrm {d}\mathcal {S}$ that the convergence events facilitate the forward $\delta q^2$ cascade, whereas divergence events facilitate the inverse cascade (Yasuda & Vassilicos Reference Yasuda and Vassilicos2018). And further, it can be inferred from the positive value of mean $\cos {\phi _u}$ that the forward cascading of $\delta q^2$ coincides also with compressing forces.

Returning to the correlation of $\varPi ^h$ with $\mathcal {T}_p^h$ in the present study, it is apparent that the role of the pressure transport is comparatively more complex. This is because $\varPi ^h$ features two distinct NLF structure functions arising from the vortex twisting and stretching mechanisms (Yan et al. Reference Yan, Li, Yu, Wang and Chen2020). The mechanisms are separated here such that $\varPi ^h=\varPi ^{h,u}-\varPi ^{h,\omega }$, where $\varPi ^{h,u}=({\partial }/{\partial r_k})\delta u_k \delta h$ (vortex twisting) and $\varPi ^{h,\omega }=\tfrac {1}{2}({\partial }/{\partial r_k})\delta \omega _k \delta q^2$ (vortex stretching). As a near-zero correlation of $\mathcal {T}_p^h$ with $|\delta \boldsymbol {\omega }| |\delta \boldsymbol {\nabla } p |$ was found, an analogous analysis to Yasuda & Vassilicos (Reference Yasuda and Vassilicos2018) may be attempted using the angle defined via

(4.2)\begin{equation} \cos{\phi_\omega} = (\widehat{\delta\boldsymbol{\omega}}\boldsymbol{\cdot}\hat{\boldsymbol{r}})(-\widehat{\delta\boldsymbol{\nabla} p}\boldsymbol{\cdot}\hat{\boldsymbol{r}}) + (\widehat{\delta\boldsymbol{\omega}}\times\hat{\boldsymbol{r}})\boldsymbol{\cdot}(-\widehat{\delta\boldsymbol{\nabla} p}\times\hat{\boldsymbol{r}}). \end{equation}

The alignment for both the dot product and cross product contributions were similarly found to be small but positive, as can be seen in figure 6(b). The meaning of this alignment, however, is not trivial in the case of helicity. Such a correlation is expected to vanish in mirror-symmetric flow for lack of a preference for any specific sense of rotation (imposed by the sign of $\widehat {\delta \boldsymbol {\omega }}\boldsymbol {\cdot }\hat {\boldsymbol {r}}$). The non-zero average of $\cos {\phi _\omega }$, increasing with $r$, is reminiscent of the scale-to-scale helicity itself (figure 2b).

To reconcile the meaning of this result, we shall clarify the direction of the helicity cascade in the following. In contrast to $\delta q^2$, the terms ‘forward cascade’ and ‘inverse cascade’ are not well-established in the context of helicity (Alexakis Reference Alexakis2017), due to the lack of positive definiteness of $\delta h$. To address this, an auxiliary quantity $H_R$ is introduced as a reference for the sense of rotation as

(4.3)\begin{equation} H_R = \unicode{x222D}_{\mathcal{V}_R} \underbrace{(\delta\boldsymbol{u}\boldsymbol{\cdot}\hat{\boldsymbol{r}})(\delta\boldsymbol{\omega}\boldsymbol{\cdot}\hat{\boldsymbol{r}})}_{\delta h_R} \,\mathrm{d}\mathcal{V}_R. \end{equation}

The forward and backward directions of the helicity cascade is then defined based on the sign of $H_R$ (i.e. ${H_R}/{\lvert H_R\rvert })$. The helicity cascading events causing $({H_R}/{\lvert H_R\rvert })\unicode{x222D} _{\mathcal {V}_R} \varPi ^{h} \,\mathrm {d}\mathcal {V}_R < 0$ are deemed to cascade forward, while $({H_R}/{\lvert H_R\rvert })\unicode{x222D} _{\mathcal {V}_R} \varPi ^{h} \mathrm {d}\mathcal {V}_R> 0$ implies inverse cascading. Similarly to $\delta q^2$, analogous relations invoking the divergence theorem may be drawn for the nonlinear transfer terms:

(4.4)\begin{equation} \unicode{x222D}_{\mathcal{V}_R} \varPi^{h,u}\,\mathrm{d}\mathcal{V}_R = \unicode{x222F}_{\partial\mathcal{V}_R} \delta \boldsymbol{u}\boldsymbol{\cdot}\hat{\boldsymbol{r}}\delta h\,\mathrm{d}\mathcal{S}\quad {\rm and}\quad \unicode{x222D}_{\mathcal{V}_R} \varPi^{h,\omega}\,\mathrm{d}\mathcal{V}_R = \frac{1}{2}\unicode{x222F}_{\partial\mathcal{V}_R} \delta \boldsymbol{\omega}\boldsymbol{\cdot}\hat{\boldsymbol{r}}\delta q^2\,\mathrm{d}\mathcal{S}. \end{equation}

It follows that events of $({\delta h H_R}/{\lvert \delta h\rvert \lvert H_R\rvert })\delta \boldsymbol {u}\boldsymbol {\cdot }\hat {\boldsymbol {r}} < 0$ and $({H_R}/{\lvert H_R\rvert })\delta \boldsymbol {\omega }\boldsymbol {\cdot }\hat {\boldsymbol {r}} > 0$ facilitate the forward helicity cascade and vice-versa for the inverse cascade.

It is important to note that the two mechanisms (i.e. stretching and twisting) cascade helicity in the same direction (either upscale or downscale) only when $\delta h \delta h_r <0$. Additionally, a positive mean value of $\cos {\phi _\omega }$ (as in figure 6b) suggests that when the signs of $H_R$ and $\delta \boldsymbol {\omega }\boldsymbol {\cdot }\hat {\boldsymbol {r}}$ are in agreement, the downscale helicity cascade couples with expanding pressure forces and the upscale helicity cascade associates with compressing pressure forces (in the averaged sense).

The correlations of the individual vortex twisting and stretching mechanisms with $\mathcal {T}_p^h$ in figure 6(c) shows a clear distinction between mechanisms. In the former, the correlation is relatively steady within the probed separation range, similar to what is seen in the context of $\delta q^2$. On the other hand, the stretching mechanism exhibits strong interactions with pressure forces (stronger than the twisting) at dissipative scales, which diminish almost entirely by the time $r$ reaches $\lambda$.

Finally, we turn out attention to the correlations of the transfer terms of the scale-to-scale enstrophy are presented in figure 7. As explained in Appendix A, noise propagation in the VK data renders correlations of $\mathcal {A}_t^{\omega ^2}$ and $\mathcal {E}^{\omega ^2}$ particularly erroneous. The results of the VK data are therefore omitted.

Figure 7. Correlation coefficients between terms of (2.8) for $\delta \omega ^2$ at various separations of JHU data (a–c).

A striking similarity with the energy and helicity budgets is immediately apparent through the anti-correlation of the unsteady transport $\mathcal {A}_t^{\omega ^2}$ and turbulent transport $\mathcal {T}^{\omega ^2}$. This confirms the random sweeping effect persists across all budgets and across all scales. The generation term $\mathcal {G}_s^{\omega ^2}$ exhibits a positive correlation with $\mathcal {E}^{\omega ^2}$ and a negative correlation with $\mathcal {D}_{\nu }^{\omega ^2}$. This implies a concurrent enstrophy generation and dissipation mechanism (Davidson et al. Reference Davidson, Morishita and Kaneda2008). This simultaneous generation and dissipation is tied to a reduction (during high dissipative events) or increase (during low dissipative events) in the diffusion of enstrophy. This is seen to persist across all scales considered. A small positive correlation between $\mathcal {D}_{\nu }^{\omega ^2}$ and $\mathcal {E}^{\omega ^2}$, akin to the energy and helicity cascades, is seen only for $r=2.5\eta$.

4.2. Correlations between budgets

We now explore possible connections between budget equations, i.e. correlations between the mechanisms involved in the budget of a given scale-by-scale structure function and those of the others, as shown in figure 8. The correlations between both the scale energy and enstrophy transfers and the helicity transfers were found to be universally of order $10^{-2}$ and are thus omitted. At first glance, this appears to contradict a wealth of research that has identified clear causal relationships between the three quantities (Bershadskii et al. Reference Bershadskii, Kit, Tsinober and Vaisburd1994; Biferale, Musacchio & Toschi Reference Biferale, Musacchio and Toschi2013; Alexakis Reference Alexakis2017; Bos Reference Bos2021), even analytically (via Lagrangian closure theory, see Inagaki Reference Inagaki2021). As discussed by Tsinober (Reference Tsinober2001), near-zero correlations are necessary, but not sufficient, to determine that two quantities are unrelated. For example, the unsteady term ${\partial u_i}/{\partial t}$ (2.1) is almost entirely decorrelated with the material acceleration ${{\rm D} u_i}/{{\rm D}t}$ simply due to the underlying anti-correlation of ${\partial u_i}/{\partial t}$ with $u_k ({\partial u_i}/{\partial x_k})$.

Figure 8. Correlation coefficients between terms of (2.4) and (2.8) (respectively transfer terms of $\delta q^2$ and $\delta \omega ^2$ on the horizontal and vertical axis) at various separations of JHU data (a–c).

In the present case, the root of the decorrelation is likely to be the fact that helicity is not positive-definite (Kurien et al. Reference Kurien, Taylor and Matsumoto2004). Alternatively, a geometric approach allows to separate the transfer quantities into the product of a (positive-definite) norm with an appropriately defined angle (Yasuda & Vassilicos Reference Yasuda and Vassilicos2018). This avenue for further insight is explored in § 5.

Let us now return to figure 8, which shows correlations between energy and enstrophy budgets. The most robust correlation is that between dissipation rate of energy $\mathcal {E}^{q^2}$ and the dissipation rate of enstrophy $\mathcal {E}^{\omega ^2}$, with a magnitude exceeding 0.5 across all considered scales. Figure 8 also shows that the energy dissipation $\mathcal {E}^{q^2}$ and enstrophy diffusion $\mathcal {D}_{\nu }^{\omega ^2}$ are negatively correlated, whereas the energy dissipation $\mathcal {E}^{q^2}$ and enstrophy generation $\mathcal {G}_s^{\omega ^2}$ (for $r=0.5\lambda$ and $r=\lambda$) are positively correlated. Together, these observations support the classical picture that the generation of scale enstrophy at small scales (Siggia Reference Siggia1981) is largely responsible for the dissipation of energy (as well as enstrophy generation, Siggia Reference Siggia1981). Finally, the signature of the random sweeping effect can be clearly seen amongst the correlations between $\mathcal {A}_t$ and $\mathcal {T}$ in figure 8, particularly at the smallest scales. Together, these mechanisms point to the interconnected nature of the simultaneous turbulence exchanges.

5.1. Application of Lamb decomposition

In the previous section, it was found that the correlation between transfers of energy and transfers of helicity was zero. There is however evidence that the magnitude of the helicity itself has an impact on the local transfer of energy. Regions of high helicity are thought to suppress nonlinear energy transfer and dissipation (Pelz et al. Reference Pelz, Yakhot, Orszag, Shtilman and Levich1985; Stepanov et al. Reference Stepanov, Golbraikh, Frick and Shestakov2015), though some results suggest otherwise (Zhou et al. Reference Zhou, Nagata, Sakai, Ito and Hayase2016). To probe more closely the connection between the helicity and the energy cascade in the present study, the Lamb decomposition is invoked on the nonlinear term of (2.1):

(5.1)\begin{equation} u_k \frac{\partial}{\partial x_k}u_i = \frac{1}{2}\frac{\partial}{\partial x_i}(u_k u_k)+\epsilon_{jki}\omega_j u_k, \end{equation}

where $\epsilon _{jki}\omega _j u_k={\boldsymbol \omega }\times {\boldsymbol u}$ is known as the Lamb vector. This quantity is central to the nonlinear turbulence cascade as maximal helicity $h={\boldsymbol \omega }\boldsymbol {\cdot }{\boldsymbol u}$, i.e. the alignment of ${\boldsymbol u}$ and ${\boldsymbol \omega }$, corresponds to a suppression of nonlinearity ${\boldsymbol \omega }\times {\boldsymbol u}$ in the Navier–Stokes equation.

A kinematic connection can be gleaned from (2.1), invoking (5.1). Similarly to the derivation of the KHMH energy (§ 2) but focusing on the nonlinear terms exclusively, it follows that

(5.2)\begin{align} &2\delta u_i \left(u_k \frac{\partial}{\partial x_k}u_i-u_k' \frac{\partial}{\partial x_k'}u_i'\right) ={-}2\varSigma u_k \epsilon_{jik}\delta \omega_j\delta u_i \nonumber\\ &\quad + \frac{\partial}{\partial r_k}(\delta q^2\delta u_k)-\frac{1}{2}\frac{\partial}{\partial X_k}(\varSigma u_k\delta q^2)+2\frac{\partial}{\partial r_i}(\delta u_i \varSigma q^2). \end{align}

After simplification, this leads to the triple product relation

(5.3)\begin{equation} \varSigma {\boldsymbol u}\boldsymbol{\cdot}(\delta{\boldsymbol \omega}\times\delta{\boldsymbol u})=\varPi^{q^2>}-\mathcal{T}^{q^2<}, \end{equation}

where $\varPi ^{q^2>}=({\partial }/{\partial r_i})(\delta u_i \varSigma q^2)$ is the nonlinear scale-to-scale transfer for the energy sum $\varSigma q^2=\varSigma u_i \varSigma u_i$, and $\mathcal {T}^{q^2<}=\tfrac {1}{2}({\partial }/{\partial X_k})(\varSigma u_k\delta q^2)$ is the usual nonlinear energy transport from (2.4). Hereafter the superscripts < and > are used for shorthand to refer to transfer of the increment and the sum quantities, respectively. A detailed derivation of this result is provided in Appendix B.

An immediate curiosity stems from the presence of the energy sum nonlinear transfer $\varPi ^{q^2>}$. The dynamical significance of the energy (or velocity) sum is a matter of some debate but has been linked to so-called ‘large-scale intermittency’ in previous studies (Sreenivasan & Antonia Reference Sreenivasan and Antonia1997; Blum et al. Reference Blum, Kunwar, Johnson and Voth2010; Chien, Blum & Voth Reference Chien, Blum and Voth2013; Carter & Coletti Reference Carter and Coletti2018). To summarise, instances when the energy sum is large and within the range of correlated scales (large-scale sweeping motions) have been shown to increase the content of the energy structure functions relative to the mean at each scale. This implies a connection between large and small scales in turbulence (Mininni et al. Reference Mininni, Alexakis and Pouquet2006; Hosokawa Reference Hosokawa2007). Equation (5.3) indicates that these sweeping motions play a central role in nonlinear energy transport. It shows that sufficiently large $\varSigma {\boldsymbol u}$ can overcome the alignment of $\delta {\boldsymbol u}$ and $\delta {\boldsymbol \omega }$ (as long as they not perfectly aligned). This could explain some conflicting results where strong dissipation is found even when these quantities are aligned and helicity is maximal (Zhou et al. Reference Zhou, Nagata, Sakai, Ito and Hayase2016).

Particularly for turbulence theory, the kinematic connection revealed by (5.2) provides novel insight into the mechanics of the energy cascade. By substituting for $\mathcal {T}^{q^2<}$ in (2.4) using (5.2) and invoking steady, homogeneous turbulence within an independent intermediate range of scales (Kolmogorov Reference Kolmogorov1941a,Reference Kolmogorovb), it is found upon ensemble averaging that

(5.4)\begin{equation} \overline{\varSigma {\boldsymbol u}\boldsymbol{\cdot}(\delta{\boldsymbol \omega}\times\delta{\boldsymbol u})}=4\overline{\epsilon^{q^2}}. \end{equation}

By virtue of homogeneity, the sum $\overline {\varPi ^{q^2>}}+\overline {\varPi ^{q^2<}}=0$ and therefore no longer features in the balance with the dissipation rate of energy. This can be shown analytically and was verified for both data sets in the present study (not shown for brevity). This demonstrates the high relevance of the triple product on the left-hand side of (5.4). It is not only a feature of the instantaneous kinematics but also central to the overall dynamics. It is perhaps unwelcome to see the velocity sum play a central role in the mean rate of energy dissipation in the inertial range, as this challenges the possibility that there exists a range of scales independent of the large-scale effects (or of the forcing). Such an observation has been previously identified in terms of triadic interactions (Yeung & Brasseur Reference Yeung and Brasseur1991), but to the authors’ knowledge this is the first such demonstration in real space. This provides direct insight on the role of large-scale intermittency (via $\varSigma {\boldsymbol u}$) on the inertial range in the cascade of energy (Sreenivasan & Antonia Reference Sreenivasan and Antonia1997).

Equation (5.4) highlights a geometric perspective of the energy dissipation, where the mean projection of the velocity sum onto the cross product of the velocity and vorticity increment balances the mean dissipation rate. By virtue of the fact that $\|\delta \boldsymbol {u}\times \delta \boldsymbol {\omega }\|^2=\|\delta \boldsymbol {u}\|^2\|\delta \boldsymbol {\omega }\|^2 - \|\delta h\|^2$, there is an implicit connection between the dissipation of energy and the scale-by-scale helicity. Although this does not necessarily connect the transfers, or dynamics, of the two quantities together, it does show that the magnitude of the scale-space helicity influences the dissipation of energy (but also, critically, depends on the alignment with sweeping motions).

Having identified the statistical significance of the triple product, we turn our attention to the instantaneous behaviour. Motivated by the kinematics of (5.3), the joint p.d.f.s of the triple product with the nonlinear energy sum transfer $\varPi ^{q^2>}$ and the nonlinear energy transport $\mathcal {T}^{q^2<}$ were computed for the VK and JHU data sets and plotted in figure 9. There is a clear positive and negative correlation between the triple product and $\varPi ^{q^2>}$ and $\mathcal {T}^{q^2<}$, respectively, with very close agreement in terms of the slope ($\pm 0.5$) of the major axis of the ellipse formed by the joint p.d.f.s. In fact, (5.2) requires the slopes be complementary. The slope, and therefore the underlying correlation, is robust across all scales considered and for both the experimental and numerical data sets. The increase in intermittency (i.e. the flatness of the joint p.d.f.s) with increasing scale is reminiscent of large-scale intermittency (Sreenivasan & Antonia Reference Sreenivasan and Antonia1997).

Figure 9. Joint p.d.f.s of the triple product of (5.3) with the nonlinear energy sum transfer $\varPi ^{q^2>}$ (a,c,e) and nonlinear energy transport $\mathcal {T}^{q^2<}$ (b,d, f) for $r=2.5\eta$ (a,b), $r=0.5\lambda$ (c,d) and $r=\lambda$ (e, f). Contour values range from $10^{-12}$ to $10^{-8}$ (counting from the outermost line).

It is of interest that the triple product reveals a similar correlation to both $\varPi ^{q^2>}$ and $\mathcal {T}^{q^2<}$ separately. This originates with the velocity sum $\varSigma {\boldsymbol u}$, featuring both in the triple product and via $\mathcal {T}^{q^2<}=\varSigma {\boldsymbol u}\boldsymbol {\cdot }({\partial }/{\partial {\boldsymbol X}})({\delta q^2}/{2})$. This establishes an immediate and scale-independent connection between the two quantities. The remaining correlation between the triple product and $\varPi ^{q^2>}$ follows as a direct consequence of (5.3). Despite their closely tied kinematics, there are important dynamic distinctions amongst the quantities involved in terms of their statistical behaviour. Under the assumption of homogeneity, the ensemble average $\overline {\mathcal {T}^{q^2<}}=0$, whereas $\overline {\varPi ^{q^2>}}=-\overline {\varPi ^{q^2<}}$, and $\overline {\varSigma {\boldsymbol u}\boldsymbol {\cdot }(\delta {\boldsymbol \omega }\times \delta {\boldsymbol u})}=4\overline {\epsilon ^{q^2}}$. This motivates the interpretation of $\varPi ^{q^2>}$ as a transfer of energy to larger scales whose average is a complement to $\overline {\varPi ^{q^2<}}$.

The establishment of an overall balance between the triple product and the dissipation of energy in the inertial range (5.4) motivates further insight into the alignment of $\varSigma {\boldsymbol u}$ and $(\delta {\boldsymbol \omega }\times \delta {\boldsymbol u})$. This is quantified using the angle defined by

(5.5)\begin{equation} \cos\theta=\frac{\varSigma {\boldsymbol u}\boldsymbol{\cdot}(\delta{\boldsymbol \omega}\times\delta{\boldsymbol u})}{\|\varSigma {\boldsymbol u}\|\,\|\delta {\boldsymbol u}\times\delta{\boldsymbol \omega}\|}. \end{equation}

The p.d.f. is presented for both data sets in figure 10. Note that an extra separation at $r=4.7\lambda$ (the maximum separation attainable using the sub-cubes extracted from the JHU database) is presented for further insight. As required by (5.4) to maintain dissipation downscale, the alignment is found to be biased towards positive values of $\cos \theta$ for all separations such that $\overline {\cos \theta }>0$. The distributions appear approximately quadratic in nature, with an inflection occurring somewhere between $2.5\eta$ and $0.5\lambda$.

Figure 10. P.d.f. of $\cos \theta$ characterising the alignment between the velocity sum and the cross product (5.3) for the VK data (a) and JHU data (b), and an extrapolation of the p.d.f. at extreme values of $\cos \theta$ for JHU data (c) ($L$ marks the integral length scale).

As the separation increases, the probability of parallel ($\cos \theta =1$) and anti-parallel ($\cos \theta =-1$) alignment is increasingly likely. Most evident from the JHU distributions (figure 10b), the increase in likelihood across separations is more rapid for anti-parallel alignment than parallel alignment. A linear regression for these points with increasing scale indicates they will be equal in probability at $r\approx 13\lambda$, corresponding almost exactly to the integral scale of the DNS data set (figure 10c). This implies that at $r=L$, the p.d.f. will be symmetric about zero and, due to the parabolic shape, with equal likelihood of parallel and anti-parallel alignment, a minimum at $\cos \theta =0$ and $\overline {\cos \theta }=0$. A similar extrapolation for the experimental data was not conclusive due to the limited number of available separations.