4.1 Unconditional statistics

As a first step towards quantifying the length scales of the velocity gradient we compute the two-point correlations of $Q_{A}$ , $Q_{S}$ and $Q_{W}$ . Note that these are effectively fourth-order statistics of the velocity gradient $\unicode[STIX]{x1D63C}$ . It should be pointed out that intense vorticity is typically organized into quasi one-dimensional tight vortex tubes, while intense strain tends to be rather two-dimensional (Moisy & Jiménez Reference Moisy and Jiménez2004). So, inevitably, the scalar two-point correlations represent these different geometries in a highly condensed and simplified fashion. Nevertheless, the auto-correlations indicate typical average length scales of the fields, while the cross-correlations indicate a typical average separation between them.

As an example figure 2(a) shows all auto-correlations as well as the cross-correlation between $Q_{S}$ and $Q_{W}$ for the flow with $Re_{\unicode[STIX]{x1D706}}=279$ . Only the region up to $80\unicode[STIX]{x1D702}$ is shown in order to emphasize the different decaying profiles. The full simulation extends up to ${\approx}2805\unicode[STIX]{x1D702}$ (see table 1). The correlation lengths associated with the correlation functions are computed using (2.5) and (2.8) and their analogues for $Q_{S}$ and $Q_{W}$ .

Clearly, $Q_{A}$ is much more shortly correlated than $Q_{W}$ which in turn is more shortly correlated than $Q_{S}$ . The short correlation length scale of $Q_{A}$ is due to cancellations between $Q_{S}$ and $Q_{W}$ , when enstrophy and strain structures overlap. This can be seen in figure 1 and is supported by the decomposition

(4.1) $$\begin{eqnarray}{\mathcal{C}}_{A}(r)=\left(\frac{\unicode[STIX]{x1D70E}_{S}}{\unicode[STIX]{x1D70E}_{A}}\right)^{2}\,{\mathcal{C}}_{S}(r)+\left(\frac{\unicode[STIX]{x1D70E}_{W}}{\unicode[STIX]{x1D70E}_{A}}\right)^{2}\,{\mathcal{C}}_{W}(r)+2\,\frac{\unicode[STIX]{x1D70E}_{S}\unicode[STIX]{x1D70E}_{W}}{\unicode[STIX]{x1D70E}_{A}^{2}}\,{\mathcal{C}}_{SW}(r).\end{eqnarray}$$

Figure 2(b) illustrates this decomposition for the flow with $Re_{\unicode[STIX]{x1D706}}=279$ . Note that by definition ${\mathcal{C}}_{SW}(r)\leqslant 0$ , while ${\mathcal{C}}_{W}(r)\geqslant 0$ and ${\mathcal{C}}_{S}(r)\geqslant 0$ . So, the auto-correlations of $Q_{W}$ and $Q_{S}$ enhance the auto-correlation of $Q_{A}$ and the cross-correlation between $Q_{S}$ and $Q_{W}$ decreases it. At the very small scales ${\mathcal{C}}_{A}$ is dominated by enstrophy correlations. The lack of correlations beyond approximately $20\unicode[STIX]{x1D702}$ is the result of a fine balance between the length scales of enstrophy and strain structures on the one hand, and the separation between them on the other. A crucial component which maintains the balance is the weighting by the respective standard deviations. In the end the appreciable $Q_{A}$ correlations are the residue from small-scale enstrophy and strain structures.

Figure 2.

Spatial correlations of the velocity-gradient invariants for $Re_{\unicode[STIX]{x1D706}}=279$ . Only the central portion of the correlations is shown to emphasize their different small-scale properties. (a) Auto-correlation function of $Q_{A}$ and its constituents – the auto-correlations of $Q_{S}$ and $Q_{W}$ and their cross-correlation. (b) The correlations from (a) rescaled by the respective standard deviations emphasizing the almost perfect agreement of the components of (4.1) beyond ${\approx}20\unicode[STIX]{x1D702}$ . The vertical lines in both panels mark the position of the respective correlation lengths as defined by (2.5) and (2.8). The full simulation domain, which extends to $2805\unicode[STIX]{x1D702}$ , is used in the calculation of these length scales.

All simulations show the same qualitative behaviour. To quantify it, we plot the $Re_{\unicode[STIX]{x1D706}}$ dependence of the associated correlation lengths in figure 3. The length scales are normalized by the Kolmogorov scale. As a reference we show the scaling of the velocity integral length scale $L_{int}^{K}=K^{3/2}/\unicode[STIX]{x1D700}$ , which agrees with the theoretical scaling of $Re_{\unicode[STIX]{x1D706}}^{3/2}$ (Pope Reference Pope2000, p. 200). Here $K$ is the kinetic energy of the velocity fluctuations and $\unicode[STIX]{x1D700}=4\unicode[STIX]{x1D708}\langle Q_{S}\rangle$ is the mean dissipation rate. As can be seen, the $Re_{\unicode[STIX]{x1D706}}$ dependence of all length scales is fitted quite well by power laws. Remarkably, even though $L_{S}$ , $L_{W}$ and $L_{SW}$ do not scale with the Kolmogorov scale (they grow with $\unicode[STIX]{x1D702}$ ), $L_{A}$ does. In fact, a direct linear fit gives $L_{A}/\unicode[STIX]{x1D702}=4.5+1.3\times 10^{-3}Re_{\unicode[STIX]{x1D706}}$ within the range of investigated Reynolds numbers, with error comparable to that of the power-law fit. This indicates that structures of $Q_{A}$ are characterized well on average by the $4.5\unicode[STIX]{x1D702}$ scale, in agreement with Lawson & Dawson (Reference Lawson and Dawson2015).

Figure 3.

Dependence of the correlation lengths of velocity-gradient correlations on $Re_{\unicode[STIX]{x1D706}}$ . The velocity integral length scale $L_{int}^{K}$ is plotted for reference as well. All length scales are normalized by the Kolmogorov scale. The symbols show the DNS results, the dashed lines the corresponding best power-law fits (in a least square sense). The scaling of $L_{int}^{K}$ agrees with the theoretical expectation. Of note is the extremely weak dependence of $L_{A}$ on $Re_{\unicode[STIX]{x1D706}}$ .

4.2 Conditional statistics

The considerations above describe coarsely a characteristic length scale of $Q_{A}$ , but disregard completely the structural differences between regions of small and large intensity or positive and negative sign. For instance, regions of smaller intensity tend to be more diffuse and less structured (Moisy & Jiménez Reference Moisy and Jiménez2004). In order to disentangle the properties of the different regions we continue by examining the $Q_{A}$ , $Q_{S}$ and $Q_{W}$ fields in the neighbourhood of strain- and enstrophy-dominated regions of different intensities using two-point conditional statistics. For simplicity, we focus on the radial properties of these regions, averaging out the directional information. This also facilitates the quantification of the non-locality of the pressure in these regions discussed in § 5.

In order to understand the structure of these fields and interpret their statistics, it is illustrative to begin with a simpler example. Consider how the $Q_{A}$ field is organized around a point in a turbulent flow dominated by enstrophy. To illustrate the kinematic relationship between strain and enstrophy we show this in appendix A for an isolated Townsend’s eddy. Here, instead we consider the structure of $Q_{A}$ on spherical shells of increasing radius around a reference point with $Q_{A}(\boldsymbol{x}_{0})\approx 45\langle Q_{W}\rangle$ in a turbulent flow with $Re_{\unicode[STIX]{x1D706}}=168$ . The reference point belongs to a vortex tube, as sketched in figure 4(a). So, for distances smaller than the vortex tube’s radius, in this case for $r\lesssim 3\unicode[STIX]{x1D702}$ , $Q_{A}$ is large and positive. The spatial distribution of $Q_{A}$ at larger distances is shown in figure 4(b–d). Figure 4(b) shows that the vortex tube is embedded in a strongly strain-dominated neighbourhood. As the radius increases further, the visualization surface grows beyond the vortex tube’s length and typical turbulence contours emerge. Thus figure 4(c,d) shows the footprints of neighbouring entangled vortex tubes and strain structures. It is evident in all panels that regions of enhanced enstrophy are neighbouring regions of enhanced strain (as dictated by the kinematic relationship between them (Ohkitani Reference Ohkitani1994)) demonstrating the correlation between the two fields on the level of individual structures. This leads to cancellations when computing e.g. the average of $Q_{A}$ at a given distance from the reference point, i.e. $\overline{Q_{A}}(\boldsymbol{x},r)$ . It decays quickly from the value at the reference point $45\langle Q_{W}\rangle$ to its overall mean of zero, albeit with some fluctuations. In particular, the decay is already significant at ${\approx}10\unicode[STIX]{x1D702}$ , where the structure of the vortex tube is still clearly identifiable.

Figure 4.

Illustration of the distribution of $Q_{A}$ on concentric shells of increasing radius centred on a vortex tube. The data are taken from the simulation with $Re_{\unicode[STIX]{x1D706}}=168$ (see table 1). The central point has $Q_{A}(\boldsymbol{x}_{0})\approx 45\langle Q_{W}\rangle$ . (a) A sketch of the intersection of the vortex tube with spherical shells. (b–d) Colour maps of the $Q_{A}$ field at distances $10\unicode[STIX]{x1D702}$ , $21\unicode[STIX]{x1D702}$ and $41\unicode[STIX]{x1D702}$ from the central point, respectively. The grey contour identifies the $Q_{A}=0$ level. The average over the shell, $\overline{Q_{A}}(\boldsymbol{x}_{0},r)$ , is given for each panel. The ringing in some of the panels is an artefact from the cubic interpolation of the Cartesian grid data to polar coordinates and does not represent inaccuracy in the computational data themselves.

In the following we test the statistical relevance of these observations by analysing the conditional averages of $\overline{Q_{A}}(\boldsymbol{x},r)$ around reference points $\boldsymbol{x}$ with different values of $Q_{A}$ , i.e. $\left<\left.\overline{Q_{A}}(r)\right|Q_{A}\right>$ . To isolate the effect of cancellations between vorticity and strain structures, we also calculate the corresponding conditional averages of $\overline{Q_{S}}(\boldsymbol{x},r)$ and $\overline{Q_{W}}(\boldsymbol{x},r)$ . The angle-averaged fields arise naturally when considering the generation of the pressure field (see (2.2)). In their own right, they serve to characterize the isotropic component of the velocity gradient structures. (See e.g. Biferale & Procaccia (Reference Biferale and Procaccia2005) for a review on the effects of anisotropy on turbulent statistics.) We condition on the local value of $Q_{A}$ to separate properties of enstrophy- and strain-dominated reference points. Note, however, that this is not equivalent to separating between regions of strong enstrophy and shear because of the strong overlap between the two (as observed experimentally by e.g. Fiscaletti et al. (Reference Fiscaletti, Westerweel and Elsinga2014) for a jet flow and Worth & Nickels (Reference Worth and Nickels2011) in a mixing tank).

Figure 5.

Radial profiles of $\overline{Q_{A}}(r)$ (a,d), $\overline{Q_{W}}(r)$ (b,e) and $\overline{Q_{S}}(r)$ (c,f) at enstrophy-dominated regions $Q_{A}>0$ (a–c) and strain-dominated regions $Q_{A}<0$ (d–f). The different colours show different intensity levels of the condition field, normalized to the respective standard deviation. The vertical dashed lines show the correlation lengths associated with the auto- or cross-correlation functions derivable from the corresponding conditional averages. The radial coordinate is normalized by the Kolmogorov length $\unicode[STIX]{x1D702}$ and the average contributions are non-dimensionalized by $\langle Q_{W}\rangle$ , so as to be directly comparable across the different panels. The data are taken from the simulation with $Re_{\unicode[STIX]{x1D706}}=168$ .

Conditional averages are an established tool to describe turbulent structures. One-point conditional averages have been used to associate low-pressure regions with intense enstrophy (Cao et al. Reference Cao, Chen and Doolen1999) and with intense enstrophy and strain (Yeung et al. Reference Yeung, Donzis and Sreenivasan2012). In the context of velocity gradients, two-point conditional averaging of $Q_{A}$ has been used to study numerically and experimentally the dominant flow structures for different types of local topologies (Lawson & Dawson Reference Lawson and Dawson2015) and in the context of a stochastic estimation model (Wilczek & Meneveau Reference Wilczek and Meneveau2014). In the former it is found that intense strain and vorticity structures of various geometries are usually in the neighbourhood of one another in agreement with previous literature. Additionally, geometrical conditioning has been used to establish a dominant structure for small-scale turbulence (Elsinga & Marusic Reference Elsinga and Marusic2010) consisting of a shear layer coincident with a pair of stretched co-rotating vortices, the core of which scales with $\unicode[STIX]{x1D702}$ . The following analysis provides complementary information to these studies. Firstly, we use conditional averaging to characterize structures of different types and intensity. Secondly, we quantify the scale of these structures along with their typical separation.

In particular, figure 5 shows the profiles of $\left<\left.\overline{Q_{A}}(r)\right|Q_{A}\right>$ , $\left<\left.\overline{Q_{W}}(r)\right|Q_{A}\right>$ and $\left<\left.\overline{Q_{S}}(r)\right|Q_{A}\right>$ for a flow with $Re_{\unicode[STIX]{x1D706}}=168$ . A selection of strain- and enstrophy-dominated conditions highlights the qualitatively different structure of these regions. The figure shows a representative example of all analysed simulations. All conditional averages are normalized by $\langle Q_{W}\rangle =-\langle Q_{S}\rangle$ as a measure of the background average. To identify how extreme an event is, the conditions are given in terms of the standard deviation of $Q_{A}$ . Note that due to the skewness of the PDF of $Q_{A}$ , the positive conditions extend to larger values. For reference, the PDFs of the velocity-gradient invariants are discussed in appendix B; in particular the standard deviations of the fields are given in table 2. In a way figure 5 can also be considered as a model-free characterization of the isotropic component of the typical profiles of vortical and strain structures of given intensity.

Figure 5(a) illustrates the case of enstrophy-dominated reference points. It fully supports the qualitative description of enstrophy-dominated regions suggested by the example of figure 4. The initial value corresponds to the reference condition, i.e. the structures associated with the reference point. For high-intensity conditions these are expected to be vortex tubes. Note, however, that low-intensity conditions correspond to both quiescent regions and regions with a strong degree of cancellation between intense strain and enstrophy structures. For all conditions the decay of the profiles indicates the length scale of the reference structures in the following sense.

Independent of the condition value, the decay is almost complete within approximately $10\unicode[STIX]{x1D702}$ ( ${\approx}2L_{A}$ ) for all conditions. So, all intense enstrophy-dominated profiles are contained within approximately the same length scale, i.e. they all decay completely before approximately the same radius. A comparison with figure 5(b,c) shows that this is not due to the underlying vortical structures alone. Both $\overline{Q_{W}}$ and $\overline{Q_{S}}$ conditional averages decay slowly for $r\gtrsim 2L_{A}$ . This can be associated with the gradual decorrelation of these fields with distance. However, on these scales, for each condition value, $\overline{Q_{W}}$ must decorrelate at the same rate as $\overline{Q_{S}}$ , so as to cancel out and leave virtually no imprint in the $\overline{Q_{A}}$ conditional averages in figure 5(a).

The $\overline{Q_{W}}$ and $\overline{Q_{S}}$ conditional averages in the slowly decaying outer regions are larger for more intense conditions. However, it is interesting to note that in relative terms (e.g. normalized to the peak of each conditional average), these regions are less prominent for more intense conditions. This reflects the notion that more intense structures are more localized than less intense ones.

Furthermore, there are regions of enhanced strain located at $r\approx L_{A}$ whose position appears to be fixed. Their peak intensity is much lower than that of the reference structure and increases with the intensity of the condition. As a result the effective size and intensity of the enstrophy-dominated regions are attenuated. A similar purely kinematic result is described in appendix A for an isolated vortex. However, in the turbulent case the surrounding strain shells are relatively weak and much closer to the core of the central vortex region. As a result they do not dominate their radial shells and the resulting enstrophy-dominated structures decay monotonically into the background. (Note that this behaviour could also be accommodated by an isolated vortex with axial instead of spherical symmetry.)

Conditioning on strain-dominated regions (cf. figure 5 d) reveals a qualitatively different behaviour. The conditional profiles change sign, unequivocally identifying strongly enstrophy-dominated regions around the reference strain-dominated structures. Here the similarity to the isolated eddy example is even stronger, as the neighbouring vorticity structures actually dominate their radial shells. The combined structures decay completely for $r\gtrsim 20\unicode[STIX]{x1D702}$ . Again, this happens almost independently of the intensity of the condition, just as for the enstrophy-dominated profiles. Considering the larger reference values of more intense structures, this leads to the conclusion that the more intense the $Q_{A}$ structures are, the more spatially localized they are. Disentangling the individual strain and enstrophy contributions shows the reason for the different behaviour of $\left<\left.\overline{Q_{A}}\right|Q_{A}<0\right>$ (cf. figure 5 e,f). The enstrophy-dominated shells also lie at $r\approx L_{A}$ for all large conditions, confirming this as the typical separation between intense strain and enstrophy structures. However, their peak intensity is much closer to the peak intensity of the reference structures – approximately half (in contrast the peak of the strain-dominated regions in figure 5(c) is approximately a fifth of the reference condition). Furthermore, the reference strain structures have much larger length scales and are much more diffuse than their enstrophy counterparts.

The correlation lengths associated with the corresponding correlation functions are also indicated in figure 5. As noted, $L_{A}$ is a good indicator for the separation between regions of extreme vorticity and strain. It also designates the length scale of $\overline{Q_{A}}$ around enstrophy-dominated points, on the one hand, and the radius of the enstrophy-dominated shell around strain-dominated points, on the other. In all cases, this behaviour is robust for the investigated range of Reynolds numbers and forcing scales and types. (Recall that $L_{A}$ scales with the Kolmogorov scale.) In contrast, both $L_{AW}$ (shown in figure 5 b,e) and $L_{AS}$ (shown in figure 5 c,f) grow much more rapidly with $\unicode[STIX]{x1D702}$ . In fact, with increasing $Re_{\unicode[STIX]{x1D706}}$ their positions slowly drift to the right with respect to the peaks of the shown profiles. This behaviour is consistent with the Reynolds number dependence of the correlation lengths (cf. figure 3), which we can now explain.

It is easy to express the correlation lengths as averages of the conditional profiles, weighted by the corresponding PDFs and condition values. For example,

(4.2) $$\begin{eqnarray}L_{AS}=\frac{1}{\unicode[STIX]{x1D70E}_{S}\unicode[STIX]{x1D70E}_{A}}\int _{0}^{\infty }\left|\int \left<\left.\overline{Q_{S}}(r)\right|Q_{A}=a\right>a\;\text{PDF}_{Q_{A}}(a)\,\text{d}a\right|\,\text{d}r.\end{eqnarray}$$

As the Reynolds number increases, the occurrence of extreme events increases, which adds weight to the more pronounced high-intensity structures through the PDF. The contributions from each conditional average also increases with $Re_{\unicode[STIX]{x1D706}}$ . This is because both the peak intensity and the length scales of the $\overline{Q_{S}}$ and $\overline{Q_{W}}$ profiles grow for each condition. However, since the intensities and length scales of the strain and vorticity profiles increase at similar rates, the resulting $\left<\left.\overline{Q_{A}}(r)\right|Q_{A}\right>$ profiles are practically unaffected. Hence the correlation length of $Q_{A}$ remains unaffected.

In summary, we find that the conditional averages of the velocity gradient invariants are dominated by kinematics not unlike that of an isolated vortex (appendix A). The central structure is associated with a surrounding shell of the opposite type, with separation of approximately $L_{A}$ . The intensity of the surrounding shell scales with the intensity of the reference structure. The type of the central structure seems to play an important role in determining the shape and position of the surrounding shell. Thus, strongly strain- and enstrophy-dominated regions consist of structures of similar characteristic scales of the order of $10\unicode[STIX]{x1D702}-20\unicode[STIX]{x1D702}$ independent of the reference intensity and $Re_{\unicode[STIX]{x1D706}}$ . They originate from underlying strain and enstrophy structures with much larger and more widely varying length scales. However, the outskirts of these structures always share similar decay profiles and hence cancel out.

Figure 6.

Average contributions to the pressure at (a) enstrophy-dominated regions and (b) strain-dominated regions from thin spherical shells of prescribed radius. The different colours show different intensity levels of the condition field, normalized to the respective standard deviation. The radial coordinate is normalized by the Kolmogorov length $\unicode[STIX]{x1D702}$ and the average contributions are non-dimensionalized by $\unicode[STIX]{x1D702}\langle Q_{W}\rangle$ . The data are taken from the simulation with $Re_{\unicode[STIX]{x1D706}}=168$ .

5.1 Conditional pressure contributions

In order to translate the structure of the velocity gradient to the pressure field, we need to examine the effect of the Poisson kernel. So, we proceed by considering the conditional averages of the contributions to the pressure field $\langle \overline{q_{A}}(r)|Q_{A}\rangle$ . Using (2.2), it is immediate that

(5.1) $$\begin{eqnarray}\langle p(\boldsymbol{x})|Q_{A}(\boldsymbol{x})\rangle \equiv \left<\left.p\right|Q_{A}\right>=\int _{0}^{\infty }\langle \overline{q_{A}}(r)|Q_{A}\rangle \,\text{d}r.\end{eqnarray}$$

Recall from (2.3) that after angle-averaging the kernel weight is proportional to $r$ . Clearly then the Poisson kernel separates the parts of $\left<\left.\overline{Q_{A}}(r)\right|Q_{A}\right>$ which decay faster from those which decay more slowly than linearly. Thus, the local extrema of $\left<\left.\overline{q_{A}}(r)\right|Q_{A}\right>$ occur at the isolated points where $\left<\left.\overline{Q_{A}}(r)\right|Q_{A}\right>$ decays as $1/r$ . Additionally, the Poisson kernel has a more pronounced effect on more intense conditions because they are associated with profiles with larger amplitudes. Moving beyond these general consideration for turbulent flows is analytically challenging. To aid the intuition the simple case of the contributions to the pressure at the centre of an isolated Townsend eddy is described in appendix A.

To address the turbulent case we consider a representative example of $\left<\left.\overline{q_{A}}(r)\right|Q_{A}\right>$ for a range of conditions for $Re_{\unicode[STIX]{x1D706}}=168$ (see figure 6). Based on the effect of the Poisson kernel, three qualitatively distinct regions can be identified for all condition types and intensities: a core, a tail and an intermediate region. The central core for all conditions is dominated by the Poisson kernel. The relatively constant $\left<\left.\overline{Q_{A}}(r)\right|Q_{A}\right>$ central profiles in this region lead to growing pressure contributions, which peak at the edge of the core. Naturally, the height of the peak increases with the intensity of the reference structure. In the analysed simulation, it occurs at ( $r\approx 2\unicode[STIX]{x1D702}$ ). So, in view of their impact on the pressure field, the cores of both enstrophy- and strain-dominated structures may be defined as the regions where the $\overline{Q_{A}}$ conditional averages decay more slowly than $1/r$ . It should be pointed out that an increase in small-scale resolution may lead to steeper gradients at these scales and consequently reduce the size of these regions. Comparing simulations 6 and 7 reveals this to be the case for the strain-dominated, but not for the enstrophy-dominated conditions.

In contrast, the tails of the profiles are dominated by the far-field asymptote of $\overline{Q_{A}}$ . Here the type of condition begins to play a role. As shown in the figure, enstrophy-dominated conditions receive almost no net pressure contributions from regions beyond ${\approx}10\unicode[STIX]{x1D702}$ . In comparison, for strain-dominated conditions this happens at ${\approx}20\unicode[STIX]{x1D702}$ . In both cases, however, on these scales the Poisson kernel does little more than slow down the decay rate of the high intensity $\overline{Q_{A}}$ conditional averages and enhance the unavoidable statistical noise. It should be noted that this result is entirely contingent on the fact that $\langle Q_{A}\rangle =0$ . If the mean were finite, the resulting pressure contributions would grow linearly with radius and thus in a physical system would be dominated by the largest scales and/or boundary conditions (unless those also cancel each other). This is in fact the case for the individual strain and enstrophy contributions to the pressure, quantified by the conditional averages of $-2r\left<\left.\overline{Q_{S}}(r)\right|Q_{A}\right>$ and $-2r\left<\left.\overline{Q_{W}}(r)\right|Q_{A}\right>$ (not shown). It is because of the global balance of $Q_{S}$ and $Q_{W}$ ( $\langle Q_{S}\rangle =-\langle Q_{W}\rangle$ ) that the dominant pressure contributions at extreme events remain relatively local.

The intermediate region in between the core and the tail of the conditional averages is where the properties of the velocity gradient and the Poisson kernel interact most interestingly and yield the bulk of the pressure contributions. In this region, for all types of conditions, larger intensity corresponds to steeper profiles. This suggests that the pressure at more intense structures is more local, independent of the type of the structure. More striking, however, are the differences between the enstrophy- and strain-dominated reference structures.

On the one hand, the enstrophy-dominated structures simply decay into the background (faster than $1/r$ ), so the Poisson kernel simply slows down that decay. It also reveals a small footprint of the surrounding strain regions at $r\approx 10\unicode[STIX]{x1D702}$ for the very extreme conditions. However its amplitude is negligible for the examined range of Reynolds numbers. Overall, this leads to the conclusion that the pressure at strongly enstrophy-dominated structures is a very local quantity, with most contributions coming from a shell of radius ${\leqslant}L_{A}$ . Furthermore, due to the Poisson kernel, the periphery and the core of the structures have comparable weights. Recall that the structure and spatial scale of the periphery is sensitively determined by the cancellation of strain and enstrophy contributions. In that sense, strongly enstrophy-dominated regions exhibit a form of self-shielding with a characteristic length scale of $L_{A}$ .

On the other hand, the pressure contributions of the strain-dominated conditional averages are determined primarily by the presence of the weakly enstrophy-dominated shell at $r\approx L_{A}$ (cf. figure 5 d). The Poisson kernel amplifies the importance of this region, so much so that it overwhelms the pressure contributions from the strain-dominated core (cf. figure 6 b). This leads to the negative pressure values at intense strain regions noted by Yeung et al. (Reference Yeung, Donzis and Sreenivasan2012), for example. As a result, the central core has a direct impact only on the magnitude of the corresponding pressure. The length scale of the pressure contributions, however, is governed by the structure of the surrounding enstrophy-dominated regions. So, in strain-dominated regions the self-shielding mechanism is unbalanced and the pressure is determined at larger length scales, which reach the inertial range. In all analysed simulations the inertial range is considered to extend to several tens of $\unicode[STIX]{x1D702}$ .

5.2 Length scale of non-locality

In order to set an upper bound on the non-locality of the pressure contributions more precisely, we introduce a concrete length scale, $r_{\unicode[STIX]{x1D6FC}}(Q_{A})$ . We refer to it as a threshold radius. Explicitly it is defined as

(5.2) $$\begin{eqnarray}r_{\unicode[STIX]{x1D6FC}}(Q_{A})=\underset{d}{\max }\left\{\left|\int _{0}^{d}\left<\left.\overline{q_{A}}(r)\right|Q_{A}\right>\,\text{d}r\right|=\unicode[STIX]{x1D6FC}\left|\left<\left.p\right|Q_{A}\right>\right|\right\}.\end{eqnarray}$$

In essence, for each condition $Q_{A}$ , we compute the size of the local neighbourhood which contributes a fraction $\unicode[STIX]{x1D6FC}$ of the conditional pressure average at the reference point. If cancellations lead to the existence of several such (nested) neighbourhoods, we take the largest one.

In the following, we consider specifically a threshold level of $70\,\%$ , i.e. $r_{0.7}(Q_{A})$ . This threshold is chosen to reduce the contamination from the statistical and numerical noise for all datasets and all condition fields. The results for all examined simulations are plotted in figure 7. The shaded regions represent an estimate of the numerical uncertainty induced by the finite cell width. In order to clarify the dependence on the condition, the plotted curves are obtained after oversampling the $\left<\left.\overline{q_{A}}(r)\right|Q_{A}\right>$ profiles by a factor of ten with respect to $r$ .

Figure 7.

Threshold radius $r_{0.7}(Q_{A})$ for a range of Reynolds numbers, for strain-dominated conditions (a) and enstrophy-dominated conditions (b). The shaded regions designate an estimate for the numerical and statistical uncertainty of the computation due to the cell width. The dashed line at $4.5\unicode[STIX]{x1D702}$ corresponds to the best fit value for the integral correlation length $L_{A}$ . The low-intensity conditions $-10\unicode[STIX]{x1D70E}_{A}<Q_{A}<\unicode[STIX]{x1D70E}_{A}$ are omitted, since for those the threshold radius is dominated by statistical and numerical uncertainties.

As anticipated, for any given Reynolds number, the threshold radius decreases as the intensity of the condition value increases. This supports and quantifies the conclusion drawn above that the pressure becomes increasingly more local in regions of extreme velocity gradients. The strong dependence on the type of conditioning is also evident.

For enstrophy-dominated conditions (see figure 7 b), $r_{0.7}(Q_{A})$ quickly decays to a few Kolmogorov scales at small condition intensities and could be crudely approximated by the correlation length $L_{A}$ for the more intense conditions. Thus, the conditional pressure average at these conditions is determined by a dissipation-scale region of the size of the reference enstrophy-dominated structure. From this it follows naturally that $r_{0.7}(Q_{A})/\unicode[STIX]{x1D702}$ should not depend strongly on the Reynolds number. While the data give some hints to a small Reynolds number dependence, the issue can be settled by improved small-scale resolution simulations. These would also be necessary to uncover any significant dependence on the intensity of the condition for the most extreme vortices.

For strain-dominated conditions (see figure 7 a), the minimum value of the threshold radius is in the tens of $\unicode[STIX]{x1D702}$ . This quantifies the importance of the surrounding enstrophy-dominated shell described above. Clearly, the Poisson kernel plays the crucial role of extending the range of significant pressure contributions to inertial scales. As a result, they can no longer be tied to a single velocity-gradient structure and its kinematics, but are determined by the spatial distribution of a multitude of such structures. Thus the threshold radius for these conditions cannot be expected to naively scale with $\unicode[STIX]{x1D702}$ . There are indications in the data that it increases with Reynolds number. However, several simulations (including those with different forcing scale and type) break the trend, suggesting a possibility of non-universal behaviour. Critically, the significant difference between the simulations with the similar $Re_{\unicode[STIX]{x1D706}}$ but different small-scale resolution indicates that, for the strain-dominated conditions, the threshold radius should be considered only as an upper bound, which may decrease with improved small-scale resolution.

For completeness, we should mention that we have omitted from consideration the conditions $-10\unicode[STIX]{x1D70E}_{A}<Q_{A}<\unicode[STIX]{x1D70E}_{A}$ . For them the threshold radius is dominated by statistical and numerical uncertainties, which is not entirely unexpected. On the one hand, at the strain-dominated end of this range, there are nearly vanishing local pressure contributions because of almost perfect cancellation between the contributions from the strain-dominated core and the surrounding enstrophy-dominated shell. This dramatically increases both statistical and numerical uncertainties in the computation of the threshold radius. On the other hand, the low-intensity conditions correspond to the large quiescent regions with ambient or slightly positive pressure (as seen in figure 1). Because these quiescent regions occupy a significant volume of the simulation domain (see appendix B), it follows from homogeneity that for them $\langle p|Q_{A}\rangle \approx \langle p\rangle =0$ . They are also not associated with well-defined structures. So, for them the pressure contributions vanish at all radii; for instance see the $Q_{A}=\pm \unicode[STIX]{x1D70E}_{A}$ profiles in figure 6. As a result, analytically the threshold radius is expected to diverge in these regions.