Correlation and decomposition framework for identifying and disentangling flow structures: canonical examples and application to isotropic turbulence

We start with the Helmholtz decomposition, which states that a sufficiently smooth (twice continuously differentiable) vector field, defined on a bounded or an unbounded domain, can be uniquely decomposed into three components : (i) an irrotational vector field $\mathbf{D}$, (ii) a solenoidal vector field $\mathbf{C}$ and (iii) a harmonic vector field $\mathbf{B}$. Applied to the velocity field $\mathbf{u}$, this can be written as

where

and

In a bounded domain with a volume $V$ and a bounding surface $S$, the three components can be written as a generalized Biot-Savart law (see for instance Wu et al. (2007))

where $\mathbf{r}$ is the position vector, from a point in the volume $V$ (or the surface $S$) to the point $\mathbf{x}$ where the integrals are being evaluated. The integrals over the volume $V$ can be considered the “near-field” contribution whereas the surface integral over the bounding surface $S$ can be considered the “far-field” contribution. Any region within the domain has a bounding surface that separates it from the rest of the domain. Therefore, the integral over this bounding surface can be seen as a representing the sum of the contributions of (i) the integral over the volume surrounding the region and (ii) the integral over the bounding surface of the whole domain.

In an unbounded (infinite) domain, the Helmholtz decomposition reduces to

provided that $\mathbf{B}$ goes to zero when $|\mathbf{r}|$ goes to infinity. This happens if the distance over which the velocity field is correlated along the surface grows slower than $|\mathbf{r}|$ when $|\mathbf{r}|$ goes to infinity.

This can be illustrated by considering an integral correlation measure $L_{\alpha}$ defined on the surface of a sphere, similarly to what was done for 3D, which is given as

where $\mathbf{x}$ is a point on the surface of the sphere. Since this calculation is confined to the surface, the correlation around each point $\mathbf{x}$ is simply a function of an angle $\alpha$ and an integration length $\Lambda$, measured along the direction (on the surface) specified by $\alpha$. In the Helmholtz decomposition, the contribution of the boundary to the velocity field (i.e. the contribution of $\mathbf{B}$ to $\mathbf{u}$) goes to zero if

for any $\mathbf{x}$, $\alpha$ and $\Lambda<\pi|\mathbf{r}|$ (see, for example, Phillips (1933)). For an infinite domain, the contribution of $\mathbf{B}$ to the velocity field could be the result of an “organized motion over an infinite distance”, which can be seen as the result of (arbitrary) choices in constructing the field (e.g. the choice of a particular frame of reference). This arbitrary artificial “coherent motion”, which is the “extrinsic motion” can be removed by making a transformation in the velocity field such that the condition given by eq. 20 becomes true for any $\mathbf{x}$, $\alpha$ and $\Lambda<\pi|\mathbf{r}|$; this condition will result in the integrals given by eq. 17c becoming equal to zero for any surface $S$ in the sphere, when the radius of the sphere goes to infinity.

When doing turbulent flow simulations with periodic boundary conditions, any arbitrary artificial “coherent motion” can be removed by not considering $\mathbf{B}$, provided that the domain is large enough to take into account any relevant “intrinsic motion”; i.e. provided that the domain is “significantly” larger than the distance over which the velocity field is correlated; this requires that the domain needs to be “significantly” larger than the integral length scale of the turbulence, which is our case. If this happens, for $|\mathbf{r}|$ larger than half the domain size the contribution of $\mathbf{B}$ to the velocity approaches a constant, which can be made zero by neglecting $\mathbf{B}$ (which is equivalent to “choosing the appropriate frame of reference”). Actually, in our case, as usually true for turbulence simulations in triperiodic domains, this constant is already zero, since the mean velocity is equal to zero; as it will be shown, in this case $\mathbf{C}$ (i.e. eq. 17b) gives a good approximation of $\mathbf{u}$ (since we deal with an incompressible flow, $\mathbf{D}=0$) for $|\mathbf{r}|$ approaching the domain size.

With these considerations, the generalized Helmholtz decomposition reduces to the simplified Biot-Savart law, applicable for incompressible flows over periodic (or infinite) domains, which is given as

The Biot-Savart law provides a way to disentangle flow structures by isolating the contributions from different vorticity regions to a local velocity structure. For instance, local and non-local vorticity contributions can be separated using this paradigm, or the vorticity field can be conditionally sampled to identify the contribution of different vorticity levels in generating velocity field structures.

Note that, even though far-field vorticity contribution, i.e. $\mathbf{B}$, may be absent, the volumetric region can also be split into an isolated local region ($V_{\mathrm{L}}$), surrounded by the non-local region ($V_{\mathrm{NL}}$), which essentially behaves as a far-field for the region $V_{\mathrm{L}}$. This leads to the consequence that, if the local region has negligible vorticity within $V_{\mathrm{L}}$, while the non-local region $V_{\mathrm{NL}}$ induces a velocity field within $V_{\mathrm{L}}$, then this velocity field must be a potential flow. This means that the local flow in $V_{\mathrm{L}}$ can be described by the gradient of a harmonic function $\psi$, i.e. $\mathbf{u}_{\mathrm{L}}=\nabla\psi$, while $\nabla^{2}\psi=0$. The non-local contributions from $V_{\mathrm{NL}}$ cannot generate vorticity within the local region $V_{\mathrm{L}}$ (as illustrated in figure 4). A last feature to note regarding the Biot-Savart law is the rapid decay (of $1/r^{2}$ over a distance $r$) of the vorticity contribution, which means that a small, isolated, vorticity region cannot extend its influence over a large distance beyond its immediate neighbourhood.

Ideas associated with the Biot-Savart law can be used to define correlation measures in order to identify, extract and disentangle structures associated with the relation between the velocity and vorticity fields. In regions of strong vorticity associated with swirling-flow in the orthogonal plane, the Lamb vector, i.e. $\boldsymbol{\omega}\times\mathbf{u}$, yields high values. Although, this is again a local quantity. We propose a correlation which utilizes this idea and extends it to a non-local form, where the vorticity at point $\mathbf{x}$ is correlated with the velocity at point $\mathbf{x}+\mathbf{r}_{i}$, leading to a three-tuple $\mathbf{H}(\mathbf{x},\Lambda)$, which, similarly to $\mathbf{L}(\mathbf{x},\Lambda)$, can be written as follows

The above correlation has a flavour of the Biot-Savart law and it allows correlating the contribution of the local vorticity $\boldsymbol{\omega}(\mathbf{x})$ with the global vorticity contribution to its neighbouring velocity field, since the velocity $\mathbf{u}(\mathbf{x}+\mathbf{r}_{i})$ can be seen as an integral result of the global vorticity field. Note that, with the above definition, $\mathbf{H}(\mathbf{x},\Lambda)$ will tend to be orthogonal to $\boldsymbol{\omega}(\mathbf{x})$, as $H_{i}$ (i.e. $H$ along the $i-$direction) will have a high magnitude when the vorticity is large and orthogonal to the $i-$direction.

Since the vorticity at a point generates flow, in a Biot-Savart sense, in the plane orthogonal to the vorticity vector, a more natural correlation definition is proposed, which takes into account this fact. This is done by correlating the vorticity along a particular direction, say $x$, to the flow in the orthogonal plane, i.e. $yz$. At each point, a velocity field is generated using the $x-$vorticity (i.e. $\omega_{x}$) in the orthogonal $yz-$plane within a circular region (along perimeters of circles of radius $0<r_{yz}\leq\Lambda$). This local velocity field is calculated with a simplification of the Biot-Savart law, by taking the cross product of the vorticity with unit vectors in the orthogonal plane ($\mathbf{r}_{yz}/|\mathbf{r}_{yz}|$), as done for the $\mathbf{H}$ correlation. This is illustrated in figure 5, where the $\omega_{x}$ vorticity component generates the velocity field shown in blue (solid lines), while the real velocity field generated from the global vorticity contributions is shown in red (dashed lines). The $\mathbf{H^{p}}$ correlation (for $\mathbf{H}-$planar) is calculated as the integral of the dot product between the vorticity-generated velocity vectors (blue) and the real velocity (red), over rings of radius $0<r_{yz}<\Lambda$. Since the length of the rings increases proportionally with the radius $r$, the integral over each ring is further divided by $r$ (i.e. $|\mathbf{r}_{yz}|$), to give an average correlation at a distance $r$, though other definitions can be used. This correlation is given by

where $\hat{i}$ is the unit vector in the $x$-direction. $H^{p}_{y}$ and $H^{p}_{z}$ are defined in a similar way. Evidently, this correlation is more computationally expensive to calculate than $\mathbf{H}$, as three planar regions need to be considered for each point in space. This requirement can be relaxed by sampling the rings $0<r\leq\Lambda$ with a chosen frequency, i.e. using every $n$-th ring such that $r\in\left\{n,2n,3n,\dotsc\right\}$.

$\mathbf{H^{p}}(\mathbf{x},\Lambda)$ can be seen as an area-integral correlation-measure of the planar (2D) version of $\mathbf{C}$ in eq. 17b (see, for e.g., Wu et al. (2007)) with $\mathbf{u}$. In other words, it correlates the contribution of $\omega_{x}$ (the vorticity at $\mathbf{x}$) to $\mathbf{C}$ at $\mathbf{x}+\mathbf{r}$ with $\mathbf{u}$ at $\mathbf{x}+\mathbf{r}$; i.e. it correlates the value of $\mathbf{u}(\mathbf{x}+\mathbf{r})$ due to $\omega_{x}(\mathbf{x})$ with the actual value of $\mathbf{u}(\mathbf{x}+\mathbf{r})$. However, contrary to, e.g., $L_{\alpha\beta}(\mathbf{x},\Lambda)$, which is a line-integral correlation-measure over a distance $\Lambda$, $\mathbf{H^{p}}(\mathbf{x},\Lambda)$ is an area-integral correlation-measure over a disc of radius $\Lambda$. Similarly, different combinations of the 2D and 3D versions of $\mathbf{C}$, and line, area and volume integral correlation-measures can be constructed, providing different “Biot-Savart perspectives” on the correlation between the vorticity field and the velocity field; however, we do not further explore these possibilities here.

We construct a 1D velocity field along a line, starting with Oseen vortices, which can be defined by a tangential velocity field and a vorticity field, given as

where $r$ is the radial distance from the vortex center, $\Gamma$ is the circulation, $\nu$ is the fluid kinematic viscosity and $t$ is the time. The Oseen vortex comprises a small core region in (near) solid body rotation, within which the velocity increases (approximately) radially as $u_{\theta}\propto r$ to its maximum value. Beyond this, there exists a (near) potential flow region (where $\omega_{z}$ is nearly zero) and $u_{\theta}\propto 1/r$. We add a noise $\zeta$ to the velocity field given by eq. 24, to generate a “structure immersed in noise”. The vortex has a certain ‘reach’, which depends on the amplitude of $\zeta$, and is defined as the distance beyond which $\zeta>u_{\theta}$. The vorticity of the Oseen vortex is calculated in Cartesian coordinates as $\omega_{z}=\nabla\times\mathbf{u}=\partial u_{y}/\partial x-\partial u_{x}/\partial y$, and not using eq. 25, due to the addition of the noise to the velocity field.

We generate the velocity field by placing the centers of two counter-rotating Oseen vortices along the $x-$axis, separated by a distance greater than the typical reach of either vortex. Along the $x-$axis the velocity only has a $y-$component, $u_{y}$, and it is this one-dimensional velocity field that we consider. Furthermore, we impose a periodicity in $u_{y}$, over a length of $N_{x}$. This field, hence, consists of a large periodic structure, with smaller sub-structures (which are associated with each of the Oseen vortices and their interaction). This creates a pattern of symmetries and anti-symmetries in the velocity and vorticity fields.

The two vortices are generated using the same parameters, which are given (in arbitrary units) as: $\Gamma=10$, $\nu=2.0$ and $t=2.0$. The length of the field is $N_{x}=500$ and the centers of the vortices are placed at $150$ and $350$, with a core solid body rotation region extending over roughly $5$ units, and an Oseen velocity pattern extending roughly $50$ units, on either side of the centers. Uniformly distributed random noise ($-1<\zeta<1$), scaled to an amplitude of $1\%$ of the maximum velocity magnitude, was added to the velocity. The velocity reduces to within $2.5\%$ of the maximum value in the range of $x<50$ and $x>450$. As can be seen in the top panel of figure 6, the velocity field consists of a “larger structure” (approximately in the range $50<x<450$), which is symmetric, except for the noise, with respect to its middle ($x\approx 250$). This larger structure contains the following sub-structures: (i) two Oseen velocity patterns around the center of each vortex (within approximately $100<x<200$ and $300<x<400$), containing a solid body rotation region and a potential flow region, and (ii) an almost uniform velocity region (between roughly $200<x<300$) due to the interaction between the two vortices.

The middle and bottom panels of figure 6 show $\mathbf{L}$ and $\mathbf{L^{s}}$, respectively (which, here, only have a $x-$component, i.e. $L_{x}$ and $L_{x}^{s}$), with the integration length spanning the entire length of the velocity field, i.e. $\Lambda=N_{x}/2$. A few features of the correlation profiles point at the nature of the correlation definitions, as well as the importance of the choice of $\Lambda$.

First, $L_{x}$ is found to have a shape similar to the function itself. If we consider the definition of $L_{x}$, it is the product between the value of the function at a point and its integral over a length, hence, if either of the two is zero, the correlation becomes zero. Therefore, the correlation will have significant, or large, values where the function itself has significant, or large, values, and the structure of the function (i.e. its symmetries and/or anti-symmetries) does not make its integral small.

When the function is the sum of several “basis functions” (in this case two “pure” Oseen vortices and a “noise”), the $L_{x}$ correlation at a point involves the product between the local values of the basis functions and the integrals of the basis functions. So, even if the structure of each of the “basis functions” makes their integral small (i.e. if $L_{x}$ of a basis function itself is small), $L_{x}$ of the total function is not necessarily small. In the particular case presented here, the function $u_{y}$ is the sum of a “pure function”, $u_{yp}$, and a “noise function”, $\zeta$. The pure function $u_{yp}$ is the sum of two “basis functions”, the Oseen vortices; the interaction between the velocity fields of these two Oseen vortices results in a larger structure with a finite integral. The integral of the noise function, $\zeta$, is itself a “noise”. Hence, the $L_{x}$ correlation of $u_{y}$ has the same shape as $u_{y}$ itself. Mathematically, this can be seen as

If either the integral of $u_{yp}$ or the integral of $\zeta$ are non-zero, $L_{x}$ will be similar to $u_{yp}$. Interestingly, even for a velocity profile which leads to the integral of $u_{yp}$ becoming zero (i.e. $L_{x}$ of the pure function is zero), the additional noise breaks the overall symmetries and anti-symmetries, such that $L_{x}$ becomes non-zero, and the shape of the pure velocity field $u_{yp}$ can be extracted from $L_{x}$. If the noise is removed from the velocity field, and the vortices are placed sufficiently far from each other such that the integral of $u_{yp}$ becomes zero, $L_{x}$, indeed, goes to zero for $\Lambda=N_{x}/2$ (not shown here).

$L^{s}_{x}$, in figure 6(c), shows different features, starting with a central peak around $x=250$, which is exactly between the two counter-rotating vortices. Although the velocity around this position is small, $L^{s}_{x}$ attains a large value since the velocity field is essentially mirrored around this point, hence being perfectly correlated (with only the noise values being different between the mirrored halves), i.e. the large peak around $x=250$ is associated with the symmetry of the larger structure around this point. At the core of the two vortices ($x=150$ and $x=350$), where $L_{x}$ becomes zero, $L^{s}_{x}$ shows a large negative peak, since the velocity field on the left and right of these points is anti-correlated up to the reach of each vortex, i.e. the large magnitude of $L_{x}^{s}$ at $x=150$ and $x=350$ is associated with the anti-symmetry of the “local sub-structures” around these points. The part of $L^{s}_{x}$ in the region outside of the larger structure (i.e. approximately $x<100$ and $x>400$), is a repetition of the $L^{s}_{x}$ profile in the center (i.e. $150<x<350$) due to the periodicity of the velocity field and the integration length spanning the entire length of the field ($\Lambda=N_{x}/2$).

Figure 7 shows the correlations for a similar velocity field, now integrated over a length of $\Lambda=35$, which corresponds approximately to the size of the sub-structures (i.e. the individual vortices). This is because within a distance of $35$ units from the center of each vortex, the velocity reduces to roughly $10\%$ of its maximum value (note that a slightly lower or higher $\Lambda$ does not change the results significantly). The $L_{x}$ and $L^{s}_{x}$ profiles show a few similarities and differences. First, at the vortex cores, where the velocity is close to zero, $L_{x}$ goes to zero, while $L_{x}^{s}$ yields strong negative peaks because the velocity is strongly anti-correlated across the core region. In the potential flow regions of the two vortices and in the region between the vortices, the general shape of the two correlations is similar, with both $L_{x}$ and $L^{s}_{x}$ yielding positive correlation values, reflecting that the velocity is mostly uniform in these regions. This shows that in the core regions of vortices $\mathbf{L}$ and $\mathbf{L^{s}}$ correlations behave very differently while in regions of roughly uniform flow they have a similar behaviour. $L^{s}_{x}$ does not identify the ‘larger structure’, which in this case has a lengthscale larger than the integration length of the correlation. This shows the importance of the choice of $\Lambda$ in identifying larger or smaller symmetries and anti-symmetries of the vector fields. We shall revisit the importance of the choice of $\Lambda$, in the context of turbulence, in section 5.3.

Note that it is not only the length of integration $\Lambda$, but also the lower and upper limits of integration which determine the symmetries and structures being identified. For instance, eq. 9, can be changed to, instead, find non-local structures between lengths $\Lambda_{1}<r_{\alpha\beta}<\Lambda_{2}$, while looking around from point $\mathbf{x}$. We show an example using the $\mathbf{L^{s}}$ correlation, as follows

Recalling the generalized correlation definition, this change in the limits of $\Lambda$ is essentially defining $\mathcal{S}_{1}$ and $\mathcal{S}_{2}$ as finite regions going from $x\in[x_{0}+\Lambda_{1},x_{0}+\Lambda_{2}]$ and $x\in[x_{0}-\Lambda_{2},x_{0}-\Lambda_{1}]$ respectively, while $x_{0}$ is a point of the region of observation $\mathcal{S}_{o}$. One example of this is shown in figure 8, with $\Lambda_{1}=75$ and $\Lambda_{2}=125$. These integration limits are such that, when $x_{0}$ corresponds to the middle of the larger structure in the velocity field (i.e. at $x=250$), $\Lambda_{1}$ and $\Lambda_{2}$ span across most of the vortex regions. The $L^{s}_{x}$ profile, consequently, shows a strong peak in between the two vortices, resulting from the larger structure comprising the two counter-rotating vortices. These particular limits of integration do not identify any significant large symmetries, as observed from other $x$ locations.

Figure 9 shows correlations $G_{x}$ and $G^{s}_{x}$, which are the vorticity field equivalents of $L_{x}$ and $L^{s}_{x}$, integrated over $\Lambda=N_{x}/2$. The $G_{x}$ correlation remains mostly zero throughout, with small, noisy fluctuations. This is because, unlike the velocity field, which yields a finite value for the intergal of $u_{y}(x)$ (over the length $\Lambda=N_{x}/2$) due to the interaction between the two vortices and the non-zero contribution from the integral of the noise term $\zeta$, the integral of $\omega_{z}$ remains nearly zero, since the vorticity is localized at the core of the two vortices and has the same magnitude, but opposite sign at the two cores; the only contribution here is from the non-zero integral of the noise term, which breaks the anti-symmetry of the vorticity field. Similarly for $L_{x}$ with $u_{y}$, the break of the anti-symmetry due to the noise, leads to the shape of $G_{x}$ being similar to the shape of $\omega_{z}$; however, since here the only contribution is from the non-zero integral of the noise term, the magnitude of $G_{x}$ is much smaller than the magnitude of $\omega_{z}$.

The $G^{s}_{x}$ correlation has a very similar behaviour to $L^{s}_{x}$, just with the opposite sign. It yields a large negative peak at the middle of the larger structure ($x=250$), associated with the anti-symmetry of the vorticity of the larger structure around this point (while $L^{s}_{x}$ has a large positive peak, associated with the symmetry of the velocity field around this point). It also shows smaller positive peaks at the core of the vortices ($x=150$ and $x=350$), around which the vorticity is high and symmetric (while $L^{s}_{x}$ has a negative peak, associated the anti-symmetry of the velocity field around the vortex core). The $G^{s}_{x}$ profile is also repeated due to the periodicity of the velocity field and the large integration length, similarly to the $L^{s}_{x}$ profile in figure 6.

Figure 10 shows the $G_{x}$ and $G^{s}_{x}$ correlations for an integration length of $\Lambda=35$. Both $G_{x}$ and $G^{s}_{x}$ show a very similar overall shape, yielding sharp positive peaks at the vortex cores, where the vorticity is high and symmetric. The $G_{x}$ correlation decays with some “noise”, in the potential flow regions corresponding to the two vortices. The $G^{s}_{x}$ correlation, here, gives a sharper and less noisy profile. This is because, at the vortex core, the vorticity values to the left and right are perfectly symmetric (apart from the noise values), which gives a large correlation $G^{s}_{x}$ value at the vortex core. Slightly moving away from the core in either direction strongly disturbs this symmetry, leading to a sharper profile for $G^{s}_{x}$ than for $G_{x}$.

Finally, figure 11 shows the $H_{x}$ correlation for the Oseen vortex-pair, integrated over lengths of $\Lambda=N_{x}/2$ (panel b) and $\Lambda=35$ (panel c). The shape of the $H_{x}$ correlation is found to be almost insensitive to the choice of $\Lambda$, while a higher $\Lambda$ increases the amplitude of the correlation. This can be understood from the construction of this correlation, which, in a Biot-Savart sense, is designed to identify regions where the angular velocity aligns with the vorticity. In this example, $H_{x}$ yields large positive values in the core region of the two vortices, since the flow around the vortex cores is generated by the vorticity at the cores. The independence of the choice of $\Lambda$ is because the influence of the local vorticity at a point $x_{0}$ rapidly decays over distance, such that at larger $\Lambda$ values, the velocity field $u_{y}(x_{0}\pm\Lambda)$ is not influenced by $\omega_{z}(x_{0})$. The larger structure in this example is induced by the sum of the two individual vortices, and is, hence, “externally generated”, which is why $H_{x}$ remains zero in the middle of the two structures. The results, for a ‘linear’ version of the $H_{x}^{p}$ correlation (instead of its planar construction), are identical, and are not additionally shown here.

The $L_{x}$ and $L^{s}_{x}$ correlations help in identifying the structure of high velocity regions, while their analogues for the vorticity field, the $G_{x}$ and $G^{s}_{x}$ correlations, help in identifying the structure of high vorticity regions. The $H_{x}$ (and $H^{p}_{x}$) correlation gives information on the vorticity-velocity structure. The similarities and differences between these correlation measures gives information on the structure of the velocity and vorticity fields. In this example, where the velocity field is “induced” by a pair of vortices, at the core of the vortices, while $L_{x}$ is close to zero, there exists a large peak in the values of $L^{s}_{x}$, $G_{x}$, $G^{s}_{x}$ and $H_{x}$. On the other hand, in the region outside of the core of the vortices where the velocity is approximately uniform, if the integration length does not allow the capture of the larger structure periodicity, $L_{x}$ and $L^{s}_{x}$ have a similar shape, while $G_{x}$, $G^{s}_{x}$ and $H_{x}$ remain close to zero.

We next test the correlations on a two-dimensional Taylor-Green flow field, which comprises a set of counter-rotating vortices. Interaction between vortex-pairs generates regions of unidirection flow, seprated by regions of swirling flow. A simplified Taylor-Green flow pattern can be generated by creating a velocity field as follows

We generate such a velocity field with $0\leq x<2\pi$ and $0\leq y<2\pi$, in arbitrary units. A uniform noise $\zeta$, of amplitude $1\%$ of the maximum velocity magnitude (which is unity), is added to the velocity field. The vorticity field, which has just one component, i.e. $\omega_{z}$, is calculated using central differences. In figure 12(a), the kinetic energy $E_{k}$ is shown, while panel (b) shows the vorticity field $\omega_{z}$, along with the flow streamlines, where the counter-rotating vortex array can be seen. We would like to highlight that the velocity field structure in Taylor-Green flow is determined by the four large vortices, which are generated by regions of diffused, relatively high vorticity (as opposed to small, localized regions of high vorticity found in turbulence). These vortices have a diameter of approximately $\pi$. Essentially, the velocity field structure consists of: (i) four large vortices with a core region of high vorticity and low kinetic energy, (ii) regions of well-aligned jet-like velocity, with high kinetic energy and low vorticity, in between the large vortices, and (iii) a stagnation flow in the center with low kinetic energy and low vorticity.

In figure 13, the correlation fields for the Taylor-Green flow are shown. The correlation two-tuples are for the $x$ and $y$ directions, with an integration length of $\Lambda=\pi/2$. Panel (a) shows $\mathbf{L}$, which is found to have a high magnitude in regions of high kinetic energy $E_{k}$ (refer to figure 12a). This is because the $\mathbf{L}$ correlation identifies velocity regions that have simultaneously a high velocity magnitude and a high degree of local vector alignment. At the vortex core regions, $|\mathbf{L}|$ yields low values. It can be noted that for this velocity field, the regions of high $|\mathbf{L}|$ are separated from regions of high vorticity, showing that jet-like coherent regions can be generated by non-local vorticity (since the vorticity within high $|\mathbf{L}|$ regions is negligible, see figure 12b).

Figure 13(b) shows $|\mathbf{L^{s}}|$, which has a very different structure than $|\mathbf{L}|$, since it identifies symmetries and anti-symmetries in the velocity field, which are not just associated with jet-like parallel streamlines. There are regions of high $|\mathbf{L^{s}}|$ associated with different types of symmetries/anti-symmetries, other than the jet-like parallel streamlines. First, the regions of highest $|\mathbf{L^{s}}|$ are found at the vortex cores, as the vortices have a large swirling region with a velocity that is perfectly anti-correlated across the vortex cores. High $|\mathbf{L^{s}}|$ regions are also found coinciding with high $|\mathbf{L}|$ regions, where the flow has a symmetry associated with jet-like parallel streamlines. These two regions of high $|\mathbf{L^{s}}|$ are separated by thin regions where $|\mathbf{L^{s}}|$ goes to zero, due to the larger symmetries in the velocity field which cancel out in the calculation of $\mathbf{L^{s}}$. There is also a region of high $|\mathbf{L^{s}}|$ at the center, which is a stagnation flow region, which reflects the anti-symmetries in the velocity field of the four large vortices that produce the stagnation region.

Panel (c) shows $|\mathbf{G}|$, which yields high values in the vortex core regions (which also are the regions of high vorticity, see figure 12b). In the regions of jet-like flow between vortex-pairs, $|\mathbf{G}|$ has values very close to zero, which is due to the negligible vorticity in these regions. Panel (d) shows $|\mathbf{G^{s}}|$, which has a very different strucure than $|\mathbf{G}|$, since it identifies the symmetries and anti-symmetries in the vorticity field across every point. First, smaller diffused region of high $|\mathbf{G^{s}}|$ are found corresponding to the vortex cores, similarly to $|\mathbf{G}|$, since the vorticity across the vortex cores is highly correlated. In between counter-rotating vortex-pairs, both in the $x$ and $y$ directions, there appear slightly elongated regions of high $|\mathbf{G^{s}}|$. This is because along lines connecting the centres of counter-rotating vortex-pairs, the vorticity field varies uniformly from positive to negative, and this sign transition coincides with the jet-like velocity region, which itself has negligible vorticity; this strong anti-symmetry in the vorticity field gives the high $|\mathbf{G^{s}}|$. Note that at the center there exists also a strong symmetry in the vorticity field along the diagonals of the square, however, this is not reflected in $\mathbf{G^{s}}$, which is a two-tuple for the $x$ and $y$ directions, along which the vorticity is zero. This indicates that the particular alignment of the structures can lead to different values of the correlation, depending on the directions chosen for calculating the correlations. However, in homogeneous isotropic turbulence, with an arbitrary alignment of the structures, this effect will not appear, in a statistical sense.

Lastly, panels (e) and (f) show the $|\mathbf{H}|$ and $|\mathbf{H^{p}}|$ fields. Both correlations have a pattern that closely resembles $|\mathbf{G}|$, with large values at the vortex core regions, since the vortices here are generated by strong vorticity values at the vortex cores. Regions of jet-like flow, which are externally induced by the interaction between the velocity fields of adjacent vortices, yield negligible values of $|\mathbf{H}|$ and $|\mathbf{H^{p}}|$.

As a final example of the correlations applied to canonical flows, we now consider a three-dimensional velocity field generated by superposing two Burgers vortices, which again generates a ‘large-scale’ structure comprising two smaller sub-structures. The Burgers vortex is an exact solution of the Navier-Stokes equation, consisting of a radial velocity component along with a tangential velocity, and can be constructed as

where $a$ represents the rate of strain, $\Gamma$ the circulation and $\nu$ the kinematic viscosity. Here $u_{z}$, $u_{r}$ and $u_{\theta}$ give velocity components in the axial, radial and tangential directions, which are converted from cylindrical to Cartesian coordinates. We isolate the vortex in space by multiplying the velocity field with a three-dimensional Gaussian function $\mathcal{G}$ to contain the Burgers vortex within a spherical region,

where $x$, $y$ and $z$ are measured with the origin placed at the center of the vortex. The value of $\sigma$ is chosen such that it creates a spherical region circumscribing the axial length of the vortex. This suppresses the strain regions generated by each vortex, far away from its center, such that the velocity field comprises primarily of two swirling-flow regions. The swirling-flow of the Burgers vortex resembles the one-dimensional Oseen vortex (as was described in Section 4.1), with a core in solid-body rotation where $u_{\theta}\propto r$, followed by a potential flow region with $u_{\theta}\propto 1/r$. A low amplitude uniform noise, $\zeta$, is added to the final velocity field, over which the correlations are subsequently calculated. Lastly, the vortices are also rotated at arbitrary azimuthal and elevation angles $(\alpha,\beta)$. This is done to change the orientation of the velocity field symmetries with respect to the orthogonal bases along which the correlations are calculated, to test the applicability of the correlation definitions for arbitrarily aligned structures, as will be encountered in turbulence fields.

We generate two vortices, on a grid of $100^{3}$, with $a=0.1$, $\nu=0.025$, $\Gamma=15$ and $\zeta=0.004$ (i.e. $\sim 1\%$ of the maximum velocity magnitude), all quantities being presented in arbitrary units. The actual values being used here are not important, as we simply intend to generate a velocity field with a Burgers vortex structure. The resulting vortex has a core region with solid body rotation up to $5$ units (grid cells) from its axis, and the velocity magnitude in the potential flow region (with swirling motion) decays to approximately $40\%$ and $10\%$ of the maximum velocity magnitude within $15$ and $30$ grid cells from the axis, respectively. Both vortices are multiplied with the Gaussian function $\mathcal{G}$, generated with $\sigma=5$. The vortices are rotated at arbitrary azimuthal and elevation angles of $(0.6,0.25)$ and $(-0.45,-0.3)$, measured in radians. These vortices are then superposed by adding their velocity fields, with their centres placed at $(40,40,40)$ and $(70,70,70)$. The resulting velocity field is shown in figure 14. Since the vortices are placed close to each other, their velocity fields begin to entangle and interact. The larger structure of the Burgers vortex-pair, however, is distinct from the Oseen vortex-pair in Section 4.1, and it does not have the same kind of symmetries and anti-symmetries.

Figure 15 shows the amplitude of all the correlations, integrated over a length of $\Lambda=12$ for the Burgers vortex-pair. The value of $\Lambda$ is large enough to encompass most of the structure of each vortex, and the results were found to remain qualitatively unchanged for $\Lambda=10$ and $\Lambda=15$, which only causes a change in the magnitude of the correlations. The features of the correlation fields strongly reflect the behaviour of their one-dimensional analogues as was presented for the Oseen vortex-pair (see figures 7 and 10, where $\Lambda$ also encompasses, roughly, the structure of each vortex). First, panels (a) and (b) show the $\mathbf{L}$ and $\mathbf{L^{s}}$ correlations. The correlation regions are well aligned with the axes of the vortices, and have a size slightly smaller than the extent of the swirling-flow region. $\mathbf{L}$ goes to zero at the vortex core where the velocity is also zero. $\mathbf{L^{s}}$ yields a large value at the vortex core, across which the flow is highly anti-correlated, surrounding which is a thin region of zero correlation, and an outer region of finite correlation corresponding to the potential flow region of the vortex. Both these correlations identify the outer swirling-flow region where the streamlines are well-aligned, and locally parallel; however, in the core region of the vortex they are very different. Due to their definitions, near the center of a strong swirling flow $\mathbf{L^{s}}$ has a large magnitude whereas $\mathbf{L}$ has a magnitude close to zero. Panels (c) and (d) show the $\mathbf{G}$ and $\mathbf{G^{s}}$ correlations, both of which yield thin, elongated correlation profiles aligned with the axes of the vortices, while the $\mathbf{G^{s}}$ correlation is sharper. This again reflects that, at the core of the vortices, the vorticity vectors are well aligned. Panels (e) and (f) show $\mathbf{H}$ and $\mathbf{H^{p}}$, which also yield strong correlation profiles at the cores of the vortices, aligned with the axes of the vortices. This is because the swirling velocity field is associated with the vorticity at the vortex core regions.

We also explored larger integration lengths, assuming a periodic field, similarly to what was done for the one-dimensional flow. We found that for larger integration lengths, of $\Lambda\approx N_{x}/2$, the correlations also begin to recognize non-local symmetries (due to periodicity) in the twin Burgers vortex velocity field, similarly to the one-dimensional Oseen vortex pair (see figures 6 and 9), which is not additionally shown here. This shows that the correlations, depending on the integration length, can identify large complex structures, containing smaller sub-structures; however, this will not be further explored here.

We note that, similar to the Oseen vortex examples, in the vortex core there exists a strong relation between $\mathbf{L^{s}}$, $\mathbf{G}$, $\mathbf{G^{s}}$, $\mathbf{H}$ and $\mathbf{H^{p}}$, with all of them having large magnitudes, whereas $\mathbf{L}$ has a magnitude close to zero. On the contrary, in the outer regions of the vortex, there exists a strong relation between $\mathbf{L^{s}}$ and $\mathbf{L}$, with both of them having high magnitudes, whereas $\mathbf{G}$, $\mathbf{G^{s}}$, $\mathbf{H}$ and $\mathbf{H^{p}}$ have a magnitude close to zero. Essentially, over the entire vortex, where the swirling flow is strong, $\mathbf{L^{s}}$ has a large magnitude, and this region of high $|\mathbf{L^{s}}|$ encloses the vortex core where $|\mathbf{L}|$ is close to zero and $|\mathbf{G}|$, $|\mathbf{G^{s}}|$, $|\mathbf{H}|$ and $|\mathbf{H^{p}}|$ are also high. This happens because the swirling flow region is self-induced by the high vorticity at the core of the vortices leading to a highly anti-correlated velocity field around the center of the vortex, where the velocity is close to zero; moving away from the center of the vortex, the vorticity decays rapidly but the swirling flow is still strong, and the velocity field, locally, is highly correlated due to the parallel streamlines. Note that this is unlike the example of the Taylor-Green flow, where the jet-like high $|\mathbf{L}|$ regions are externally-induced by the interaction of vortices, leading to symmetries and anti-symmetries in the vorticity field in this region, which can result in high magnitudes of $\mathbf{G^{s}}$ even with low magnitudes of $\mathbf{G}$. The similarities and differences between the different correlation definitions play a key role in the subsequent analysis of turbulence field structures.

Overall, we find that the correlation definitions are adept at identifying typical velocity and vorticity field patterns, also in fields arbitrarily aligned with respect to directions along which the correlations are evaluated.

For this study, we use a dataset from DNS simulations of homogeneous isotropic turbulence, for which the Navier-Stokes equations with a body force $\mathbf{F}$ (as given below) are solved numerically

Turbulence is generated in a periodic box by means of low wavenumber forcing, which is divergence-free by construction and is concentrated over a range of Fourier modes. It is of the form given by Biferale et al. (2011), and has properties similar to that devised by Alvelius (1999) and Ten Cate et al. (2006), which can be written as

The forcing is stochastic (white noise) in time, which is achieved by varying each $\phi(k)$ randomly, and the force is distributed over a small range of wavenumers, given by $k_{a}\leq k\leq k_{b}$ (for this study we fix $k_{a}=1,k_{b}=8$), and the amplitude $A(k)$ of each of these wavenumbers is a Gaussian distribution in Fourier space, centered around a central forcing wavenumber $k_{f}$, given as