2.2.1 Topological classification of streamlines

Figure 1.

Classification of local three-dimensional streamlines into non-degenerate topologies in the $Q$–$R$ plane (Chong et al. Reference Chong, Perry and Cantwell1990) for incompressible turbulence. The curved solid lines are discriminant $D=0$ lines.

The topological classification of local streamline structure in incompressible flows can be achieved with only the VGT invariants (Chong et al. Reference Chong, Perry and Cantwell1990),

(2.3a,b)$$\begin{eqnarray}Q=-{\textstyle \frac{1}{2}}\unicode[STIX]{x1D608}_{ij}\unicode[STIX]{x1D608}_{ji}\quad \text{and}\quad R=-{\textstyle \frac{1}{3}}\unicode[STIX]{x1D608}_{ij}\unicode[STIX]{x1D608}_{jk}\unicode[STIX]{x1D608}_{ki}.\end{eqnarray}$$

The $R=0$ and the discriminant $D=Q^{3}+(27/4)R^{2}=0$ lines divide the $Q$–$R$ plane into four regions, each representing a topology, as depicted in figure 1. These four non-degenerate three-dimensional topologies are stable focus stretching (SFS), unstable focus compression (UFC), unstable-node/saddle/saddle (UN/S/S) and stable-node/saddle/saddle (SN/S/S). This classification groups together streamline shapes that are ‘topologically equivalent’ or geometrically homeomorphic. In other words, different shapes related to each other by affine transformations are categorized as the same topology. A representative streamline shape of each topology, based on a canonical form of VGT, is shown in figure 1.

The topological description in the $Q$–$R$ plane does not uniquely describe the geometric shape of streamlines. This is best illustrated by the following example of an elliptic (closed) two-dimensional streamline flow. A VGT of the form (Blaisdell & Shariff Reference Blaisdell and Shariff1996),

(2.4)$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D63C}=\left[\begin{array}{@{}cc@{}}0 & -\unicode[STIX]{x1D6FE}-e\\ \unicode[STIX]{x1D6FE}-e & 0\end{array}\right],\quad \text{where }0<|e|<|\unicode[STIX]{x1D6FE}| & & \displaystyle\end{eqnarray}$$

describes local elliptic streamlines in the $x_{1}$–$x_{2}$ plane with major axis along the $x_{1}$ direction. It is easy to recognize that in terms of velocity gradient (VG) magnitude ($A\equiv \sqrt{\unicode[STIX]{x1D608}_{ij}\unicode[STIX]{x1D608}_{ij}}$) and invariants, the parameters $\unicode[STIX]{x1D6FE}$ and $e$ are

(2.5a,b)$$\begin{eqnarray}\unicode[STIX]{x1D6FE}=\sqrt{\frac{1}{2}+\frac{Q}{A^{2}}},\quad e=\sqrt{\frac{1}{2}-\frac{Q}{A^{2}}}.\end{eqnarray}$$

It may be recalled here that $R=0$ for all two-dimensional streamlines. The shape of an ellipse is defined by the aspect ratio $E$, which is the ratio of major to minor axes of the ellipse. It can be shown that the aspect ratio of the elliptic streamlines is (Blaisdell & Shariff Reference Blaisdell and Shariff1996)

(2.6)$$\begin{eqnarray}E=\sqrt{\frac{\unicode[STIX]{x1D6FE}+e}{\unicode[STIX]{x1D6FE}-e}}=\sqrt{\frac{\sqrt{{\displaystyle \frac{1}{2}}+{\displaystyle \frac{Q}{A^{2}}}}+\sqrt{{\displaystyle \frac{1}{2}}-{\displaystyle \frac{Q}{A^{2}}}}}{\sqrt{{\displaystyle \frac{1}{2}}+{\displaystyle \frac{Q}{A^{2}}}}-\sqrt{{\displaystyle \frac{1}{2}}-{\displaystyle \frac{Q}{A^{2}}}}}}.\end{eqnarray}$$

It is evident from (2.6) that for the same value of $Q$, different values of VG magnitude $A$ results in different aspect ratios, i.e. different shapes, of the local streamlines. For example, for the same value of $Q$, $A^{2}=2Q$ represents circular streamlines while $A^{2}\gg Q$ represents an infinitely elongated ellipse. Clearly, $Q$ does not uniquely describe the aspect ratio. On the other hand the variable

(2.7)$$\begin{eqnarray}q\equiv \frac{Q}{A^{2}}\end{eqnarray}$$

uniquely determines the aspect ratio of the ellipse, i.e.

(2.8)$$\begin{eqnarray}E(q)=\sqrt{\frac{\sqrt{{\textstyle \frac{1}{2}}+q}+\sqrt{{\textstyle \frac{1}{2}}-q}}{\sqrt{{\textstyle \frac{1}{2}}+q}-\sqrt{{\textstyle \frac{1}{2}}-q}}}.\end{eqnarray}$$

Therefore, in this two-dimensional case, $q$ is the only shape parameter and it uniquely characterizes the streamline shape.

The velocity components at any location ($x_{1},x_{2}$) on a streamline can be obtained using (2.2) and (2.4) as follows:

(2.9)$$\begin{eqnarray}\left.\begin{array}{@{}c@{}}u_{1}=\unicode[STIX]{x1D608}_{1k}x_{k}=-{\displaystyle \frac{A}{\sqrt{2}}}\left(\sqrt{{\displaystyle \frac{1}{2}}+q}+\sqrt{{\displaystyle \frac{1}{2}}-q}\right)x_{2},\\ u_{2}=\unicode[STIX]{x1D608}_{2k}x_{k}={\displaystyle \frac{A}{\sqrt{2}}}\left(\sqrt{{\displaystyle \frac{1}{2}}+q}-\sqrt{{\displaystyle \frac{1}{2}}-q}\right)x_{1}.\end{array}\right\}\end{eqnarray}$$

Clearly, each of the velocity components scale by a factor of $A$. Therefore, $A$ only influences the speed along the streamline, the shape of which is defined by $q$. From (2.9), the major and minor axis lengths of any given streamline are

(2.10)$$\begin{eqnarray}\left.\begin{array}{@{}c@{}}L_{maj}(A,q)\propto {\displaystyle \frac{1}{A}}\times \left(\sqrt{{\displaystyle \frac{1}{2}}+q}-\sqrt{{\displaystyle \frac{1}{2}}-q}\right)^{-1},\\ L_{min}(A,q)\propto {\displaystyle \frac{1}{A}}\times \left(\sqrt{{\displaystyle \frac{1}{2}}+q}+\sqrt{{\displaystyle \frac{1}{2}}-q}\right)^{-1}.\end{array}\right\}\end{eqnarray}$$

If there are two geometrically similar ellipses of same aspect ratio $E(q)$ and different VG magnitudes – $A_{1}$ and $A_{2}$ – , then one can be scaled to the other by the ratio,

(2.11)$$\begin{eqnarray}\frac{L_{maj1}(A_{1},q)}{L_{maj2}(A_{2},q)}=\frac{L_{min1}(A_{1},q)}{L_{min2}(A_{2},q)}=\frac{A_{2}}{A_{1}}.\end{eqnarray}$$

Therefore, the scale factor of an elliptic streamline is inversely proportional to the VG magnitude $A$.

This example illustrates that the shape of the ellipse is not uniquely defined by $Q$ as it combines the streamline shape and scale information. Similarly in three-dimensional flow, the invariants $Q$ and $R$ cannot uniquely define the streamline shape or scale. Any point in the $Q$–$R$ plane can represent multiple streamline shapes, which are homeomorphic but not necessarily similar.

2.2.2 Geometric description of streamlines

From (2.2), it is evident that all elements of $\unicode[STIX]{x1D608}_{ij}$ must be known to fully describe local streamline geometry. The locally linearized velocity vector field is given by,

(2.12)$$\begin{eqnarray}u_{i}=\unicode[STIX]{x1D608}_{ij}x_{j}.\end{eqnarray}$$

Since a streamline is always tangential to the velocity, the equation of a streamline can be obtained as follows:

(2.13)$$\begin{eqnarray}\text{d}\boldsymbol{s}\times \boldsymbol{u}=0\quad \text{where}\;\text{d}\boldsymbol{s}=\text{d}x_{1}\hat{i} +\text{d}x_{2}\,{\hat{j}}+\text{d}x_{3}\hat{k}.\end{eqnarray}$$

Here, $\text{d}\boldsymbol{s}$ is the infinitesimal arclength vector along the streamline. Therefore,

(2.14)$$\begin{eqnarray}(u_{3}\text{d}x_{2}-u_{2}\text{d}x_{3})\hat{i} +(u_{1}\text{d}x_{3}-u_{3}\text{d}x_{1}){\hat{j}}+(u_{2}\text{d}x_{1}-u_{1}\text{d}x_{2})\hat{k}=0.\end{eqnarray}$$

Now, setting each vector component to zero, we obtain the following differential equations:

(2.15a-c)$$\begin{eqnarray}\frac{\text{d}x_{2}}{\text{d}x_{3}}=\frac{u_{2}}{u_{3}}=\frac{\unicode[STIX]{x1D608}_{2j}x_{j}}{\unicode[STIX]{x1D608}_{3k}x_{k}},\quad \frac{\text{d}x_{3}}{\text{d}x_{1}}=\frac{u_{3}}{u_{1}}=\frac{\unicode[STIX]{x1D608}_{3j}x_{j}}{\unicode[STIX]{x1D608}_{1k}x_{k}},\quad \frac{\text{d}x_{1}}{\text{d}x_{2}}=\frac{u_{1}}{u_{2}}=\frac{\unicode[STIX]{x1D608}_{1j}x_{j}}{\unicode[STIX]{x1D608}_{2k}x_{k}}.\end{eqnarray}$$

The above differential equations can be integrated to obtain the equations describing streamlines.

Our goal is to derive a geometric description that separates shape and size features. We start by factorizing the VGT $\unicode[STIX]{x1D63C}$ into its magnitude $A$ and normalized VGT or VG structure tensor $\unicode[STIX]{x1D657}$ (Das & Girimaji Reference Das and Girimaji2019) such that

(2.16)$$\begin{eqnarray}\unicode[STIX]{x1D608}_{ij}=A\unicode[STIX]{x1D623}_{ij}.\end{eqnarray}$$

We now seek the streamline structure corresponding to the $\unicode[STIX]{x1D623}_{ij}$-field. We define a local velocity field given by

(2.17)$$\begin{eqnarray}u_{i}^{\ast }\equiv \unicode[STIX]{x1D623}_{ij}x_{j}.\end{eqnarray}$$

The corresponding $\unicode[STIX]{x1D623}_{ij}$-streamlines are given by,

(2.18)$$\begin{eqnarray}\text{d}\boldsymbol{s}^{\ast }\times \boldsymbol{u}^{\ast }=0\quad \text{where}\;\text{d}\boldsymbol{s}^{\ast }=\text{d}x_{1}^{\ast }\hat{i} +\text{d}x_{2}^{\ast }\,{\hat{j}}+\text{d}x_{3}^{\ast }\hat{k},\end{eqnarray}$$

where $\text{d}\boldsymbol{s}^{\ast }$ is the infinitesimal arclength vector along the $\unicode[STIX]{x1D623}_{ij}$-streamline. Using (2.15), (2.16) and (2.18), we can write the following identities involving $\unicode[STIX]{x1D608}_{ij}$- and $\unicode[STIX]{x1D623}_{ij}$-streamlines:

(2.19)$$\begin{eqnarray}\frac{\text{d}x_{\unicode[STIX]{x1D6FC}}^{\ast }}{\text{d}x_{\unicode[STIX]{x1D6FD}}^{\ast }}=\frac{u_{\unicode[STIX]{x1D6FC}}^{\ast }}{u_{\unicode[STIX]{x1D6FD}}^{\ast }}=\frac{\unicode[STIX]{x1D623}_{\unicode[STIX]{x1D6FC}i}x_{i}}{\unicode[STIX]{x1D623}_{\unicode[STIX]{x1D6FD}j}x_{j}}\equiv \frac{\unicode[STIX]{x1D608}_{\unicode[STIX]{x1D6FC}k}x_{k}}{\unicode[STIX]{x1D608}_{\unicode[STIX]{x1D6FD}l}x_{l}}=\frac{\text{d}x_{\unicode[STIX]{x1D6FC}}}{\text{d}x_{\unicode[STIX]{x1D6FD}}}\quad \forall \;\;\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD}=1,2,3\text{ and }\unicode[STIX]{x1D6FC}\neq \unicode[STIX]{x1D6FD}.\end{eqnarray}$$

Thus, all the internal structure features, including ratios of distances and alignments, of $\unicode[STIX]{x1D608}_{ij}$- and $\unicode[STIX]{x1D623}_{ij}$-streamlines are identical. Therefore, the shape parameters of the two sets of streamlines must be the same. It is now clear from (2.19) that all the $\unicode[STIX]{x1D623}_{ij}$-elements are required in order to determine the streamline shape.

From (2.12) and (2.17), at any location ($\boldsymbol{x}=\boldsymbol{x}_{0}$) of a streamline, it can be shown that

(2.20)$$\begin{eqnarray}\boldsymbol{u}=\unicode[STIX]{x1D63C}\,\boldsymbol{x}_{0}=A\unicode[STIX]{x1D657}\boldsymbol{x}_{0}=A\boldsymbol{u}^{\ast }\quad \;\Longrightarrow \;\quad \boldsymbol{u}=A\boldsymbol{u}^{\ast }.\end{eqnarray}$$

Therefore, it is clear that the velocities of $\unicode[STIX]{x1D608}_{ij}$- and $\unicode[STIX]{x1D623}_{ij}$-streamlines only differ in magnitude by a factor of $A$. As shown for the case of two-dimensional elliptic streamlines, this implies that the streamline scale factor is inversely proportional to $A$. From the analysis so far, we conclude the following.

1. (i) The shape features of the streamline geometry are entirely contained in $\unicode[STIX]{x1D623}_{ij}$.

2. (ii) All independent elements of $\unicode[STIX]{x1D623}_{ij}$ are required to completely describe the streamline geometric shape.

3. (iii) The VG magnitude $A$ determines the streamline scale factor and is inversely proportional to the scale factor.

Now we examine the properties of normalized VGT ($\unicode[STIX]{x1D657}$) in order to understand the shape characteristics of local streamline geometry. The second and third invariants of $\unicode[STIX]{x1D657}$,

(2.21a,b)$$\begin{eqnarray}q=-\frac{1}{2}\unicode[STIX]{x1D623}_{ij}\unicode[STIX]{x1D623}_{ji}=\frac{Q}{A^{2}}\quad \text{and}\quad r=-\frac{1}{3}\unicode[STIX]{x1D623}_{ij}\unicode[STIX]{x1D623}_{jk}\unicode[STIX]{x1D623}_{ki}=\frac{R}{A^{3}},\end{eqnarray}$$

classify streamline topology as before, and have the added advantage of separating shape from scale. The other advantage is that $\unicode[STIX]{x1D623}_{ij}$-elements are bounded as follows:

(2.22a,b)$$\begin{eqnarray}-\sqrt{{\textstyle \frac{2}{3}}}\leqslant \unicode[STIX]{x1D623}_{ij}\leqslant \sqrt{{\textstyle \frac{2}{3}}}\quad \forall \;\;i=j\quad \text{and}\quad -1\leqslant \unicode[STIX]{x1D623}_{ij}\leqslant 1\quad \forall \;\;i\neq j.\end{eqnarray}$$

We now seek the smallest set of independent $\unicode[STIX]{x1D623}_{ij}$-elements or parameters required to characterize the normalized VGT and hence the local streamline shape. The tensor $\unicode[STIX]{x1D657}$ can be decomposed into its symmetric and anti-symmetric counterparts as follows:

(2.23)$$\begin{eqnarray}\unicode[STIX]{x1D657}=\unicode[STIX]{x1D668}+\unicode[STIX]{x1D66C},\quad \text{where }\unicode[STIX]{x1D634}_{ij}=(\unicode[STIX]{x1D623}_{ij}+\unicode[STIX]{x1D623}_{ji})/2\quad \text{and}\quad \unicode[STIX]{x1D638}_{ij}=(\unicode[STIX]{x1D623}_{ij}-\unicode[STIX]{x1D623}_{ji})/2.\end{eqnarray}$$

Here, $\unicode[STIX]{x1D634}_{ij}$ is the normalized strain rate tensor and $\unicode[STIX]{x1D638}_{ij}$ is the normalized rotation rate tensor. Since the orientation of streamlines with respect to the laboratory frame of reference is immaterial for shape description, the tensor components can be considered in any coordinate system of choice to simplify geometric shape description. Therefore, for present purposes, $\unicode[STIX]{x1D657}$ can be expressed in the principal frame of $\unicode[STIX]{x1D668}$ without any loss of generality as

(2.24)$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D657}=\left[\begin{array}{@{}ccc@{}}a_{1} & 0 & 0\\ 0 & a_{2} & 0\\ 0 & 0 & a_{3}\end{array}\right]+\left[\begin{array}{@{}ccc@{}}0 & -\unicode[STIX]{x1D714}_{3} & \unicode[STIX]{x1D714}_{2}\\ \unicode[STIX]{x1D714}_{3} & 0 & -\unicode[STIX]{x1D714}_{1}\\ -\unicode[STIX]{x1D714}_{2} & \unicode[STIX]{x1D714}_{1} & 0\end{array}\right],\quad \text{where }a_{1}\geqslant a_{2}\geqslant a_{3}. & & \displaystyle\end{eqnarray}$$

Here, $a_{i}$ are the normalized strain rates, i.e. eigenvalues of tensor $\unicode[STIX]{x1D668}$. For incompressible flow, $a_{1}({>}0)$ is the most expansive strain rate, $a_{3}({<}0)$ is the most compressive strain rate and $a_{2}$ can be positive, negative or zero. The corresponding eigenvectors ($\boldsymbol{E}_{a_{1}}$, $\boldsymbol{E}_{a_{2}}$ and $\boldsymbol{E}_{a_{3}}$) are mutually orthogonal and constitute the principal directions of the symmetric tensor $\unicode[STIX]{x1D668}$. Further, $\unicode[STIX]{x1D714}_{i}$ are the components of normalized vorticity vector ($\unicode[STIX]{x1D74E}$) along the strain rate eigendirections (note that the unnormalized vorticity vector is given by $2A\unicode[STIX]{x1D74E}$). Therefore, these six parameters – $a_{1}$, $a_{2}$, $a_{3}$, $\unicode[STIX]{x1D714}_{1}$, $\unicode[STIX]{x1D714}_{2}$ and $\unicode[STIX]{x1D714}_{3}$ – completely define the normalized VGT and thence the geometric shape of the local streamlines.

In a previous study, Chakraborty, Balachandar & Adrian (Reference Chakraborty, Balachandar and Adrian2005) describe the VGT in terms of a scale factor and four parameters. In that study, the scale factor is the largest strain rate eigenvalue and the four shape parameters are relative vorticity magnitude ($a$), strain-field parameter ($\unicode[STIX]{x1D709}$) and orientation angles of vorticity vector with respect to strain rate eigendirections ($\unicode[STIX]{x1D703}$ and $\unicode[STIX]{x1D719}$). Then, they proceed to reduce the system to three parameters – $\Vert \unicode[STIX]{x1D6FA}\Vert /\Vert S\Vert$, $\unicode[STIX]{x1D709}$ and $\unicode[STIX]{x1D713}$ – that completely determine the VGT eigenvalues, but not the entire geometric shape. Different local streamline geometries are characterized in the phase space of $\Vert \unicode[STIX]{x1D6FA}\Vert /\Vert S\Vert -\unicode[STIX]{x1D713}$ at specific values of $\unicode[STIX]{x1D709}$. It must be noted that one of the parameters describing the streamline shapes is unbounded: $\Vert \unicode[STIX]{x1D6FA}\Vert /\Vert S\Vert \in (0,\infty )$. In the present study, we seek an alternate framework where all the shape parameters are mathematically bounded, thus providing a compact space for characterizing streamline shapes. The Frobenius norm of VGT ($A$) is selected as the scale factor due to its inherent advantage of directly representing pseudo-dissipation. As described in Das & Girimaji (Reference Das and Girimaji2019), using $A$ as the scale factor also leads to ease of VGT modelling.

We begin by reducing the six parameters defining VGT in (2.24) into the smallest independent set. First, we apply the incompressibility condition,

(2.25)$$\begin{eqnarray}a_{1}+a_{2}+a_{3}=0\quad \;\Longrightarrow \;\quad a_{3}=-(a_{1}+a_{2}).\end{eqnarray}$$

Then, due to normalization of VGT, we have,

(2.26)$$\begin{eqnarray}\unicode[STIX]{x1D623}_{ij}\unicode[STIX]{x1D623}_{ij}=\unicode[STIX]{x1D634}_{ij}\unicode[STIX]{x1D634}_{ij}+\unicode[STIX]{x1D638}_{ij}\unicode[STIX]{x1D638}_{ij}=a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+2(\unicode[STIX]{x1D714}_{1}^{2}+\unicode[STIX]{x1D714}_{2}^{2}+\unicode[STIX]{x1D714}_{3}^{2})=1.\end{eqnarray}$$

These constraints reduce the system to a total of four functionally independent parameters, which completely specify $\unicode[STIX]{x1D657}$ in the principal frame of $\unicode[STIX]{x1D668}$ and therefore determine the exact local streamline shape. It is expeditious to choose the frame-independent invariants, $q$ and $r$, as two of the four parameters.

Following the work of Ashurst et al. (Reference Ashurst, Kerstein, Kerr and Gibson1987), which demonstrates a preferential alignment of vorticity vector with the intermediate strain rate eigenvector in a turbulent flow field, we choose the other two parameters to be the intermediate strain rate eigenvalue $a_{2}$ and the vorticity component along the intermediate strain rate eigendirection $|\unicode[STIX]{x1D714}_{2}|$. From (2.21), we have

(2.27)$$\begin{eqnarray}-{\textstyle \frac{1}{2}}\unicode[STIX]{x1D623}_{ij}\unicode[STIX]{x1D623}_{ji}={\textstyle \frac{1}{2}}(\unicode[STIX]{x1D638}_{ij}\unicode[STIX]{x1D638}_{ij}-\unicode[STIX]{x1D634}_{ij}\unicode[STIX]{x1D634}_{ij})=q.\end{eqnarray}$$

Note that a positive $q$ implies rotation-dominated flow while a negative $q$ represents strain-dominated flow. Using (2.26) and (2.27),

(2.28)$$\begin{eqnarray}\unicode[STIX]{x1D634}_{ij}\unicode[STIX]{x1D634}_{ij}=a_{1}^{2}+a_{2}^{2}+a_{3}^{2}={\textstyle \frac{1}{2}}-q.\end{eqnarray}$$

Equations (2.25) and (2.28) can be used to show that

(2.29a,b)$$\begin{eqnarray}a_{1}={\textstyle \frac{1}{2}}(-a_{2}+\sqrt{1-3a_{2}^{2}-2q})\quad \text{and}\quad a_{3}={\textstyle \frac{1}{2}}(-a_{2}-\sqrt{1-3a_{2}^{2}-2q}).\end{eqnarray}$$

Therefore, $a_{1}$ and $a_{3}$ may be calculated from the four parameter set. Next, the third invariant in (2.21) can be expanded as follows:

(2.30)$$\begin{eqnarray}r=-{\textstyle \frac{1}{3}}\unicode[STIX]{x1D623}_{ij}\unicode[STIX]{x1D623}_{jk}\unicode[STIX]{x1D623}_{ki}=-{\textstyle \frac{1}{3}}(\unicode[STIX]{x1D634}_{ij}\unicode[STIX]{x1D634}_{jk}\unicode[STIX]{x1D634}_{ki}+3\unicode[STIX]{x1D634}_{ij}\unicode[STIX]{x1D638}_{jk}\unicode[STIX]{x1D638}_{ki})\quad \;\Longrightarrow \;\quad r=-a_{1}a_{2}a_{3}-a_{1}\unicode[STIX]{x1D714}_{1}^{2}-a_{2}\unicode[STIX]{x1D714}_{2}^{2}-a_{3}\unicode[STIX]{x1D714}_{3}^{2}.\end{eqnarray}$$

From (2.26) and (2.27), we also obtain,

(2.31)$$\begin{eqnarray}\unicode[STIX]{x1D638}_{ij}\unicode[STIX]{x1D638}_{ij}=2(\unicode[STIX]{x1D714}_{1}^{2}+\unicode[STIX]{x1D714}_{2}^{2}+\unicode[STIX]{x1D714}_{3}^{2})={\textstyle \frac{1}{2}}+q.\end{eqnarray}$$

Solving (2.30) and (2.31) leads to

(2.32)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D714}_{1}=\pm \frac{1}{2\sqrt{2}}\sqrt{(1+2q-4\unicode[STIX]{x1D714}_{2}^{2})-\frac{8a_{2}^{3}+8r-a_{2}(3-2q-12\unicode[STIX]{x1D714}_{2}^{2})}{\sqrt{1-3a_{2}^{2}-2q}}}=f(q,r,a_{2},\unicode[STIX]{x1D714}_{2}), & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

(2.33)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D714}_{3}=\pm \frac{1}{2\sqrt{2}}\sqrt{(1+2q-4\unicode[STIX]{x1D714}_{2}^{2})+\frac{8a_{2}^{3}+8r-a_{2}(3-2q-12\unicode[STIX]{x1D714}_{2}^{2})}{\sqrt{1-3a_{2}^{2}-2q}}}=g(q,r,a_{2},\unicode[STIX]{x1D714}_{2}). & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

Equations (2.29), (2.32) and (2.33) show that $a_{1}$, $a_{3}$, $|\unicode[STIX]{x1D714}_{1}|$ and $|\unicode[STIX]{x1D714}_{3}|$ can be completely and uniquely determined in terms of $q$, $r$, $a_{2}$ and $|\unicode[STIX]{x1D714}_{2}|$, thus completely specifying the tensor $\unicode[STIX]{x1D657}$. Therefore, each combination of $q$, $r$, $a_{2}$ and $|\unicode[STIX]{x1D714}_{2}|$ represents a unique geometric shape. These four quantities are now designated as the shape parameters.

This four-parameter system is consistent with that of Chakraborty et al. (Reference Chakraborty, Balachandar and Adrian2005), as previously mentioned. Each of their four parameters can be expressed in terms of the shape parameters as follows:

(2.34)$$\begin{eqnarray}\left.\begin{array}{@{}c@{}}a\equiv {\displaystyle \frac{|\unicode[STIX]{x1D714}|}{2a_{1}}}={\displaystyle \frac{\sqrt{1+2q}}{2a_{1}}}={\displaystyle \frac{\sqrt{1+2q}}{-a_{2}+\sqrt{1-3a_{2}^{2}-2q}}},\\ \unicode[STIX]{x1D709}\equiv -{\displaystyle \frac{2a_{2}}{a_{1}}}=-{\displaystyle \frac{2a_{2}}{a_{1}}}=-{\displaystyle \frac{4a_{2}}{-a_{2}+\sqrt{1-3a_{2}^{2}-2q}}},\\ \unicode[STIX]{x1D703}=\cos ^{-1}\left({\displaystyle \frac{2\unicode[STIX]{x1D714}_{1}}{\sqrt{1+2q}}}\right)=\cos ^{-1}\left({\displaystyle \frac{2f(q,r,a_{2},\unicode[STIX]{x1D714}_{2})}{\sqrt{1+2q}}}\right),\\ \unicode[STIX]{x1D719}=\tan ^{-1}\left({\displaystyle \frac{\unicode[STIX]{x1D714}_{3}}{\unicode[STIX]{x1D714}_{2}}}\right)=\tan ^{-1}\left({\displaystyle \frac{g(q,r,a_{2},\unicode[STIX]{x1D714}_{2})}{\unicode[STIX]{x1D714}_{2}}}\right).\end{array}\right\}\end{eqnarray}$$

As mentioned earlier, the advantage of the current approach is that the shape is completely defined in a compact parameter space.

Figure 2.

Summary of the key points of different frameworks for studying local flow streamline structure.

The streamline shape and its scale factor together constitute the complete geometry of the local streamlines. Figure 2 summarizes the important distinctions between topological and geometric descriptions of streamlines.

2.2.3 Kinematic bounds of shape parameters

We now seek to establish the bounds of the shape parameters. Let us first determine the bounds of the invariant parameter $q$. From (2.28) and (2.31), we can write

(2.35a,b)$$\begin{eqnarray}\unicode[STIX]{x1D634}_{ij}\unicode[STIX]{x1D634}_{ij}={\textstyle \frac{1}{2}}-q\geqslant 0\quad \text{and}\quad \unicode[STIX]{x1D638}_{ij}\unicode[STIX]{x1D638}_{ij}={\textstyle \frac{1}{2}}+q\geqslant 0\end{eqnarray}$$

leading to

(2.36)$$\begin{eqnarray}-{\textstyle \frac{1}{2}}\leqslant q\leqslant {\textstyle \frac{1}{2}}.\end{eqnarray}$$

Applying the conditions $a_{1}\geqslant a_{2}\geqslant a_{3}$ in (2.29) and solving for $a_{2}$ in the resulting inequality equations, we obtain

(2.37)$$\begin{eqnarray}\displaystyle -\sqrt{\frac{1-2q}{12}}\leqslant a_{2}\leqslant \sqrt{\frac{1-2q}{12}}. & & \displaystyle\end{eqnarray}$$

The most expansive strain rate ($a_{1}$) is non-negative by definition and attains its maximum value when $a_{2}$ is minimum, thus,

(2.38)$$\begin{eqnarray}\displaystyle 0\leqslant a_{1}\leqslant \sqrt{\frac{1-2q}{3}}. & & \displaystyle\end{eqnarray}$$

Similarly, the most compressive strain rate ($a_{3}$) has the following bounds:

(2.39)$$\begin{eqnarray}\displaystyle -\sqrt{\frac{1-2q}{3}}\leqslant a_{3}\leqslant 0. & & \displaystyle\end{eqnarray}$$

From (2.31) it is seen that all vorticity components have identical bounds,

(2.40)$$\begin{eqnarray}|\unicode[STIX]{x1D714}_{i}|^{2}\leqslant \frac{q}{2}+\frac{1}{4}\quad \;\Longrightarrow \;\quad -\sqrt{\frac{q}{2}+\frac{1}{4}}\leqslant \unicode[STIX]{x1D714}_{i}\leqslant \sqrt{\frac{q}{2}+\frac{1}{4}}\quad \forall \;\;i=1,2,3.\end{eqnarray}$$

Determining the bounds of the third invariant $r$ is quite involved and the steps are not displayed here. Substituting the upper bounds of $\unicode[STIX]{x1D714}_{1}$ and $\unicode[STIX]{x1D714}_{3}$ (2.40) into (2.32) and (2.33) and applying the bounds of $a_{2}$ and $|\unicode[STIX]{x1D714}_{2}|$ ((2.37) and (2.40)) leads to the following inequality:

(2.41)$$\begin{eqnarray}-\frac{1+q}{3}\left(\frac{1-2q}{3}\right)^{1/2}\leqslant r\leqslant \frac{1+q}{3}\left(\frac{1-2q}{3}\right)^{1/2}.\end{eqnarray}$$

This represents the kinematic bounds of $r$ for a given value of $q$ and therefore the boundary of the realizable region of $q$–$r$ plane. The extreme values of $r$ occur at $q=0$. Therefore, the absolute bounds of $r$ (also derived by Wang et al. (Reference Wang, Szalay, Aragón-Calvo, Neyrinck and Eyink2014) in a different context) are

(2.42)$$\begin{eqnarray}\displaystyle -{\textstyle \frac{\sqrt{3}}{9}}\leqslant r\leqslant {\textstyle \frac{\sqrt{3}}{9}}. & & \displaystyle\end{eqnarray}$$

Equations (2.36), (2.37), (2.40) and (2.41) define the kinematic bounds of the shape parameters – $q$, $a_{2}$, $|\unicode[STIX]{x1D714}_{2}|$ and $r$. Further constraints may be possible and will be investigated in future work.

2.2.4 Characterization of geometric shape in the $q$–$r$ plane

Figure 3.

Streamline shapes represented by different points, lines and regions of the$q$–$r$ plane.

First, we classify the topologies in $q$–$r$ space in a manner similar to that in $Q$–$R$ space (Chong et al. Reference Chong, Perry and Cantwell1990), as shown in figure 3. The $r=0$ and discriminant $d=q^{3}+(27/4)r^{2}=0$ lines divide the plane into four non-degenerate topology types. A schematic representation of the general streamline shapes belonging to these topologies are depicted in figure 3. The difference here is that $q$–$r$ provides a mathematically bounded phase plane and each ($q,r$) combination represents a unique streamline shape. For the sake of completeness, we reiterate this classification in the context of the present framework.

In focal streamlines ($d>0$), $\unicode[STIX]{x1D657}$ has one real ($\unicode[STIX]{x1D706}_{2}$) and two complex conjugate ($\unicode[STIX]{x1D706}_{1,3}$) eigenvalues.

1. (i) Stable focus stretching or SFS streamlines ($r<0$ or $\unicode[STIX]{x1D706}_{2}>0$) spiral towards a stable focus while stretching out of the focal plane.

2. (ii) Unstable focus compression or UFC streamlines ($r>0$ or $\unicode[STIX]{x1D706}_{2}<0$) spiral away from the centre while being compressed into the focal plane.

In nodal streamlines ($d<0$), $\unicode[STIX]{x1D657}$ has three distinct real eigenvalues ($\unicode[STIX]{x1D706}_{1}>\unicode[STIX]{x1D706}_{2}>\unicode[STIX]{x1D706}_{3}$) and three solution planes (not necessarily orthogonal).

1. (iii) Stable node/saddle/saddle streamlines consist of a stable node ($\unicode[STIX]{x1D706}_{2}<0$ or $r<0$) in one plane and saddle nodes in two planes.

1. (iv) Unstable node/saddle/saddle streamlines consist of an unstable node ($\unicode[STIX]{x1D706}_{2}>0$ or $r>0$) in one plane and saddle nodes in two planes.

Table 1.

Description of streamline shapes at points/lines marked in the $q$–$r$ plane in figure 3.

The focus is now on the complete description of various streamline shapes associated with different locations in the $q$–$r$ plane. We examine the streamline shapes at several lines and points of geometric significance in the $q$–$r$ plane as marked in figure 3. The corresponding streamline shape descriptions are listed in table 1, along with a schematic representation of each shape. The details of the various shapes are further discussed below.

Two-dimensional streamlines: the $r=0$ line represents two-dimensional planar flow streamlines. At every point along this line the intermediate eigenvalue of $\unicode[STIX]{x1D657}$ is zero ($\unicode[STIX]{x1D706}_{2}=0$) but the intermediate strain rate eigenvalue ($a_{2}$) is not necessarily zero. In the event that the vorticity vector is perfectly aligned with the intermediate strain rate eigenvector ($\boldsymbol{E}_{a_{2}}$), $a_{2}$ is zero. The specific shapes at different $q$ values along this line are as follows:

1. (i) Elliptic: the upper half of the $r=0$ line represents closed elliptic streamlines or centers (Kaplan (Reference Kaplan1958) figure 11-10) with the aspect ratio of the ellipse dependent on the $q$-value as given in (2.8). Here, $\unicode[STIX]{x1D657}$ has one real and two purely imaginary eigenvalues ($\unicode[STIX]{x1D706}_{2},\unicode[STIX]{x1D706}_{1,3}=\pm \text{i}\unicode[STIX]{x1D706}_{i}$).

2. (ii) Pure rotation: the top-most point in the plane, (i.e. $q=1/2$, $r=0$), represents circular streamlines undergoing pure or solid-body rotation. A pure rotation flow is elliptic flow with aspect ratio 1. Here, the strain rate eigenvalues are zero, i.e. $a_{1}=a_{2}=a_{3}=0$, and the normalized VGT, $\unicode[STIX]{x1D657}$, is only composed of vorticity/rotation rate tensor ($\unicode[STIX]{x1D66C}$).

3. (iii) Pure shear: the entire $q=0$ line represents streamlines with equal contributions of strain and rotation ($\unicode[STIX]{x1D634}_{ij}\unicode[STIX]{x1D634}_{ij}=\unicode[STIX]{x1D638}_{ij}\unicode[STIX]{x1D638}_{ij}$). The $r=0$ point on this line is of particular significance since at this point all the eigenvalues of $\unicode[STIX]{x1D657}$ are zero and therefore it represents only shear deformation of the local fluid element. The eigenvalues of $\unicode[STIX]{x1D668}$ and $\unicode[STIX]{x1D66C}$ are individually non-zero due to the contribution of shear to each of these tensors. Thus, the origin of the $q$–$r$ plane represents pure shear parallel streamlines (Kaplan (Reference Kaplan1958) figure 11-11).

4. (iv) Saddle: any point on the lower half ($q<0$) of the $r=0$ line has two real equal and opposite eigenvalues ($\unicode[STIX]{x1D706}_{1,3}=\pm \unicode[STIX]{x1D706}$) of $\unicode[STIX]{x1D657}$. These points represent open hyperbolic streamlines, constituted by a saddle point with compression in one eigendirection and expansion in the other (Kaplan (Reference Kaplan1958) figure 11-4). These eigenvectors of $\unicode[STIX]{x1D657}$ are in general oblique and become progressively orthogonal as $q$ approaches its lower limit.

5. (v) Orthogonal saddle: at the bottom-most point ($q=-1/2$, $r=0$) on this line, there is no vorticity and the two real eigenvectors of $\unicode[STIX]{x1D657}$ are perpendicular to each other. This results in a two-dimensional orthogonal saddle with the compressing streamlines perpendicular to the expanding streamlines.

Pure strain: the entire bottom-most line ($q=-1/2$) represents three-dimensional nodal streamlines with three orthogonal solution planes ($\unicode[STIX]{x1D657}$-eigenvectors) of compression and expansion. This is due to the fact that along this line streamlines have zero vorticity ($\unicode[STIX]{x1D74E}=0$ and $\unicode[STIX]{x1D638}_{ij}=0$) and thus, $\unicode[STIX]{x1D657}$ ($=\unicode[STIX]{x1D668}$) is a symmetric tensor. The $q=-1/2$ line, therefore, represents pure-strain streamlines.

Star-node/saddle/saddle: the zero discriminant lines demarcating the focal streamlines from nodal streamlines also have a specific streamline shape. At any point on the left $d=0$ line, the tensor $\unicode[STIX]{x1D657}$ has one positive real eigenvalue, $\unicode[STIX]{x1D706}_{1}$, and two equal negative real eigenvalues, $\unicode[STIX]{x1D706}_{2}=\unicode[STIX]{x1D706}_{3}=-\unicode[STIX]{x1D706}_{1}/2$. This results in a stable symmetrical node or star-node, i.e. straight streamlines directed towards the critical point (see Kaplan (Reference Kaplan1958) figure 11-9a), in one of the three eigenvector planes and saddles in the other two eigenvector planes. Similarly, on the right $d=0$ line, $\unicode[STIX]{x1D657}$ has two equal positive eigenvalues, $\unicode[STIX]{x1D706}_{1}=\unicode[STIX]{x1D706}_{2}=-\unicode[STIX]{x1D706}_{3}/2$, thus representing an unstable symmetrical node or star-node (straight streamlines directed away from the critical point) in one of the solution planes and saddles in the other two.

Axisymmetric compression/expansion: a special case of the above mentioned shape occurs at the points of intersection of the $d=0$ lines with the pure-strain line, i.e. at the corner points ($q=-1/2,r=\pm 1/3\sqrt{6}$) of the $q$–$r$ plane. Due to the orthogonality of the eigenvector planes, the bottom-left corner of the plane represents axisymmetric compression accompanied by twice as strong expansion perpendicular to it, forming an elongated tube-like streamline structure. The bottom-right corner represents axisymmetric expansion with twice as strong compression perpendicular to it, forming a flatter disk-like streamline geometry.

Orthogonal focal stretching/compression: next we emphasize on the significance of the left and right boundaries of the $q$–$r$ plane as given by (2.41). When $r$ is at its lower bound for a given $q$ (left boundary), $\unicode[STIX]{x1D74E}$ is perfectly aligned with the most expansive strain rate eigenvector ($\boldsymbol{E}_{a_{1}}$) provided $a_{1}$ is equal to its upper kinematic limit for that $q$ value (or $a_{2}$ is equal to its lower kinematic limit, implying $a_{2}=a_{3}=-a_{1}/2$). In other words, when

(2.43a-c)$$\begin{eqnarray}\unicode[STIX]{x1D714}_{1}=\unicode[STIX]{x1D714}=\sqrt{\frac{1}{4}+\frac{q}{2}},\quad \unicode[STIX]{x1D714}_{2}=\unicode[STIX]{x1D714}_{3}=0\quad \text{and}\quad a_{2}=-\sqrt{\frac{1-2q}{12}}\end{eqnarray}$$

solving (2.32) yields,

(2.44)$$\begin{eqnarray}r=-\frac{1+q}{3}\left(\frac{1-2q}{3}\right)^{1/2},\end{eqnarray}$$

which is the left boundary of the $q$–$r$ plane. This result is important as it will be shown in § 4.3 that in a turbulent flow field, $a_{1}$ achieves its maximum value along the left boundary and therefore $\unicode[STIX]{x1D74E}$ is indeed most likely aligned with $\boldsymbol{E}_{a_{1}}$ along the left boundary of the plane. This line therefore represents SFS streamlines stretching in a direction perpendicular to the plane of rotation/spiralling. Similarly, the right boundary of the $q$–$r$ plane represents perfect alignment of $\unicode[STIX]{x1D74E}$ with the most compressive strain rate eigenvector ($\boldsymbol{E}_{a_{3}}$) provided $a_{3}$ is at its most negative limit for a given $q$ value (i.e. $a_{1}=a_{2}=-a_{3}/2$). This line represents UFC streamlines with compression perpendicular to the focal plane. Again, the DNS data substantiates this result.

It is evident that shape-defining geometric properties, such as strain rate eigenvalues and alignment of vorticity with strain rate eigendirections, vary in a specific manner across the $q$–$r$ plane. There is scope for further characterization of this variation within the non-degenerate topologies of the $q$–$r$ plane, which will be pursued in future work.

5.5.1 Inertial effects

The $Q$–$R$ plane: the inertial CMTs in the $Q$–$R$ plane, as depicted in figure 10(a), move from left to right in the plane, rendering stable topologies unstable. This is in line with the CMTs obtained from the restricted Euler solution (Cantwell Reference Cantwell1992) shown by Martín et al. (Reference Martín, Dopazo and Valiño1998a), Ooi et al. (Reference Ooi, Martín, Soria and Chong1999) and Chevillard et al. (Reference Chevillard, Meneveau, Biferale and Toschi2008). Some of the trajectories tend to asymptote to a line above the right $D=0$ line (invariant line of the restricted Euler solution) due to the absence of the isotropic pressure term. This line is also different from the invariant line of Burger’s equation, shown in Bikkani & Girimaji (Reference Bikkani and Girimaji2007), since the incompressibility condition is not enforced in that case.

The $q$–$r$ plane: the $q,\!r$-evolution due to inertial effects, as given by the following terms from (2.47) and (2.48):

(5.3a,b)$$\begin{eqnarray}\left(\frac{\text{d}q}{\text{d}t}\right)_{I}=A(-3r+2q\unicode[STIX]{x1D623}_{ij}\unicode[STIX]{x1D623}_{ik}\unicode[STIX]{x1D623}_{kj}),\quad \left(\frac{\text{d}r}{\text{d}t}\right)_{I}=A(2q^{2}+3r\unicode[STIX]{x1D623}_{ij}\unicode[STIX]{x1D623}_{ik}\unicode[STIX]{x1D623}_{kj}),\end{eqnarray}$$

is plotted in figure 10(b) for $Re_{\unicode[STIX]{x1D706}}=225$. Since the inertial terms only depend on $\unicode[STIX]{x1D657}$, the CMTs for other $Re_{\unicode[STIX]{x1D706}}$ cases are identical and not displayed separately. As observed for the $Q$–$R$ plane (Martín et al. Reference Martín, Dopazo and Valiño1998a), the origin of the $q$–$r$ plane also appears to be a degenerate saddle point. Clearly, in this compact and bounded phase space, the left boundary acts as a repeller and the right boundary acts as an attractor of the system. Referring back to figure 3, it is evident that inertia, whilst acting in isolation, causes the intermediate strain rate to become more positive and the vorticity vector to be more aligned with the most compressive strain rate eigendirection.

Figure 10.

Inertial CMTs in (a) $Q$–$R$ plane and (b) $q$–$r$ plane for $Re_{\unicode[STIX]{x1D706}}=225$. Background contours indicate the speed of the trajectory at each point, normalized by Kolmogorov time scale.

Figure 11.

Pressure CMTs in (a) $Q$–$R$ plane and (b) $q$–$r$ plane for $Re_{\unicode[STIX]{x1D706}}=225$. Background contours indicate the speed of the trajectory at that point, normalized by Kolmogorov time scale. The dashed lines separate the three types of pressure CMTs in $q$–$r$ space.

5.5.2 Pressure effects

The $Q$–$R$ plane: figure 11(a) shows the $Q$–$R$ CMTs due to pressure action. As opposed to the anisotropic pressure Hessian contribution to $Q$–$R$ evolution illustrated in previous studies (Chevillard et al. Reference Chevillard, Meneveau, Biferale and Toschi2008; Johnson & Meneveau Reference Johnson and Meneveau2016), the complete pressure Hessian term has been plotted here. It shows that pressure action opposes the inertial action.

The $q$–$r$ plane: the $q$–$r$ CMTs based on the pressure terms,

(5.4a,b)$$\begin{eqnarray}\left(\frac{\text{d}q}{\text{d}t}\right)_{{\mathcal{P}}}=A(-\unicode[STIX]{x1D629}_{ij}(\unicode[STIX]{x1D623}_{ji}+2q\unicode[STIX]{x1D623}_{ij})),\quad \left(\frac{\text{d}r}{\text{d}t}\right)_{{\mathcal{P}}}=A\left(-\frac{4}{3}q^{2}-\unicode[STIX]{x1D629}_{ij}(\unicode[STIX]{x1D623}_{ki}\unicode[STIX]{x1D623}_{jk}+3r\unicode[STIX]{x1D623}_{ij})\right)\end{eqnarray}$$

considered in isolation from the other terms in (2.47) and (2.48), are plotted in figure 11(b) for the $Re_{\unicode[STIX]{x1D706}}=225$ case. It is clear that there are three types of pressure CMTs. (i) Conditional mean trajectories of rotation-dominated focal streamlines near the top of the plane travel directly from the right boundary to the left boundary with very high speeds. (ii) Conditional mean trajectories in the middle region of the plane are repelled from a small region close to the right boundary and are attracted towards a small region in the left boundary. These relatively slow-moving CMTs tend to traverse along a line in the upper UFC region, get deflected by what appears to be a degenerate saddle point at origin and converges to the attractor in the lower SFS region. (iii) The strain-dominated streamlines near the bottom of the plane evolve reasonably fast, straight from the right to left boundary. Overall, pressure action causes UFC streamlines with vorticity along most compressive strain rate (right boundary of the plane) to change towards SFS streamlines with vorticity aligned along the most expansive strain rate (left boundary of the plane). In general the pressure trajectories oppose the inertial trajectories except the type-ii CMTs in the middle region of UFC topology, where pressure contribution aligns with the inertial contribution. The isotropic pressure term ($\text{d}q/\text{d}t=0$, $\text{d}r/\text{d}t=-4Aq^{2}/3$) simply drives the streamline shapes from right to left in straight horizontal lines. All the additional features of the pressure CMTs stem from the anisotropic pressure term. Pressure $q,\!r$-CMTs at other Reynolds numbers (not displayed) are nearly identical to the present case and therefore effect of pressure on local streamline shape can be deemed nearly independent of Reynolds number at high enough $Re_{\unicode[STIX]{x1D706}}$.

Figure 12.

Pressure-inertial CMTs in (a) $Q$–$R$ and (b) $q$–$r$ plane for $Re_{\unicode[STIX]{x1D706}}=225$. Background contours indicate the speed of the trajectory at that point, normalized by Kolmogorov time scale.

5.5.3 Pressure and inertial effects

In incompressible flows, the role of pressure is to counter inertial action in a manner that the velocity field is divergence free. As the action of pressure is a response to inertial effect on the velocity field, it may be useful to examine the combined outcome of the two processes.

The $Q$–$R$ plane: the $Q$–$R$ CMTs due to pressure and inertia are plotted in figure 12(a) for $Re_{\unicode[STIX]{x1D706}}=225$. All trajectories appear to be repelled from the origin and moving outwards in clockwise direction. This indicates that pressure and inertia together increases the VG magnitude, evolving the streamline topology outward in the $Q$–$R$ plane.

The $q$–$r$ plane: the $q$–$r$ CMTs of pressure and inertial processes are plotted in figure 12(b). The trajectories suggest a very different picture. The pressure-inertial $q$–$r$ CMTs are attracted towards the origin, which is in contrast to the pressure-inertial $Q$–$R$ CMTs. The reason for this difference is that the $Q$–$R$ trajectories predominantly reflect the growth in magnitude (altering the scale of streamline structure) and are unable to distinguish the effects on streamline shape.

There are two attractors in the $q$–$r$ phase space – the origin, representing pure shear, and the $q=-1/2$ line, representing pure strain. The CMTs spiral clockwise towards both the attractors. The basin of attraction of pure-shear attractor spans almost the entire $q$–$r$ plane, with only a few CMTs in the slender outer region of extremely low population density that asymptote to the pure-strain attractor. The pressure-inertial CMTs are exactly identical in other Reynolds number cases (not displayed). Therefore, we conclude that pressure responds to inertial action in a manner that most of the geometric shapes of streamlines evolve towards pure-shear geometry irrespective of the Reynolds number of the flow.

Figure 13.

Viscous CMTs in (a) $Q$–$R$ plane for $Re_{\unicode[STIX]{x1D706}}=225$, (b) $q$–$r$ plane for $Re_{\unicode[STIX]{x1D706}}=225$, (c) $q$–$r$ plane for $Re_{\unicode[STIX]{x1D706}}=385$, (d) $q$–$r$ plane for $Re_{\unicode[STIX]{x1D706}}=588$. Background contours indicate the speed of the trajectory at that point, normalized by Kolmogorov time scale.

5.5.4 Viscous effects

The $Q$–$R$ plane: the viscous contribution to $Q$–$R$ evolution is plotted in figure 13(a) and is in accordance with the results of Chevillard et al. (Reference Chevillard, Meneveau, Biferale and Toschi2008) and Johnson & Meneveau (Reference Johnson and Meneveau2016). Viscous action directs all trajectories in the $Q$–$R$ plane towards the origin. The evolution rate diminishes as the trajectories approach the origin. This reflects the damping effect of viscous action, which is to reduce the VG magnitudes of all streamline topologies alike. However, the effect of viscosity on evolution of geometric shape cannot be surmised from the $Q$–$R$ CMTs.

The $q$–$r$ plane: the viscous contribution to $q$–$r$ evolution, based on the following terms from (2.47) and (2.48):

(5.5a,b)$$\begin{eqnarray}\left(\frac{\text{d}q}{\text{d}t}\right)_{V}=-A\unicode[STIX]{x1D70F}_{ij}(\unicode[STIX]{x1D623}_{ji}+2q\unicode[STIX]{x1D623}_{ij}),\quad \left(\frac{\text{d}r}{\text{d}t}\right)_{V}=-A\unicode[STIX]{x1D70F}_{ij}(\unicode[STIX]{x1D623}_{ki}\unicode[STIX]{x1D623}_{jk}+3r\unicode[STIX]{x1D623}_{ij}),\end{eqnarray}$$

are plotted in figure 13(b–d) for $Re_{\unicode[STIX]{x1D706}}=225$, 385 and 588. Again, important features not apparent in the $Q$–$R$ CMTs are evident here. The viscous $q$–$r$ CMTs demonstrate that aside from reducing VG magnitude, the viscous action has a distinct influence on the geometric shape of the local streamlines. There exists a clearly defined repelling manifold in the lower UFC region. The only attracting manifold in this system is the $q=-1/2$ line, representing pure-strain streamlines with mutually orthogonal stretching and compression. Starting at the repelling manifold, viscous CMTs either follow a short path directly towards the attractor through UN/S/S streamline topology or take a longer path through SFS followed by SN/S/S and UN/S/S streamlines. Clearly, the viscous evolution of $q$–$r$ depends on its precise point of origin. Overall, viscous action directs streamlines away from strain-dominated vortex compression towards pure strain. Even though the magnitude of the viscous contribution is much lower than that of the pressure-inertial contribution, viscous action alters the pressure-inertial contribution to $q$–$r$ evolution by reducing the domain of attraction of pure-shear attractor and expanding that of pure-strain attracting manifold. The viscous action exhibits slight variation with $Re_{\unicode[STIX]{x1D706}}$ resulting in the weak $Re_{\unicode[STIX]{x1D706}}$-dependence of the CMT separatrix observed in figure 8. This will be examined in detail in a future study using higher resolution DNS data.