2 Overview

To understand how to incorporate the positivity constraint into an analysis of $\unicode[STIX]{x1D63E}$ , we consider first the simpler situation of a scalar real variable $x$ that is restricted to be positive, $x\in \mathbb{R}_{{>}0}$ . In the present case, we could think of the squared length of a polymer molecule. Let us think about moving around in $\mathbb{R}_{{>}0}$ according to the differential equation

(2.1) $$\begin{eqnarray}\frac{\text{d}s}{\text{d}t}=c,\end{eqnarray}$$

where $s(t)$ is the arclength along the half-line $x>0$ and $c$ is a real constant. With this equation, the arclength is just linear in time, $s(t)=ct$ . So far, we have not specified the relation between an infinitesimal increment of arclength $\text{d}s$ and an infinitesimal increment $\text{d}x$ . If we use the Euclidean relationship $\text{d}s=\text{d}x$ , then, with initial condition $x=x(0)>0$ , (2.1) integrates to $s(t)=x(t)-x(0)=ct$ . Clearly, if $c<0$ , then for large enough time, $x(t)$ will become negative, i.e. it will leave the set that it is supposed to be restricted to.

Now, imagine that instead of using the Euclidean expression $\text{d}s=\text{d}x$ , we measure arclength with the formula

(2.2) $$\begin{eqnarray}\text{d}s=\frac{1}{x}\,\text{d}x.\end{eqnarray}$$

Since it is position-dependent, this expression is an example of a non-Euclidean, and specifically a Riemannian arclength formula. It measures the relative distance between $x$ and $x+\text{d}x$ . Now, (2.1) integrates to

(2.3) $$\begin{eqnarray}s(t)=\log \frac{x(t)}{x(0)}=ct.\end{eqnarray}$$

Thus, $x(t)=x(0)\exp (ct)>0$ for all $x(0)>0$ and any $c$ : if we move according to this arclength formula, we will never leave the set $\mathbb{R}_{{>}0}$ . Intuitively, we can understand this by observing that, with distance metric (2.2), $\text{d}s/\text{d}x$ diverges as $x$ decreases: zero is infinitely far from any positive number. Finally, for any two positive numbers $p$ and $q$ , we can always set $x(0)=p$ and find $c$ and $t$ such that $x(t)=q$ . The distance $d_{R}(p,q)$ between these points is $|s(t)|$ , which from (2.3) is simply

(2.4) $$\begin{eqnarray}d_{R}(p,q)=\left|\text{log}\frac{q}{p}\right|=|\text{log}q-\log p|.\end{eqnarray}$$

Logarithmic intensity and distance formulae are widely used, from the decibel scale for sound, to the Richter scale for earthquake intensity, to the pH scale for hydrogen ion concentration in aqueous solutions.

For positive definite matrices, closely analogous results have been developed (Lang Reference Lang1999). For $\unicode[STIX]{x1D653}>0$ , (2.2) generalizes to

(2.5) $$\begin{eqnarray}\text{d}s=(\text{Tr}(\unicode[STIX]{x1D653}^{-1}\,\text{d}\unicode[STIX]{x1D653})^{2})^{1/2},\end{eqnarray}$$

and the Riemannian distance between positive definite matrices $\unicode[STIX]{x1D64B}$ and $\unicode[STIX]{x1D64C}$ is given by a relation analogous to (2.4),

(2.6) $$\begin{eqnarray}d_{R}(\unicode[STIX]{x1D64B},\unicode[STIX]{x1D64C})=\Vert \text{log}(\unicode[STIX]{x1D64B}^{-1}\boldsymbol{\cdot }\unicode[STIX]{x1D64C})\Vert _{F},\end{eqnarray}$$

where $\Vert \unicode[STIX]{x1D653}\Vert _{F}=(\text{Tr}(\unicode[STIX]{x1D653}^{\text{T}}\boldsymbol{\cdot }\unicode[STIX]{x1D653}))^{1/2}$ is the Frobenius norm of  $\unicode[STIX]{x1D653}$ .

Hameduddin et al. (Reference Hameduddin, Meneveau, Zaki and Gayme2018) apply the distance formula, (2.6), and some related results to the conformation tensor in viscoelastic flow. The squared Riemannian distance $\unicode[STIX]{x1D705}$ between the instantaneous conformation $\unicode[STIX]{x1D63E}(t)$ and the mean conformation $\overline{\unicode[STIX]{x1D63E}}$ is simply

(2.7) $$\begin{eqnarray}\unicode[STIX]{x1D705}=d_{R}^{2}(\overline{\unicode[STIX]{x1D63E}},\unicode[STIX]{x1D63E}(t))=\Vert \text{log}(\overline{\unicode[STIX]{x1D63E}}^{-1}\boldsymbol{\cdot }\unicode[STIX]{x1D63E}(t))\Vert _{F}^{2}.\end{eqnarray}$$

The conformation tensor at any instant can be transformed to $\unicode[STIX]{x1D642}(t)=\overline{\unicode[STIX]{x1D63E}}^{-1/2}\boldsymbol{\cdot }\unicode[STIX]{x1D63E}(t)\boldsymbol{\cdot }\overline{\unicode[STIX]{x1D63E}}^{-1/2}$ . Under this transformation, $\overline{\unicode[STIX]{x1D63E}}$ becomes the identity $\unicode[STIX]{x1D644}$ and (2.7) can be rewritten $\unicode[STIX]{x1D705}=d_{R}^{2}(\unicode[STIX]{x1D644},\unicode[STIX]{x1D642})$ . A measure of anisotropy of the instantaneous configuration relative to the mean can also be defined, $\unicode[STIX]{x1D709}=\text{inf}_{a}d_{R}^{2}(a\overline{\unicode[STIX]{x1D63E}},\unicode[STIX]{x1D63E}(t))=\text{inf}_{a}d_{R}^{2}(a\unicode[STIX]{x1D644},\unicode[STIX]{x1D642}(t))$ . That is, $\unicode[STIX]{x1D709}$ is the distance between $\unicode[STIX]{x1D642}$ and the nearest isotropic tensor. Finally, based on the geometric interpretation of the product of the eigenvalues of a tensor as a measure of its ‘volume’, Hameduddin et al. (Reference Hameduddin, Meneveau, Zaki and Gayme2018) define a logarithmic volume ratio $\unicode[STIX]{x1D701}=\log \det \unicode[STIX]{x1D63E}-\log \det \overline{\unicode[STIX]{x1D63E}}$ .

Figure 1. Contours of Riemannian distance $\unicode[STIX]{x1D705}$ between instantaneous and mean conformation tensors at planes of constant wall-normal position $y^{+}$ in viscoelastic channel flow: (a) $y^{+}=15$ (near the bottom wall) and (b) $y^{+}=180$ (at the centreline). Reproduced with permission from Hameduddin et al. (Reference Hameduddin, Meneveau, Zaki and Gayme2018, figure 6(c,d)).

To illustrate the usefulness of these new measures of polymer conformation, Hameduddin et al. (Reference Hameduddin, Meneveau, Zaki and Gayme2018) present results from direct numerical simulations of flow of a polymer solution in a plane channel under conditions where the polymer leads to 60 % drag reduction relative to the Newtonian flow. Figure 1 shows instantaneous contours of the distance measure $\unicode[STIX]{x1D705}$ (2.7) at two wall-normal positions. A number of important features of the flow become apparent in this representation. First, the value of $\unicode[STIX]{x1D705}$ is generally much larger at the centreline than near the wall. Recalling that this Riemannian distance is a relative distance measure, we see immediately that the relative magnitude of the fluctuations near the centre is much larger than near the wall. A plot of the analogous results for $\Vert \unicode[STIX]{x1D63E}^{\prime }\Vert _{F}$ would not reveal this information. We also see that regions of constant $\unicode[STIX]{x1D705}$ are much more elongated in the flow ( $x$ ) direction near the wall than at the centre, revealing important differences in the anisotropy of the spatial structure of the flow. Another interesting result that would not arise from the usual Euclidean analysis is that the anisotropy $\unicode[STIX]{x1D709}$ of the conformation tensor relative to the mean displays, to a very good approximation, a logarithmic variation with distance from the wall. Logarithmic scalings in wall turbulence may indicate self-similarity, so this observation is worthy of further examination.

3 Future

The Riemannian quantities introduced by Hameduddin et al. (Reference Hameduddin, Meneveau, Zaki and Gayme2018) provide a new view of polymer dynamics during turbulent flow. Strongly time-dependent flows of complex fluids arise in many other contexts as well, and the new approach described here may also lend insights into these flows. Many other complex fluids including droplet suspensions (Barthes-Biesel & Rallison Reference Barthes-Biesel and Rallison1981) and blood (Moyers-Gonzalez, Owens & Fang Reference Moyers-Gonzalez, Owens and Fang2008) are also treated with conformation tensor models, so these are also natural candidates for analysis with the approach described here. Even in the context of Newtonian flow, positive definite tensors appear in the form of the convective momentum flux, $\unicode[STIX]{x1D70C}\boldsymbol{v}\boldsymbol{v}$ . Subgrid-scale modelling of this quantity is of central interest in large-eddy simulations of turbulent flows (Meneveau & Katz Reference Meneveau and Katz2000). Perhaps the geometric decomposition introduced here can lead to new ways to make simplified models for complex flows.