3.1 Evolution equations in the Reynolds-filtered case

In this section, we will consider the case when the nominal tensor is obtained using Reynolds filtering. We choose to restrict our attention to Reynolds filters but the development can be generalized to other filters, e.g. for applications in large-eddy simulations (LES). We thus have

(3.8) $$\begin{eqnarray}\displaystyle \overline{\unicode[STIX]{x1D63E}}=\langle \unicode[STIX]{x1D63E}\rangle . & & \displaystyle\end{eqnarray}$$

By the properties (2.7), the associated $\overline{\unicode[STIX]{x1D641}}$ satisfies $\langle \overline{\unicode[STIX]{x1D641}}\rangle =\overline{\unicode[STIX]{x1D641}}$ . Applying the averaging operation to (3.6) yields

(3.9) $$\begin{eqnarray}\displaystyle \langle [\unicode[STIX]{x1D642}]_{\unicode[STIX]{x1D64D}}\rangle =\unicode[STIX]{x1D644}, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D64D}(\boldsymbol{x},t)$ is the rotation tensor field given in (3.4). Henceforth, we will restrict the rotation tensor field so that

(3.10) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D64D}=\langle \unicode[STIX]{x1D64D}\rangle . & & \displaystyle\end{eqnarray}$$

By (2.7), we then have $\langle \unicode[STIX]{x1D642}\rangle =\unicode[STIX]{x1D644}$ .

The Reynolds decomposition is applied to $p$ and $\boldsymbol{u}$ while $\unicode[STIX]{x1D63E}$ is decomposed using (3.6) with $\overline{\unicode[STIX]{x1D63E}}$ defined according to (3.8). We thus have

(3.11a-c ) $$\begin{eqnarray}\displaystyle p=\overline{p}+p^{\prime },\quad \boldsymbol{u}=\overline{\boldsymbol{u}}+\boldsymbol{u}^{\prime },\quad \unicode[STIX]{x1D63E}=[\unicode[STIX]{x1D642}]_{\overline{\unicode[STIX]{x1D641}}}, & & \displaystyle\end{eqnarray}$$

where $\overline{\boldsymbol{u}}=\langle \boldsymbol{u}\rangle$ and $\overline{p}=\langle p\rangle$ and the primes denote fluctuating quantities obtained via the Reynolds decomposition. In general, $\overline{p}=\overline{p}(\boldsymbol{x},t)$ , $\overline{\boldsymbol{u}}=\overline{\boldsymbol{u}}(\boldsymbol{x},t)$ , $\overline{\unicode[STIX]{x1D63E}}=\overline{\unicode[STIX]{x1D63E}}(\boldsymbol{x},t)$ . Note that $\overline{\unicode[STIX]{x1D641}}\neq \langle \unicode[STIX]{x1D641}\rangle$ , in general.

Following the standard procedure, we can then decompose the momentum equation as follows:

(3.12) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}_{t}\overline{\boldsymbol{u}}+\overline{\boldsymbol{u}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\overline{\boldsymbol{u}} & = & \displaystyle -\unicode[STIX]{x1D735}\overline{p}+\frac{\unicode[STIX]{x1D6FD}}{Re}\unicode[STIX]{x1D6E5}\overline{\boldsymbol{u}}+\frac{1-\unicode[STIX]{x1D6FD}}{Re}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\overline{\unicode[STIX]{x1D64F}}-\unicode[STIX]{x1D735}\boldsymbol{\cdot }\overline{\boldsymbol{u}^{\prime }\boldsymbol{u}^{\prime }},\end{eqnarray}$$

(3.13) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}_{t}\boldsymbol{u}^{\prime }+\overline{\boldsymbol{u}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}^{\prime }+\boldsymbol{u}^{\prime }\boldsymbol{\cdot }\unicode[STIX]{x1D735}\overline{\boldsymbol{u}} & = & \displaystyle -\unicode[STIX]{x1D735}p^{\prime }+\frac{\unicode[STIX]{x1D6FD}}{Re}\unicode[STIX]{x1D6E5}\boldsymbol{u}^{\prime }+\frac{1-\unicode[STIX]{x1D6FD}}{Re}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\unicode[STIX]{x1D64F}^{\prime }-\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\boldsymbol{u}^{\prime }\boldsymbol{u}^{\prime })^{\prime },\end{eqnarray}$$

where $\overline{\unicode[STIX]{x1D64F}}=\langle \unicode[STIX]{x1D64F}\rangle$ , $\unicode[STIX]{x1D64F}^{\prime }=\unicode[STIX]{x1D64F}-\overline{\unicode[STIX]{x1D64F}}$ , $\overline{\boldsymbol{u}^{\prime }\boldsymbol{u}^{\prime }}=\langle \boldsymbol{u}^{\prime }\boldsymbol{u}^{\prime }\rangle$ and $(\boldsymbol{u}^{\prime }\boldsymbol{u}^{\prime })^{\prime }=\boldsymbol{u}^{\prime }\boldsymbol{u}^{\prime }-\overline{\boldsymbol{u}^{\prime }\boldsymbol{u}^{\prime }}$ .

The precise form of $\overline{\unicode[STIX]{x1D64F}}$ , which appears in the mean momentum equation in (3.13), depends on the constitutive model used. In the Oldroyd-B model, $\overline{\unicode[STIX]{x1D64F}}$ only depends on  $\overline{\unicode[STIX]{x1D641}}$ :

(3.14) $$\begin{eqnarray}\displaystyle \overline{\unicode[STIX]{x1D64F}}=\frac{1}{Wi}(\overline{\unicode[STIX]{x1D641}}\boldsymbol{\cdot }\overline{\unicode[STIX]{x1D641}}^{\mathsf{T}}-\unicode[STIX]{x1D644}). & & \displaystyle\end{eqnarray}$$

In models that are nonlinear in $\unicode[STIX]{x1D63E}$ , the fluctuating tensor $\unicode[STIX]{x1D642}$ cannot be eliminated or factored out of $\overline{\unicode[STIX]{x1D64F}}$ . For example, in the FENE-P model, $\overline{\unicode[STIX]{x1D64F}}$ can be expressed as a series in which the dominant term is equal to (3.14) while the remaining terms depend on higher-order moments of $\unicode[STIX]{x1D642}$ . In general, we have

(3.15) $$\begin{eqnarray}\displaystyle \overline{\unicode[STIX]{x1D64F}}=\frac{1}{Wi}[\langle \unicode[STIX]{x1D707}_{0}\rangle \unicode[STIX]{x1D644}+\overline{\unicode[STIX]{x1D641}}\boldsymbol{\cdot }\langle \unicode[STIX]{x1D707}_{1}\unicode[STIX]{x1D642}+\unicode[STIX]{x1D707}_{2}\unicode[STIX]{x1D642}\boldsymbol{\cdot }\overline{\unicode[STIX]{x1D641}}^{\mathsf{T}}\boldsymbol{\cdot }\overline{\unicode[STIX]{x1D641}}\boldsymbol{\cdot }\unicode[STIX]{x1D642}\rangle \boldsymbol{\cdot }\overline{\unicode[STIX]{x1D641}}^{\mathsf{T}}]. & & \displaystyle\end{eqnarray}$$

Substituting (3.11) into (2.3) and applying the filtering operation $\langle \cdot \rangle$ defined in (2.6) yields the following equations for $\overline{\unicode[STIX]{x1D63E}}$ :

(3.16) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}_{t}\overline{\unicode[STIX]{x1D60A}}_{ij}+\overline{u}_{k}\unicode[STIX]{x2202}_{k}\overline{\unicode[STIX]{x1D60A}}_{ij}-(\overline{\unicode[STIX]{x1D60A}}_{ik}\unicode[STIX]{x2202}_{k}\overline{u}_{j}+\overline{\unicode[STIX]{x1D60A}}_{jk}\unicode[STIX]{x2202}_{k}\overline{u}_{i})+\overline{\unicode[STIX]{x1D61B}}_{ij} & = & \displaystyle -\unicode[STIX]{x2202}_{k}\Bigl[\overline{\unicode[STIX]{x1D60D}}_{ip}\overline{\unicode[STIX]{x1D60D}}_{qj}^{\mathsf{T}}\underbrace{\langle \unicode[STIX]{x1D60E}_{pq}u_{k}^{\prime }\rangle }_{(\text{a})}\Bigr]\nonumber\\ \displaystyle & & \displaystyle +\,\overline{\unicode[STIX]{x1D60D}}_{ip}\overline{\unicode[STIX]{x1D60D}}_{qk}^{\mathsf{T}}\underbrace{\langle \unicode[STIX]{x1D60E}_{pq}\unicode[STIX]{x2202}_{k}u_{j}^{\prime }\rangle }_{(\text{b})}+\overline{\unicode[STIX]{x1D60D}}_{jp}\overline{\unicode[STIX]{x1D60D}}_{qk}^{\mathsf{T}}\underbrace{\langle \unicode[STIX]{x1D60E}_{pq}\unicode[STIX]{x2202}_{k}u_{i}^{\prime }\rangle }_{(\text{c})}.\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

Term (a) is the averaged turbulent transport and terms (b), (c) describe the mean stretching and rotation of the polymer arising due to the gradients in the fluctuating velocity field. The right-hand side of (3.16) is the cumulative effect of the turbulent fluctuations on the mean balance. The mean balance (3.16) can also be written as

(3.17) $$\begin{eqnarray}\displaystyle \overline{\unicode[STIX]{x1D648}}=2\,\text{sym}\,\{\overline{\unicode[STIX]{x1D640}}+\langle \unicode[STIX]{x1D642}\boldsymbol{\cdot }\unicode[STIX]{x1D640}^{\prime }\rangle -[(\unicode[STIX]{x2202}_{t}\overline{\unicode[STIX]{x1D641}}^{\mathsf{T}})+(\overline{\boldsymbol{u}}+\langle \unicode[STIX]{x1D642}\boldsymbol{\cdot }\boldsymbol{u}^{\prime }\rangle )\boldsymbol{\cdot }\unicode[STIX]{x1D735}\overline{\unicode[STIX]{x1D641}}^{\mathsf{T}}]\boldsymbol{\cdot }\overline{\unicode[STIX]{x1D641}}^{-\mathsf{T}}\}-\langle \boldsymbol{u}^{\prime }\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D642}\rangle , & & \displaystyle\end{eqnarray}$$

where $\overline{\unicode[STIX]{x1D640}}=\langle \unicode[STIX]{x1D640}\rangle$ , $\unicode[STIX]{x1D640}^{\prime }=\unicode[STIX]{x1D640}-\overline{\unicode[STIX]{x1D640}}$ , $\overline{\unicode[STIX]{x1D648}}=\langle \unicode[STIX]{x1D648}\rangle$ , and

(3.18) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D640} & \equiv & \displaystyle \overline{\unicode[STIX]{x1D641}}^{\mathsf{T}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}\boldsymbol{\cdot }\overline{\unicode[STIX]{x1D641}}^{-\mathsf{T}}\end{eqnarray}$$

(3.19) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D648} & \equiv & \displaystyle \overline{\unicode[STIX]{x1D641}}^{-1}\boldsymbol{\cdot }\unicode[STIX]{x1D64F}\boldsymbol{\cdot }\overline{\unicode[STIX]{x1D641}}^{-\mathsf{T}}.\end{eqnarray}$$

Here, the tensors $\unicode[STIX]{x1D640}$ and $\unicode[STIX]{x1D648}$ serve as modified velocity gradient and polymer stress tensors. Note that the invariants of $\unicode[STIX]{x1D735}\boldsymbol{u}$ and $\unicode[STIX]{x1D640}$ coincide. The equation (3.17) shows that the mean modified stress, $\overline{\unicode[STIX]{x1D648}}$ , is a function of the mean velocity gradient, $\overline{\unicode[STIX]{x1D640}}$ , the mean stretching due to turbulent velocity gradients $\langle \unicode[STIX]{x1D642}\boldsymbol{\cdot }\unicode[STIX]{x1D640}^{\prime }\rangle$ , and the turbulent advection of the fluctuating polymer deformation, $\langle \boldsymbol{u}^{\prime }\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D642}\rangle$ . Additionally, new terms appear due to the time-rate of change of $\overline{\unicode[STIX]{x1D641}}$ , and modified advection of $\overline{\unicode[STIX]{x1D641}}$ .

We can find the evolution equation for $\unicode[STIX]{x1D642}$ by substituting (3.11) into (2.3) and simplifying, which yields

(3.20) $$\begin{eqnarray}\displaystyle \frac{\text{D}\unicode[STIX]{x1D642}}{\text{D}t}=2\,\text{sym}(\unicode[STIX]{x1D642}\boldsymbol{\cdot }\unicode[STIX]{x1D646})-\unicode[STIX]{x1D648}, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D646}$ is given by

(3.21) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D646}\equiv \unicode[STIX]{x1D640}-\left(\overline{\unicode[STIX]{x1D641}}^{-1}\boldsymbol{\cdot }\frac{\text{D}\overline{\unicode[STIX]{x1D641}}}{\text{D}t}\right)^{\mathsf{T}}, & & \displaystyle\end{eqnarray}$$

and represents the modified velocity gradient augmented with an additional stretching that arises due to the decomposition. The expression (3.20) is more general than considered here; an equivalent expression can be derived when the nominal tensor, $\overline{\unicode[STIX]{x1D63E}}$ , is not equal to the mean conformation tensor.

The higher-dimensional nature of $\unicode[STIX]{x1D642}$ makes the quantification of the fluctuating turbulent polymer deformation a more difficult task. We will examine the elastic potential energy as a method to evaluate this deformation in the next subsection and then introduce more general scalar characterizations of $\unicode[STIX]{x1D642}$ in the next section.

3.2 Elastic energy and its relation to $\unicode[STIX]{x1D642}$

The turbulent mean polymer configuration is not the thermodynamic equilibrium state, and thus $\unicode[STIX]{x1D642}$ alone is not sufficient to fully determine thermodynamic quantities such as the elastic potential energy, $\unicode[STIX]{x1D700}_{\unicode[STIX]{x1D713}}(\unicode[STIX]{x1D63E})$ . For example, Beris & Edwards (Reference Beris and Edwards1994) define $\unicode[STIX]{x1D700}_{\unicode[STIX]{x1D713}}(\unicode[STIX]{x1D63E})$ for an Oldroyd-B model as

(3.22) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D700}_{\unicode[STIX]{x1D713}}(\unicode[STIX]{x1D63E})=\int _{\unicode[STIX]{x1D6FA}}\unicode[STIX]{x1D713}\,\text{tr}(\overline{\unicode[STIX]{x1D63E}}\boldsymbol{\cdot }\unicode[STIX]{x1D642})\,\text{d}^{3}\boldsymbol{x}, & & \displaystyle\end{eqnarray}$$

where we have rewritten the expression in terms of $\unicode[STIX]{x1D642}$ and $\overline{\unicode[STIX]{x1D63E}}$ by setting $\overline{\unicode[STIX]{x1D641}}=\overline{\unicode[STIX]{x1D63E}}^{1/2}$ and using the cyclic property of the trace to obtain $\text{tr}\,\unicode[STIX]{x1D63E}=\text{tr}(\overline{\unicode[STIX]{x1D63E}}\boldsymbol{\cdot }\unicode[STIX]{x1D642})$ . Here, $\unicode[STIX]{x1D6FA}$ is the spatial domain, and the scalar function $\unicode[STIX]{x1D713}(\boldsymbol{x})$ is proportional to the polymer elastic constant times the elasticity density.

The mean elastic potential, $\langle \unicode[STIX]{x1D700}_{\unicode[STIX]{x1D713}}(\unicode[STIX]{x1D63E})\rangle$ , for the Oldroyd-B model has the convenient property that it can be written solely in terms of the mean conformation tensor: $\langle \unicode[STIX]{x1D700}_{\unicode[STIX]{x1D713}}(\unicode[STIX]{x1D63E})\rangle =\unicode[STIX]{x1D700}_{\unicode[STIX]{x1D713}}(\overline{\unicode[STIX]{x1D63E}})$ . However, the contribution of $\unicode[STIX]{x1D642}$ in $\unicode[STIX]{x1D700}_{\unicode[STIX]{x1D713}}(\unicode[STIX]{x1D63E})$ cannot be fully separated from that of $\overline{\unicode[STIX]{x1D63E}}$ because $\text{tr}\,\unicode[STIX]{x1D63C}\boldsymbol{\cdot }\unicode[STIX]{x1D63D}\neq \text{tr}\,\unicode[STIX]{x1D63C}\,\text{tr}\,\unicode[STIX]{x1D63D}$ . Nonetheless, insight into the role of the different contributions can be obtained by using a trace inequality proved by Mori (Reference Mori1988), which yields

(3.23a,b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D700}_{\unicode[STIX]{x1D713}_{3}}(\unicode[STIX]{x1D642})\leqslant \unicode[STIX]{x1D700}_{\unicode[STIX]{x1D713}}(\unicode[STIX]{x1D63E})\leqslant \unicode[STIX]{x1D700}_{\unicode[STIX]{x1D713}_{1}}(\unicode[STIX]{x1D642}),\quad \unicode[STIX]{x1D713}_{i}\equiv \unicode[STIX]{x1D713}\unicode[STIX]{x1D70E}_{i}(\overline{\unicode[STIX]{x1D63E}}), & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D70E}_{i}(\unicode[STIX]{x1D63C})$ denotes the $i$ th largest eigenvalue of a tensor $\unicode[STIX]{x1D63C}$ . In terms of the bounds in (3.23), the contribution of $\overline{\unicode[STIX]{x1D63E}}$ to $\unicode[STIX]{x1D700}_{\unicode[STIX]{x1D713}}(\unicode[STIX]{x1D63E})$ is equivalent to a modification of the local elasticity density or the elastic constant.

In other constitutive models, the contribution to the elastic potential energy from the mean polymer deformation is more difficult to separate. For example, the elastic potential energy for the FENE-P model (Beris & Edwards Reference Beris and Edwards1994) is

(3.24) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D700}_{\unicode[STIX]{x1D713}}(\unicode[STIX]{x1D63E};L_{\max })=-\int _{\unicode[STIX]{x1D6FA}}\unicode[STIX]{x1D713}L_{\max }^{2}\log \left(1-\frac{\text{tr}(\overline{\unicode[STIX]{x1D63E}}\boldsymbol{\cdot }\unicode[STIX]{x1D642})}{L_{\max }^{2}}\right)\text{d}^{3}\boldsymbol{x}, & & \displaystyle\end{eqnarray}$$

where $L_{\max }$ , $\unicode[STIX]{x1D6FA}$ and $\unicode[STIX]{x1D713}$ are as defined before. Here, the mean elastic potential energy, $\langle \unicode[STIX]{x1D700}_{\unicode[STIX]{x1D713}}(\unicode[STIX]{x1D63E};L_{\max })\rangle$ , cannot be separated from $\unicode[STIX]{x1D642}$ as in the Oldroyd-B model because, according to (3.24), $\langle \unicode[STIX]{x1D700}_{\unicode[STIX]{x1D713}}(\unicode[STIX]{x1D63E})\rangle \neq \unicode[STIX]{x1D700}_{\unicode[STIX]{x1D713}}(\overline{\unicode[STIX]{x1D63E}})$ . However, we can again bound $\unicode[STIX]{x1D700}_{\unicode[STIX]{x1D713}}(\unicode[STIX]{x1D63E};L_{\max })$ as

(3.25a,b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D700}_{\unicode[STIX]{x1D713}}(\unicode[STIX]{x1D642};L_{\max ,3})\leqslant \unicode[STIX]{x1D700}_{\unicode[STIX]{x1D713}}(\unicode[STIX]{x1D63E};L_{\max })\leqslant \unicode[STIX]{x1D700}_{\unicode[STIX]{x1D713}}(\unicode[STIX]{x1D642};L_{\max ,1}),\quad L_{\max ,i}\equiv L_{\max }/(\unicode[STIX]{x1D70E}_{i}(\overline{\unicode[STIX]{x1D63E}}))^{1/2}. & & \displaystyle\end{eqnarray}$$

In terms of the bounds in (3.25), the contribution of $\overline{\unicode[STIX]{x1D63E}}$ in $\unicode[STIX]{x1D700}_{\unicode[STIX]{x1D713}}(\unicode[STIX]{x1D63E};L_{\max })$ is equivalent to a modification of the local polymer maximum extensibility.

Elastic energy may itself be insufficient to fully characterize the polymer deformation. For example, in both Oldroyd-B and FENE-P models, the elastic energy is equal for all conformation tensors that are given by

(3.26) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D63E}=[\text{diag}(\unicode[STIX]{x1D6FC}_{1}+\unicode[STIX]{x1D6FF},\unicode[STIX]{x1D6FC}_{2}-\unicode[STIX]{x1D6FF},\unicode[STIX]{x1D6FC}_{3})]_{\unicode[STIX]{x1D64C}},\quad 0\leqslant \unicode[STIX]{x1D6FF}<\unicode[STIX]{x1D6FC}_{2}, & & \displaystyle\end{eqnarray}$$

where $\text{diag}(\unicode[STIX]{x1D719}_{1},\unicode[STIX]{x1D719}_{2},\unicode[STIX]{x1D719}_{3})$ denotes a diagonal tensor with the $i$ th diagonal component given by $\unicode[STIX]{x1D719}_{i}$ , and $\unicode[STIX]{x1D6FC}_{1}\geqslant \unicode[STIX]{x1D6FC}_{2}\geqslant \unicode[STIX]{x1D6FC}_{3}>0$ and $\unicode[STIX]{x1D64C}\in \boldsymbol{S}\boldsymbol{O}_{3}$ are fixed. Even though the trace is fixed, the volume of the deformation ellipsoid changes with $\unicode[STIX]{x1D6FF}$ , and is given by

(3.27) $$\begin{eqnarray}\displaystyle \text{det}\,\unicode[STIX]{x1D63E}=\unicode[STIX]{x1D6FC}_{1}\unicode[STIX]{x1D6FC}_{2}\unicode[STIX]{x1D6FC}_{3}+(\unicode[STIX]{x1D6FC}_{1}-\unicode[STIX]{x1D6FC}_{2})\unicode[STIX]{x1D6FC}_{3}\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6FC}_{3}. & & \displaystyle\end{eqnarray}$$

In addition, since the governing equations are not Hamiltonian (Beris & Edwards Reference Beris and Edwards1994), the elastic potential energy only provides a partial characterization of the dynamics underlying the polymer deformation. Due to the above limitations of the elastic energy, and its dependence on the choice of the particular constitutive model, we instead develop an approach to characterizing the polymer deformation using the inherent geometric structure underlying $\unicode[STIX]{x1D642}$ . This approach, introduced in the next section, is mathematically rigorous and can be applied to any positive definite tensor.

4.1 Geodesic curves and distances between positive definite tensors

The set $\boldsymbol{P}\boldsymbol{o}\boldsymbol{s}_{3}$ is a Cartan–Hadamard manifold: it is a simply connected, geodesically complete Riemannian manifold with seminegative curvature (Lang Reference Lang2001). We summarize this characterization in the present section in order to develop a notion of distances between positive definite tensors that will be used to formulate appropriate scalar measures of the fluctuating conformation tensor $\unicode[STIX]{x1D642}$ . Details on the Riemannian structure of $\boldsymbol{P}\boldsymbol{o}\boldsymbol{s}_{3}$ and theorems leading to the results used in this section are presented in the appendix.

Consider two matrices $\unicode[STIX]{x1D653},\unicode[STIX]{x1D654}\in \boldsymbol{P}\boldsymbol{o}\boldsymbol{s}_{3}$ . In the set of all curves along the manifold $\boldsymbol{P}\boldsymbol{o}\boldsymbol{s}_{3}$ connecting $\unicode[STIX]{x1D653}$ and $\unicode[STIX]{x1D654}$ , there exists a unique curve that minimizes the distance between $\unicode[STIX]{x1D653}$ and $\unicode[STIX]{x1D654}$ with respect to the Riemannian metric on $\boldsymbol{P}\boldsymbol{o}\boldsymbol{s}_{3}$ , i.e. there exists a $\unicode[STIX]{x1D64B}(r)$ with $\unicode[STIX]{x1D64B}(0)=\unicode[STIX]{x1D653}$ and $\unicode[STIX]{x1D64B}(1)=\unicode[STIX]{x1D654}$ that uniquely minimizes the distance traversed along the manifold between $\unicode[STIX]{x1D653}$ and $\unicode[STIX]{x1D654}$ . We call this curve the geodesic curve along the manifold and it is given by

(4.3) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D653}\#_{r}\unicode[STIX]{x1D654}=[([\unicode[STIX]{x1D654}]_{\unicode[STIX]{x1D653}^{-1/2}})^{r}]_{\unicode[STIX]{x1D653}^{1/2}},\quad 0\leqslant r\leqslant 1. & & \displaystyle\end{eqnarray}$$

The geodesic distance associated with the geodesic curve between $\unicode[STIX]{x1D653}$ and $\unicode[STIX]{x1D654}$ is the minimum separation between them along the manifold and is given by

(4.4) $$\begin{eqnarray}\displaystyle d(\unicode[STIX]{x1D653},\unicode[STIX]{x1D654})=\left[\mathop{\sum }_{i=1}^{3}(\log \unicode[STIX]{x1D70E}_{i}(\unicode[STIX]{x1D653}^{-1}\boldsymbol{\cdot }\unicode[STIX]{x1D654}))^{2}\right]^{1/2}=\sqrt{\text{tr}\log ^{2}(\unicode[STIX]{x1D653}^{-1/2}\boldsymbol{\cdot }\unicode[STIX]{x1D654}\boldsymbol{\cdot }\unicode[STIX]{x1D653}^{-1/2})}. & & \displaystyle\end{eqnarray}$$

The distance, $d(\unicode[STIX]{x1D653},\unicode[STIX]{x1D654})$ , is affine invariant, i.e. $d(\unicode[STIX]{x1D653},\unicode[STIX]{x1D654})=d([\unicode[STIX]{x1D653}]_{\unicode[STIX]{x1D63C}},[\unicode[STIX]{x1D654}]_{\unicode[STIX]{x1D63C}})$ for all $\unicode[STIX]{x1D63C}\in \boldsymbol{G}\boldsymbol{L}_{3}$ .

Geodesic curves and distances along a Riemannian manifold are analogous to straight lines and distances in Euclidean space. In the case of $\boldsymbol{P}\boldsymbol{o}\boldsymbol{s}_{3}$ , the analogy can be taken quite far because $\boldsymbol{P}\boldsymbol{o}\boldsymbol{s}_{3}$ is geodesically complete; a geodesic connecting any two points on the manifold parametrized by $r$ can be arbitrarily extended letting $r\in \mathbb{R}$ . For example, $\unicode[STIX]{x1D653}\#_{r}\unicode[STIX]{x1D654}$ with $r\in [0,a]$ , is a geodesic between $\unicode[STIX]{x1D653}\#_{0}\unicode[STIX]{x1D654}$ and $\unicode[STIX]{x1D653}\#_{a}\unicode[STIX]{x1D654}$ for all $[0,a]\subseteq \mathbb{R}$ . Furthermore, for each $r>0$ , we have

(4.5) $$\begin{eqnarray}\displaystyle d(\unicode[STIX]{x1D653},\unicode[STIX]{x1D653}\#_{r}\unicode[STIX]{x1D654})=r\,d(\unicode[STIX]{x1D653},\unicode[STIX]{x1D654}). & & \displaystyle\end{eqnarray}$$

We now illustrate the geodesic distance derived above using two specific examples.

1. (i) Isotropic tensors: Let $\unicode[STIX]{x1D653}=a\unicode[STIX]{x1D644}$ and $\unicode[STIX]{x1D654}=b\unicode[STIX]{x1D644}$ be elements of the one-dimensional sub-manifold of $\boldsymbol{P}\boldsymbol{o}\boldsymbol{s}_{3}$ consisting of the isotropic tensors. The geodesic path joining $\unicode[STIX]{x1D653}$ and $\unicode[STIX]{x1D654}$ is given by

   (4.6) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D653}\#_{r}\unicode[STIX]{x1D654}=(a^{1-r}b^{r})\unicode[STIX]{x1D644} & & \displaystyle\end{eqnarray}$$
   and the geodesic distance is given by $d(\unicode[STIX]{x1D653},\unicode[STIX]{x1D654})=\sqrt{3}\log (b/a)$ .

   Notice that $a^{1-r}b^{r}$ , which appears in (4.6), is a generalized geometric mean of $a$ and $b$ with the classical definition realized at $r=1/2$ . It can be shown that a similar interpretation is admissible when $\unicode[STIX]{x1D653}$ and $\unicode[STIX]{x1D654}$ are not isotropic (Bhatia Reference Bhatia2015). This fact has formed the basis of efforts to formulate alternative definitions of statistical quantities such as means and covariances so that they conform to the geometric structure of $\boldsymbol{P}\boldsymbol{o}\boldsymbol{s}_{3}$ (Pennec, Fillard & Ayache Reference Pennec, Fillard and Ayache2006; Fletcher & Joshi Reference Fletcher and Joshi2007).

2. (ii) Tensors differing by a rotation: Consider $\unicode[STIX]{x1D653}$ and $\unicode[STIX]{x1D654}=[\unicode[STIX]{x1D653}]_{\unicode[STIX]{x1D64D}}$ for $\unicode[STIX]{x1D64D}\in \boldsymbol{S}\boldsymbol{O}_{3}$ . The geodesic joining $\unicode[STIX]{x1D653}$ and $\unicode[STIX]{x1D654}$ is given by

   (4.7) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D653}\#_{r}\unicode[STIX]{x1D654}=[([\unicode[STIX]{x1D653}]_{\unicode[STIX]{x1D653}^{-1/2}\boldsymbol{\cdot }\unicode[STIX]{x1D64D}})^{r}]_{\unicode[STIX]{x1D653}^{1/2}}. & & \displaystyle\end{eqnarray}$$
   The distance between $\unicode[STIX]{x1D653}$ and $\unicode[STIX]{x1D654}$ is then bounded as
   (4.8) $$\begin{eqnarray}\displaystyle 0\leqslant d(\unicode[STIX]{x1D653},\unicode[STIX]{x1D654})\leqslant \sqrt{3}\min \,\left\{\max _{i}\left\{\left|\log \left(\frac{\unicode[STIX]{x1D70E}_{i}(\unicode[STIX]{x1D654})}{\unicode[STIX]{x1D70E}_{3}(\unicode[STIX]{x1D653})}\right)\right|\right\},\max _{i}\left\{\left|\log \left(\frac{\unicode[STIX]{x1D70E}_{1}(\unicode[STIX]{x1D654})}{\unicode[STIX]{x1D70E}_{i}(\unicode[STIX]{x1D653})}\right)\right|\right\}\right\}, & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$
   where the lower bound is achieved for $\unicode[STIX]{x1D64D}=\unicode[STIX]{x1D644}$ . The upper bound in (4.8) suggests that a differential rotation of a second-order tensor, $\unicode[STIX]{x1D653}$ , leads to an excursion along $\boldsymbol{P}\boldsymbol{o}\boldsymbol{s}_{3}$ with a path length that depends on the anisotropy of $\unicode[STIX]{x1D653}$ . For isotropic $\unicode[STIX]{x1D653}$ , the path length is zero and it otherwise increases with increasing anisotropy.

The geometry of $\boldsymbol{P}\boldsymbol{o}\boldsymbol{s}_{3}$ and the properties discussed above are next used to define scalar measures that characterize the turbulent fluctuations in $\unicode[STIX]{x1D642}$ .

4.2 Scalar measures of the fluctuating conformation tensor

In this subsection we introduce scalar measures that can be used to quantify the fluctuating polymer deformation represented by $\unicode[STIX]{x1D642}$ . In what follows, we will denote the matrix logarithm of $\unicode[STIX]{x1D642}$ as $\boldsymbol{{\mathcal{G}}}$ , i.e.

(4.9) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D642}=\mathop{\sum }_{k=0}^{\infty }\frac{\boldsymbol{{\mathcal{G}}}^{k}}{k!}\equiv \text{e}^{\boldsymbol{{\mathcal{G}}}}. & & \displaystyle\end{eqnarray}$$

The matrix logarithm is guaranteed to exist and is unique since $\unicode[STIX]{x1D642}$ is positive definite. A key point to note is that the eigenvalues of $\boldsymbol{{\mathcal{G}}}$ are the logarithms of the eigenvalues of $\unicode[STIX]{x1D642}$ .

4.2.1 Logarithmic volume ratio, $\unicode[STIX]{x1D701}$

Let $\unicode[STIX]{x1D6E4}_{i}=\unicode[STIX]{x1D70E}_{i}(\unicode[STIX]{x1D642})$ , for $i=1,2,3$ , be the eigenvalues of $\unicode[STIX]{x1D642}$ . Then $\log (\text{det}\,\unicode[STIX]{x1D642})=\log (\prod _{i=1}^{3}\unicode[STIX]{x1D6E4}_{i})=\sum _{i=1}^{3}\log \unicode[STIX]{x1D6E4}_{i}$ . We thus define the logarithmic volume ratio of the fluctuation, $\unicode[STIX]{x1D701}$ , as

(4.10) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D701}\equiv \text{tr}\,\boldsymbol{{\mathcal{G}}}=\log (\text{det}\,\unicode[STIX]{x1D642})=\log \left(\frac{\text{det}\,\unicode[STIX]{x1D63E}}{\text{det}\,\overline{\unicode[STIX]{x1D63E}}}\right). & & \displaystyle\end{eqnarray}$$

When $\unicode[STIX]{x1D701}=0$ , the mean and the instantaneous conformation tensors have the same volume; when $\unicode[STIX]{x1D701}$ is negative (positive), the instantaneous conformation tensor has a smaller (larger) volume than the volume of the mean. The logarithm ensures that there is no asymmetry between compressions and expansions with respect to the mean.

4.2.2 Squared distance from the mean, $\unicode[STIX]{x1D705}$

When $\unicode[STIX]{x1D63E}=\overline{\unicode[STIX]{x1D63E}}$ , we have $\unicode[STIX]{x1D642}=\unicode[STIX]{x1D644}$ . When $\unicode[STIX]{x1D63E}\neq \overline{\unicode[STIX]{x1D63E}}$ , we wish to consider the (appropriately defined) shortest distance between $\unicode[STIX]{x1D644}$ and $\unicode[STIX]{x1D642}$ as a measure of the magnitude of the fluctuation. The shortest path between $\unicode[STIX]{x1D644}$ and $\unicode[STIX]{x1D642}$ along the manifold $\boldsymbol{P}\boldsymbol{o}\boldsymbol{s}_{3}$ is given by the geodesic along $\boldsymbol{P}\boldsymbol{o}\boldsymbol{s}_{3}$ connecting $\unicode[STIX]{x1D644}$ and $\unicode[STIX]{x1D642}$ ,

(4.11) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D644}\#_{r}\unicode[STIX]{x1D642}=\unicode[STIX]{x1D642}^{r}=\text{e}^{\boldsymbol{{\mathcal{G}}}r}. & & \displaystyle\end{eqnarray}$$

The squared geodesic distance associated with this path is then

(4.12) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D705}\equiv \text{tr}\,\boldsymbol{{\mathcal{G}}}^{2}=d^{2}(\unicode[STIX]{x1D644},\unicode[STIX]{x1D642})=\mathop{\sum }_{i=1}^{3}(\log \unicode[STIX]{x1D6E4}_{i})^{2}, & & \displaystyle\end{eqnarray}$$

where (4.12) follows from (4.4). Using (4.4), one can verify that $d^{2}(\unicode[STIX]{x1D644},\unicode[STIX]{x1D642})=d^{2}(\unicode[STIX]{x1D644},\unicode[STIX]{x1D642}^{-1})$ and thus the squared distance measure treats both expansions and compressions with respect to the mean similarly. The affine-invariance property, furthermore, ensures that

(4.13) $$\begin{eqnarray}\displaystyle d^{2}(\unicode[STIX]{x1D644},\unicode[STIX]{x1D642})=d^{2}([\unicode[STIX]{x1D644}]_{\unicode[STIX]{x1D63C}},[\unicode[STIX]{x1D642}]_{\unicode[STIX]{x1D63C}}) & & \displaystyle\end{eqnarray}$$

for all $\unicode[STIX]{x1D63C}\in \boldsymbol{G}\boldsymbol{L}_{3}$ . In particular, with $\unicode[STIX]{x1D63C}=\overline{\unicode[STIX]{x1D641}}$ , we obtain

(4.14) $$\begin{eqnarray}\displaystyle d^{2}(\unicode[STIX]{x1D644},\unicode[STIX]{x1D642})=d^{2}(\overline{\unicode[STIX]{x1D63E}},[\unicode[STIX]{x1D642}]_{\overline{\unicode[STIX]{x1D641}}})=d^{2}(\overline{\unicode[STIX]{x1D63E}},\unicode[STIX]{x1D63E})=d^{2}(\overline{\unicode[STIX]{x1D63E}}^{-1},\unicode[STIX]{x1D63E}^{-1}), & & \displaystyle\end{eqnarray}$$

which exhibits the highly desirable property that the squared distance between $\unicode[STIX]{x1D63E}$ and $\overline{\unicode[STIX]{x1D63E}}$ is equal to the squared distance between $\unicode[STIX]{x1D644}$ and $\unicode[STIX]{x1D642}$ . A further consequence of the affine-invariance property is that $d^{2}(\unicode[STIX]{x1D644},\unicode[STIX]{x1D642})$ is independent of the choice of the rotation $\unicode[STIX]{x1D64D}\in \boldsymbol{S}\boldsymbol{O}_{3}$ in (3.4).

The path between $\overline{\unicode[STIX]{x1D63E}}$ and $\unicode[STIX]{x1D63E}$ along $\boldsymbol{P}\boldsymbol{o}\boldsymbol{s}_{3}$ is given by

(4.15) $$\begin{eqnarray}\displaystyle \overline{\unicode[STIX]{x1D63E}}\#_{r}\unicode[STIX]{x1D63E}=[([\unicode[STIX]{x1D63E}]_{\overline{\unicode[STIX]{x1D63E}}^{-1/2}})^{r}]_{\overline{\unicode[STIX]{x1D63E}}^{1/2}}, & & \displaystyle\end{eqnarray}$$

which reduces to

(4.16) $$\begin{eqnarray}\displaystyle \overline{\unicode[STIX]{x1D63E}}\#_{r}\unicode[STIX]{x1D63E}=[\unicode[STIX]{x1D642}^{r}]_{\overline{\unicode[STIX]{x1D641}}} & & \displaystyle\end{eqnarray}$$

when $\unicode[STIX]{x1D64D}=\unicode[STIX]{x1D644}$ in (3.4). The choice $\unicode[STIX]{x1D64D}=\unicode[STIX]{x1D644}$ is then natural in the sense that it allows the path along the manifold between $\overline{\unicode[STIX]{x1D63E}}$ and $\unicode[STIX]{x1D63E}$ , whose distance is a measure of the fluctuation, to be described using only $\overline{\unicode[STIX]{x1D641}}$ and $\unicode[STIX]{x1D642}$ .

We next consider realizability in the $(\unicode[STIX]{x1D701},\unicode[STIX]{x1D705})$ plane. Since $\boldsymbol{{\mathcal{G}}}$ is symmetric, its eigenvalues must be real, i.e. the eigenvalues must together belong in $\mathbb{R}^{3}$ . In the $\mathbb{R}^{3}$ space of eigenvalues of $\boldsymbol{{\mathcal{G}}}$ , surfaces of constant $\unicode[STIX]{x1D701}$ are planes, and surfaces of constant $\unicode[STIX]{x1D705}$ are spheres. A particular choice of $\unicode[STIX]{x1D701}$ and $\unicode[STIX]{x1D705}$ is realizable only if the plane and sphere intersect. The coordinates along the intersecting circle of the sphere satisfy

(4.17) $$\begin{eqnarray}\displaystyle \cos \unicode[STIX]{x1D703}\sin \unicode[STIX]{x1D719}+\sin \unicode[STIX]{x1D703}\sin \unicode[STIX]{x1D719}+\cos \unicode[STIX]{x1D719}=\frac{\unicode[STIX]{x1D701}}{\sqrt{\unicode[STIX]{x1D705}}},\quad \unicode[STIX]{x1D703}\in [0,2\unicode[STIX]{x03C0}),\,\unicode[STIX]{x1D719}\in [0,\unicode[STIX]{x03C0}], & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D719}$ is the inclination angle and $\unicode[STIX]{x1D703}$ is the azimuthal angle, in a spherical coordinate representation of $\mathbb{R}^{3}$ . The physically realizable region in the $(\unicode[STIX]{x1D705},\unicode[STIX]{x1D701})$ plane is thus given by

(4.18) $$\begin{eqnarray}\displaystyle -\sqrt{3\unicode[STIX]{x1D705}}\leqslant \unicode[STIX]{x1D701}\leqslant \sqrt{3\unicode[STIX]{x1D705}}. & & \displaystyle\end{eqnarray}$$

When $\unicode[STIX]{x1D705}=(1/3)\unicode[STIX]{x1D701}^{2}$ , the circle of intersection reduces to a point and $\boldsymbol{{\mathcal{G}}}$ consequently has only two independent tensor invariants, $\unicode[STIX]{x1D701}$ and $\unicode[STIX]{x1D705}$ . The angles that maximize the left-hand side of (4.17) are given by $\unicode[STIX]{x1D703}_{\max }=\unicode[STIX]{x03C0}/4$ , $\unicode[STIX]{x1D719}_{\max }=\arctan \sqrt{2}$ . At $(\unicode[STIX]{x1D703},\unicode[STIX]{x1D719})=(\unicode[STIX]{x1D703}_{\max },\unicode[STIX]{x1D719}_{\max })$ , it is readily verified that $\unicode[STIX]{x1D705}=(1/3)\unicode[STIX]{x1D701}^{2}$ and also that the eigenvalues of $\boldsymbol{{\mathcal{G}}}$ are all equal. Thus $\boldsymbol{{\mathcal{G}}}$ , and hence $\unicode[STIX]{x1D642}$ , is isotropic at the realizability bounds.

4.2.3 Anisotropy index, $\unicode[STIX]{x1D709}$

Following the approach taken by Moakher & Batchelor (Reference Moakher and Batchelor2006), we define the anisotropy index, $\unicode[STIX]{x1D709}$ , of $\unicode[STIX]{x1D642}$ as the squared geodesic distance between $\unicode[STIX]{x1D642}$ and the closest isotropic tensor,

(4.19) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D709}\equiv \inf _{a}\,d^{2}(a\unicode[STIX]{x1D644},\unicode[STIX]{x1D642})=\inf _{a}\,\text{tr}\,(\boldsymbol{{\mathcal{G}}}-(\log a)\unicode[STIX]{x1D644})^{2}. & & \displaystyle\end{eqnarray}$$

By differentiation, we find that $a^{3}=\prod _{i=1}^{3}\unicode[STIX]{x1D70E}_{i}(\unicode[STIX]{x1D642})=\text{det}\,\unicode[STIX]{x1D642}$ is a minimizing stationary point of (4.19) and hence the closest isotropic tensor to $\unicode[STIX]{x1D642}$ along $\boldsymbol{P}\boldsymbol{o}\boldsymbol{s}_{3}$ is $(\sqrt[3]{\text{det}\,\unicode[STIX]{x1D642}})\unicode[STIX]{x1D644}$ . We then have

(4.20) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D709}=d^{2}((\sqrt[3]{\text{det}\,\unicode[STIX]{x1D642}})\unicode[STIX]{x1D644},\unicode[STIX]{x1D642})=\unicode[STIX]{x1D705}-{\textstyle \frac{1}{3}}\unicode[STIX]{x1D701}^{2}. & & \displaystyle\end{eqnarray}$$

Notice that $\unicode[STIX]{x1D712}=0$ if and only if $\unicode[STIX]{x1D701}^{2}=3\unicode[STIX]{x1D705}$ . But since we already showed that $\boldsymbol{{\mathcal{G}}}$ , and hence $\unicode[STIX]{x1D642}$ , are isotropic at the bound $\unicode[STIX]{x1D701}^{2}=3\unicode[STIX]{x1D705}$ , it follows that $\unicode[STIX]{x1D709}=0$ only for isotropic tensors.

Batchelor et al. (Reference Batchelor, Moakher, Atkinson, Calamante and Connelly2005) first introduced the index $\sqrt{\unicode[STIX]{x1D709}}$ for characterizing positive definite diffusion tensors measured in magnetic resonance imaging. The index is analogous to the ‘fractional anisotropy index’ that is commonly used in turbulence and which provides the Euclidean distance to the closest isotropic tensor,

(4.21) $$\begin{eqnarray}\displaystyle \frac{\Vert \unicode[STIX]{x1D642}-(\text{tr}\,\unicode[STIX]{x1D642}/3)\unicode[STIX]{x1D644}\Vert _{F}}{\Vert \unicode[STIX]{x1D642}\Vert _{F}}, & & \displaystyle\end{eqnarray}$$

where $\Vert \unicode[STIX]{x1D63C}\Vert _{F}=\text{tr}\,(\unicode[STIX]{x1D63C}^{\mathsf{T}}\boldsymbol{\cdot }\unicode[STIX]{x1D63C})$ indicates the Frobenius norm of matrix $\unicode[STIX]{x1D63C}$ . A review of anisotropy measures is available in Moakher & Batchelor (Reference Moakher and Batchelor2006).

The three scalar measures presented above can be used together to obtain a better understanding of the fluctuations in the conformation tensor. The logarithmic volume ratio, $\unicode[STIX]{x1D701}$ , is positive (negative) for volumetric expansions (contractions) with respect to the mean. However, $\unicode[STIX]{x1D701}=0$ does not necessarily imply no deformation, since $\text{det}(\text{det}(\unicode[STIX]{x1D642})\,\unicode[STIX]{x1D63C})=\text{det}(\unicode[STIX]{x1D642})$ for all $\unicode[STIX]{x1D63C}$ with unit determinant. The squared geodesic distance to the identity, $\unicode[STIX]{x1D705}$ , helps distinguish such cases since $\unicode[STIX]{x1D705}=0$ only when $\unicode[STIX]{x1D642}=\unicode[STIX]{x1D644}$ ( $\unicode[STIX]{x1D63E}=\overline{\unicode[STIX]{x1D63E}}$ ). Finally, the anisotropy index, $\unicode[STIX]{x1D709}$ , provides a quantification of the deviation of the shape of the polymer from the shape of the mean conformation tensor because it is a measure of the distance from $\unicode[STIX]{x1D642}$ to the closest isotropic tensor, or equivalently, the minimizing distance between $\unicode[STIX]{x1D63E}$ and $a\overline{\unicode[STIX]{x1D63E}}$ over all $a>0$ .

We next derive evolution equations for the scalar measures presented above and for the particular case when $\overline{\unicode[STIX]{x1D63E}}=\langle \unicode[STIX]{x1D63E}\rangle$ .

4.3 Evolution equations for $\unicode[STIX]{x1D701}$ , $\unicode[STIX]{x1D705}$ and $\unicode[STIX]{x1D709}$

Since $\unicode[STIX]{x1D701}$ , $\unicode[STIX]{x1D705}$ and $\unicode[STIX]{x1D709}$ are scalar characterizations of $\unicode[STIX]{x1D642}$ , one need only evolve $\unicode[STIX]{x1D642}$ (or equivalently, $\unicode[STIX]{x1D63E}$ ) to obtain the field-valued $\unicode[STIX]{x1D701}$ , $\unicode[STIX]{x1D705}$ and $\unicode[STIX]{x1D709}$ . Nevertheless, it is of interest to mathematically evaluate the evolution equations of these scalar measures separately in order to find the quantities that contribute to their dynamics.

Using (3.20) and the relationship $\text{tr}\,\boldsymbol{{\mathcal{G}}}^{n}=\sum _{i=1}^{3}(\log \unicode[STIX]{x1D6E4}_{i})^{n}$ , we can derive the following equations for the fluctuating scalar measures:

(4.22a-c ) $$\begin{eqnarray}\displaystyle \frac{\text{D}\unicode[STIX]{x1D701}}{\text{D}t}=\text{tr}\,\boldsymbol{{\mathcal{D}}},\quad \frac{1}{2}\frac{\text{D}\unicode[STIX]{x1D705}}{\text{D}t}=\text{tr}\,(\boldsymbol{{\mathcal{D}}}\boldsymbol{\cdot }\boldsymbol{{\mathcal{G}}}),\quad \frac{1}{2}\frac{\text{D}\unicode[STIX]{x1D709}}{\text{D}t}=\text{tr}\,(\boldsymbol{{\mathcal{D}}}\boldsymbol{\cdot }\text{dev}\,\boldsymbol{{\mathcal{G}}}), & & \displaystyle\end{eqnarray}$$

where $\text{dev}\,\boldsymbol{{\mathcal{G}}}=\boldsymbol{{\mathcal{G}}}-(\text{tr}\,\boldsymbol{{\mathcal{G}}}/3)\unicode[STIX]{x1D644}$ is the deviatoric part of $\boldsymbol{{\mathcal{G}}}$ , and $\boldsymbol{{\mathcal{D}}}$ is defined as

(4.23) $$\begin{eqnarray}\displaystyle \boldsymbol{{\mathcal{D}}} & \equiv 2\,\text{sym}\,\unicode[STIX]{x1D646}-\unicode[STIX]{x1D648}\boldsymbol{\cdot }\text{e}^{-\boldsymbol{{\mathcal{G}}}}. & \displaystyle\end{eqnarray}$$

The derivation of (4.22), omitted here for brevity, closely follows the procedure used by Vaithianathan & Collins (Reference Vaithianathan and Collins2003) to obtain evolution equations for the continuous eigendecomposition of $\unicode[STIX]{x1D63E}$ .

The Cauchy–Schwarz inequality can be used to show that

(4.24a,b ) $$\begin{eqnarray}\displaystyle \left|\frac{\text{D}\sqrt{\unicode[STIX]{x1D705}}}{\text{D}t}\right|\leqslant \Vert \boldsymbol{{\mathcal{D}}}\Vert _{F},\quad \left|\frac{\text{D}\sqrt{\unicode[STIX]{x1D709}}}{\text{D}t}\right|\leqslant \Vert \boldsymbol{{\mathcal{D}}}\Vert _{F}. & & \displaystyle\end{eqnarray}$$

The bounds in (4.24) illustrate the role of the stretching and relaxation balance, $\boldsymbol{{\mathcal{D}}}$ , in bounding the growth of $\unicode[STIX]{x1D705}$ and $\unicode[STIX]{x1D709}$ . In the Reynolds-filtered case, $\boldsymbol{{\mathcal{D}}}$ can be simplified using (3.17) so that

(4.25) $$\begin{eqnarray}\displaystyle \boldsymbol{{\mathcal{D}}} & = & \displaystyle 2\,\text{sym}(\unicode[STIX]{x1D640}^{\prime }-\langle \unicode[STIX]{x1D642}\boldsymbol{\cdot }\unicode[STIX]{x1D640}^{\prime }\rangle -(\boldsymbol{u}^{\prime }-\langle \unicode[STIX]{x1D642}\boldsymbol{\cdot }\boldsymbol{u}^{\prime }\rangle )\boldsymbol{\cdot }\unicode[STIX]{x1D735}\overline{\unicode[STIX]{x1D641}}^{\mathsf{T}}\boldsymbol{\cdot }\overline{\unicode[STIX]{x1D641}}^{-\mathsf{T}})+\langle \boldsymbol{u}^{\prime }\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D642}\rangle \nonumber\\ \displaystyle & & \displaystyle -\,(\unicode[STIX]{x1D648}\boldsymbol{\cdot }\unicode[STIX]{x1D642}^{-1}-\overline{\unicode[STIX]{x1D648}}),\end{eqnarray}$$

which shows that the turbulence intensity of the fluctuating conformation tensor, as measured by $\unicode[STIX]{x1D705}$ , is not directly affected by the mean velocity gradient tensor $\unicode[STIX]{x1D735}\overline{\boldsymbol{u}}$ . The contribution of $\unicode[STIX]{x1D735}\overline{\boldsymbol{u}}$ to $\unicode[STIX]{x1D705}$ is captured indirectly through $\overline{\unicode[STIX]{x1D641}}$ , which is determined based on the mean balance.

According to (4.25), the tensor $\boldsymbol{{\mathcal{D}}}$ consists of a stretching component: $2\,\text{sym}(\unicode[STIX]{x1D640}^{\prime }-\langle \unicode[STIX]{x1D642}\boldsymbol{\cdot }\unicode[STIX]{x1D640}^{\prime }\rangle )$ , a component that arises due to gradients in $\overline{\unicode[STIX]{x1D641}}$ and represents modified advection of $\overline{\unicode[STIX]{x1D641}}$ : $-2\,\text{sym}[(\boldsymbol{u}^{\prime }-\langle \unicode[STIX]{x1D642}\boldsymbol{\cdot }\boldsymbol{u}^{\prime }\rangle )\boldsymbol{\cdot }\unicode[STIX]{x1D735}\overline{\unicode[STIX]{x1D641}}]$ , a component that comprises mean advection of $\unicode[STIX]{x1D642}$ by the fluctuating velocity field: $\langle \boldsymbol{u}^{\prime }\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D642}\rangle$ and finally a component that resembles a fluctuating relaxation contribution: $-(\unicode[STIX]{x1D648}\boldsymbol{\cdot }\unicode[STIX]{x1D642}^{-1}-\overline{\unicode[STIX]{x1D648}})$ .

4.3.1 Reynolds filtering the evolution equations

As a first-order statistical characterization of the fluctuating quantities, $\unicode[STIX]{x1D701}$ , $\unicode[STIX]{x1D705}$ and $\unicode[STIX]{x1D709}$ , we will consider their filtered or averaged values,

(4.26a-c ) $$\begin{eqnarray}\displaystyle \overline{\unicode[STIX]{x1D701}}\equiv \langle \unicode[STIX]{x1D701}\rangle ,\quad \overline{\unicode[STIX]{x1D705}}\equiv \langle \unicode[STIX]{x1D705}\rangle ,\quad \overline{\unicode[STIX]{x1D709}}\equiv \langle \unicode[STIX]{x1D709}\rangle . & & \displaystyle\end{eqnarray}$$

Reynolds filtering (4.22) and using the expression (4.25), we obtain the filtered evolution equations for $\unicode[STIX]{x1D701}$ and $\unicode[STIX]{x1D705}$ . The filtered equation for $\unicode[STIX]{x1D701}$ is given by

(4.27) $$\begin{eqnarray}\displaystyle \left\langle \frac{\text{D}\unicode[STIX]{x1D701}}{\text{D}t}\right\rangle & = & \displaystyle -2\,\text{tr}\,\text{sym}(\langle \unicode[STIX]{x1D642}\boldsymbol{\cdot }\unicode[STIX]{x1D640}^{\prime }\rangle -\langle \unicode[STIX]{x1D642}\boldsymbol{\cdot }\boldsymbol{u}^{\prime }\rangle \boldsymbol{\cdot }\unicode[STIX]{x1D735}\overline{\unicode[STIX]{x1D641}}^{\mathsf{T}}\boldsymbol{\cdot }\overline{\unicode[STIX]{x1D641}}^{-\mathsf{T}})\nonumber\\ \displaystyle & & \displaystyle -\,\text{tr}(-\langle \boldsymbol{u}^{\prime }\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D642}\rangle +\langle \unicode[STIX]{x1D648}\boldsymbol{\cdot }\unicode[STIX]{x1D642}^{-1}\rangle -\overline{\unicode[STIX]{x1D648}}),\end{eqnarray}$$

where each term can be compared to those in (4.25) that were described in the previous subsection. Similarly, the filtered equation for $\unicode[STIX]{x1D705}$ is given by

(4.28) $$\begin{eqnarray}\displaystyle \frac{1}{2}\left\langle \frac{\text{D}\unicode[STIX]{x1D705}}{\text{D}t}\right\rangle & = & \displaystyle 2\,\text{tr}\langle \text{sym}(\unicode[STIX]{x1D640}^{\prime }-\overline{\unicode[STIX]{x1D641}}^{-1}\boldsymbol{\cdot }\boldsymbol{u}^{\prime }\boldsymbol{\cdot }\unicode[STIX]{x1D735}\overline{\unicode[STIX]{x1D641}})\boldsymbol{\cdot }\boldsymbol{{\mathcal{G}}}\rangle -\text{tr}\,\langle \unicode[STIX]{x1D648}\boldsymbol{\cdot }\unicode[STIX]{x1D642}^{-1}\boldsymbol{\cdot }\boldsymbol{{\mathcal{G}}}\rangle \nonumber\\ \displaystyle & & \displaystyle -\,\text{tr}\{[2\,\text{sym}(\langle \unicode[STIX]{x1D642}\boldsymbol{\cdot }\unicode[STIX]{x1D640}^{\prime }\rangle -\langle \unicode[STIX]{x1D642}\boldsymbol{\cdot }\boldsymbol{u}^{\prime }\rangle \boldsymbol{\cdot }\unicode[STIX]{x1D735}\overline{\unicode[STIX]{x1D641}}^{\mathsf{T}}\boldsymbol{\cdot }\overline{\unicode[STIX]{x1D641}}^{-\mathsf{T}})-\langle \boldsymbol{u}^{\prime }\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D642}\rangle -\overline{\unicode[STIX]{x1D648}}]\boldsymbol{\cdot }\langle \boldsymbol{{\mathcal{G}}}\rangle \}.\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

The equation for $\unicode[STIX]{x1D709}$ , which we omit here for brevity, can be similarly derived.

5.1 Mean profiles and comparisons with the laminar profiles

The statistics presented in this section were obtained by averaging in space and over 750 time units, and by exploiting the symmetry of the flow about the centreline. Halving the number of samples maintained the trends and caused only minor deviations in the statistics, with no impact on the conclusions.

The mean streamwise velocity profile is shown in figure 2 as a function of $y^{+}=Re_{\unicode[STIX]{x1D70F}}(y+1)$ , where $Re_{\unicode[STIX]{x1D70F}}$ is always taken to be the turbulent frictional Reynolds number given in table 2. Also shown are the von Kármán log law and Virk’s maximum drag reduction asymptote. The mean velocity lies in between these two lines, indicating a drag-reduced state. Using Dean’s correlation for the skin friction (Dean Reference Dean1978), we obtain a friction Reynolds number of approximately 284 for a Newtonian flow with $Re=4667$ and thus the drag reduction percentage is

(5.1) $$\begin{eqnarray}\displaystyle \text{DR}\,\%\equiv \left[1-\left(\frac{Re_{\unicode[STIX]{x1D70F}}}{Re_{\unicode[STIX]{x1D70F}}|_{Newtonian}}\right)^{2}\right]\times 100=59.8\,\%. & & \displaystyle\end{eqnarray}$$

The non-zero components of $\langle \unicode[STIX]{x1D63E}\rangle$ calculated for the same parameter values, are shown in figure 3. All the components of $\langle \unicode[STIX]{x1D63E}\rangle$ are even functions of $y$ except $\langle \unicode[STIX]{x1D60A}_{xy}\rangle$ , which is an odd function of $y$ . The streamwise stretch $\langle \unicode[STIX]{x1D60A}_{xx}\rangle$ is four times larger than the laminar case (not shown) near the wall. It is also an order of magnitude larger than $\langle \unicode[STIX]{x1D60A}_{yy}\rangle$ and $\langle \unicode[STIX]{x1D60A}_{zz}\rangle$ . The remaining normal components of the conformation tensor are also larger in the turbulent case than the laminar: figure 3 shows that $\max _{y}\langle \unicode[STIX]{x1D60A}_{yy}\rangle \approx 45$ and $\max _{y}\langle \unicode[STIX]{x1D60A}_{zz}\rangle \approx 120$ , while $\unicode[STIX]{x1D60A}_{yy}=\unicode[STIX]{x1D60A}_{zz}\approx 1$ throughout the channel when the flow is laminar. The peak values of each of the components $\langle \unicode[STIX]{x1D60A}_{xx}\rangle$ , $\langle \unicode[STIX]{x1D60A}_{yy}\rangle$ and $\langle \unicode[STIX]{x1D60A}_{zz}\rangle$ occur at different locations in the channel. Figure 3 also shows that $\langle \unicode[STIX]{x1D60A}_{zz}\rangle \geqslant \langle \unicode[STIX]{x1D60A}_{yy}\rangle$ throughout the channel. The trends above are consistent with those reported in the literature (Dallas, Vassilicos & Hewitt Reference Dallas, Vassilicos and Hewitt2010).

Figure 3.

Mean conformation tensor profiles from a FENE-P drag-reduced channel flow simulation. (a) $\overline{\unicode[STIX]{x1D60A}}_{xx}$ . (b) The solid line with star symbols (— * —) is $\overline{\unicode[STIX]{x1D60A}}_{yy}$ , the dashed line with square symbols (– – ▫ – –) is $\overline{\unicode[STIX]{x1D60A}}_{zz}$ and the dashed-dot line with circle symbols (– - ○ – -) is $\overline{\unicode[STIX]{x1D60A}}_{xy}$ . The remaining components of the mean conformation tensor are 0. Note that the symbols in (b) are identifiers and are thus only a small subset of all the data points used in the line plots.

Figure 4.

Scalar measures applied to nominal conformation tensors, in equivalent dimensions, plotted as functions of $y^{+}$ . (a) The solid line with star symbols (— * —) is $\log \det \,\overline{\unicode[STIX]{x1D63E}}$ , the logarithmic volume; the dashed line with square symbols (– – ▫ – –) is $d(\unicode[STIX]{x1D644},\overline{\unicode[STIX]{x1D63E}})$ , the geodesic distance between $\overline{\unicode[STIX]{x1D63E}}$ and $\unicode[STIX]{x1D644}$ on $\boldsymbol{P}\boldsymbol{o}\boldsymbol{s}_{3}$ . (b) Anisotropy index, $\sqrt{(d(\unicode[STIX]{x1D644},\overline{\unicode[STIX]{x1D63E}}))^{2}-(1/3)(\log \det \,\overline{\unicode[STIX]{x1D63E}})^{2}}$ , which is the geodesic distance from the closest isotropic tensor. For both (a) and (b), black lines are for $\overline{\unicode[STIX]{x1D63E}}=\langle \unicode[STIX]{x1D63E}\rangle$ and grey lines are when $\overline{\unicode[STIX]{x1D63E}}$ is equal to the FENE-P laminar conformation tensor. The red dotted line (- - - - - -) in (b) is $-1.375\log y^{+}+7.925$ . Note that the symbols in (a) are identifiers and are thus only a small subset of all the data points used in the line plots.

Figure 4(a) shows the logarithmic volume of the mean and laminar conformation tensors along with the distance from the origin on the manifold of positive definite tensors. The figure shows that both the logarithmic volume, $\log \det \,\overline{\unicode[STIX]{x1D63E}}$ , and the distance from the origin, $d(\unicode[STIX]{x1D644},\overline{\unicode[STIX]{x1D63E}})$ are larger in the turbulent case compared to the laminar. Furthermore, these two quantities are monotonically decreasing in the laminar case but have peaks around $y^{+}\approx 60$ in the turbulent case. Both quantities in the two cases asymptote to a constant at locations very close to the wall, $y^{+}\leqslant 2$ . The weak growth in $d(\unicode[STIX]{x1D644},\overline{\unicode[STIX]{x1D63E}})$ in the turbulent case despite a rapid increase in $\log \det \,\overline{\unicode[STIX]{x1D63E}}$ is due to the increase in isotropy (sphericity) as we move away from the wall, since for a given volume the tensor closest to $\unicode[STIX]{x1D644}$ is an isotropic tensor. In figure 3 we see that the mean normal stretches in the $y$ and $z$ directions in the turbulent case peak between $y^{+}\approx 40$ and $y^{+}\approx 70$ and are at least an order of magnitude larger than the stretches in the laminar case, where $\overline{\unicode[STIX]{x1D60A}}_{yy}=\overline{\unicode[STIX]{x1D60A}}_{zz}\approx 1$ . The term $\overline{\unicode[STIX]{x1D60A}}_{xx}$ is decreasing towards the channel centre in both the turbulent and laminar case but is accompanied by an increase in $\overline{\unicode[STIX]{x1D60A}}_{yy}$ , $\overline{\unicode[STIX]{x1D60A}}_{zz}$ in the turbulent flow which leads to increased isotropy for locations sufficiently removed from the wall.

Figure 4(b) shows the anisotropy, the geodesic distance to the closest isotropic tensor, of the mean and laminar conformation tensors. The anisotropy index is approximately constant in the vicinity of the wall for both the laminar and turbulent cases, and decays away from the wall. In the turbulent flow, the decay starts very close to the wall – approximately three friction units away from the wall – and then shows a remarkable logarithmic decay that proceeds all the way to very close to the centreline where it sharply turns and forms a stationary point. The increased isotropy in the turbulent case may be explained by the fact that, although more stretching occurs in this case, the stretching in the cross-stream directions is much larger than in the laminar case. Overall, this leads to a more isotropic mean conformation tensor.

Figure 5.

Wall-parallel $(x,z)$ planes of isocontours of instantaneous $\text{I}_{\unicode[STIX]{x1D642}}=\text{tr}\,\unicode[STIX]{x1D642}$ : (a) $y^{+}=15$ and (b) $y^{+}=180$ (centreline).

5.2 Invariants of the fluctuating conformation tensor

In order to motivate the scalar measures proposed in the present work, we consider the invariants of $\unicode[STIX]{x1D642}$ as alternatives in this subsection. Figure 5 shows isocontours of instantaneous $\text{I}_{\unicode[STIX]{x1D642}}$ at a given time at two wall-parallel planes, $y^{+}=15$ and $y^{+}=180$ (centreline). The isocontours of instantaneous $\text{II}_{\unicode[STIX]{x1D642}}$ and $\text{III}_{\unicode[STIX]{x1D642}}$ are qualitatively similar to those of $\text{I}_{\unicode[STIX]{x1D642}}$ and are thus not shown here. The instantaneous $\text{I}_{\unicode[STIX]{x1D642}}$ can vary over several orders of magnitude. As a result, obtaining reliable statistics for the invariants is challenging. We found that the peak root-mean-square (r.m.s.) of the invariants (not shown) are at least an order of magnitude larger than the corresponding mean values. This large spread in the instantaneous invariants of $\unicode[STIX]{x1D642}$ suggests that $\log \unicode[STIX]{x1D642}$ is a more appropriate quantity to consider, and reinforces the need for the geometrically consistent scalar measures introduced in § 4.

Figure 6.

Wall-parallel $(x,z)$ planes of isocontours of instantaneous (a,b) logarithmic volume ratio, $\unicode[STIX]{x1D701}$ , (c,d) geodesic distance from the identity, $\unicode[STIX]{x1D705}$ and (e,f) anisotropy index, $\unicode[STIX]{x1D709}$ . (a,c,e) $y^{+}=15$ , (b,d,f) $y^{+}=180$ (centreline).

5.3 The scalar measures: $\unicode[STIX]{x1D701}$ , $\unicode[STIX]{x1D705}$ and $\unicode[STIX]{x1D709}$

Figure 6 shows isocontours of instantaneous values of $\unicode[STIX]{x1D701}$ , $\unicode[STIX]{x1D705}$ and $\unicode[STIX]{x1D709}$ for two wall-parallel planes, $y^{+}=15$ and $y^{+}=180$ . As a comparison, the fluctuating tensor $\unicode[STIX]{x1D60A}_{xx}^{\prime }$ obtained by the Reynolds decomposition (1.2) and normalized by the local $\overline{\unicode[STIX]{x1D60A}}_{xx}$ is shown in figure 7. We normalized $\unicode[STIX]{x1D60A}_{xx}^{\prime }$ so that the fluctuations near the wall could be compared to those at the centreline, since $\overline{\unicode[STIX]{x1D60A}}_{xx}$ differs by an order of magnitude between the two locations. The isocontours of $\unicode[STIX]{x1D60A}_{xx}^{\prime }/\overline{\unicode[STIX]{x1D60A}}_{xx}$ , and in particular the negative values, are difficult to interpret since the correspondence to a physical deformation or a mathematical metric is unclear.

Figure 6(a,b) shows the logarithmic volume ratio, $\unicode[STIX]{x1D701}$ . This quantity is the logarithm of $\text{III}_{\unicode[STIX]{x1D642}}$ , which itself is qualitatively similar to $\text{I}_{\unicode[STIX]{x1D642}}$ and hence we observe a strong visual resemblance between figures 5 and 6(a,b). The colour scale in the former is logarithmic and is thus consistent with the linear scale in figure 6. Both figures 6(a) and 6(b) have predominantly negative values, indicating that the instantaneous volume is smaller than the volume of the mean conformation. We also find regions of very high $\unicode[STIX]{x1D701}$ adjacent to regions of very low values, especially at the centreline. This is partially a result of the lack of diffusion in the polymers since there is no direct mechanism for smoothing out shocks in the conformation tensor field.

Figure 7.

Wall-parallel $(x,z)$ planes of isocontours of instantaneous $\unicode[STIX]{x1D60A}_{xx}^{\prime }/\overline{\unicode[STIX]{x1D60A}}_{xx}$ at (a) $y^{+}=15$ , (b) $y^{+}=180$ (centreline), where $\overline{\unicode[STIX]{x1D60A}}_{xx}(y^{+}=15)=2.22\times 10^{3}$ and $\overline{\unicode[STIX]{x1D60A}}_{xx}(y^{+}=180)=1.02\times 10^{2}$ . The limits of the divergent colour map are set at the planar maxima and minima of $\unicode[STIX]{x1D60A}_{xx}^{\prime }/\overline{\unicode[STIX]{x1D60A}}_{xx}$ .

Another important effect is that of memory: polymers are stretched near the wall where the shear is significant, and are then transported out to the centreline. If the half-channel transit time, the ratio of the channel half-height to the r.m.s. wall-normal velocity, is smaller than the polymer relaxation time, we expect to observe a footprint of the near-wall stretching all the way out to the centreline. In the present case, the relaxation time is two to three times the half-channel transit time. Since material points that are initially close are exponentially diverging in a turbulent flow (see Johnson et al. Reference Johnson, Hamilton, Burns and Meneveau2017, for a study of Lagrangian stretching in Newtonian turbulent channel flow), it is unsurprising to find adjacent regions of strongly and weakly stretched polymers.

The reductionist explanation given above is useful for a basic understanding but is insufficient to account for other observed features of the flow. For example, the present considerations would suggest that the streamwise elongated shape of the isocontours of $\unicode[STIX]{x1D701}$ near the wall would lead to a similar shape at the centreline. However, this is manifestly not the case. Instead, the $\unicode[STIX]{x1D701}$ field appears to generate, on the whole, highly curved isocontours at the centre of the channel.

The measure $\unicode[STIX]{x1D701}$ does not distinguish between volume-preserving deformations. For example, $\unicode[STIX]{x1D701}$ does not distinguish between $\unicode[STIX]{x1D63E}$ and $(\det \,\unicode[STIX]{x1D63E})\unicode[STIX]{x1D63C}$ for any $\unicode[STIX]{x1D63C}$ with determinant $=1$ . In particular, $\unicode[STIX]{x1D701}=0$ , does not imply $\unicode[STIX]{x1D63E}=\overline{\unicode[STIX]{x1D63E}}$ . In order to identify regions where $\unicode[STIX]{x1D63E}=\overline{\unicode[STIX]{x1D63E}}$ is true, and quantify the deviation when it is not, we use the squared distance away from the origin ( $\unicode[STIX]{x1D644}$ ) along the manifold, $\unicode[STIX]{x1D705}$ . Figure 6(c,d) shows isocontours of instantaneous $\unicode[STIX]{x1D705}$ . Most of the conformation tensor field is significantly far away, in the sense of distance along the manifold, from $\overline{\unicode[STIX]{x1D63E}}$ . However, regions where $\overline{\unicode[STIX]{x1D63E}}$ is a good representation of $\unicode[STIX]{x1D63E}$ are interspersed between regions where $\unicode[STIX]{x1D705}$ is large. This behaviour is true both in the near-wall region as well as the channel centre but more so in the latter. In contrast, it is well known that kinetic energy fluctuations are weakest at the centreline in a Newtonian channel flow. A different behaviour for the polymers is unsurprising since, due to the strong memory effect, $\unicode[STIX]{x1D63E}$ at each point is strongly dependent on the Lagrangian path that is obtained by a pull-back of the particular Eulerian point of interest.

Figure 6(e,f) shows isocontours of instantaneous $\unicode[STIX]{x1D709}$ , the anisotropy index. This index shows how close the shape of instantaneous conformation tensor is to the shape of the mean conformation tensor, irrespective of volumetric changes. The visual resemblance of $\unicode[STIX]{x1D705}$ and $\unicode[STIX]{x1D709}$ suggests that deformations to the mean conformation are largely anisotropic, or in other words, lead to shape change.

Figure 8.

Mean scalar measures, plotted in equivalent dimensions, as functions of $y^{+}$ . The solid line with star symbols (— * —) is minus the mean volume ratio, $-\overline{\unicode[STIX]{x1D701}}=-\langle \unicode[STIX]{x1D701}\rangle$ ; the dashed line with square symbols (– – ▫ – –) is the square-root of the mean geodesic distance from the identity, $\overline{\unicode[STIX]{x1D705}}^{1/2}=\sqrt{\langle \unicode[STIX]{x1D705}\rangle }$ ; the dashed-dot line with circle symbols (– - ○ – -) is the square-root of the mean anisotropy index, $\overline{\unicode[STIX]{x1D709}}^{1/2}=\sqrt{\langle \unicode[STIX]{x1D709}\rangle }$ . Red dotted lines (- - - - - -) are logarithmic fits to the profiles: the fit to the $\overline{\unicode[STIX]{x1D705}}^{1/2}$ profile (– – ▫ – –) is given by $0.725\log y^{+}+2.15$ , the fits to the $\overline{\unicode[STIX]{x1D709}}^{1/2}$ profile (– - ○ – -) are given by $0.65\log y^{+}+1.20$ and $-0.9\log y^{+}+7.6$ . Note that the symbols are identifiers and are thus only a small subset of all the data points used in the line plots.

Figure 8 shows the mean values of $\unicode[STIX]{x1D701}$ , $\unicode[STIX]{x1D705}$ and $\unicode[STIX]{x1D709}$ in dimensions of distance along the manifold. These statistics were generated using 225 convective time units. We checked for convergence by halving the number of samples. This process led to only minor deviations in the results, with no material significance to the discussion that follows. As was inferred earlier, the average logarithmic volume ratio is negative throughout the channel and is monotonically decreasing towards the centreline where it becomes roughly constant, similar to the behaviour very near the wall $y^{+}\lesssim 2$ . This behaviour of the mean logarithmic volume being smaller than the volume of the mean is consistent with the ‘swelling’ problem associated with the arithmetic mean of positive definite tensors that has been previously reported in the literature (Arsigny et al. Reference Arsigny, Fillard, Pennec and Ayache2007). It also suggests that, although an arithmetic mean of the conformation tensor may be unavoidable for modelling in the averaged equations, it may not be the most representative conformation tensor for deducing the most likely physical deformation of the polymers. A more extensive study, beyond the scope of the present work, is required to determine better alternatives to the arithmetic mean.

The square-root mean squared distance from the origin along the manifold, $\overline{\unicode[STIX]{x1D705}}^{1/2}$ , is logarithmically increasing up to close to the centreline but peaks at $y^{+}\approx 100$ . The anisotropy, $\overline{\unicode[STIX]{x1D709}}^{1/2}$ , shows logarithmic increase over a small range $3\lesssim y^{+}\lesssim 20$ , peaking at $y^{+}\approx 60$ , but then shows a logarithmic decrease towards the centreline. The logarithmic behaviour in these quantities, especially in $\overline{\unicode[STIX]{x1D705}}^{1/2}$ where the behaviour extends over a significant range, resembles the behaviour that appears in the mean velocity as well as in the statistical moments and two-point correlations of the velocity fluctuations in wall-bounded shear flows (Meneveau & Marusic Reference Meneveau and Marusic2013; Yang, Marusic & Meneveau Reference Yang, Marusic and Meneveau2016).

If we momentarily accept the simplified picture of polymers being deformed closer to the wall in an ‘active’ region and then passively transported out to the centreline of the channel, then these results indicate that the active region of the channel exists all the way up to $y^{+}\approx 100$ . Beyond this region, the stretching of polymers weakens and thus $\overline{\unicode[STIX]{x1D705}}^{1/2}$ decreases monotonically. The active region involves a region of logarithmic increase, and so we can write $\overline{\unicode[STIX]{x1D705}}^{1/2}=a_{1}\int y^{-1}\,\text{d}y+a_{0}$ for constants $a_{1}$ and $a_{0}$ . If the stretching at each wall-normal station is actually additive, the polymers undergo deformation that on average leads to a diminishing increment in the deformation. This behaviour is consistent with the velocity gradients weakening with wall-normal distance. At $y^{+}\approx 100$ , the velocity gradients then either weaken or act on such long time scales that polymers relax quickly enough not to retain any additional deformation.

Figure 9.

Joint probability density functions (JPDF) of the logarithmic volume ratio, $\unicode[STIX]{x1D701}$ , and the geodesic distance from the identity, $\unicode[STIX]{x1D705}$ , at four different wall-normal locations: (a) $y^{+}=2$ , (b) $y^{+}=15$ , (c) $y^{+}=100$ and (d) $y^{+}=180$ (centreline). Dotted lines (- - - - -) are isocontours of the anisotropy index, $\unicode[STIX]{x1D709}$ . The thick red dashed line (– – –) denotes the realizability bound, $\unicode[STIX]{x1D705}=(1/3)\unicode[STIX]{x1D701}^{2}$ , derived in (4.18), which coincides with the zero anisotropy index isocontour ( $\unicode[STIX]{x1D709}=0$ ).

In order to quantify the fluctuations observed in figure 6 in more detail, we calculated the joint probability density function (JPDF) for $\unicode[STIX]{x1D701}$ and $\unicode[STIX]{x1D705}$ using 12 snapshots evenly spaced over 120 convective time units. The JPDFs, at four different wall-normal locations, are shown in figure 9 along with isocontours of $\unicode[STIX]{x1D709}$ , which is purely a function of $\unicode[STIX]{x1D701}$ and $\unicode[STIX]{x1D705}$ .

The JPDFs are non-zero primarily on the lower half of the realizability region, which is consistent with the isocontours in figure 6(a,b) and $\overline{\unicode[STIX]{x1D701}}<0$ throughout the channel in figure 8. In addition, the isocontours tend to concentrate along the isotropy line (thick red dashed line) that was derived in (4.18) as a realizability bound. However, with the exception of the centreline, the most probable $(\unicode[STIX]{x1D701},\unicode[STIX]{x1D705})$ are located away from the isotropy line. This implies that the most likely conformation tensor away from the centreline does not have the same shape as the local mean conformation tensor. Although the most likely conformation tensor at the centreline assumes the shape of the mean conformation tensor, the JPDF at the centreline occupies a greater area in the $(\unicode[STIX]{x1D701},\unicode[STIX]{x1D705})$ plane than at $y^{+}=2$ or $y^{+}=15$ , which implies a greater degree of uncertainty. In addition, the most likely conformation tensor, as determined by the peak of the JPDF, is further away from the mean than at any other wall-normal location.

The JPDF indicate that the most intermittent region of the flow, determined by the most extreme excursions away from the identity on $\boldsymbol{P}\boldsymbol{o}\boldsymbol{s}_{3}$ , do not occur near the wall or at the centreline. This can be seen in figure 9, where the JPDF at $y^{+}=100$ shows events with up to $\unicode[STIX]{x1D705}=100$ . This behaviour is consistent with the peak $\overline{\unicode[STIX]{x1D705}}$ occurring away from the centreline in figure 8.

Figure 10.

Root-mean-square profiles of $\unicode[STIX]{x1D63E}^{\prime }$ , $[\unicode[STIX]{x1D60A}_{ij}]_{rms}=\sqrt{\langle \unicode[STIX]{x1D60A}_{ij}^{2}\rangle -\langle \unicode[STIX]{x1D60A}_{ij}\rangle ^{2}}$ based on the Reynolds decomposition (1.2). (a) $[\unicode[STIX]{x1D60A}_{xx}]_{rms}$ . (b) The solid line with star symbols (— * —) is $[\unicode[STIX]{x1D60A}_{yy}]_{rms}$ , the dashed line with square symbols (– – ▫ – –) is $[\unicode[STIX]{x1D60A}_{zz}]_{rms}$ and the dashed-dot line with circle symbols (– - ○ – -) is $[\unicode[STIX]{x1D60A}_{xy}]_{rms}$ . Note that the symbols in (b) are identifiers and are thus only a small subset of all the data points used in the line plots.

Finally the r.m.s. of $\unicode[STIX]{x1D63E}^{\prime }$ , defined according to the Reynolds decomposition (1.2), is shown in figure 10 for comparison to the present approach. The protocol used to obtain these quantities was the same as that used to obtain the mean conformation tensor in figure 3. The r.m.s. values of $\unicode[STIX]{x1D63E}^{\prime }$ are of similar magnitude to the components of $\langle \unicode[STIX]{x1D63E}\rangle$ . In fact, the peak r.m.s. of $\unicode[STIX]{x1D60A}_{yy}$ , $\unicode[STIX]{x1D60A}_{zz}$ and $\unicode[STIX]{x1D60A}_{xy}$ are larger than their respective mean values. Interestingly, the peak fluctuating deformation found at $y^{+}\approx 100$ using our present framework is not discernible from the r.m.s. fluctuations. Different components of the r.m.s. tensor peak at different locations in the channel with $[\unicode[STIX]{x1D60A}_{yy}]_{rms}$ showing a peak that is closest to $y^{+}=100$ . The r.m.s. quantities only show the component-wise behaviour of the conformation tensor, and hence are not indicative of the total polymer deformation. A more appropriate quantity to evaluate in the context of $\unicode[STIX]{x1D63E}^{\prime }$ would then be the JPDF of all six independent components of $\unicode[STIX]{x1D63E}^{\prime }$ . Owing to high-dimensionality, this characterization is more difficult to both calculate and analyse. The scalar measures suggested in the present work provide a good alternative to such a characterization.