Representation of the Reynolds stress tensor through quadrant analysis for a near-neutral atmospheric surface layer flow

Abstract

To disseminate the role of the eddy motions on the anisotropic states of the Reynolds stress tensor ($\varvec{B}$), we devise a novel methodology based on the quadrant analysis, where the distributions of the invariants of $\varvec{B}$ are studied on a $u^{\prime }$–$w^{\prime }$ quadrant plane, with $u^{\prime }$ and $w^{\prime }$ being the turbulent fluctuations in the streamwise and vertical velocities. We apply this methodology to a near-neutral atmospheric surface layer (ASL) flow, derived from a field experiment dataset having multi-level turbulence measurements. The results show that in a near-neutral ASL flow, the anisotropic states of $\varvec{B}$ are determined by the distribution of the streamwise and cross-stream velocity variances ($\sigma _{u}^2$ and $\sigma _{v}^2$) on the $u^{\prime }$–$w^{\prime }$ quadrant plane. By studying the contour maps of the invariants of $\varvec{B}$ on the $u^{\prime }$–$w^{\prime }$ quadrant plane, we discover three distinct zones with elliptical boundaries in the $u^{\prime }$–$w^{\prime }$ plane, across which the anisotropic states of the eddy motions evolve. We find that the eddy motions which occur

• 1. Inside the inner elliptical zone display rod-like anisotropy being determined by $\sigma _{v}^2$,

• 2. Within the annular zone between the inner and outer ellipses display pancake-like anisotropy being determined by both $\sigma _{u}^2$ and $\sigma _{v}^2$,

• 3. Outside the outer elliptical zone display rod-like anisotropy being determined by $\sigma _{u}^2$.

We also notice that the distinction between these three zones in the $u^{\prime }$–$w^{\prime }$ plane is prominent at the lowest measurement level, but becomes progressively indistinguishable as the measurement height increases.

Appendices

Appendix 1: The normalization factor while constructing $\varvec{B}(\hat{u},\hat{w})$

The normalization factor used in the construction of $\varvec{B}(\hat{u}_{\mathrm{bin}}(i),\hat{w}_{\mathrm{bin}}(j))$ (Eq. 26) is the turbulent kinetic energy contained in the grid {$\hat{u}_{\mathrm{bin}}(i)$, $\hat{w}_{\mathrm{bin}}(j)$}, rather than the total kinetic energy e of the half-hour run. The advantage of this normalization is it makes $\varvec{B}(\hat{u},\hat{w})$ trace-free, and so its anisotropic states can be studied by using the two invariants only. Also since we are examining the anisotropic states of the eddy motions contained within the grid {$\hat{u}_{\mathrm{bin}}(i)$, $\hat{w}_{\mathrm{bin}}(j)$}, it makes physically more sense to normalize $\varvec{B}(\hat{u}_{\mathrm{bin}}(i),\hat{w}_{\mathrm{bin}}(j))$ by the turbulent kinetic energy contained in that grid, rather than by e. This is very similar to the spectral version of $\varvec{B}$ (Eq. 12) as defined in Liu et al. [17], where they normalized $\varvec{B}(\kappa )$ by the energy contained at each $\kappa$.

One question that may arise, whether we would have obtained the original invariants of $\varvec{B}$ ($\xi$ and $\eta$) by integrating the functions $\xi (\hat{u},\hat{w})/d\hat{u}d\hat{w}$ and $\eta (\hat{u},\hat{w})/d\hat{u}d\hat{w}$ over the $\hat{u}$–$\hat{w}$ quadrant plane, if we had normalized $\varvec{B}(\hat{u},\hat{w})$ by the total kinetic energy e. The major problem in that case would be the trace of $\varvec{B}(\hat{u},\hat{w})$ will not be zero, thus introducing an additional invariant to study its anisotropic states. Apart from that problem, even in that case the integration of the functions $\xi (\hat{u},\hat{w})/d\hat{u}d\hat{w}$ and $\eta (\hat{u},\hat{w})/d\hat{u}d\hat{w}$ would not yield $\xi$ and $\eta$ of $\varvec{B}$. Mathematically it can be shown that, if $\varvec{B}(\hat{u},\hat{w})$ was normalized by e, then

$$\begin{aligned} \varvec{B}={\sum _{j=1}^{50}\sum _{i=1}^{50}} \varvec{B}(\hat{u}_{\mathrm{bin}}(i),\hat{w}_{\mathrm{bin}}(j)). \end{aligned}$$

(32)

Now combining Eqs. 2 with 8, and Eqs. 3 with 7, we can write,

$$\begin{aligned} 6\xi ^3=\mathrm {Tr}\left[ {\left( {{\sum _{j=1}^{50} \sum _{i=1}^{50}} \varvec{B}(\hat{u}_{\mathrm{bin}}(i), \hat{w}_{\mathrm{bin}}(j))}\right) }^3 \right] , \end{aligned}$$

(33)

and

$$\begin{aligned} 6\eta ^2=\mathrm {Tr}\left[ {\left( {{\sum _{j=1}^{50} \sum _{i=1}^{50}} \varvec{B}(\hat{u}_{\mathrm{bin}}(i), \hat{w}_{\mathrm{bin}}(j))}\right) }^2 \right] . \end{aligned}$$

(34)

By following the Einstein summation convention, we can expand the terms in Eqs. 33 and 34 as,

$$\begin{aligned} 6\xi ^3=\mathrm {Tr}\left[ \varvec{B}(\hat{u}_{\mathrm{bin}}(i), \hat{w}_{\mathrm{bin}}(j))\ \varvec{B} (\hat{u}_{\mathrm{bin}}(j), \hat{w}_{\mathrm{bin}}(k)) \ \varvec{B}(\hat{u}_{\mathrm{bin}}(k), \hat{w}_{\mathrm{bin}}(i))\right] , \end{aligned}$$

(35)

and

$$\begin{aligned} 6\eta ^2=\mathrm {Tr}\left[ \varvec{B}(\hat{u}_{\mathrm{bin}}(i), \hat{w}_{\mathrm{bin}}(j)) \varvec{B} (\hat{u}_{\mathrm{bin}}(j),\hat{w}_{\mathrm{bin}}(i)) \right] . \end{aligned}$$

(36)

Since the cross-terms in the Eqs. 35 and 36 do not vanish, the invariants $\xi$ and $\eta$ are simply not the sums of the invariants of $\varvec{B}(\hat{u}_{\mathrm{bin}}(i),\hat{w}_{\mathrm{bin}}(j))$, and so the integration of the functions $\xi (\hat{u},\hat{w})/d\hat{u}d\hat{w}$ and $\eta (\hat{u},\hat{w})/d\hat{u}d\hat{w}$ do not yield the values of $\xi$ and $\eta$ of $\varvec{B}$.

Appendix 2: Analytical expressions of the Reynolds stress invariants $\xi$ and $\eta$

From the Cayley-Hamilton theorem [4], the characteristic polynomial of $\varvec{B}$ can be written as,

$$\begin{aligned} {\varvec{B}}^{3}-{\mathrm {Tr}(\varvec{B}}) {\varvec{B}}^{2} +\frac{1}{2}[{({\mathrm {Tr}(\varvec{B}}))}^2 -\mathrm {Tr}({\varvec{B}}^{2})]{\varvec{B}} -\det (\varvec{B})I_{3,3}=0_{3,3}, \end{aligned}$$

(37)

where $I_{3,3}$ and $0_{3,3}$ are the identity and null matrices of the order $3 \times 3$ respectively. The coefficients of the characteristic polynomial of $\varvec{B}$ are the invariants of $\varvec{B}$ (I, II, and III), since these quantities are conserved under any coordinate transformation. Given the fact that $\varvec{B}$ has zero trace this makes $I=0$, and the other two invariants of Eq. 37 (II and III) can be written as,

$$\begin{aligned} II=-\frac{1}{2} \mathrm {Tr}({\varvec{B}}^{2}), \end{aligned}$$

(38)

and

$$\begin{aligned} III=\det (\varvec{B}). \end{aligned}$$

(39)

It is to note that, since $\varvec{B}$ is symmetric ($\varvec{B}_{ij}=\varvec{B}_{ji}$) Eq. 38 can also be written as,

$$\begin{aligned} II=-\frac{1}{2} (\varvec{B:B}), \end{aligned}$$

(40)

where $\varvec{B:B}$ is the Frobenius inner product between the $\varvec{B}$s. Now from the matrix algebra [5], the $\det (\varvec{B})$ can be written as,

$$\begin{aligned} \det (\varvec{B})=\frac{1}{6}[{({\mathrm {Tr}(\varvec{B}}))}^3 -3\mathrm {Tr}({\varvec{B}}^{2})\mathrm {Tr}(\varvec{B}) +2\mathrm {Tr}({\varvec{B}}^{3})]. \end{aligned}$$

(41)

Since $\mathrm {Tr}(\varvec{B})=0$, by combining Eqs. 39 and 41 we can write,

$$\begin{aligned} III=\frac{1}{3} \mathrm {Tr}({\varvec{B}}^{3}). \end{aligned}$$

(42)

By converting the invariants II and III to $\xi$ and $\eta$ from Eqs. 7 and 8, we can write,

$$\begin{aligned} 6\eta ^{2} &= \left( \frac{\overline{u^{\prime 2}}}{2e} - \frac{1}{3}\right) ^{2} + \left( \frac{\overline{v^{\prime 2}}}{2e} - \frac{1}{3}\right) ^{2} + \left( \frac{\overline{w^{\prime 2}}}{2e} - \frac{1}{3}\right) ^{2} \nonumber \\&\quad +\,2\left[ \left( \frac{\overline{{u}^{\prime }{v}^{\prime }}}{2e}\right) ^{2} + \left( \frac{\overline{{v}^{\prime }{w}^{\prime }}}{2e}\right) ^{2} + \left( \frac{\overline{{u}^{\prime }{w}^{\prime }}}{2e}\right) ^{2}\right] , \end{aligned}$$

(43)

and

$$\begin{aligned} 6\xi ^{3} &= \left( \frac{\overline{u^{\prime 2}}}{2e} - \frac{1}{3}\right) ^{3} + \left( \frac{\overline{v^{\prime 2}}}{2e} - \frac{1}{3}\right) ^{3} + \left( \frac{\overline{w^{\prime 2}}}{2e} - \frac{1}{3}\right) ^{3} \nonumber \\&\quad +\,3\left( \frac{\overline{{u}^{\prime }{v}^{\prime }}}{2e}\right) ^{2} \left[ \frac{\overline{u^{\prime 2}}}{2e} + \frac{\overline{v^{\prime 2}}}{2e} - \frac{2}{3}\right] \nonumber \\&\quad +\, 3\left( \frac{\overline{{v}^{\prime }{w}^{\prime }}}{2e}\right) ^{2} \left[ \frac{\overline{v^{\prime 2}}}{2e} + \frac{\overline{w^{\prime 2}}}{2e} - \frac{2}{3}\right] \nonumber \\&\quad +\,3\left( \frac{\overline{{u}^{\prime }{w}^{\prime }}}{2e}\right) ^{2} \left[ \frac{\overline{u^{\prime 2}}}{2e} + \frac{\overline{w^{\prime 2}}}{2e} - \frac{2}{3}\right] \nonumber \\&\quad +\,6\left( \frac{\overline{u^{\prime }v^{\prime }}}{2e}\right) \left( \frac{\overline{v^{\prime }w^{\prime }}}{2e}\right) \left( \frac{\overline{u^{\prime }w^{\prime }}}{2e}\right) , \end{aligned}$$

(44)

respectively, by expanding $\mathrm {Tr}({\varvec{B}}^{2})$ and $\mathrm {Tr}({\varvec{B}}^{3})$ in terms of the components of $\varvec{B}$. From Fig. 9, we notice that the components ${\overline{{u}^{\prime }{v}^{\prime }}}/{2e}$, ${\overline{{v}^{\prime }{w}^{\prime }}}/{2e}$, and ${\overline{{u}^{\prime }{w}^{\prime }}}/{2e}$ contribute negligibly to $\varvec{B}$. We can thus simplify Eqs. 43 and 44 as,

$$\begin{aligned} 6\eta ^{2} \approx \left( \frac{\overline{u^{\prime 2}}}{2e} -\frac{1}{3}\right) ^{2} + \left( \frac{\overline{v^{\prime 2}}}{2e} -\frac{1}{3}\right) ^{2} + \left( \frac{\overline{w^{\prime 2}}}{2e} -\frac{1}{3}\right) ^{2}, \end{aligned}$$

(45)

and

$$\begin{aligned} 6\xi ^{3} \approx \left( \frac{\overline{u^{\prime 2}}}{2e} -\frac{1}{3}\right) ^{3} + \left( \frac{\overline{v^{\prime 2}}}{2e} -\frac{1}{3}\right) ^{3} + \left( \frac{\overline{w^{\prime 2}}}{2e} -\frac{1}{3}\right) ^{3}. \end{aligned}$$

(46)

Fig. 9

The vertical profiles of each component of $\varvec{B}$ such as, the principal stresses a$\overline{{u^{\prime }}^2}/2e$, $\overline{{v^{\prime }}^2}/2e$, and $\overline{{w^{\prime }}^2}/2e$, and the cross terms b$\overline{u^{\prime }w^{\prime }}/2e$, $\overline{v^{\prime }w^{\prime }}/2e$, and $\overline{u^{\prime }v^{\prime }}/2e$. The vertical profiles are averaged with the error bars showing one standard deviation from the mean

Appendix 3: Construction of a conditionally sampled $\varvec{B}$

In Fig. 4b, d and in Fig. 7d–f, we show the Lumley triangle where the invariants of $\varvec{B}$ are plotted for the conditionally sampled eddy motions. We can write the elements of $\varvec{B}$ for any conditionally sampled eddy motions as,

$$\begin{aligned} \varvec{B}_{11}(C) &= \frac{{(\sum {{u^{\prime }}^2})}_{C}}{{(\sum {{u^{\prime }}^2})}_{C}+{(\sum {{v^{\prime }}^2})}_{C} +{(\sum {{w^{\prime }}^2})}_{C}} - \frac{1}{3}\nonumber \\ \varvec{B}_{12}(C) &= \frac{{(\sum {u^{\prime }v^{\prime }})}_{C}}{{(\sum {{u^{\prime }}^2})}_{C}+{(\sum {{v^{\prime }}^2})}_{C} +{(\sum {{w^{\prime }}^2})}_{C}}\nonumber \\ \varvec{B}_{13}(C) &= \frac{{(\sum {u^{\prime }w^{\prime }})}_{C}}{{(\sum {{u^{\prime }}^2})}_{C}+{(\sum {{v^{\prime }}^2})}_{C} +{(\sum {{w^{\prime }}^2})}_{C}}\nonumber \\ \varvec{B}_{21}(C) &= \varvec{B}_{12}(C) \nonumber \\ \varvec{B}_{22}(C) &= \frac{{(\sum {{v^{\prime }}^2})}_{C}}{{(\sum {{u^{\prime }}^2})}_{C}+{(\sum {{v^{\prime }}^2})}_{C} +{(\sum {{w^{\prime }}^2})}_{C}} - \frac{1}{3}\nonumber \\ \varvec{B}_{23}(C) &= \frac{{(\sum {v^{\prime }w^{\prime }})}_{C}}{{(\sum {{u^{\prime }}^2})}_{C}+{(\sum {{v^{\prime }}^2})}_{C} +{(\sum {{w^{\prime }}^2})}_{C}} \nonumber \\ \varvec{B}_{31}(C) &= \varvec{B}_{13}(C) \nonumber \\ \varvec{B}_{32}(C) &= \varvec{B}_{23}(C) \nonumber \\ \varvec{B}_{33}(C) &= \frac{{(\sum {{w^{\prime }}^2})}_{C}}{{(\sum {{u^{\prime }}^2})}_{C}+{(\sum {{v^{\prime }}^2})}_{C} +{(\sum {{w^{\prime }}^2})}_{C}} - \frac{1}{3}, \end{aligned}$$

(47)

where C is the condition under which the eddy motions are sampled, which for example can be those motions located inside or outside the HH (Eq. 30), or within any of the three elliptical zones as defined in Eq. 31. It is to be noted here that similar to Eq. 26 we normalize the elements of $\varvec{B}(C)$ in Eq. 47 by the turbulent kinetic energy contained within these conditionally sampled eddy motions, for the same reasons as discussed earlier.

We can write the invariants of $\varvec{B}(C)$ as,

$$\begin{aligned} \eta (C)={\left( \frac{\lambda _{1C}^2+\lambda _{2C}^2 +\lambda _{3C}^2}{6}\right) }^{\frac{1}{2}}, \end{aligned}$$

(48)

and

$$\begin{aligned} \xi (C)={\left( \frac{\lambda _{1C}^3+\lambda _{2C}^3 +\lambda _{3C}^3}{6}\right) }^{\frac{1}{3}}, \end{aligned}$$

(49)

where $\lambda _{1C}$, $\lambda _{2C}$, and $\lambda _{3C}$ are the eigenvalues of $\varvec{B}(C)$. These invariants $\eta (C)$ and $\xi (C)$ are plotted on the Lumley triangle as shown in Fig. 4b, d and in Fig. 7d–f.