5.1.1 Turbulent kinetic energy balance

The transport equation of turbulent kinetic energy is (refer to Tennekes & Lumley Reference Tennekes and Lumley1972),

where $\unicode[STIX]{x1D634}_{ij}$ and $\unicode[STIX]{x1D61A}_{ij}$ are the fluctuating and mean strain rates respectively,

In (5.1), $C_{k}$ denotes the convection due to mean flow, $\unicode[STIX]{x03C0}_{k}$ represents the pressure work, $T_{k}$ corresponds to the transport by fluctuations, $D_{k}$ stands for viscous diffusion, the production of turbulent kinetic energy by mean flow is $P_{k}$ and the dissipation of turbulent kinetic energy is denoted by  $\unicode[STIX]{x1D716}_{k}$ .

Figure 6.

(a) Evolution of turbulent kinetic energy along the grid-element centreline. (b) Turbulent kinetic energy budget along the grid-element centreline (all terms are normalised by $U_{\infty }^{3}/t_{0}$ ).

The streamwise evolution of turbulent kinetic energy along the grid-element centreline is shown in figure 6(a). In the production region, the turbulent kinetic energy increases from zero to a peak value. This increase is gradual for $x/x^{\ast }<0.2$ where the flow is mostly irrotational, and steeper for $0.2\leqslant x/x^{\ast }\leqslant 0.4$ where the turbulence is developing. The kinetic energy decreases in the decay region (i.e. for $x/x^{\ast }>0.5$ ).

The budget of turbulent kinetic energy along the grid-element centreline is depicted in figure 6(b) where all the terms in (5.1) are normalised by $U_{\infty }^{3}/t_{0}$ . For statistically stationary flows, as the one considered in this paper, the transient term on the left-hand side is zero. Figure 6(b) shows that the grid-element centreline turbulent kinetic energy budget is markedly different from some previously reported budgets for boundary layer (Mansour, Kim & Moin Reference Mansour, Kim and Moin1988) and backward facing step (Le, Moin & Kim Reference Le, Moin and Kim1997). The distinguishing feature here is the absence of production term, $P_{k}$ . The turbulent kinetic energy along the grid-element centreline is not the result of mean-flow gradients producing turbulent fluctuations, as in wall-bounded shear flows. In the region $0\leqslant x/x^{\ast }<0.2$ , the rise of energy is due mainly to pressure work. As we show in the next section, this part of the grid-element centreline is dominated by irrotational flow. Around $x/x^{\ast }\approx 0.2$ , the transport by turbulent fluctuations becomes the term responsible for the turbulent kinetic energy growth. This is the direct result of wakes starting to reach the grid-element centreline and meet around that location and thereby transporting turbulent kinetic energy towards the grid-element centreline. From $x/x^{\ast }\approx 0.2$ to the end of the production region, transport by fluctuations is counteracted by viscous dissipation, pressure work and mean advection. At the start of the decay region, the mean-flow advection term changes its sign and is mainly balanced by viscous dissipation in most of the decay region. The budget analysis reveals that the flow along the grid-element centreline is initially irrotational and becomes turbulent due to lateral infiltration of turbulent wakes from the grid-element bars.

Figure 7.

(a) Evolution of turbulent kinetic energy along the bar centreline. (b) Turbulent kinetic energy budget along the bar centreline (all terms are normalised by $U_{\infty }^{3}/t_{0}$ ).

The turbulent kinetic energy profile along the bar centreline is given in figure 7(a). The turbulent intensity increases up to $x/x^{\ast }\approx 0.13$ along the bar centreline due to the deformation of the Reynolds stress tensor by the mean velocity gradients (production term, $P_{k}$ ) and the pressure work, $\unicode[STIX]{x03C0}_{k}$ (refer to figure 7 b). Notice that $P_{k}$ is now the dominant term, but $\unicode[STIX]{x03C0}_{k}$ is still important. The contribution of $P_{k}$ would have been even stronger if the budget analysis was performed along the core of the shear layer emanating from the top (or the bottom) face of the bar. This region has the strongest strain rate, refer to figure 3. Near the bar, $P_{k}$ and $\unicode[STIX]{x03C0}_{k}$ are balanced by the sum of turbulent transport and dissipation. Further downstream, the mean turbulent kinetic energy starts to decay as the contribution of pressure work and mean velocity gradients decrease. In the far downstream (i.e. after $x/x^{\ast }>0.4$ ), the mean flow carries the turbulent kinetic energy, and it is balanced by dissipation. Figures 6(b) and 7(b) also show that the mean turbulent kinetic energy equation is properly balanced in our simulations.

5.1.2 Mean enstrophy balance

The transport equation for mean enstrophy $\langle \unicode[STIX]{x1D714}_{i}\unicode[STIX]{x1D714}_{i}\rangle$ is given by (refer to Tennekes & Lumley Reference Tennekes and Lumley1972),

where $\unicode[STIX]{x1D714}_{i}$ and $\unicode[STIX]{x1D6FA}_{i}$ are the components of the vorticity vector of the fluctuating and time-averaged fields respectively and are given by

where $\unicode[STIX]{x1D716}_{ijk}$ is the Levi-Civita symbol. In (5.4) $C_{\unicode[STIX]{x1D714}}$ is the advection by mean flow, $G_{\unicode[STIX]{x1D714}}$ represents the production by mean field, $T_{\unicode[STIX]{x1D714}}$ is the transport by turbulent fluctuations, $P_{\unicode[STIX]{x1D714}}$ is the enstrophy production by stretching of fluctuating strain rate, $\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D714}}$ corresponds to production (or removal) of enstrophy by stretching (or squeezing) due to mean strain rate, $\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D714}}$ is the mixed production term while $\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D714}}$ and $\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D714}}$ represent viscous diffusion and dissipation of enstrophy respectively.

Figure 8.

(a) Evolution of mean enstrophy along the grid-element centreline. (b) Budget of mean enstrophy along the grid-element centreline (all terms are normalised by $(U_{\infty }/t_{0})^{3}$ ).

The profile of mean enstrophy along the grid-element centreline is depicted in figure 8(a). In the lee of the grid-element ( $x/x^{\ast }<0.2$ ) the enstrophy is almost zero. Therefore, the flow in this part of the grid-element centreline maybe considered irrotational, which also implies non-turbulent. The mean enstrophy increases slowly after $x/x^{\ast }>0.2$ and it continues to increase until the turbulence peak at $x/x^{\ast }=0.5$ . Further downstream in the decay region, the enstrophy continuously decreases.

Further physical insight on the observed variation of enstrophy can be gained from the budget of enstrophy (figure 8 b). Every term in equation (5.4) is normalised by $(U_{\infty }/t_{0})^{3}$ . The budget demonstrates that all the terms are nearly zero for the irrotational flow part of the grid-element centreline (i.e. for $x/x^{\ast }<0.2$ ). Moving further from this location, the first term in space to contribute to the generation of mean enstrophy is the turbulent transport term, $T_{\unicode[STIX]{x1D714}}$ . The physical mechanism, therefore, is clear: vorticity, which is produced at the grid-element surface due to the no-slip boundary condition, is first shed into bar wakes and then transported towards the grid-element centreline through the bar wakes that disperse vorticity in the lateral directions. Vorticity reaches the grid-element centreline at $x/x^{\ast }\approx 0.2$ . Interestingly, the skewness and flatness (shown in figure 5) are close to their Gaussian values of 0 and 3 respectively exactly at this point, and deviate from them further downstream in the production region. In particular, the skewness becomes negative which indicates the presence of strong negative fluctuations. These decelerations correlate with the intermittent interaction between the two wakes in the part of production region where $x/x^{\ast }\approx 0.2$ (see Melina, Bruce & Vassilicos (Reference Melina, Bruce and Vassilicos2016) for further information on this point).

Shortly after the action of $T_{\unicode[STIX]{x1D714}}$ , the other terms in the transport equation start to become active. For example, after $x/x^{\ast }=0.25$ , the fluctuating strain field stretches the (small amount of) existing transported vorticity causing further production of enstrophy ( $P_{\unicode[STIX]{x1D714}}$ term). Both $T_{\unicode[STIX]{x1D714}}$ , $P_{\unicode[STIX]{x1D714}}$ are positive leading to the very rapid growth of $\langle \unicode[STIX]{x1D714}_{i}\unicode[STIX]{x1D714}_{i}\rangle$ (figure 8 a). These two terms are balanced by advection due to mean flow and viscous dissipation.

Turbulent and mean transport terms grow in the production region but are reduced to small values at the peak location. Further downstream in the decay region, $P_{\unicode[STIX]{x1D714}}$ and $\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D714}}$ become the dominant terms that balance each other. It is worth mentioning here that the terms involving mean velocity gradients do not contribute to the transport of enstrophy anywhere on the grid-element centreline even in the production region, although this region is highly inhomogeneous.

Figure 9.

(a) Evolution of mean enstrophy along the bar centreline. (b) Budget of mean enstrophy along the bar centreline (all terms are normalised by $(U_{\infty }/t_{0})^{3}$ ).

Figure 9 illustrates the dynamics of fluctuating enstrophy along the bar centreline. The mean enstrophy profile (figure 9 a) appears to be qualitatively similar to that of the bar centreline turbulent kinetic energy with values peaking at the same location $x/x^{\ast }\approx 0.13$ . Figure 9(b) reveals that once the vorticity is shed into the bar wake, it is continuously stretched by the fluctuating strain rate throughout the bar centreline. Close to the bar, the increase in mean enstrophy is mostly because of the vortex stretching ( $P_{\unicode[STIX]{x1D714}}$ term). Mechanisms such as the stretching of vorticity by mean strain ( $\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D714}}$ ), and the production due to mean vorticity gradients ( $G_{\unicode[STIX]{x1D714}}$ ) also contribute to the growth of enstrophy in the near-bar region although their contribution is small compared to that of the vortex stretching. Further downstream, the vortex stretching term is in balance with the dissipation of enstrophy, and all other terms are negligible (figure 9 b). Although the mean and turbulent transport terms are the significant budget terms along the grid-element centreline, their contribution to the mean enstrophy budget is minimum along the bar centreline. This budget analysis also shows that the mixed production term ( $\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D714}}$ ) does not contribute to the transport of mean enstrophy along both the grid-element centreline and the bar centreline.

Tennekes & Lumley (Reference Tennekes and Lumley1972) predicted that $P_{\unicode[STIX]{x1D714}}$ and $\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D714}}$ are the dominant terms in (5.4). Their prediction appears to be valid on the bar centreline. On the grid-element centreline, however, our DNS reveals that the turbulent and mean transport terms are also equally important, in particular in the production region. All other terms are predicted to be significantly smaller by Tennekes & Lumley (Reference Tennekes and Lumley1972) and our simulation confirms that they are indeed negligible.

The balancing of mean enstrophy equation (figures 8 b and 9 b), while not absolutely perfect, is definitely within acceptable limits. Our choice of low Reynolds number has been essential in achieving this and it is the first time such a point balance is reported in a study of a spatially developing flow. For channel flow it has been reported by Sandham & Tsinober (Reference Sandham and Tsinober2000).

5.1.3 Mean strain-product balance

The transport equation for mean strain product $\langle \unicode[STIX]{x1D634}_{ij}\unicode[STIX]{x1D634}_{ij}\rangle$ is (refer to Tsinober Reference Tsinober2009),

where $C_{s}$ is the mean-flow advection, $\unicode[STIX]{x1D6FC}_{s}$ is the production by mean strain field, $T_{s}$ is the turbulent transport, $\unicode[STIX]{x1D6FD}_{s}$ is the production due to stretching by mean strain rate, $\unicode[STIX]{x1D708}_{s}$ is the mixed production term, $P_{s}$ is the production due to stretching by fluctuating strain field (also known as the strain self-amplification term). The enstrophy production term ( $\unicode[STIX]{x1D714}_{s}$ ) also appears in the transport equation of mean strain product but with a negative sign and premultiplied by ( $-1/4$ ), $\unicode[STIX]{x03C0}_{s}$ is the strain pressure Hessian correlation term and $\unicode[STIX]{x1D716}_{s}$ is the viscous dissipation term.

Figure 10.

(a) Evolution of mean strain product along the grid-element centreline. (b) Budget of mean strain product along the grid-element centreline (all terms are normalised by $(U_{\infty }/t_{0})^{3}$ ).

Figure 10(a) depicts the profile of mean strain product along the grid-element centreline. The profile of $0.5\langle \unicode[STIX]{x1D714}_{i}\unicode[STIX]{x1D714}_{i}\rangle$ is also plotted in the same figure. Although the flow in the vicinity of grid-element ( $x/x^{\ast }<0.2$ ) does not contain any significant enstrophy, it does possess strain (figure 10 a). The dissipation of turbulent kinetic energy is directly proportional to the strain product, $\unicode[STIX]{x1D716}_{k}=2\unicode[STIX]{x1D708}\langle \unicode[STIX]{x1D634}_{ij}\unicode[STIX]{x1D634}_{ij}\rangle$ , so this is a region of mostly irrotational dissipation. The strain increases rapidly after $x/x^{\ast }>0.2$ until the peak region, and then decreases in the decay region. Note that $0.5\langle \unicode[STIX]{x1D714}_{i}\unicode[STIX]{x1D714}_{i}\rangle \approx \langle \unicode[STIX]{x1D634}_{ij}\unicode[STIX]{x1D634}_{ij}\rangle$ is not valid in the production region (see also table 3). Recall that in homogeneous turbulence it is straightforward to prove analytically that $0.5\langle \unicode[STIX]{x1D714}_{i}\unicode[STIX]{x1D714}_{i}\rangle =\langle \unicode[STIX]{x1D634}_{ij}\unicode[STIX]{x1D634}_{ij}\rangle$ . The dominance of strain product over enstrophy in the production region is also observed in the low blockage grid-element simulations of Zhou et al. (Reference Zhou, Nagata, Sakai, Ito and Hayase2016b ).

Figure 11.

Budget of mean strain product for $x/x^{\ast }<0.2$ along the grid-element centreline.

Figure 10(b) shows the budget of mean strain product along the grid-element centreline. Each term is normalised by $(U_{\infty }/t_{0})^{3}$ . Unlike the budget of enstrophy, some of the terms are active in the region of $x/x^{\ast }<0.2$ . It is of interest to magnify this region to see what really contributes to the generation of small-scale strain where there is no vorticity. A magnified view of figure 10(b) for $x/x^{\ast }<0.2$ is shown in figure 11. The first term to become active along the grid-element centreline is the strain pressure Hessian. This term is balanced by nothing else than the convection due to mean flow for $x/x^{\ast }<0.14$ . The pressure Hessian acts on the fluid elements to produce strain in the region where enstrophy is absent. It is clearly a non-local effect. Therefore, the mean strain-product equation for $x/x^{\ast }<0.14$ can be simplified as,

The importance of the pressure Hessian term in the vicinity of the grid-element was observed by Zhou et al. (Reference Zhou, Nagata, Sakai, Ito and Hayase2015) when the conditional mean trajectories of the $Q$ , $R$ invariants in the $Q{-}R$ plane were computed. It was not however directly identified as the dominant generating mechanism for strain in the lee of the grid-element, as we show in the present paper.

Once strain is produced by pressure Hessian, it is then transported by fluctuations as can be seen from figure 11, where the $T_{s}$ term becomes active after $x/x^{\ast }\approx 0.14$ , acting to increase $\langle \unicode[STIX]{x1D634}_{ij}\unicode[STIX]{x1D634}_{ij}\rangle$ . This transport, however, is not due to the lateral fluctuations, but due to the streamwise fluctuations which are generated by the pressure work mechanism, as discussed in § 5.1.1. The increase of small-scale strain leads to turning on of the strain self-amplification which starts being very significant from $x/x^{\ast }\approx 0.2$ . Therefore, equation (5.7) can be simplified for $0.14<x/x^{\ast }<0.2$ as,

Looking now at figure 10(b), the strain pressure Hessian term eventually becomes a sink after $x/x^{\ast }\approx 0.25$ . Around that location, enstrophy has been brought towards the grid-element centreline as discussed earlier and the presence of small-scale strain brought about by the dynamics in (5.8) and (5.9) initiates the vortex stretching term as seen in figures 10(b) and 8(b). From then on, vortex stretching and dissipation by viscosity become sinks of $\langle \unicode[STIX]{x1D634}_{ij}\unicode[STIX]{x1D634}_{ij}\rangle$ . Therefore, for $0.25<x/x^{\ast }<0.5$ , the transport of mean strain product is due to transport by fluctuations and strain self-amplification which are balanced by the sum of strain pressure Hessian, dissipation by viscosity, vortex stretching and convection due to mean flow. This can be written in equation form as,

This analysis of mean strain product suggests that there are two subzones in the production region. One is the ‘strain-dominant’ zone where only strain is effectively present, and the other one is located adjacent to it further downstream where both strain and vorticity coexist.

In the decay region, the pressure Hessian term becomes zero and transport due to fluctuations becomes a homogenising consuming term. The mean convection term and the strain self-amplification term are balanced by enstrophy production (stretching), transport by fluctuations and dissipation. Therefore, the transport equation of mean strain product in the decay region can be written as,

The mean velocity gradient terms such as $\unicode[STIX]{x1D6FC}_{s}$ , $\unicode[STIX]{x1D6FD}_{s}$ and $\unicode[STIX]{x1D708}_{s}$ do not contribute to the transport of mean strain product even in the inhomogeneous production region. The same was observed in the transport equation for enstrophy.

Figure 12.

(a) Evolution of mean strain product along the bar centreline. (b) Budget of mean strain product along the bar centreline (all terms are normalised by $(U_{\infty }/t_{0})^{3}$ ).

Unlike the grid-element centreline, strain and vorticity coexist throughout the bar centreline (see figure 12 a). The dynamics of the strain product and its budget along the bar centreline, as shown in figure 12(b), is similar to that of the bar centreline mean enstrophy discussed in the previous section. Here also, the turbulent stretching of fluctuating strain is the most dominant term, while the gradient production and stretching by mean strain rate contribute only marginally to the production of mean strain product. Compared to the enstrophy budget, the only difference is the presence of the strain pressure Hessian correlation term that acts as a source in a near-bar region, and eventually becomes a sink further downstream. Far downstream, strain self-amplification is balanced by viscous dissipation.

Here also, the balancing of budgets in the mean strain product is not perfect, yet the percentage error is minimal which makes these results acceptable. We repeat that this is the first time that such numerically well-balanced equations are presented in a study of this kind.

We close this subsection by discussing the effect of aspect ratio AR on the near field. As mentioned in § 2, the AR of the examined grid-element is 0.14, while in classical grids it is equal to 1. It is intuitive to expect that the AR of the bar will influence the near field, both in terms of size of the production region as well as the maximum r.m.s. velocity in the wake. To estimate the size of the production region, we can employ a similarity argument. Gomes-Fernandes et al. (Reference Gomes-Fernandes, Ganapathisubramani and Vassilicos2012) defined a modified wake-interaction length scale, denoted as ${x^{\ast }}^{\prime }$ , that was found to collapse the spatial variation of the normalised turbulent intensities for various grids. The definition of ${x^{\ast }}^{\prime }$ uses the drag coefficient ( $C_{D}$ ) of the bar, which is sensitive to AR. Bearman & Trueman (Reference Bearman and Trueman1972) measured the $C_{D}$ values of rectangular cylinders of different AR values. $C_{D}$ increases with AR, reaches a maximum value of 3 around $AR\approx 0.6$ , and then is reduced. This trend is found to be true also for low Reynolds number transitional flows (refer to Norberg Reference Norberg1993). Interestingly, for $AR=1.0$ and 0.14, the $C_{D}$ values are found to be very similar, $C_{D}\approx 2$ . Consequently, the modified wake-interaction length scale ${x^{\ast }}^{\prime }$ of the classical grid-element with $AR=1.0$ will be very close to one for the current grid-element. Based on this similarity argument, we expect the size of the production region for the two ARs to be close to each other. To the best of our knowledge, there are no similarity variables to predict the maximum r.m.s. values. Therefore one needs to perform additional simulations in order to confirm the prediction of the above similarity argument for the size of the production region as well as to compute the maximum turbulent intensity.

5.4.1 $Q$ – $R$ diagrams

In this section we study the joint PDFs of $Q{-}R$ , i.e. $Q{-}R$ diagrams. To interpret $Q{-}R$ diagrams, one needs to rewrite (5.14) as $3R=-\unicode[STIX]{x1D634}_{ij}\unicode[STIX]{x1D634}_{jk}\unicode[STIX]{x1D634}_{ki}-\unicode[STIX]{x1D714}_{i}\unicode[STIX]{x1D714}_{j}\unicode[STIX]{x1D634}_{ij}/4-\unicode[STIX]{x1D714}_{i}\unicode[STIX]{x1D714}_{j}\unicode[STIX]{x1D634}_{ij}/2$ . This reveals that $3R$ is the difference between two production terms: the term $-\unicode[STIX]{x1D634}_{ij}\unicode[STIX]{x1D634}_{jk}\unicode[STIX]{x1D634}_{ki}-\unicode[STIX]{x1D714}_{i}\unicode[STIX]{x1D714}_{j}\unicode[STIX]{x1D634}_{ij}/4$ (the average of which appears in the equation for $\langle \unicode[STIX]{x1D634}_{ij}\unicode[STIX]{x1D634}_{ij}\rangle /2$ , equation (5.7)) and the term $\unicode[STIX]{x1D714}_{i}\unicode[STIX]{x1D714}_{j}\unicode[STIX]{x1D634}_{ij}/2$ (the average of which appears in the equation for $\langle \unicode[STIX]{x1D714}_{i}\unicode[STIX]{x1D714}_{i}\rangle /4$ , i.e. (5.4) multiplied by $1/2$ on both sides). The reason for considering the production of $\unicode[STIX]{x1D714}_{i}\unicode[STIX]{x1D714}_{i}/4$ rather than $\unicode[STIX]{x1D714}_{i}\unicode[STIX]{x1D714}_{i}/2$ lies in the fact that $Q=\unicode[STIX]{x1D714}_{i}\unicode[STIX]{x1D714}_{i}/4-\unicode[STIX]{x1D634}_{ij}\unicode[STIX]{x1D634}_{ij}/2$ . Hence in a $Q$ – $R$ plot, the upper left quadrant (where $Q>0$ and $R<0$ ) represents events in which $\unicode[STIX]{x1D714}_{i}\unicode[STIX]{x1D714}_{i}/4$ dominates over $\unicode[STIX]{x1D634}_{ij}\unicode[STIX]{x1D634}_{ij}/2$ and the production of $\unicode[STIX]{x1D714}_{i}\unicode[STIX]{x1D714}_{i}/4$ also dominates over the production of $\unicode[STIX]{x1D634}_{ij}\unicode[STIX]{x1D634}_{ij}/2$ . In the upper right quadrant (where $Q>0$ and $R>0$ ) $\unicode[STIX]{x1D714}_{i}\unicode[STIX]{x1D714}_{i}/4$ dominates over $\unicode[STIX]{x1D634}_{ij}\unicode[STIX]{x1D634}_{ij}/2$ but the production of $\unicode[STIX]{x1D634}_{ij}\unicode[STIX]{x1D634}_{ij}/2$ dominates over the production of $\unicode[STIX]{x1D714}_{i}\unicode[STIX]{x1D714}_{i}/4$ . The lower left quadrant (where $Q<0$ and $R<0$ ) represents events in which $\unicode[STIX]{x1D634}_{ij}\unicode[STIX]{x1D634}_{ij}/2$ dominates over $\unicode[STIX]{x1D714}_{i}\unicode[STIX]{x1D714}_{i}/4$ and where the production of $\unicode[STIX]{x1D634}_{ij}\unicode[STIX]{x1D634}_{ij}/2$ also dominates over the production of $\unicode[STIX]{x1D714}_{i}\unicode[STIX]{x1D714}_{i}/4$ ; and finally, the lower right quadrant (where $Q<0$ and $R>0$ ) includes events where $\unicode[STIX]{x1D634}_{ij}\unicode[STIX]{x1D634}_{ij}/2$ dominates over $\unicode[STIX]{x1D714}_{i}\unicode[STIX]{x1D714}_{i}/4$ , but the production of $\unicode[STIX]{x1D714}_{i}\unicode[STIX]{x1D714}_{i}/4$ dominates over the production of $\unicode[STIX]{x1D634}_{ij}\unicode[STIX]{x1D634}_{ij}/2$ . In many turbulent flows the $Q{-}R$ diagram has a teardrop shape (Chong, Perry & Cantwell Reference Chong, Perry and Cantwell1990; Tsinober Reference Tsinober2009). This shape reflects strong correlations between $Q<0$ and $R>0$ on one hand and between $Q>0$ and $R<0$ on the other.

Table 3.

Values of the normalised mean $\langle Q\rangle$ and $\langle R\rangle$ at different locations along the grid-element centreline.

The values of normalised $\langle Q\rangle$ and $\langle R\rangle$ are shown in table 3. It can be seen from the table that the values of $\langle Q\rangle$ are not zero in the region $x/x^{\ast }\leqslant 0.35$ , making it absolutely clear that this region is not homogeneous, even for small-scale statistics. The table shows that, on average, strain product dominates over enstrophy in this region as already seen in § 5.1. Strain is the first small-scale velocity gradient quantity generated along the grid-element centreline by the mechanism discussed in § 5.1. Note, however, that this strain dominance progressively disappears once vorticity is brought to the grid-element centreline. By the end of the production region, the average of both invariants is 0, which is the expected result for homogeneous turbulence.

Figure 16.

Joint PDF of $Q$ and $R$ at different locations along the grid-element centreline: (a)  $x/x^{\ast }=0.1$ , (b)  $x/x^{\ast }=0.25$ , (c)  $x/x^{\ast }=0.35$ , (d)  $x/x^{\ast }=0.5$ , (e)  $x/x^{\ast }=0.75$ , (f)  $x/x^{\ast }=0.95$ . The isocontours range from $10^{1}$ to $10^{-2}$ .

Figure 16 shows the JPDF of $Q$ and $R$ at six different locations. The blue line in this figure is the locus of points where $D=0$ (5.15). The invariants are normalised by the mean strain product at that location. In the region where the wakes have not met and the flow is irrotational ( $x/x^{\ast }=0.1$ ), the shape of the $Q{-}R$ diagram is very different from the usual teardrop shape, and contains no positive $Q$ values (figure 16 a). This observation is consistent with and strengthens the previous observation that the flow in the initial part of the grid-element centreline is mostly irrotational, and $Q<0$ at all $x/x^{\ast }<0.2$ . In this region, the VGT is almost equal to the strain-rate tensor, which is a symmetric tensor, and therefore all the eigenvalues are real. This explains why all isolines are located below the $D=0$ curve, in an area where $D<0$ . The shape of JPDF at $x/x^{\ast }<0.1$ is similar to the one reported for the non-turbulent side of turbulent/non-turbulent interfaces (da Silva & Pereira Reference da Silva and Pereira2008). The shape of figure 16(a) also ascertains that the flow in the initial part of the grid-element centreline is indeed non-turbulent.

Although the wakes have already met by $x/x^{\ast }=0.25$ and the energy spectrum exhibits a short-range approximate $-5/3$ power-law exponent, the $Q{-}R$ diagram, as shown in figure 16(b), does not show its usual shape. However, it is interesting to note that $Q<0$ already correlates with $R>0$ with a visible tail in the $Q{-}R$ diagram at $x/x^{\ast }=0.25$ . This is also the point on the grid-element centreline where strain self-amplification and vortex stretching just started being significant, and the turbulence is still developing, non-Gaussian and inhomogeneous. The undeveloped shape in the $Q{-}R$ diagram in figure 16(b) for $x/x^{\ast }=0.25$ is due to the fact that the strain still dominates enstrophy. It should be noted that the undeveloped $Q{-}R$ diagram in the production region reported by Gomes-Fernandes et al. (Reference Gomes-Fernandes, Ganapathisubramani and Vassilicos2014) (see their figure 17a) correlates with an indistinguishable or slight dominance of vortex stretching over vortex compression while our results show a clear dominance of vortex stretching over compression.

From $x/x^{\ast }=0.35$ onwards the $Q{-}R$ diagrams evolve towards their usual teardrop shape as $Q$ starts to take both positive and negative values, and the correlation between $Q>0$ and $R<0$ starts to appear as seen in figure 16(c). This observation suggests that the preference for vortex stretching increases from the production to the decay region.

In this study, the $Q{-}R$ diagram is observed to attain its usual shape around $x/x^{\ast }=0.5$ i.e. at the end of the production region, as seen in figure 16(d). Our results agree with the numerical results of Laizet et al. (Reference Laizet, Vassilicos and Cambon2013) that the $Q{-}R$ diagram obtains its teardrop shape by the end of the production region and well downstream of the place where the $-5/3$ spectra and the $2/3$ structure functions first appear. Zhou et al. (Reference Zhou, Nagata, Sakai, Suzuki, Ito, Terashima and Hayase2014a , Reference Zhou, Nagata, Sakai, Ito and Hayase2015) obtained the teardrop shape closer to the grid-element, at $x/x^{\ast }=0.16$ . This discrepancy could be due to the difference noticed in the behaviour of skewness and flatness which is the result of different blockage ratios in these two studies (0.11 as opposed to 0.2 in the present study). Also, this is the first study that reports the appearance of a teardrop shape in the $Q{-}R$ diagram for a very low $Re_{\unicode[STIX]{x1D706}}$ homogeneous fully developed turbulence (see figures 16 e and 16 f).

5.4.2 $Q_{s}{-}R_{s}$ diagrams and strain-rate eigenvalue ratios

We assume that the strain-rate eigenvalues are $\unicode[STIX]{x1D706}_{i}(i=1,2,3)$ with $\unicode[STIX]{x1D706}_{1}>\unicode[STIX]{x1D706}_{2}>\unicode[STIX]{x1D706}_{3}$ and $\unicode[STIX]{x1D706}_{1}+\unicode[STIX]{x1D706}_{2}+\unicode[STIX]{x1D706}_{3}=0$ for an incompressible flow. To interpret $Q_{s}{-}R_{s}$ diagrams, note that $R_{s}$ can be written in terms of $\unicode[STIX]{x1D706}_{i}$ as $R_{s}=-\unicode[STIX]{x1D706}_{1}\unicode[STIX]{x1D706}_{2}\unicode[STIX]{x1D706}_{3}$ . One might therefore expect some tendency for sheet-like vortex structures if $R_{s}>0$ (two extensive strain-rate eigenvalues) and some tendency for tube-like vortex structures if $R_{s}<0$ (two compressive strain-rate eigenvalues).

Defining $a=\unicode[STIX]{x1D706}_{2}/\unicode[STIX]{x1D706}_{1}$ , the following equation can be obtained:

The flow geometry can be defined in terms of ratios of the eigenvalues of strain-rate tensor. The ratio $\unicode[STIX]{x1D706}_{1}\,\text{:}\,\unicode[STIX]{x1D706}_{2}\,\text{:}\,\unicode[STIX]{x1D706}_{3}=2\,\text{:}\,-1\,\text{:}\,-1$ ( $a=-1/2$ ) corresponds to axisymmetric contraction, $1\,\text{:}\,0\,\text{:}\,-1$ ( $a=0$ ) is for a two-dimensional flow, $3\,\text{:}\,1\,\text{:}\,-4$ ( $a=1/3$ ) represents biaxial stretching, and $1\,\text{:}\,1\,\text{:}\,-2$ ( $a=1$ ) is for axial stretching (da Silva & Pereira Reference da Silva and Pereira2008).

Figure 17.

Joint PDF of $Q_{s}$ and $R_{s}$ at different locations along the grid-element centreline: (a)  $x/x^{\ast }=0.1$ , (b)  $x/x^{\ast }=0.25$ , (c)  $x/x^{\ast }=0.35$ , (d)  $x/x^{\ast }=0.5$ , (e)  $x/x^{\ast }=0.75$ , (f)  $x/x^{\ast }=0.95$ . The isocontours range from $10^{1}$ to $10^{-2}$ . The four lines from the left to the right in each panel correspond to $\unicode[STIX]{x1D706}_{1}\,\text{:}\,\unicode[STIX]{x1D706}_{2}\,\text{:}\,\unicode[STIX]{x1D706}_{3}=2\,\text{:}\,-1\,\text{:}\,-1$ , $1\,\text{:}\,0\,\text{:}\,-1$ , $3\,\text{:}\,1\,\text{:}\,-4$ and $1\,\text{:}\,1\,\text{:}\,-2$ .

Figure 17 shows the JPDF of $Q_{s}$ and $R_{s}$ in the production and decay regions. There is no real preference for positive or negative $R_{s}$ in the irrotational flow region as shown in figure 17(a). The contour lines align with the ratio of $1\,\text{:}\,0\,\text{:}\,-1$ at $x/x^{\ast }=0.1$ which shows that the irrotational flow is largely two-dimensional. From $x/x^{\ast }=0.25$ onwards the JPDF starts showing a tendency towards positive $R_{s}$ though the preference is very small at $x/x^{\ast }=0.25$ (figure 17 b). From $x/x^{\ast }=0.35$ onwards the JPDF looks similar to what is observed for homogeneous isotropic turbulence with larger probabilities occurring between $\unicode[STIX]{x1D706}_{1}\,\text{:}\,\unicode[STIX]{x1D706}_{2}\,\text{:}\,\unicode[STIX]{x1D706}_{3}=3\,\text{:}\,1\,\text{:}\,-4$ and  $1\,\text{:}\,1\,\text{:}\,-2$ . Here also, the universal characteristics of $Q_{s}$ – $R_{s}$ diagram is obtained for a very low $Re_{\unicode[STIX]{x1D706}}$ homogeneous turbulence of our decay region (see figures 17 e and 17 f).

Figure 18.

(a) Mean trajectory of $Q_{s}$ and $R_{s}$ in the phase map. Triangles represent the production region and circles the decay region. (b) Eigenvalue ratios $\unicode[STIX]{x1D706}_{1}\,\text{:}\,\unicode[STIX]{x1D706}_{2}\,\text{:}\,\unicode[STIX]{x1D706}_{3}$ along the grid-element centreline.

Figure 18 shows the trajectory of $\langle Q_{s}\rangle$ and $\langle R_{s}\rangle$ in their associated phase map. In figure 18(a), the triangles represent locations in the production region as one moves downstream along the grid-element centreline. The value of $\langle R_{s}\rangle$ grows from being negligibly small in the near-grid-element region to being slightly above $0.1\langle \unicode[STIX]{x1D634}_{ij}\unicode[STIX]{x1D634}_{ij}\rangle$ at $x/x^{\ast }\approx 0.4$ then dropping again as the turbulence peak location is approached. The circles represent locations in the decay region. This phase map can also be plotted in terms of $x/x^{\ast }$ as shown in figure 18(b). The first observation from these figures is that the average invariant values are in the quadrant of $Q_{s}<0$ and $R_{s}>0$ , and that they are mostly on the line $a=1$ for $0\leqslant x/x^{\ast }<0.4$ , and mostly on or near the line $a=1/3$ for the decay region. The transition between axial stretching to biaxial stretching occurs in the region of $0.4<x/x^{\ast }<0.5$ . The eigenvalue ratio in this transitional region is close to $2\,\text{:}\,1\,\text{:}\,-3$ . Note that the most probable eigenvalue ratios obtained in most previous studies are $3\,\text{:}\,1\,\text{:}\,-4$ and $2\,\text{:}\,1\,\text{:}\,-3$ (da Silva & Pereira Reference da Silva and Pereira2008).

5.5.1 Eigenvalues of strain-rate tensor and their contribution to enstrophy production

The statistics of the strain-rate eigenvalues are analysed in this section to shed more light onto the mechanisms of enstrophy production and strain self-amplification. This statistical analysis can also help to explain the discrepancy observed in the PDF of $\unicode[STIX]{x1D714}_{i}\unicode[STIX]{x1D714}_{j}\unicode[STIX]{x1D634}_{ij}$ with respect to the PIV study of Gomes-Fernandes et al. (Reference Gomes-Fernandes, Ganapathisubramani and Vassilicos2014).

Table 4.

Values of $\langle \unicode[STIX]{x1D706}_{i}^{3}\rangle$ ( $i=1,2,3$ ) normalised by the mean strain self-amplification at different stations along the grid-element centreline. The field experiment data are taken from Kholmyansky, Tsinober & Yorish (Reference Kholmyansky, Tsinober and Yorish2001), and the homogeneous isotropic turbulence data are taken from the periodic DNS of Onishi, Baba & Takahashi (Reference Onishi, Baba and Takahashi2011) for $Re_{\unicode[STIX]{x1D706}}=80$ .

Table 4 gives the values of $\langle \unicode[STIX]{x1D706}_{i}^{3}\rangle$ ( $i=1,2,3$ ) normalised by $-\langle \unicode[STIX]{x1D634}_{ij}\unicode[STIX]{x1D634}_{jk}\unicode[STIX]{x1D634}_{ki}\rangle$ which is the positive source term in the strain-product equation as shown in figure 10(b). From table 4 it can be seen that both $\langle \unicode[STIX]{x1D706}_{1}^{3}\rangle$ and $\langle \unicode[STIX]{x1D706}_{2}^{3}\rangle$ are positive i.e. there are two extensive strain-rate eigenvalues, an universal feature of three-dimensional turbulent flows. The tendency of the intermediate eigenvalue to be statistically more positive than negative, i.e. $\langle \unicode[STIX]{x1D706}_{2}^{3}\rangle >0$ and $\langle \unicode[STIX]{x1D706}_{2}\rangle >0$ , was predicted for homogeneous isotropic turbulence by Betchov (Reference Betchov1956), first observed in the DNS of Ashurst et al. (Reference Ashurst, Kerstein, Kerr and Gibson1987) and then confirmed by many other studies (e.g. Tsinober et al. Reference Tsinober, Kit and Dracos1992; She, Jackson & Orszag Reference She, Jackson and Orszag1991; Su & Dahm Reference Su and Dahm1996). This universal turbulence result is found to be valid even for this very low $Re_{\unicode[STIX]{x1D706}}$ turbulence. Since $-\langle \unicode[STIX]{x1D634}_{ij}\unicode[STIX]{x1D634}_{jk}\unicode[STIX]{x1D634}_{ki}\rangle =-(\langle \unicode[STIX]{x1D706}_{1}^{3}\rangle +\langle \unicode[STIX]{x1D706}_{2}^{3}\rangle +\langle \unicode[STIX]{x1D706}_{3}^{3}\rangle )$ , the positive sign of $-\langle \unicode[STIX]{x1D634}_{ij}\unicode[STIX]{x1D634}_{jk}\unicode[STIX]{x1D634}_{ki}\rangle$ originates from $-\langle \unicode[STIX]{x1D706}_{3}^{3}\rangle$ . Therefore, the strain self-amplification process is associated, in a time-averaged sense, with a fluid mechanical picture involving stretching and compressing velocity gradients into sheets. The values reported in table 4 approximate those of homogeneous isotropic turbulence in the decay region, although there seems to be a deviation from the measurements taken in very high Reynolds number field experiments. In the production region, the values of $\langle \unicode[STIX]{x1D706}_{i}^{3}\rangle$ normalised by $-\langle \unicode[STIX]{x1D634}_{ij}\unicode[STIX]{x1D634}_{jk}\unicode[STIX]{x1D634}_{ki}\rangle$ are similar to those reported by Gomes-Fernandes et al. (Reference Gomes-Fernandes, Ganapathisubramani and Vassilicos2014) for $i=2$ but up to an order of magnitude smaller for $i=1,3$ . The main difference between our production region and that of Gomes-Fernandes et al. (Reference Gomes-Fernandes, Ganapathisubramani and Vassilicos2014) is the Reynolds number, which is lower in our case, and the presence of irrotational flow, which is prominent in our case given the turbulence generated by all the extra fractal iterations on the fractal grid of Gomes-Fernandes et al. (Reference Gomes-Fernandes, Ganapathisubramani and Vassilicos2014). These fractal iterations seem to greatly increase the intensity of the small-scale stretching and compressing actions. Also, the downstream evolution of the values given in table 4 qualitatively agrees with that of Zhou et al. (Reference Zhou, Nagata, Sakai, Ito and Hayase2016a ).

Figure 19.

PDF of strain-rate eigenvalues at different locations along the grid-element centreline: (a)  $x/x^{\ast }=0.1$ , (b)  $x/x^{\ast }=0.25$ , (c)  $x/x^{\ast }=0.35$ , (d)  $x/x^{\ast }=0.5$ , (e)  $x/x^{\ast }=0.75$ , (f)  $x/x^{\ast }=0.95$ .

Attention is now turned to the PDFs of the strain-rate eigenvalues. As observed in figure 19(a), in the irrotational part of the grid-element centreline, the PDFs of $\unicode[STIX]{x1D706}_{1}$ and $\unicode[STIX]{x1D706}_{3}$ are near mirror images of each other, and the PDF of $\unicode[STIX]{x1D706}_{2}$ is symmetric around 0. As the enstrophy production and strain self-amplification depend on the strain-rate eigenvalues linearly (5.20) and to the power of three (5.21) respectively, any small symmetry breaking in the PDFs of the strain-rate eigenvalues will be more pronounced on the PDF of strain self-amplification than on the PDF of the enstrophy production. This is the reason why the PDF of enstrophy production is perfectly symmetric, and the PDF of strain self-amplification is close to (but not perfectly) symmetric in the irrotational flow region (figures 14 a and 15 a).

The PDF of $\unicode[STIX]{x1D706}_{2}$ becomes positively skewed at $x/x^{\ast }=0.25$ (figure 19 b) where the turbulence is inhomogeneous and the small-scale terms have just started developing, which is also a characteristic feature of small-scale turbulence (Betchov Reference Betchov1975; Ashurst et al. Reference Ashurst, Kerstein, Kerr and Gibson1987; Ganapathisubramani, Lakshminarasimhan & Clemens Reference Ganapathisubramani, Lakshminarasimhan and Clemens2008; Gomes-Fernandes et al. Reference Gomes-Fernandes, Ganapathisubramani and Vassilicos2014). In order to satisfy the incompressibility constraint i.e. $\unicode[STIX]{x1D706}_{1}+\unicode[STIX]{x1D706}_{2}+\unicode[STIX]{x1D706}_{3}=0$ , the PDF of $\unicode[STIX]{x1D706}_{3}$ becomes negatively skewed thus breaking the mirror image symmetry with $\unicode[STIX]{x1D706}_{1}$ . Qualitatively similar behaviour is observed in all other locations as seen in figure 19(c–f). It should be noted here that the study of Gomes-Fernandes et al. (Reference Gomes-Fernandes, Ganapathisubramani and Vassilicos2014) observed symmetry in the PDF of $\unicode[STIX]{x1D706}_{2}$ in the inhomogeneous production region. Our study, however, clearly shows that $\unicode[STIX]{x1D706}_{2}$ is positively skewed. This is related to the discrepancy observed in the skewness of the PDF of enstrophy production which depends on the skewness of the PDF of $\unicode[STIX]{x1D706}_{2}$ . The statistics of $\unicode[STIX]{x1D706}_{2}$ are further analysed at the end of this section. The tendency for the intermediate strain-rate eigenvalue to skew towards positive is found to be true for homogeneous turbulence of $Re_{\unicode[STIX]{x1D706}}$ as low as 33 (figure 19 f).

Figure 20.

PDF of strain-rate eigenvalues contribution to enstrophy production at different locations along the grid-element centreline: (a)  $x/x^{\ast }=0.1$ , (b)  $x/x^{\ast }=0.25$ , (c)  $x/x^{\ast }=0.35$ , (d)  $x/x^{\ast }=0.5$ , (e)  $x/x^{\ast }=0.75$ , (f)  $x/x^{\ast }=0.95$ .

The contribution of each eigenvalue and eigenvector to the enstrophy production is shown in table 5. Note that $\langle \unicode[STIX]{x1D714}^{2}\unicode[STIX]{x1D706}_{1}\cos ^{2}(\unicode[STIX]{x1D74E},\boldsymbol{e}_{\mathbf{1}})\rangle >\langle \unicode[STIX]{x1D714}^{2}\unicode[STIX]{x1D706}_{2}\cos ^{2}(\unicode[STIX]{x1D74E},\boldsymbol{e}_{\mathbf{2}})\rangle >0$ whereas $\langle \unicode[STIX]{x1D714}^{2}\unicode[STIX]{x1D706}_{3}\cos ^{2}(\unicode[STIX]{x1D74E},\boldsymbol{e}_{\mathbf{3}})\rangle <0$ . This observation suggests that the most extensive eigenvector of the strain-rate tensor is the most responsible for the positive enstrophy production. The ratio $\langle \unicode[STIX]{x1D714}^{2}\unicode[STIX]{x1D706}_{1}\cos ^{2}(\unicode[STIX]{x1D74E},\boldsymbol{e}_{\mathbf{1}})\rangle /\langle \unicode[STIX]{x1D714}^{2}\unicode[STIX]{x1D706}_{2}\cos ^{2}(\unicode[STIX]{x1D74E},\boldsymbol{e}_{\mathbf{2}})\rangle$ decreases from its value near the grid-element to its value at the position of the turbulence peak. The ratio $-\langle \unicode[STIX]{x1D714}^{2}\unicode[STIX]{x1D706}_{1}\cos ^{2}(\unicode[STIX]{x1D74E},\boldsymbol{e}_{\mathbf{1}})\rangle /\langle \unicode[STIX]{x1D714}^{2}\unicode[STIX]{x1D706}_{3}\cos ^{2}(\unicode[STIX]{x1D74E},\boldsymbol{e}_{\mathbf{3}})\rangle$ in the production and decay regions is 1.61 and 1.95 respectively. This indicates that the strength of vortex stretching over compression is marginally weaker in the production region compared to the decay region. All three time-averaged quantities in table 5 appear to have a tendency towards approaching the reference HIT values in our restricted decay region, even though they do not quite reach them. Also, this tendency towards reaching a universal value is faster in our simulation compared to the values reported in Zhou et al. (Reference Zhou, Nagata, Sakai, Ito and Hayase2016a ). Although the value of $\langle \unicode[STIX]{x1D714}^{2}\unicode[STIX]{x1D706}_{2}\cos ^{2}(\unicode[STIX]{x1D74E},\boldsymbol{e}_{\mathbf{2}})\rangle$ in the production region is closer to the value reported in Gomes-Fernandes et al. (Reference Gomes-Fernandes, Ganapathisubramani and Vassilicos2014), the other two components are smaller. The reason for this discrepancy is the pronounced irrotational region in our flow configuration which results in weaker stretching/compressing events compared to Gomes-Fernandes et al. (Reference Gomes-Fernandes, Ganapathisubramani and Vassilicos2014) as noted earlier.

Table 5.

Contribution of strain-rate eigenvalues to enstrophy production along the grid-element centreline. The values are normalised by mean enstrophy production. The field experiment data are taken from Kholmyansky et al. (Reference Kholmyansky, Tsinober and Yorish2001), grid experiment data are from Tsinober (Reference Tsinober2009) and the homogeneous isotropic turbulence data are from Onishi et al. (Reference Onishi, Baba and Takahashi2011).

The PDFs of the individual components of enstrophy production are depicted in figure 20. All the components are small close to the grid-element (figures 20 a and 20 b) because of the small values of vorticity. The PDF of $\unicode[STIX]{x1D714}^{2}\unicode[STIX]{x1D706}_{2}\cos ^{2}(\unicode[STIX]{x1D74E},\boldsymbol{e}_{\mathbf{2}})$ is nearly symmetric only at $x/x^{\ast }=0.1$ as seen in figure 20(a). The positively skewed PDF of $\unicode[STIX]{x1D706}_{2}$ in all other locations (figure 19 b–f) is expected to result into positively skewed PDFs of $\unicode[STIX]{x1D714}^{2}\unicode[STIX]{x1D706}_{2}\cos ^{2}(\unicode[STIX]{x1D74E},\boldsymbol{e}_{\mathbf{2}})$ as indeed observed in figure 20(b–f). The tails of the PDFs of $\unicode[STIX]{x1D714}^{2}\unicode[STIX]{x1D706}_{1}\cos ^{2}(\unicode[STIX]{x1D74E},\boldsymbol{e}_{\mathbf{1}})$ and $\unicode[STIX]{x1D714}^{2}\unicode[STIX]{x1D706}_{3}\cos ^{2}(\unicode[STIX]{x1D74E},\boldsymbol{e}_{\mathbf{3}})$ are wider in the production region but they become more narrow in the decay region. This indicates that strong and rare vortex stretching and compressing events are more prevalent in the production region than in the decay region.

5.5.2 Analysis of the intermediate eigenvalue

As observed in table 4, the normalised extensive ( $\unicode[STIX]{x1D706}_{1}$ ) and compressive ( $\unicode[STIX]{x1D706}_{3}$ ) strain-rate eigenvalues vary significantly along the grid-element centreline, while the intermediate one ( $\unicode[STIX]{x1D706}_{2}$ ) remains almost constant. Moreover, the statistical behaviour of this particular eigenvalue is observed to be different in the production region when compared to the PIV study of Gomes-Fernandes et al. (Reference Gomes-Fernandes, Ganapathisubramani and Vassilicos2014). This also causes a discrepancy in the PDF of enstrophy production as the statistical characteristics of $\unicode[STIX]{x1D714}_{i}\unicode[STIX]{x1D634}_{ij}\unicode[STIX]{x1D714}_{j}$ depend on $\unicode[STIX]{x1D706}_{2}$ . These factors prompt further analysis of  $\unicode[STIX]{x1D706}_{2}$ .

Figure 21.

Joint PDF of the normalised intermediate strain-rate eigenvalue with $Q_{s}$ and $Q_{w}$ at $x/x^{\ast }=0.1$ , 0.25 and 0.95 along the grid-element centreline. The isocontours range from $10^{-1}$ to $10^{-3}$ .

Although various normalisations are available (Ganapathisubramani et al. Reference Ganapathisubramani, Lakshminarasimhan and Clemens2008), we follow Ashurst et al. (Reference Ashurst, Kerstein, Kerr and Gibson1987) and normalise $\unicode[STIX]{x1D706}_{2}$ as

With this normalisation, $\unicode[STIX]{x1D706}_{2}^{\ast }$ is in the interval $[-1,+1]$ owing to the incompressibility constraint. The extreme values $+1,-1$ represent axisymmetric expansion and contraction respectively.

The joint PDFs of $\unicode[STIX]{x1D706}_{2}^{\ast }$ with $Q_{s}$ ( $=\unicode[STIX]{x1D634}_{ij}\unicode[STIX]{x1D634}_{ij}/2$ ) and $Q_{w}$ ( $=\unicode[STIX]{x1D714}_{i}\unicode[STIX]{x1D714}_{i}/4$ ) are shown in figure 21 for three locations along the grid-element centreline: close to the grid-element, location where the $-5/3$ slope in energy spectrum is first observed for the inhomogeneous turbulence, and in the decay region. As can be seen, $\unicode[STIX]{x1D706}_{2}^{\ast }$ is indeed bounded by $\pm 1$ , and this gives further confidence in our data and the procedure followed in normalising and computing the joint PDF. Close to the grid-element (where $x/x^{\ast }=0.1$ , see figure 21 a), the JPDFs of $\unicode[STIX]{x1D706}_{2}^{\ast }-Q_{s}$ and $\unicode[STIX]{x1D706}_{2}^{\ast }-Q_{w}$ are slightly skewed towards positive values of ${\unicode[STIX]{x1D706}_{2}}^{\ast }$ . This indicates that although $\langle \unicode[STIX]{x1D706}_{2}\rangle \approx 0$ , the normalised eigenvalue $\langle \unicode[STIX]{x1D706}_{2}^{\ast }\rangle >0$ . Further downstream, the JPDFs become more positively skewed (figures 21 b and 21 c). Notice that the asymmetry is more pronounced in the $\unicode[STIX]{x1D706}_{2}^{\ast }{-}Q_{s}$ plane compared to $\unicode[STIX]{x1D706}_{2}^{\ast }-Q_{w}$ at $x/x^{\ast }=0.25$ . This could be perhaps due to the fact that at this location strain strongly dominates enstrophy (see table 3).

It is interesting to notice that in the decay region where the turbulence is most homogeneous, the JPDF peaks at $\unicode[STIX]{x1D706}_{2}^{\ast }$ close to 0.5, in very good agreement with previous studies including some which concerned periodic turbulence (Betchov Reference Betchov1975; Ashurst et al. Reference Ashurst, Kerstein, Kerr and Gibson1987; Ganapathisubramani et al. Reference Ganapathisubramani, Lakshminarasimhan and Clemens2008; Gomes-Fernandes et al. Reference Gomes-Fernandes, Ganapathisubramani and Vassilicos2014).

5.5.3 Alignment of vorticity vector and strain-rate eigenvectors

Finally, we discuss the geometric alignments between the vorticity vector and the eigenvectors of the strain-rate tensor as they influence the enstrophy production term (see (5.20)).

Figure 22.

PDF of absolute value of the cosine of the angle between the vorticity vector and the eigenvectors of the strain-rate tensor at different locations along the grid-element centreline: (a)  $x/x^{\ast }=0.25$ , (b)  $x/x^{\ast }=0.35$ , (c)  $x/x^{\ast }=0.5$ , (d)  $x/x^{\ast }=0.75$ , (e)  $x/x^{\ast }=0.95$ .

Figure 22 shows the PDFs of the absolute value of cosine of the angle between the vorticity vector and the strain-rate eigenvectors. These alignments are considered at all locations except at $x/x^{\ast }=0.1$ where the magnitude of vorticity is very small. At $x/x^{\ast }=0.25$ , where the turbulence is inhomogeneous and strain dominates enstrophy, both the extensive and intermediate strain-rate eigenvectors align with the vorticity vector, although the alignment is weak. This behaviour differs from the universal behaviour of strong alignment between the vorticity vector with the intermediate strain-rate eigenvector in turbulent flows (Ashurst et al. Reference Ashurst, Kerstein, Kerr and Gibson1987; Kerr Reference Kerr1987). Further downstream along the grid-element centreline, the alignment of both the extensive strain-rate eigenvectors with vorticity becomes stronger within the inhomogeneous production region (figures 22 b and 22 c). This evidence suggests that, both the positive strain-rate eigenvectors act effectively to produce enstrophy in the production region, and can explain the rapid growth of the vortex stretching term from $x/x^{\ast }=0.25$ to the turbulence peak location (see figure 8 a). The universal alignment behaviour is encountered only in the decay region (figures 22 d and 22 e) even though the local $Re_{\unicode[STIX]{x1D706}}$ there is only between 30 and 40. In this region, the compressive strain-rate eigenvector aligns perpendicularly to vorticity and the extensive strain-rate eigenvector is indifferent to vorticity. Zhou et al. (Reference Zhou, Nagata, Sakai, Ito and Hayase2016a ) also find that the alignment with the intermediate eigenvector is gradual.

Why is the alignment behaviour different in the production region? As explained in § 5.1.3, vorticity is produced at the grid-element bars, is shed in to the wake, and then it is transported towards the grid-element centreline by turbulent fluctuations. There it finds itself in an already developed fluctuating strain field which has been produced due to the action of the pressure Hessian. In other words, most of the strain field at about $x/x^{\ast }=0.2$ or so on the grid-element centreline has not been created by the vorticity, and the two fields are uncoupled. Further downstream, vorticity and strain start to interact but this interaction is gradual and is completed only in the decay region, where the alignments show the universal behaviour.

This scenario is consistent with the findings of Hamlington, Schumacher & Dahm (Reference Hamlington, Schumacher and Dahm2008a ). Following Jiménez (Reference Jiménez1992), these authors decomposed the fluctuating strain field at every point in the domain into two components: one (local) component produced by the local vorticity field around the point in question and another (non-local or background) component produced by vorticity further away. Their numerical simulations in forced, homogeneous isotropic turbulence at $Re_{\unicode[STIX]{x1D706}}=107$ demonstrated that while the intermediate strain-rate eigenvector of the combined strain aligns with the vorticity vector, it is the most extensive eigenvector of the background strain component that aligns with vorticity. Therefore, when most of the strain is not produced by the vorticity, as is the case in much of the production region, it acts as background strain and the alignment differs from the well-known universal behaviour. As a result, the extensional strain-rate eigenvector is more strongly aligned with vorticity.

In the present study, fluctuating vorticity is absent for $x/x^{\ast }<0.2$ , so the alignment cannot be computed. Once vorticity is present, the results show equal tendency for alignment between the vorticity vector and the two positive strain-rate eigenvectors. The above analysis suggests that the already existing and dominant background strain (which is the result of non-local pressure) is responsible for the alignment of the most extensive strain-rate eigenvector with vorticity and that this effect coexists in this region with the alignment of the vorticity with its own strain field’s intermediate eigenvector. Indeed, Hamlington, Schumacher & Dahm (Reference Hamlington, Schumacher and Dahm2008b ) found that the intermediate eigenvector of the vorticity-induced strain tensor aligns very strongly with vorticity. As fluid elements move downstream towards the peak of turbulence intensity and into the decay region, the effect of the vorticity on the strain field becomes stronger and eventually dominates over the background strain, resulting in the universal alignment behaviour.

We close this section by comparing our alignment results with those of the experimental study of Gomes-Fernandes et al. (Reference Gomes-Fernandes, Ganapathisubramani and Vassilicos2014). They found the universal alignment behaviour even in the middle of the production region. It has to be borne in mind, however, that there are two significant differences compared to our set-up. Firstly, they studied a space-filling fractal grid and not a single square grid-element. The fractal grid has small and thin bars located close to and around the grid-element centreline that shed wakes which meet much closer to the grid-element compared to the large-scale wakes in our case. Secondly, the free-stream turbulence intensity was 2.8 % for the streamwise velocity component and 4.4 % for the spanwise velocity component in their experiment. These two differences indicate that vorticity was already present much closer to the grid-element in their case compared to ours, and therefore strain and vorticity were very early much more closely coupled. This explains why they see the universal alignment even in the first location they studied, which was in the middle of the production region.