3.2.1. High-speed and low-speed large-scale streamwise motions

The instantaneous flow fields shown in figure 7 visually demonstrate the positive skewness of the streamwise velocity fluctuations in the outer layer. The positive skewness of streamwise velocity fluctuations (figure 8) strongly suggests that high-speed and low-speed motions have different properties, especially in the outer layer.

The two-point correlations in figure 9 demonstrate large-scale coherence. However, they do not shed any light on the nature of the large-scale coherence, i.e. whether low-speed or high-speed motions are responsible for large-scale streamwise coherence. It is also unclear from the instantaneous velocities in figure 7 whether the length scales of high-speed motions are greater/less than the length scales of low-speed motions. It is investigated here by calculating the two-point conditional correlations of the streamwise velocity fluctuations. The two-point conditional correlations of the streamwise velocity fluctuations are defined as

(3.2)\begin{equation} R^{CP}_{u_2 u_2} = \frac{\langle u_2(x_2,x_3) > 0\,|\, u_2(x_2 + \Delta x_2,x_3 + \Delta x_3)\rangle}{\sigma_{u_2} \,|\, \sigma_{u_2} > 0}, \end{equation}

and

(3.3)\begin{equation} R^{CN}_{u_2 u_2} = \frac{\langle u_2(x_2,x_3) < 0\,|\, u_2(x_2 + \Delta x_2,x_3 + \Delta x_3)\rangle}{\sigma_{u_2} \,|\, \sigma_{u_2} < 0}, \end{equation}

for positive and negative streamwise velocity fluctuations, respectively, similar to the two-point conditional correlations of Lee & Sung (Reference Lee and Sung2011) and Sillero et al. (Reference Sillero, Jiménez and Moser2014).

The two-point streamwise–spanwise conditional correlation coefficients of the high-speed (positive) and low-speed (negative) streamwise velocity fluctuations at four different wall-normal locations are shown in figure 10. The two-point conditional correlation coefficients shown in the figure demonstrate that the high-speed and low-speed streamwise velocity fluctuations exhibit large-scale streamwise coherence and can form streamwise-elongated motions. The streamwise-elongated regions of positive correlations are flanked by regions of negative correlations, exhibiting similarities to the streamwise–spanwise correlation shown in figure 9. This again suggests the possibility of alternative regions of high-speed and low-speed large-scale streamwise motions, similar to the LSMs in the logarithmic layer of wall-bounded turbulence (Ganapathisubramani et al. Reference Ganapathisubramani, Hutchins, Hambleton, Longmire and Marusic2005; Hutchins & Marusic Reference Hutchins and Marusic2007; Baltzer et al. Reference Baltzer, Adrian and Wu2013; Kevin et al. Reference Kevin, Monty and Hutchins2019). Both the conditional correlations contrast the observations of Hattori et al. (Reference Hattori, Tsuji, Nagano and Tanaka2006) regarding unstratified NCBLs. In that study, regions of negative two-point correlation coefficient (no conditional sampling applied) did not flank a region of positive two-point correlation coefficient (no conditional sampling applied) in the spanwise direction. The ‘X-shaped’ correlation pattern is also visible in the outer layer, implying the meandering nature of the LSMs. As the pattern is consistent for positive and negative streamwise velocity fluctuations, it can be inferred that both the positive and negative LSMs meander appreciably (see § 3.2.2 on the meandering of LSMs).

Figure 10.

Streamwise–spanwise two-point conditional correlation coefficient map at (a) $x_1 / \delta _{bl} = 0.04$(b) $x_1 / \delta _{bl} = 0.1$, (c) $x_1 / \delta _{bl} = 0.3$ and (d) $x_1 / \delta _{bl} = 0.6$. Red contours represent $R^{CP}_{u_2 u_2}$ and black contours represent $R^{CN}_{u_2 u_2}$. The contour lines are $-$0.05, 0.05, 0.15 and 0.25. Negative values are represented using dashed contours.

The two-point correlations of the positive streamwise fluctuations are biased towards the upstream direction. In contrast, the two-point correlations of the negative streamwise fluctuations are biased towards the downstream direction. This could be due to the asymmetric presence of different quadrant events in the outer layer of the buoyancy layer, similar to the asymmetry observed in turbulent boundary layers (Lee & Sung Reference Lee and Sung2011). It should be noted that the positive streamwise fluctuations are biased towards the upstream direction, and the negative streamwise fluctuations are biased towards the downstream direction in turbulent boundary layers, opposite to what is observed in the present case.

In figure 10, the spatial extent of the negative correlation coefficient in the outer layer is larger than the regions of the negative correlation coefficient in the inner layer. Figure 11 shows the positive two-point conditional correlations’ streamwise ($l_{x_2}$) and spanwise length scales ($l_{x_3}$). The length scales are identified as regions where the two-point correlation coefficients are greater than 0.05, which aligns with the definition used in (Hutchins, Hambleton & Marusic Reference Hutchins, Hambleton and Marusic2005; Monty et al. Reference Monty, Stewart, Williams and Chong2007; Lee & Sung Reference Lee and Sung2011, Reference Lee and Sung2013; Lee Reference Lee2017). The streamwise length shown in figure 11(a) increases with increasing wall-normal distance. This behaviour differs from what is observed in turbulent boundary layers, where the large-scale streamwise coherence, in the form of two-point correlations, peaks in the logarithmic layer and reduces in the wake region (Ganapathisubramani et al. Reference Ganapathisubramani, Hutchins, Hambleton, Longmire and Marusic2005; Lee & Sung Reference Lee and Sung2011; Sillero et al. Reference Sillero, Jiménez and Moser2014). It is conjectured that the interaction of the boundary layer flow with the flow reversal region generates large-scale coherence even at the edge of the boundary layer; however, it is not explored further as this interaction is outside the scope of the current study. The streamwise length of the positive streamwise velocity fluctuations is smaller than the negative streamwise velocity fluctuations, suggesting that the negative streamwise velocity fluctuations tend to form longer structures than the positive streamwise velocity structures. Despite the variation in the streamwise length scales between the positive and negative streamwise velocity fluctuations, it should be noted that the streamwise length scales at all the wall-normal distances investigated are always greater than the boundary layer thickness.

Figure 11.

Wall-normal variation of the length scales of the two-point correlations. (a) Streamwise length scales, (b) spanwise length scales and (c) the ratio of the streamwise to the spanwise length scales; AV corresponds to the two-point correlation of the streamwise velocity fluctuations, PV corresponds to the conditional two-point correlation of positive streamwise velocity fluctuations and NV corresponds to the conditional two-point correlation of negative streamwise velocity fluctuations.

Figure 11(b) shows the spanwise length scales. The spanwise length of positive two-point correlations is smaller than the streamwise length of positive two-point correlations, consistent with the earlier discussion. Also, unlike the streamwise length scales, the spanwise length scale does not monotonically increase with increasing wall-normal distance.

The spanwise length is highest close to the wall, drops to a minimum at around $x_1 = 0.2 \delta _{bl}$, and increases once again with increasing wall-normal distance. The spanwise length of the negative streamwise velocity fluctuations is greater than that of the positive streamwise velocity fluctuations until $x_1 < 0.5 \delta _{bl}$. However, the trend reverses at $x_1 > 0.5 \delta _{bl}$ with the spanwise length of the positive streamwise velocity fluctuations becoming greater than the spanwise length of the negative streamwise velocity fluctuations. It is conjectured that differences in the mean shear in the inner and the outer layers of the buoyancy layer induce changes to the spanwise scales of LSMs.

The ratio of the streamwise length to the spanwise length of the two-point correlations ($l_{x_2}/l_{x_3}$) is shown in figure 11(c). Close to the wall, $l_{x_2}/l_{x_3}$ exhibits a minimum and increases rapidly until $x_1 \leq 0.3 \delta _{bl}$. Also, close to the wall, the plot is almost identical for positive and negative streamwise velocity fluctuations, suggesting a similarity. The ratio of the streamwise length to the spanwise length approximately attains a constant value in the outer layer for the two-point correlation of the unconditional streamwise velocity fluctuations and the two-point conditional correlation of the negative streamwise velocity perturbations. However, the ratio decreases with increasing wall-normal distance for positive streamwise velocity fluctuations, implying that the aspect ratio of the LSMs of the positive streamwise velocity fluctuations differs from the aspect ratio of the LSMs of the negative streamwise velocity fluctuations. It should be noted that the ratio is always greater than 1 for both positive and negative streamwise velocity fluctuations, signifying that the streamwise length of the two-point correlations is always greater than the spanwise length across the entire thickness of the boundary layer. This strongly implies that the LSMs are anisotropic, agreeing with the observations of Hattori et al. (Reference Hattori, Tsuji, Nagano and Tanaka2006) regarding unstratified vertical NCBLs.

Also, the viscous forces are not dominant in most of the outer layer at $Re = 800$ (regions where velocity variances and Reynolds shear stress are dominant – see § 3.1). This is demonstrated in Appendix A using a force balance approach, similar to the one employed in canonical wall turbulence (Fife et al. Reference Fife, Wei, Klewicki and McMurtry2005; Wei et al. Reference Wei, Fife, Klewicki and McMurtry2005). This suggests that the results presented regarding the LSMs in the outer layer only have marginal low-Reynolds number effects.

3.2.2. Meandering of large-scale streamwise motions

From the streamwise-spanwise two-point correlations shown in figures 9 and 10, especially in the outer layer, it is clear that the large-scale streamwise motions meander significantly in the spanwise direction. The streamwise velocity fluctuations shown in figure 7 also reveal that the LSMs are not perfectly aligned in the streamwise direction and are offset diagonally, exhibiting signs of meandering motion.

In this section, motivated by the analysis of Sillero et al. (Reference Sillero, Jiménez and Moser2014) and de Silva et al. (Reference de Silva, Kevin, Baidya, Hutchins and Marusic2018), the meandering of LSMs is investigated in relation to the fluctuating spanwise velocity. To this end, the two-point correlation coefficients of the streamwise velocity fluctuations conditioned on the signs of streamwise and spanwise velocity fluctuations are calculated. The two-point conditional correlations of high-speed and low-speed motions of the streamwise velocity fluctuations are analogous to the two-point conditional correlations shown in (3.2) and (3.3), except that they are also conditioned based on the sign of spanwise velocity fluctuations. It should be noted that no thresholding is applied, and the conditional correlation only depends on the sign of streamwise and spanwise velocity fluctuations. It is identical to the conditional two-point correlation used by Sillero et al. (Reference Sillero, Jiménez and Moser2014) and de Silva et al. (Reference de Silva, Kevin, Baidya, Hutchins and Marusic2018). The conditional correlation is defined as

(3.4)\begin{equation} R^{CPW}_{u_2 u_2} = \frac{\langle u_2(x_2,x_3) u_2(x_2 + \Delta x_2,x_3 + \Delta x_3)\rangle \,|\,u_3 > 0}{\sigma^2_{u_2} \,|\,u_3 > 0}, \end{equation}

for positive spanwise velocity fluctuations and

(3.5)\begin{equation} R^{CNW}_{u_2 u_2} = \frac{\langle u_2(x_2,x_3) u_2(x_2 + \Delta x_2,x_3 + \Delta x_3)\rangle \,|\, u_3 < 0}{\sigma^2_{u_2} \,|\,u_3 < 0}, \end{equation}

for negative spanwise velocity fluctuations.

The streamwise–spanwise two-point correlations of streamwise velocity fluctuations conditioned on the signs of streamwise and spanwise velocity fluctuations are shown in figures 12 and 13. The high-speed and low-speed streamwise velocity correlations exhibit spanwise drift in the inner and outer layers. This statistical signature is similar to what is observed by de Silva et al. (Reference de Silva, Kevin, Baidya, Hutchins and Marusic2018) in the logarithmic region of the turbulent boundary layer, implying that the meandering of the LSMs in the turbulent buoyancy layer is related to spanwise velocity fluctuations.

Figure 12.

Streamwise–spanwise two-point conditional correlation coefficient $R^{CPW}_{u_2 u_2}$ of fast-moving structures at (a,e) $x_1 / \delta _{bl} = 0.04$ (b,f) $x_1 / \delta _{bl} = 0.1$, (c,g) $x_1 / \delta _{bl} = 0.3$ and (d,h) $x_1 / \delta _{bl} = 0.6$. Red contours in (a–d) represent $R^{CPW}_{u_2 u_2}$ of positive $u_3$ and black contours in (e–h) represent $R^{CPW}_{u_2 u_2}$ of negative $u_3$. The contour lines are $-$0.05, 0.05, 0.15 and 0.25. Negative values are represented using dashed contours.

Figure 13.

Streamwise–spanwise two-point conditional correlation coefficient $R^{CNW}_{u_2 u_2}$ of slow-moving structures at (a,e) $x_1 / \delta _{bl} = 0.04$ (b,f) $x_1 / \delta _{bl} = 0.1$, (c,g) $x_1 / \delta _{bl} = 0.3$ and (d,h) $x_1 / \delta _{bl} = 0.6$. Red contours (a–d) represent $R^{CNW}_{u_2 u_2}$ of positive $u_3$ and black contours (e–h) represent $R^{CNW}_{u_2 u_2}$ of negative $u_3$. The contour lines are $-$0.05, 0.05, 0.15 and 0.25. Negative values are represented using dashed contours.

In terms of the correlations presented here, spanwise velocity fluctuations affect both high-speed and low-speed LSMs equally. The orientation of the two-point conditional correlation coefficients is similar across the thickness of the boundary layer, implying that the meandering due to spanwise velocity fluctuations is consistent. However, this does not indicate that the interaction mechanisms are the same. For fast-moving streamwise velocity fluctuations (shown in figure 12), positive spanwise velocity fluctuations induce a positive spanwise drift (oriented to the right in the streamwise direction of the flow). In contrast, negative spanwise velocity fluctuations cause a negative spanwise drift (oriented to the left in the streamwise direction of the flow). The opposite is the case for the slow-moving velocity fluctuations (shown in figure 13). Positive spanwise velocity fluctuations induce a negative spanwise drift, and negative spanwise velocity fluctuations cause a positive spanwise drift in low-speed LSMs. This suggests that positive and negative spanwise velocity fluctuations interact differently with high-speed and low-speed motions.

For the given thresholding of spanwise velocities fluctuations, it is interesting to note that the spanwise drift of the LSMs is more significant in the turbulent buoyancy layer than the turbulent boundary layer. In turbulent boundary layers, solely in terms of two-point correlations, for the given threshold, the spanwise drift is significant at the edge of the boundary layer. It is only marginal in the logarithmic layer (de Silva et al. Reference de Silva, Kevin, Baidya, Hutchins and Marusic2018). In the present case, a high degree of preferential orientation of the two-point conditional streamwise correlations is observed at all the four wall-normal locations shown in figures 12 and 13. This suggests that the spanwise velocity fluctuations in the turbulent buoyancy layer can induce a more significant spanwise drift to the LSMs than the turbulent boundary layer.

3.2.3. Wall-normal coherence

The two-point correlations above reveal LSMs’ streamwise and spanwise coherence in the turbulent buoyancy layer. In this section, the two-point wall-normal correlation is calculated to investigate the streamwise velocity fluctuations’ wall-normal coherence. The two-point wall-normal correlation is calculated using

(3.6)\begin{equation} R^{W}_{u_2 u_2} = \frac{\langle u_2(x_{1 ref},x_2) u_2(x_1,x_2 + \Delta x_2)\rangle}{\sigma_{u_2(x_{1 ref})} \sigma_{u_2(x_1)} }, \end{equation}

where $\sigma _{u_2(x_{1ref})}$ is the standard deviation at a reference wall-normal location and $\sigma _{u_2(x_1)}$ is the standard deviation at a wall-normal location $x_1$.

Figure 14 shows the statistically averaged structure arising from the two-point wall-normal correlation of the streamwise velocity fluctuations. The reference height is chosen as $x_1 = 0.065 \delta _{bl}$, which is the wall-normal location where the mean streamwise velocity is maximum. This also corresponds to the location in the boundary layer where the mean shear is zero.

Figure 14.

Wall-normal–streamwise two-point correlation coefficient ($R^{W}_{u_2 u_2}$) map of streamwise velocity fluctuations. The black contour lines are equally spaced from 0.1 to 0.9. The red cross represents the reference location, and the black dashed line separates the inner and outer layers. Gravity ($\boldsymbol {g}$) acts in the negative $x_2$ direction (downwards), and the fluid flows in the positive $x_2$ direction (upwards).

It is evident from the figure that the streamwise velocity fluctuations exhibit coherence over significant wall-normal distances. The positive two-point correlation coefficient contours are present for almost the entire boundary layer thickness. Large-scale coherence is also present in the downstream and upstream directions, along with substantial wall-normal correlation. This implies that, statistically, large-scale structures are responsible for turbulence in the turbulent buoyancy layer. It should be noted that the presence of non-zero values for two-point correlations across significant wall-normal distances is also observed in unstratified turbulent NCBLs (Tsuji et al. Reference Tsuji, Nagano and Tagawa1992; Abedin et al. Reference Abedin, Tsuji and Lee2012; Nakao et al. Reference Nakao, Hattori and Suto2017).

The differences in the shape of the statistical structure shown in figure 14 and the wall-normal statistical structure of canonical wall-bounded turbulence are immediately evident. In canonical wall-bounded turbulence, a ramp-like structure inclined in the downstream direction of the flow is observed, which is believed to be the consequence of a packet of hairpin-like vortex structures (Christensen & Adrian Reference Christensen and Adrian2001; Ganapathisubramani et al. Reference Ganapathisubramani, Hutchins, Hambleton, Longmire and Marusic2005; Adrian Reference Adrian2007; Marusic & Adrian Reference Marusic and Adrian2012; Baltzer et al. Reference Baltzer, Adrian and Wu2013; Hwang et al. Reference Hwang, Lee, Sung and Zaki2016). In the turbulent buoyancy layer, a ramp-like structure inclined in the downstream direction is observed in the inner layer. In contrast, the ramp-like structure is inclined in the upstream direction in the outer layer. The differences between the wall-normal coherence of the LSMs in the turbulent buoyancy layer and canonical wall-bounded turbulence can be attributed to the presence of different vortex structures due to the change in the sign of the mean shear in the outer layer (Nakao et al. Reference Nakao, Hattori and Suto2017). At this stage, the role of hairpin vortices or similar in NCBL transition and turbulence is not fully understood (Pallares et al. Reference Pallares, Vernet, Ferre and Grau2010; Abramov et al. Reference Abramov, Smirnov and Goryachev2014; Nakao et al. Reference Nakao, Hattori and Suto2017; Zhao et al. Reference Zhao, Lei and Patterson2017, Reference Zhao, Lei and Patterson2019) and no definitive conclusions can be drawn that relate hairpin-like vortex structures observed in prior studies to the two-point correlations discussed in this study.

To quantify the tilt of the two-point correlation shown in figure 14, the structure inclination angle (Carper & Porté-Agel Reference Carper and Porté-Agel2004; Marusic & Heuer Reference Marusic and Heuer2007; Chauhan et al. Reference Chauhan, Hutchins, Monty and Marusic2013; Deshpande, Monty & Marusic Reference Deshpande, Monty and Marusic2019) is calculated using the velocity maximum as the reference position. It is calculated as

(3.7)\begin{equation} \theta = \arctan(x_1^* / \Delta s), \end{equation}

where $\theta$ is the structural inclination angle, $x_1^*$ is the absolute wall-normal separation distance between the velocity maximum and the location of interest, and $\Delta s$ is the streamwise spatial separation corresponding to the peak of correlation $R^{W}_{u_2 u_2}$. Note that $\Delta s$ can take both positive and negative values, with positive values indicating a spatial separation in the downstream direction and negative values indicating a spatial separation in the upstream direction.

Figure 15 shows the structure inclination angle at different wall-normal locations in the turbulent buoyancy layer. A single dominant inclination angle is absent, agreeing with figure 14. In the inner layer, $\theta$ is positive and is around $24^\circ$. In the outer layer, $\theta$ is approx. $55^\circ$ at small separation distances and decays to approx. $-25 ^\circ$ with increasing wall-normal distance. Here, positive values of $\theta$ indicate that $\Delta s$ is positive and that the structure is inclined in the downstream direction. On the other hand, negative values of $\theta$ indicate that $\Delta s$ is negative and that the structure is inclined in the upstream direction.

Figure 15.

Variation of structure inclination angle $\theta$ with respect to the wall-normal location. The red dashed line indicates the reference position, which is also the location of the velocity maximum.

The structures responsible for the large-scale wall-normal coherence are also visible in the instantaneous flow fields. Figure 16 shows the instantaneous streamwise velocity field across the mid-span of the domain at three different times (the exact time instant is irrelevant as the flow fields represent developed turbulence that is statistically homogeneous in the streamwise ($x_2$) and spanwise ($x_3$) directions). It is clear from the figure that the large-scale flow structures spanning almost the entire boundary layer thickness populate the buoyancy layer. This is reminiscent of the large-scale eddies observed in unstratified vertical NCBLs (Fujii Reference Fujii1959; Vliet & Liu Reference Vliet and Liu1969; Tsuji & Nagano Reference Tsuji and Nagano1988b; Hattori et al. Reference Hattori, Tsuji, Nagano and Tanaka2006; Abedin et al. Reference Abedin, Tsuji and Kim2017), highlighting the qualitative similarities between the two flows. Close to the wall ($x_1 \lessapprox 0.2$), the velocity flow structures are inclined in the downstream direction, while away from the wall ($x_1 \gtrapprox 0.2$), the flow structures are tilted in the upstream direction, agreeing with the two-point correlation and structure inclination angle plots in figures 14 and 15, respectively.

Figure 16.

Three different snapshots of the instantaneous streamwise velocity ($\tilde {u}_2$) field at the mid-span of the domain. Gravity ($\boldsymbol {g}$) acts in the negative $x_2$ direction, and the fluid flows in the positive $x_2$ direction. Only a portion of the entire domain is shown.

3.2.4. Role of LSMs in turbulence production

In turbulent buoyancy layers, the Reynolds shear stress and the streamwise turbulent heat flux are responsible for the production of TKE (Giometto et al. Reference Giometto, Katul, Fang and Parlange2017; Maryada et al. Reference Maryada, Armfield, Dhopade and Norris2022). The shear production ($P_S$) and buoyancy flux ($P_B$) are defined as

(3.8a)\begin{gather} P_S ={-} \langle u_1 u_2 \rangle \frac{\partial \bar{u_2}}{\partial x_1}, \end{gather}

(3.8b)\begin{gather}P_B = \frac{2}{Re} \langle u_1 \vartheta \rangle. \end{gather}

The wall-normal variation of the shear production and the buoyancy flux are shown in figure 17. It is clear from the figure that both shear and buoyancy are dominant producers of TKE in the vertical buoyancy layer. Most of the TKE is produced in the outer layer at the Prandtl number and Reynolds number investigated, consistent with the DNS results of Giometto et al. (Reference Giometto, Katul, Fang and Parlange2017). It should be noted that despite the flow being driven by buoyancy, shear still dominates over buoyancy in terms of TKE production. The relative dominance of shear over buoyancy was also observed in unstratified NCBLs and transitional buoyancy layers having a Prandtl number of $0.71$ (Janssen & Armfield Reference Janssen and Armfield1996; Maryada et al. Reference Maryada, Armfield, Dhopade and Norris2022). Similarities between the current flow at $Pr = 0.71$ and the flow at $Pr = 1$ (Giometto et al. Reference Giometto, Katul, Fang and Parlange2017) are also immediately evident, where shear dominates TKE production. This is in contrast to transitional unstratified vertical NCBLs at $Pr = 7.0$ where buoyancy dominates (Zhao et al. Reference Zhao, Lei and Patterson2017).

Figure 17.

Wall-normal variation of the production of TKE due to shear ($P_S$) and buoyancy ($P_B$). The black dot-dashed vertical line represents the wall-normal location of the velocity maximum, which is used to demarcate the inner and the outer layers.

In the inner layer, the shear production is positive in a layer close to the wall. In a region that is bounded by this layer and the velocity maximum, the shear production is negative. It indicates counter-gradient turbulent flux, a characteristic feature of turbulent buoyancy layers (Giometto et al. Reference Giometto, Katul, Fang and Parlange2017). In this region, the buoyancy flux is positive and is responsible for most of the production of TKE. In the outer layer, $P_S$ mostly dominates over $P_B$ until the edge of the boundary layer. Near the edge of the boundary layer, at $x_1 > 0.6 \delta _{bl}$, $P_B$ is negative, suggesting that buoyancy at these wall-normal locations is responsible for the destruction of TKE. The dual role of buoyancy flux in producing and destroying TKE was also observed in transitional buoyancy layers (Maryada et al. Reference Maryada, Armfield, Dhopade and Norris2022).

Figure 17 only indicates the wall-normal locations where the TKE production is dominant. The two-point cross-correlations of $u_1$ and $u_2$ ($R^{CC}_{u_1 u_2}$), and $u_2$ and $\vartheta$ ($R^{CC}_{u_2 \vartheta }$) are calculated to understand the momentum and heat transfer associated with the LSMs. These are defined as

(3.9a)\begin{gather} R^{CC}_{u_1 u_2} = \frac{\langle u_1(x_2,x_3) u_2(x_2 + \Delta x_2,x_3 + \Delta x_3)\rangle}{\sigma_{u_1} \sigma_{u_2}}, \end{gather}

(3.9b)\begin{gather}R^{CC}_{u_2 \vartheta} = \frac{\langle u_2(x_2,x_3) \vartheta(x_2 + \Delta x_2,x_3 + \Delta x_3)\rangle}{\sigma_{u_2} \sigma_{\vartheta}}. \end{gather}

The two-point cross-correlations of Reynolds shear stress and streamwise turbulent heat flux at $x_1 = 0.04 \delta _{bl}$ and $x_1 = 0.3 \delta _{bl}$ are shown in figure 18. The wall-normal location of $x_1 = 0.04 \delta _{bl}$ represents the wall-normal location in the inner layer where the buoyancy variance is significant. In comparison, $x_1 = 0.3 \delta _{bl}$ represents the wall-normal location in the outer layer where the wall-normal velocity variance and Reynolds shear stress are significant (see figure 4).

Figure 18.

Streamwise–spanwise two-point cross-conditional correlation coefficient map of Reynolds shear stress $R^{CC}_{u_1 u_2}$ and streamwise turbulent heat flux $R^{CC}_{u_2 \vartheta }$ at (a,b) $x_1 / \delta _{bl} = 0.04$ and (c,d) $x_1 / \delta _{bl} = 0.3$. Black contours in (a,c) represents $R^{CC}_{u_1 u_2}$ and red contours in (b,d) represents $R^{CC}_{u_2 \vartheta }$. The contour lines are $-$0.05, 0.05, 0.15 and 0.25. Negative values are represented using dashed contours.

It is evident from figure 18(a) that, in the inner layer, the two-point cross-correlation of the Reynolds shear stress does not extend to large streamwise and spanwise distances, implying the absence of large-scale coherence. This suggests that the high-speed or low-speed LSMs are not strongly correlated with the wall-normal velocity fluctuations.

In figure 18(c), large-scale coherence of Reynolds shear stress is observed in the outer layer, albeit with a shorter length scale than streamwise velocity fluctuations (see figure 9). The two-point cross-correlation of Reynolds shear stress in a turbulent boundary layer also has a shorter streamwise length scale than the streamwise length scale of the two-point correlation of streamwise velocity fluctuations (Sillero et al. Reference Sillero, Jiménez and Moser2014).

The two-point correlation map in figure 18(c) shows a region of positive correlation flanked by regions of negative correlation. This implies that the high-speed streamwise velocity fluctuations are correlated with positive wall-normal velocity fluctuations, and the low-speed streamwise velocity fluctuations are correlated with negative wall-normal velocity fluctuations. Therefore, the high-speed fluid in the outer layer comprises an upwash flow, while the low-speed fluid in the outer layer comprises a downwash flow.

The mean Reynolds shear stress is negative in the inner layer and positive in the outer layer of the vertical buoyancy layer (see figure 4b), and the positive value of the cross-correlation near $\Delta x_2 = \Delta x_3 = 0$ is consistent with the sign of the Reynolds shear stress in the outer layer.

If a decomposition of the Reynolds shear stress is made according to quadrant analysis (Wallace Reference Wallace2016), it would imply that the high-speed LSMs are composed of $Q1$ quadrant events ($u_1 > 0, u_2 > 0$) and the low-speed LSMs are composed of $Q3$ quadrant events ($u_1 < 0, u_2 < 0$). It should be stressed that this is opposite to what is observed in canonical wall-bounded turbulence, where $Q2$ ($u_1 < 0, u_2 > 0$) and $Q4$ ($u_1 > 0, u_2 < 0$) events dominate the flow field (Ganapathisubramani et al. Reference Ganapathisubramani, Hutchins, Hambleton, Longmire and Marusic2005; Dennis & Nickels Reference Dennis and Nickels2011; Lee & Sung Reference Lee and Sung2011; Hwang et al. Reference Hwang, Lee, Sung and Zaki2016; Wallace Reference Wallace2016).

Large-scale streamwise coherence is observed in the two-point cross-correlation maps of streamwise turbulent heat flux in both the inner and outer layers, evident in figure 18(b,d). The streamwise length scale of the streamwise turbulent heat flux is shorter than the streamwise length scale of streamwise velocity fluctuations. Similar to the two-point correlation of the Reynolds shear stress, $R^{CC}_{u_2 \vartheta }$ is biased in the downstream direction.

Regions of negative streamwise two-point cross-correlation surround a region of positive two-point correlation. This indicates that the positive streamwise velocity fluctuations are correlated with positive buoyancy perturbations (higher temperature than the mean buoyancy field). Negative streamwise velocity fluctuations are correlated with negative buoyancy perturbations (lower temperature than the mean buoyancy field). Decomposing the streamwise turbulent heat flux using quadrant analysis would imply that the high-speed motions are composed of $Q1$ quadrant events ($u_1 > 0, \vartheta > 0$) and the low-speed LSMs comprised $Q3$ quadrant events ($u_1 < 0, \vartheta < 0$).

The positive value of the cross-correlation near $\Delta x_2 = \Delta x_3 = 0$ is consistent with the ensembled averaged streamwise turbulent heat flux, which is positive across most of the turbulent buoyancy layer and is only negative at its edge (see figure 4b).

By comparing the cross-correlations of the Reynolds shear stress and streamwise turbulent heat flux, it is clear that the high-speed LSMs are composed of upwash flow having relatively higher temperatures, and the low-speed LSMs are composed of downwash flow having relatively lower temperatures.

To better understand this distribution of Reynolds shear stress and streamwise turbulent heat flux, the instantaneous streamwise velocity fluctuations overlayed with intense events of Reynolds shear stress and streamwise turbulent heat flux are shown in figure 19. The visualisation is similar to the one used by Dennis & Nickels (Reference Dennis and Nickels2011), who investigated LSMs in a turbulent boundary layer. In the inner layer, the intense events of Reynolds shear stress and streamwise turbulent heat flux are marginally better correlated with low-speed motions (figure 19a,c), suggesting that low-speed motions are the dominant contributors of Reynolds shear stress and turbulent heat flux. High-speed motions also contribute to Reynolds shear stress and streamwise turbulent heat flux but do not carry significant portions of them. The distribution of Reynolds shear stress is similar to that of Reynolds shear stress in turbulent boundary layers (Dennis & Nickels Reference Dennis and Nickels2011).

Figure 19.

Instantaneous intense positive (red) and negative (blue) streamwise velocity fluctuations overlayed with contours (black) of intense events of Reynolds shear stress and streamwise turbulent heat flux at (a,c) $x_1 / \delta _{bl} = 0.04$ and (b,d) $x_1 / \delta _{bl} = 0.3$. The red contours correspond to $u_2 > 0.25 \bar {{u_2}_m}$ and the blue contours correspond to $u_2 < -0.25 \bar {{u_2}_m}$ where $\bar {{u_2}_m}$ is the maximum mean flow velocity. The black contour lines in (a,b) correspond to the contours of $\langle u_1 u_2 \rangle > 2 \sigma _{\langle u_1 u_2 \rangle }$ and the black contour lines in (c,d) correspond to the contours of $\langle u_2 \vartheta \rangle > 2 \sigma _{\langle u_1 \vartheta \rangle }$, where $\sigma$ is the standard deviation. Gravity acts in the negative $x_2$ direction, and the fluid flows in the positive $x_2$ direction.

The distribution of Reynolds shear stress and streamwise turbulent heat flux in the outer layer (figure 19b,d) reveals the striking differences between the turbulent buoyancy layer and canonical wall-bounded turbulence. In the outer layer, the extreme events of the Reynolds shear stress and the streamwise turbulent heat flux are strongly correlated with high-speed LSMs, implying that the high-speed LSMs are the dominant contributors of Reynolds shear stress and streamwise turbulent heat flux in the outer layer. This differs from canonical wall-bounded turbulence where low-speed LSMs are shown to be dynamically more important (Dennis & Nickels Reference Dennis and Nickels2011). Quadrant analysis (Wallace Reference Wallace2016) also demonstrates that the $Q1$ events ($u_1, u_2 > 0$ and $u_2, \vartheta > 0$) contribute around $60\,\%$ of mean Reynolds shear stress and around $65\,\%$ of mean streamwise turbulent heat flux at $x_1 = 0.3 \delta _{bl}$. As the high-speed LSMs mostly comprised $Q1$ events, it can be presumed that the high-speed LSMs are responsible for most of the production of TKE in the outer layer. This is qualitatively similar to unstratified turbulent NCBLs, where $Q1$ events in the outer layer significantly contribute to Reynolds shear stress (Hattori et al. Reference Hattori, Tsuji, Nagano and Tanaka2006).