3.1. First- and second-order turbulence statistics

In figure 3, we provide instantaneous flow visualization in the streamwise–spanwise plane (panels a–f) and spanwise–wall-normal plane (panels g–l) at $z/z_i = 0.225$ and $x/z_i = 2.6$, respectively. The panels include annotations for the width and centre of the elevated heat flux strips, which are defined in the caption. We have shown arbitrarily selected instantaneous realizations for cases A1.25 (figure 3a–c,g–i) and C6 (figure 3d–f ,j–l), which are the limiting cases addressed here with respect to $\beta = Q_0^H/Q_0^L$ and $\lambda = \ell _H/\ell _L$, as per table 1.

Figure 3.

Instantaneous realizations of first-order flow quantities selected at an arbitrary computational time, $t_i$, during LES. Panels (a–c,g–i) and (d–f, j–l) correspond to cases A1.25 and C6, respectively. From left to right, columns (a,d,g,j), (b,e,h,k) and (c, f,i,l) correspond to $\tilde {u}({\boldsymbol x},t_i)$, $\tilde {w}({\boldsymbol x},t_i)$ and $\tilde {\theta }({\boldsymbol x},t_i)$, respectively. Panels (a–f) and (g–l) show instantaneous realizations in the streamwise–spanwise plane at elevation $z/z_i = 0.225$, and the spanwise–wall-normal plane at streamwise location $x/z_i = 2.6$, respectively ($x/z_i = 2.6$ denoted with solid vertical lines on (a–f); $z/z_i = 0.225$ denoted with solid horizontal lines on (g–l)). Dashed lines indicate locations of spanwise heterogeneity; solid horizontal lines on (a–f) and solid vertical lines on (g–l) denote centre of warm strip (see also figure 1).

In the streamwise–spanwise plane, elevated heat flux manifests as an added drag, evident from the relative streamwise momentum deficit aloft the warm strip (figures 3a,d). This relative streamwise momentum deficit corresponds to upwelling (figures 3b,e). The flow response is a product of elevated heat fluxes, which manifest as elevated potential temperature above the strips (figures 3c, f). This flow response appears to be, at least based on instantaneous realizations, consistent with changes in $\beta$ and $\lambda$: variation in relative strip width and heat flux ratio does not alter the underlying flow physics. In complementary prior work wherein the roughness of adjacent strips was varied, downwelling occurs above the relatively more rough strips (figure 1(a) and Anderson et al. Reference Anderson, Barros, Christensen and Awasthi2015). Here, however, elevated heat injection above the warm strips manifests as a buoyancy force that drives upwelling.

In the spanwise–wall-normal plane, we observe that the relative streamwise momentum deficits in figures 3(a,d) occupy the extent of the flow (figures 3g,j). These streamwise momentum deficits exist as ‘plumes’ above the elevated heat flux regions. This flow response is concurrent with the instantaneous upwelling above $Q_0^H$, as per figures 3(h,k). Figures 3(i,l) show potential temperature, where we observe the greatest wall-normal thermal gradients above $Q_0^H$. Recall (1.3), (1.4) and (1.11), where rudimentary deductions related to the thermal torque illustrate how spanwise thermal gradients can sustain a streamwise secondary flow. Inspection of figures 3(j,l) shows how positive and negative thermal gradients across the warm strips yield a thermal torque responsible for sustenance of the secondary flows, which are evident from inspection of figures 3(g,h,j,k).

The instantaneous flow response is also exhibited in the Reynolds-averaged statistics, as reported in figure 4. Panels (a,c) and (b,d) show Reynolds-averaged streamwise velocity and potential temperature, respectively, with vector maps, $\{ \langle \tilde {v} \rangle _{xt}(y,z), \langle \tilde {w} \rangle _{xt}(y,z) \}$, superimposed to highlight secondary flow structure. As per figure 3, we show only results for the limiting parameters captured with cases A1.25 and C6; this structural flow response was consistent across the $\lambda$ and $\beta$ parameter space considered – although relative secondary flow intensity varied monotonically, as will be reviewed in subsequent results – and for brevity visualizations for intermediate cases are omitted.

Figure 4.

Reynolds-averaged first-order flow quantities in the spanwise–wall-normal plane at streamwise location $x/z_i = 2.6$, where (a,b) and (c,d) correspond to cases A1.25 and C6, respectively. From left to right, columns (a,c) and (b,d) correspond to $\langle \tilde {u} \rangle _t({\boldsymbol x})$ and $\langle \tilde {\theta } \rangle _t ({\boldsymbol x})$, respectively, where corresponding vector maps, $\{ \langle \tilde {v} \rangle _{xt}(y,z), \langle \tilde {w} \rangle _{xt}(y,z) \}$, are superimposed to demonstrate vortical structure. Dashed lines indicate locations of spanwise heterogeneity; solid vertical lines denote centre of warm strip (see also figure 1). Grey curves denote spanwise profiles of outer-normalized Obukhov length, $-L/z_i$, recovered a posteriori.

The Reynolds-averaged contours are structurally consistent with the instantaneous observations: elevated wall-normal thermal gradients above the relatively warm strip impose buoyancy fluxes that induce upwelling over the depth of the flow, introducing the relative streamwise momentum deficit over the depth of the flow and sustaining the secondary flow. Alternatively, this can be interpreted via the thermal torque due to spanwise thermal gradients, as per (1.3) and (1.4). Note, too, that the spatial extent of the mean secondary cells is roughly equivalent between case A1.25 (figures 4a,b) and case C6 (figures 4c,d) – roughly $z_i$-scale counter-rotating secondary cells flanking the centreline of the warm strips – yet the width of the high heat flux strips varies significantly between the cases. This result is similar conceptually to prior work on variation of the width of high roughness strips (figure 1(a) and Willingham et al. Reference Willingham, Anderson, Christensen and Barros2013; Anderson et al. Reference Anderson, Barros, Christensen and Awasthi2015).

In order to consolidate presentation of first-order Reynolds-averaged flow statistics into a single figure, we have elected to show vertical profiles of dispersive fluctuations of streamwise velocity, vertical velocity and temperature in figures 5(a–c), 5(d–f) and 5(g–i), respectively. The profiles are shown above the centre of the high (dashed lines) and low (solid lines) heat flux regions, respectively, as annotated on the figure ordinate labels. In each panel, the effect of varying $\beta$ is shown for $\lambda = 0.2$ (figures 5a,d,g), $\lambda = 0.5$ (figures 5b,e,h) and $\lambda = 2$ (figures 5c, f,i), where the outer-normalized Obukhov length is superimposed, for reference.

Figure 5.

Vertical profiles of large-scale spatial variability relative to the underlying Reynolds-averaged distribution (dispersive fluctuation) shown above the high (dashed lines) and low (solid lines) heat flux regions, where the former and latter are defined with $\delta \tilde {\boldsymbol u}^H(z)/u_\tau = (\langle \tilde {\boldsymbol u} \rangle _{xt} (y^H,z) - \langle \tilde {\boldsymbol u} \rangle _{xyt} (z))/u_\tau$ and $\delta \tilde {\boldsymbol u}^L(z)/u_\tau = (\langle \tilde {\boldsymbol u} \rangle _{xt} (y^L,z) - \langle \tilde {\boldsymbol u} \rangle _{xyt} (z))/u_\tau$, respectively, where (a–c) and (d–f) show streamwise and vertical velocity, respectively. Panels (g–i) show dispersive potential temperature, where $\delta \tilde {\theta }^H(z)/\theta _\tau = (\langle \tilde {\theta } \rangle _{xt} (y^H,z) - \langle \tilde {\theta } \rangle _{xyt} (z))/u_\tau$ and $\delta \tilde {\theta }^L(z)/u_\tau = (\langle \tilde {\theta } \rangle _{xt} (y^L,z) - \langle \tilde {\theta } \rangle _{xyt} (z))/u_\tau$. Here, $y^H$ and $y^L$ are discrete spanwise locations coincident with the centre of the high and low heat flux regions, respectively (see also figure 1). Panels (a,d,g), (b,e,h) and (c, f,i) correspond to $\lambda = 1/5$, $1/2$ and 2, respectively. In all panels, the outer-normalized Obukhov length, $-L/z_i$, is denoted by a vertical grey band.

Elevated heat fluxes above the high flux region manifest as an ‘added drag’, inducing a relative streamwise momentum deficit (figures 5a–c) and a relative vertical momentum excess (figures 5d–f) associated with plumes of low streamwise momentum fluid throughout the domain. This corresponds with elevated temperatures throughout the depth of the flow (figures 5g–i). This flow construct provides underlying sustenance of the secondary cells, as conservation of mass necessitates concurrent streamwise momentum excess and vertical momentum deficit above the relatively low heat flux region (solid lines). This ‘downwelling’ provides an ongoing supply of relatively cool fluid, as observed in figures 5(g–i). The broad response in all quantities to variable $\beta$ is, by itself, preliminary evidence that spanwise heterogeneity in surface heat flux forcing and the resultant thermal torque provide distinctly different mechanistic sustenance for Reynolds-averaged secondary flows (i.e. Prandtl's secondary flow of the third kind).

In all panels of figure 5, it is clear that with increasing $\beta$ – i.e. increasingly dramatic spanwise gradient in thermal forcing – the relative dispersive fluctuations increase monotonically. The response to $\lambda$ is somewhat less evident, particularly for momentum (figures 5a–f). The exception is temperature, where we observe monotonically declining temperature above the high heat flux region with increasing $\lambda$. This result is a consequence of the underlying surface attributes: since cumulative heat flux is equivalent across all cases, even with variable $\beta$, and because $\lambda = 0.2$ and $\lambda = 2$ correspond to relatively narrow and wide high heat flux strips, respectively, it follows that for $\lambda = 0.2$ a relatively elevated heat flux is necessitated across the high heat flux area. Likewise, for $\lambda = 2$, it follows that a relatively lesser heat flux is needed over the high heat flux region to accomplish the same values of $\beta$. The inverse relationship is true, too, for temperature across the low heat flux region.

For $\lambda = 0.2$, elevated temperature over the high heat flux regions induces significant upwelling (figure 5d), which declines monotonically with decreasing temperature (figures 5e, f). As would be expected, maximum values in dispersive vertical velocity fluctuation occur above the Obukhov length (vertical grey bar), which highlights the transition between shear- and thermal-dominated production of turbulent kinetic energy. The kinks in the dispersive fluctuations in the temperature profiles near $z/z_i = 1$ are due to the increase in temperature in the entrainment zone and are evidence that the inversion strength over the high and low heat flux regions differs from that of the planar average.

To further the analysis, we now shift our attention to higher-order turbulence statistics. Figure 6 shows salient second-order turbulence statistics for cases A1.25 (panels a–c), B2 (panels d–f) and C6 (panels g–i), where the vertical dashed and solid lines denote heat flux transitions and high heat flux centreline, respectively; we show streamwise–wall-normal turbulent stresses, $-\langle u^\prime w^\prime \rangle _t$ (panels a,d,g), turbulent transport of potential temperature, $-\langle w^\prime \theta ^\prime \rangle _t$ (panels b,e,h), and turbulent kinetic energy (panels c, f,i). In the interests of brevity, we have not presented all second-order statistics. The horizontal lines in figure 6 show $-L/z_i$, where $L$ is the Obukhov length, which collapses independent parameters governing the ensemble mean wall-normal gradient of streamwise velocity (Obukhov Reference Obukhov1946). The Obukhov length physically represents the wall-normal elevation at which the dominant turbulent kinetic energy production mechanism transitions from mechanical shear to buoyancy:

(3.1)\begin{equation} L(y) = \frac{-u_\tau(y)^3\,\varTheta_0}{\kappa g\,Q_0(y)},\quad \mathrm{where}\ u_\tau(y) = \left[\frac{\tau^w}{\rho}\,(\hat{\boldsymbol n}(y) : \hat{\boldsymbol n}(y)) \right]^{1/2}. \end{equation}

Here, $\hat {\boldsymbol n}(y,z) = \langle \tilde {\boldsymbol u} \rangle _{xt}(y,z)/\langle \tilde {\boldsymbol u} : \tilde {\boldsymbol u})^{1/2} \rangle _{xt}(y,z)$ is the unit vector based on the velocity vector, $\kappa$ is the von Kármán constant, and $Q_0(y) = Q_0^L$ or $Q_0^H$, depending on location across the surface (see also figure 1 for illustration of heat flux variation). For unstable flows such as those considered in this article (table 1), $-L/z_i$ provides a literal visualization of the mechanisms responsible for $k$ production. In this sense, superposition of $(-z_i/L)^{-1}$ is instructive for the present cases.

Figure 6.

Contours of turbulent stresses for cases A1.25 (a–c), B2 (d–f) and C6 (g–i) in the spanwise–wall-normal plane at streamwise location $x/z_i = 2.6$. From left to right, columns show $-\langle u^\prime w^\prime \rangle _t$ (a,d,g), $\langle w^\prime \theta ^\prime \rangle _t$ (b,e,h), and $k = (1/2)\langle {\boldsymbol u}^\prime : {\boldsymbol u}^\prime \rangle _t$ (c, f,i). Dashed lines indicate locations of spanwise heterogeneity, solid vertical lines denote centre of warm strip (see also figure 1). Grey curves denote spanwise profiles of outer-normalized Obukhov length, $-L/z_i$, recovered a posteriori.

Recall that a homogeneous roughness is prescribed for the present cases, yet figures 6(a,d,g) demonstrate that $-\langle u^\prime w^\prime \rangle _t$ exhibits large-scale spanwise heterogeneity across the surface. This highlights how buoyancy-driven secondary flows redistribute $k$ throughout the domain, such that the high and low heat flux regions manifest physically as relatively more and less rough, respectively. Here, $(-z_i/L)^{-1}$ is lowest over the $Q_0^H$ region, indicating that production of $k$ by buoyancy is larger over this region, relative to mechanical shear, even given the larger $-\langle u^\prime w^\prime \rangle _t$ across the high heat flux strip. In contrast, across the low heat flux region, $(-z_i/L)^{-1}$ rises dramatically, illustrating the relative dominance of mechanical shear production across the $Q_0^L$ region.

Figures 6(b,e,h) show the corresponding vertical turbulent flux of potential temperature, $\langle w^\prime \theta ^\prime \rangle _t$. The spatial distributions of $\langle w^\prime \theta ^\prime \rangle _t$ resemble $-\langle u^\prime w^\prime \rangle _t$; note, however, that the colourbar limits for (a,d,g) and (b,e,h) are $-0.1 \lesssim z/z_i \lesssim 1.5$ and $-2 \lesssim z/z_i \lesssim 6$, respectively, which confounds visual inference on the value of $(-z_i/L)^{-1}$. Thus although the $-\langle u^\prime w^\prime \rangle _t$ and $\langle w^\prime \theta ^\prime \rangle _t$ contours are similar, in fact the relative difference in magnitude illustrates why $-z_i/L$ declines across the $Q_0^H$ strip.

Finally, figures 6(c, f,i) provide spatial contours of turbulent kinetic energy for the cases noted in the caption. The panels show plumes of elevated $k$ above the $Q_0^H$ strips. The plume structure is consistent with those for $-\langle u^\prime w^\prime \rangle _t$ and $\langle w^\prime \theta ^\prime \rangle _t$, which is consistent with the underlying production mechanisms as per (1.6).

The figures demonstrate that second-order statistics increase monotonically with $\beta$, evidenced by relative increases in the magnitudes of $-\langle u^\prime w^\prime \rangle _t$, $\langle w^\prime \theta ^\prime \rangle _t$ and $k$ over the high heat flux region. Note, however, that the spatial distributions in these quantities do not respond materially to $\lambda$. This is consistent with the Reynolds-averaged first-order statistics shown in figures 4 and 5, where spanwise heat flux variability yields counter-rotating $z_i$-scale secondary cells centred about the centreline of the $Q_0^H$ region.

3.2. Interpreting secondary flows from turbulent kinetic energy transport

In this subsection, the inertia-dominated (Reynolds-averaged) $k$ transport equation (1.6) is used to discern the underlying physics responsible for the secondary flow processes presented in § 3.1. This follows from Hinze (Reference Hinze1967), who used $k$ transport to characterize secondary flow processes in turbulent duct flow, arguing that secondary flow advection is necessitated by an imbalance in production and dissipation, such that ‘turbulence poor’ fluid is entrained by regions of ‘turbulence rich’ fluid. This argument has been used to infer secondary transport in roughness-driven secondary flows in hydraulic flows (Wang & Cheng Reference Wang and Cheng2005; Vermaas et al. Reference Vermaas, Uijttewall and Hoitink2011) and engineering roughness (Anderson et al. Reference Anderson, Barros, Christensen and Awasthi2015). Others have used the Reynolds-averaged transport equation for in-plane vorticity (Hoagland Reference Hoagland1960), but balancing this equation is problematic. In particular, large residuals usually are found when calculating terms in the mean streamwise vorticity budget, regardless of whether one calculates it from experimental or numerical simulation data. This may be due to the spontaneous reversals of the rotational direction of secondary cells that complicate averaging (Anderson Reference Anderson2019), although the underlying reason is not abundantly clear. We therefore declined this avenue in favour of the $\langle k \rangle _t$ balance equation (1.6) (Madabhushi & Vanka Reference Madabhushi and Vanka1991; Anderson et al. Reference Anderson, Barros, Christensen and Awasthi2015).

As summarized in (1.6), transport of Reynolds-averaged $\langle k \rangle _t$ is regulated by the interplay between production and dissipation, where the former occurs via mechanical shear ($\mathcal {P}_{S}$) and buoyancy ($\mathcal {P}_{B}$), while the latter is quantified here for the inertia-dominated (Reynolds-averaged) form of (1.6), such that $\epsilon = \langle - \boldsymbol{\mathsf{T}} : \boldsymbol{\mathsf{S}} \rangle _{xt}$. Figure 7 shows contours for constituent terms within (1.6) for cases A1.25 (panels a,b,g,h), B2 (panels c,d,i,j) and C6 (panels e, f,k,l), where the colour flood contours and isocontour values are defined in the figure caption.

Figure 7.

Constituent terms of the $\langle k \rangle _t$ transport equation for cases A1.25 (a,b,g,h), B2 (c,d,i,j) and C6 (e, f,k,l). Panels show shear production, $\mathcal {P}_{S}$ (a,c,e), buoyancy production, $\mathcal {P}_{B}$ (b,d, f), advection, $\langle \tilde {\boldsymbol u} \rangle _t \boldsymbol {\cdot } \boldsymbol {\nabla } \langle k\rangle _t$ (g,i,k), and dissipation (as it appears in (1.6)), $- \epsilon = - \langle - \boldsymbol{\mathsf{T}} : \boldsymbol{\mathsf{S}} \rangle _{xt}$ (h,j,l); see also (1.6). Included in the panels are isocontours of vertical velocity with values $\tilde {w}({\boldsymbol x},t)/u_\tau = 0.2$ (red) and $-0.2$ (blue). Dashed lines indicate locations of spanwise heterogeneity; solid vertical lines denote centre of warm strip (see also figure 1). Horizontal line denotes spanwise profile of outer-normalized Obukhov length, $-L/z_i$, recovered a posteriori (see also (3.1)).

In the results shown – which, we emphasize, are also representative of intermediate table 1 cases – we see a consistent pattern of upwelling and downwelling over the high and low heat flux surface regions, respectively. Comparison of shear-driven $\langle k \rangle _t$ production $\mathcal {P}_{S}$ (figures 7a,e,i) shows similar patterns, although $\lambda$ changes substantially across these cases. We see a consistent ‘plume’ of elevated $\mathcal {P}_{S}$ across the centre of the $Q_0^H$ region, and this pattern is ostensibly insensitive to $\lambda$. This result contrasts against the superimposed profiles of outer-normalized Obukhov length $-L/z_i$, which show that shear driven-production is relatively weakest across the warm strips. Noting that $(-z_i/L)^{-1} \rightarrow 0$ and $(-z_i/L)^{-1} \rightarrow \infty$ for free convection and canonical channel turbulence, respectively, large-scale spatial spanwise variability in $(-z_i/L)^{-1}$ highlights the distinct ways that spanwise heat flux heterogeneity affects the resultant flow processes. Recall, however, that turbulent transport of temperature is relatively greater across the warm strip, as per figures 6(b,e,h). With this, the dominant production mechanism above the $Q_0^H$ strip transitions from mechanical shear (figures 7a,c,e) to buoyancy (figures 7b,d, f).

Advection of Reynolds-averaged turbulent kinetic energy, $\mathcal {A}_k = - \langle \tilde {\boldsymbol u} \rangle _t \boldsymbol {\cdot }\boldsymbol {\nabla } \langle k \rangle _t$, is shown in figures 7(g,i,k) (see also (1.6) and accompanying text). The advection contours provide a unique avenue to interpret the resultant secondary flow: across the centre of the $Q_0^H$ strip, $- \langle \tilde {\boldsymbol u} \rangle _t \boldsymbol {\cdot } \boldsymbol {\nabla } \langle k \rangle _t \rightarrow - \langle \tilde {w} \rangle _t\partial _z \langle k \rangle _t$. Here, vertical velocity is positive, and the vertical gradient of turbulent kinetic energy is negative; thus advection is necessarily positive given the relative imbalance between the production and dissipation of turbulence (Hinze Reference Hinze1967; Anderson et al. Reference Anderson, Barros, Christensen and Awasthi2015). This deduction illustrates that since $\partial _z \langle k \rangle _t > 0$ (as per figures 6c, f,i), $\langle \tilde {w} \rangle _t$ must also be positive, which is reported above the centre of the warm strip (figures 3b,e,h,k, 4 and 5d,e, f).

The negative of Reynolds-averaged dissipation, which serves as the principal sink term in the $\langle k \rangle _t$ transport equation, $-\epsilon = -\langle - \boldsymbol{\mathsf{T}} : \boldsymbol{\mathsf{S}} \rangle _{xt}$, is shown in figures 7(h,j,l). For the non-Hamiltonian conservative systems considered herein, $\epsilon$ is everywhere positive; thus $-\epsilon$ is negative definite and serves as a sink of $\langle k \rangle _t$. The spatial distribution of $-\epsilon$ reaches its maximum magnitude in the roughness sublayer and within the base of the shear-dominated $\langle k \rangle _t$ production region above $Q_0^H$.

When characterizing mechanisms responsible for turbulent secondary flow, Hinze (Reference Hinze1967) introduced the notion of ‘turbulence poor’ and ‘turbulence rich’ fluid, where the former is necessarily entrained by the latter, setting in motion the secondary flow direction. Later, Anderson et al. (Reference Anderson, Barros, Christensen and Awasthi2015) used this argument to discern underlying mechanisms responsible for roughness-driven secondary flows in channel flows over surfaces depicted by figure 1(a). For such flows, figure 8(a) illustrates how a pronounced production–dissipation imbalance across the high-roughness strip, $z_0^H$, necessitates downwelling of low-$\langle k \rangle _t$ fluid from directly above (dark red region in figure 8a). This resultant mean flow yields a large-scale undulation in $\langle \tilde {u} \rangle _t$, such that a mean excess and deficit in $\langle \tilde {u} \rangle _t$ occurs above $z_0^H$ and $z_0^L$, respectively. This has previously been defined as low- and high-momentum pathways (LMP, HMP) (Barros & Christensen Reference Barros and Christensen2014).

Figure 8.

Schematic illustrating problem parameters addressed in this study (see also table 1 for quantitative details). Panel (a) shows the case of roughness-driven secondary flows, where the turbulent torque term and production–dissipation imbalance above the relatively high-roughness region, $z_0^H$, induce downwelling and formation of high- and low-momentum pathways as shown (Willingham et al. Reference Willingham, Anderson, Christensen and Barros2013; Barros & Christensen Reference Barros and Christensen2014; Anderson et al. Reference Anderson, Barros, Christensen and Awasthi2015); upwelling occurs above the relatively low-roughness region, $z_0^L$. In contrast, for the cases considered herein, shear-driven $k$ production is also concentrated above the relatively elevated flux region, but this region is itself capped by a region of buoyancy production (b). This additional $k$ production necessitates lateral entrainment of low-$k$ fluid, reversing the secondary flow direction such that an LMP forms over $Q_0^H$. The outer-normalized Obukhov lengths for the spanwise heterogeneous (grey) and homogeneous (black) heat flux cases are illustrated in (b); shown also in (b) are $\ell _H$ and representative $\ell _d$.

In contrast, the case of spanwise heterogeneous heat fluxes – as per figure 8(b) – yields a complete reversal in secondary flow direction as compared to the roughness heterogeneity case. This is quantified by spanwise variability of the Obukhov length, where $-L/z_i$ exhibits a relative minimum above $Q_0^H$ due to elevated shear (dark red region in figure 8b). Above $-L/z_i$, however, buoyancy continues active production of turbulent kinetic energy (yellow region in figure 8b). As such, a relative production–dissipation imbalance persists over the depth of the flow, and ‘turbulence poor’ fluid can not readily be entrained from aloft. In fact, the production–dissipation imbalance can be resolved only via lateral entrainment, thereby reversing the mean secondary flow polarity relative to the roughness-driven case and resulting in formation of an LMP above $Q_0^H$. We have provided annotations on figure 8 for the locations of HMPs and LMPs (Barros & Christensen Reference Barros and Christensen2014), which summarizes polarity reversal due to thermal torque.

3.3. Reynolds-averaged flow attributes

In this final subsection, we focus on aggregating bulk statistics from the suite of cases used in this study and summarized in table 1. Combining bulk results in this manner provides an avenue to identify underlying trends in turbulence statistics with variation in $\beta$ and $\lambda$. We elected to provide profiles showing spanwise distribution of the similarity parameter, $-L/z_i$, and compensated circulation (discussion to follow). These results complement quantities already presented in preceding sections.

Figures 9(a), 9(b) and 9(c) provide spanwise profiles of $-L/z_i$ for cases in groups A, B and C, respectively, where specific cases are summarized in the panel legend. In all panels, we have superimposed the reference value, $(-z_i/L)^{-1} = 0.26$, for case  H, which is the homogeneous heat flux case. For all table 1 cases, the area-averaged heat flux, $Q_0 = \int _{\mathrm {d}^2 {\boldsymbol x}} Q_0 ({\boldsymbol x})\,\mathrm {d}^2 {\boldsymbol x} / \int _{\mathrm {d}^2 {\boldsymbol x}}\,\mathrm {d}^2 {\boldsymbol x}$, is equivalent (including case  H). Figure 9 illustrates monotonic trends in the deviation in $(-z_i/L)^{-1}$ relative to the values for a homogeneous heat flux: as $\beta$ increases, shear-driven production of turbulence is relatively weaker and stronger above $Q_0^H$ and $Q_0^L$, respectively. As per (3.1), the Obukhov length is based on the prescribed heat flux and corresponding surface friction velocity; the ‘step-wise’ profile in $(-z_i/L)^{-1}$ is thus a product of the prescribed heat fluxes as per figure 1(b).

Figure 9.

Spanwise profiles of Obukhov length for all cases, recovered a posteriori from LES (profiles defined in panel legends). Dashed lines indicate locations of spanwise heterogeneity; solid vertical lines denote centre of warm strip (see also figure 1). Horizontal dashed line denotes $-L/z_i$ for case H.

In addition to figure 9, figure 10 shows datapoints for compensated circulation:

(3.2)\begin{equation} {\varGamma}^* = \frac{{\varGamma}}{w_\star z_i} = \int_{\mathrm{d}^2 {\boldsymbol x}} |\langle \tilde{\omega}_x \rangle_{xt} (y,z)| \,\mathrm{d}^2 {\boldsymbol x}, \end{equation}

where the absolute value of Reynolds-averaged streamwise vorticity, $|\langle \tilde {\omega }_x \rangle _{xt} (y,z)|$, is used to quantify secondary flow intensity (Yang & Anderson Reference Yang and Anderson2017). Without computation based on the absolute value of streamwise vorticity, ${\varGamma }^* \rightarrow 0$ due to the presence of Reynolds-averaged counter-rotating secondary cells. Figure 10 shows ${\varGamma }^*$ against $\mathcal {H}$ (1.11), where the symbols are defined in the figure. Figure 10 is shown with ordinate range $0 \leq \varGamma ^* \leq 15$, where the lower limit, $\varGamma ^* = 0$, is an important benchmark for assessing the influence of spanwise heterogeneity: for spanwise spacing, $s/z_i = 0$ and $s/z_i = \infty$, there is no differential heat flux to induce thermal torque, thus $\varGamma ^* \rightarrow 0$ is expected in the Reynolds-averaged limit. Between these limits, however, the preceding results have illustrated how thermal torque sustains $z_i$-scale secondary cells (figures 3, 4, 5, 6 and 7).

Figure 10.

Datapoints for compensated circulation (see (3.2)) for the cases considered in this study (with the exception of case  H, for which ${\varGamma }^* = 0$; see also table 1). Values of ${\varGamma }^*$ are shown against $\mathcal {H} ({u_\tau }/{w_\star }) = \varOmega _{B} / \varOmega _{A}$ (abscissa; see table 1 for case attributes) for the values of $\lambda$ denoted in the figure.

The figure 10 results can be understood readily via consideration of the scaling arguments presented in § 1.1. Since $\langle \tilde {\omega }_x \rangle _{xt} \sim w_\star /\ell _H$ (see (1.9)) and $\mathrm {d}^2 {\boldsymbol x} \sim \ell _H \ell _d \sim \ell _H^2 (w_\star /u_\tau )$ (see (1.8)), it follows that $\varGamma ^* \sim \ell _d/z_i$. (Recall that $\ell _H$ and $\ell _d$ correspond to the heterogeneity length scale and the depth of the secondary cells, respectively.) Therefore, given that $z_i$ presents only small variations between the cases analysed in this work (see table 1), potential changes in compensated circulation should then be mostly attributed to changes in $\ell _d$. However, from the results presented earlier, in figures 3 and 7, it is found that $\ell _d$ does not significantly respond to changes in $\beta$ and $\lambda$. This explains the almost-constant behaviour of compensated circulation as a function of $\varOmega _{B} / \varOmega _{A}$, as shown in figure 10. Furthermore, this result is clear evidence that in all cases considered in this work, the thermal torque provides sustenance for the balance of Reynolds-averaged streamwise vorticity, and hence they are examples of Prandtl's secondary flow of the third kind.