3.1. Procedure to obtain the p.d.f.s

To understand the velocities of dominant motions within the boundary layers, we now study the instantaneous p.d.f.s of $U=\sqrt {u^2+v^2}$, the magnitude of the horizontal velocities within the local boundary layers, where $u$ and $v$ are the fluid velocities in the $x$ and $y$ directions, respectively. These instantaneous p.d.f.s were estimated from the instantaneous PIV velocity fields in horizontal planes at heights $h_m < \delta _{nc} \ll \delta _{pb}$, given in table 1, at each $Ra_w$. An in-house code generated the instantaneous p.d.f.s using the following iterative procedure on each instantaneous spatial velocity field, separated temporally by $t_f$, from a set having 1399 such velocity fields; 1–8 such sets, separated temporally by $210$ s, were used at each $Ra_w$ to span at least $10$ LSF times.

At any instant, an initial p.d.f. was first obtained from the spatial velocity field using 50 bins of equal sizes as

(3.1)\begin{equation} \mathrm{p.d.f.}=\frac{N_U}{s b}, \end{equation}

where $N_U$ is the number of occurrences of $U$ in each bin, $s$ is the sample size of $U$, and $b$ is the bin size. We then proceed to estimate the total number of dominant peaks occurring in the p.d.f.s, so as to classify these p.d.f.s in the following way. In this calculation, since we are interested in the velocities of the dominant motions, peaks with $N_U$ less than 10 $\%$ of the $N_U$ of the highest peak are neglected. The minimum allowed separation between the peaks was kept equal to $\frac{1}{4}(V_{L}/V_{bl}-1)$, to minimise the possibility of counting peaks due to fluctuations as dominant peaks. Here, $V_L$ is the shear on the local boundary layers, estimated as the velocity of longitudinal motion of plumes by Gunasegarane & Puthenveettil (Reference Gunasegarane and Puthenveettil2014), given through the corresponding Reynolds number

(3.2)\begin{equation} Re_{L}=V_{L}H/\nu=0.55\,Ra_w^{4/9}\,Pr^{-2/3}. \end{equation}

Here, $V_{bl}$ is the characteristic velocity in the local natural convection boundary layer (NCBL), given through the corresponding Reynolds number

(3.3)\begin{equation} Re_{bl}=V_{bl}H/\nu=1.88\,Ra_w^{1/3}\,Pr^{-0.98} \end{equation}

(Vipin Reference Vipin2013; Gunasegarane & Puthenveettil Reference Gunasegarane and Puthenveettil2014). The rarely occurring fluctuations at the tails of the p.d.f.s, similar to that at $U/V_{bl}\approx 7.75$ in figure 2(e), are also neglected in this estimation.

Figure 2. Instantaneous p.d.f.s of the dimensionless horizontal velocity magnitudes and the corresponding instantaneous spatial distributions of horizontal velocity vectors overlaid over the dimensionless horizontal velocity magnitudes at three $Ra_w$. Values of $(Ra_w,\varGamma,Re_F)$ for $\ast$ are: (a,b) $(2.76 \times 10^5,10,34.42)$; (c,d) $(1.31 \times 10^7,4.3,320)$; (e,f) $(6.05 \times 10^8,1.71,3148)$. Values for $\circ$ are: (a) $(1.31\times 10^7,4.3,320)$; (e) $(1.10\times 10^8,2.5,1169)$. In insets, values for $\ast$ are: (a) $(1.66\times 10^6,10,82.86)$; (e) $(2\times 10^7,4.3,401.6)$. The values of $V_{L}/V_{bl}$ corresponding to $\ast$ and $\circ$ are shown by dashed lines and dash-dotted lines, respectively, in (a,c,e). The error bars in (a,c,e) show the maximum uncertainties $7.1\,\%$, 2.5 % and $4\,\%$ at $U/V_{bl}=0.6$, $2$ and $3$, respectively. Details of error analysis can be found in Appendix C.

If the number of maxima detected by this procedure was greater than two, then the p.d.f. was recalculated by reducing the number of bins by one in successive iterations until the number of maxima became less than or equal to two. The iteration was stopped when the number of bins became 10, before two or fewer dominant peaks were detected. In such a case, peaks with frequencies less than $11\,\%$ of the frequency of the highest peak in the p.d.f. were neglected, and the above iterative process, with the initial number of bins equal to 50, was repeated. After this process, in the rare instances where the number of bins was still ${\leqslant }10$ with the number of peaks being ${>}2$, the cut-off frequency was further increased to $12\,\%$ and the process repeated until a smooth p.d.f. with number of bins $\geqslant$10 and number of peaks $\leqslant$2 was obtained.

3.2. Types of p.d.f.s

In the present study, $Ra_w$ was increased from the lowest value $Ra_w=2.76\times 10^5$ to the highest value $Ra_w=6.05\times 10^8$ by changing the temperature difference $\Delta T$ and the cell height $H$. To increase $Ra_w$, (i) the cell height was kept constant and $\Delta T$ was increased, or (ii) both $\Delta T$ and $H$ were increased. Then, with increasing $Ra_w$, the aspect ratio $\varGamma =L/H$ in the experiments could either be a constant for the first case, or decrease monotonically for the second case. As shown in table 1, the aspect ratio changed from $10$ at $Ra_w=2.76\times 10^5$ to 1.71 at $Ra_w=6.05\times 10^8$. However, in both of these cases, since the Reynolds number based on the LSF velocity $V_F$ scaled as $Re_F\sim Ra_w^{4/9}\varGamma ^{-0.57}$ (see (B1)), $V_F$ increased monotonically with increase in $Ra_w$, whether this increase in $Re_F$ occurred due to the increase in $\Delta T$ at a fixed $\varGamma$ or due to a decreasing $\varGamma$, shown in table 1. This increasing $V_F$ with increasing $Ra_w$ changed the shapes of the p.d.f.s and the locations of their peaks. So the change in the p.d.f. shapes and their peaks in the present study, shown in figures 2, 3 and 4, was occurring primarily due to increase in $Re_F$, whether this increase in $Re_F$ occurred due to increasing $Ra_w$ and/or due to decreasing $\varGamma$.

Figure 3. Instantaneous bimodal p.d.f.s of dimensionless horizontal velocity magnitudes, and the corresponding instantaneous spatial distributions of dimensionless horizontal velocity vectors, overlaid over the dimensionless horizontal velocity magnitudes, at intermediate and high $Ra_w$: (a,b) $(Ra_w,\varGamma,Re_F) = (1.66 \times 10^6$,10,82.86); (c,d) $(Ra_w,\varGamma,Re_F) = (6.05 \times 10^8,1.71,3148)$. The error bars in (a,c) show the maximum uncertainties of 3.8$\%$ and $4.9\,\%$ at $U/V_{bl}=1.04$ and $3.2$, respectively. Details of error analysis can be found in Appendix C.

Figure 4. Instantaneous, type C p.d.f.s of dimensionless horizontal velocity magnitudes for two different $Ra_w$ at fixed $\varGamma$: (a) ($Ra_w,\varGamma, Re_F)=(2.21 \times 10^8,2,1800)$; (b) $(Ra_w,\varGamma,Re_F) = (4.04 \times 10^8,2,2557)$. The error bars show the maximum uncertainties 2.1 $\%$ and $7\,\%$ at $U/V_{bl}=3.66$ and $1.97$, respectively. Details of error analysis can be found in Appendix C.

Three different types of instantaneous p.d.f.s, obtained using the above procedure, at $Ra_w=2.76\times 10^5, 1.31\times 10^7, 6.05\times 10^8$, are shown in figures 2(a), 2(c) and 2(e), respectively. The corresponding horizontal velocity fields in a horizontal plane at height $h_m$ are shown in figures 2(b), 2(d) and 2(f), respectively. P.d.f.s in figures 2(a) and 2(c) are unimodal with peaks at $U/V_{bl}=1.06$ and 7.6, respectively, while the p.d.f. in figure 2(c) is bimodal with peaks at $U/V_{bl}=2$ and 3.1. The dashed vertical lines in figures 2(a), 2(c) and 2(e) show the values of the ratio $V_L/V_{bl}$ of the shear velocity $V_L$ (see (3.2)) and the boundary layer velocity $V_{bl}$ (see (3.3)) at the corresponding $Ra_w$. We choose $V_L$ as the relevant shear velocity, rather than $V_F$ of (1.1), since $V_L$ is measured closer to the plate at distances of the order of $h_m$, at which the present velocity fields are measured. The values of $V_{L}$ and $V_{bl}$ for the range of $Ra_w$ in the present study are listed in table 1.

It needs to be noted that at each $Ra_w$, all three types of p.d.f.s shown in figure 2 were observed. However, as we discuss in § 3.4, the relative occurrence of these types of p.d.f.s changed with increasing $Ra_w$. At the lowest $Ra_w=2.76\times 10^5$, the p.d.f.s obtained were predominantly ($45\,\%$) of a unimodal nature with a peak at $U/V_{bl}\approx 1$, similar to that shown in figure 2(a); hereinafter, we refer to these as type A p.d.f.s. The most frequently occurring horizontal velocities at the lowest $Ra_w$ are then of the order of $V_{bl}$, the NCBL velocity, given by (3.3). The spatial distribution of the velocities corresponding to these type A p.d.f.s, as shown in figure 2(b), shows a converging nature towards various randomly oriented lines.

At the highest near-plate Rayleigh number $Ra_w=6.05\times 10^8$ also, we observed predominantly ($71\,\%$) unimodal p.d.f.s that were, however, peaking mostly at values greater than $V_{L}/V_{bl}$, as shown in figure 2(e) for a specific instant; hereinafter, we refer to these types of p.d.f.s as type C. The most frequently occurring horizontal velocities at the largest $Ra_w$ are then of the order of, or greater than, $V_{L}$. The spatial distribution of velocities shown in figure 2(f), corresponding to these type C p.d.f.s, shows a more aligned nature, with larger magnitudes of velocity vectors compared to those with type A p.d.f.s at $Ra_w=2.76\times 10^5$ in figure 2(b).

In contrast to p.d.f.s at low and high $Ra_w$, for an intermediate $Ra_w=1.31\times 10^7$, we obtain mostly (73 %) p.d.f.s of a bimodal nature. As shown in figure 2(c) for a specific instant, the first peaks of these bimodal p.d.f.s occur in between $U/V_{bl}\approx 1$ and $V_{L}/V_{bl}$, while the second peaks occur at a value greater than or equal to $V_{L}/V_{bl}$. Hereinafter, the bimodal distributions of this form are referred to as type B. Unimodal p.d.f.s are also observed at these intermediate $Ra_w$, but at only approximately 21 % of the instants. Most of the unimodal p.d.f.s observed at this intermediate $Ra_w$ peak at $U/V_{bl}>1+\frac{2}{3}(V_{L}/V_{bl}-1)$ or $U/V_{bl} < 1+\frac{1}{3}(V_{L}/V_{bl}-1)$. Horizontal velocities corresponding to the peaks of the former are much greater in magnitude compared to the velocities corresponding to the peaks of the latter. Further, the former type of p.d.f. is skewed to the left, similar to the p.d.f. shown in figure 2(e), while the latter type of p.d.f. is skewed to the right, similar to the p.d.f. shown in figure 2(a). For these reasons, we classify these unimodal p.d.f.s as type C and type A, respectively. Less than $1\,\%$ of the total number of p.d.f.s at this intermediate $Ra_w$ show a unimodal nature while peaking at $1+\frac{1}{3}(V_{L}/V_{bl}-1)< U/V_{bl}<1+\frac{2}{3}(V_{L}/V_{bl}-1)$. For this very small number of unimodal p.d.f.s, we consider those peaking at $U/V_{bl}< 1+\frac{1}{2}(V_{L}/V_{bl}-1)$ to be of type A. An instance of such a p.d.f. that occurs at the intermediate $Ra_w=1.31\times 10^7$ is shown with the symbol $\circ$ in figure 2(a), which peaks at $U/V_{bl}=1+0.4(V_L/V_{bl}-1)$. From these $1\,\%$ p.d.f.s at intermediate $Ra_w$, those peaking at $U/V_{bl}>1+\frac{1}{2}(V_{L}/V_{bl}-1)$ are considered to be of type C; the p.d.f. shown by $\circ$ in figure 2(e) is an example of such a case.

3.3. Area affected by shear

We now study the evolution of areas of such shear affected regions with $Ra_w$. At any instant, for bimodal p.d.f.s, we calculate the ratios of areas corresponding to the first mode ($A_{bl}$) and the second mode ($A_{sh}$) to the total area $(A)$. In unimodal p.d.f.s of type A and type C, a similar splitting of the total area $A$ into areas strongly affected by shear ($A_{sh}$) and areas mildly affected by shear ($A_{bl}$) can be achieved as well, by using appropriate thresholds, as described below. In figure 2(b), which is the velocity field corresponding to a p.d.f. of type A shown in figure 2(a), the green region around $(-10, 20)$ mm has velocity vectors of aligned nature with velocity magnitudes greater than $V_{L}$. The velocity vectors in this region are of a nature similar to that in regions corresponding to the second mode (red/yellow regions in figure 2d) in bimodal p.d.f.s, or to those in most of the regions corresponding to type C p.d.f.s (figure 2f). Hence when unimodal p.d.f.s of type A occur, we separate the regions strongly affected by shear from the regions mildly affected by shear using a threshold at $U / V_{bl}>V_{L}/V_{bl}$. Since the velocities below $U/V_{bl}=1$ are not the shear velocities due to the LSF (see (3.2)), when p.d.f.s of type C occur, we separate the areas mildly affected by shear from the areas strongly affected by shear using the threshold $U/V_{bl}<1$. In this way, at each $Ra_w$, we calculate the instantaneous area ratios $A_{bl} /A$ and $A_{sh} /A$ at all the instants, using the appropriate threshold from the above three thresholds, depending upon the type of p.d.f. that occurs at each instant.

Figure 5 shows the variation of $\langle A_{bl}/A\rangle$ and $\langle A_{sh}/A\rangle$ with $Ra_w$, where $\langle \cdot \rangle$ indicates the mean of the instantaneous area ratios over all the instants at each $Ra_w$. The values of $\langle A_{bl}/A\rangle$ decrease with increase in $Ra_w$, while those of $\langle A_{sh}/A\rangle$ increase with increase in $Ra_w$; increase in $Ra_w$ then increases the areas strongly affected by shear due to the LSF. This increase is drastic in the range $1.66\times 10^6\leqslant Ra_w \leqslant 1.10\times 10^8$, with approximately $20\,\%$ of the area strongly affected by shear in the lower limit, changing to $90\,\%$ of the area strongly affected by shear in the higher limit; quite small variations are seen in the extent of areas strongly affected by shear beyond this range of $Ra_w$. The increase of $\langle A_{sh}/A\rangle$ shows a trend opposite to the decrease of $\langle A_{bl}/A\rangle$, as expected, since $\langle A_{bl}/A\rangle + \langle A_{sh}/A\rangle =1$. As shown in figure 5, the dependence of $\langle A_{sh}/A\rangle$ on $Ra_w$ can be approximated well by

(3.4)\begin{equation} \langle A_{sh}/A\rangle=1-(a\,{\rm e}^{b Ra_w} + c\,{\rm e}^{d Ra_w}), \end{equation}

where $a=0.7229$, $b =-0.1416\times 10^{-7}$, $c = 0.09798$ and $d = -0.5553\times 10^{-11}$. The relation (3.4) implies that at $Ra_w\approx 10^{12}$, $\langle A_{sh}/A\rangle =1$; the whole area is then affected by shear. However, this specific limiting value of $Ra_w$ needs to be verified further with measurements at higher $Ra_w$ since (3.4) is not accurate beyond $Ra_w=6.05\times 10^8$.Since $A_{sh}+A_{bl}=A$, the corresponding relation for $\langle A_{bl}/A\rangle$ will be

(3.5)\begin{equation} \langle A_{bl}/A\rangle=1-\langle A_{sh}/A\rangle, \end{equation}

with $\langle A_{sh}/A\rangle$ given by (3.4).

Figure 5. Variation of the mean fraction of areas mildly affected by shear $A_{bl}/A$, and that of areas strongly affected by shear $\langle A_{sh}/A\rangle$, with $Ra_w$. Red squares indicate $\langle A_{sh}/A\rangle$; blue circles indicate $\langle A_{bl}/A\rangle$; red solid line uses (3.4); blue dashed line uses (3.5). The error bars show the estimated errors $10.8\,\%$ in $A_{bl}/A$ and $31.6\,\%$ in $A_{sh}/A$ at $Ra_w=6.05\times 10^{8}$ and $2.76\times 10^{5}$, respectively. Details of error analysis can be found in Appendix C.

3.4. Number of occurrences of p.d.f.s

As mentioned earlier, at each $Ra_w$, all three types of p.d.f.s occur with varying relative occurrences. The percentages of the numbers of type A, B and C p.d.f.s in the total number of p.d.f.s, at each of the five different $Ra_w$, are shown in figure 6. At the smallest $Ra_w=2.76\times 10^5$ in figure 6, the p.d.f.s are mostly type A and type B, with the number of type A p.d.f.s ($46\,\%$) being roughly equal to the number of type B p.d.f.s ($53\,\%$). The type A p.d.f.s peak at $U/V_{bl}\sim 1$ at this $Ra_w$ (figure 2a). Even though type B p.d.f.s are dominant at this $Ra_w$, since these type B p.d.f.s were similar to that in figure 3(a), the probabilities of occurrence of the first peak velocities were much higher than those of the second peak velocities. The velocities at this $Ra_w$ were then mostly of the order of $V_{bl}$ (see (3.3)).

Figure 6. Variation of the percentages of type A, type B and type C p.d.f.s in the total number of p.d.f.s, with $Ra_w$. The total numbers of p.d.f.s at each $Ra_w$ are given in table 2. Blue circles indicate type A; magenta diamonds indicate type B; red squares indicate type C; blue dashed line for $3080\,Ra_w^{-1/3}$; magenta solid line for $14.8\,Ra_w^{1/9}-0.0145\,Ra_w^{-4/9}$; red dash-dotted line for $0.008\,Ra_w^{4/9}$; magenta dashed line for $13\,Ra_w^{1/9}$; magenta dotted line for $23\,000\,Ra_w^{-1/3}$. The error bars show the errors $7.1\,\%$ in type A p.d.f.s, $18.7\,\%$ in type B p.d.f.s, and $23.6\,\%$ in type C p.d.f.s, at $Ra_w=2.76\times 10^{5}$. Details of error analysis can be found in Appendix C.

As $Ra_w$ increases from $2.76\times 10^5$ to $1.31\times 10^7$, the percentages of both type B and type C p.d.f.s increase from $53\,\%$ to $73\,\%$, and from $1\,\%$ to $6\,\%$, respectively, at the expense of type A p.d.f.s, which decrease from $46\,\%$ to $21\,\%$. With increase in $Ra_w$, the first peaks of the type B p.d.f.s shift from $U/V_{bl}\sim 1$ at the lowest $Ra_w$ (figure 3a), through $U/V_{bl}\sim 2$ at the intermediate $Ra_w$ (figure 2c), to $U/V_{bl}\approx 3$ at higher $Ra_w$ (figure 3c). Similarly, the type A p.d.f.s – which peaked at $U/V_{bl}=1$ at low $Ra_w$ – shift with increase in $Ra_w$, to peak at $U/V_{bl}=1.83$, as shown in figure 2(a). The local NCBLs, which contribute to the type A p.d.f.s and the first peak of the type B p.d.f.s, could then be changing to a mixed convection type with increase in $Ra_w$.

Comparing the last three $Ra_w$ in figure 6, we see that the percentage of type C p.d.f.s increases from $6\,\%$ to $70\,\%$ with increase in $Ra_w$ from $1.31\times 10^7$ to $6.05\times 10^8$, at the expense of both the type A and type B p.d.f.s. Further, as shown in figure 2(e), peaks of type C p.d.f.s shift from $U/V_{bl}=5$ at an intermediate $Ra_w=1.10\times 10^8$ to $U/V_{bl}=7.6$ at the highest $Ra_w=6.05\times 10^8$. At the highest $Ra_w=6.05\times 10^8$, the shear velocities caused by the LSF dominate the boundary layer flow field even though approximately $30\,\%$ of the p.d.f.s are still type A and type B, which have their sole/first modes showing mixed convection nature. Figure 5 shows that these mixed convection regions that are mildly affected by shear are limited to approximately $10\,\%$ of the area at the highest $Ra_w$.

We now look at the scaling of percentages of types of p.d.f.s with $Ra_w$. As shown in figure 6, as $Ra_w$ increases, the percentages of type A peaks decrease from 46 % at $Ra_w=2.76\times 10^5$ and become negligible (3 %) at the highest $Ra_w$. The figure shows that the percentages of type A peaks scale as

(3.6)\begin{equation} N_A=3080\,Ra_w^{-1/3}, \end{equation}

in the same way as $\lambda _0$ (see (1.5)) scales with $Ra_w$. At the same time, the percentages of type C peaks increase from $1\,\%$ at $Ra_w=2.76\times 10^5$ and become the highest ($70\,\%$) at the highest $Ra_w$, scaling as

(3.7)\begin{equation} N_C=0.008\,Ra_w^{4/9}, \end{equation}

in the same way as the shear velocities caused by the LSF $V_{L}$ (see (3.2)) scale with $Ra_w$. The variation of percentages of type B peaks with $Ra_w$ is non-monotonic, as shown in figure 6, and can be fitted with a combination of increasing and decreasing power-law functions, of the form

(3.8)\begin{equation} N_B=14.8\,Ra_w^{1/9}-0.0145\,Ra_w^{-4/9}. \end{equation}

Interestingly, as shown in figure 6, for the first three $Ra_w$, this relation has an effective exponent $1/9=4/9-1/3$, a sum of the power-law exponents of the variations of percentages of type C and type A p.d.f.s, respectively. For the last two $Ra_w$, the effective power-law exponent for the variation of the percentages of type B p.d.f.s is $-1/3$.

3.5. Experimental scaling of peak velocities

At each $Ra_w$, from various instants, using all the second peaks of the bimodal p.d.f.s of type B, and the sole peaks of the unimodal p.d.f.s of type C, whichever occurs at each instant, we calculate the mean velocity $V_2$, and the corresponding Reynolds number $Re_2=V_2H/\nu$. In estimating $V_2$, we neglect the instants when type A p.d.f.s occur. Figure 7 shows the variation of $Re_2$ with $Ra_w$, along with (3.2), with the error bars showing the 2$\sigma$ variation of the $V_2$ values; it is clear that $Re_2$ scales in the same way as $Re_{L}$ given by (3.2), with a small increase in the prefactor. Similarly, at each instant, we calculate the mean from the first peak of the bimodal p.d.f.s and the unimodal peak of type A p.d.f.s, whichever occurs at that instant, for each $Ra_w$. We denote this mean velocity as $V_1$, with the corresponding Reynolds number as $Re_1=V_1H/\nu$; p.d.f.s of type C are skipped in this calculation. Figure 7 shows the variation of $Re_1$ with $Ra_w$; interestingly, $Re_1\sim Ra_w^{0.38}$, the same dependence as that of $Nu$ on $Ra_w$ observed by He et al. (Reference He, Funfschilling, Nobach, Bodenschatz and Ahlers2012) at high $Ra_w$. The variation of $Re_1$ does not follow the $Ra_w^{1/3}$ scaling of NCBL velocities (3.3), which are expected to occur in the absence of external shear; neither does it follow the scaling of $Re_{L}$ in (3.2) due to LSF. As we have seen in § 3.2, the peaks of type A p.d.f.s and the first peak of the type B p.d.f.s occur at values larger than $V_{bl}$ with increasing $Ra_w$, as expected due to the forcing of the boundary layers by the increasing LSF strength. We hence expect the deviation of $Re_1$ from (3.3) to be due to the shear generated by the LSF changing the characteristic velocities within the boundary layer. Based on this expectation, we now look at the expected theoretical variation of the characteristic velocity within NCBLs when these local boundary layers are forced by the shear due to the LSF velocity $V_{F}$.

Figure 7. Variation of the mean Reynolds numbers based on the first peak velocities of type B p.d.f.s and the sole peak of the type A p.d.f.s ($Re_1$), and that based on the second peak velocities of type B p.d.f.s and the sole peak of the type C p.d.f.s ($Re_2$), with $Ra_w$.

4.1. Scaling relation for horizontal velocity

We consider a two-dimensional laminar NCBL on a horizontal surface, forced by an external shear due to an LSF of strength $V_{F}$, as shown in figure 8. The external shear at the edge of the boundary layer is of strength $AV_{F}$, where $A=u_{y=\delta }/V_F$ is the dimensionless shear at the edge of the boundary layers. We consider local boundary layers that occur on either side of the plumes on the hot plate, unlike the global, single boundary layer that is considered in many flux scaling theories (Grossmann & Lohse Reference Grossmann and Lohse2000; Ahlers et al. Reference Ahlers, Grossman and Lohse2009; Chilla & Schumacher Reference Chilla and Schumacher2012; Scheel & Schumacher Reference Scheel and Schumacher2017). These local boundary layers are an order thinner than the global boundary layers ($\delta _{nc}/\delta _{pb}=0.2$ at $Ra=1.21\times 10^9$), which, due to their larger thickness, are more likely to be affected by the turbulent bulk. These local boundary layers on the hot plate become unstable to turn upwards and form the plumes. Since plumes are the outcome of the instability of these thin local boundary layers, on either side, it is natural to expect that before they become unstable, these boundary layers will be laminar. Further, successful scaling laws for the mean spacing between such line plumes (Theerthan & Arakeri Reference Theerthan and Arakeri1998, Reference Theerthan and Arakeri2000; Puthenveettil & Arakeri Reference Puthenveettil and Arakeri2005), their total lengths (Puthenveettil et al. Reference Puthenveettil, Gunasegarane, Agrawal, Schmeling, Bosbach and Arakeri2011) and their mean dynamics (Gunasegarane & Puthenveettil Reference Gunasegarane and Puthenveettil2014) have all been obtained assuming steady two-dimensional laminar NCBLs (Rotem & Classen Reference Rotem and Classen1969; Pera & Gebhart Reference Pera and Gebhart1973a) feeding these plumes. More importantly, the observed vertical distributions of fluctuations of velocities and temperature near the hot plate have been predicted well by a model that assumes laminar boundary layers giving rise to laminar plumes (Theerthan & Arakeri Reference Theerthan and Arakeri1998). This means that the observed fluctuations in velocity and temperature near the hot plate are created by the spatial averaging of a spatially non-uniform field consisting of many laminar local boundary layers giving rise to many plumes, as well as by the lateral motion of such plumes (Shevkar & Puthenveettil Reference Shevkar and Puthenveettil2022). Also, as an a posteriori justification, the scaling laws obtained in the present study using two-dimensional steady laminar local boundary layer equations, forced by shear, match well with our measurements in turbulent RBC shown later, in figure 10. For these reasons, we expect that our assumption of a laminar mixed convection nature of the local boundary layers that occurs on either side of the plumes on the plate in turbulent convection, which we use to find the scaling of peak velocities in areas mildly affected by shear, to be valid, for at least $Ra \lesssim 10^9$. The integral boundary layer equations

(4.1)\begin{equation} \frac{\partial}{\partial x} \int_0^{\delta} u^2\,{\rm d}z-A V_{F}\,\frac{\partial}{\partial x}\int_0^\delta u\,{\rm d}z -g\beta\int_0^\delta\frac{\partial }{\partial x} \int_0^\delta(T-T_B)\,{\rm d}z \, {\rm d}z+\nu \left.\frac{\partial u}{\partial z}\right|_{z=0}=0\end{equation}

and

(4.2)\begin{equation} \frac{\partial}{\partial x} \int_0^{\delta_T} u(T-T_B)\,{\rm d}z+\alpha \left. \frac{\partial T}{\partial z} \right|_{z=0}=0, \end{equation}

Figure 8. Schematic of the local boundary layer being forced by an external shear due to the LSF for $Pr>1$: (a) side view; (b) top view, and (c) view along line $A$–$A$. Dash-dotted line indicates horizontal velocity profile within an NCBL; dashed line indicates horizontal velocity profile within a PBBL; solid line indicates horizontal velocity profile due to forcing by external shear of NCBL velocities.

derived as (A8) and (A13), relate the horizontal velocity and temperature distributions within such boundary layers.

By using the characteristic scale for the driving temperature difference and the scales for the lengths in the horizontal and vertical directions in (4.1) and (4.2), with the external shear given by $AV_{F}$, we now obtain scaling relations for $U_c$, the characteristic horizontal velocities within NCBLs forced by the shear due to the LSF. The characteristic scale of temperature difference in (4.1) and (4.2) is $\Delta T_w$. The vertical length scale in (4.1) is $\delta$, while that in (4.2) is $\delta _T$, as shown in figure 8. Since these boundary layers become unstable at a horizontal distance to result in line plumes, the appropriate characteristic horizontal distance is half of the mean plume spacing $\lambda$, given by (1.6). However, for the present range of $Ra_w$, the second term in (1.6) becomes at least an order smaller than the first term and can hence be neglected; then $\lambda \approx \lambda _0$. Using these characteristic scales, the order of magnitude balances of (4.1) and (4.2) become

(4.3)\begin{equation} \frac{U_c^2 \delta}{\lambda_0/2}-\frac{A V_{F}U_c\delta}{\lambda_0/2}-\frac{g\beta\,\Delta T_w\, \delta^2}{\lambda_0/2}+\frac{\nu U_c}{\delta}\sim0 \end{equation}

and

(4.4)\begin{equation} \frac{U_c\,\Delta T_w\,\delta_T}{\lambda_0/2}\sim\frac{\alpha\,\Delta T_w}{\delta_T}. \end{equation}

We now assume

(4.5)\begin{equation} \frac{\delta}{\delta_T}=C_2\,Pr^{n}, \end{equation}

where $C_2$ and $n$ are constants. Eliminating $\delta$ from (4.3) using (4.5) and (4.4), we obtain the scaling relation for $U_c$ in terms of $Pe_{\lambda }={U_c\lambda _0}/{2\alpha }$ – the Péclet number based on $\lambda _0$ – as

(4.6)\begin{equation} 8E\,\frac{Pe_{\lambda}^{5/2}}{Ra_{\lambda}}-\epsilon\,\frac{4A}{Pr^n}\,Pe_{\lambda}^{3/2}-C_2\sim 0, \end{equation}

where

(4.7)\begin{equation} \epsilon=Re_{\lambda}/Ra_{\lambda}, \end{equation}

with

(4.8)\begin{equation} Re_{\lambda}=V_{F}\lambda_0/\nu \end{equation}

being the shear Reynolds number based on $\lambda _0$, $Ra_{\lambda }=g\beta \, \Delta T_w\,\lambda _0^3/\nu \alpha$ being the Rayleigh number based on $\lambda _0$, and

(4.9)\begin{equation} E(Pr)=\frac{C_2^2\,Pr^{2n-1}+1}{C_2^2\,Pr^{3n}}. \end{equation}

4.2. Perturbation solution

The values of $Re_{\lambda }$, $Ra_{\lambda }$ and $\epsilon$ for the range of $Ra_w$ in the present study are listed in table 3. With increase in $Ra_w$, $Ra_{\lambda }$ remains of order $10^5$, while $Re_{\lambda }$ changes by a factor 7 from 30 to 209, so that $\epsilon$ increases by an order from $10^{-4}$ to $10^{-3}$; $\epsilon$ is then a small parameter. We can then find the solution of (4.6) as a power series in $\epsilon$ of the form

(4.10)\begin{equation} Pe_{\lambda}=Pe_{\lambda_0}+\epsilon\,Pe_{\lambda_1}+\epsilon^2\,Pe_{\lambda_2}+O(\epsilon^3). \end{equation}

Substituting (4.10) in (4.6) and comparing the coefficients of similar powers of $\epsilon$, we obtain

(4.11)\begin{gather} Pe_{\lambda_0}=\left( \frac{C_2}{8E}\right)^{2/5} Ra_{\lambda}^{2/5}, \end{gather}

(4.12)\begin{gather} Pe_{\lambda_1}=\frac{1}{5E\,Pr^n}\,A\,Ra_{\lambda} \end{gather}

and

(4.13)\begin{equation} Pe_{\lambda_2}=\frac{0.069}{C_2^{2/5}E^{8/5}\,Pr^{2n}}\,A^2\,Ra_{\lambda}^{8/5}. \end{equation}

Substituting (4.11)–(4.13) back in (4.10), we obtain

(4.14)\begin{equation} Pe_{\lambda}=\left( \frac{C_2}{8E}\right)^{2/5} Ra_{\lambda}^{2/5}+\frac{1}{5E\,Pr^n}\,A\,Re_{\lambda}+ \frac{0.069}{C_2^{2/5}E^{8/5}\,Pr^{2n}}\,\frac{(A\, Re_{\lambda})^2}{Ra_{\lambda}^{2/5}}+O(\epsilon^3). \end{equation}

Using (1.5) to replace $\lambda _0$ appearing in $Re_{\lambda }$ and $Ra_{\lambda }$ in (4.14), then using the definitions of $Ra_w$, $Re_{bl}$, $Re$ and $Z_w$, and dropping the second-order terms in $Re_{\lambda }$, we obtain the Reynolds number based on the characteristic velocity in the local NCBL, forced externally by the LSF, as

(4.15)\begin{equation} Re_{bs}=\frac{U_cH}{\nu}=\left(\frac{C_2}{E}\right)^{2/5}Re_{bl}+\frac{0.4}{E\,Pr^{n+1}}\,A\,Re+O(\epsilon^2), \end{equation}

where $Re_{bl}$ is given by (3.3). Using (1.1) for $Re$ in (4.15) will not capture the effect of $\varGamma$ on the LSF strength, which is strong in the present case since $\varGamma$ changes from $10$ to $1.71$ with increase in $Ra$ in the present experiments. To account for the dependence of $\varGamma$ on $Re$, we hence measure the LSF velocity in the present experiments and obtain the corresponding relation (B1) for $Re(Ra, Pr, \varGamma )$ in Appendix B, which is then used in (4.15).

Table 3. Values of the Rayleigh and Reynolds numbers based on the mean plume spacing, along with the values of the perturbation parameter for the present study.

As per (4.15), the velocities in the local boundary layers, in the presence of LSF, are then a linear perturbation of the NCBL velocities by the large-scale velocities, the amount of perturbation being dependent on the $Pr$. This perturbation also depends on the values of shear at the edge of these local boundary layers, given by the dimensionless shear

(4.16)\begin{equation} A=\frac{u|_{z=\delta}}{V_{F}}=2\,\frac{\delta}{\delta_{pb}}-\left(\frac{\delta}{\delta_{pb}}\right)^2, \end{equation}

obtained as (A5) in Appendix A, using the von Kármán velocity profile. Since $V_{F}$, $\delta _{pb}$ and $\delta$ in (4.16) are all functions of $Ra_w$ and $Pr$, $A$ is also a function of $Ra_w$ and $Pr$. We now find this expression for the dimensionless shear ($A$) at the edge of the velocity boundary layer. Since $A$, given by (4.16), is a function of $\delta$, whose dependence on $Ra_{\lambda }$ and $Re_{\lambda }$ is unknown, we first proceed to find the dependence of $\delta$ on $Ra_{\lambda }$ and $Re_{\lambda }$.

4.3. Velocity boundary layer thicknesses for small shear

Replacing $\delta _T$ in (4.4) with $\delta$ using (4.5), and substituting the resulting expression for $U_c$ in (4.3), we obtain

(4.17)\begin{equation} \frac{4}{C_2^2}\,\xi^5+2A\,Pr^{2n}\,\epsilon \xi^2 -\frac{C_2^{2}\,Pr^{5n}\,E}{Ra_{\lambda}}\sim0, \end{equation}

where

(4.18)\begin{equation} \xi=\frac{\delta}{\lambda_0}=\frac{Ra_{\delta}^{1/3}}{C_1\,Pr^{n_1}} \end{equation}

is the dimensionless local velocity boundary layer thickness when these boundary layers are perturbed by the LSF, with $Ra_{\delta }=g\beta \,\Delta T_w\,\delta ^3/\nu \alpha$ being the Rayleigh number based on $\delta$ and $\Delta T_w$. Let the solution for (4.17) be a power series in $\epsilon$ of the form

(4.19)\begin{equation} \xi=\xi_0+\epsilon \xi_1+O(\epsilon^2). \end{equation}

Substituting (4.19) in (4.17) and comparing the coefficients of different powers of $\epsilon$, we get

(4.20)\begin{equation} \xi_0=0.76\,Pr^n\left( \frac{C_2^4E}{ Ra_{\lambda}}\right)^{1/5} \end{equation}

and

(4.21)\begin{equation} \xi_1=-0.17A\left( \frac{C_2\,Ra_{\lambda}}{E }\right)^{2/5}. \end{equation}

Thus the perturbation solution of (4.17), obtained by substituting (4.20) and (4.21) in (4.19), after neglecting higher-order terms, is

(4.22)\begin{equation} \xi \sim 0.76\,Pr^n\left( \frac{C_2^4E}{Ra_{\lambda}}\right)^{1/5}-0.17\left( \frac{C_2}{E}\right)^{2/5}A\,\frac{Re_{\lambda}}{Ra_{\lambda}^{3/5}}+O(\epsilon^2). \end{equation}

Rewriting (4.22) in terms of $Ra_{\delta }$ by using (4.18) and (1.5), we get

(4.23)\begin{equation} Ra_{\delta}^{1/3}=\frac{\delta}{Z_w} \sim 36.1\,Pr^{n+0.1}\left( \frac{C_2^4E}{Ra_{\lambda}}\right)^{1/5}-\frac{8.1}{Pr^{-0.1}}\left( \frac{C_2}{E}\right)^{2/5}A\,\frac{Re_{\lambda}}{Ra_{\lambda}^{3/5}}+O(\epsilon^2) \end{equation}

as the expression for the critical, local dimensionless boundary layer thickness, when these boundary layers are forced by an external shear due to the LSF. The first term on the right-hand side of (4.23) is the critical dimensionless boundary layer thickness in the absence of external shear, while the second term is the perturbation due to the external shear. For no external shear, $Re_{\lambda }\rightarrow {0}$, ${\lambda }\rightarrow {\lambda _0}$ and $n=0.5$ (Puthenveettil et al. Reference Puthenveettil, Gunasegarane, Agrawal, Schmeling, Bosbach and Arakeri2011). Then, using (1.5) and (1.3), for no external shear, (4.23) becomes

(4.24)\begin{equation} Ra_{\delta}\sim {45.1}{\left(C_2^4E\right)^{3/5}} Pr^{1.62}, \end{equation}

which, being constant for a given fluid, is of the same form as the marginal stability condition proposed by Howard (Reference Howard1964).

Figure 9 shows the variation of $\xi$ with $Re_{\lambda }$ when $A=1$ for three different cases $(Pr, Ra_{\lambda })=(0.7, 9.63\times 10^4)$, $(1, 1.07\times 10^5)$, $(6, 1.83\times 10^5)$, as given by the numerical solution of (4.17). A similar plot, but for the $Pr=1$ case alone, was shown by Puthenveettil & Harsha (Reference Puthenveettil and Harsha2010). It needs to be noted that in RBC, $Re_{\lambda }$, $Ra_{\lambda }$ and $A$ are all actually interdependent through the LSF, and none of these can be fixed independent of others. Figure 9 could then be considered to show the expected variation of $\xi$ when a small shear is forced externally and independently of the fixed $Ra_w$, as occurs in the case of mixed convection that causes negligible change of $\lambda$ from its no-shear value $\lambda _0$. Figure 9 also shows the variations predicted by the perturbation solutions (4.22) at $A=1$ for $(Pr, Ra_{\lambda })=(1,1.07\times 10^5)$. The perturbation solution (4.22) is a good approximation for the variation of $\xi$ predicted by (4.17) for $Re_{\lambda }< 250$. The figure shows that for $Re_{\lambda }<400$, $\xi$ is a function of $Pr$ and takes larger values for higher $Pr$; however, $\xi$ becomes a very weak function of $Pr$ for $Re_{\lambda }>400$.

Figure 9. Variation of the critical, local dimensionless boundary layer thickness (4.18) with the shear Reynolds number based on the mean plume spacing (4.8) at $Pr=1$ and $Ra_{\lambda }=1.07\times 10^5$. Symbols and lines are: blue circles, numerical solution of (4.17); solid black line, (4.22); cyan dashed line, NCBL; magenta dots, PBBL; dash-dotted line, (D5); green diamonds, numerical solution of (4.17) at $Pr=0.7$ and $Ra_{\lambda }=9.63\times 10^4$; red crosses, numerical solution of (4.17) at $Pr=6$ and $Ra_{\lambda }=1.83\times 10^5$. The inset shows the variations of $I$ (see (4.26)) and $J$ (see (4.27)) with $Pr$ for $n=0.5$ and $\varGamma =1.71$: blue triangles, $I(Pr)$; red squares, $J(Pr)$; blue dotted line, $0.09\,Pr^{0.14}$. In computing solutions of (4.22) and (D5), terms of orders $\epsilon ^2$ and $\epsilon _1^2$ were neglected, respectively.

For large $Re_{\lambda }$, figure 9 shows that the variation of $\xi$ is similar to that in the Blasius boundary layer. However, at large $Re_{\lambda }$, $\lambda$ will vary from $\lambda _0$ (Shevkar et al. Reference Shevkar, Gunasegarane, Mohanan and Puthenveettil2019), hence the actual variation of $\xi$ will deviate from that shown in figure 9. The singular perturbation solution of (4.17) for large shear, given in Appendix D, shown by the dash-dotted line in figure 9, matches the numerical solution of (4.17) and the variation for PBBLs for large $Re_{\lambda }$. Figure 9 also shows that at $Pr=1$, the value of $\xi$ decreases by only $10.5\,\%$ as $Re_{\lambda }$ increases from $10^0$ to $10^2$, while it decreases by $57\,\%$ for further increase in $Re_{\lambda }$ from $10^2$ to $10^3$. Similarly, at $Pr=6$, the value of $\xi$ decreases by $20\,\%$ and $62\,\%$ for the increases in $Re_{\lambda }$ from $10^0$ to $10^2$, and from $10^2$ to $10^3$, respectively. So at any $Pr$, the deviation of the values of $\xi$ from its natural convection value in (4.20) is less, approximately, for $Re_{\lambda }\leqslant 100$. For $Re_{\lambda }>100$, the value of $\xi$ decreases more rapidly at higher $Pr$ with increase in $Re_{\lambda }$ so as to match the variation given by the large shear value $\xi _{s_0}$ (see (D3)), which again is independent of $Pr$. A similar, larger increase of $\lambda$ at larger $Pr$ has been found by Shevkar et al. (Reference Shevkar, Gunasegarane, Mohanan and Puthenveettil2019) at a given $Ra_w$ and $Re$. Similarly, in sheared convection, with increasing wall shear velocity, Blass et al. (Reference Blass, Zhu, Verzicco, Lohse and Stevens2020) found a more pronounced increase of $Nu$ at larger $Pr$, which could be due to this larger decrease of $\delta$ with increasing shear at higher $Pr$.

For $Re_{\lambda }<30$, at any $Pr$, as shown in figure 9, $\xi$ remains approximately independent of $Re_{\lambda }$, which is the expected variation for a pure NCBL. This constancy of $\xi$ for small shear can be quantified by rewriting (4.22) in terms of $Ra_w$. Using $\lambda _0$ from (1.5) in (4.22), and then using (1.3) and (B1), we obtain

(4.25)\begin{equation} \xi \sim I-J A\,Ra_w^{1/9}+O(\epsilon^2), \end{equation}

where

(4.26)\begin{equation} I(Pr)=C_3\,Pr^{n-{3}/{50}}\,E^{1/5} \end{equation}

and

(4.27)\begin{equation} J(Pr)=\frac{C_4 \varGamma^{-0.57}}{Pr^{3/4}\,E^{2/5}}, \end{equation}

with $C_3=(C_2^4/4C_1^3)^{1/5}$ and $C_4=0.27(C_2/C_1^2)^{2/5}$.

For $Pr>1$, since the velocity boundary layer thickness is more than the thermal boundary layer thickness, in (4.5), $n$ is positive and less than 1 (Shishkina et al. Reference Shishkina, Stevens, Grossmann and Lohse2010); we assume that $n=0.5$, the same value as in the no-shear case (Puthenveettil et al. Reference Puthenveettil, Gunasegarane, Agrawal, Schmeling, Bosbach and Arakeri2011), for the present case of small shear. The inset of figure 9 shows that for $n=0.5$, $I(Pr)$ (see (4.26)) is more than an order larger than $J(Pr)$ (see (4.27)). Then since $A<1$ (see (A5) and figure 8), the second term in (4.25), which has a weak dependence on $Ra_w$, gives negligible contribution to $\xi$ when $Pr>1$, for the range of $Ra_w$ in the present study; (4.25) then implies that

(4.28)\begin{equation} \xi \sim I(Pr). \end{equation}

The relation (4.28) implies that in the presence of weak shear, $\delta$ and $\lambda _0$ have the same functional dependence on $Ra_w$, so that their ratio is only a function of $Pr$; a similar dependence occurs in the no-shear case, as has been shown by Puthenveettil et al. (Reference Puthenveettil, Gunasegarane, Agrawal, Schmeling, Bosbach and Arakeri2011). As shown in the inset of figure 9, for $n=0.5$, the rational function (4.26) can be approximated by a simple power law, $I(Pr)\sim Pr^{0.14}$; (4.28) then becomes

(4.29)\begin{equation} \xi\sim C_5\,Pr^{0.14}, \end{equation}

where $C_5=0.09$. Rearranging (4.29) using $\lambda _0$ from (1.5), we get

(4.30)\begin{equation} \frac{\delta}{H}=C_6\,Ra_w^{-1/3}\,Pr^{0.24}, \end{equation}

where $C_6=C_1C_5=4.28$. The $Pr$ dependence of $\delta$ in (4.30) is the same as that obtained by Lam et al. (Reference Lam, Shang, Zhou and Xia2002) for the range $6\leqslant Pr\leqslant 1027$ in RBC.

4.4. Dimensionless shear ($A$) at the edge of the boundary layers

Now that the expression (4.30) for $\delta$, the NCBL thickness forced by small shear, has been obtained, we substitute (4.30) and (1.4) using (B1) in (A5), and simplify, to obtain the dimensionless shear velocity acting on the edge of the local boundary layers:

(4.31)\begin{equation} A=\chi(2-\chi), \end{equation}

where

(4.32)\begin{equation} \chi=\frac{\delta}{\delta_{pb}}=\frac{1.37C_6}{\varGamma^{0.285}\,Ra_w^{1/9}\,Pr^{0.09}}. \end{equation}

Interestingly, (4.31) and (4.32) show that for a given $\varGamma$ and $Pr$, within the limit given by (E4), the dimensionless shear at the edge of the local NCBL decreases with increase in $Ra_w$ as $A\sim (2Ra_w^{1/9}-1)/Ra_w^{2/9}$. In the present study, since $\varGamma$ and $Ra_w$ increase much more than the decrease of $Pr$ with increasing $Ra_w$, $A$ decreases from 0.87 at the lowest $Ra_w$, to 0.71 at the highest $Ra_w$. However, the velocity at the edge of these boundary layers, due to the shear by the LSF, is $u|_{y=\delta }=A V_F$ from (A5), where $V_F\sim Ra_w^{4/9}$ from (B1). Hence the actual shear forcing at the edge of these local boundary layers increases with $Ra$ as $u|_{y=\delta }\sim Ra_w^{1/3}-Ra_w^{2/9}$, which has an approximate power-law dependence $Ra^{0.35}$ in the range $10^7< Ra_w<10^{10}$. As per the present analysis, since $V_{bl}$ scales as $Ra_w^{1/3}$ (see (3.3)) with increasing $Ra_w$, the local NCBL would eventually become dominated by shear. However, the present analysis, limited to low shears, does not consider the change in plume spacing due to shear given by (1.6), as well as the change in boundary layer thickness with shear (see (4.30)). The above conclusion of local NCBLs becoming shear-dominant at large $Ra_w$ has to be explored further by extending the present analysis to larger shears by including the changes in the plume spacing and the boundary layer thicknesses at larger shear.

4.5. Reynolds number based on the characteristic velocity $U_c$

We now obtain the effect of external shear caused by the LSF on the velocities within the boundary layers. Substituting (4.31) and (B1) in (4.15), and simplifying, we obtain

(4.33)\begin{equation} Re_{bs}=Re_{bl}\left(\left(\frac{C_2}{E} \right)^{2/5}+\frac{F}{\varGamma^{0.855}}\,(2-\chi)\right) \end{equation}

as the Reynolds number based on the characteristic velocity close to the hot plate in RBC when the local boundary layers are forced by the LSF. Here,

(4.34)\begin{equation} F(Pr)=\frac{0.46\,C_6}{E\,Pr^{1.28}}, \end{equation}

with $\chi$ being given by (4.32), $Re_{bl}$ by (3.3), and $E$ by (4.9), and we have chosen $n=0.5$ in (4.5), the value in the no-shear case (Puthenveettil et al. Reference Puthenveettil, Gunasegarane, Agrawal, Schmeling, Bosbach and Arakeri2011). Comparing (4.33) and (4.15), we can rewrite (4.33) as

(4.35)\begin{equation} Re_{bs}=Re_b+Re_s, \end{equation}

i.e. the sum of a buoyant contribution and a shear perturbation, where

(4.36)\begin{equation} Re_b=\left(\frac{C_2}{E}\right)^{2/5}Re_{bl}=1.07\,Ra_w^{1/3}\,Pr^{-0.38} \end{equation}

and

(4.37)\begin{equation} Re_s=\frac{0.4}{E\,Pr^{1.5}}\,A\,Re=\frac{1.27}{\varGamma^{0.855}}\,Ra_w^{1/3}\,Pr^{-0.76}\,(2-\chi), \end{equation}

with $Re_{bl}$, $Re$ and $A$ given by (3.3), (B1) and (4.31), respectively.

Equation (4.35) gives the variation of the dimensionless boundary layer velocity as a function of $Ra_w$, $Pr$ and $\varGamma$ in RBC for small shear by the LSF. The variation predicted by (4.35), shown as the solid line in figure 10, matches the variation of $Re_1$, the Reynolds number based on the sole peaks of type A p.d.f.s and the first peaks of type B p.d.f.s for a reasonable value $C_2=0.72$. The single-peak type A p.d.f.s occurred at the low shear at low $Ra_w$, while the first peaks of type B p.d.f.s were due to the low-shear regions at all $Ra_w$. The deviation of the data point at the lowest $Ra_w$ in figure 10 is expected to occur since (1.4) may not be valid for $Ra_w<10^7$.

Figure 10. Variation of Reynolds number, based on the mean velocity from the sole peaks of type A p.d.f.s and the first peak of type B p.d.f.s, with $Ra_w$: black circles, (4.35) with $Re_{bs}=Re_1$; black solid line, $1.0\,Ra_w^{1/3}$. Bottom right inset shows the variation of $Re_{bs}$ (see (4.33)) with $Ra_w$ at $Pr=1$: blue solid line, $\varGamma =1$; green solid line, $\varGamma =2$; magenta solid line, $\varGamma =5$; red solid line, $\varGamma =20$; black dashed line, $Ra_w^{0.375}$; cyan dashed line, $Ra_w^{1/3}$. Top left inset shows variation of $Re_{bs}$ (see (4.33)) with $Ra_w$ at $\varGamma =1$: blue solid line, $Pr=1$; green solid line, $Pr=5$; magenta solid line, $Pr=25$; red solid line, $Pr=100$; black dashed line, $Ra_w^{0.375}$; cyan dashed line, $Ra_w^{1/3}$. Bottom left inset shows the variation of the dimensionless shear velocity $\chi$ (see (4.32)) with $Ra_w$: green solid line, $\varGamma =1$; black dashed line, $\varGamma =10$.

It is interesting to see the effective power-law dependence of $Re_{bs}$ on $Ra_w$ corresponding to (4.35). The top left inset of figure 10 shows the variation of $Re_{bs}$ with $Ra_w$ given by (4.35) at $\varGamma =1$ for $1\leqslant Pr \leqslant 100$. At $Pr=1$, a power law $Re_{bs}\sim Ra_w^{0.375}$ approximates (4.35) for $10^7\leqslant Ra_w \leqslant 3.3\times 10^{8}$. As $Pr$ is increased from 1 to 100, the exponent of $Ra_w$ is decreased from 0.375 to $1/3$; $Re_{bs}$ then scales with $Ra_w$ in the same way as $Re_{bl}$ scales with $Ra_w$ (see (3.3)). Therefore, the present analysis suggests that for $Pr\geqslant 100$, the effect of external forcing on the local NCBLs become negligible close to the hot plate. Similarly, the bottom right inset of figure 10 shows that with increase in $\varGamma$, the dependence of $Re_{bs}$ on $Ra_w$ becomes weaker, and for $\varGamma \geqslant 20$, $Re_{bs}\sim Ra_w^{1/3}$. It is interesting to note that at $Pr=1$ and $\varGamma =1$, $Re_{bs}$ shows the same power law of $Nu$ versus $Ra_w$ observed at large $Ra_w$ (He et al. Reference He, Funfschilling, Nobach, Bodenschatz and Ahlers2012), which has been proposed as a sign of the ultimate regime, even though this inference has been disputed by Urban et al. (Reference Urban, Hanzelka, Kralik, Musilova, Srnka and Skrbek2012). Considering that at low $Ra_w$, the scaling of $Re_b$ in (4.36) (or $Re_{bl}$ in (3.3)) with $Ra_w$ is the same as that of $Nu$ with $Ra_w$, the power-law exponent 0.375 for the dependence of $Re_{bs}$ on $Ra_w$ at higher $Ra_w$ is analogous to what occurs at low $Ra_w$; the reason for such an interesting occurrence needs to be investigated.