3.1. Wall-shear stress and velocity reconstruction in the dynamic frame

The local wall-shear stress can be calculated using the vertical gradients of the horizontal velocity components at the centre of the bottom plate. Figure 1(b,c) shows samples of the instantaneous and mean profiles of the horizontal velocity components at $Ra = 1.01 \times 10^9$, respectively. We see that the linear sections of the near-wall BL profiles can be accurately resolved by the present measurement. The velocity gradients $(\partial u / \partial z)|_{z = 0}$ and $(\partial v / \partial z)|_{z = 0}$ are obtained by a least-square fitting of the corresponding linear section close to the wall surface, and the wall-shear stress $\boldsymbol {\tau }$ is calculated as

(3.1)\begin{equation} \boldsymbol{\tau} = (\tau_x, \tau_y) = \left(\left. \mu \frac{\partial u}{\partial z} \right|_{z = 0}, \left. \mu \frac{\partial v}{\partial z} \right|_{z = 0} \right), \end{equation}

wherein $\mu$ is the dynamic viscosity.

Figure 2 shows the joint probability density function (p.d.f.) of the two wall-shear stress components at three different Rayleigh numbers. We can see that there exist strong fluctuations in $\tau _y$ which are comparable in magnitude with $\tau _x$, despite the fact that the convection cell has been tilted along the $x$ direction. As $Ra$ increases, the joint p.d.f. becomes more extended for both components. The probability of $\tau _{x}$ against the orientation of the mean LSC, i.e. $\tau _{x} < 0$, also becomes higher as $Ra$ increases. This may originate from the complicated 3-D motions of the LSC in the cylindrical cell including meandering, torsional and sloshing motions.

Figure 2.

Joint p.d.f. of normalised wall-shear stress components at Rayleigh numbers (a) $Ra = 1.82 \times 10^8$, (b) $Ra = 1.01 \times 10^9$ and (c) $Ra = 5.26 \times 10^9$.

Given the observed strong fluctuations of the local wall-shear stress, the single horizontal velocity component in the mean LSC plane may not be sufficient to characterise the instantaneous viscous BL. With the measured full velocity profile, we propose to reconstruct the near-wall velocity using a dynamic wall-shear frame based on the measured instantaneous wall-shear stress (hereafter we refer to this as the dynamic frame, with quantities obtained in this frame marked with superscript $d$). The direction (projection angle) of the wall-shear stress $\phi$ is defined as

(3.2)\begin{equation} \phi \equiv {\rm{arctan}}{(\tau_y / \tau_x)}, \end{equation}

and the corresponding transformation matrix $R$ as

(3.3)\begin{equation} R = \left[\begin{array}{ccc} {\cos \phi} & {\sin \phi} & 0 \\ {-\sin \phi} & {\cos \phi} & 0 \\ 0 & 0 & 1 \end{array}\right]. \end{equation}

The velocity vector in the dynamic frame $(u^d, v^d, w^d)$ can thus be transformed from the laboratory frame $(u, v, w)$ following $R$. The vertical velocity remains unchanged after the transformation, i.e. $w^d = w$.

The viscous BL thicknesses in the respective frames $\delta _u$ and $\delta _u^d$ are obtained using the slope method (see figure 1c), i.e. the distance at which the extrapolated linear section intersects the maximum velocity (Xin et al. Reference Xin, Xia and Tong1996),

(3.4a)\begin{gather} \delta_u \equiv \frac{U_m}{({\partial U}/{\partial z})|_{z = 0}}, \end{gather}

(3.4b)\begin{gather}\delta_u^d \equiv \frac{U_m^d}{({\partial U^d}/{\partial z})|_{z = 0}}, \end{gather}

wherein $U \equiv \langle u \rangle _{t}$ and $U^d \equiv \langle u^d \rangle _{t}$ are the time-averaged velocities, the operator $\langle \,\cdot \, \rangle _t$ denotes the time-averaging, and $U_m \equiv \max (U)$ and $U_m^d \equiv \max (U^d)$ are the maximum velocities. The difference between $\delta _u$ and $\delta _u^d$ results from the inapplicability of the projection equation for the maximum horizontal velocity, i.e. $u_m \ne \cos (\phi ) u_m^d$ ($u_m \equiv \max (u)$, $u_m^d \equiv \max (u^d)$). The rotating transformation is strictly satisfied only in the linear section of the velocity profile, i.e. $u = \cos (\phi ) u^d$ in the linear section of the $u(z)$ profile. Above the linear section, $u(z)$ and $v(z)$ may present different shapes, resulting in $u \ne \cos (\phi ) u^d$ and thus $u_m \ne \cos (\phi ) u_m^d$. For example, for the instantaneous velocity profiles in figure 1(b), $\delta _u = 2.541$ mm while $\delta _u^d = 2.309$ mm.

Figure 3(a) plots the viscous BL thickness as a function of the Rayleigh number, indicating a thinner viscous BL in the dynamic frame than in the laboratory frame. Whereas the power-law scaling exponents with $Ra$ in two frames are essentially the same. As indicated in table 1, the scaling exponent of $\delta _u$ with $Ra$ has a larger value (i.e. a smaller magnitude) in Xin et al. (Reference Xin, Xia and Tong1996), Xin & Xia (Reference Xin and Xia1997) and Lam et al. (Reference Lam, Shang, Zhou and Xia2002), which is probably related to the dynamics of the LSC in the levelled cylindrical convection cell. Figure 3(b) plots the Reynolds number $Re = U_m H / \nu$ ($Re^d = U_m^d H / \nu$) as a function of $Ra$ in both frames, which characterises the intensity of the LSC. Fitting a power law to the data points yields $Re \sim Ra^{0.45}$ and $Re^d \sim Ra^{0.46}$ in the two frames, respectively. We also calculate the scaling of the viscous BL thickness with the Reynolds number by applying the data points in figure 3. The results of $\delta _u \sim Re^{-0.48}$ and $\delta _u^d \sim (Re^d)^{-0.47}$ basically satisfy the prediction of the classical Prandtl–Blasius BL, i.e. $\delta _u \sim Re^{-0.5}$. Our results are consistent with the fact that, for moderate Rayleigh number range, the viscous BL is still laminar albeit with fluctuations (Tilgner, Belmonte & Libchaber Reference Tilgner, Belmonte and Libchaber1993; Verzicco & Camussi Reference Verzicco and Camussi2003; Xia et al. Reference Xia, Sun and Zhou2003; Funfschilling & Ahlers Reference Funfschilling and Ahlers2004; Stevens et al. Reference Stevens, Zhou, Grossmann, Verzicco, Xia and Lohse2012; Scheel & Schumacher Reference Scheel and Schumacher2014). In the recent BL model of Ching et al. (Reference Ching, Leung, Zwirner and Shishkina2019) for turbulent RBC, the scaling of the viscous BL thickness is also proportional to $Re^{-0.5}$.

Figure 3.

(a) The viscous BL thickness $\delta _u$ and $\delta _u^d$ as a function of $Ra$. The fitted power-laws to the respective data are $\delta _u = 291 Ra^{-0.22 \pm 0.01}$ mm (blue dashed line) and $\delta _u^d = 241 Ra^{-0.21 \pm 0.01}$ mm (red solid line). (b) The Reynolds number $Re = U_m H / \nu$ and $Re^d = U_m^d H / \nu$ based on the maximum horizontal velocity as a function of $Ra$. The fitted power-laws of the respective data are $Re = 0.149 Ra^{0.45 \pm 0.01}$ (blue dashed line) and $Re^d = 0.136 Ra^{0.46 \pm 0.01}$ (red solid line).

Table 1.

Scaling exponents of the viscous BL thickness $\delta _u$ ($\delta _u^d$) and the Reynolds number $Re$ ($Re^d$) as a function of the Rayleigh number $Ra$. The coefficients $A$ and $B$, and scaling exponents $\beta$ and $\gamma$ are determined from the power laws $\delta _u = A Ra^{\beta }$ ($\delta _u^d = A^d Ra^{\beta }$) and $Re = B Ra^{\gamma }$ ($Re^d = B^d Ra^{\gamma }$). The coefficients $A$ and $A^d$ are in units of millimetres. The control parameters $Ra$, $Pr$, the cell geometry and the aspect ratio $\varGamma$ are also listed. The sources are (a) Xin et al. (Reference Xin, Xia and Tong1996), (b) Xin & Xia (Reference Xin and Xia1997), (c) Lam et al. (Reference Lam, Shang, Zhou and Xia2002), (d) Sun et al. (Reference Sun, Cheung and Xia2008), (e) Li et al. (Reference Li, Shi, Du Puits, Resagk, Schumacher and Thess2012), (f) Wei & Xia (Reference Wei and Xia2013), (g) Scheel & Schumacher (Reference Scheel and Schumacher2014) (h) Zou et al. (Reference Zou, Zhou, Chen, Bao, Chen and She2019) and (i) present work. Abbreviations are: direct numerical simulation (DNS); experiment (EXP). The tilt angle of the convection cell is considered to be zero when not explicitly mentioned in the respective sources.

3.2. The eddy viscosity and the mean velocity profile

Figure 4 plots time traces of velocity fluctuation components ${u'}^d$ and $w'$ in the dynamic frame at different wall distances. Deep inside the viscous BL at $z / \delta _u^d = 0.2$, the magnitudes of ${u'}^d$ and $w'$ are small, and a close inspection of the two time traces reveals a negative correlation in time. At a higher position of $z / \delta _u^d = 0.6$, the velocity fluctuation components ${u'}^d$ and $w'$ have larger magnitudes and their negative correlation is more significant. Outside the viscous BL at $z / \delta _u^d = 1.5$, the magnitudes of ${u'}^d$ and $w'$ become larger, which is due to the intense plume emissions outside the BL. Meanwhile, the (negative) correlation between ${u'}^d$ and $w'$ is less significant once outside the viscous BL.

Figure 4.

Time traces of velocity components ${u'}^d$ and $w'$ at different distances from the bottom plate at $Ra = 1.01 \times 10^9$, (a) $z / \delta _u^d = 0.2$, (b) $z / \delta _u^d = 0.6$, (c) $z / \delta _u^d = 1.5$.

Taking the Reynolds decomposition of the velocity field and the 2-D BL approximation, we obtain the momentum equation of the horizontal velocity component as

(3.5)\begin{equation} U \frac{\partial U}{\partial x} + W \frac{\partial U}{\partial z} ={-} \frac{\partial P}{\rho \partial x} - \frac{\partial \langle u'w' \rangle_t}{\partial z} + \nu \frac{\partial^2 U}{\partial z^2}, \end{equation}

wherein $P$ is the mean pressure, $u'$ and $w'$ are the velocity fluctuations in the horizontal and vertical directions, i.e. $u = U + u'$ and $w = W + w'$.

Since the magnitudes of the molecular momentum diffusion and the turbulent momentum diffusion are much larger than other terms in (3.5) inside the viscous BL, the balance of these two terms in the laboratory frame and in the dynamic frame are written as

(3.6a)\begin{gather} \nu \frac{\partial^2 U}{\partial z^2} = \frac{\partial \langle u' w'\rangle_t}{\partial z}, \end{gather}

(3.6b)\begin{gather}\nu \frac{\partial^2 U^d}{\partial z^2} = \frac{\partial \langle {u'}^d w'\rangle_t}{\partial z}, \end{gather}

wherein $u'^d$ is the velocity fluctuation in the dynamic frame following $u^d = U^d + {u'^d}$.

The turbulent eddy viscosity $\nu _t$ and $\nu _t^d$ can be defined as

(3.7a)\begin{gather} \langle u'w' \rangle_t \equiv{-}\nu_t \frac{\partial U}{\partial z}, \end{gather}

(3.7b)\begin{gather}\langle {u'^d}{w'} \rangle_t \equiv{-}\nu_t^d \frac{\partial U^d}{\partial z}. \end{gather}

Figures 5(a) and 5(b) plot profiles of the normalised eddy viscosity $\nu _t / \nu$ and $\nu _t^d / \nu$ in the laboratory frame and the dynamic frame, respectively. In the laboratory frame (figure 5a), the eddy viscosity profiles present no universal behaviour as the curves at low $Ra$ lie below the high $Ra$ cases. On the contrary, the profiles for $\nu _t^d / \nu$ in the dynamic frame (figure 5b) all collapse on a single master curve. A power-law fitting of all the data points within the viscous BL thickness, i.e. $z \le \delta _u^d$, yields

(3.8)\begin{equation} \nu_t^d / \nu = 0.81 {(z / \delta_u^d)}^{3.10 \pm 0.05}. \end{equation}

Figure 5.

(a,b) Profiles of the normalised eddy viscosity $\nu _t / \nu$ and $\nu _t^d / \nu$ as a function of the normalised wall distance $z / \delta _u$ and $z / \delta _u^d$ for different $Ra$ in (a) the laboratory frame and (b) the dynamic wall-shear frame. The solid line in panel (b) indicates the fitted power-law relation using all data points within the viscous BL thickness.

The scaling exponent $\zeta = 3.10 \pm 0.05$ ($\nu _t^d \sim z^{\zeta }$) is close to those measured in the near-wall region of a quasi-2-D vertical thin disk (Xu et al. Reference Xu, Wang, He, Wang, Schumacher, Huang and Tong2021). This indicates that near the bottom plate, the effects of meandering/torsional/sloshing motions of the LSC are well-eliminated in the dynamic frame, and the viscous BL in 3-D and 2-D configurations seem to follow a universal dynamical process. The no-slip boundary condition and the continuity equation lead to $\langle u' w' \rangle _t = \partial \langle u' w' \rangle _t / \partial z = \partial ^2 \langle u' w' \rangle _t / \partial z^2 = 0$, which indicates that the leading order of the Reynolds shear stress is greater than or equal to three. Therefore, the concern scaling exponent being equal to three is not an exact theoretical result. Our result that the scaling exponent being $3.10\pm 0.05$ indicates that, within the experimental uncertainty, the higher-order terms (larger than three) also make a non-negligible contribution. We also remark that the scaling exponent for the eddy viscosity is close to that for the eddy thermal diffusivity (Shishkina et al. Reference Shishkina, Horn, Wagner and Ching2015; Schumacher et al. Reference Schumacher, Bandaru, Pandey and Scheel2016; Ching et al. Reference Ching, Leung, Zwirner and Shishkina2019).

The formal solution of (3.6b) and (3.7b) in the dynamic wall-shear frame is

(3.9)\begin{equation} \tilde U^d(\xi^d) = \int_{0}^{\xi^d} \frac{1}{1 + \nu_t^d(s) / \nu} \,{\rm d}s, \end{equation}

wherein $\xi ^d = z / \delta _u^d$ is the similarity variable, and $\tilde U^d = U^d / U_m^d$ is the normalised horizontal velocity. Therefore, the fitted eddy viscosity relation in (3.8) can be used to calculate the mean velocity profile via (3.9).

Figure 6 plots the profiles of the normalised velocity as a function of the normalised wall distance for different Rayleigh numbers in the dynamic frame. Since the eddy viscosity profiles do not overlap in the laboratory frame (see figure 5a), a corresponding general theoretical profile of the mean velocity cannot be obtained. The mean horizontal velocity profile following (3.8) and (3.9) is shown as the solid curve in figure 6, indicating a good agreement with the measured data points in the near-wall region. The deviation of the experimental data from the theoretical curve at positions far away from the wall may be partly attributed to the contributions by those neglected mean convection and pressure-gradient terms in (3.5). Additionally, considering the 3-D nature of the flow field in the cylindrical cell, the proposed reconstruction scheme based on the instantaneous wall-shear stress may be ineffective in the region far away from the wall.

Figure 6.

Profiles of the normalised horizontal velocity $\tilde {U}^d$ as a function of the normalised wall distance $\xi ^d$ for different $Ra$ in the dynamic wall-shear frame. The solid curve represents the theoretical solution following (3.8) and (3.9).

3.3. The wall energy dissipation rate

The 3-D measurement of the viscous BL allows us to directly obtain the wall energy dissipation rate at the centre of the bottom plate $\varepsilon _{w}$ as

(3.10)\begin{equation} \varepsilon_{w} = \nu \left[ \left( \left. \frac{\partial u}{\partial z} \right|_{z =0} \right)^2 + \left( \left. \frac{\partial v}{\partial z} \right|_{z =0} \right)^2 \right]. \end{equation}

The other terms in the velocity gradient tensor vanish at $z = 0$ considering the no-slip boundary condition and the continuity equation.

Figure 7(a) shows the mean wall dissipation rate $\langle \varepsilon _w \rangle _t$ as a function of $Ra$. For reference, we plot in the same figure the indirectly measured mean dissipation rate at the cell centre $\langle \varepsilon _c \rangle _t$ (Ni et al. Reference Ni, Huang and Xia2011), as indicated by the black circles. The reference mean dissipation rate at the cell centre $\langle \varepsilon _c \rangle _t$ is corrected following $\langle \varepsilon _c \rangle _t \sim Pr^{1.15}$ to account for the variation of $Pr$ between 4.4 and 6.1 in Ni et al. (Reference Ni, Huang and Xia2011). For the parameter range explored in this study, the directly measured wall dissipation $\langle \varepsilon _w \rangle _t$ is over one order of magnitude larger than the centre reference $\langle \varepsilon _c \rangle _t$. The power-law scaling exponent on $Ra$ at the bottom plate ($\langle \varepsilon _{w} \rangle _t / (\nu ^3 H^{-4}) = 9.65 \times 10^{-2} Ra^{1.25 \pm 0.02}$) is smaller than that obtained in the cell centre ($\langle \varepsilon _c \rangle _t \sim Ra^{1.55 \pm 0.02}$). We also note that the measured scaling exponent of the wall dissipation is smaller than the value of 1.5 predicted by the Prandtl–Blasius laminar BL theory. We attribute this deviation to the strong fluctuations in the instantaneous wall-shear stress (see figure 2). The fitted lines for wall dissipation and the bulk dissipation intersect at $Ra = 2.2 \times 10^{14}$, which may be used to infer the transition to the ‘ultimate regime’ at which turbulent flow in the bulk starts to take over the entire convection cell and a new regime of heat transfer may emerge (Kraichnan Reference Kraichnan1962).

Figure 7.

(a) The mean wall energy dissipation rate $\langle \varepsilon _w \rangle _t$ as a function of $Ra$. The fitted power-law of the wall dissipation rate (magenta dashed line) is $\langle \varepsilon _{w} \rangle _t / (\nu ^3 H^{-4}) = 9.65 \times 10^{-2} Ra^{1.25 \pm 0.02}$. The mean dissipation rate by Ni et al. (Reference Ni, Huang and Xia2011) at the cell centre $\langle \varepsilon _c \rangle _t$ is included and corrected following $\langle \varepsilon _c \rangle _t \sim Pr^{1.15}$. (b) The shear Reynolds number $Re_s = \delta _u U_m / \nu$ and $Re_s^d = \delta _u^d U_m^d / \nu$ as a function of $Ra$. The fitted power-laws of the respective data are $Re_s = 0.229 Ra^{0.23 \pm 0.01}$ (blue dashed line) and $Re_s^d = 0.174 Ra^{0.24 \pm 0.01}$ (red solid line).

Another way to estimate the onset of ‘ultimate regime’ is based on the instability of the viscous BL. Figure 7(b) plots the shear Reynolds number $Re_s = \delta _u U_m / \nu$ ($Re_s^d = \delta _u^d U_m^d / \nu$) as a function of $Ra$. The scaling exponents in two frames are similar, while the value of $Re_s^d$ in the dynamic frame is slightly larger than $Re_s$ in the laboratory frame. We tentatively estimate the onset $Ra_c$ at which the BL becomes unstable (i.e. the turbulent BL is expected to occur) following the critical Reynolds number $Re_s = 420$ (Landau & Lifschitz Reference Landau and Lifschitz1987; Ahlers et al. Reference Ahlers, Grossmann and Lohse2009). The resulting onset Rayleigh number is $Ra_c = 1.9 \times 10^{14}$ and $Ra_c = 6.1 \times 10^{13}$ in the static frame and the dynamic frame, respectively. We remark that these values are close to the preceding estimation of $Ra_c = 2.2 \times 10^{14}$ based on the bulk dissipation becoming dominant using the wall dissipation. He et al. (Reference He, Funfschilling, Nobach, Bodenschatz and Ahlers2012) reported a transition region covering $10^{13} \lesssim Ra_c \lesssim 5 \times 10^{14}$ and our estimates using both methods fall within this range. It is also worth mentioning that $Ra_c$ has been reported to be strongly dependent on the aspect ratio $\varGamma$ of the convection cell (Roche et al. Reference Roche, Gauthier, Kaiser and Salort2010; He, Bodenschatz & Ahlers Reference He, Bodenschatz and Ahlers2020; Ahlers et al. Reference Ahlers2022).