2.1. Equations and parameters

We solve the three-dimensional equations of motion of thermal convection with the Boussinesq approximation. We consider a fluid of depth $H$, with a kinematic viscosity $\nu _f$ and a thermal diffusivity $\kappa _f$ at constant mass density $\rho _f$. The specific heat at constant pressure $c_{p,f}$ and the thermal conductivity $\lambda _{f}=\rho _f c_{p,f}\kappa _f$ are given. The equations for non-dimensional fluid velocity ${\boldsymbol {u}}({\boldsymbol {x}},t)$ and fluid temperature $T({\boldsymbol {x}},t)$ are given by

(2.1)\begin{gather} \boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{u} = 0, \end{gather}

(2.2)\begin{gather}\frac{\partial \boldsymbol{u}}{\partial t}+(\boldsymbol{u}\boldsymbol{\cdot}\boldsymbol{\nabla})\boldsymbol{u}=-\boldsymbol{\nabla} p+\sqrt{\frac{Pr}{Ra}}\nabla^{2} \boldsymbol{u}+T \boldsymbol{e}_z, \end{gather}

(2.3)\begin{gather}\frac{\partial T}{\partial t}+(\boldsymbol{u}\boldsymbol{\cdot}\boldsymbol{\nabla}){T}=\frac{1}{\sqrt{Ra Pr}}\nabla^{2}T, \end{gather}

with the unit vector in vertical direction $\boldsymbol {e}_z$. The kinematic pressure field is denoted by $p({\boldsymbol {x}},t)$. The equations (2.1)–(2.3) are made dimensionless by the fluid depth $H$, the free-fall velocity $U_f=(g\alpha \tilde {\varTheta } H)^{1/2}$, and the characteristic temperature scale $\tilde {\varTheta }$ for the specific thermal boundary conditions (details are given below). Here, $\alpha$ is the thermal expansion coefficient and $g$ is the acceleration due to gravity. For clarity in our notation, we set a tilde for the temperature scale with a physical dimension.

For turbulent RBC there are three main input parameters which control the system; the aspect ratio $\varGamma$ which is fixed to 1/2 here, the Prandtl number $Pr$ and the Rayleigh number $Ra$. The specific definition of $Ra$, which quantifies the strength of thermal driving of the turbulence, depends on the prescribed boundary conditions, and is given by

(2.4)\begin{equation} Ra=\frac{\alpha g H^3}{\nu_f \kappa_f} \tilde{\varTheta}. \end{equation}

In all three cases studied here, we consider the equilibrium state of pure conductive heat transfer as the initial condition. It takes the form

(2.5)\begin{equation} T= 1 - z, \quad \textrm{for}\ 0 < z < 1. \end{equation}

The specification of (2.1)–(2.3) is completed once the conditions at all boundaries are specified. We either impose the thermal boundary conditions directly at the top and bottom interfaces of the fluid volume as seen in figure 1(a) or at the bottom/top of the solid plates as sketched in figure 1(b) for the conjugate heat transfer case. This is specified in the following in detail as well as the other remaining boundary conditions.

Figure 1.

A schematic showing different boundary conditions and parameters in Rayleigh–Bénard convection in a cylindrical cell of aspect ratio $\varGamma =D/H=1/2$. The working fluid has a constant kinematic viscosity ($\nu _f$), thermal conductivity ($\lambda _f$) and thermal diffusivity ($\kappa _f$). $(a)$ The constant-temperature (Dirichlet BC) or constant heat flux is imposed at the lower and upper boundaries. $(b)$ The cylinder cell is bounded by conductive plates having a thickness of $h_s$, thermal conductivity ($\lambda _s$) and thermal diffusivity ($\kappa _s$) with constant temperature at the boundaries. Values at the vertical coordinate axes and for the temperature or its derivative are non-dimensional.

2.2. Constant temperature and heat flux at bottom and top plates

The typical and most frequently studied thermal BC at the bottom and top interfaces is that of a constant temperature, which is known as the Dirichlet boundary condition. These Dirichlet BCs introduce a temperature scale $\tilde {\varTheta } = \Delta \tilde {T}$ which is the temperature difference between bottom hot and top cold plates. The non-dimensional form of these thermal BCs is

(2.6a,b)\begin{equation} T \mid_{z=0}= 1 \quad\textrm{and}\quad T \mid_{z=1}= 0. \end{equation}

Consequently, the Rayleigh number follows as $Ra=\alpha g \Delta \tilde {T} H^3/\nu _f \kappa _f$, which we will be called the RBC case in the following. A detailed overview of the simulations can be found in table 2.

Table 2.

Parameters of the different spectral element simulations for Rayleigh numbers $Ra=10^7$ and $10^8$ with the convection cell of aspect ratio $\varGamma =1/2$. We list additionally the Prandtl number $Pr$, the total number of spectral elements $N_{e}$, the number of spectral elements in the vertical direction $N_{e,z}$ and the number of grid planes inside the thermal boundary layer. Furthermore, the Nusselt number $Nu$ (see § 3 for definition), the thermal boundary layer thickness $\delta _T=1/(2 Nu)$, the root mean square velocity taken in the whole fluid volume with respect to all three velocity components $u_{{rms},V}$, the Reynolds number $Re$, and the total integration time $t_{total}$ in units of the free-fall time $t_f=H/U_f$ are listed. This is the time interval for which the flow is in a statistically stationary regime and the analysis is conducted on the basis of three-dimensional snapshots.

The second choice is to impose constant vertical heat flux (CHF) at the top and bottom boundaries, $\partial \tilde {T}/ \partial \tilde {z}=-|\beta |$, which is known as the Neumann boundary condition. In this limit, the dimensionless thermal BCs are

(2.7)\begin{equation} \frac{\partial T}{\partial z}= -1 \quad \text{on}\ z =0 \ \textrm{and}\ z =1. \end{equation}

The corresponding temperature scale is now $\tilde {\varTheta }=|\beta | H$. Therefore, the typical length, velocity and temperature scales are given by, $H, \sqrt {\alpha g |\beta | H^2}$ and $|\beta | H$, respectively. The Rayleigh number in this case ($Ra_q$) and its relation to the Rayleigh number for the RBC case are given by

(2.8a,b)\begin{equation} Ra_q = \frac{\alpha g |\beta| H^4}{\nu_f \kappa_f}, \quad Ra_q=Ra\frac{|\beta| H}{\Delta \tilde{T}}= Ra Nu , \end{equation}

as discussed by Otero et al. (Reference Otero, Wittenberg, Worthing and Doering2002), Verzicco & Sreenivasan (Reference Verzicco and Sreenivasan2008) and Johnston & Doering (Reference Johnston and Doering2009). We denote this second case by CHF (see again table 2). Consequently, one can find $\boldsymbol {u}_{CHF}= \sqrt {Nu}\,\boldsymbol {u}_{RBC}$ and $T_{CHF}=T_{RBC} Nu$. Here, $Nu$ is the Nusselt number, the dimensionless measure of turbulent heat transfer which will be defined in § 3.

2.3. Fluid bounded by finitely thick plates at bottom and top

In order to study the boundary layer structure with more realistic conditions and get as close as possible to laboratory set-up (see e.g. Verzicco & Camussi Reference Verzicco and Camussi2003; Zürner et al. Reference Zürner, Schindler, Vogt, Eckert and Schumacher2019), we also studied the fluid bounded by two finitely thick copper plates from above and below. Now Dirichlet BCs are applied at the top of the upper plate and the bottom of the lower plate, as shown in figure 1(b). The solid plates have the same thickness $h_s=0.2H$ for the present case with a thermal diffusivity $\kappa _s$ and a thermal conductivity $\lambda _s=\rho _s c_{p,s}\kappa _s$. For the following, it is more convenient to define the ratio of solid-to-fluid thickness as

(2.9)\begin{equation} h=\frac{h_s}{H}. \end{equation}

In this case, the heat conduction equations in the solid parts, $-h< z< 0$ and $1<z<1+h$, for the temperature $T_s$ have to be coupled to the advection–diffusion equation (2.3) in the fluid domain , $0 < z < 1$. This configuration is denoted as the conjugated heat transfer (CHT) case. We thus have to solve the additional heat conduction equation in the plates which is given by

(2.10)\begin{equation} \frac{\partial \tilde{T}_s}{\partial \tilde{t}}=\kappa_s \tilde{\boldsymbol{\nabla}}^2 \tilde{T}_s \quad\mathrm{for}\ -h_s <z<0 \ \mathrm{and}\ H<z<H+h_s. \end{equation}

Moreover, we require the continuity of temperature and heat flux at the solid–fluid interfaces,

(2.11)\begin{gather} \left.\tilde{T}_s\right|_{\tilde{z}=0,H}=\left.\tilde{T}_f\right|_{\tilde{z}=0,H}, \end{gather}

(2.12)\begin{gather}\left.\lambda_s\frac{\partial \tilde{T}_s}{\partial \tilde{z}}\right|_{\tilde{z}=0,H}=\lambda_f \left. \frac{\partial \tilde{T}_f}{\partial \tilde{z}}\right|_{\tilde{z}=0,H}. \end{gather}

In the CHT case, we have consequently applied constant temperatures

(2.13a,b)\begin{equation} \tilde{T}(\tilde{z}=-h_s) =\tilde{T}(\tilde{z}=0)+\delta \tilde{T} \quad\mathrm{and}\quad \tilde{T}(\tilde{z}=H+h_s) =\tilde{T}(\tilde{z}=H)-\delta \tilde{T}. \end{equation}

The temperature drop across both solid plates varies with $Pr$ and is adjusted at the boundaries for each case such that the (dimensionless) average temperature (but not locally in time or space) $\langle T(z=0)\rangle _{A,t}\approx 1$ and $\langle T(z=1)\rangle _{A,t}\approx 0$, respectively, where $\langle \cdot \rangle _{A,t}$ stands for an average over the circular cross-section $A$ and time $t$. When the convection cell is filled with GaInSn, the outer imposed temperature increment is $\delta T_1= 2.51$ for $Ra=10^7$ and 4.99 for $Ra= 10^8$; in case of air it is $\delta T_2 = 4.72$ at $Ra=10^7$ (see also figure 2). The actual Rayleigh numbers for the cases CHT1, CHT2 and CHT3 thus deviate from the RBC case references by 16 %, 13 % and 9 %, respectively. That means that the solid plates are sensitive to the effective thermal conductivity ($\lambda _{eff}= Nu \lambda _s$) rather than the molecular thermal conductivity. The mean temperature profiles for the CHT case in solid–fluid–solid domain are plotted in figure 2 at $Ra=10^7$ with $Pr=0.7$ and $Pr=0.033$. The solid domain is shaded with grey colour and the fluid domain with yellow.

Figure 2.

Mean dimensionless temperature profiles in the CHT setting which are compared here for $Ra=10^7$, $\varGamma =1/2$, $Pr=0.7$ (red solid line) and $Pr=0.033$ (blue solid line). Temperatures have to be adjusted at the outer solid boundaries such that the mean temperatures at the bottom and top interfaces reach $\langle T(z=0) \rangle _{A,t}\approx 1$ and $\langle T(z=1)\rangle _{A,t}\approx 0$, respectively. The solid plates are shaded in grey. The yellow box indicates the range of the mean temperature profile in the corresponding RBC cases.

In all cases studied here, the computational domain is a cylindrical container of aspect ratio $\varGamma =1/2$ with $x,y \in [-0.25,0.25]$ and $z\in [0,1]$. At all walls, no-slip boundary conditions $\boldsymbol {u}=0$ for the fluid are applied. The sidewalls are adiabatic, i.e. the normal derivative of the temperature field vanishes (${\partial T}/{\partial \boldsymbol {n}}=0$). Therefore, the RBC, CHF and CHT cases are directly comparable to each other. Similar to the RBC case the reference temperature in the CHT case $\tilde {\varTheta }=\Delta \tilde {T}$ is applied across the characteristic scale $H$ which results in the same Rayleigh number definition as in the RBC case.

2.4. Numerical method

We numerically solve (2.1)–(2.3) in the fluid domain along with the heat equation (2.10) in both solid plates using the Nek5000 spectral element code optimized and adapted to our problem (Fischer Reference Fischer1997; Deville, Fischer & Mund Reference Deville, Fischer and Mund2002; Scheel, Emran & Schumacher Reference Scheel, Emran and Schumacher2013). The code employs second-order time stepping using a backward difference formula. The entire set of equations is transformed into a weak formulation and discretized with a particular choice of spectral basis functions (Fischer Reference Fischer1997; Deville et al. Reference Deville, Fischer and Mund2002), where Lagrangian interpolation polynomials are expanded into series of Legendre polynomials. In each of the $N_e$ elements all turbulent fields are expanded by $N$th-order polynomials with respect to each spatial direction. In the present study, seventh-order polynomials are taken for all cases such that we have 8 Gauss–Lobatto–Legendre collocation grid points in each direction on each element. The spectral element mesh is non-uniform and becomes finer towards all walls, see e.g. Scheel et al. (Reference Scheel, Emran and Schumacher2013) for a horizontal element arrangement. We list the number of grid points within the thermal boundary layer $N_{bl}$ of each case in table 2. For the case of conjugate heat transfer, the number of spectral elements in each plate is $N_{e,z}^s=20$ with respect to the vertical direction. The element height is uniform there, the horizontally element shape and thus the resolution corresponds to that in the working fluid volume. We have also verified that our flows are adequately resolved according to the criteria summarized by Scheel et al. (Reference Scheel, Emran and Schumacher2013). Further details on each of the six production runs are summarized in table 2.

As an example, instantaneous temperature isosurface plots for $Ra= 10^7$ are shown in figure 3. We plot temperature iso-surfaces of RBC, CHF and CHT for $Pr=0.7$ in figure 3(a–c) and for $Pr=0.033$ in figure 3(d–f), respectively. At first glance it is clear that the mean flow is similar in all six cases, the size of the isosurfaces changes with $Pr$ however. We observe that the surfaces appear less smooth for the case at lower Prandtl number which indicates a higher magnitude of velocity fluctuations. As seen in table 2, the latter are enhanced by a factor of 2 for the liquid metal case in comparison to convection in air. Also visible are the differences of the temperature contours close to the top plate (see figure 3b,e) which suggest differences in the thermal boundary layers.

Figure 3.

Instantaneous isosurface plots of the temperature field $T$ for different boundary conditions, constant-temperature RBC, constant-heat-flux CHF and conjugate heat transfer CHT, all at $Ra=10^7$ and $\varGamma =1/2$. Panels (a–c) are for $Pr=0.7$ and (d–f) for $Pr=0.033$. $(a)$ Case RBC1. $(b)$ Case CHF1. $(c)$ Case CHT1. $(d)$ Case RBC2. $(e)$ Case CHF2. $(f)$ Case CHT2. The temperature legend holds for all six panels.

4.1. Temperature field

It is well known that the mean temperature profile in thermal convection changes basically across a thin thermal boundary layer with a non-dimensional thickness $\delta _T = 1/(2Nu)$, while it remains nearly constant in the bulk region. In figure 6, we display the mean temperature profiles $\langle T(z)\rangle _{A,t}$ for $Pr= 0.7$ (see figure 6a) and $Pr=0.033$ (see figure 6b), both at $Ra=10^7$. Here, $\langle \cdot \rangle _{A,t}$ stands for averaging in horizontal planes $A$ and time $t$. It is worth mentioning that the mean temperature profiles for the case of CHF are normalized ($T_{CHF} = T_{RBC} Nu$). The approximate thermal boundary layer thickness of the three runs is highlighted in grey in both panels and denoted by $\delta _T$ in figure 6. The plotted temperature profiles are very similar with a small deviations close to the edge of thermal boundary layers, which is somewhat better visible for $Pr=0.7$. All cases show a finite negative slope in the bulk region which indicates that the temperature field is not perfectly mixed by the turbulence in the slender cell.

Figure 6.

Averaged profiles of normalized temperature $\langle T(z)\rangle _{A,t}/\varTheta$ along the vertical axis $z$ for $Ra=10^7$ and different Prandtl numbers: $(a)$ $Pr=0.7$, $(b)$ $Pr=0.033$. Different thermal boundary conditions are indicated in the legend.

In order to compare the profiles in the vicinity of the bottom plate for different cases better, we plot rescaled mean temperature $\theta (y^*)$ profiles in figure 7 for all 9 simulation runs. Following Shishkina & Thess (Reference Shishkina and Thess2009) and Scheel & Schumacher (Reference Scheel and Schumacher2016), we apply

(4.1)\begin{equation} \theta(y^*) = 2 - 2 \langle T(y^*) \rangle _{A,t} \quad \mathrm{with}\ y^* = z/ \delta_{T}. \end{equation}

The dotted line indicates the slope of one of the profiles close to the wall. We see the profiles are $Pr$-dependent. The profiles which show a deviation further away from the wall are those for the CHT cases (in comparison to the other cases). Recall that the mean plate temperature can fluctuate. Numerical (e.g. by Verzicco & Sreenivasan Reference Verzicco and Sreenivasan2008) and experimental studies (e.g. by Huang et al. Reference Huang, Wang, Xi and Xia2015) found that the plume emissions are stronger for constant temperature than for constant-heat-flux boundary conditions.

Figure 7.

Scaled mean temperature profiles $\theta (y^*)$ versus $y^* = z/\delta _T$, see (4.1): $(a)$ for $Ra=10^7$ and $Pr = 0.7$, $(b)$ for $Ra=10^7$, $10^8$ and $Pr = 0.033$. The legend denotes the 9 different cases, see also table 2.

The statistics of the corresponding temperature fluctuations, $T^{\prime }=T-\langle T(z)\rangle _{A,t}$, are compared in figure 8 by means of the probability density functions (PDFs) taken at different subvolumes in the convection cell. We defined the subvolumes with $x,y \in [-0.25,0.25]$ at the bottom $z\in [0,0.2]$, bulk $z\in [0.4,0.6]$ and top $z\in [0.8,1]$. All PDFs of the RBC cases have the strongest positive tail at the bottom and the strongest negative tail at the top which are caused by plume detachments that seem to be more pronounced as for the other thermal boundary conditions. These differences are found in the far tails (see red curves). The distributions are always symmetric in the cell centre which is expected. The distributions remain close to the Gaussian and tend to be even sub-Gaussian. Significant differences with a variation of $Ra$ or $Pr$ cannot be identified. The temperature statistics at low Prandtl numbers is particularly determined by coarse plume structures and geometric constraints in the slender cell. It can be seen that $Ra=10^8$ (see figure 8g–i). The PDFs differ less of each other in the centre, the PDF profiles with imposed constant heat flux are closer to those of the conjugate heat transfer set-up.

Figure 8.

Probability density functions of the temperature fluctuations with different boundary conditions: (a–c) for $Ra=10^7, Pr=0.7$, (d–f) for $Ra=10^7, Pr=0.033$ and (g–i) for $Ra=10^8$ and $Pr=0.033$. Data are taken for subvolume with $x,y \in [-0.25,0.25]$ at the bottom $z\in [0,0.2]$ in panels (a,d,g), bulk $z\in [0.4,0.6]$ in panels (b,e,h) and top $z\in [0.8,1]$ in panels (c,f,i). The temperature fluctuations are normalized by the corresponding root mean square values. The legends in each row hold for all three panels. The Gaussian distribution is added to each figure as a black dashed-dotted line.

4.2. Velocity field

Next, we compare the vertical root-mean-square (r.m.s.) profiles obtained from all three components of the velocity field and their vertical velocity components for different $Ra, Pr$ and boundary conditions. Note again that $u_{rms}(CHF)=\sqrt {Nu}\, u_{rms}(RBC)$. The corresponding r.m.s. values are given by

(4.2a,b)\begin{equation} u_{rms}(z)=\sqrt{\langle u_x^2 +u_y^2 +u_z^2 \rangle_{A,t}} \quad\mathrm{and}\quad w_{rms}(z)=\sqrt{\langle u_z^2 \rangle_{A,t}} . \end{equation}

Figure 9 displays our findings. Due to the no-slip BC at the top and bottom plates the fluctuation profiles are zero at the wall. The flow in the bulk region is driven and dominated by a thermal plumes which feed the LSC flow. Thus the velocity field rises in a short distance at the plate from zero to that of the LSC amplitude across the viscous boundary layer thickness. It can be seen that the level of velocity fluctuations for both the three-dimensional $u_{rms}(z)$ and the vertical $w_{rms}(z)$ is significantly enhanced for the lower $Pr$ at the same $Ra$. The coarser thermal plumes (which are in line with a thicker thermal boundary layer) drive the fluid turbulence starting from a larger scale and extend the inertial cascade range such that the flow Reynolds number is enhanced. The magnitude of both fluctuation profiles decreases at fixed $Pr$ for the larger of both $Ra$. In figure 9(b), we also observe a local maximum in all profiles at the edge of thermal BLs, where thermal plumes detach from the plates and rise into the bulk of the cell. We found that for the RBC and CHF cases $u_{rms}(z)$ is practically similar in the boundary layers. The differences between all three cases remain small and do not exceed a few per cent in magnitude.

Figure 9.

Root-mean-square velocity profiles for all 9 simulation runs. $(a)$ Vertical velocity component root-mean-square profile $w_{rms}(z)$. $(b)$ Total root-mean-square velocity profile $u_{rms}(z)$. The legend holds for both (a,b). See table 2 for the labelling. Since the statistical fluctuations are symmetric with respect to the midplane at $z=0.5$, we plot the arithmetic average of the profiles of the lower and upper halves.

5.1. Local thermal boundary layer thickness

The strongest local temperature gradient magnitudes will generally be also confined to the boundary layers near the bottom and top plates. The mean thickness of boundary layer can be defined in several ways. The standard way with $\delta _T=1/(2 Nu)$ was discussed in § 3. The alternative way is via a refined local analysis. Such detailed numerical studies of local thermal boundary layer thicknesses in turbulent RBC in cylindrical cells with aspect ratio one were reported in Scheel & Schumacher (Reference Scheel and Schumacher2014), where the inverse of the magnitude of the vertical temperature derivative at the plate, a local slope, was associated with a local thermal boundary layer thickness, given in dimensionless form by

(5.1)\begin{equation} \lambda_T(x,y)=\left.\frac{1}{2}\right| \left.\frac{\partial T}{\partial z} \right|_{z=0,1}^{-1}. \end{equation}

Equation (5.1) gives the inverse local slope of the temperature at each point of the plate. To shed more light on the effect of thermal BCs on the local boundary layer scales, we display contours of instantaneous plots of $\lambda _T(x,y,0)$ for RBC and CHT cases in figure 10. The left column is for RBC and the right column for CHT. Figure 10(a–d) plots data for $Ra=10^7$ and figure 10(e,f) for $Ra=10^8$. The first row shows the case of $Pr=0.7$ while the second and third rows display the analysis for $Pr=0.033$. In all cases, we observe a spatial variation across the plate with largest values close to the sidewalls where the thermal plumes detach and rise preferentially with the LSC. As the Rayleigh number is increased to $10^8$, the structures become more filamented. It is interesting to see that the range of values of the local thicknesses is about the same for both RBC cases in figures 10(a) and 10(c) at first glance and that the area fraction of high amplitude events is larger for the lower Prandtl number which results in a larger mean value. Also observed is a slightly larger spatial variability in cases of the lower Prandtl number which is in line with the more complex wall stress field structure as analysed for example by Schumacher et al. (Reference Schumacher, Bandaru, Pandey and Scheel2016). For the CHT cases, the heat injection depends on the ratio of the conductivities of the fluid and the plate. Thus this case should be more prone to Prandtl-number variations as visible in figures 10(b) and 10(d) at the same $Ra$ number.

Figure 10.

Snapshots of the local thermal boundary layer scale $\lambda _T(x,y)$, as given by (5.1) at the bottom plate. Fixed-temperature boundary condition is applied in: (a,b) RBC1 and CHT1 at $Pr=0.7$; (c,d) RBC 2 and CHT2 at $Pr=0.033$; (e,f) RBC3 and CHT3 at $Pr=0.033$ and $Ra=10^8$. The colour bars indicate the magnitude of $\lambda _T$ in each case. All local boundary layer thicknesses are measured in units of $H$.

The mean value of the local thicknesses $\lambda _T(x,y)$ follows as an ensemble average and is given by

(5.2)\begin{equation} \langle \lambda_T \rangle=\int_{0}^{\infty}\lambda_T\,p(\lambda_T)\,\textrm{d}\lambda_T=\frac{1}{2}\left\langle \left| \frac{\partial T}{\partial z} \right|_{z=0,1}^{-1}\right\rangle, \end{equation}

where $p(\lambda _T)$ is the PDF of the local scales $\lambda _T(x,y)$. When comparing this mean thickness with the standard $\delta _T=1/(2 Nu)$, we obtain the following: for RBC1 $(\langle \lambda _T\rangle ,\delta _T)=(0.043,0.029)$, for case RBC2 $(0.067,0.047)$, and for RBC3 $(0.036,0.023)$. Furthermore, for case CHT1 we get $(0.027,0.023)$, for CHT2 $(0.043,0.041)$ and finally for CHT3 $(0.021,0.020)$. It is seen that the values of $\langle \lambda _T\rangle$ are somewhat overestimated in the RBC case as already found in Scheel & Schumacher (Reference Scheel and Schumacher2014). The values agree consistently for the three CHT cases as we match the fluxes at the plate–fluid interfaces.

In figure 11, we plot the results of the rescaled PDFs of the local BL scales for the RBC and CHT cases as given in (5.1) and (5.2). The distribution for the CHF case would be a delta function, $p(\lambda _T/\langle \lambda _T\rangle )=\delta (\lambda _T/\langle \lambda _T\rangle -1)$. The data are taken over the whole cross-section of the plate and for long sequences of snapshots. We see in both cases that the dependence on the Prandtl number remains small when data are compared for each set of boundary conditions. The PDFs remain practically similar even though the tails are slightly extended only for the lower $Pr$. A main difference is the shift of the maximum of the PDF from $\lambda _T/\langle \lambda _T\rangle <1$ in RBC to $\lambda _T/\langle \lambda _T\rangle \approx 1$ in CHT. This is in line with a fatter tail in the RBC case that indicates a much stronger variability of the local scales. An additional increase of the Rayleigh number from $10^7$ to $10^8$ in the RBC set-up leads to a significantly fatter tail indicating enhanced fluctuations of the local BL thickness. The shape and the trend with Rayleigh number are found to agree with Scheel & Schumacher (Reference Scheel and Schumacher2014). This is the first significant difference when finitely thick plates are applied instead of the idealized thin ones.

Figure 11.

Probability density functions of the local thermal boundary layer scale $\lambda _T(x,y)$ given in units the corresponding mean thickness $\langle \lambda _T\rangle$. Data are combined for the three RBC and CHT cases. The notation corresponds to table 2. The PDFs are obtained over a sequence of snapshots and the whole bottom and top plates. The vertical dashed line is for $\lambda _T/\langle \lambda _T\rangle =1$.

5.2. Local viscous boundary layer thickness

Similarly one can proceed for the viscous boundary layer thickness of the velocity field. The local velocity boundary scale $\lambda _v(x,y)$ and its mean value $\langle \lambda _v(x,y)\rangle$ are defined as

(5.3a,b)\begin{equation} \lambda_v(x,y)= u_{rms} \left| \frac{\partial \boldsymbol{u}^{(2)}}{\partial z} \right|_{z=0,1}^{-1} \quad\mathrm{and}\quad \langle \lambda_v \rangle = u_{rms}\left\langle \left| \frac{\partial \boldsymbol{u}^{(2)}}{\partial z} \right|_{z=0,1}^{-1}\right\rangle, \end{equation}

where $\boldsymbol {u}^{(2)}=(u_x,u_y)$ and zero derivatives are excluded. In figure 12(a,b) we plot the results of the PDFs of $\lambda _v$ and in figure 12(c,d) we replot the data rescaled with their respective means $\langle \lambda _v\rangle$ as given in (5.3a,b). Figure 12(a,c) for $Pr=0.7$, figure 12(b,d) for $Pr=0.033$. Comparing the results in figure 12(a,b), we find that the velocity boundary layer thicknesses are on average smaller for the liquid metal case than for air which is in line with the larger Reynolds number reported for these cases in table 2. Interestingly, for given $Ra$ and $Pr$ numbers, we find that the local velocity boundary layer distributions with respect to $\lambda _v$ for RBC and CHT collapse, and that they are shifted in comparison to the CHF cases. As can also be seen, all re-scaled PDFs in figure 12(c,d) collapse well and are in good agreement with the findings in Scheel & Schumacher (Reference Scheel and Schumacher2014).

Figure 12.

Probability density functions (PDFs) of the local outer velocity boundary layer thickness $\lambda _v$ of all simulation data for $Ra=10^7, \varGamma =1/2$: (a,c) for $Pr=0.7$ and (b,d) for $Pr=0.033$. In the (c,d) cases, we scaled each PDF by its mean value as given in (5.3a,b). Data are obtained over a sequence of snapshots and the whole bottom and top plates. See table 2 for the legend name. The behaviour of the data at $Ra=10^8$ is similar and thus not displayed. All distributions collapse again when rescaled by the mean thickness.