Boundary layer analysis in turbulent Rayleigh-Bénard convection in air: Experiment versus simulation

We report measurements and numerical simulations of the three-dimensional velocity and temperature fields in turbulent Rayleigh-Bénard convection in air. Highly resolved velocity and temperature measurements inside and outside the boundary layers have been directly compared with equivalent data obtained in direct numerical simulations (DNSs). This comparison comprises a set of two Rayleigh numbers at $\mathrm{Ra}=3\times {10}^9$ and $3\times {10}^{10}$ and a fixed aspect ratio; this is the ratio between the diameter and the height of the Rayleigh-Bénard cell of $\Gamma =1$. We find that the measured velocity data are in excellent agreement with the DNS results while the temperature data slightly differ. In particular, the measured mean temperature profile does not show the linear trend as seen in the DNS data, and the measured gradients at the wall are significantly higher than those obtained from the DNS. Both viscous and thermal boundary layer thickness scale with respect to the Rayleigh number as ${\delta }_v\sim {\mathrm{Ra}}^{-0.24}$ and ${\delta }_{\theta }\sim {\mathrm{Ra}}^{-0.24}$, respectively.

Figure 1

Description of the experiment. (a) Sketch of “Barrel of Ilmenau” with the new inset cell of $2.5$ m height and $2.5$ m diameter. In this paper we present the results at center and side window 1, 2, and 3. (b) Setup of the 3D-laser Doppler anemometry measurement, which is mounted above the cooling plate. $u$, $v$, and $w$ are the desired velocity components in Cartesian coordinates.

Figure 2

Orientation of the horizontal velocity vector. (a) Experiment and (b) DNS data are taken at the center line. Times from the DNS have been recalculated to the experimental ones.

Figure 3

Sketch of the arrays with the measurement locations. Probe array entitled center is located at the center line ($r$, $\varphi$) $=$ (0, 0), array 1 at ($0.88$ R, 0), array 2 at ($0.88$ R, $\pi$), and array 3 at ($0.88$ R, $3\pi /2$).

Figure 4

Profiles of the mean horizontal velocity (a,b) and the standard deviation (c,d) measured in the experiment (closed circles) and obtained from the DNS (open circles) at $\mathrm{Ra}=3\times {10}^9$ (a,c) and $\mathrm{Ra}=3\times {10}^{10}$ (b,d). The dashed lines in (a) and (b) represent the velocity field of a laminar flat plate BL according to Blasius [9]. The insets of (a) and (b) show the entire mean velocity profile in logarithm scale, and the insets of (c) and (d) show the near-wall region of the BL fluctuations.

Figure 5

Profiles of the wall-normal velocity (a,b) and the standard deviation (c,d) measured in the experiment (closed circles) and obtained from the DNS (open circles) at $\mathrm{Ra}=3\times {10}^9$ (a,c) and $\mathrm{Ra}=3\times {10}^{10}$ (b,d). The insets show the near-wall region of the BL.

Figure 6

Profiles of the mean temperature (a,b) and the standard deviation (c,d) measured in the experiment (closed circles) and obtained from the DNS (open circles) at $\mathrm{Ra}=3\times {10}^9$ (a,c) and $\mathrm{Ra}=3\times {10}^{10}$ (b,d). The insets show the entire mean temperature profile in logarithm scale. Here ${\vartheta }_b$ and ${\vartheta }_{cp}$ denote the mean bulk temperature and the surface temperature of the cooling plate.

Figure 7

Displacement thickness of the viscous (a) and thermal (b) boundary layers versus $\mathrm{Ra}$. Experimental results are displayed as closed symbols, DNS data points are open symbols. The solid lines in each of the graphs correspond to power laws ${\delta }_{v,d}/H=0.66{\mathrm{Ra}}^{-0.24}$ and ${\delta }_{\theta ,d}/H=0.76{\mathrm{Ra}}^{-0.24}$, respectively.

Figure 8

Thickness of the viscous (a) and thermal (b) boundary layers versus $\mathrm{Ra}$ according to the slope method. Experimental results are displayed as closed symbols, DNS data points are open symbols. The solid lines in each of the graphs correspond to power laws ${\delta }_{v,s}/H=0.90{\mathrm{Ra}}^{-0.24}$ and ${\delta }_{\theta ,s}/H=0.42{\mathrm{Ra}}^{-0.24}$, respectively.

Figure 9

Shear Reynolds number ${\mathrm{Re}}_{\mathrm{s}}$ versus $\mathrm{Ra}$ from experiment (closed circles) and DNS (open circles). The solid line is the fit to all data.

Figure 10

Mean horizontal velocity profiles at side window 1, 2, and 3, which are located at $r=0.88$ R and $\varphi =0$, $\pi$, and $3\pi /2$; see Fig. 3. (a) Profiles of the measured data at $\mathrm{Ra}=2.88\times {10}^{10}$, at window 1 (circle), window 2 (triangle), and window 3 (star). (b) Profiles of the DNS data at $\mathrm{Ra}=3\times {10}^{10}$ at array 1 (circle), array 2 (triangle), and array 3 (star).

Figure 11

Mean wall-normal velocity profiles at side window 1, 2, and 3. (a) Profiles of the measured data at $\mathrm{Ra}=2.88\times {10}^{10}$, at window 1 (circle), window 2 (triangle), and window 3 (star). There is a clear pair of upwelling and downwelling mean velocities. (b) Profiles of the numerical data at $\mathrm{Ra}=3\times {10}^{10}$ at array 1 (circle), array 2 (triangle), and array 3 (star).