Observation of a link between energy dissipation rate and oscillation frequency of the large-scale circulation in dry and moist Rayleigh-Bénard turbulence

In this study both the small- and large-scale flow properties of turbulent Rayleigh-Bénard convection are investigated. Experiments are carried out using the $\mathrm{\Pi }$ chamber (aspect ratio $\mathrm{\Gamma }=2$) for Rayleigh number range $\mathrm{Ra}\sim {10}^8–{10}^9$ and Prandtl number $\mathrm{Pr}\approx 0.7$. Furthermore, experiments are run for dry and wet conditions, i.e., top and bottom surfaces of the chamber are dry and wet, respectively. For wet conditions we further distinguish between conditions with and without the presence of sodium chloride aerosol particles which, if supersaturated conditions are achieved, lead to cloud droplet formation. We therefore refer to these conditions as moist and cloudy, respectively. We see that the addition of water vapor influences the turbulent flow. In all cases, the turbulent kinetic energy dissipation rates increase with increasing temperature difference, but the slopes are different for wet and dry convection. We do not observe a clear difference between moist and cloudy convection due to low liquid water content. A similar lack of collapse with Ra is observed for the characteristic oscillations of the large-scale circulation. We observe that the first normalized characteristic oscillation frequency increased with increasing temperature difference, i.e., increasing Ra, for all conditions considered, but the slopes are different for wet and dry convection with again no clear difference between moist and cloudy convection. It turns out that the sloshing or torsional mode of the large-scale circulation and the turbulent flow or energy dissipation rate seem to be influenced by the same mechanism additional to the effect of buoyancy alone. These observational results provide supporting evidence that the large-scale circulation is insensitive to phase composition or interfacial physics and rather depends only on the strength of the turbulence.

Figure 1

(a) Sketch of the cloud chamber with one door being open and the cylindrical thermal panel being in place [12]. (b) Picture of the instrumentation inside the cylinder. The IR absorption hygrometer, the sonic anemometer, and the thermistor array were placed such that the distance between the instruments is as small as possible in order to avoid cross influence between the instruments but to ensure possible cross correlations.

Figure 2

(a) Vertical velocity plus running mean, (b) kinetic energy dissipation rate (4 min average), (c) temperature measured in the center of the chamber, (d) water vapor mass density, (e) supersaturation based on the water vapor partial pressure (${\mathrm{H}}_2\mathrm{O}$ analyzer measurements) and the saturation vapor pressure (based on the RTD closest to the ${\mathrm{H}}_2\mathrm{O}$ analyzer), and (f) LWC based on PDI data (black squares) and based on the water vapor concentration difference before and after cloud formation (dashed green line).

Figure 3

(a) Buoyancy as a function of temperature difference for all conditions investigated. (b) Rayleigh number as a function of temperature difference for all conditions investigated.

Figure 4

Peak position ${\tau }_p$ as a function of $r$. The dotted line shows the fitted function ${\tau }_p=\alpha r$, with $\alpha =7.9\times {10}^{-3}$ $\mathrm{s}{\mathrm{mm}}^{-1}$.

Figure 5

Second-order velocity structure function (solid line) including the $r_E^{2/3}$ fit (dashed line) for the $\mathrm{\Delta }T=8$ K moist convection case.

Figure 6

Thermal and kinetic energy dissipation rate as a function of (a) temperature difference and (b) Ra. Dissipation rates are based on second-order structure function calculated from the vertical velocity component as measured by the sonic anemometer and calculated from the temperature data obtained from the thermistor array. The elliptic model has been applied since the mean velocity is on the same order as the rms value.

Figure 7

Dimensionless kinetic energy dissipation rate ${\tilde{\varepsilon }}_{\mathrm{bulk}}$ as a function of Ra. For comparison, dissipation rates determined by Scheel and Schumacher [23] obtained for their full cylinder volume $V$ (red closed triangles) and a subvolume $V_b$ (red closed circles) are presented too. Scaling laws were fitted to the data; determined exponents are given in the text.

Figure 8

Average kinetic energy dissipation rate of the bulk as a function of bulk scaling according to, e.g., Chillà and Schumacher [2]. For the blue symbols, $H$ is the height of the $\mathrm{\Pi }$ chamber, which is 1 m. The dashed line represents the 1:1 line. In order to obtain the 1:1 fit, $H$ has to be decreased to 0.08 m, which corresponds to the integral length $l$ (red symbols). A scaling with ${\mathrm{Re}}^3$ was also obtained in [23] for Pr = 0.7 and Ra between ${10}^6$ and ${10}^{10}$, which also showed that the kinetic energy dissipation rate in the bulk dominates.

Figure 9

(a) Frequency power spectra of the $w$ component, (b) $T$ measured close to the ${\mathrm{H}}_2\mathrm{O}$ analyzer, and (c) raw water vapor absorptance for the wet RB convection experiment ($T_0=10{}^{\circ }\mathrm{C}$ and $\mathrm{\Delta }T=8$ K). The prominent peak near $f_0=0.013$ Hz corresponds to similar oscillation frequencies (for the same Ra) measured in previous studies (see [7]).

Figure 10

(a) Oscillation frequency $f_0$ and (b) normalized oscillation frequency $f_0{H}^2/\kappa$ for different temperature gradients (i.e., different Rayleigh numbers). Here $H$ is the chamber height and $\kappa$ is the thermal diffusivity of air or the diffusivity of water vapor in air. The solid lines in (b) represent power-law fits (dry convection, $f_0{H}^2/\kappa =0.017{\mathrm{Ra}}^{0.50}$; wet convection, $f_0{H}^2/\kappa =0.636{\mathrm{Ra}}^{0.33}$). The red line represents the power-law fit determined by Qiu et al. [8] for RB convection experiments in water ($f_0{H}^2/\kappa =0.167{\mathrm{Ra}}^{0.47}$).

Figure 11

Normalized oscillation frequency $f_0{H}^2/\kappa$ for different temperature gradients as a function of the average turbulent kinetic energy dissipation rate in the bulk ${\varepsilon }_{\mathrm{bulk}}$. A power-law fit has been applied: $f_0{H}^2/\kappa \sim {({\varepsilon }_{\mathrm{bulk}}{l}^4/{\nu }^3)}^{0.28}$.