Experimental velocity study of non-Boussinesq Rayleigh-Bénard convection

The effects of strongly varying fluid properties, beyond the validity range of the so-called Boussinesq approximation, were experimentally studied in Rayleigh-Bénard (RB) convection. Two experiments were conducted in the same cubical RB convection cell at similar Rayleigh and Prandtl numbers. In one experiment water was used as working fluid and the imposed temperature difference between the top and bottom plates of the cell was such to ensure non-Boussinesq conditions. In the other experiment, taken as a reference for Boussinesq conditions, methanol was used as working fluid, allowing a smaller temperature difference between the plates. In both experiments the instantaneous and time-averaged flow fields were determined experimentally in a vertical cross section of the cell by using particle image velocimetry. Results show a non-Boussinesq effect that manifests itself as an increase of the time-averaged horizontal velocity component close to the bottom wall of the cell and as a global top-bottom asymmetry of the velocity field. This is an experimental study of the whole velocity field of RB convection at non-Boussinesq conditions.

Figure 1

Sketch of a vertical section of the Rayleigh-Bénard cell.

Figure 2

Mean velocities $\left(\right|\overline{u}\left|\right)$ contour plots in a vertical cross section at half-depth of the cell $\left(Z/H=0.5\right)$. $\left(\mathrm{a}\right)$ Non-Boussinesq: $\mathrm{Ra}=6.91\times {10}^8, \mathrm{Pr}=4.33$; $\left(\mathrm{b}\right)$ Boussinesq: $\mathrm{Ra}=6.74\times {10}^8, \mathrm{Pr}=7.24$.

Figure 3

Mean velocities $\left(\right|\overline{u}\left|\right)$ contour plots in a vertical cross section at half-depth of the cell $\left(Z/H=0.5\right)$. Velocity values normalized by $U_{\text{LSC}\left(\text{GL}\right)}$ in panels $\left(\mathrm{a}\right)$ and $\left(\mathrm{c}\right)$ and by $U_{\text{LSC}\left(P\right)}$ in panels $\left(\mathrm{b}\right)$ and $\left(\mathrm{d}\right)$. Panels $\left(\mathrm{a}\right)$ and $\left(\mathrm{b}\right)$ Non-Boussinesq: $\mathrm{Ra}=6.91\times {10}^8, \mathrm{Pr}=4.33$; panels $\left(\mathrm{c}\right)$ and $\left(\mathrm{d}\right)$ Boussinesq: $\mathrm{Ra}=6.74\times {10}^8, \mathrm{Pr}=7.24$.

Figure 4

Comparison between non-Boussinesq and Boussinesq mean velocity [panels $\left(\mathrm{a}\right)$ and $\left(\mathrm{c}\right)]$ and velocity fluctuations rms [panels $\left(\mathrm{b}\right)$ and $\left(\mathrm{d}\right)]$ at half-depth of the cell $\left(\mathrm{Z}/H=0.5\right)$. Velocity values normalized by $U_{\text{LSC}\left(\text{GL}\right)}$. Panels $\left(\mathrm{a}\right)$ and $\left(\mathrm{b}\right)$: horizontal component along the line $X/H=0.5$. Panels $\left(\mathrm{c}\right)$ and $\left(\mathrm{d}\right)$: vertical component along the line $Y/H=0.5$. Non-Boussinesq: $\mathrm{Ra}=6.91\times {10}^8, \mathrm{Pr}=4.33$; non-Boussinesq $2^{\mathrm{nd}}$: $\mathrm{Ra}=6.84\times {10}^8, \mathrm{Pr}=4.33$; Boussinesq: $\mathrm{Ra}=6.74\times {10}^8, \mathrm{Pr}=7.24$; Boussinesq $2^{\mathrm{nd}}$: $\mathrm{Ra}=6.77\times {10}^8, \mathrm{Pr}=7.24$.

Figure 5

Comparison between non-Boussinesq and Boussinesq mean velocity [panels $\left(\mathrm{a}\right)$ and $\left(\mathrm{c}\right)]$ and velocity fluctuations rms [panels $\left(\mathrm{b}\right)$ and $\left(\mathrm{d}\right)]$ at half-depth of the cell $\left(Z/H=0.5\right)$. Velocity values normalized by $U_{\text{LSC}\left(P\right)}$. Panels $\left(\mathrm{a}\right)$ and $\left(\mathrm{b}\right)$: horizontal component along the line $X/H=0.5$. Panels $\left(\mathrm{c}\right)$ and $\left(\mathrm{d}\right)$: vertical component along the line $Y/H=0.5$. Non-Boussinesq: $\mathrm{Ra}=6.91\times {10}^8, \mathrm{Pr}=4.33$; non-Boussinesq $2^{\mathrm{nd}}$: $\mathrm{Ra}=6.84\times {10}^8, \mathrm{Pr}=4.33$; Boussinesq: $\mathrm{Ra}=6.74\times {10}^8, \mathrm{Pr}=7.24$; Boussinesq $2^{\mathrm{nd}}$: $\mathrm{Ra}=6.77\times {10}^8, \mathrm{Pr}=7.24$.

Figure 6

Turbulent kinetic energy in a vertical cross section at half-depth of the cell $\left(Z/H=0.5\right)$. Values normalized by $U_{\text{LSC}\left(\text{GL}\right)}$ in panels $\left(\mathrm{a}\right)$ and $\left(\mathrm{c}\right)$ and by $U_{\text{LSC}\left(P\right)}$ in panels $\left(\mathrm{b}\right)$ and $\left(\mathrm{d}\right)$. Panels $\left(\mathrm{a}\right)$ and $\left(\mathrm{b}\right)$: Non-Boussinesq: $\mathrm{Ra}=6.91\times {10}^8, \mathrm{Pr}=4.33$. Panels $\left(\mathrm{c}\right)$ and $\left(\mathrm{d}\right)$: Boussinesq: $\mathrm{Ra}=6.74\times {10}^8, \mathrm{Pr}=7.24$.

Figure 7

Turbulent kinetic energy in a vertical cross section at half-depth of the cell $\left(Z/H=0.5\right)$ (repeated experiments). Values normalized by $U_{\text{LSC}\left(\text{GL}\right)}$ in panels $\left(\mathrm{a}\right)$ and $\left(\mathrm{c}\right)$ and by $U_{\text{LSC}\left(P\right)}$ in panels $\left(\mathrm{b}\right)$ and $\left(\mathrm{d}\right)$. Panels $\left(\mathrm{a}\right)$ and $\left(\mathrm{b}\right)$: non-Boussinesq $2^{\mathrm{nd}}$: $\mathrm{Ra}=6.84\times {10}^8, \mathrm{Pr}=4.33$. Panels $\left(\mathrm{c}\right)$ and $\left(\mathrm{d}\right)$: Boussinesq $2^{\mathrm{nd}}$: $\mathrm{Ra}=6.77\times {10}^8, \mathrm{Pr}=7.24$.

Figure 8

One-line plots of the turbulent kinetic energy values at several heights $Y/H$ ($Y/H=0.1,0.2,0.3,0,7,0.8,0.9$) along the axis $X/H$ at $Z/H=0.5$. Values normalized by $U_{\text{LSC}\left(\text{GL}\right)}.$ (a), (b) Non-Boussinesq: $\mathrm{Ra}=6.9\times {10}^8, \mathrm{Pr}=4.33$; non-Boussinesq $2^{\mathrm{nd}}$: $\mathrm{Ra}=6.84\times {10}^8, \mathrm{Pr}=4.33$. (c), (d) Boussinesq: $\mathrm{Ra}=6.7\times {10}^8, \mathrm{Pr}=7.24$; Boussinesq $2^{\mathrm{nd}}$: $\mathrm{Ra}=6.77\times {10}^8, \mathrm{Pr}=7.24$.

Figure 9

One-line plots of the turbulent kinetic energy values at several heights $Y/H$ ($Y/H=0.1,0.2,0.3,0,7,0.8,0.9$) along the axis $X/H$ at $Z/H=0.5$. Values normalized by $U_{\text{LSC}\left(P\right)}.$ (a), (b) Non-Boussinesq: $\mathrm{Ra}=6.9\times {10}^8, \mathrm{Pr}=4.33$; non-Boussinesq $2^{\mathrm{nd}}$: $\mathrm{Ra}=6.84\times {10}^8, \mathrm{Pr}=4.33$. (c), (d) Boussinesq: $\mathrm{Ra}=6.7\times {10}^8, \mathrm{Pr}=7.24$; Boussinesq $2^{\mathrm{nd}}$: $\mathrm{Ra}=6.77\times {10}^8, \mathrm{Pr}=7.24$.