Flux scaling in Rayleigh Bénard convection: a local boundary layer analysis
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arXiv
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| Natura: | Preprint |
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2024
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| author | Shevkar, Prafulla P. Puthenveettil, Baburaj A. |
| author_facet | Shevkar, Prafulla P. Puthenveettil, Baburaj A. |
| contents | We study the effect of shear due to the large scale flow (LSF) on the heat flux in Rayleigh B'enard convection for a range of near-plate Rayleigh numbers $8\times 10^7 \leq Ra_w \leq 5\times 10^{14}$, by studying its effect on the local boundary layers (BLs) on either sides of the plumes, which are much thinner than the global shear BL created by the LSF velocity $V_F$. Considering these local BLs forced externally by the LSF, we obtain a fifth order algebraic equation for the local boundary layer thicknesses. Solving these equations numerically using $Re$ relations for aspect ratios $Γ=1$ and 0.5, we obtain the variation of the local BL thicknesses with the longitudinal distance for various $Ra_w$. We find that the average shear acting on the edges of these local BLs ($\overline{u|}_{z=δ}$) increases as $\overline{u|}_{z=δ} \sim Ra_w^{1/3}$ for $8\times 10^7\leq Ra_w \leq 10^{12}$ at $Γ=1$, and as $\overline{u|}_{z=δ} \sim Ra_w^{0.38}$ for $1\times 10^{11}\leq Ra_w \leq 5\times 10^{14}$ at $Γ=0.5$. We then estimate the average local thermal BL thickness to find the global Nusselt number $Nu$.We find that $Nu\sim Ra_w^m$, where $m\approx 0.327$ for $8\times 10^7 \leq Ra_w \leq 1\times 10^{12}$ at $Γ=1$, and $m=0.33$ for $1\times10^{11}\leq Ra_w \leq 5\times10^{14}$ at $Γ=0.5$. Inspite of the increasing shear on these BLs with increasing $Ra_w$, we then surprisingly obtain the classical 1/3 scaling of flux since the shear forcing acting on those BLs remains sub-dominant compared to the NCBL velocities ($V_{bl}$) within these BLs, upto $Ra_w\leq 5\times10^{14}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_00930 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Flux scaling in Rayleigh Bénard convection: a local boundary layer analysis Shevkar, Prafulla P. Puthenveettil, Baburaj A. Fluid Dynamics We study the effect of shear due to the large scale flow (LSF) on the heat flux in Rayleigh B'enard convection for a range of near-plate Rayleigh numbers $8\times 10^7 \leq Ra_w \leq 5\times 10^{14}$, by studying its effect on the local boundary layers (BLs) on either sides of the plumes, which are much thinner than the global shear BL created by the LSF velocity $V_F$. Considering these local BLs forced externally by the LSF, we obtain a fifth order algebraic equation for the local boundary layer thicknesses. Solving these equations numerically using $Re$ relations for aspect ratios $Γ=1$ and 0.5, we obtain the variation of the local BL thicknesses with the longitudinal distance for various $Ra_w$. We find that the average shear acting on the edges of these local BLs ($\overline{u|}_{z=δ}$) increases as $\overline{u|}_{z=δ} \sim Ra_w^{1/3}$ for $8\times 10^7\leq Ra_w \leq 10^{12}$ at $Γ=1$, and as $\overline{u|}_{z=δ} \sim Ra_w^{0.38}$ for $1\times 10^{11}\leq Ra_w \leq 5\times 10^{14}$ at $Γ=0.5$. We then estimate the average local thermal BL thickness to find the global Nusselt number $Nu$.We find that $Nu\sim Ra_w^m$, where $m\approx 0.327$ for $8\times 10^7 \leq Ra_w \leq 1\times 10^{12}$ at $Γ=1$, and $m=0.33$ for $1\times10^{11}\leq Ra_w \leq 5\times10^{14}$ at $Γ=0.5$. Inspite of the increasing shear on these BLs with increasing $Ra_w$, we then surprisingly obtain the classical 1/3 scaling of flux since the shear forcing acting on those BLs remains sub-dominant compared to the NCBL velocities ($V_{bl}$) within these BLs, upto $Ra_w\leq 5\times10^{14}$. |
| title | Flux scaling in Rayleigh Bénard convection: a local boundary layer analysis |
| topic | Fluid Dynamics |
| url | https://arxiv.org/abs/2409.00930 |