Bounds on buoyancy driven flows with Navier-slip conditions on rough boundaries
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arXiv
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866917800228945920 |
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| author | Bleitner, Fabian Nobili, Camilla |
| author_facet | Bleitner, Fabian Nobili, Camilla |
| contents | We consider two-dimensional Rayleigh-Bénard convection with Navier-slip and fixed temperature boundary conditions at the two horizontal rough walls described by the height function $h$. We prove rigorous upper bounds on the Nusselt number $\text{Nu}$ which capture the dependence on the curvature of the boundary $κ$ and the (non-constant) friction coefficient $α$ explicitly. If $h\in W^{2,\infty}$ and $κ$ satisfies a smallness condition with respect to $α$, we find $$ \text{Nu}\lesssim \text{Ra}^{\frac{1}{2}}+\|κ\|_{\infty}\,,$$ where $\text{Ra}$ is the Rayleigh number, which agrees with the predicted Spiegel-Kraichnan scaling when $κ=0$. This bound is obtained via local regularity estimates in a small strip at the boundary. When $h\in W^{3,\infty}$, the functions $κ$ and $α$ are sufficiently small in $L^{\infty}$ and the Prandtl number $\Pr$ is sufficiently large, we prove upper bounds using the background field method, which interpolate between $\text{Ra}^{\frac{1}{2}}$ and $\text{Ra}^{\frac{5}{12}}$ with non-trivial dependence on $α$ and $κ$. These bounds agree with the result in Drivas et al (2022 Phil. Trans. R. Soc. A 380 20210025) for flat boundaries and constant friction coefficient. Furthermore, in the regime $\Pr\geq \text{Ra}^{\frac 57}$, we improve the $\text{Ra}^{\frac 12}$-upper bound, showing $$\text{Nu}\lesssim_{α,κ}\text{Ra}^{\frac{3}{7}}\,,$$ where $\lesssim_{α,κ}$ hides an additional dependency of the implicit constant on $α$ and $κ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_00226 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Bounds on buoyancy driven flows with Navier-slip conditions on rough boundaries Bleitner, Fabian Nobili, Camilla Analysis of PDEs Mathematical Physics Fluid Dynamics 80A19, 76R10, 35Q35 We consider two-dimensional Rayleigh-Bénard convection with Navier-slip and fixed temperature boundary conditions at the two horizontal rough walls described by the height function $h$. We prove rigorous upper bounds on the Nusselt number $\text{Nu}$ which capture the dependence on the curvature of the boundary $κ$ and the (non-constant) friction coefficient $α$ explicitly. If $h\in W^{2,\infty}$ and $κ$ satisfies a smallness condition with respect to $α$, we find $$ \text{Nu}\lesssim \text{Ra}^{\frac{1}{2}}+\|κ\|_{\infty}\,,$$ where $\text{Ra}$ is the Rayleigh number, which agrees with the predicted Spiegel-Kraichnan scaling when $κ=0$. This bound is obtained via local regularity estimates in a small strip at the boundary. When $h\in W^{3,\infty}$, the functions $κ$ and $α$ are sufficiently small in $L^{\infty}$ and the Prandtl number $\Pr$ is sufficiently large, we prove upper bounds using the background field method, which interpolate between $\text{Ra}^{\frac{1}{2}}$ and $\text{Ra}^{\frac{5}{12}}$ with non-trivial dependence on $α$ and $κ$. These bounds agree with the result in Drivas et al (2022 Phil. Trans. R. Soc. A 380 20210025) for flat boundaries and constant friction coefficient. Furthermore, in the regime $\Pr\geq \text{Ra}^{\frac 57}$, we improve the $\text{Ra}^{\frac 12}$-upper bound, showing $$\text{Nu}\lesssim_{α,κ}\text{Ra}^{\frac{3}{7}}\,,$$ where $\lesssim_{α,κ}$ hides an additional dependency of the implicit constant on $α$ and $κ$. |
| title | Bounds on buoyancy driven flows with Navier-slip conditions on rough boundaries |
| topic | Analysis of PDEs Mathematical Physics Fluid Dynamics 80A19, 76R10, 35Q35 |
| url | https://arxiv.org/abs/2301.00226 |