Thermal Convection over Fractal Surfaces

TL;DR

This study uses numerical simulations to analyze how fractal boundary roughness influences heat transfer and flow dynamics in Rayleigh-Bénard convection, revealing increased heat transport with rougher surfaces.

Contribution

It introduces a detailed numerical investigation of fractal boundary effects on convection, quantifying how roughness degree impacts heat transfer scaling laws.

Findings

Heat transport scaling exponent increases with roughness.

Reynolds number scales as Ra^{0.57} regardless of roughness.

Heat transfer and flow are insensitive to roughness realization.

Abstract

We use well resolved numerical simulations with the Lattice Boltzmann Method to study Rayleigh-B\'enard convection in cells with a fractal boundary in two dimensions for $Pr = 1$ and $Ra \in \left[10^7, 10^{10}\right]$. The fractal boundaries are functions characterized by power spectral densities $S(k)$ that decay with wavenumber, $k$, as $S(k) \sim k^{p}$ ($p < 0$). The degree of roughness is quantified by the exponent $p$ with $p < -3$ for smooth (differentiable) surfaces and $-3 \le p < -1$ for rough surfaces with Hausdorff dimension $D_f=\frac{1}{2}(p+5)$. By computing the exponent $\beta$ in power law fits $Nu \sim Ra^{\beta}$, where $Nu$ and $Ra$ are the Nusselt and the Rayleigh numbers for $Ra \in \left[10^8, 10^{10}\right]$, we observe that heat transport scaling increases with roughness over the top two decades of $Ra \in \left[10^8, 10^{10}\right]$. For $p$ $= -3.0$, $-2.0$ and $-1.5$ we find $\beta = 0.288 \pm 0.005, 0.329 \pm 0.006$ and $0.352 \pm 0.011$, respectively. We also observe that the Reynolds number, $Re$, scales as $Re \sim Ra^{\xi}$, where $\xi \approx 0.57$ over $Ra \in \left[10^7, 10^{10}\right]$, for all $p$ used in the study. For a given value of $p$, the averaged $Nu$ and $Re$ are insensitive to the specific realization of the roughness.