Hyperviscosity stabilisation of the RBF-FD solution to natural convection

TL;DR

This work investigates hyperviscosity as a stabilisation mechanism for a meshless RBF-FD solution to natural convection in the turbulent regime. The authors implement a higher-order dissipation term $(-1)^{1-\alpha}\gamma \Delta^\alpha u$ with $\alpha=3$ and a scaling $\gamma=c h^{2\alpha}$, applying it to the momentum and/or heat equations in a coupled buoyancy-driven Navier–Stokes and heat transfer system solved on scattered nodes. They show that hyperviscosity extends stable Rayleigh numbers up to $\text{Ra} \approx 10^8$ (with convergence close to reference solutions at $\text{Ra}=10^6$) and that spectral analysis reveals a shift of the eigenspectrum into the stable region when applied to the momentum (or both momentum and heat) equations; however, no universal $\gamma$ works across all $\text{Ra}$ ranges, underscoring the need for adaptive parameter tuning. The results indicate hyperviscosity is a viable stabilization approach for meshless RBF-FD natural-convection simulations, matching reference solutions on scattered nodes while maintaining reasonable computational cost at $\alpha=3$.

Abstract

The numerical stability of fluid flow is an important topic in computational fluid dynamics as fluid flow simulations usually become numerically unstable in the turbulent regime. Many mesh-based methods have already established numerical dissipation procedures that dampen the effects of the unstable advection term. When it comes to meshless methods, the prominent stabilisation scheme is hyperviscosity. It introduces numerical dissipation in the form of a higher-order Laplacian operator. Many papers have already discussed the general effects of hyperviscosity and its parameters. However, hyperviscosity in flow problems has not yet been analyzed in depth. In this paper, we discuss the effects of hyperviscosity on natural convection flow problems as we approach the turbulent regime.

Figures (4)

• Figure 1: Convergence of the average Nusselt number on the left wall with an increasing number of computational nodes for $\text{Ra}=10^6$. The line labelled as *-Hyperviscosity represents the results with additional hyperviscosity stabilisation for the specific equation.
• Figure 2: Eigenvalue spectra of the inverse of implicit differentiation matrix $\mathbf{A}^{-1}$ at $t=3 \cdot 10^{-4}$ for $\text{Ra}=10^8$ with spatial discretisation density of $\Delta x = 0.009$ ($N \approx 1.2 \cdot 10^4$). The black line shows a unit circle.
• Figure 3: The average left wall Nusselt number in the steady state for a range of Rayleigh numbers. The line labelled as *-Hyperviscosity represents the results with additional hyperviscosity stabilisation for the specific equation.
• Figure 4: Velocity and temperature field of the stationary solution (left) at $Ra=10^7$ and $t=0.1$. Magnitudes of the hyperviscosity operator applied to the velocity field(center) and the temperature field(right).