Attached and separated rotating flow over a finite height ridge

TL;DR

This work addresses how rotation influences boundary-layer flow over topography, specifically a finite-height ridge, and the transition from attached to separated to unsteady states. It advances a stabilised equal-order finite-element method with Continuous Interior Penalty to solve the full rotating Navier–Stokes equations across shallow and deep flow regimes, complemented by boundary-layer theory that couples an inviscid outer solution, a viscous inner layer, and a shallow-flow limit. Key findings show that with sufficiently small $Ro$, the boundary layer remains attached up to $Re\sim10^6$, while increasing $Re$ at fixed $Ro$ induces a steady separation bubble and, at higher $Re$, loss of steady solutions; deep flow introduces inertial waves and lowers the separation threshold, and simulations over a cylinder reproduce experimental trends including the division between steady and unsteady regimes around $Ro\cdot Re_{\text{crit}} \approx 275$. Overall, the paper provides a robust numerical framework that links theory, high-$Re$ simulations, and experiments to elucidate geophysical rotating flow over topography, showing rotation can stabilize attached flow even at very large Reynolds numbers.

Abstract

This paper discusses the effect of rotation on the boundary layer in high Reynolds number flow over a ridge using a numerical method based on stabilised finite elements that captures steady solutions up to Reynolds number of order $10^6$. The results are validated against boundary layer computations in shallow flows and for deep flows against experimental observations reported in Machicoane et al. (Phys. Rev. Fluids, 2018). In all cases considered the boundary layer remains attached, even at large Reynolds numbers, provided the Rossby number of the flow is sufficiently small. At any fixed Rossby number the flow detaches at sufficiently high Reynolds number to form a steady recirculating region in the lee of the ridge. At even higher Reynolds numbers no steady flow is found. This disappearance of steady solutions closely reproduces the transition to unsteadiness seen in the laboratory.

Figures (14)

• Figure 1: Flow over a ridge. We assume a constant inflow $U$ in horizontal direction, while the geometry rotates around the vertical $z$-axis.
• Figure 2: The shear stress (red) in the $x$-direction at the wall and the boundary-layer displacement thickness (blue) as functions of position $x$ for a Gaussian ridge of maximum fractional depth $\Delta=0.5$. (a) $\text{Ro}=0.5.$ (b) $\text{Ro}=1.1.$
• Figure 3: Profiles of $U$, the $x$-component of velocity, as a function of the scaled normal coordinate, $\zeta$ for a Gaussian ridge of maximum fractional depth $\Delta=0.5$ and $\text{Ro}=1.1$, as in figure \ref{['f:wsd']}(b). The red curve gives the profile at $x=0.63$ corresponding to the minimum in the wall shear and the blue curve gives the profile at $x=4.23$ corresponding to the minimum in the displacement thickness.
• Figure 4: The wall shear stress for $\delta=0.1$ and $Ro=0.5$ (left) and $Ro=1.1$ (right) for different $Re$, compared to the boundary layer calculations for $\delta\to 0, Re \to \infty$.
• Figure 5: The wall shear stress for $\delta=0.1$ and $Ro=1.2$ (left) and $Ro=1.5$ (right) for various $Re$.