# The minimal-span channel for rough-wall turbulent flows

## TL;DR

This paper investigates the minimal-span channel approach for simulating rough-wall turbulent flows, optimizing domain size, boundary conditions, and computational cost to accurately capture near-wall dynamics with reduced resources.

## Contribution

It introduces a refined minimal-span channel framework, including domain length criteria, slip wall boundary conditions, and a cost estimation method for efficient simulations of rough-wall turbulence.

## Key findings

- Optimal streamwise domain length is three times the spanwise width or 1000 viscous units.
- Half-height channels with slip walls replicate near-wall behavior with fewer grid points.
- A cost model relates simulation duration to statistical uncertainty in roughness function.

## Abstract

Roughness predominantly alters the near-wall region of turbulent flow while the outer layer remains similar. This makes it a prime candidate for the minimal-span channel, which only captures the near-wall flow by restricting the spanwise channel width to be of the order of a few hundred viscous units. Recently, Chung et al. (J. Fluid Mech., vol. 773, 2015, pp. 418-431) showed that a minimal-span channel can accurately characterise the hydraulic behaviour of roughness. Following this, we aim to investigate the fundamental dynamics of the minimal-span channel framework with an eye towards further improving performance. The streamwise domain length of the channel is investigated with the minimum length found to be three times the spanwise width or 1000 viscous units, whichever is longer. A half-height (open) channel with slip wall is shown to reproduce the near-wall behaviour seen in a standard channel, but with half the number of grid points. Next, a forcing model is introduced into the outer layer of a half-height channel to reduce the high streamwise velocity. Finally, an investigation is conducted to see if varying the roughness Reynolds number with time is a feasible method for obtaining the full hydraulic behaviour of a rough surface. An empirical costing argument is developed to determine the cost in terms of CPU hours of minimal-span channel simulations a priori. This argument involves counting the number of eddy lifespans in the channel, which is then related to the statistical uncertainty of the streamwise velocity. For a given statistical uncertainty in the roughness function, this can then be used to determine the simulation run time. Following this, a finite-volume code with a body-fitted grid is used to determine the roughness function for square-based pyramids using the above insights. Good agreement with the literature for the same roughness geometry is observed.