Rotation of an ellipse in a shear flow

Comparison with the reference data by Aidun et al., J. Fluid Mech. (373), 1998 is described, who simulated an rotating ellipse in a shear flow.

Configuration

We consider a rectangular domain (\(l_x = 1, l_y = 2\)), in which an elliptic object (whose major and minor axes are \(0.10\) and \(0.05\), respectively) is positioned at \(x = 0.5, y = 1.0\). An analytical Couette flow profile is given initially, while the particle is at rest with its major axis being parallel to the wall-normal direction (\(\theta_z \left( t = 0 \right) = 0\)). The liquid viscosity is adjusted so that the bulk Reynolds number \(Re = l_x \Delta U_{walls} / \nu\) leads \(25\), which corresponds to \(Re_p = 1\) (see Aidun et al., J. Fluid Mech. (373), 1998 for the definition of the particle-based Reynolds number). Three spatial resolutions, from \(48\) to \(96\) grids int the wall-normal direction, are considered.

In practice, the configuration is specified as follows (\(48\) grids per domain):

#!/bin/bash

## directory name containing flow fields to restart
# export dirname_restart="output/save/stepxxxxxxxxxx"

## durations
# maximum duration (in free-fall time)
export timemax=1.0e+1
# maximum duration (in wall time [s])
export wtimemax=6.0e+2
# logging rate (in free-fall time)
export log_rate=1.0e-2
# logging after (in free-fall time)
export log_after=0.0e+0
# save rate (in free-fall time)
export save_rate=5.0e+1
# save after (in free-fall time)
export save_after=0.0e+0
# statistics collection rate (in free-fall time)
export stat_rate=1.0e-1
# statistics collection after (in free-fall time)
export stat_after=2.0e+3

## domain
# domain lengths
export ly=2.0e+0
# number of grids
export itot=48
export jtot=96

## dt safe factors, adv and dif
export safefactor_adv=0.9
export safefactor_dif=0.9

## physical parameters
export Re=2.5e+1
# export Fr=1.e+0

## external forcing in y direction
export extfrcy=0.

mpirun -n 2 --oversubscribe ./a.out

Results

Note

Conducted by GitHub Actions.

OS   : "Ubuntu 22.04.3 LTS"
Date : Sat Feb 10 13:46:26 UTC 2024
Hash : bc64913d53d88051808a651acdffa80b0fbaada0

OS   : "Ubuntu 22.04.3 LTS"
Date : Sat Feb 10 13:46:39 UTC 2024
Hash : bc64913d53d88051808a651acdffa80b0fbaada0

OS   : "Ubuntu 22.04.3 LTS"
Date : Sat Feb 10 13:48:17 UTC 2024
Hash : bc64913d53d88051808a651acdffa80b0fbaada0

The final flow field is shown below. Note that the picture is transposed, i.e., in the picture, \(x\) and \(y\) directions in the simulation are shown as the vertical and horizontal directions, respectively.

../../_images/snapshot2.png

The colour denotes the size of the stream-wise velocity, while the center gray object is the immersed particle.

Angular velocity of the particle \(\omega_z\) as a function of time is shown below. Note that, although the reference data is obtained with \(160\) grid points (\(320\) lattice nodes) in the wall-normal direction, we limit the resolutions to lower values to reduce the computational cost.

../../_images/result2.png