alpha.md

Figures and Tables

Introduction

Figure: Original (left) and modified cross sections of the snake during its gliding phase. In addition to the original geometry, we also looked at geometries missing one or two lips.

Figure: Front views of the typical mesh used to compute the flow around a snake cylinder. We show the mesh around a snake cylinder with both lips at $AoA=35^o$. (a) is a view of the near-body wake region; (b) is a zoom near the surface of the cylinder.

LES of the flow over a circular cylinder at Re=3900

Figure: History of the drag and lift coefficients for a circular cylinder at $Re=3900$. We compare the curves obtained with the base grid and the coarser grid.

Figure: Mean pressure coefficient on the surface of the circular cylinder at $Re=3900$. The surface pressure is averaged along the spanwise direction and in time (between $100$ and $150$ non-dimensional time units of flow simulation).

Figure: Mean streamwise velocity along the centerline at $y/D=0$ behind a circular cylinder at $Re=3900$. The velocity profile is averaged along the spanwise direction and over time between $50$ and $150$ time units.

Figure: Mean velocity profiles along the crossflow direction at several locations along the streamwise direction behind a circular cylinder at Re = 3900. Profiles are reported at $x/D = 1.06$, $1.54$, $2.02$, $4.0$, $7.0$, $10.0$. The velocity components are averaged along the spanwise direction and over time (between $50$ and $150$ non-dimensional time units of flow simulation).

Figure: Isosurfaces of the second invariant of the velocity gradient tensor $Q=0.2$ at an instant in time in the near-body wake region. The isosurfaces are colored with the streamwise vorticity $\omega_x$ values.

Figure: Snapshot of the spanwise vorticity field $\omega_x$ in the near-body wake of a circular cylinder at $Re = 3900$. The snapshot is taken at mid-spanwise after $135$ non-dimensional time units of flow simulation

Figure: Filled contour of the streamwise velocity $u_x$ and crossflow velocity $u_y$ in the $x/z$ plane at $y/D = 0$ behind a circular cylinder at $Re = 3900$. The gray area shows a projection of the circular cylinder in the $x/z$ plane.

Results

Mean force coefficients

Figure: Mean lift (top) and drag (bottom) coefficients at Reynolds numbers $1000$, $2000$, and $3000$ versus the angle of attack of a snake cross-section with both lips, only the front lip or the back lip, and no lips. All values are averaged along the spanwise direction and in time (between $100$ and $200$ non-dimensional time units of flow simulation).

Figure: Mean lift-to-drag ratios at Reynolds numbers $1000$, $2000$, and $3000$ versus the angle of attack of a snake cross-section with both lips, only the front lip or the back lip, and no lips. All values are averaged along the spanwise direction and in time (between $100$ and $200$ non-dimensional time units of flow simulation).

Figure: Mean Strouhal number at Reynolds numbers $1000$, $2000$, and $3000$ versus the angle of attack of a snake cross-section with both lips, only the front lip or the back lip, and no lips. The Strouhal number is computed from the lift curve and averaged in time (between $100$ and $200$ non-dimensional time units of flow simulation).

$Re=2000$ and $AoA=35^o$

Figure: Mean surface pressure coefficient on the cylinder for all four cross-sections at $Re=2000$ and $AoA=35^o$. The surface pressure is averaged along the spanwise direction and in time (between $100$ and $200$ non-dimensional time units of flow simulation).

Figure: Mean streamwise velocity along the centerline at $y=0$ for the original and modified snake cross-sections at $Re=2000$ and $AoA=35^o$. The velocity profile is averaged along the spanwise direction and in time (between $100$ and $200$ non-dimensional time units of flow simulation).

Figure: Vertical profiles of the mean streamwise velocity deficit at several locations along the streamwise direction behind a cylinder, comparing the four cross-sections at $Re=2000$ and $AoA=35^o$. The streamwise velocity is averaged along the spanwise direction and in time (between $100$ and $200$ non-dimensional time units of flow simulation).

Figure: Isosurfaces $Q=0.1$ of the second invariant of the velocity gradient tensor ($Q$-criterion) at an instant in time in the near-body wake region. The isosurfaces are colored with the streamwise vorticity $\omega_x$ values.

Figure: Filled contour of the streamwise velocity $u_x$ in the $x/z$ plane at $y/c = 0$ behind snake cylinders at $Re=2000$ with $AoA=35^o$. The four lips configurations are represented: (a) both lips, (b) front lip, (c) back lip, and (d) no lips. The gray area shows a projection of the snake cylinder in the $x/z$ plane.

Figure: Filled contour of the crossflow velocity $u_y$ in the $x/z$ plane at $y/c = 0$ behind snake cylinders at $Re=2000$ with $AoA=35^o$. The four lips configurations are represented: (a) both lips, (b) front lip, (c) back lip, and (d) no lips. The gray area shows a projection of the snake cylinder in the $x/z$ plane.

Figure: Filled contour of the spanwise vorticity field $\omega_z$ in the $x/y$ plane at $z/c = 0$ behind snake cylinders at $Re=2000$ with $AoA=35^o$. The four lips configurations are represented: (a) both lips, (b) front lip, (c) back lip, and (d) no lips.

$Re=2000$ and $AoA=25^o$

Figure: Mean surface pressure coefficient on the cylinder for all four cross-sections at $Re=2000$ and $AoA=25^o$. The surface pressure is averaged along the spanwise direction and in time (between $100$ and $200$ non-dimensional time units of flow simulation).

Figure: Mean streamwise velocity along the centerline at $y=0$ for the original and modified snake cross-sections at $Re=2000$ and $AoA=25^o$. The velocity profile is averaged along the spanwise direction and in time (between $100$ and $200$ non-dimensional time units of flow simulation).

Figure: Vertical profiles of the mean streamwise velocity deficit at several locations along the streamwise direction behind a cylinder, comparing the four cross-sections at $Re=2000$ and $AoA=25^o$. The streamwise velocity is averaged along the spanwise direction and in time (between $100$ and $200$ non-dimensional time units of flow simulation).

Figure: Filled contour of the spanwise vorticity field $\omega_z$ in the $x/y$ plane at $z/c = 0$ behind snake cylinders at $Re=2000$ with $AoA=35^o$. The four lips configurations are represented: (a) both lips, (b) back lip, and (c) no lips.