Inertial flow past a finite-length axisymmetric cylinder of aspect ratio 3: Effect of the yaw angle

We study the flow past a finite-length yawed three-dimensional cylinder by a finite-volume–fictitious-domain method. We validate our non-boundary-fitted method against boundary-fitted numerical results for a finite-length cylinder whose axis is parallel to the streamwise direction. Drag and lift forces exerted on the cylinder and vortex shedding onset and frequency are carefully analyzed. Satisfactory agreement with published results gives strong confidence in the numerical methodology provided the boundary layer is accurately resolved. Then we carry out a detailed study of the flow past a yawed cylinder of aspect ratio $L/D=3$ (where $L$ is the cylinder length and $D$ is the cylinder diameter) at moderate Reynolds numbers $\left(25\le \text{Re}\le 250\right)$. We show that the wake patterns and the associated transitions depend strongly on Re and the yaw angle $\theta$ with respect to the streamwise direction. Various regimes are encountered including a standing-eddy pattern, steady shedding of one or two pairs of counterrotating vortices, periodic shedding of counterrotating vortices and unsteady shedding of hairpin-shaped vortices. The steady shedding of one or two pairs of counterrotating vortices prevails in the range of parameters studied. Hydrodynamic forces exerted on the cylinder are well approximated by laws derived in the Stokes flow regime, even for moderate Reynolds numbers. For the highest Reynolds numbers $\left(\text{Re}=150,200,250\right)$ we show that the forces slowly depart from the Stokes-based laws. Simple modification of these laws is proposed, yielding a satisfactory match with the numerical results. An accurate way to compute the torque is also proposed based on the normal force to the cylinder.

Figure 1

Sketch of the computational domain.

Figure 2

Wake pattern past an $L/D=1$ cylinder aligned with the streamwise direction at $\text{Re}=360$. Ninety-six cells are distributed along the cylinder diameter. The wake is visualized using the $Q$ criterion. Isosurfaces of $Q={10}^{-3}$ are shown and are colored by the longitudinal vorticity ranging from $-0.2$ to 0.2.

Figure 3

Wake patterns past an $L/D=3$ cylinder as a function of $\theta$ and Re. Colored areas represent regimes by symmetry and unsteadiness: blue, steady axisymmetric wake; yellow, steady symmetric wake with respect to the $(x,y)$ plane; purple, steady wake with two symmetry planes $(x,y)$ and $(x,z)$; green, periodic wake with one symmetry plane $(x,y)$; red, periodic wake with one symmetry plane $(x,z)$; brown, periodic shedding of quasisymmetric double-sided hairpin vortices; and cyan, three-dimensional unsteady wakes. The symbols denote the following: $\blacksquare$, steady standing eddy or toroidal vortex; $\times$, steady shedding of one counterrotating vortex pair; +, steady shedding of two counterrotating vortex pairs; $\square$, steady shedding of two symmetric counterrotating vortex pairs; $▵$, periodic shedding of two counterrotating vortex pairs [the $(x,y)$ symmetry plane]; $▿$, periodic shedding of one counterrotating vortex pair [the $(x,y)$ symmetry plane]; $\odot$, periodic shedding of two counterrotating vortex pairs [the $(x,z)$ symmetry plane]; $▴$, periodic shedding of single-sided hairpin-shaped vortices; $•$, periodic shedding of symmetric double-sided hairpin-shaped vortices [the $(x,z)$ symmetry plane]; $*$, unsteady shedding of asymmetric double-sided hairpin-shaped vortices; and $⧫$, periodic shedding of quasisymmetric double-sided hairpin vortices.

Figure 4

Two pairs of asymmetric counterrotating vortices at $(\theta ={75}^{\circ },\text{Re}=100)$. (a) $Q=0.01$ isosurfaces are colored by the longitudinal vorticity ranging from $-0.2$ to 0.2. The dark lines starting at the center of the cylinder indicate the cylinder axis and the streamwise direction. (b) Streamlines in a $(y,z)$ plane perpendicular to the $x$ streamwise direction located at $5D$ from the $x$ coordinate of the cylinder mass center.

Figure 5

Periodic shedding of two asymmetric counterrotating vortex pairs at $(\theta ={80}^{\circ },\text{Re}=150)$. The time interval between two frames is $\mathrm{\Delta }t=10$. Here $Q=0.004$ isosurfaces are colored by the longitudinal vorticity ranging from $-0.2$ to 0.2. The dark lines starting at the center of the cylinder indicate the cylinder axis and the streamwise direction.

Figure 6

Periodic shedding of single-sided hairpin vortices at $(\theta ={75}^{\circ },\text{Re}=150)$. Here $Q=0.004$ isosurfaces are colored by the longitudinal vorticity ranging from $-0.2$ to 0.2. The dark lines starting at the center of the cylinder indicate the cylinder axis and the streamwise direction. Shown on top is the view in the $(x,y)$ plane and on the bottom is the view in the $(x,z)$ plane, from the bottom.

Figure 7

Force and torque coefficient diagrams for the periodic shedding of single-sided hairpin vortices at $(\theta ={75}^{\circ },\text{Re}=150)$: (a) $C_D-C_{Ly}$ and (b) $C_D-C_{Tz}$.

Figure 8

Unsteady shedding of asymmetric double-sided hairpin vortices at $(\theta ={85}^{\circ },\text{Re}=175)$. The time interval between two frames is $\mathrm{\Delta }t=5$. Here $Q=0.008$ isosurfaces are colored by the longitudinal vorticity ranging from $-0.2$ to 0.2. The dark lines starting at the center of the cylinder indicate the cylinder axis and the streamwise direction.

Figure 9

Force and torque coefficient diagrams for the 3D unsteady regime dominated by the shedding of asymmetric double-sided hairpin vortices at $(\theta ={85}^{\circ },\text{Re}=175)$.

Figure 10

Unsteady shedding of asymmetric double-sided hairpin vortices at $(\theta ={85}^{\circ },\text{Re}=175)$. The time interval between two frames is $\mathrm{\Delta }t=5$. Here $Q=0.004$ isosurfaces are colored by the longitudinal vorticity ranging from $-0.2$ to 0.2. The dark lines starting at the center of the cylinder indicate the cylinder axis and the streamwise direction.

Figure 11

Force and torque coefficient diagrams for the 3D unsteady regime dominated by the shedding of asymmetric double-sided hairpin vortices at $(\theta ={75}^{\circ },\text{Re}=200)$.

Figure 12

Strouhal number St as a function of the yaw angle $\theta$ at $\text{Re}=150$ (+), $\text{Re}=175$ ($\times$), $\text{Re}=200$ ($*$), and $\text{Re}=250$ ($⊡$): (a) oscillations in the $y$ vertical direction and (b) oscillations in the $z$ horizontal direction. The meaning of the error bars can be found in Sec. 2.

Figure 13

Accordion regime or periodic shedding of quasisymmetric double-sided hairpin vortices at $(\theta ={60}^{\circ },\text{Re}=200)$. Here $Q=0.008$ isosurfaces are colored by the longitudinal vorticity ranging from $-0.2$ to 0.2. The dark lines starting at the center of the cylinder indicate the cylinder axis and the streamwise direction.

Figure 14

The $C_D-C_{Lz}$ diagrams for the periodic shedding of quasisymmetric double-sided hairpin vortices at $\theta ={60}^{\circ }$: (a) $\text{Re}=200$ and (b) $\text{Re}=250$.

Figure 15

Plot of ${\text{St}}_z$ as a function of Re for $\theta ={60}^{\circ }$ (+) and $\theta ={65}^{\circ }$ ($\blacksquare$). The meaning of the error bars can be found in Sec. 2.

Figure 16

Original periodic shedding of single-sided hairpin vortices at $(\theta ={55}^{\circ },\text{Re}=250)$. Here $Q=0.008$ isosurfaces are colored by the longitudinal vorticity ranging from $-0.2$ to 0.2. The dark lines starting at the center of the cylinder indicate the cylinder axis and the streamwise direction.

Figure 17

Sketch of the force and torque exerted on a yawed cylinder.

Figure 18

Perpendicular and parallel force coefficients for $\text{Re}=50$: , numerical results; , Eqs. (2) and (3).

Figure 19

Pressure as a function of the distance along the perimeter of a rectangle contained in the $(x,y)$ plane for different value of $\theta$: $\theta ={90}^{\circ }$, ; $\theta =0^{\circ }$, ; $\theta ={30}^{\circ }$, ; and linear theory for $\theta ={30}^{\circ }$, .

Figure 20

Perpendicular and parallel force coefficients for $\text{Re}=250$: , numerical results; , Eqs. (2) and (3); , Eqs. (13) and (14); and , Eqs. (15) and (16). The error bars indicate the magnitude of the unsteady modes.

Figure 21

Position of the hydrodynamic center normalized by the length as a function of the yaw angle: , $\text{Re}=50$; , $\text{Re}=100$; , $\text{Re}=150$; , $\text{Re}=200$; , $\text{Re}=250$; , Eq. (21) for $\text{Re}=50$; and , Eq. (22) for $\text{Re}=250$.

Figure 22

Drag, lift, and torque coefficients for (a)–(c) $\text{Re}=25$, (d)–(f) $\text{Re}=50$, (g)–(i) $\text{Re}=75$, and (j)–(l) $\text{Re}=100$: , numerical results; , empirical relation (10) of Rosendahl [40]; , Eqs. (4) and (5) for the force and Eq. (20) based on (21) and (2) for the torque; and , Eqs. (11) and (12).

Figure 23

Drag, lift, and torque coefficients for (a)–(c) $\text{Re}=150$, (d)–(f) $\text{Re}=200$, and (g)–(i) $\text{Re}=250$: , numerical results; , empirical relation (10) of Rosendahl [40]; , Eqs. (4) and (5); , IP; and , Eqs. (17) and (18) for the force and Eq. (20) based on (22) and (15) for the torque. The error bars indicate the magnitude of the unsteady modes.

Figure 24

Plot of $C_{Lz}$ as a function of time (top) and power spectra density (PSD) of $C_{Lz}$ (bottom) for $L/D=3$, $\text{Re}=150$, and $\theta ={90}^{\circ }$.

Figure 25

Plot of $C_{Lz}$ as a function of time (top) and power spectra density of $C_{Lz}$ (bottom) for $L/D=3$, $\text{Re}=250$, and $\theta ={75}^{\circ }$.

Figure 26

Two steady symmetric counterrotating vortex pairs at $(\theta ={90}^{\circ },\text{Re}=100)$. Here $Q=0.01$ isosurfaces are colored by the longitudinal vorticity ranging from $-0.1$ to 0.1. The dark lines starting at the center of the cylinder indicate the cylinder axis and the streamwise direction.

Figure 27

Periodic shedding of two symmetric counterrotating pairs at ($\theta ={90}^{\circ },Re=125$). The time interval between two frames is $\mathrm{\Delta }t=5$. Here $Q=0.004$ isosurfaces are colored by the longitudinal vorticity ranging from $-0.2$ to 0.2. The dark lines starting at the center of the cylinder indicate the cylinder axis and the streamwise direction.

Figure 28

The $C_D-C_{Lz}$ diagrams for the symmetric shedding of double-sided hairpin vortices at ($\theta ={90}^{\circ }$): (a) $\text{Re}=125$, (b) $\text{Re}=150$, and (c) $\text{Re}=175$.

Figure 29

Periodic shedding of double-sided horizontal hairpin vortices at $(\theta ={90}^{\circ },\text{Re}=150)$. Here $Q=0.008$ isosurfaces are colored by the longitudinal vorticity ranging from $-0.2$ to 0.2. The dark lines starting at the center of the cylinder indicate the cylinder axis and the streamwise direction.

Figure 30

Strouhal number as a function of Re. The meaning of the error bars can be found in Sec. 2.

Figure 31

Three-dimensional unsteady regime dominated by the shedding of horizontal double-sided hairpin vortices at $(\theta ={90}^{\circ },\text{Re}=250)$. The time interval between two frames is $\mathrm{\Delta }t=5$. Here $Q=0.008$ isosurfaces are colored by the longitudinal vorticity ranging from $-0.2$ to 0.2. The dark lines starting at the center of the cylinder indicate the cylinder axis and the streamwise direction.

Figure 32

Force and torque coefficient diagrams for the 3D unsteady regime dominated by the shedding of horizontal double-sided hairpin vortices at $(\theta ={90}^{\circ },\text{Re}=250)$.

Figure 33

One steady counterrotating vortex pair at $(\theta ={45}^{\circ },\text{Re}=75)$. (a) $Q=0.004$ isosurfaces are colored by the longitudinal vorticity ranging from $-0.2$ to 0.2. The dark lines starting at the center of the cylinder indicate the cylinder axis and the streamwise direction. (b) Streamlines in a $(y,z)$ plane perpendicular to the $x$ streamwise direction located at $5D$ from the $x$ coordinate of the cylinder mass center.

Figure 34

Periodic shedding of one counterrotating vortex pair at $(\theta ={65}^{\circ },\text{Re}=135)$. The time interval between two frames is $\mathrm{\Delta }t=5$. Here $Q=0.004$ isosurfaces are colored by the longitudinal vorticity ranging from $-0.2$ to 0.2. The dark lines starting at the center of the cylinder indicate the cylinder axis and the streamwise direction.

Figure 35

Steady toroidal vortex downstream of a $\theta =0^{\circ }$ cylinder at $\text{Re}=50$. Streamline pattern $(x,y)$ plane perpendicular to the $z$ transverse containing the cylinder axis colored by the dimensionless axial velocity (dark blue, $u_x=0$; red, $u_x=1$).

Figure 36

Recirculation length $l_r$ of an aligned cylinder with the flow direction with respect to the square root of the Reynolds number: +, $L/D=3$ cylinder; $\times$, $L/D=1$ cylinder (see the SM [18]). The solid lines correspond to fits of our numerical results of the $a{\text{Re}}^{1/2}+b$ form, where $a$ and $b$ are the fitting parameters.