Hydrodynamic torque on a slender cylinder rotating perpendicularly to its symmetry axis

Using fully-resolved simulations, we examine the torque experienced by a finite-length circular cylinder rotating steadily perpendicularly to its symmetry axis. The aspect ratio $\chi$, i.e., the ratio of the length of the cylinder to its diameter, is varied from 1 to 15. In the creeping-flow regime, we employ the slender-body theory to derive the expression of the torque up to order 4 with respect to the small parameter $1/ln\left(2\chi \right)$. Numerical results agree well with the corresponding predictions for $\chi \gtrsim 3$. We introduce an ad hoc modification in the theoretical prediction to fit the numerical results obtained with shorter cylinders, and a second modification to account for the increase of the torque resulting from finite inertial effects. In strongly inertial regimes, a prominent wake pattern made of two pairs of counter-rotating vortices takes place. Nevertheless the flow remains stationary and exhibits two distinct symmetries, one of which implies that the contributions to the torque arising from the two cylinder ends are identical. We build separate empirical formulas for the contributions of pressure and viscous stress to the torque provided by the lateral surface and the cylinder ends. We show that, in each contribution, the dominant scaling law may be inferred from simple physical arguments. This approach eventually results in an empirical formula for the rotation-induced torque valid throughout the range of inertial regimes and aspect ratios considered in the simulations.

Figure 1

Sketch of the computational domain (not to scale). The cyan area represents the region over which the grid is uniform.

Figure 2

Low-Reynolds-number torque on a rotating finite-length cylinder ($T$), normalized by the torque on a rotating sphere with the same volume ($T_s$), as a function of the aspect ratio $\chi$. Dotted, dash-dotted-dotted, dash-dotted, and dashed lines: predictions of first-, second-, third-, and fourth-order slender-body approximations, respectively; solid line: semiempirical formula (1), $•$: numerical results for $\text{Re}=0.05$. The region $\chi \lesssim 3$ is magnified in the inset to clarify the behavior of the various approximations for short cylinders.

Figure 3

Ratio of the pressure ($T_p$) to the shear stress ($T_{\mu }$) contributions to the torque for $\text{Re}=0.05$. Solid line: empirical formula (2), $•$: numerical results.

Figure 4

Torque on a rotating finite-length cylinder ($T$), normalized by the torque on a rotating sphere with the same volume ($T_s$), as a function of the Reynolds number in the range $0.1\le \text{Re}\le 10$. $•$: numerical results; dashed line: creeping-flow prediction (1); solid line: semiempirical formula (3). The slight difference between numerical results and predictions of (1) for $\text{Re}\ll 1$ is already present in Fig. 1 but is magnified here, owing to the chosen origin of the vertical axis.

Figure 5

Wake past a rotating cylinder with $\chi =3$ for $\text{Re}=220$. (a) Side view from a fixed $z$-plane (see Fig. 6 for the exact definition of the coordinate system); (b) front view from a fixed $x$-plane. The wake is visualized using the $Q$-criterion [21]. The 0.001 isosurface of $Q$ is colored with the magnitude of the normalized transverse vorticity ${\omega }_z/\mathrm{\Omega }$.

Figure 6

Sketch of the flow configuration near the rotating cylinder, especially the distribution of $u_y$ (the $y$-component of the fluid velocity) close to the ends. Colors help identify the flow symmetries.

Figure 7

Contributions to the torque coefficient arising from the lateral surface as a function of Re. (a) Shear-induced viscous contribution ($C_{T\mu l}$); (b) pressure contribution ($C_{Tpl}$). $▴$: $\chi =2$, $▾$: $\chi =3$, $•$: $\chi =5$, $\blacksquare$: $\chi =8$. Solid line: empirical fits (4) and (5).

Figure 8

Contributions to the torque coefficient arising from each of the cylinder ends as a function of Re. (a) Shear-induced viscous contribution ($C_{T\mu e}$); (b) pressure contribution ($C_{Tpe}$). $▴$: $\chi =2$, $▾$: $\chi =3$, $•$: $\chi =5$, $\blacksquare$: $\chi =8$. Solid line: empirical fits (6) and (7).

Figure 9

Torque coefficient ($C_T$) as a function of Re for cylinders with various aspect ratios. $▴$: $\chi =2$, $▾$: $\chi =3$, $•$: $\chi =5$, $\blacksquare$: $\chi =8$. Solid line: empirical fit (8).

Figure 10

Ratio of the pressure ($T_p$) to shear stress ($T_{\mu }$) contributions to the torque as a function of the cylinder aspect ratio for different Reynolds numbers. $▴$: $\text{Re}=20$, $▾$: $\text{Re}=60$, $•$: $\text{Re}=140$, $\blacksquare$: $\text{Re}=240$; solid line: empirical formula (9).

Figure 11

Torque coefficient ($C_T$) as a function of Re for cylinders with various aspect ratios. Numerical results for: $\chi =3$ ($▾$) and $\chi =10$ ($•$). Dashed line: creeping-flow prediction (1); dash-dotted line: finite-Re semiempirical formula (3); solid line: empirical inertial fit (8); red and orange lines refer to $\chi =3$ and $\chi =10$, respectively.

Figure 12

Successive orders of the Stokeslet density distribution $f_y\left(x\right)$ along the cylinder. The blue, red, and green lines correspond to $f_y^1$, $f_y^2$, and $f_y^3$, respectively.