Flow structure and loads over inclined cylindrical rodlike particles and fibers

The flow past a fixed finite-length circular cylinder, the axis of which makes a nonzero angle with the incoming stream, is studied through fully resolved simulations, from creeping-flow conditions to strongly inertial regimes. The investigation focuses on the way the body aspect ratio $\chi$ (defined as as the length-to-diameter ratio), the inclination angle $\theta$ with respect to the incoming flow, and the Reynolds number Re (based on the cylinder diameter) affect the flow structure past the body and therefore the hydrodynamic loads acting on it. The configuration $\theta =0^{\circ }$ (where the cylinder is aligned with the flow) is first considered from creeping-flow conditions up to $\text{Re}=400$, with aspect ratios up to 20 (10) for $\text{Re}\le 10$ ($\text{Re}\ge 10$). In the low-to-moderate Reynolds-number regime ($\text{Re}\lesssim 5$), influence or the aspect ratio, inclination (from $0^{\circ }$ to ${30}^{\circ }$), and inertial effects are examined by comparing numerical results for the axial and transverse force components and the spanwise torque with theoretical predictions based on the slender-body approximation, possibly incorporating finite-Reynolds-number corrections. Semiempirical models based on these predictions and incorporating finite-length and inertial corrections extracted from the numerical data are derived. For large enough Reynolds numbers ($\text{Re}\gtrsim {10}^2$), separation takes place along the upstream part of the lateral surface of the cylinder, deeply influencing the surface stress distribution. Numerical results are used to build empirical models for the force components and the torque, valid for moderately inclined cylinders ($\left|\theta \right|\lesssim {30}^{\circ }$) of arbitrary aspect ratio up to $\text{Re}\approx 300$ and matching those obtained at low-to-moderate Reynolds number.

Figure 1

Scheme of the computational domain in an azimuthal plane (not to scale).

Figure 2

Drag on a finite-length cylinder aligned with the flow direction, normalized by the drag $F_{ds}$ of a sphere of same volume. Dotted, dashed-dotted, and dashed lines: predictions of second-, third-, and fourth-order slender-body approximations, respectively; solid line: semiempirical formula (1); $•$: numerical results of [26]; $\blacksquare$: experimental results of [27]; $▾$: present numerical results for $\text{Re}=0.05$.

Figure 3

Radial pressure distribution on the upstream end of the cylinder for $\text{Re}=0.05$. $•$: $\chi =3$; $★$: $\chi =5$; $+$: $\chi =7$; $▾$: $\chi =10$; $▴$: $\chi =20$. The pressure is assumed to be zero in the far field.

Figure 4

Contributions to the drag force. (a) Pressure contribution; (b) ratio of the pressure to the shear stress contributions. $▾$: numerical results for $\text{Re}=0.05$; solid line: low-Re prediction (2) for a prolate spheroid.

Figure 5

Influence of inertial effects on the drag of finite-length cylinders aligned with the upstream flow. The drag is normalized by that of a sphere of same volume. $▾$: numerical results; solid line: semiempirical prediction (5); dashed line: theoretical prediction (3); dotted line: predictions (1) (upper line) and (3) with $\text{Re}=0$ (lower line).

Figure 6

Streamlines pattern for $\chi =2$ colored with the magnitude of the axial velocity (from $-0.25$ to 1). (a) $\text{Re}=140$; (b) $\text{Re}=300$.

Figure 7

Variation of the length $l_r$ of the toroidal eddy vs the Reynolds number past cylinders with different aspect ratios; $l_r$ is measured along the symmetry axis, from the downstream end of the cylinder to the point at which the axial velocity changes sign. $▴$: $\chi =1$ from [23]; $▾$: $\chi =2$; $•$: $\chi =3$; $\times$: $\chi =4$; $★$: $\chi =5$; $+$: $\chi =7$; solid line: $l_r\left(\text{Re}\right)=0.083{\text{Re}}^{1/2}$.

Figure 8

Pressure drag coefficient $C_p$ as a function of Re for different aspect ratios. (a) Upstream end; (b) downstream end; (c) sum of the two contributions. $▾$: $\chi =2$; $\times$: $\chi =4$; $+$: $\chi =7$; $▴$: $\chi =10$; solid line in (a): empirical fit (6); solid line in (b): empirical fit (7); solid line in (c): sum (6)$+$(7) for $\chi =2$ (upper line) and $\chi =10$ (lower line).

Figure 9

Drag coefficient as a function of Re for different aspect ratios. (a) Viscous contribution $C_{\mu }$; (b) total drag coefficient $C_d$. $▾$: $\chi =2$; $\times$: $\chi =4$; $+$: $\chi =7$; $▴$: $\chi =10$; for each aspect ratio, the solid lines in (a) and (b) represent the fits (8) and (9), respectively.

Figure 10

Schematic of the force components on an inclined cylinder. In this configuration, the inertial torque is negative, tending to rotate the cylinder clockwise.

Figure 11

Parallel force component $F_{\parallel }$ vs the inclination angle $\theta$ for $•$: $\chi =3$, $★$: $\chi =5$, $▾$: $\chi =10$. Dashed line: prediction (12) based on the numerical value of $F_{\parallel }^{\theta =0^{\circ }}$; solid line: prediction (12) based on the semiempirical approximation (5) for $F_{\parallel }^{\theta =0^{\circ }}$.

Figure 12

Perpendicular force component $F_{\perp }$ as a function of $\theta$ for $•$: $\chi =3$, $★$: $\chi =5$, $▾$: $\chi =10$. Solid line: Stokes law (13) based on the semiempirical creeping-flow estimate (C2) for $F_{\perp }^{\theta ={90}^{\circ }}\left(\chi \right)$; dashed line: same with $F_{\perp }^{\theta ={90}^{\circ }}$ based on the semiempirical finite-Re estimate (C4). In (a), the two estimates overlap for $\chi =3$; in (b), the prediction for $\chi =10$ based on (C2) overlaps with that for $\chi =5$ based on (C4).

Figure 13

Torque as a function of $\theta$ for $•$: $\chi =3$, $★$: $\chi =5$, $▾$: $\chi =10$. Solid line: asymptotic prediction (17) for low-but-finite Re; dashed line: approximate fit (18) for $\chi =5$ and 10.

Figure 14

Streamlines in the symmetry plane $z=0$ (left) and in the plane $y=D/2$ tangent to the lateral surface along the top generatrix (right) for a cylinder with $\chi =5$ at $\text{Re}=140$. Streamlines are colored according to the magnitude and sign of the axial velocity from $-0.25$ (deep blue) to $+1$ (deep red). (a), (b) $\theta =5^{\circ }$; (c), (d) $\theta ={15}^{\circ }$; (e), (f) $\theta ={30}^{\circ }$.

Figure 15

Three-dimensional streamlines past an inclined cylinder with $\chi =5$, $\theta ={15}^{\circ }$, and $\text{Re}=300$.

Figure 16

Parallel and perpendicular force coefficients ($C_{\parallel },C_{\perp }$) vs $\theta$ for $•$: $\chi =3$, $★$: $\chi =5$, $+$: $\chi =7$. Dashed lines: Stokes laws (12) and (13) based on the value of $C_{\parallel }^{\theta =0^{\circ }}(\chi =3)$ and $C_{\perp }^{\theta ={90}^{\circ }}(\chi \gg 1)$ at the relevant Reynolds number; solid line: approximate fit (20). The yellow bullets in (c), (d) and (e), (f) refer to the results of [13] for $\chi =3$ at $\text{Re}=75$ and 250, respectively.

Figure 17

Contributions to $C_{\perp }(\chi ,\theta ,\text{Re})$ at $\text{Re}=20$ (blue) and $\text{Re}=300$ (red) for $•$: $\chi =3$, $★$: $\chi =5$, +: $\chi =7$. (a) Pressure on the lateral surface; (b) viscous stress on the lateral surface; (c), (d) viscous stress on the upstream and downstream ends, respectively.

Figure 18

Contributions to $C_{\parallel }(\chi ,\theta ,\text{Re})$ at $\text{Re}=20$ (blue), $\text{Re}=80$ (green), and $\text{Re}=300$ (red) for $•$: $\chi =3$, $★$: $\chi =5$, +: $\chi =7$. (a), (b) Pressure on the upstream and downstream ends, respectively; (c) viscous stress on the lateral surface.

Figure 19

Parallel force coefficient $C_{\parallel }(\chi ,\theta ,\text{Re})$ vs $\theta$ for $•$: $\chi =3$, $★$: $\chi =5$, +: $\chi =7$. Solid line: empirical fit (22).

Figure 20

Torque coefficient vs $\theta$ for $•$: $\chi =3$, $★$: $\chi =5$, +: $\chi =7$. The yellow bullets in (b) and (c) refer to the results of [13] for $\chi =3$ at $\text{Re}=75$ and 250, respectively.

Figure 21

Contributions to $C_t(\chi ,\theta ,\text{Re})$ at $\text{Re}=20$ (blue), $\text{Re}=80$ (green), and $\text{Re}=300$ (red) for $•$: $\chi =3$, $★$: $\chi =5$, +: $\chi =7$. (a), (b) Pressure on the upstream end and the lateral surface, respectively; (c), (d) viscous stress on the upstream end and the lateral surface, respectively.

Figure 22

Torque coefficient $C_t(\chi ,\theta ,\text{Re})$ vs $\theta$ for $•$: $\chi =3$, $★$: $\chi =5$, +: $\chi =7$. Solid line: empirical fit (23).

Figure 23

Extrapolated predictions at $\text{Re}=5$ of fits derived in the fully inertial regime for $•$: $\chi =3$, $★$: $\chi =5$, +: $\chi =7$. (a) $C_{\parallel }$ and fit (22); (b) $C_{\perp }$ and fit (20); (c) $C_t$ and fit (23).

Figure 24

Drag on a finite-length cylinder held perpendicular to the flow direction, normalized by the drag $F_{ds}$ of a sphere of same volume. Dotted, long-dashed, and dashed lines: slender-body approximation (C1) truncated at second, third, and fourth order, respectively; solid line: semiempirical formula (C2); $•$: numerical results of [39]; $\blacksquare$: experimental results of [27]; $▾$: experimental results of [40].

Figure 25

Influence of inertial effects on the drag of finite-length cylinders held perpendicular to the upstream flow. The drag is normalized by that of a sphere of same volume. Solid line: semiempirical prediction (C4); dashed line: prediction (C2); $•$: numerical results of [12].