Wake of inertial waves of a horizontal cylinder in horizontal translation

We analyze theoretically and experimentally the wake behind a horizontal cylinder of diameter $d$ horizontally translated at constant velocity $U$ in a fluid rotating about the vertical axis at a rate $\mathrm{\Omega }$. Using particle image velocimetry measurements in the rotating frame, we show that the wake is stabilized by rotation for Reynolds number $\mathrm{Re}=Ud/\nu$ much larger than in a nonrotating fluid. Over the explored range of parameters, the limit of stability is $\mathrm{Re}\simeq (275\pm 25)/\mathrm{Ro}$, with $\mathrm{Ro}=U/2\mathrm{\Omega }d$ the Rossby number, indicating that the stabilizing process is governed by the Ekman pumping in the boundary layer. At low Rossby number, the wake takes the form of a stationary pattern of inertial waves, similar to the wake of surface gravity waves behind a ship. We compare this steady wake pattern to a model, originally developed by Johnson [E. R. Johnson, J. Fluid Mech. 120, 359 (1982)], assuming a free-slip boundary condition and a weak streamwise perturbation. Our measurements show quantitative agreement with this model for $\mathrm{Ro}\lesssim 0.3$. At larger Rossby number, the phase pattern of the wake is close to the prediction for an infinitely small line object. However, the wake amplitude and phase origin are not correctly described by the weak-streamwise-perturbation model, calling for an alternative model for the boundary condition at moderate rotation rate.

Figure 1

Propagation of the two symmetric wave packets, carrying wave number $k$, emitted at time $t-\tau$ by a horizontal line object (invariant along ${\mathbf{e}}_y)$ in translation at constant velocity $\mathbf{U}=U{\mathbf{e}}_x$ in a fluid rotating at a rate $\mathbf{\Omega }=\mathrm{\Omega }{\mathbf{e}}_z$. At time $t$, the line object is at point $B$, the upper wave packet at point $C(s=+)$, and the lower wave packet at $C(s=-)$. At time $t-\tau$, the line object is at point $A$. At locations $C(s=\pm )$, fluid particles describe a circular translation in planes tilted at an angle $\theta \left(k\right)$, with orientation given by the vector $-s\mathbf{k}\left(s\right)$. This circular translation motion propagates along wave vector $\mathbf{k}$ at the phase velocity $c_{\phi }=\sigma /k=Usin\theta$. Here $(X=x-Ut,Z=z)$ is the system of coordinates attached to the moving object.

Figure 2

(a) Energy propagation angle in the frame of the fluid at rest $\theta$ [Eq. (10), blue line] and in the frame of the moving object $\alpha$ [Eq. (14), red line] as a function of the Rossby number ${\mathrm{Ro}}_k=Uk/2\mathrm{\Omega }$ based on the wave number. (b) Plot of $\alpha$ as a function of $\theta$.

Figure 3

Lines of constant phase ${\phi }_0=2\pi n$ [Eqs. (17) and (18)] with $n\in [0;14]$ in the wake of inertial waves of a line source (of axis ${\mathbf{e}}_y)$ translated at velocity $U$ along ${\mathbf{e}}_x$ in an inviscid fluid under rotation at a rate $\mathrm{\Omega }$ about ${\mathbf{e}}_z$. Coordinates are normalized by ${\lambda }_0=U\pi /\mathrm{\Omega }$, the wake wavelength along the translation axis.

Figure 4

(a) Vertical velocity profile $u_z\left(X\right)/U$ at $Z=0_{+}$ used as the boundary condition (50) in the weak-streamwise-perturbation approximation. (b) Corresponding spectrum $\mathfrak{Im}\left[{\widehat{q}}_1\left(k_x\right)\right]/RU$ [Eq. (51)]. We report with vertical dashed lines the critical horizontal wave-vector component $k_0=1/\mathrm{Ro}d$, above which waves are vertically evanescent for three values $(\mathrm{Ro}=0.01$, 0.1, and 1) of the Rossby number $\mathrm{Ro}=U/2\mathrm{\Omega }d$.

Figure 5

Wake structure computed from the weak-streamwise-perturbation model (31, 32, 33) for a cylinder of diameter $d$, invariant along the $y$ direction. Arrows show the in-plane velocity components $(u_x,u_z)$ in the reference frame of the cylinder and the colormap the vertical velocity component $u_z$ normalized by $U$. White lines show the projection of streamlines in the vertical plane. In (a) and (b) the dash-dotted lines show the radiation angles ${\alpha }_{\mathrm{zero}}^n$, associated with the first roots $k_{x,\mathrm{zero}}^{n}R$ of the spectrum, along which no energy is present. In (c) and (d) these angles of zero amplitude do not exist since the roots of the spectrum ${\widehat{q}}_1\left(k_x\right)$ correspond to evanescent waves. In (a)–(d) we also show with dashed lines a few lines of constant phase of the far-field wake (37, 38, 39). We actually plot the parametric curve (41) and (42) for ${\phi }_s+\pi /2=\pi ,2\pi ,3\pi ,4\pi ,...,9\pi$ (from right to left) corresponding to the local maxima and minima of $u_z$ in the far-field theory (39) for the spectrum (51) [the spectrum (51) being imaginary introduces the $+\pi /2$ phase shift].

Figure 6

Wake structure, for the same values of $\mathrm{Ro}$ and $\mathrm{Re}$ as in Fig. 5, predicted in the far-field approximation (37, 38, 39) of the weak-streamwise-perturbation model [spectrum (51)] for a cylinder of diameter $d$, invariant along the $y$ direction. The layout is the same as in Fig. 5.

Figure 7

Experimental setup. A motor coupled by a belt to a translation rail drives a horizontal cylinder, of diameter $d$ and length 65 cm, at a constant velocity $U{\mathbf{e}}_x$ in a parallelepipedic water-filled tank mounted on a platform rotating at a rate $\mathrm{\Omega }$ about $z$. The PIV measurements are performed in the rotating frame, in a vertical region of $47\times 35{\mathrm{cm}}^2$ (green area). Here $L_x=150$ cm, $L_y=80$ cm, and $H=65$ cm.

Figure 8

Stability diagram of the cylinder wake in the $(\mathrm{Re},\mathrm{Ro})$ plane: $\circ$, steady wake; $\blacksquare$, periodic vortex shedding; and $\square$, turbulent wake. The solid line delimits the transition to unsteadiness, $\mathrm{Ro}=275/\mathrm{Re}$. The vertical dash-dotted line shows the classical instability threshold in a nonrotating fluid, $\mathrm{Re}=47$.

Figure 9

Close-up view of the experimental wake of a cylinder for (a) $\mathrm{Ro}=2.4,\mathrm{Re}=105$, (b) $\mathrm{Ro}=0.32,\mathrm{Re}=460$, (c) $\mathrm{Ro}=0.01,\mathrm{Re}=52$, and (d) $\mathrm{Ro}=0.08,\mathrm{Re}=410$, for which the wake is steady. Each panel reports the time-averaged velocity field in the cylinder reference frame: Arrows show the in-plane velocity components $(u_x,u_z)$ and the colormap shows the vertical velocity component $u_z$ normalized by $U$. White lines show streamlines of the in-plane velocity field.

Figure 10

Instantaneous fields of the vertical velocity component $u_z$ normalized by $U$ for several couples $(\mathrm{Re},\mathrm{Ro})$ close to the onset of nonstationarity: (a), (c), (e), and (g) steady wakes and (b), (d), (f), and (h) unsteady wakes. We show as dashed lines theoretical lines of constant phase (41) and (42) of the far-field wake (37, 38, 39) for ${\phi }_s+\pi /2=\pi ,2\pi ,3\pi$, and $4\pi$, from right to left.

Figure 11

Instantaneous fields of the vertical velocity component $u_z$ normalized by $U$ for a fixed Reynolds number $\mathrm{Re}=840$ and a decreasing Rossby number: (a) $\text{Ro}=19.7$, (b) $\text{Ro}=4.92$, and (c) $\text{Ro}=1.97$. The cylinder diameter is $d=10$ mm.

Figure 12

Experimental wake for increasing $\mathrm{Ro}$ and nearly constant $\mathrm{Re}\left(21\le \mathrm{Re}\le 52\right)$. Each panel shows the time-averaged velocity field in the reference frame of the cylinder: Arrows show the in-plane velocity components $(u_x,u_z)$ and the colormap shows the vertical velocity component $u_z$ normalized by $U$. White lines are streamlines of the in-plane velocity field; assuming the flow invariance along $y$, these lines are the projection in the vertical plane of the real streamlines. Dashed lines are the predicted lines of constant phase (41) and (42) of the far-field wake for ${\phi }_s+\pi /2=\pi ,2\pi ,3\pi ,4\pi$, and $5\pi$.

Figure 13

Normalized axial $u_x/U$ (dotted lines) and vertical $u_z/U$ (solid lines) velocity profiles at $Z/d=-3$ as a function of $X/d$ for several values of $(\mathrm{Ro},\mathrm{Re})$ for the experiments (black lines) and for the prediction based on the weak-streamwise-perturbation approximation (red lines, denoted by WSP in the legend).

Figure 14

(a) Amplitude $\mathrm{\Delta }{u}_z/U$ of the maximum vertical velocity oscillation at $Z/d=-3$ as a function of $\mathrm{Ro}$ for all the experiments with $15\lesssim \mathrm{Re}\lesssim 100$. For each data point (squares), the prediction based on the weak-streamwise-perturbation approximation for the same $(\mathrm{Ro},\mathrm{Re})$ values is reported with a circle. (b) Corresponding ratio $\mathrm{\Delta }{u}_x/\mathrm{\Delta }{u}_z$ of the axial to vertical maximum velocity oscillation at $Z/d=-3$. The straight line shows the expected asymptotic behavior at low $\mathrm{Ro},\mathrm{\Delta }{u}_x/\mathrm{\Delta }{u}_z\simeq k_{x,\mathrm{extr}}^{1}d\text{Ro}$. In both figures, the dashed and dash-dotted lines show the theoretical predictions of the weak-streamwise-perturbation model for $\mathrm{Re}=15$ and $\mathrm{Re}=100$, respectively.

Figure 15

Lines of constant phase, computed from the viscous model (B2) and (B3) [red lines, for ${\phi }_s+\pi /2=(2n+1)\pi$ and for the three values (a) $\mathrm{Re}\mathrm{Ro}=1$, (b) $\mathrm{Re}\mathrm{Ro}=10$, and (c) $\mathrm{Re}\mathrm{Ro}=100]$ and its inviscid limit (41) and (42) [black lines, for ${\phi }_s+\pi /2=(2n+1)\pi$ and for $\mathrm{Re}\mathrm{Ro}=\infty$ in which case ${\phi }_s={\phi }_0-\pi /4]$, for $n\in [0;10]$. The background color shows the vertical velocity of the corresponding prediction by the weak-streamwise-perturbation model of Sec. 2b2 for a cylinder of diameter $d$ and Rossby number $\mathrm{Ro}=U/2\mathrm{\Omega }d=0.1$.

Figure 16

Wake structure predicted in the uniform far-field approximation (C1, C2, C3), with the same parameters and layout as in Figs. 5 and 6.