Wavelength selection of vortex ripples in an oscillating cylinder: The effect of curvature and background rotation

We present results of laboratory experiments on the formation, evolution, and wavelength selection of vortex ripples. These ripples formed on a sediment bed at the bottom of a water-filled oscillating cylindrical tank mounted on top of a rotating table. The table is made to oscillate sinusoidally in time, while a constant background rotation was added for some experiments. The changes in bed thickness are measured using a light attenuation technique. It was found that the wavelength normalized with the excursion length depends on both a Reynolds number and the Strouhal number. This differs from straight or annular geometries where the wavelength is proportional to the excursion length. The flow in an oscillating cylinder has the peculiarity that it develops a secondary flow in the radial direction that depends on the excursion length. The effect of this secondary circulation is evident in the radial transport for small values of the Strouhal number or in the orientation of the ripples for strong enough background rotation. Additionally, ripples in an oscillating cylinder present a rich dynamic behavior where the number of ripples can oscillate even with constant forcing parameters.

Figure 1

Trace of the dimensionless shear stress (${\tau }_r^{\prime }$ vs. ${\tau }_{\theta }^{\prime }$) at the bottom for an oscillating disk with Sr $=0.5848$ and Ro $=\infty$ (solid line) and Sr $=0.5848$, Ro $=1.25$ (dashed line). The arrows indicate the direction of time.

Figure 2

Magnitude of the shear stress above an oscillating disk as a function of the Strouhal number (Sr) without background rotation. The symbols represent the amplitude of the total shear stress $|{\tau }^{\prime }|$ ($\circ$), the amplitude of the shear stress in the azimuthal direction ${\tau }_{\theta }$ ($+$), and the amplitude of the shear stress in the radial direction ${\tau }_r$ ($\times$). The dashed line represents the value of the total shear stress for a von Karman flow, and the dotted line represents the shear stress for a Stokes boundary layer.

Figure 3

Schematic side view of the experimental setup.

Figure 4

Raw images (left) and reconstructed bed thicknes (right) for an experiment with PS particles, Re = 24 000, Sr = 2.20, and Ro $=\infty$ at two different times $t=250$ s (top) and $t=5000$ s (bottom).

Figure 5

Example of the evolution of the number of ripples as a function of time at a cross-section at $r=0.23$ cm. (a) Changes in bed elevation as a function of time. (b) Fourier transform for every cross section of the bed elevation (gray-scale) and number of ripples (black-and-white bullets). Results are from the experiment with PS particles, Re = 24 000, Sr = 2.20, and Ro $=\infty$.

Figure 6

Changes in sediment bed thickness between $t=0$ and $t=5000$ s for the experiments with PS particles, Re = 24 000, Sr = 0.31, and $\mathrm{Ro}=\infty$.

Figure 7

Shear stress for a Stokes boundary layer at the critical radius $R_c$ [see Eq. (14)] normalized with the critical shear stress for sediment motion ${\tau }_c$ as a function of the Strouhal number. Squares denote the experiments with PMMA particles and bullets those with PS particles. The error bars denote the error in determining the critical shear stress. The horizontal dashed line represents ${\tau }_{s}t(r=R_c)/{\tau }_c=1$ and the vertical dashed line represents Sr = 0.64.

Figure 8

Changes in the bed elevation as a function of time and angular position at a cross-section at $r=0.23$ cm for an experiment with PS particles with Re = 37 500, Sr = 1.26, and Ro = 2.

Figure 9

Changes in sediment bed thickness between $t=0$ and $t=5000$ s for the experiments with background rotation. All experiments were done with PS particles. The values of the nondimensional parameters are (a) Ro = 8, (b) Ro = 4, (c) Ro = 2, with Re = 24 000 and Sr = 1.26, and (d) Ro = 1, with Re = 18 000 and Sr = 1.7.

Figure 10

Ripple wavelength $\lambda$ normalized with the radius $R$ as a function of ${\text{Re}}_L$. The experiments with PMMA particles are denoted by squares, and the experiments with PS particles by bullets. The triangles denote experiments with PS particles and background rotation. The error bars denote the wavelength plus or minus one ripple. The color denotes the bed shear stress parameter given by Eq. (8).

Figure 11

Ripple wavelength $\lambda$ normalized with the excursion length $L$ as a function of ${\text{Re}}_L^{0.42}\text{Sr}$. The experiments with PMMA particles are denoted by squares, and the experiments with PS particles by bullets. The diamond symbols correspond to experiments with PMMA particles in an annular configuration. The triangles correspond to experiments with PS particles and background rotation. The marker's color denotes the Strouhal number Sr. The dashed line denotes the “2/3-law.”

Figure 12

Evolution of the number of ripples as a function of time at a cross-section at $r=0.23$ cm for one 4.5-h-long experiment and for one 5-h-long experiment both with PMMA particles. Upper row: Re = 37 500, Sr = 0.76. Lower row: Re = 37 500, Sr = 0.85. Left column: Changes in the bed elevation. Right column: Fourier transform of the bed thickness (gray scale) and number of ripples counted (black-and-white bullets).

Figure 13

Difference in the bed elevation as a function of time and angular position at a cross-section at $r=0.23$ cm for two experiments with forcing changing conditions in an (a) annular configuration and (b) a cylindrical configuration. Both experiments were carried out with PMMA particles. Initially Re = 37500 and Sr = 0.63. During the period in between the dashed lines, Sr is set to Sr = 0.85. After this period, the value of Sr is returned to its original value.