Experiments relating to the flow induced by a vibrating quartz tuning fork and similar structures in a classical fluid

We report on an experimental study of the behavior of a number of commercially available quartz tuning forks oscillating in a classical cryogenic fluid, in the form of either liquid helium I or gaseous helium, extending our previous studies [M. Blazkova et al. Phys. Rev. E 75, 025302 (2007)]. Measurements of the damping of the oscillations allowed us to deduce the drag on the prong of a fork, as a function of the velocity with which the prong moves, for various sizes of fork and various oscillation frequencies. Transitions to turbulent flow have been identified, and the dependence of the critical velocity, expressed as a dimensionless critical Keulegan-Carpenter number, on the dimensionless Stokes number has been established. These measurements have not allowed us to visualize the flow, so we have carried out visualization experiments with oscillating rods in water, the rod dimensions, and the frequencies of oscillation, being chosen so that the relevant dimensionless parameters are similar to those for the prongs of the forks. Some information about the nature of the instability that leads to turbulence has been obtained in this way, and the results for the critical Keulegan-Carpenter number for the rods in water have been compared with values for the tuning forks in a cryogenic fluid.

Figure 1

Schematic sketch of a quartz tuning fork and a micrograph of the surface of tuning fork A1 (the marker is $25\mu \text{m}$ long).

Figure 2

(Color online) Upper left panel: the observed drag coefficient plotted against Keulegan-Carpenter number for a 32 kHz fork of type B1. Lower left panel: the same drag coefficient multiplied by velocity (normalized to unity)—this quantity indicates a departure from linearity more clearly. The (blue) dashed-dotted line indicates the fully turbulent drag. The right panels show detailed view of the departures from linear regime indicated by the (black) dashed-dotted line. In all panels the 5% departure criterion is marked by the (red) dashed line and the critical Keulegan-Carpenter numbers are marked with vertical black dotted lines.

Figure 3

(Color online) The critical Keulegan-Carpenter number $K_C^{\text{crit}}$ plotted against Stokes number $\beta$ for different forks as indicated, vibrating in normal liquid ${}^4\text{H}\text{e}$ and in cold pressurized helium gas at liquid-nitrogen temperature. The solid (red) line represents the expected instability for circular cylinders and the (blue) dashed line represents the observed square-root behavior as indicated.

Figure 4

(Color online) The critical Keulegan-Carpenter numbers, $K_C^{\text{crit}}$ and $K_C^{\text{crit}}\left(0.05\right)$, plotted against Stokes number for different forks as indicated, vibrating in normal liquid ${}^4\text{H}\text{e}$ and in cold pressurized helium gas at liquid-nitrogen temperature. The (red) solid line represents the expected instability for circular cylinders; the other lines are the individual observed square-root dependences.

Figure 5

(Color online) Schematic diagram of the apparatus used to visualize the flow produced by an oscillating bar in water. For a detailed description, see the text.

Figure 6

(Color online) A photograph of the (brass or stainless steel) rods of square cross section that we have studied. They have cross sections of $5\times 5$, $3\times 3$, $2\times 2$ (with rounded corners), $2\times 2$, and $1\times 1{\text{cm}}^2$.

Figure 7

(Color online) Left: the detail of the trimmed edge (after the second trimming) of the $3\times 3{\text{cm}}^2$ brass rod. Right: the detail of the roughened gold-plated surface of the $2\times 2{\text{cm}}^2$ brass rod.

Figure 8

(Color online) A photograph of two brass cylinders of square cross section $2\times 2{\text{cm}}^2$. The surface of the lower one was roughened by soldering small brass shavings to it; this surface was then electrochemically cleaned and gold plated.

Figure 9

A photograph showing a typical pattern of vortices produced by an oscillating rod of cross section $3\times 3{\text{cm}}^2$ at a velocity amplitude well above the transition.

Figure 10

(Color online) The critical Keulegan-Carpenter number plotted against Stokes number for various rods of square cross section oscillating in water. All the data were obtained by visualization, and relate to the first appearance of vortices being shed by the rod about 0.5 cm below the upper edge. The data in the main graph were obtained with the Baker $p\text{H}$ technique; those in the inset were obtained by the Kalliroscope technique. In the main graph, filled (blue) circles, filled (orange) squares, filled (red) triangles, and (green) crosses represent the data observed with rods of $1\times 1$, $2\times 2$, $3\times 3$, and $5\times 5{\text{cm}}^2$ square cross sections, respectively; (magenta) stars and (black) triangles show how the critical Keulegan-Carpenter number changes when the sharp edges of the $3\times 3{\text{cm}}^2$ rod are trimmed successively. The inset shows the critical Keulegan-Carpenter number for the $2\times 2{\text{cm}}^2$ rod, obtained with the Kalliroscope technique before [upper (dark green) symbols] and after [lower (cyan) symbols] roughening of the surface. The dotted straight line corresponds to $K_C^{\text{crit}}=17{\beta }^{-1/2}$ (c.f. Fig. 3) and the solid line is $K_C^{\text{crit}}=7.5{\beta }^{-1/2}$.